From aa2c565d121d6175862219384209ceea6d942b09 Mon Sep 17 00:00:00 2001 From: John Maddock Date: Sun, 21 May 2006 15:45:37 +0000 Subject: [PATCH] Initial commit. [SVN r2955] --- doc/HyperComplex.bib | 225 -- doc/HyperComplex.tex | 1542 ---------- doc/Jamfile.v2 | 17 + doc/beta.qbk | 108 + doc/big_whole.html | 724 ----- doc/boost-build.jam | 3 + doc/concepts.qbk | 66 + doc/equations/beta.xml | 517 ++++ doc/equations/beta1.png | Bin 0 -> 537 bytes doc/equations/beta2.png | Bin 0 -> 1107 bytes doc/equations/beta3.png | Bin 0 -> 1314 bytes doc/equations/beta4.png | Bin 0 -> 560 bytes doc/equations/beta5.png | Bin 0 -> 667 bytes doc/equations/beta6.png | Bin 0 -> 631 bytes doc/equations/beta7.png | Bin 0 -> 512 bytes doc/equations/beta8.png | Bin 0 -> 477 bytes doc/equations/erf.xml | 72 + doc/equations/erf1.png | Bin 0 -> 531 bytes doc/equations/erf2.png | Bin 0 -> 332 bytes doc/equations/error.xml | 41 + doc/equations/error1.png | Bin 0 -> 452 bytes doc/equations/fraction1.png | Bin 0 -> 563 bytes doc/equations/fraction2.png | Bin 0 -> 517 bytes doc/equations/fraction3.png | Bin 0 -> 518 bytes doc/equations/fraction4.png | Bin 0 -> 602 bytes 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++++ test/erf_inv_data.ipp | 105 + test/erf_large_data.ipp | 305 ++ test/erf_small_data.ipp | 162 + test/erfc_inv_data.ipp | 105 + test/gamma_inv_big_data.ipp | 136 + test/gamma_inv_data.ipp | 206 ++ test/gamma_inv_small_data.ipp | 235 ++ test/ibeta_data.ipp | 511 ++++ test/ibeta_large_data.ipp | 1222 ++++++++ test/ibeta_small_data.ipp | 512 ++++ test/igamma_big_data.ipp | 298 ++ test/igamma_med_data.ipp | 712 +++++ test/igamma_small_data.ipp | 264 ++ test/log1p_expm1_test.cpp | 593 ++++ test/modulo_test.cpp | 206 -- test/powm1_sqrtp1m1_test.cpp | 1603 ++++++++++ test/test_beta.cpp | 123 + test/test_beta_hooks.hpp | 81 + test/test_cbrt.cpp | 97 + test/test_classify.cpp | 136 + test/test_erf.cpp | 230 ++ test/test_erf_hooks.hpp | 105 + test/test_gamma.cpp | 464 +++ test/test_gamma_data.ipp | 582 ++++ test/test_gamma_hooks.hpp | 176 ++ test/test_ibeta.cpp | 310 ++ test/test_roots.cpp | 297 ++ tools/beta_data.cpp | 72 + tools/cbrt_data.cpp | 68 + tools/erf_data.cpp | 202 ++ tools/ibeta_data.cpp | 294 ++ tools/igamma_data.cpp | 155 + tools/lanczos_generator.cpp | 4351 +++++++++++++++++++++++++++ tools/ntl.diff | 135 + tools/ntl_rr_lanczos.hpp | 903 ++++++ 159 files changed, 47182 insertions(+), 6007 deletions(-) delete mode 100644 doc/HyperComplex.bib delete mode 100644 doc/HyperComplex.tex create mode 100644 doc/Jamfile.v2 create mode 100644 doc/beta.qbk delete mode 100644 doc/big_whole.html create mode 100644 doc/boost-build.jam create mode 100644 doc/concepts.qbk create mode 100644 doc/equations/beta.xml create mode 100644 doc/equations/beta1.png create mode 100644 doc/equations/beta2.png create mode 100644 doc/equations/beta3.png create mode 100644 doc/equations/beta4.png create mode 100644 doc/equations/beta5.png create mode 100644 doc/equations/beta6.png create mode 100644 doc/equations/beta7.png create mode 100644 doc/equations/beta8.png create mode 100644 doc/equations/erf.xml create mode 100644 doc/equations/erf1.png create mode 100644 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doc/roadmap.qbk create mode 100644 doc/roots.qbk create mode 100644 doc/series.qbk create mode 100644 doc/test_data.qbk create mode 100644 doc/tgamma.qbk create mode 100644 test/Jamfile.v2 create mode 100644 test/beta_exp_data.ipp create mode 100644 test/beta_med_data.ipp create mode 100644 test/beta_small_data.ipp delete mode 100644 test/big_whole_test.cpp delete mode 100644 test/cayley_element_test.cpp create mode 100644 test/cbrt_data.ipp create mode 100644 test/erf_data.ipp create mode 100644 test/erf_inv_data.ipp create mode 100644 test/erf_large_data.ipp create mode 100644 test/erf_small_data.ipp create mode 100644 test/erfc_inv_data.ipp create mode 100644 test/gamma_inv_big_data.ipp create mode 100644 test/gamma_inv_data.ipp create mode 100644 test/gamma_inv_small_data.ipp create mode 100644 test/ibeta_data.ipp create mode 100644 test/ibeta_large_data.ipp create mode 100644 test/ibeta_small_data.ipp create mode 100644 test/igamma_big_data.ipp create mode 100644 test/igamma_med_data.ipp create mode 100644 test/igamma_small_data.ipp create mode 100644 test/log1p_expm1_test.cpp delete mode 100644 test/modulo_test.cpp create mode 100644 test/powm1_sqrtp1m1_test.cpp create mode 100644 test/test_beta.cpp create mode 100644 test/test_beta_hooks.hpp create mode 100644 test/test_cbrt.cpp create mode 100644 test/test_classify.cpp create mode 100644 test/test_erf.cpp create mode 100644 test/test_erf_hooks.hpp create mode 100644 test/test_gamma.cpp create mode 100644 test/test_gamma_data.ipp create mode 100644 test/test_gamma_hooks.hpp create mode 100644 test/test_ibeta.cpp create mode 100644 test/test_roots.cpp create mode 100644 tools/beta_data.cpp create mode 100644 tools/cbrt_data.cpp create mode 100644 tools/erf_data.cpp create mode 100644 tools/ibeta_data.cpp create mode 100644 tools/igamma_data.cpp create mode 100644 tools/lanczos_generator.cpp create mode 100644 tools/ntl.diff create mode 100644 tools/ntl_rr_lanczos.hpp diff --git a/doc/HyperComplex.bib b/doc/HyperComplex.bib deleted file mode 100644 index 9faee05df..000000000 --- a/doc/HyperComplex.bib +++ /dev/null @@ -1,225 +0,0 @@ -%% This BibTeX bibliography file was created using BibDesk. -%% http://bibdesk.sourceforge.net/ - - -%% Created for Daryle Walker at 2005-08-13 07:09:33 -0400 - - -%% Saved with string encoding ASCII (TeX) - - - -@electronic{Inc.:it, - Address = {450 Phillip Street, Waterloo, ON, N2L SJ2, Canada}, - Date-Added = {2005-08-13 07:03:51 -0400}, - Date-Modified = {2005-08-13 07:09:22 -0400}, - Edition = {version V, release 5.1}, - Editor = {Waterloo Maple Inc.}, - Note = {A commercial computer-aided mathematics software}, - Title = {Maple}, - Url = {http://www.maplesoft.com}} - -@electronic{Holin:lq, - Author = {Hubert Holin and Didier Vidal}, - Date-Added = {2005-08-13 06:59:47 -0400}, - Date-Modified = {2005-08-13 07:02:44 -0400}, - Note = {available for licencing from the author, {\copyright}}, - Title = {Horizontal-preserving quaternions}} - -@url{Weisstein:ym, - Annote = {Was by E. Weisstein, but was redirected to a page that included a suggested cite listing. -}, - Author = {Eric W. Weisstein}, - Date-Added = {2005-08-13 06:50:54 -0400}, - Date-Modified = {2005-08-13 06:58:24 -0400}, - Publisher = {MathWorld---A Wolfram Web Resource}, - Title = {Cayley Number}, - Url = {http://mathworld.wolfram.com/CayleyNumber.html}} - -@article{Sudbery:1979ei, - Author = {A. Sudbery}, - Date-Added = {2005-08-13 06:46:18 -0400}, - Date-Modified = {2005-08-13 06:47:44 -0400}, - Journal = {Proceedings of the Cambridge Philosophical Society}, - Title = {Quaternionic Analysis}, - Volume = {85}, - Year = {1979}} - -@article{Salamon:1982sg, - Author = {S. Salamon}, - Date-Added = {2005-08-13 06:42:04 -0400}, - Date-Modified = {2005-08-13 06:44:22 -0400}, - Journal = {Inventiones Mathematica}, - Number = {1}, - Pages = {143--171}, - Title = {Quaternionic Kaeler Manifolds}, - Volume = {67}, - Year = {1982}} - -@article{Pletincks:1989ds, - Author = {D. Pletincks}, - Date-Added = {2005-08-13 06:36:48 -0400}, - Date-Modified = {2005-08-13 06:38:45 -0400}, - Journal = {The Visual Computer, an International Journal of Computer Graphics}, - Number = {1/2}, - Pages = {2--13}, - Title = {Quaternion calculus as a basic tool in computer graphics}, - Volume = {5}, - Year = {1989}} - -@article{Pichon:1973oh, - Address = {Paris}, - Author = {G. Pichon}, - Date-Added = {2005-08-13 06:30:53 -0400}, - Date-Modified = {2005-08-13 06:34:48 -0400}, - Journal = {Collection M\'{e}thodes}, - Pages = {31}, - Publisher = {Hermann}, - Title = {Groupes de Lie, representations lin\'{e}aires et applications}, - Year = {1973}} - -@unpublished{Pham:1996gf, - Author = {F. Pham}, - Date-Added = {2005-08-13 06:23:13 -0400}, - Date-Modified = {2005-08-13 06:26:04 -0400}, - Note = {early version, unpublished, presented at the Journ\'{e}es Non Standard, Paris, 1996-12-14 and perhaps elsewhere}, - Title = {Une d\'{e}finition non standard de l'exponentielle (variation sur un th\`{e}me de Pierre Cartier)}, - Year = {1996}} - -@booklet{Misner:1973es, - Address = {New York}, - Author = {C.W. Misner and K.S. Thone and J.A. Wheeler}, - Chapter = {41---Spinors}, - Date-Added = {2005-08-13 06:16:27 -0400}, - Date-Modified = {2005-08-13 06:23:28 -0400}, - Publisher = {W.H. Freeman and Company}, - Title = {Gravitation}, - Year = {1973}} - -@book{Lelong-Ferrand:1978gq, - Author = {J. Lelong-Ferrand and J.M. Arnaudi\`{e}s}, - Date-Added = {2005-08-13 06:08:32 -0400}, - Date-Modified = {2005-08-13 06:12:02 -0400}, - Edition = {3eme \'{e}dition}, - Publisher = {Dunod Universit\'{e}}, - Title = {Cours de math\`{e}matiques, Tome 1, Alg\`{e}bre}, - Year = {1978}} - -@article{Leborgne:1982hf, - Author = {D. Leborgne}, - Date-Added = {2005-08-13 06:04:45 -0400}, - Date-Modified = {2005-08-13 06:06:39 -0400}, - Journal = {Math\'{e}matiques}, - Publisher = {P.U.F.}, - Title = {Calcul diff\'{e}rentiel et g\'{e}ometrie}, - Year = {1982}} - -@book{Lang:1971vf, - Author = {S. Lang}, - Date-Added = {2005-08-13 05:58:26 -0400}, - Date-Modified = {2005-08-13 06:02:25 -0400}, - Edition = {Revised printing: January 1971}, - Publisher = {Addisson-Wesley Publishing Company}, - Series = {Addison-Wesley Series in Mathematics}, - Title = {Algebra}, - Year = {1971}} - -@article{Khoan:1972ld, - Author = {V. K. Khoan}, - Date-Added = {2005-08-13 05:53:27 -0400}, - Date-Modified = {2005-08-13 05:57:05 -0400}, - Journal = {Cours et exercices r\'{e}solus maitrise de math\'{e}matiques certificat C2}, - Publisher = {Vuibert}, - Title = {Distributions Analyse de Fourier Op\'{e}rateurs aux D\'{e}riv\'{e}es Partielles, tome 1}, - Year = {1972}} - -@article{Howell:1975ft, - Author = {T. Howell and J.C. Lafon}, - Date-Added = {2005-08-13 05:50:14 -0400}, - Date-Modified = {2005-08-13 05:51:43 -0400}, - Journal = {Cornell Computer Science TR 75-245}, - Month = {June}, - Title = {The complexity of the quaternion product}, - Year = {1975}} - -@article{Gurlebeck:1989mb, - Author = {K. G\"{u}rlebeck and W. Sp\"{o}ssig}, - Date-Added = {2005-08-13 05:46:31 -0400}, - Date-Modified = {2005-08-13 05:48:44 -0400}, - Journal = {International Series of Numerical Mathematics}, - Publisher = {Birkha\"{u}ser}, - Title = {Quaternion Analysis and Elliptical Boundary Problems}, - Volume = {89}, - Year = {1989}} - -@article{Exbrayat:1971pk, - Author = {J. M. Exbrayat and P. Mazet}, - Date-Added = {2005-08-13 05:06:22 -0400}, - Date-Modified = {2005-08-13 05:44:53 -0400}, - Journal = {Notions Modernes de Math\'{e}matiques}, - Publisher = {Hatier-Universit\'{e}}, - Title = {alg\`{e}bre 1, Notions Fondamentales de la Th\'{e}orie des Ensembles}, - Year = {1971}} - -@webpage{Dixon:ex, - Annote = {The web site was originally , but it has a redirect page to , of which I trimmed the default key. -}, - Author = {G. Dixon}, - Date-Added = {2005-08-13 04:59:19 -0400}, - Date-Modified = {2005-08-13 05:02:52 -0400}, - Url = {http://www.7stones.com}} - -@article{Dixon:1994rh, - Author = {G. Dixon}, - Date-Added = {2005-08-13 04:54:24 -0400}, - Date-Modified = {2005-08-13 04:56:21 -0400}, - Journal = {Mathematics and Its Applications}, - Publisher = {Kluwer Academic Publishers}, - Title = {Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics}, - Year = {1994}} - -@article{Casteljau:1987bc, - Author = {P. de Casteljau}, - Date-Added = {2005-08-13 04:47:03 -0400}, - Date-Modified = {2005-08-13 04:50:13 -0400}, - Journal = {Trait\'{e} des nouvelles technologies, s\'{e}rie math\'{e}matiques appliqu\'{e}es}, - Publisher = {Hermes}, - Title = {Les quaternions}, - Year = {1987}} - -@book{Bourbaki:do, - Author = {N. Bourbaki}, - Date-Added = {2005-08-13 04:41:28 -0400}, - Date-Modified = {2005-08-13 04:42:11 -0400}, - Title = {Espaces Vectoriels Topologiques}} - -@book{Bourbaki:sc, - Author = {N. Bourbaki}, - Date-Added = {2005-08-13 04:39:45 -0400}, - Date-Modified = {2005-08-13 04:41:18 -0400}, - Title = {Alg\`{e}bre}} - -@book{Berger:1990ce, - Author = {M. Berger}, - Date-Added = {2005-08-13 04:35:48 -0400}, - Date-Modified = {2005-08-13 04:36:40 -0400}, - Publisher = {Nathan}, - Title = {G\'{e}ometrie 1}, - Year = {1990}} - -@inbook{Beardon:1979ek, - Author = {A.F. Beardon}, - Date-Added = {2005-08-13 04:29:25 -0400}, - Date-Modified = {2005-08-13 04:33:48 -0400}, - Publisher = {John Wiley \& Sons, A Wiley-Interscience Publication}, - Title = {Complex Analysis, The Argument Principle in Analysis and Topology}, - Year = {1979}} - -@article{Godbillon:1971th, - Author = {C. Godbillon}, - Date-Added = {2005-08-13 04:12:45 -0400}, - Date-Modified = {2005-08-13 04:35:21 -0400}, - Journal = {Collection M\'{e}thodes}, - Publisher = {Hermann}, - Title = {\'{E}l\'{e}ments de Topologie Alg\'{e}brique}, - Year = {1971}} \ No newline at end of file diff --git a/doc/HyperComplex.tex b/doc/HyperComplex.tex deleted file mode 100644 index 792ceb95a..000000000 --- a/doc/HyperComplex.tex +++ /dev/null @@ -1,1542 +0,0 @@ -\documentclass{book} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{url} -\usepackage{xspace} -\usepackage{dsfont} -\usepackage[colorlinks=true,% - linkcolor=blue,% - anchorcolor=blue,% - citecolor=blue,% - pagecolor=blue,]{hyperref} - -\title{% - The Quaternionic Exponential\\ - (and beyond)% -} -\author{% - Hubert HOLIN% - \medskip\\% - \small\url{mailto:Hubert.Holin@Bigfoot.com}\\ - \small\url{http://www.bigfoot.com/~Hubert.Holin}% -} - -\date{08/12/1999} % plus the errata and addenda from 23/03/2001 - -\newtheorem{theorem}{Theorem}[section] -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{scholie}[theorem]{Scholie} -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{corollary}[theorem]{Corollary} - -\newcommand{\funcDef}[5]{ - \begin{array}{rccc} - {#1}: & {#2} & \to & {#4} \\ - & {#3} & \mapsto & {#5} - \end{array} -} -\newcommand{\Funcify}[1]{\mathop{\mathrm{#1}}} -% Usually \xspace does a good job when deciding whether to add a space or not. -% In the rare case that you want to force \xspace to not add a space append a -% pair of braces after the command. E.g., \Fc{N}{} is typeset like -% \ensuremath{\mathfrak{N}}. -\newcommand{\Bl}[1]{\ensuremath{\mathds{#1}}\xspace} -\newcommand{\Fc}[1]{\ensuremath{\mathcal{#1}}\xspace} -\newcommand{\Ck}[1]{\ensuremath{\mathfrak{#1}}\xspace} -\newcommand{\fcT}{\Fc{T}} -\newcommand{\fcN}{\Fc{N}} -\newcommand{\Trace}[1]{\ensuremath{{\fcT}_{\Bl{#1}}}} -\newcommand{\Norm}[1]{\ensuremath{{\fcN}_{\Bl{#1}}}} -\newcommand{\fnRe}{\Funcify{Re}} -\newcommand{\fnIm}{\Funcify{Im}} -\newcommand{\fnUr}{\Funcify{Ur}} -\newcommand{\fnVect}{\Funcify{Vect}} -\newcommand{\fnId}{\Funcify{Id}} -\newcommand{\fnSinc}{\Funcify{sinc}} -\newcommand{\fnSinhc}{\Funcify{sinhc}} -\newcommand{\fnSincPi}{{\fnSinc}_{\pi}} -\newcommand{\fnSinhcPi}{{\fnSinhc}_{\pi}} -\newcommand{\setN}{\Bl{N}} -\newcommand{\setZ}{\Bl{Z}} -\newcommand{\setQ}{\Bl{Q}} -\newcommand{\setR}{\Bl{R}} -\newcommand{\setC}{\Bl{C}} -\newcommand{\setH}{\Bl{H}} -\newcommand{\setO}{\Bl{O}} -\newcommand{\setX}{\Bl{X}} -\newcommand{\xsetA}{\Bl{A}} -\newcommand{\xsetE}{\Bl{E}} -\newcommand{\xsetF}{\Bl{F}} -\newcommand{\setS}{\Bl{S}} -\newcommand{\cBasis}{\Fc{C}} -\newcommand{\cMatrix}{\Fc{M}} - -% a comma that allows linebreaks - used for overlong tuples -% (A perfectionist would probably imitate the url package's implementation and -% add potential line breaks after math punctuation characters. But I am not -% familiar enough with low level TeX programming and the following gets the -% job done.) -% The optional argument is passed on to \linebreak. Default is \lbc[1]. -\newcommand{\lbc}[1][1]{,\linebreak[#1]} - -\newcommand{\sst}{\scriptstyle} - -% Special spacing for the "C++" logo -\newcommand{\Cpp}{C\nolinebreak % - \hspace{-0.05em}\raisebox{0.4ex}{\tiny\bf +}\nolinebreak % - \hspace{-0.1em}\raisebox{0.4ex}{\tiny\bf +}\xspace} - -\begin{document} -\maketitle -\tableofcontents - -\chapter*{Motivation} -\addcontentsline{toc}{chapter}{Motivation} -\label{part:N} - -I felt the need to take a closer look at quaternions when, some time -back, I was looking for new applications to Harthong-Reeb circles (on -which I was working at the time), and came across~\cite{Pletincks:1989ds}. -That paper, on one hand, did indicate one potential application -for that method, but, on the other hand, alluded to some odd -constructions involving quaternions, the validity of which was -propitiously left in the shadows. The present text is therefore a -compilation of many well-known but apparently scattered results about -quaternions (and related entities), as well as some new developments, -notably the explicit formula for the quaternionic exponential (and -friends). Incidentally, these results enables one to solve the problem -found in~\cite{Pletincks:1989ds}, but without the unsalvageable -constructions. - -\chapter{Quaternions redux} -\label{part:I} - -\section{What to find here} -\label{part:I.1} - -This chapter only contains a quick-and-dirty (but sufficient for most -uses) presentation of the quaternions, along with their most classical -properties, inspired very largely by~\cite{Leborgne:1982hf}, -\cite{Lelong-Ferrand:1978gq} and~\cite{Berger:1990ce}. -This approach, however, obscures the deep relationship which -links the quaternions, the complex and real numbers and more exotic -things known as octonions; this relationship will be the thrust of the -next chapter. - -It should be said that other important uses of quaternions exist - (\cite{Gurlebeck:1989mb},~\ldots), but that they will not be touched -upon here. As well, quaternionic analysis (\cite{Sudbery:1979ei}) -and geometry (\cite{Salamon:1982sg}), though perhaps not as vibrant -as their complex counterparts, do keep evolving; though these -usually involve fairly sophisticated mathematical machinery, -very nice results can also be had with very elementary ones - (\cite{Casteljau:1987bc},~\ldots). All are beyond the scope of -this article, however. - -\section{The nature of the Beast} -\label{part:I.2} - -Let $\setH = \setR^{4}$ with the usual four-dimensional vector -space structure over \setR. We define $e = (1,0,0,0)$, $i = -(0,1,0,0)$, $j = (0,0,1,0)$ and $k = (0,0,0,1)$. - -The first important thing we need is a multiplication, denoted $*$, -which we \emph{define\/} to be a (non-commutative) \setR-bilinear -operation on \setH such that $i*i = j*j = k*k = -e$, $i*j = --(j*i) = k$, $j*k = -(k*j) = i$, $k*i = -(i*k) = j$ and $e$ is its -neutral element. - -The second important thing we need is the \emph{conjugation\/} on -\setH (and we will usually denote by $\bar{q}$ the conjugate of -$q$) which we define by $(\alpha, \beta, \gamma, \delta) \mapsto -(\alpha, -\beta, -\gamma, -\delta)$. Important properties are that -$\overline{q * q'} = \overline{q'} * \overline{q}$, that $\bar{e} = e$, -that $q * \bar{q} = \bar{q} * q \in \setR \cdot e$ and that $q + -\bar{q} \in \setR \cdot e$. Actually $q * \bar{q} = 0$ if and only -if $q = 0$, as is easily seen. - -A straightforward verification then shows that $(\setH, +, *, -\cdot)$ is an effectively non-commutative, but associative, -\setR-algebra with unit $e$, and that $[\setR \to -\setH, x \mapsto (x, 0, 0, 0)]$ and $[\setC \to \setH, z -\mapsto (\fnRe z, \fnIm z, 0, 0)]$ are algebra -homomorphisms, bijective from their sources onto their images. The -image of the conjugate of a complex number is also seen to be the -conjugate (in \setH) of the image of that complex, by the above -function. We will therefore assimilate \setH to a superset of - (both) \setR and \setC, and identify $e$ with $1$ and $i$ -with its complex counterpart. We see at once that the operations we -have defined on \setH extend their counterparts on \setC -and \setR. The multiplication can then be memorized thru the -well-known formula: -\[ -i*i = j*j = k*k = i*j*k = -1 -\] - -It is important to notice that given any quaternion $q$ and any -\emph{real\/} number $x$, we always have $q * x = x * q = x \cdot q$. - -We will usually write a quaternion under the form $q = \alpha + \beta i -+ \gamma j + \delta k$ with $\alpha$, $\beta$, $\gamma$ and $\delta$ -reals, omitting the ``$\cdot$'' when multiplying a quaternion by a real -number (as per the vector space structure). We will also omit -the ``$*$'' when multiplying a quaternion by a real number, from the -left as well as from the right. When no confusion may arise, we will do -away entirely with the ``$*$''. - -With the above notations, the conjugate of $q = \alpha + \beta i + -\gamma j + \delta k$ will then simply be $\bar{q} = \alpha - \beta i - -\gamma j - \delta k$. - -Looking at \setH as a 4-dimensional \setR-vector space, it -is easy to see the usual euclidian scalar product is equal to the -following: -\begin{eqnarray*} -(p | q) & = & (p + q)\overline{(p + q)} - p\bar{p} - q\bar{q} \\ - & = & \frac{1}{2} (p\bar{q} + q\bar{p}) \\ - & = & \frac{1}{2} (\bar{p}q + \bar{q}p) -\end{eqnarray*} - -All the same, the usual euclidian norm on $\setR^{4}$, coincides -with $[q \mapsto \|q\| = \sqrt{q * \bar{q}}]$, and of course $(q | q) = -\|q\|^{2} = \alpha^{2} + \beta^{2} + \gamma^{2} + \delta^{2}$. Note -that, if $q \neq 0$ then $q^{-1} = {(q * \bar{q})}^{-1} * \bar{q} = -\bar{q} * {(q * \bar{q})}^{-1}$. For the quaternions, we will also use a -notation compatible with real and complex numbers and define $|q|$ as -$\|q\|$ (of course, if $q$ is actually complex, $|q|$ has exactly the -value of the modulus of $q$). - -It is important to remember that $(\setH, +, *, \cdot, ||)$ is a -Banach \setR-algebra. The norm is better than what we might -expect, though, as we have $|p * q| = |p| |q|$ instead of just $|p * q| -\leq |p| |q|$.\footnote{We ask for a norm of a Banach algebra to verify -$|p * q| \leq |p| |q|$ and not just that there exists some positive $k$ -such that $|p * q| \leq k |p| |q|$. The superior case for the norm, $|p -* q| = |p| |q|$, also applies to real numbers, complex numbers and -octonions.} - -We will call the real and unreal parts of quaternion, respectively, -$\fnRe q = \frac{1}{2} (q + \bar{q})$ and $\fnUr q = -\frac{1}{2} (q - \bar{q})$. We will say that a quaternion is pure if -its real part is zero. For a complex number, the quaternionic real part -is what is already known as the complex real part, and the unreal part -is just the imaginary part multiplied by $i$. - -\section{Quaternions' kin} -\label{part:I.3} - -As we have just seen, quaternions are related to both real numbers and -complex numbers. As we shall see in some details in the next chapter, -quaternions are actually part of an infinite family of -sets\footnote{Actually, several families, but we will focus on just one -here; for others, see~\cite{Dixon:1994rh}.} which we will call the -Cayley ladder, some of which we will introduce here as we will have some -need of them for our purposes. - -First relative in that family, beyond the quaternions, are the -octonions. We denote by \setO the set $\setR^{8}$, with its -usual vector space structure on \setR, we identify $1 = -(1,0,0,0,0,0,0,0)$, $i = (0,1,0,0,0,0,0,0)$, $j = (0,0,1,0,0,0,0,0)$ and -$k = (0,0,0,1,0,0,0,0)$ and we define $e' = (0,0,0,0,1,0,0,0)$, $i' = -(0,0,0,0,0,1,0,0)$, $j' = (0,0,0,0,0,0,1,0)$ and $k' = -(0,0,0,0,0,0,0,1)$. We now consider \setO to be a super-set of -\setH. We can now define a multiplication on \setO by the -following table (the value at line $n$ and column $m$ is -the product of the element in the left column by the element in the top -row; for instance $i * i' = -e'$): - -\begin{align*} - \begin{array}{c|cccccccc} - & 1 & i & j & k & e' & i' & j' & k' \\ - \hline - 1 & 1 & i & j & k & e' & i' & j' & k' \\ - i & i & -1 & k & -j & i' & -e' & -k' & j' \\ - j & j & -k & -1 & i & j' & k' & -e' & -i' \\ - k & k & j & -i & -1 & k' & -j' & i' & -e' \\ - e' & e' & -i' & -j' & -k' & -1 & i & j & k \\ - i' & i' & e' & -k' & j' & -i & -1 & -k & j \\ - j' & j' & k' & e' & -i' & -j & k & -1 & -i \\ - k' & k' & -j' & i' & e' & -k & -j & i & -1 - \end{array} -\end{align*} -Other presentations, perhaps more useful, exist (\cite{Dixon:ex}). -This multiplication still has a unit ($1$), but is no longer associative - (for instance $i' * (e' * j) = +k \neq -k = (i' * e') * j$). Real -numbers still commute with every octonion. We define a conjugation by -$\overline{\alpha + \beta i + \gamma j + \delta k + \epsilon e' + \zeta -i' + \eta j' + \theta k'} = \alpha - \beta i - \gamma j - \delta k - -\epsilon e' - \zeta i' - \eta j' - \theta k'$, a scalar product and a -norm which, as with the quaternions turn out to be exactly the euclidian -scalar product and euclidian norm on $\setR^{8}$. Again, we have -just extended the quaternionic operations. As with complex numbers and -quaternions, we have $|o * o'| = |o| |o'|$ for any two octonions $o$ and -$o'$, and an octonion $o$ is invertible if and only if it is non-zero, -and then $o^{-1} = \frac{1}{|o|^{2}} \bar{o}$. - -Beyond even the octonions, we find $\setR^{16}$, which appears not -to have any agreed-upon name. We shall here call them hexadecimalions, -and denote the set by \setX (after the C/\Cpp notation\ldots). We -have the usual vector space structure on \setR, we identify -$1,\ldots, k'$ with -$(1\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc -0\lbc 0\lbc 0\lbc 0), \ldots\lbc -(0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 1\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc -0\lbc 0\lbc 0)$ respectively, and define -$e''$, $i''$, $j''$, $k''$, $e'''$, $i'''$, $j'''$, $k'''$ as -$(0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 1\lbc 0\lbc 0\lbc 0\lbc -0\lbc 0\lbc 0\lbc 0), \ldots\lbc (0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc -0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 0\lbc 1)$ respectively. We define a -multiplication on \setX as explicited in the next chapter, for -which $1$ is still a unit and for which reals commute with every -hexadecimalion. We define as well a conjugation, a scalar product and a -norm (for details, see next chapter), which once again coincide with the -euclidian scalar product and euclidian norm on $\setR^{16}$. These -all extend the octonionic case. However, the product has even fewer -properties than in the octonionic case (the algebra is no longer even -alternative\footnote{We will define this in the next chapter.}, as for -instance $(i + e''') * ((i + e''') * j) = -2j + 2k \neq -2j = ((i + -e''') * (i + e''')) * j$), and the norm is not even an algebra norm any -longer, as for instance ${\|(i + j'') * (e' + k''')\|}^{2} = 8 > 4 = -{\|i + j''\|}^{2} {\|e' + k'''\|}^{2}$. - -\section{Quaternions and rotations} -\label{part:I.4} - -It is pleasant to think that perhaps the relationship between -quaternions and rotations has been stumbled upon while running a -check-list of classical constructs on the then-newly discovered -quaternions. At any rate, the easiest way to explain that link is thru -interior automorphisms. - -More precisely, given a \emph{non-zero\/} quaternion $q = \alpha + \beta i -+ \gamma j + \delta k$, we can consider the interior automorphism: -\begin{align*} - \funcDef{\lambda_{q}}{\setH}{p}{\setH}{(q)p(q^{-1})} -\end{align*} - -These objects have several fundamental properties: $\lambda_{q q'} = -\lambda_{q} \circ \lambda_{q'}$ and $\lambda_{q}(q) = q$, $\lambda_{q}$ -leaves \setR invariant (since reals commute with all -quaternions), and $\lambda_{q}$ respects the norm on \setH. - -It is interesting to see $\lambda_{q}$ as an \setR-linear -function on \setH. As it preserves the norm, it preserves the -scalar product, and hence $\lambda_{q} \in \mathrm{O}(4, \setH)$. -Then, as it leaves \setR globally invariant, it must leave its -orthogonal (i.e.\ the unreals) globally invariant. - -Consider now the matrix of $\lambda_{q}$; expressed in the canonical -basis $\cBasis = (1, i, j, k)$ that matrix is: -\begin{align*} - \cMatrix(\lambda_{q}, \cBasis, \cBasis) = - \tfrac{1}{{\|q\|}^{2}} - \begin{bmatrix} - \sst \alpha^2+\beta^2+\gamma^2+\delta^2 & - \sst 0 & - \sst 0 & - \sst 0 \\ - \sst 0 & - \sst \alpha^2+\beta^2-\gamma^2-\delta^2 & - \sst -2\alpha\delta+2\beta\gamma & - \sst +2\alpha\gamma+2\beta\delta \\ - \sst 0 & - \sst +2\alpha\delta+2\beta\gamma & - \sst \alpha^2-\beta^2+\gamma^2-\delta^2 & - \sst -2\alpha\beta+2\gamma\delta \\ - \sst 0 & - \sst -2\alpha\gamma+2\beta\delta & - \sst +2\alpha\beta+2\gamma\delta & - \sst \alpha^2-\beta^2-\gamma^2+\delta^2 - \end{bmatrix} -\end{align*} - -It is quite obvious\footnote{We will note $\mathrm{M}(U, n, m)$ the set of -matrices, $n$ rows by $m$ columns, with elements in $U$.} that -$\Fc{Q}:[\setR^{4} - \{0\} \to \mathrm{M}(\setR, 4, 4); q -\mapsto \cMatrix(\lambda_{q}, \cBasis, \cBasis)]$ is -continuous, and a group homomorphism. As we have seen, -$\Fc{Q}(\setR^{4} - \{0\}) \subset \mathrm{O}(4, \setR)$, -and as $\Fc{Q}(1) = \mathrm{I}_{4}$, the identity -matrix\footnote{More generally, we will denote by $\mathrm{I}_{n}$ the -identity matrix on $\setR^{n}$.} on $\setH = \setR^{4}$, -$\Fc{Q}(\setR^{4} - \{0\})$ must actually be included in the -connected component of $\mathrm{I}_{4}$ in $\mathrm{O}(4, \setR)$, -and that is $\mathrm{SO}(4, \setR)$, i.e., $\lambda_{q}$ -is a rotation on $\setR^{4}$, and hence on \setR, where it -is the identity $\mathrm{I}_{1}$, and thus must also be a rotation on -$\{0\} \times \setR^{3}$, i.e.\ the unreals. We can find a -far simpler (if somewhat tedious) proof of that by simply computing the -determinant of $\cMatrix(\lambda_{q}, \cBasis, \cBasis)$, -which of course turns out to be $1$ (also see next section)\ldots. - -We can therefore extract a rotation matrix on $\setR^{3}$ from -$\cMatrix(\lambda_{q}, \cBasis, \cBasis)$: -\begin{align*} - \rho_{q} = \tfrac{1}{\alpha^2+\beta^2+\gamma^2+\delta^2} - \begin{bmatrix} - \sst \alpha^2+\beta^2-\gamma^2-\delta^2 & - \sst -2\alpha\delta+2\beta\gamma & - \sst +2\alpha\gamma+2\beta\delta \\ - \sst +2\alpha\delta+2\beta\gamma & - \sst \alpha^2-\beta^2+\gamma^2-\delta^2 & - \sst -2\alpha\beta+2\gamma\delta \\ - \sst -2\alpha\gamma+2\beta\delta & - \sst +2\alpha\beta+2\gamma\delta & - \sst \alpha^2-\beta^2-\gamma^2+\delta^2 - \end{bmatrix} -\end{align*} - -Let us introduce $\Fc{R}:[\setR^{4} - \{0\} \to -\mathrm{M}(\setR, 3, 3); q \mapsto \rho_{q}]$. It is trivial to see -that $\Fc{Q}$ and $\Fc{R}$ are both $C^{\infty }$ (because -they are rational). It is important to note that they are both -\setR-homogeneous of degree $0$, which means that given any -non-zero real number $x$, $\lambda_{q}$ and $\lambda_{xq}$ are -identical, and therefore yield identical rotations (i.e.\ -$\rho_{q} = \rho_{xq}$). - -A fundamental result is that $\Fc{R}$ is surjective. There are at -least two well-known ways to prove this. - -The easiest way also has the advantage of being completely constructive: -we just compute the elements of the rotation $\rho_{q}$. - -This is possible because we always know one invariant vector. Indeed - (as an immediate consequence of $\lambda_{q}(q) = q$): -\[ - \rho_{q} - \left[ \begin{array}{c} \beta \\ \gamma \\ \delta \end{array} \right] - = - \left[ \begin{array}{c} \beta \\ \gamma \\ \delta \end{array} \right] -\] - -Furthermore, the angle, $\theta \in [0; \pi]$, is given by considering -the trace of $\rho_{q}$: -\[ - 1 + 2 \cos \theta = \frac{3\alpha^2 - \beta^2 - \gamma^2 - \delta^2} - {\alpha^2 + \beta^2 + \gamma^2 + \delta^2} -\] - -We now exploit the homogeneity of $\Fc{R}$, which implies that -$\Fc{R}(\setH - \{0\}) = \Fc{R}(\setS^{3})$, and -therefore that we can restrict our search to \emph{unit\/} quaternions. -For unit quaternions, the trace relation simplifies to $1 + \cos \theta -= 2 \alpha^{2}$. - -Therefore, the identity rotation $\mathrm{I}_{3}$ is associated with $q -= \pm 1$ (which we already knew), and these unit quaternions only. - -Let $(\vec{\imath}, \vec{\jmath}, \vec{k})$ be the canonical basis of -$\setR^{3}$. Consider now a rotation $r \neq \mathrm{I}_{3}$ - (hence $\theta \in ]0; \pi]$), it possesses a unique rotation axis, and -a unique unit vector $\vec{u} = x \vec{\imath} + y \vec{\jmath} + z \vec{k}$ -directing that axis such that $r(\vec{a}) = (\cos \theta)[\vec{a} - -(\vec{u} \cdot \vec{a}) \vec{u}] + (\sin \theta)(\vec{u} \wedge \vec{a}) -+ (\vec{u} \cdot \vec{a}) \vec{u}$ for all $\vec{a} \in \setR^{3}$. - It follows that $r$ is associated with the two unit quaternions -\[ - q = \pm \left[ \begin{array}{c} - \cos \frac{\theta}{2} \\ - x \sin \frac{\theta}{2} \\ - y \sin \frac{\theta}{2} \\ - z \sin \frac{\theta}{2} - \end{array} \right] -\] -and these two unit quaternions only. - -The second method is non-constructive, but has the advantage of -highlighting the regularity of the connection between rotations and -quaternions, which is harder to read using the first method. - -We once again exploit the homogeneity of $\Fc{R}$ and use unit -quaternions. Given that we know that in fact -$\Fc{R}(\setS^{3}) \subset \mathrm{SO}(3, \setR)$, we can -consider $\Fc{R}|^{\mathrm{SO}(3, \setR)}_{\setS^{3}}$ -which is $C^{\infty }$ (because it is rational). It is slightly -tedious, but possible, to prove that in fact -$\Fc{R}|^{\mathrm{SO}(3, \setR)}_{\setS^{3}}$ is a local -diffeomorphism at $1$. It is also a group homomorphism (stemming from -the fact that $\lambda_{q q'} = \lambda_{q} \circ \lambda_{q'}$). Since -in a connected topological group, every neighborhood of the neutral -element is a generator of the whole group (\cite{Pichon:1973oh}), -$\Fc{R}|^{\mathrm{SO}(3, \setR)}_{\setS^{3}}$ is -surjective upon the connected component of $\mathrm{I}_{3} = -\Fc{R}(1)$ in $\mathrm{SO}(3, \setR)$, i.e.\ upon -$\mathrm{SO}(3, \setR)$, and of course is everywhere a local -diffeomorphism (though it is of course not a global diffeomorphism). - -Combining these two approaches, one finds a global $C^{\infty -}$-diffeomorphism between $\mathrm{SO}(3, \setR)$ and -$\setR\mathrm{P}^{3}$ (which is nothing more than $\setS^{3}$ -where every couple of opposite points have been identified). - -Another thing worth noting is that $\Fc{R}|^{\mathrm{SO}(3, -\setR)}_{\setS^{3}}$ is more than just a locally diffeomorphic -bijection. If we call $\sigma_{\setS^{3}}$ the positive Borel -measure on $\setS^{3}$ induced by $\setH = \setR^{4}$ and -$\sigma_{\mathrm{SO}(3, \setR)}$ that induced on $\mathrm{SO}(3, -\setR)$ by $\mathrm{M}(\setR, 3, 3)$ (by assimilation of the -rotations with their matrix in the canonical basis of $\setR^{3}$), -seen as $\setR^{9}$, then we can compute\footnote{A fact that is -supposed to be found, but is not, in~\cite{Misner:1973es}.} that -$\Fc{R}* \sigma_{\mathrm{SO}(3, \setR)} = 16 \sqrt{2} -\sigma_{\setS^{3}}$. Furthermore, $\Fc{R}|^{\mathrm{SO}(3, -\setR)}_{\setS^{3}}$ actually has no critical point. - -\section{Miscellany} -\label{part:I.5} - -As we have seen, the main power of the quaternions is their ability to -pleasantly parameter $\mathrm{SO}(3, \setR)$. It should be said -that what is, perhaps their greatest strengths in this regard, with -respect to other parameterization of $\mathrm{SO}(3, \setR)$ such -as Euler angles, is that $\Fc{R}|^{\mathrm{SO}(3, -\setR)}_{\setS^{3}}$ has no critical points (no ``Gimbal -Lock''), and that the composition of rotations is extremely simple to -compute in terms of the parameter. Also and they can be shown to allow -interpolations of orientations under constraints (such has having one -axis stay ``horizontal''). - -Quaternions also allow a nice parameterization of $\mathrm{SO}(4, -\setR)$ (\cite{Berger:1990ce} the application $\setS^{3} -\times \setS^{3} \to \mathrm{SO}(4, \setR), (s, r) \mapsto [q -\mapsto sq\bar{r}]$ is a continuous group homomorphism, surjective, with -kernel $\{(1, 1), (-1, -1)\}$). - -Quaternions have other uses, though. For instance, they can be also be -used to parameter $\mathrm{SU}(2, \setC)$. More precisely, an -isomorphism exists between $\{0\} \times \setS^{3}$ and -$\mathrm{SU}(2, \setC)$ (consider, the application -\begin{align*} -\funcDef{\Psi} - {\setH}{q = \alpha + \beta i + \gamma j + \delta k} - {\mathrm{M}(\setC, 2, 2)}{\left[ \begin{array}{cc} - u & -\bar{v} \\ - v & \bar{u} \\ - \end{array} \right]} -\end{align*} -with $u = \alpha + \delta i$ and $v = \gamma + \beta i$ is a ring -isomorphism from \setH on a sub-ring of $\mathrm{M}(\setC, -2, 2)$, which induces an isomorphism). There are also applications of -quaternions to the Riemann sphere (\cite{Lelong-Ferrand:1978gq}). - -It should be mentioned that research exists to find more efficient -algorithm for the product of quaternions (\cite{Howell:1975ft}), -but has so far not reached a conclusion, one way or the other. - -Given the power of the quaternions, the question naturally arises as to -whether something similar can be done for rotations on spaces of higher -dimensions (the multiplication being commutative on the reals and -complex numbers, interiors automorphisms are just the identity). The -answer to that question is partly positive, but it should be now stated -that the right tool, in general, for that problem turns out to be -Clifford algebras rather than Cayley algebras. - -When we turn to the octonions, the multiplication is not only not -associative, it is no longer even associative. Fortunately, the -sub-algebra engendered by any two elements (and the unity) is still -associative, and therefore interior automorphism do not depend on the -order in which the products are carried out. The interesting fact is -that, as with the quaternions, the interior automorphisms leave -\setR invariant, and induce a rotation, on $\setR^{7}$ this -time. The catch is that $\mathrm{SO}(7, \setR)$ is a -21-dimentional manifold, whereas the interior automorphisms we just -described only have 7 degrees of freedom. In short, we do not get all -the rotations on $\setR^{7}$ by this method. It is still useful, -though, for theoretical purposes. - -Beyond the even the octonions, the hexadecimalions have two different -flavors of interior automorphism, $p \mapsto ((q)p)(q^{-1})$ and $p -\mapsto (q)(p(q^{-1}))$, neither of which is, in general, a rotation (on -either $\setR^{16}$ or $\setR^{15}$). The average of the two -is not a rotation either, by the way\ldots. - -Interior automorphisms having apparently reached the limits of their -usefulness, we turn now to something else, with the same objects. It -turns out that we can find rotations with even simpler constructions! - -Let $x = \alpha \in \setR$, then $\mathrm{M}_{x} = \cMatrix([y -\mapsto xy], 1, 1) = \cMatrix([y \mapsto yx], 1, 1) = [\alpha]$, -hence ${}^{\mathrm{t}}\mathrm{M}_{x}\mathrm{M}_{x} = |x| -\mathrm{I}_{1}$, and $\det \mathrm{M}_{x} = x$. Therefore if $|x| = 1$, -we find that $\mathrm{M}_{x} \in \mathrm{O}(1, \setR)$, and we of -course get all two elements of $\mathrm{O}(1, \setR)$ that -way\ldots but $\mathrm{M}_{x} \in \mathrm{SO}(1, \setR)$ only if $x -= 1$! Obviously, given $x \in \setR$ and $x' \in \setR$, -$\mathrm{M}_{x x'} = \mathrm{M}_{x} \mathrm{M}_{x'} = \mathrm{M}_{x'} -\mathrm{M}_{x}$. - -Let $c = \alpha + \beta i \in \setC$, then $\mathrm{M}_{c} = -\cMatrix([z \mapsto cz], (1, i), (1, i)) = \cMatrix([z \mapsto -zc], (1, i), (1, i)) = \left[ \begin{array}{cc} - \alpha & -\beta \\ - +\beta & \alpha -\end{array} \right]$, hence ${}^{\mathrm{t}}\mathrm{M}_{c}\mathrm{M}_{c} -= |c| \mathrm{I}_{2}$, and $\det \mathrm{M}_{c} = \alpha^{2} + -\beta^{2}$. Therefore if $|c| = 1$, $\mathrm{M}_{c} \in \mathrm{SO}(2, -\setR)$, and we get all rotations on $\setR^{2}$ that way, as -is well-known. And given $c \in \setC$ and $c' \in \setC$, we -still have $\mathrm{M}_{c c'} = \mathrm{M}_{c} \mathrm{M}_{c'} = -\mathrm{M}_{c'} \mathrm{M}_{c}$. - -Let now $q = \alpha + \beta i + \gamma j + \delta k \in \setH$, -then: -\[ - \mathrm{M}^{G}_{q} - = \cMatrix([p \mapsto qp], (1, i, j, k), (1, i, j, k)) - = \left[ \begin{array}{cccc} - \alpha & -\beta & -\gamma & -\delta \\ - +\beta & \alpha & -\delta & +\gamma \\ - +\gamma & +\delta & \alpha & -\beta \\ - +\delta & -\gamma & +\beta & \alpha - \end{array} \right] -\] -and -\[ - \mathrm{M}^{D}_{q} - = \cMatrix([p \mapsto pq], (1, i, j, k), (1, i, j, k)) - = \left[ \begin{array}{cccc} - \alpha & -\beta & -\gamma & -\delta \\ - +\beta & \alpha & +\delta & -\gamma \\ - +\gamma & -\delta & \alpha & +\beta \\ - +\delta & +\gamma & -\beta & \alpha - \end{array} \right] -,\] -hence ${}^{\mathrm{t}} \mathrm{M}^{G}_{q} \mathrm{M}^{G}_{q} = -{}^{\mathrm{t}} \mathrm{M}^{D}_{q} \mathrm{M}^{D}_{q} = |q| -\mathrm{I}_{4}$, and $\det \mathrm{M}^{G}_{q} = \det \mathrm{M}^{D}_{q} -= {(\alpha^{2} + \beta^{2} + \gamma^{2} + \delta^{2})}^{2}$. Therefore if -$|q| = 1$, $\mathrm{M}^{G}_{q} \in \mathrm{SO}(4, \setR)$ and -$\mathrm{M}^{D}_{q} \in \mathrm{SO}(4, \setR)$, but we only get a -tiny fraction of $\mathrm{SO}(4, \setR)$ that way. - -This, of course can be used as an alternate proof that the interior -automorphisms on the quaternions actually induce rotations on -$\setR^{4}$. - -It is interesting to note that given $q \in \setH$ and $q' \in -\setH$, we still have $\mathrm{M}^{G}_{q q'} = \mathrm{M}^{G}_{q} -\mathrm{M}^{G}_{q'}$ and $\mathrm{M}^{D}_{q q'} = \mathrm{M}^{D}_{q'} -\mathrm{M}^{D}_{q}$, though we now sometimes have $\mathrm{M}^{G}_{q} -\mathrm{M}^{G}_{q'} \neq \mathrm{M}^{G}_{q'} \mathrm{M}^{G}_{q}$ and -$\mathrm{M}^{D}_{q'} \mathrm{M}^{D}_{q} \neq \mathrm{M}^{D}_{q} -\mathrm{M}^{D}_{q'}$. - -Turning to the octonions, let $o = \alpha + \beta i + \gamma j + -\delta k + \epsilon e' + \zeta i' + \eta j' + \theta k' \in -\setO$, then: -\begin{multline*} - \mathrm{M}^{G}_{o} = - \cMatrix([o' \mapsto o o'], (1, i, j, k, e', i', j', k'), - (1, i, j, k, e', i', j', k'))\\ - = - \begin{bmatrix} - \alpha & -\beta & -\gamma & -\delta - & -\epsilon & -\zeta & -\eta & -\theta \\ - +\beta & \alpha & -\delta & +\gamma - & -\zeta & +\epsilon & +\theta & -\eta \\ - +\gamma & +\delta & \alpha & -\beta - & -\eta & -\theta & +\epsilon & +\zeta \\ - +\delta & -\gamma & +\beta & \alpha - & -\theta & +\eta & -\zeta & +\epsilon \\ - +\epsilon & +\zeta & +\eta & +\theta - & \alpha & -\beta & -\gamma & -\delta \\ - +\zeta & -\epsilon & +\theta & -\eta - & +\beta & \alpha & +\delta & -\gamma \\ - +\eta & - \theta & -\epsilon & +\zeta - & +\gamma & -\delta & \alpha & +\beta \\ - +\theta & +\eta & -\zeta & -\epsilon - & +\delta & +\gamma & -\beta & \alpha - \end{bmatrix} -\end{multline*} -and -\begin{multline*} - \mathrm{M}^{D}_{o} - = \cMatrix([o' \mapsto o' o], (1, i, j, k, e', i', j', k'), - (1, i, j, k, e', i', j', k'))\\ - = - \begin{bmatrix} - \alpha & -\beta & -\gamma & -\delta - & -\epsilon & -\zeta & -\eta & -\theta \\ - +\beta & \alpha & +\delta & -\gamma - & +\zeta & -\epsilon & -\theta & +\eta \\ - +\gamma & -\delta & \alpha & +\beta - & +\eta & +\theta & -\epsilon & -\zeta \\ - +\delta & +\gamma & -\beta & \alpha - & +\theta & -\eta & +\zeta & -\epsilon \\ - +\epsilon & -\zeta & -\eta & -\theta - & \alpha & +\beta & +\gamma & +\delta \\ - +\zeta & +\epsilon & -\theta & +\eta - & -\beta & \alpha & -\delta & +\gamma \\ - +\eta & + \theta & +\epsilon & -\zeta - & -\gamma & +\delta & \alpha & -\beta \\ - +\theta & -\eta & +\zeta & +\epsilon - & -\delta & -\gamma & +\beta & \alpha - \end{bmatrix}\,, -\end{multline*} -hence ${}^{\mathrm{t}} \mathrm{M}^{G}_{o} \mathrm{M}^{G}_{o} = -{}^{\mathrm{t}} \mathrm{M}^{D}_{o} \mathrm{M}^{D}_{o} = |o| -\mathrm{I}_{8}$, and $\det \mathrm{M}^{G}_{o} = \det \mathrm{M}^{D}_{o} -= (\alpha^{2} + \beta^{2} + \gamma^{2} + \delta^{2} + \epsilon^{2} + -\zeta^{2} + \eta^{2} + \theta^{2})^{4}$. Therefore if $|o| = 1$, -$\mathrm{M}^{G}_{o} \in \mathrm{SO}(8, \setR)$ and -$\mathrm{M}^{D}_{o} \in \mathrm{SO}(8, \setR)$. Again, we only get -a very tiny fraction of $\mathrm{SO}(8, \setR)$ that way. - -Also, and contrary to the case for the real numbers, the complex numbers -and the quaternions, in general $\mathrm{M}^{G}_{o o'} \neq -\mathrm{M}^{G}_{o} \mathrm{M}^{G}_{o'}$ and $\mathrm{M}^{D}_{o o'} \neq -\mathrm{M}^{D}_{o'} \mathrm{M}^{D}_{o}$, due to the non-associativity of -the product on \setO. For instance, $i' e' = -i$, but -$\mathrm{M}^{G}_{i'} \mathrm{M}^{G}_{e'} \neq \mathrm{M}^{G}_{-i}$. - -If we try to do the same thing with hexadecimalions, we find that -neither -$\cMatrix([l \mapsto hl]\lbc% -(1\lbc i\lbc j\lbc k\lbc e'\lbc i'\lbc j'\lbc k'\lbc e''\lbc -i''\lbc j''\lbc k''\lbc e'''\lbc i'''\lbc j'''\lbc k''')\lbc% -(1\lbc i\lbc j\lbc k\lbc e'\lbc i'\lbc j'\lbc k'\lbc -e''\lbc i''\lbc j''\lbc k''\lbc e'''\lbc i'''\lbc j'''\lbc k'''))$ -nor its right-hand version -are rotation in general, even if $\|l\| = 1$. That trail ends here as -well! - -\chapter{Building the Quaternions} -\label{part:II} - -\section{What to find here} -\label{part:II.1} - -This chapter, except for Section~\ref{part:II.5}, only consists of -well-known classical results (\cite{Bourbaki:sc},~\cite{Lang:1971vf}, -\ldots). Some have been slightly restated (usually with -simplifications) from their original sources, but hardly anything new is -presented here. In case the sources disagree on definitions,~\cite{Bourbaki:sc} -will take precedence. - -\section[Cayley \& alternate algebras]{Cayley algebra, alternative -algebra} -\label{part:II.2} - -Some of the structures we will be considering will not even be -associative. To save what may be, a weaker structure, which is -interesting in its own right is presented first. An algebra -\xsetE is said to be \emph{alternative\/} if the following trilinear -application, known as the \emph{associator\/} of \xsetE, is -alternating (which means its value is zero if two of its arguments are -identical): -\begin{align*} - \funcDef{\mathrm{a}} - {\xsetE \times \xsetE \times \xsetE}{(x, y, z)} - {\xsetE}{x * (y * z) - (x * y) * z} -\end{align*} - -This notion is interesting as, though an alternative algebra is not as -wieldy as an associative algebra, it is such that every sub-algebra -engendered by any two elements \emph{is\/} associative. It also implies that an -alternative algebra is a \emph{division\/} algebra (which means that for any $x -\in \xsetE, x \neq 0$, the applications $\xsetE \to \xsetE; -y \mapsto x * y$ and $\xsetE \to \xsetE; y \mapsto y * x$ are -bijective, or that elements are ``simplifiable''). In particular the -inverse of a non-zero element (if it exists) is unique in such an -algebra. - -The meat of this chapter is the following structure. - -Let \xsetA be a commutative ring, and \xsetE an algebra over -\xsetA, not necessarily commutative or associative, but having a -unit element $e$ (remember that since \xsetE is an -\xsetA-algebra, then $(\forall \lambda \in \xsetA) (\forall x -\in \xsetE) \lambda \cdot x = (\lambda \cdot e) * x = x * (\lambda -\cdot e)$). - -A \emph{conjugation\/} over \xsetE is any (there may be none) -bijective, \xsetA-linear, function $\sigma: \xsetE \to -\xsetE$ such that: -\begin{enumerate} -\item $\sigma(e) = e$. -\item $(\forall (x, y) \in \xsetE^{2}) \sigma(x * y) = \sigma(y) * - \sigma(x)$ (\emph{beware\/} the inversion of $x$ and $y$!). -\item $(\forall x \in \xsetE) (x + \sigma(x)) \in \xsetA \cdot - e$ and $(\forall x \in \xsetE) (x * \sigma(x)) \in \xsetA - \cdot e$. -\end{enumerate} - -These properties imply\footnote{$(x + \sigma(x)) \in \xsetA \cdot e -\Rightarrow x * \sigma(x) = x * (x + \sigma(x)) - x * x = (x + -\sigma(x)) * x - x * x = \sigma(x) * x$.} $(\forall x \in \xsetE) x -* \sigma(x) = \sigma(x) * x$, and\footnote{Given $x \in \xsetE$, -there exists $\alpha \in \xsetA$ such that $x + \sigma(x) = \alpha -\cdot e$; the \xsetA-linearity of $\sigma$ then implies $\sigma(x) -+ \sigma \circ \sigma(x) = \sigma(x + \sigma(x)) = \alpha \cdot -\sigma(e)$, and finally, $\sigma(e) = e$.} $(\forall x \in \xsetE) -\sigma \circ \sigma(x) = x$. - -We will also write $\bar{x}$ for $\sigma(x)$. - -If \xsetE is such an algebra, and if $\sigma$ is a conjugation -over \xsetE, the structure $(\xsetE, +, *, \cdot, \sigma)$ is -said to be a cayley algebra over \xsetA. On such a structure, it -is convenient to consider the cayley trace and cayley norm (an -unfortunate misnomer as it is actually quadratic\ldots), defined -respectively by $\Trace{\xsetE}(x) = x + \sigma(x)$ and -$\Norm{\xsetE}(x) = x * \sigma(x)$. - -Note that if $(\xsetE, +, *)$ has no zero divisors, for instance if -it is a field, then $\Norm{\xsetE}(x) = 0$ if and only if $x -= 0$. - -We have the important relations: -\begin{itemize} -\item $\Trace{\xsetE}(\sigma(x)) - = \Trace{\xsetE}(x)$ -\item $\Norm{\xsetE}(\sigma(x)) - = \Norm{\xsetE}(x)$ -\item $\fcT(x * y) = \fcT(y * x)$ -\item $\Trace{\xsetE}(x * \sigma(y)) - = \Trace{\xsetE}(y * \sigma(x)) - = \Trace{\xsetE}(x) * \Trace{\xsetE}(y) - - \Trace{\xsetE}(x * y) - = \Norm{\xsetE}(x + y) - \Norm{\xsetE}(x) - - \Norm{\xsetE}(y)$ -\end{itemize} - -It is interesting to note that $\fcT(x * y) = \fcT(y * x)$ -regardless of whether or not \xsetE is associative or commutative. - For the cayley norm, no such broad result seem to -hold;\footnote{Indeed, we have seen that such an equality does not hold -for hexadecimalions!} however if \xsetE is alternative, then we -also have $\Norm{\xsetE}(x * y) = \Norm{\xsetE}(x) -\Norm{\xsetE}(y)$. - -Finally, the following lemma will be useful for our purposes: -\begin{lemma}[Complexo\"{\i}d]\label{lemma:complexoid} - Given $x \in \xsetE$, $\fnVect_{\xsetA}(e, x)$, - the \xsetA-module spanned by $x$ and $e$, is stable for $*$; - it is a sub-cayley algebra of \xsetE which is associative and - commutative. If $x \not\in \xsetA \cdot e$, let - $y = \alpha \cdot e + \beta \cdot x$, - $\mathrm{M}^{G}_{y} = \cMatrix([u \mapsto y * u], (e, x), (e, x))$ - and $\mathrm{M}^{D}_{y} = \cMatrix([u \mapsto u * y], (e, x), (e, x))$; - then (with $\mathrm{T} \cdot e = \Trace{\xsetE}(x)$ and - $\mathrm{N} \cdot e = \Norm{\xsetE}(x)$) - \[ - \mathrm{M}_{y} - = \mathrm{M}^{G}_{y} - = \mathrm{M}^{D}_{y} - = \left[ \begin{array}{cc} - \alpha & -\beta \mathrm{N} \\ - \beta & \beta \mathrm{T} + \alpha - \end{array} \right] - .\] - Given $z \in \fnVect(e, x)$, we have - $\mathrm{M}_{y * z} - = \mathrm{M}_{y} \mathrm{M}_{z} - = \mathrm{M}_{z} \mathrm{M}_{y} - = \mathrm{M}_{z * y}$. - - This is a simple consequence of the fact that - $x * x = \mathrm{T} \cdot x - \mathrm{N} \cdot e$, with - $\mathrm{T} \cdot e = \Trace{\xsetE}(x)$ and - $\mathrm{N} \cdot e = \Norm{\xsetE}(x)$! -\end{lemma} - -This lemma allows us, in particular, to define unambiguously the $n$th -power, with $n \in \setN$, of any $x \in \xsetE$ by the usual -recursion rules, we will write the result, as usual $x^{n}$. It also -trivially induces the following scholie: -\begin{scholie}[Powers]\label{scholie:powers} - Given $x \in \xsetE$, - and $n \in \setN$, - $x^{n} \in \fnVect_{\xsetA}(e, x)$, - and $\Norm{\xsetE}(x^{n}) = {(\Norm{\xsetE}(x))}^{n}$. -\end{scholie} - -\section{The Cayley doubling procedure} -\label{part:II.3} - -It should be noted that this is simply the plain vanilla version of the -doubling process;\footnote{The general procedure involves abitrary -coefficients which parameterize the operations.} it will suffice here, -however. - -Let \xsetA be a commutative ring, and $(\xsetE, +, *, \cdot, -\sigma)$ a cayley algebra over \xsetA, not necessarily commutative -or associative, with unit element $e$. Let $\xsetF = \xsetE -\times \xsetE$ and $e_{\xsetF} = (e, 0) \in \xsetF$; -furthermore, let: -\[ -\begin{array}{c} - \funcDef{+_{\xsetF}} - {\xsetF \times \xsetF}{((x, y), (x', y'))} - {\xsetF}{(x + x', y + y')} \\ - \funcDef{*_{\xsetF}} - {\xsetF \times \xsetF}{((x, y), (x', y'))} - {\xsetF}{(x * x' - \overline{y'} * y, y * \overline{x'} + y' * x)} \\ - \funcDef{\cdot_{\xsetF}} - {\xsetA \times \xsetF}{(\lambda, (x, y))} - {\xsetF}{(\lambda \cdot x, \lambda \cdot y)} \\ - \funcDef{\sigma_{\xsetF}} - {\xsetF}{(x, y)} - {\xsetF}{(\sigma(x), -y)} -\end{array} -\] - -\begin{proposition}[Structure]\label{proposition:structure} - $(\xsetF, +_{\xsetF}, *_{\xsetF}, \cdot_{\xsetF})$ is - an \xsetA-algebra, with unit $e_{\xsetF}$, and - $\sigma_{\xsetF}$ is a conjugation over \xsetF; \xsetF - is associative if and only if \xsetE is both associative and - commutative; \xsetF is alternative if and only if \xsetE - is associative. Furthermore, $\Trace{\xsetF}((x, y)) - = \Trace{\xsetE}(x)$ and $\Norm{\xsetF}((x, y)) - = \Norm{\xsetE}(x) + \Norm{\xsetE}(y)$. -\end{proposition} - -Keep in mind that since \xsetF is also an \xsetA-algebra -then $(\forall \lambda \in \xsetA) (\forall (x, y) \in \xsetF) -\lambda \cdot_{\xsetF} (x, y) = (\lambda \cdot_{\xsetF} -e_{\xsetF}) *_{\xsetF} (x, y) = (x, y) *_{\xsetF} (\lambda -\cdot_{\xsetF} e_{\xsetF})$. It is interesting to note that, if -\xsetE is associative, we still have $\Norm{\xsetF}((x, -y) *_{\xsetF} (x', y')) = \Norm{\xsetF}((x, y)) -\Norm{\xsetF}((x', y'))$, even if \xsetF is not -associative. - -Given the proposition, we can (and will) identify \xsetE with -$\xsetE \times \{0_{\xsetE}\}$. Alternatively, we can identify -\xsetF with a superset of \xsetE. It is also possible to -identify \xsetA with a subset of \xsetE (and hence of -\xsetF as well), in that case we have noted that all elements of -\xsetA commute with all elements of \xsetE, for the -multiplication in \xsetE, as well as with all elements of -\xsetF, for the multiplication in \xsetF, even though -\xsetE or \xsetF might not be commutative. With this -identification, $\fcT$ and $\fcN$ have value in -\xsetA. - -\section{% - \texorpdfstring{% - \setR, \setC, \setH, \setO, \setX, \dots% - }{% - R, C, H, O, X, \dots% - }% -} -\label{part:II.4} - -We now consider $\xsetA = \setR$ and $\xsetE = \setR$, with $\sigma(x) -= x$ and $e = 1$, then $\Norm{\setR}(x) = x^{2}$ is always -positive (and zero if and only $x = 0$, as \setR is a field). When we -build \xsetF as above, we get exactly \setC, and $\sigma_{\xsetF}$ is -the usual conjugation on \setC. We define $i = (0; 1)$, and as stated -earlier, we identify \setR with $\setR \times \{0\}$. As is well -known, \setC is a commutative field, in particular, real numbers -commute with complex numbers. Due to our identifications, -$\Trace{\setC}$ and $\Norm{\setC}$ have values in \setR, -and actually, if $z = x + i y$ then $\Norm{\setC}(z) = -\Norm{\setR}(x) + \Norm{\setR}(y) = x^{2} + y^{2} = -|z|^{2} \geq 0$, and $\Norm{\setC}(z) = 0$ if and only if $z = -0$. We lose some of the original properties of \setR as we build -\setC, for instance we lose the existence of an order compatible with -the multiplication; we do get new and interesting properties at the same -time, of course. - -LetÕs do the doubling again, this time with $\xsetA = \setR$ and -$\xsetE = \setC$, with the usual conjugation, and this time we get -exactly \setH, the conjugation being the same as defined earlier, given -the definition of $j = (0; 1)$ and $k = (0; i)$, and the identification -of \setC with $\setC \times \{0\}$. Once again, we note that, as -predicted, for quaternion multiplication, real numbers commute with -quaternions, though some quaternions do not commute (for instance $i * j -\neq j * i$). As already stated \setH is a (non-commutative) field. -Once again, due to our new identifications, \Trace{H} and \Norm{H} have -values in \setR, and actually, \Norm{H} is always positive and -$\Norm{H}(q) = 0$ if and only if $q = 0$. We keep losing original -properties, most notably the commutativity, when we go from \setC to -\setH, but the new properties we gain, notably the link with rotations -in $\setR^{3}$, which we saw earlier, still makes it worthwhile. We -also see that $\Trace{H}(q) = 2 \fnRe q$ and $\Norm{H}(q) = -\|q\|^{2} = |q|^{2}$, as defined earlier. - -There being not such thing as too much of a good thing, letÕs do the -doubling once again, this time with $\xsetA = \setR$ and $\xsetE = -\setH$, and the conjugation just built on \setH. What the process -yields this time is known as the set of (Cayley) octonions, whose symbol -is \setO. We, as is now usual, identify \setH with $\setH \times -\{0\}$. Yet again, we note that, for octonion multiplication, real -numbers commute with octonions, though some octonions do not commute (as -some quaternions already do not commute). Yet again, due to our new -identifications, \Trace{O} and \Norm{O} have values in \setR, and -actually, \Norm{O} is always positive and $\Norm{O}(o) = 0$ if and only -if $o = 0$. The situation keeps deteriorating, though, as this time the -algebra is not associative anymore (but it is still associative). -Octonions do have uses, apart from being an example of a non-associative -algebra. They can be used to find a basis of non-vanishing vector field -on $\setS^{7}$ (the euclidian unit sphere in $\setR^{8}$), in the same -way quaternions can be used to find one on $\setS^{3}$, and complexes -are used to find one on $\setS^{1}$. They also see use in theoretical -physics (\cite{Dixon:1994rh}). Octonions still are a division -algebra, and non-zero octonions $O$ have ${[\Norm{O}(O)]}^{-1} -\sigma_{\setO}(O)$ for inverse. Despite the non-associativity of the -multiplication, we still have $\Norm{O}(o * o') = \Norm{O}(o) -\Norm{O}(o')$, since the multiplication \emph{is\/} associative on \setH. - -We can keep doubling \emph{ad nauseam}, but things really get unwieldy. -At the stage after octonions, the hexadecimalions, \setX, the algebra -is not even alternating. This author does not know of any use the -ulterior echelons may have been put to, if any. - -\section[The full Cayley ladder]{The full Cayley ladder all at once} -\label{part:II.5} - -One might wonder if the whole doubling procedure might be ``carried out -to infinity''. As it turns out, it can, after a fashion. We will -present here a special version\footnote{That is, the object built by our -``plain vanilla'' version of the Cayley doubling procedure.} of the -global object, for simplicity. - -Let \xsetA be a commutative ring, whose unit element will be called $e$. - -Let us call $\Fc{A}_{0} = \xsetA$ and $\sigma_{0}$ the identity over -\xsetA. It is quite obvious that $(\xsetA, +, \times, \times, \sigma)$ -is a cayley algebra over \xsetA. Using the doubling procedure, we build -$\Fc{A}_{1} = \xsetA \times \xsetA$ and $\sigma_{1}$, and by induction -we build $\Fc{A}_{n}$ and $\sigma_{n}$ for all $n \in \setN$. - -Consider $\xsetA[X]$ the set of polynomials (in one indeterminate $X$) -with coefficients in \xsetA. We already have an \xsetA-algebra -structure, which we will denote by $(\xsetA[X], +, \cdot, \times)$, and -is the usual commutative algebra. We readily identify $\Fc{A}_{0} = -\xsetA$ with constant polynomials, through an homomorphism of -\xsetA-modules $\Fc{J}_{0}$. It is trivial to see that $\Fc{A}_{n}$ -identifies with polynomials of degree less or equal to $2^{n} - 1$, thru -the trivial \xsetA-modules isomorphism $\Fc{J}_{n}$. Let us call -$I_{n}: \Fc{A}_{n} \to \Fc{A}_{n + 1}, x \mapsto (x, 0)$ the canonical -identification. Then $(\forall n \in \setN) \Fc{J}_{n + 1} \circ I_{n} -= \Fc{J}_{n}$, which means our identifications are all coherent. - -So every element of every rung of the Cayley ``ladder,'' build by -successively doubling the preceding rung and begun by \xsetA, \emph{a -finite number of times}, can be identified uniquely with some polynomial -with coefficients in \xsetA, and conversely every element of $\xsetA[X]$ -can be seen a some unique element of the Cayley ladder. As the -multiplication we will define differs, in general, from the polynomial -multiplication, we will choose a new symbol for our construction. - -Let $\Ck{C}(\xsetA)$ be some set equipotent to $\xsetA[X]$, the set of -polynomials in one indeterminate $X$ over \xsetA, thru a bijection -$\Ck{I}: \Ck{C}(\xsetA) \mapsto \xsetA[X]$. This bijection induces an -\xsetA-module on $\Ck{C}(\xsetA)$, from $(\xsetA[X], +, \cdot)$, which -we will denote by $(\Ck{C}(\xsetA), +, \cdot)$, and we identify -$\Fc{A}_{n}$ with $\Ck{I}^{-1}(\Fc{J}_{n}(\Fc{A}_{n}))$. - -We will now define a multiplication on $\Ck{C}(\xsetA)$, which we will -denote by ``$*$''. Let $\Ck{p} \in \Ck{C}(\xsetA)$ and $\Ck{q} \in -\Ck{C}(\xsetA)$; let $P = \Ck{I}(\Ck{p})$ and $Q = \Ck{I}(\Ck{q})$, then -there exists (at least) one $n \in \setN$ such that $P \in -\Fc{J}_{n}(\Fc{A}_{n})$ and $Q \in \Fc{J}_{n}(\Fc{A}_{n})$. We chose -the smallest such $n$. We now find the only $p_{n} \in \Fc{A}_{n}$ such -that $P = \Fc{J}_{n}(p_{n})$ and the only $q_{n} \in \Fc{A}_{n}$ such -that $Q = \Fc{J}_{n}(q_{n})$. Finally $\Ck{I}(\Ck{p} * \Ck{q}) = -\Fc{J}_{n}(p_{n} *_{\Fc{A}_{n}} q_{n})$. We note that for all $n' > n$, -we do have $P \in \Fc{J}_{n'}(\Fc{A}_{n'})$ and $Q \in -\Fc{J}_{n'}(\Fc{A}_{n'})$ and there are unique $p_{n'} \in \Fc{A}_{n'}$ -such that $P = \Fc{J}_{n'}(p_{n'})$ and $q_{n'} \in \Fc{A}_{n'}$ such -that $Q = \Fc{J}_{n'}(q_{n'})$, but thanks to the coherence of the -identifications we also have $\Fc{J}_{n}(p_{n} *_{\Fc{A}_{n}} q_{n}) = -\Fc{J}_{n'}(p_{n'} *_{\Fc{A}_{n'}} q_{n'})$. - -It is easy to verify that $(\Ck{C}(\xsetA), +, \cdot, *)$ is an -\xsetA-algebra. However, in general $\Ck{I}(\Ck{p} * \Ck{q}) \neq -\Ck{I}(\Ck{p}) \times \Ck{I}(\Ck{q})$. For instance if $\xsetA = -\setR$ then $X^{1} = \Ck{I}(i)$, $X^{2} = \Ck{I}(j)$, and $X^{3} = -\Ck{I}(k)$, and thus $\Ck{I}(i) = X^{1} \neq X^{5} = X^{2} \times X^{3} -= \Ck{I}(j) \times \Ck{I}(k)$. So \Ck{I} is \emph{not}, in general, an -algebra isomorphism between $(\Ck{C}(\xsetA), +, \cdot, *)$ and -$(\xsetA[X], +, \cdot, \times)$, as stated earlier. - -We likewise define the conjugation $\sigma$, and the cayley trace and -``norm,'' over $\Ck{C}(\xsetA)$ through the identifications -$\Fc{J}_{n}$, with values in $\Fc{A}_{0}$. It is now easy to check that -$(\Ck{C}(\xsetA), +, \cdot, *, \sigma)$ is a cayley algebra over \xsetA - (usually not commutative or associative), which, through the -identifications, contains all the rungs of the cayley doubling procedure -starting with \xsetA. Elements of \xsetA commute with all elements of -$\Ck{C}(\xsetA)$, for $*$. - -We will shortly use the fact that if $\Ck{I}(\Ck{p}) = \alpha_{0} + -\alpha_{1} X + \cdots + \alpha_{2^{n} - 1} X^{2^{n} - 1} = -\Fc{J}_{n}(p_{n})$, then $\Ck{I}(\Ck{p} * \Ck{p}) = \Fc{J}_{n}(p_{n} -*_{\Fc{A}_{n}} p_{n}) = (\alpha^{2}_{0} - (\alpha^{2}_{1} + \cdots + -\alpha^{2}_{2^{n} - 1})) + 2 \alpha_{1} \alpha_{0} X + \cdots + 2 -\alpha_{2^{n} - 1} \alpha_{0} X^{2^{n} - 1}$; this is simply proved by -recurrence. - -As a first example of Cayley ladders, let us consider -$(\Ck{C}(\frac{\setZ}{2 \setZ}), +, \times, *, \fnId)$. It is a -commutative and associative cayley algebra, the conjugation being the -identity on $\Ck{C}(\frac{\setZ}{2 \setZ})$; however it has zero -divisors, as for instance $\Ck{I}^{-1}(1 + X) * \Ck{I}^{-1}(1 + X) = 0$, -but if $\Ck{p} \in \Ck{C}(\frac{\setZ}{2 \setZ})$ and $\Ck{I}(\Ck{p})$ -has an odd number of 1 then $\Ck{I}(\Ck{p} * \Ck{p}) = 1$. - -The second, perhaps more interesting example, is $\Ck{C}(\setR)$, which -we have actually used already. In that case we can see that if $\Ck{a} -\in \Ck{C}(\setR)$, then $\Norm{\Ck{C}(\setR)}(\Ck{a})$ is always -positive, and $\Norm{\Ck{C}(\setR)}(\Ck{a}) = 0$ if and only if $\Ck{a} -= 0$; furthermore, $\Ck{a} \neq 0 \Rightarrow -[{(\Norm{\Ck{C}(\setR)}(\Ck{a}))}^{-1} \sigma(\Ck{a})] * \Ck{a} = \Ck{a} -* [{(\Norm{\Ck{C}(\setR)}(\Ck{a}))}^{-1} \sigma(\Ck{a})] = 1$. It is -also possible in this case to compute square roots! Indeed, let $x \in -\Fc{A}_{n}$, with $\Fc{J}_{n}(x) = A_{0} + A_{1} X + \cdots + A_{2^{n} - -1} X^{2^{n} - 1}$; we seek $y \in \Fc{A}_{n}$ with $\Fc{J}_{n}(y) = -\alpha_{0} + \alpha_{1} X + \cdots + \alpha_{2^{n} - 1} X^{2^{n} - 1}$ -such that $y * y = x$. This amounts to solving, in $\setR^{2^{n}}$ the -system: -\[ -\left\{ -\begin{array}{rcl} - \alpha^{2}_{0} - (\alpha^{2}_{1} + \cdots - + \alpha^{2}_{2^{n} - 1}) & = & A_{0} \\ - 2 \alpha_{1} \alpha_{0} & = & A_{1} \\ - & \vdots & \\ - 2 \alpha_{2^{n} - 1} \alpha_{0} & = & A_{2^{n} - 1} -\end{array} -\right. -\] -This system is easily solved by considering first the case $x = 0$, for -which there is a unique solution $y = 0$, second the subcase $x \in -\setR^{*}_{+}$ for which there are exactly two solutions given by $y = -\pm \sqrt{x}$, third the subcase $x \not\in \setR$ (if $n \geq 1$, of -course) for which there are also exactly two solution given by -$\Fc{J}_{n}(y) = \alpha_{0} + \frac{A_{1}}{2 \alpha_{0}} X + \cdots + -\frac{A_{2^{n} - 1}}{2 \alpha_{0}} X^{2^{n} - 1}$ with $\alpha_{0} = \pm -\sqrt{\frac{\fnRe x + |x|}{2}}$, and finally the case $x \in -\setR^{*}_{-}$, for which the solutions are all the $y \in \Fc{A}_{n}$ -such that $\Fc{R}(y) = 0$ and $|y| = |x|$. - -This means that the solutions to $y^{2} = x$, where $x \not\in -\setR_{-}$ are the same in every rung of the real Cayley ladder (that -is, there are exactly two, opposite solutions, belonging to the same -rung), and the solution to $y^{2} = 0$ is always $y = 0$, in whatever -rung of the Cayley ladder. However, solutions to $y^{2} = x$ for $x \in -\setR^{*}_{-}$ differ depending upon the precise rung: in \setR there -is no solution, in \setC there are exactly two, opposite, solutions, in -\setH and above there is an innumerable number of solutions (full -spheres)! - -Note that in any case a $y$ such that $y^{2} = x$ commutes with $x$, but -that two such solutions need not commute with each other! - -At least two topologies are interesting to consider on $\Ck{C}(\setR)$: -the norm topology induced by the square root of \fcN, the Cayley -``norm'' on $\Ck{C}(\setR)$ (we will write $\|c\| = \sqrt{\fcN(c)}$), -which we will call \fcT, and the strict inductive limit topology - (\cite{Khoan:1972ld}) defined by the rungs $\Ck{C}_{n} = -\Fc{J}_{n}(\Fc{A}_{n})$ of $\Ck{C}(\setR)$ on which we consider the -norms $q_{n} = 2^{\sup(0, n - 2)} \|\cdot\|$, which we will call -$\Trace{\infty}$. - -The problem with $\|\cdot\|$ is that it is not an algebra norm, as -evidenced by the hexadecimalions. Furthermore, $(\Ck{C}(\setR), -\fcT)$ is not complete, its completion being $\ell^{2}(\setR)$ with -its usual topology. - -On the other hand, $(\Ck{C}(\setR), \Trace{\infty})$ is complete, and -the product is (trivially) separately continuous (\cite{Bourbaki:do}), -but it is not known if it is continuous. - -For both topologies, any finite-dimensional vector space is closed and -the restriction to that vector space is just the usual (euclidian) -topology. - -At any rate, given $x \in \Ck{C}(\setR)$, the Powers Scholie proves -that $(\fnVect_{\setR}(1, x), \| \|)$ is a commutative -\setR-Banach algebra (of dimension 1 if and only if $x \in \setR$). - -As a final thought, since $\Ck{C}(\xsetA)$ is an \xsetA-Cayley algebra, -we can perform the Cayley doubling procedure on it! And again, and so -on and so forth\ldots. We can actually perform an infinity of doubling -as above, and embed all these doublings in what, essentially, is -$\xsetA[X, Y]$. And then we can start all over again\ldots. As we can -readily see, there is no ``ultimate'' step\ldots. What seems to be -going on is that we can build an object for any \emph{finite\/} ordinal - (\cite{Exbrayat:1971pk}), and we have built an object, -which we have called $\Ck{C}(\xsetA)$, for the first infinite ordinal -$\omega$. We have then seen that the doubling of $\Ck{C}(\xsetA)$ -yields the object corresponding to $\omega^{*}$ (the successor of -$\omega$). The next infinite ordinal with no predecessor ($2 \omega$) -corresponds to $\xsetA[X, Y]$. Further on (corresponding to -$\omega^{2}$), we find the set of polynomials in an indeterminate number -of indeterminates (i.e.\footnote{Recall that if $X$ is a -mono\"{\i}d (~\cite{Bourbaki:sc}) and $Y$ is a set, $X^{[Y]}$ is -the set of functions from $Y$ to $X$ which take values different from -the neutral element of $X$ only for a finite numbers of elements of -$Y$.}\ $\xsetA^{[\setN^{[\setN]}]}$). It is not clear, however, in -which way we can extend the construction to any set of ordinals - (i.e.\ there is no clear transfinite ``recurrence formula''). - -\chapter{The Exponential} -\label{part:III} - -\section{What to find here} -\label{part:III.1} - -This chapter is mostly designed to prove the explicit formula for the -exponential in $\Ck{C}(\setR)$, and give several related results. As -far as I known, these results are new. - -There are many notions of the exponential, and many ways to see several -of them. These, of course, agree when various different definitions can -be put forward for the same object to be exponentiated. We will be -concerned here mainly with the analystÕs point of view, and define the -exponential of quaternions thru the use of the usual power series - (\cite{Beardon:1979ek},~\ldots). It is known that the approach -detailed in~\cite{Pham:1996gf} could also be used, at least for -quaternions, though I believe it would then be necessary to derive the -power series representation (or the intermediary differential -representation we will also use) to achieve our present goal. It -remains to be seen if it can also be carried over to the whole of -$\Ck{C}(\setR)$. - -\section{Definition} -\label{part:III.2} - -Given $x \in \Ck{C}(\setR)$, we will call \emph{exponential\/} of $x$, -and we will write $\exp x$ the element of $\Ck{C}(\setR)$ given by -$\sum^{+\infty}_{n = 0} \frac{x^{n}}{n!}$. The unambiguity and -existence of $\exp x$ is given by the fact that -$(\fnVect_{\setR}(1, x), \| \|)$ is a commutative \setR-Banach -algebra, as we have said earlier. This, of course agrees with the -definition on \setR and \setC. We must bear in mind that $\exp x \in -\fnVect_{\setR}(1, x)$. - -We see at once that $(\forall x \in \Ck{C}(\setR)) \exp \overline{x} = -\overline{\exp x}$. The exponential is continuous when restricted to -each rung of $\Ck{C}(\setR)$, and has its values into the same rung (we -will give a more precise result later on). - -\section{Links with differentiation} -\label{part:III.3} - -Differentiating a function of one or several quaternions (or higher in -the Cayley ladder) is quite problematic. Of course, since -$\fnVect_{\setR}(1, x)$ is commutative, there is no ambiguity in -defining $\frac{f(y) - f(x)}{y - x}$ if $y \in \fnVect_{\setR}(1, -x)$, and we can therefore differentiate -$\exp|_{\fnVect_{\setR}(1, x)}$ with respect to some $y \in -\fnVect_{\setR}(1, x)$ and find that it is once again -$\exp|_{\fnVect_{\setR}(1, x)}$. - -It is more fruitful, however, to differentiate a function of a -\emph{real\/} variable, with values in some topological \setR-vector -space. - -Let us therefore consider, for some $x \in \Ck{C}(\setR)$, the function -$e_{x}: [\setR \to \Ck{C}(\setR), t \mapsto \exp tx]$. It is clear -that $e_{x}$ takes its values in $\fnVect_{\setR}(1, x)$, is -differentiable and $e'_{x}(t) = x e_{x}(t) = e_{x}(t) x$, and of course -$e_{x}(0) = 1$. This, of course proves that $e_{x}$ is the unique -solution to $f' = x f, f(0) = 1$ in $\cBasis^{1}(\setR, -\fnVect_{\setR}(1, x))$, the set of one-time continuously -differentiable functions from \setR to $\fnVect_{\setR}(1, x)$. -Given any rung \Fc{E} of $\Ck{C}(\setR)$ such that $x \in \Fc{E}$, -$e_{x}$ is still the unique solution to $f' = x f, f(0) = 1$ in -$\cBasis^{1}(\setR, \Fc{E})$. - -The perhaps surprising phenomenon is when we consider the equation $f' = -x f, f(0) = \gamma$ in $\cBasis^{1}(\setR, \Fc{E})$ for some rung \Fc{E} -of $\Ck{C}(\setR)$ such that $x \in \Fc{E}$, and $\gamma \in \Fc{E}$. -If $\Fc{E} = \setR$ or $\Fc{E} = \setC$, then of course the solution -is $e_{x}(t) \gamma$, and it turns out this is still true if $\Fc{E} = -\setH$, because of the associativity of the quaternionic product (this, -actually, is how one can navigate the unit sphere of the quaternions, -which is useful for interpolating between orientations, and was the -problem under examination in~\cite{Pletincks:1989ds}). It is -interesting to note that this is \emph{still\/} true if $\Fc{E} = \setO$ -because of the alternative nature of that algebra. This stops to be -true with hexadecimalions, however. Indeed, consider $x = i + e'''$ and -$\gamma = j$, and let $\mathrm{g}(t) = e_{x}(t) \gamma$. We will -shortly see that $e_{i + e'''}(\frac{\pi \sqrt{2}}{4}) = -\frac{\sqrt{2}}{2} (i + e''')$, from which we can deduce -$\mathrm{g}(\frac{\pi \sqrt{2}}{4}) = \frac{\sqrt{2}}{2} (i + e''') j$ -and $\mathrm{g}'(\frac{\pi \sqrt{2}}{4}) = ((i + e''') -\frac{\sqrt{2}}{2} (i + e''')) j = -\sqrt{2} j$ whereas $(i + e''') -\mathrm{g}(\frac{\pi}{2}) = (i + e''') (\frac{\sqrt{2}}{2} (i + e''') j) -= \sqrt{2}(-j + k''')$, and therefore $\mathrm{g}'(\frac{\pi}{2}) \neq -(i + e''') \mathrm{g}(\frac{\pi}{2})$. Numerical integration procedures -will yield the solution to the differential equation, and therefore not -the exponential function, unless care has been taken to chose the -starting point correctly. - -\section[The exponential formula]{The closed formula for the exponential -in $\Ck{C}(\setR)$} -\label{part:III.4} - -We now give the main result of this work. Note that it is closed only -in that we assume the exponential and classical trigonometric functions -on \setR to be givens.\footnote{A family of special functions will be -of interest here, that of the ``Sinus Cardinal'' functions, defined for -some parameter $a \in \setR^{*}_{+}$ by $\fnSinc_{a}: [\setR \to -\setR, x \mapsto \frac{\sin \pi x / a}{\pi x / a}]$. We will, by -similitude, define the ``Hyperbolic Sinus Cardinal'' family of functions -defined for some parameter $a \in \setR^{*}_{+}$ by -$\fnSinhc_{a}: [\setR \to \setR, x \mapsto \frac{\sinh \pi x / -a}{\pi x / a}]$. These functions are entire functions on all of -\setR.} - -\begin{theorem}[Exponential]\label{theorem:exponential} -If $x \in \Ck{C}(\setR)$ then $\exp x = \mathrm{e}^{\fnRe x} -[\cos \|\fnUr x\| + (\fnSincPi \|\fnUr x\|) - (\fnUr x)]$. - -Let $y \in \Ck{C}(\setR)$ such that $\fnRe y = 0, \|y\| = 1$; -then $y^{2} = [\fcT(y) - \bar{y}] y = -\fcN(y) = -1$. Therefore, in -$\fnVect_{\setR}(1, y)$ computations are carried out exactly as -in \setC, with $y$ taking the place of $i$. More precisely, $[\setC -\to \fnVect_{\setR}(1, y), a + ib \mapsto a + by]$ is a Banach -isomorphism. - -Let now $x \in \Ck{C}(\setR)$. If $x \in \setR$, we see -the result is trivially true. Assume, then that $x \not\in \setR$, and -let $\hat{x} = \frac{\fnUr x}{\|\fnUr x\|}$. Then -$\fnRe \hat{x} = 0, \|\hat{x}\| = 1, x = \fnRe x + -\|\fnUr x\| \hat{x}$, and of course $\fnVect_{\setR}(1, x) -= \fnVect_{\setR}(1, \hat{x})$. The previous identification then -allows us to find $\exp x = \mathrm{e}^{\fnRe x} [\cos -\|\fnUr x\| + (\sin \|\fnUr x\|) \hat{x}]$. -\end{theorem} - -As an example, we have, as announced earlier, $\exp \frac{\pi -\sqrt{2}}{4} (i + e''') = \frac{\sqrt{2}}{2} (i + e''')$. - -\section[Exponential properties]{Some properties of the exponential and -further consequences} -\label{part:III.5} - -We compute at once $\|\exp x\| = \mathrm{e}^{| \fnRe x|}$. - -As should be expected when we lose the benefit of commutativity, the -exponential of a sum is in general different from the product of the -exponentials; for instance we have $(\exp i) (\exp j) = {(\cos 1)}^{2} + i -(\sin 1) (\cos 1) + j (\sin 1) (\cos 1) + k {(\sin 1)}^{2}$ whereas $\exp -(i + j) = (\cos \sqrt{2}) + \frac{\sqrt{2}}{2} (i + j)$. We also see -immediately that the exponential is not injective on any rung \Fc{E} of -$\Ck{C}(\setR)$ containing \setC, as it is already not injective on -\setC! We, however also lose the periodicity when \Fc{E} contains -\setH, as the periods would make an additive subgroup of \Fc{E} but the -solutions of $\exp x = 1$ on \setH are exactly the set $2 \pi . \setN -. \{0\} \times \setS^{2}$ (with $\setS^{2}$ the unit sphere of -$\setR^{3}$); the rest is number theory (and trying to fit square pegs -into round holes). We have more details on the surjectivity of the -exponential: -\begin{corollary}[surjectivity]\label{corollary:surjectivity} -If \Fc{E} is a rung of $\Ck{C}(\setR)$ containing \setC, then the -exponential is a surjection from \Fc{E} onto $ \Fc{E} -\{0\}$. - -We first note that given any $x \in \Ck{C}(\setR)$, $\|\exp x\| = -\mathrm{e}^{| \fnRe x|}$ proves that the exponential never take -the value 0 on $\Ck{C}(\setR)$. - -Let now $y \in \Fc{E}, y \neq 0$. If $y \in \setR$ we know we can -solve our problem (in \setR if $y > 0$, in \setC if $y < 0$). Assume -therefore that $y \not\in \setR$. We can find $\rho \in \setR$ such -that $\mathrm{e}^{\rho} = \|y\|$. Let $\tilde{y} = \frac{y}{\|y\|}$; -$\|\tilde{y}\| = 1$ and $\tilde{y} \not\in \setR$, so let $\hat{y} = -\frac{\fnUr \tilde{y}}{\|\fnUr \tilde{y}\|}$, so that -$\tilde{y} = \fnRe \tilde{y} + \|\fnUr \tilde{y}\| -\hat{y}$, $\fnRe \tilde{y} \neq 0$ and -${(\fnRe \tilde{y})}^{2} + {\|\fnUr \tilde{y}\|}^{2} = 1$. -Let $\theta \in ]0; \pi[$; the unique number such that $\cos \theta = -\fnRe \tilde{y}$ and $\sin \theta = \|\fnUr \tilde{y}\|$. -We see that $\exp (\rho + \theta \hat{y}) = y$. -\end{corollary} - -We can likewise find closed formul\ae\ for other interesting entire -functions (defining $\cos x = \sum^{+\infty}_{n = 0} \frac{{(-1)}^{n} x^{2 -n}}{(2 n)!}$, $\sin x = \sum^{+\infty}_{n = 0} \frac{{(-1)}^{n} x^{2 n + -1}}{(2 n + 1)!}$, $\cosh x = \sum^{+\infty}_{n = 0} \frac{x^{2 n}}{(2 -n)!}$, $\sinh x = \sum^{+\infty}_{n = 0} \frac{x^{2 n + 1}}{(2 n + -1)!}$), to wit: -\[ - \begin{array}{rcl} - \cos x & = & (\cos \fnRe x) (\cosh \|\fnUr x\|) - - (\sin \fnRe x) (\fnSinhcPi \|\fnUr x\|) - (\fnUr x) \\ - \sin x & = & (\sin \fnRe x) (\cosh \|\fnUr x\|) - + (\cos \fnRe x) (\fnSinhcPi \|\fnUr x)\|) - (\fnUr x) \\ - \cosh x & = & (\cosh \fnRe x) (\cos \|\fnUr x\|) - + (\sinh \fnRe x) (\fnSincPi \|\fnUr x\|) - (\fnUr x) \\ - \sinh x & = & (\sinh \fnRe x) (\cos \|\fnUr x\|) - + (\cosh \fnRe x) (\fnSincPi \|\fnUr x\|) - (\fnUr x) - \end{array} -\] -and of course many other such. - -\section{Conclusion} -\label{part:III.6} - -We have found a closed formula for the exponential, for quaternions, -octonions, and beyond. - -An interesting application of this formula is navigation on the unit -sphere of the quaternions, leading to an algorithm for the interpolation -of orientations, but which, in general, does not preserve the -horizontal. This can also be achieved, however, and has been -implemented by the author and a colleague (\cite{Holin:lq}). - -\appendix -\chapter{Addenda} -\label{part:A} - -\section{More structure} -\label{part:A.1} - -It is interesting to note that \setC, \setH and \setO are (left) -vector spaces over \setC. - -The basis of \setH as a left \setC-vector space is $(1, j)$, and the -basis of \setO as a left \setC-vector space is $(1, j, e', j')$. -However, if $q = \alpha + \beta i + \gamma j + \delta k \in \setH$, -then $q = (\alpha + \beta i) + (\gamma + \delta i) j$, but if $o = -\alpha + \beta i + \gamma j + \delta k + \epsilon e' + \zeta i' + \eta -j' + \theta k' \in \setO$ then $o = (\alpha + \beta i) + (\gamma + -\delta i) j + (\epsilon + \zeta i) e' + (\eta - \theta i) j'$ (note the -minus sign in the last factor). - -If we write $q = \Gamma + \Delta j$, with $\Delta \in \setC$ and -$\Gamma \in \setC$, then $\overline{q} = \overline{\Gamma} - \Delta j$, -and if we also have $p = \mathrm{A} + \mathrm{B} j$, with $\mathrm{A} -\in \setC$ and $\mathrm{B} \in \setC$, then $p q = (\mathrm{A} \Gamma -- \mathrm{B} \overline{\Delta}) + (\mathrm{A} \Delta + \mathrm{B} -\overline{\Gamma}) j$. In particular, if $z \in \setC$ then $j z = -\overline{z} j$. - -Things break down when we want to consider \setO as a structure over -\setH, however. Indeed, there is no widely-accepted generalization of -vector field where the role of the scalars is taken by a non-commutative -structure, as is the case with \setH as most interesting properties of -vector spaces fail to remain true in that case, in general (though by -requiring the scalars to be merely a commutative ring instead of a full -blown field, quite a few properties remain true; this structure is known -as a module). - -However, if $o = \alpha + \beta i + \gamma j + \delta k + \epsilon e' + -\zeta i' + \eta j' + \theta k' \in \setO$, then it is also true that $o -= (\alpha + \beta i + \gamma j + \delta k) + (\epsilon + \zeta i + \eta -j + \theta k) e'$. - -\section{More Geometry} -\label{part:A.2} - -Another interesting way to see \setH is as $\setR \times \setR^{3}$. -In this case, if $q_{1} = (t_{1}, V_{1}) \in \setR \times \setR^{3}$ -and $q_{2} = (t_{2}, V_{2}) \in \setR \times \setR^{3}$, then the -quaternionic product can be expressed as $q_{1} q_{2} = (t_{1} t_{2} - -V_{1} \cdot V_{2}, t_{1} V_{2} + t_{2} V_{1} + V_{1} \wedge V_{2})$, -with ``$\cdot$'' the scalar product on $\setR^{3}$ and ``$\wedge$'' the -vector product in $\setR^{3}$. - -\section[Quaternions for rotations]{Finding the quaternions for a given -rotation of $\setR^{3}$} -\label{part:A.3} - -If we are given the rotation in term of vector and angle in $[0; +\pi]$, -then this has been solved in the main text (if the angle is in $[-\pi; -0]$, we just take the opposite of both the angle and vector; the -identity and its opposite are trivial to solve). The opposite -quaternion is also a solution, of course. - -If we are simply given a rotation matrix, then, essentially, we first -find its elements (vector and angle), and use the procedure above. To -find an invariant vector, we simply solve the linear system which -defines them. For the angle, we first find its cosine using the trace. -Then we build a vector orthogonal to the invariant vector we found - (always possible starting from one of the canonical basis vector and -using some classical orthonormalization procedure) to check the sign of -the angle. - -If we are given a succession of rotation, it may be advantageous in -applications to chose among the successions of pairs of opposite -solutions that for which the distance between successive quaternions is -the smallest. - -\section{More rotations} -\label{part:A.4} - -We have considered $[\setH \to \setH, p \mapsto p q]$ and $d_q = -[\setH \to \setH, p \mapsto q p]$ in Chapter~\ref{part:I} and seen, -through some amount of computation, that they gave rise to rotations on -$\setR^{4}$, when $|q| = 1$. - -We present here another take on the same subject, aimed at giving -effective methods of parameterizing $\mathrm{SO}(4, \setR)$. - -It is also interesting to consider $g_q = [\setH \to \setH, p \mapsto -p \overline{q}]$, as both $g_q$ and $d_q$ are \setC-linear operators on -\setH, which trivially verify $g_{q q'} = g_q \circ g_{q'}$ and $d_{q -q'} = d_q \circ d_{q'}$. We can also very simply verify that $(g_{q}(p) -| g_{q}(p')) = \|q\|^{2} (p | p') = (d_{q}(p) | d_{q}(p'))$. Obviously -$g_{1}$ and $d_{1}$ are both the identity on \setH. Considering them -now as \setR-linear operators on \setH, we see that their determinant -in the canonical basis must stay of the same sign on $\setS^{3}$, hence -must stay positive (since $1 \in \setS^{3}$), therefore must be always -equal to 1 on $\setS^{3}$. - -Hence $[\setS^{3} \to \mathrm{SO}(4, \setR), q \mapsto g_{q}]$ and -$[\setS^{3} \to \mathrm{SO}(4, \setR), q \mapsto d_{q}]$ are both -group homeomorphism, sending 1 to the identity. - -Using the same kind of topological argument as in the case of -$\mathrm{SO}(3, \setR)$, we get the parameterization of $\mathrm{SO}(4, -\setR)$ that we announced (\cite{Berger:1990ce},~\cite{Godbillon:1971th}) -simply by considering $(p, q) \mapsto d_{p} \circ g_{q}$, -save for the determination of the kernel. We will, however, aim -here for a more constructive approach to surjectivity. - -Given $r$ a rotation of $\setR^{4}$, we seek $p \in \setS^{3}$ and $q -\in \setS^{3}$ such that for any quaternion $s$ we have $p s -\overline{q} = r(s)$. Hence, by applying that to $s = 1$, we find that -necessarily $p = r(1) q$ (since $\overline{q} = q^{-1}$ as $|q| = 1$). -Therefore we are led to solve $q s \overline{q} = \rho(s)$ for all $s$, -with $\rho(s) = \overline{r(1)} r(s)$ (as $|r(t)| = |t|$ for all -quaternion $t$, since $r$ is a rotation, and hence $|r(1)| = 1$). But -then $\rho(\mu) = \overline{r(1)} r(\mu) = \mu \overline{r(1)} r(1) = -\mu$ for all $\mu \in \setR$, which means \setR is invariant, and we -know that $\rho$ is a rotation $\setR^{4}$, as the composition of $r$, -which is one by hypothesis, and the multiplication on the left by a unit -quaternion, which is one also as we have seen in the main text. This -means we are simply back to solving on $\setS^{3}$ the equation -$\rho_{q} = \cMatrix(\rho, \cBasis, \cBasis)$ (with $q \mapsto \rho_{q}$ as -presented in Chapter~\ref{part:I}). - -Finally, given two unit quaternions $p = \alpha + \beta i + \gamma j + -\delta k$ and $q = \epsilon + \zeta i + \eta j + \theta k$, the rotation -matrix on $\setR^{4}$ is given explicitly by: -\begin{align*} - \begin{bmatrix} - \sst \alpha \epsilon + \beta \zeta + \gamma \eta + \delta \theta & - \sst +\alpha \zeta - \beta \epsilon - \gamma \theta + \delta \eta & - \sst +\alpha \eta + \beta \theta - \gamma \epsilon - \delta \zeta & - \sst +\alpha \theta - \beta \eta + \gamma \zeta - \delta \epsilon \\ - \sst -\alpha \zeta + \beta \epsilon - \gamma \theta + \delta \eta & - \sst \alpha \epsilon + \beta \zeta - \gamma \eta - \delta \theta & - \sst -\alpha \theta + \beta \eta + \gamma \zeta - \delta \epsilon & - \sst +\alpha \eta + \beta \theta + \gamma \epsilon + \delta \zeta \\ - \sst -\alpha \eta + \beta \theta + \gamma \epsilon - \delta \zeta & - \sst +\alpha \theta + \beta \eta + \gamma \zeta + \delta \epsilon & - \sst \alpha \epsilon - \beta \zeta + \gamma \eta - \delta \theta & - \sst -\alpha \zeta - \beta \epsilon + \gamma \theta + \delta \eta \\ - \sst -\alpha \theta - \beta \eta + \gamma \zeta + \delta \epsilon & - \sst -\alpha \eta + \beta \theta - \gamma \epsilon + \delta \zeta & - \sst +\alpha \zeta + \beta \epsilon + \gamma \theta + \delta \eta & - \sst \alpha \epsilon - \beta \zeta - \gamma \eta + \delta \theta - \end{bmatrix} -\end{align*} - -\bibliographystyle{alpha} -\bibliography{HyperComplex} -\end{document} diff --git a/doc/Jamfile.v2 b/doc/Jamfile.v2 new file mode 100644 index 000000000..150f48476 --- /dev/null +++ b/doc/Jamfile.v2 @@ -0,0 +1,17 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +xml math : math.qbk ; +boostbook standalone + : + math + : + nav.layout=none + navig.graphics=0 + ; + + diff --git a/doc/beta.qbk b/doc/beta.qbk new file mode 100644 index 000000000..f924868b2 --- /dev/null +++ b/doc/beta.qbk @@ -0,0 +1,108 @@ +[section The Beta Function] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T beta(T a, T b); + + }} // namespaces + +[h4 Description] + +The beta function is defined by: + +[$../equations/beta1.png] + +[$../graphs/beta.png] + +And for small values: + +[$../graphs/beta-small.png] + +There are effectively two versions of this function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the mantissa +of T correspond to a certain __lanczos. In practice any built in +floating point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp. + +[h4 Accuracy] + +The following table shows peek errors for various domains of input arguments, +plus comparisons to other open source implementations where available. Note that +only results for the widest floating point type on the system are given as +narrower types have __zero_error. + +[table Peek Errors In the Beta Function +[[Mantissa Size] [Platform and Compiler] [Errors in range\n0.4 < a,b < 100] [Errors in range\n1e-6 < a,b < 36]] +[[53] [Win32, Visual C++ 8] [Peek=99 Mean=22\n\n(GSL Peek=1178 Mean=238)\n(Cephes=1612)] [Peek=10.7 Mean=2.6\n\n(GSL Peek=12 Mean=2.0)\n(Cephes=174)]] +[[64] [Red Hat Linux IA32, g++ 3.4.4] [Peek=112.1 Mean=26.9] [Peek=15.8 Mean=3.6]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peek=61.4 Mean=19.5] [Peek=12.2 Mean=3.6]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=42.03 Mean=13.94] [Peek=9.8 Mean=3.1]] +] + +Note that the worst errors occur when a or b are large, and that +when this is the case the result is very close to zero, so absolute +errors will be very small. + +[h4 Testing] + +A mixture of spot tests of exact values, and randomly generated test data are +used: the test data was computed using NTL::RR at 1000-bit precision. + +[h4 Implementation] + +Traditional methods of evaluating the beta function either involve evaluating +the gamma functions directly, or taking logarithms and then +exponentiating the result. However, the former is prone to overflows +for even very modest arguments, while the latter is prone to cancellation +errors. As an alternative, if we regard the gamma function as a white-box +containing the __lanczos then we can combine the power terms: + +[$../equations/beta2.png] + +which is almost the ideal solution, however almost all of the error occurs +in evaluating the power terms when /a/ or /b/ are large. If we assume that /a > b/ +then the larger of the two power terms can be reduced by a factor of /b/, which +immediately cuts the maximum error in half: + +[$../equations/beta3.png] + +This may not be the final solution, but it is very competitive compared to +other implementation methods. + +The generic implementation - where no __lanczos approximation is available - is +implemented in a very similar way to the generic version of the gamma function. +Again in order to avoid numerical overflow the power terms that prefix the series and +continued fraction parts are collected together into: + +[$../equations/beta8.png] + +where la, lb and lc are the integration limits used for a, b, and a+b. + +There are a few special cases worth mentioning: + +When /a/ or /b/ are less than one, we can use the recurrence relations: + +[$../equations/beta4.png] + +[$../equations/beta5.png] + +To move to a more favorable region where they are both greater than 1. + +In addition: + +[$../equations/beta7.png] + + +[endsect] + diff --git a/doc/big_whole.html b/doc/big_whole.html deleted file mode 100644 index 45d47c1bb..000000000 --- a/doc/big_whole.html +++ /dev/null @@ -1,724 +0,0 @@ - - - -Boost Arbitrary-length Whole-number Library - - -

c++boost.gif (8819 bytes)Arbitrary-length Whole-number Library

- -

The headers boost/math/big_whole_core.hpp and boost/math/big_whole.hpp cover the definition of and operations for unlimited-length nonnegative integers (only memory constraints should limit the numbers supported).

- -

Contents

- -
    -
  1. Contents
    2. -
  2. Rationale
    3. -
  3. Core Definition and Routines -
    4. -
  4. Additional Routines -
    5. -
  5. References
    6. -
  6. Credits -
    7. -
- -

Rationale

- -

Various "bignum" classes provide objects that model numbers beyond the limits of the built-in numeric types. The class provided in this library gives a portable implementation of nonnegative integers (also known as "whole" numbers). The implementation tries to lean towards good performance, but it's probably not as good as the various highly-refined "bignum" types available.

- -

Core Definition and Routines

- -

The boost/math/big_whole_core.hpp header contains the declaration for the unlimited-length nonnegative integer type and declarations for various core routines.

- -

Header Synopsis

- -
-#include <limits>  // for std::numeric_limits
-
-namespace boost
-{
-namespace math
-{
-
-class big_whole;
-
-void  swap( big_whole &a, big_whole &b );
-
-big_whole  and_not( big_whole const &a, big_whole const &b );
-
-big_whole  abs( big_whole const &w );
-short      sgn( big_whole const &w );
-
-big_whole  operator !( big_whole const &w );
-big_whole  operator +( big_whole const &w );
-big_whole  operator -( big_whole const &w );
-big_whole  operator ++( big_whole &w, int );
-big_whole  operator --( big_whole &w, int );
-
-bool  operator ==( big_whole const &lhs, big_whole const &rhs );
-bool  operator !=( big_whole const &lhs, big_whole const &rhs );
-bool  operator < ( big_whole const &lhs, big_whole const &rhs );
-bool  operator > ( big_whole const &lhs, big_whole const &rhs );
-bool  operator <=( big_whole const &lhs, big_whole const &rhs );
-bool  operator >=( big_whole const &lhs, big_whole const &rhs );
-
-big_whole  operator &( big_whole const &lhs, big_whole const &rhs );
-big_whole  operator |( big_whole const &lhs, big_whole const &rhs );
-big_whole  operator ^( big_whole const &lhs, big_whole const &rhs );
-
-big_whole  operator >>( big_whole const &value, big_whole const &shift );
-big_whole  operator <<( big_whole const &value, big_whole const &shift );
-
-big_whole  operator +( big_whole const &augend, big_whole const &addend );
-big_whole  operator -( big_whole const &minuend, big_whole const &subtrahend );
-
-big_whole &  operator  &=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator  |=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator  ^=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator >>=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator <<=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator  +=( big_whole &lhs, big_whole const &rhs );
-big_whole &  operator  -=( big_whole &lhs, big_whole const &rhs );
-
-}
-}
-
-namespace std
-{
-
-template < >  class numeric_limits< boost::math::big_whole >;
-
-}
-
- -

Unlimited-Length Nonnegative Integer Type

- -

Objects from the boost::math::big_whole class represent unlimited-length nonnegative integers. Even though the implemenation is advanced, objects from this type can be passed around like any other value-based type. The usual mathematical functions and operators are provided. These objects have a binary (i.e. radix-2) representation, so various bit-twiddling functions and operators are also provided.

- -
-#include <boost/cstdint.hpp>  // for boost::uintmax_t
-#include <cstddef>            // for std::size_t
-#include <valarray>           // for std::valarray
-
-class boost::math::big_whole
-{
-public:
-    // Lifetime management
-    big_whole();
-    big_whole( big_whole const &other );
-
-    big_whole( boost::uintmax_t v );
-
-    explicit  big_whole( std::valarray<bool> const &b );
-    explicit  big_whole( std::valarray<std::size_t> const &i );
-
-    // Object-mutating operations
-    void  swap( big_whole &other );
-
-    void  assign( big_whole const &other );
-    void  assign( boost::uintmax_t v );
-
-    void  reconfigure( std::valarray<bool> const &b );
-    void  reconfigure( std::valarray<std::size_t> const &i );
-
-    // Value-accessing operations
-    uintmax_t                   to_uintmax() const;
-    std::valarray<bool>         to_bit_vector() const;
-    std::valarray<std::size_t>  to_bit_indices() const;
-
-    bool  is_even() const;
-
-    // Bit-twiddling operations
-    void  reset();
-    void  reset( std::size_t from, std::size_t to );
-    void  reset( std::size_t i );
-
-    void  set( std::size_t from, std::size_t to );
-    void  set( std::size_t i );
-
-    void  flip( std::size_t from, std::size_t to );
-    void  flip( std::size_t i );
-
-    void  bit_assign( std::size_t from, std::size_t to, bool value );
-    void  bit_assign( std::size_t i, bool value );
-
-    void  bits_assign( std::size_t from, std::size_t to, big_whole const &values );
-
-    // Bit-inspecting operations
-    std::size_t  length() const;
-
-    std::size_t  count() const;
-    bool         any() const;
-    bool         none() const;
-
-    bool       test( std::size_t i ) const;
-    big_whole  tests( std::size_t from, std::size_t to ) const;
-
-    big_whole  reverse( std::size_t cap ) const;
-    big_whole  reverse() const;
-
-    std::size_t  search_tested_bit( std::size_t from, bool value, bool increase_indices ) const;
-    std::size_t  next_tested_bit( std::size_t from, bool value ) const;
-    std::size_t  previous_tested_bit( std::size_t from, bool value ) const;
-    std::size_t  next_set_bit( std::size_t from ) const;
-    std::size_t  next_reset_bit( std::size_t from ) const;
-    std::size_t  previous_set_bit( std::size_t from ) const;
-    std::size_t  previous_reset_bit( std::size_t from ) const;
-
-    // Self-operator mutators
-    void  not_self();
-    void  same_self();
-    void  negate_self();
-
-    // Object-accessing operations
-    int  compare( big_whole const &other ) const;
-
-    bool  intersects( big_whole const &other ) const;
-
-    std::size_t  scale() const;
-
-    // Operators
-    big_whole &  operator =( big_whole const &rhs );
-
-    operator bool_type() const;
-
-    big_whole &  operator ++();
-    big_whole &  operator --();
-
-};
-
- -

Constructors

- -

Note: the automatically-defined destructor is used.

- -
-
big_whole(); -
- Effects: sets the internal state such that the stored numerical value is zero (using a minimal amount of memory)
- Rationale: standard default-state creation - -
big_whole( big_whole &other ); -
- Effects: sets the internal state such that the stored numerical value is the same as the numerical value stored in other
- Rationale: standard creation via copying - -
big_whole( boost::uintmax_t v ); -
- Effects: sets the internal state such that the stored numerical value is the same as v
- Rationale: creation via conversion - -
explicit big_whole( std::valarray<bool> const &b ); -
- Effects: sets the internal state such that the stored numerical value has the same bit pattern as b
- Rationale: creation via bit-pattern - -
explicit big_whole( std::valarray<std::size_t> const &i ); -
- Effects: sets the internal state such that the stored numerical value has its set bits at the positions given by i
- Rationale: creation via set-bit-location specification -
- -

Observer Member Functions

- -
-
boost::uintmax_t to_uintmax() const; -
- Returns: the numeric value stored in *this, reduced modulo 2std::numeric_limits< boost::uintmax_t >::digits
- Rationale: explicit reverse conversion - -
std::valarray<bool> to_bit_vector() const; -
- Returns: the bit pattern for the numeric value stored in *this
- Rationale: break down to a bit pattern - -
std::valarray<std::size_t> to_bit_indices() const; -
- Returns: the places of the set bits for the numeric value stored in *this
- Rationale: break down to a set-bit index list - -
bool is_even() const; -
- Returns: whether the numeric value stored in *this is even (or odd if not)
- Note: same as *this % 2 == 0, but is probably implemented more efficiently
- Rationale: simple even/odd test - -
std::size_t length() const; -
- Returns: the smallest nonnegative integer, n, such that 2n is greater than the numeric value stored in *this (returns 0 if *this represents zero)
- Note: same as this->to_bit_vector().size() or (for nonzero length) 1 + this->to_bit_indices().max(), but is probably implemented more efficiently
- Rationale: size of bit-pattern vector - -
std::size_t count() const; -
- Returns: the number of set bits in the numeric value stored in *this
- Note: same as this->to_bit_indices().size(), but is probably implemented more efficiently
- Rationale: size of set-bit index list - -
bool any() const; -
- Returns: whether any bits in the numeric value stored in *this are set
- Note: same as this->count() > 0, but is probably implemented more efficiently
- Rationale: reversed zero test - -
bool none() const; -
- Returns: whether no bits in the numeric value stored in *this are set
- Note: same as !this->any()
- Rationale: zero test - -
bool test( std::size_t i ) const; -
- Returns: whether the bit corresponding to the 2i place for the numeric value stored in *this is set
- Rationale: inspect a specific bit - -
big_whole tests( std::size_t from, std::size_t to ) const; -
- Precondition: from <= to
- Returns: states of the bits corresponding between the 2from and 2to places (inclusive) for the numeric value stored in *this; equivalent to (*this % 2to + 1) / 2from
- Rationale: inspect a set of bits at once - -
big_whole reverse( std::size_t cap ) const; -
- Returns: a copy of the lowest cap + 1 bits (i.e. apply a modulo 2cap + 1) of *this in reverse order (e.g. for n ranging from 0 to cap, the bit in the 2n place now appears in the 2cap - n place)
- Rationale: generate bit-pattern palidromes - -
big_whole reverse() const; -
- Returns: 0 if *this represents zero; otherwise, this->reverse( this->length() - 1 )
- Rationale: shorthand for the most common case of bit-order reversal (minimal amount for all significant bits) - -
std::size_t search_tested_bit( std::size_t from, bool value, bool increase_indices ) const; -
- Returns: if increase_indices is TRUE, then this->next_tested_bit( from, value ), else this->previous_tested_bit( from, value )
- Rationale: most general bit-searching routine - -
std::size_t next_tested_bit( std::size_t from, bool value ) const; -
- Returns: the index of the first bit after from that is equal to value, 0 if such a bit cannot be found
- Rationale: forward searching for a specific bit value - -
std::size_t previous_tested_bit( std::size_t from, bool value ) const; -
- Returns: the index of the first bit before from that is equal to value, the maximum std::size_t value if such a bit cannot be found
- Rationale: backward searching for a specific bit value - -
std::size_t next_set_bit( std::size_t from ) const; -
- Returns: this->next_tested_bit( from, true )
- Rationale: forward searching for a TRUE bit - -
std::size_t next_reset_bit( std::size_t from ) const; -
- Returns: this->next_tested_bit( from, false )
- Rationale: forward searching for a FALSE bit - -
std::size_t previous_set_bit( std::size_t from ) const; -
- Returns: this->previous_tested_bit( from, true )
- Rationale: backward searching for a TRUE bit - -
std::size_t previous_reset_bit( std::size_t from ) const; -
- Returns: this->previous_tested_bit( from, false )
- Rationale: backward searching for a FALSE bit - -
int compare( big_whole const &other ) const; -
- Returns: zero if *this and other represent the same value, a positive number if *this is the greater, and a negative number if *this is the lesser (i.e. other is the greater)
- Rationale: equality/ordered comparison core routine - -
bool intersects( big_whole const &other ) const; -
- Returns: whether there is at least one bit place that is set in both *this and other
- Note: same as (*this & other) != 0, but is probably implemented more efficiently
- Rationale: simple bit-set intersection test - -
std::size_t scale() const; -
- Returns: zero if *this represents zero, otherwise the smallest nonnegative integer x such that *this >> x is odd
- Rationale: find number of trailing zeroes -
- -

Mutating Member Functions

- -
-
void swap( big_whole &other ); -
- Effects: *this takes ownership of other's state; other takes ownership of the state *this had before this member function call
- Rationale: standard class-type-specific swap member-function refinement - -
void assign( big_whole const &other ); -
- Effects: sets the internal state to a copy of other's state
- Postconditions: *this == other
- Rationale: member-function repackaging of copy operation - -
void assign( uintmax_t v ); -
- Effects: sets the internal state to give a numeric value equivalent to v
- Postconditions: *this == big_whole( v )
- Rationale: converting copy operation - -
void reconfigure( std::valarray<bool> const &b ); -
- Effects: sets the internal state to give a numeric value that has the bit pattern given by b
- Postconditions: *this == big_whole( b )
- Rationale: re-configuring copy operation with a bit pattern - -
void reconfigure( std::valarray<std::size_t> const &i ); -
- Effects: sets the internal state to give a numeric value that has only the bit positions given by i set
- Postconditions: *this == big_whole( i )
- Rationale: re-configuring copy operation with a list of the set bits - -
void reset();
- void reset( std::size_t from, std::size_t to );
- void reset( std::size_t i ); -
- Effects: changes the internal state so some set of the bits of the represented value become zero; the zero-argument version affects all bits, the two-argument version affects the bits between the 2from and 2to places (inclusive), the one-argument version affects the bit at the 2i place
- Rationale: specific-bit(s) assignment, unsetting - -
void set( std::size_t from, std::size_t to );
- void set( std::size_t i ); -
- Effects: changes the internal state so some set of the bits of the represented value become one; the two-argument version affects the bits between the 2from and 2to places (inclusive), the one-argument version affects the bit at the 2i place
- Rationale: specific-bit(s) assignment, setting - -
void flip( std::size_t from, std::size_t to );
- void flip( std::size_t i ); -
- Effects: changes the internal state so some set of the bits of the represented value become inverted (zeroes become ones, ones become zeroes); the two-argument version affects the bits between the 2from and 2to places (inclusive), the one-argument version affects the bit at the 2i place
- Rationale: specific-bit(s) assignment, inverting - -
void bit_assign( std::size_t from, std::size_t to, bool value );
- void bit_assign( std::size_t i, bool value ); -
- Effects: changes the internal state so some set of the bits of the represented value become value; the three-argument version affects the bits between the 2from and 2to places (inclusive), the two-argument version affects the bit at the 2i place
- Rationale: specific-bit(s) assignment, user-defined constant value - -
void bits_assign( std::size_t from, std::size_t to, big_whole const &values ); -
- Precondition: from <= to
- Effects: changes the internal state so the set of bits between the 2from and 2to places (inclusive) of the represented value become (the lowest to - from + 1 bits of) values; in other words, *this ^= (((*this / 2from) ^ values) * 2from) % 2to + 1
- Rationale: specific-bit(s) assignment, user-defined specific values - -
void not_self(); -
- Effects: changes the internal state so a value of zero becomes one and any nonzero value becomes zero
- Rationale: doing x = !x without the copying - -
void same_self(); -
- Effects: none (may secretly optimize storage....)
- Rationale: doing x = +x without the copying - -
void negate_self(); -
- Effects: changes the internal value to its additive inverse
- Throws: std::range_error if the inverse cannot be represented
- Note: since this type can only store nonnegative values, all stored positive values will throw; the sole representable nonpositive value, zero, is the only one that does not throw (but it transforms to itself)
- Rationale: doing x = -x without the copying -
- -

Operators

- -
-
big_whole & operator =( big_whole const &rhs ); -
- Effects: calls this->assign( rhs )
- Returns: *this
- Rationale: standard copy-assignment operation - -
operator bool_type() const; -
- Returns: if this->any(), then some non-zero/non-null/true value, otherwise a zero/null/false value
- Note: bool_type is an unspecified built-in type that supports Boolean semantics and should have a minimum of common non-Boolean semantics (this probably excludes bool since that type [mistakenly?] fully participates in integral and/or numeric operations)
- Rationale: Boolean (output) conversion - -
big_whole & operator ++(); -
- Effects: increases the internal value by one
- Returns: *this
- Rationale: pre-increment - -
big_whole & operator --(); -
- Effects: decreases the internal value by one
- Returns: *this
- Throws: std::range_error if the internal value is currently zero (because the next lower value, -1, cannot be represented by this type)
- Rationale: pre-decrement -
- -

Non-Member Functions & Operators

- -
-void
-boost::math::swap( boost::math::big_whole &a, boost::math::big_whole &b );
-
- -

Effects: calls a.swap( b )

- -

Postconditions: a has the state that b had before the function call; b has the state a had before the function call

- -

Rationale: exchanges the state of two objects; can use in standard(-like) algorithms with ADL

- -
-boost::math::big_whole
-boost::math::and_not( boost::math::big_whole const &a, boost::math::big_whole const &b );
-
- -

Returns: an object with a set of bits such that a particular bit is set only if the corresponding bit in a is set and the corresponding bit in b is unset

- -

Note: friendly with big_whole

- -

Rationale: a method to compute a & ~b without the need for the bitwise-complement operator (big_whole cannot have that operator because it gives a result approaching infinity for any input)

- -
-boost::math::big_whole
-boost::math::abs( boost::math::big_whole const &w );
-
- -

Returns: the absolute value of w

- -

Note: since big_whole only supports nonnegative values, this function always returns a copy of w

- -

Rationale: finding the absolute value with a function profile used in the standard

- -
-short
-boost::math::sgn( boost::math::big_whole const &w );
-
- -

Returns: -1, 0, or +1 depending on w being negative, zero, or positive, respectively

- -

Note: since big_whole only supports nonnegative values, this function only returns zero or one

- -

Rationale: finding the sign of a number

- -

Operators

- -
-
boost::math::big_whole boost::math::operator !( boost::math::big_whole const &w ); -
- Returns: 1 if w represents zero, 0 otherwise
- Note: calls big_whole::not_self() - -
boost::math::big_whole boost::math::operator +( boost::math::big_whole const &w ); -
- Returns: w
- Note: calls big_whole::same_self() - -
boost::math::big_whole boost::math::operator -( boost::math::big_whole const &w ); -
- Returns: the additive inverse of w, if it is representable
- Throws: std::range_error if the inverse cannot be represented (which is true for any nonzero value since big_whole can only store nonnegative values)
- Note: calls big_whole::negate_self() - -
boost::math::big_whole boost::math::operator ++( boost::math::big_whole &w, int );
- boost::math::big_whole boost::math::operator --( boost::math::big_whole &w, int ); -
- Effects: for operator $$, calls w.operator $$()
- Returns: a copy of w before the change - -
bool boost::math::operator ==( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs );
- bool boost::math::operator !=( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs );
- bool boost::math::operator <( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs );
- bool boost::math::operator >( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs );
- bool boost::math::operator <=( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs );
- bool boost::math::operator >=( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs ); -
- Returns: whether the given relation applies to lhs and rhs
- Note: calls big_whole::compare() - -
boost::math::big_whole boost::math::operator &( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs ); -
- Returns: an object produced by a bitwise-AND combination of the bits in lhs and rhs (a bit in the result is set if both the corresponding bits in lhs and rhs are set, otherwise the bit is unset)
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator |( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs ); -
- Returns: an object produced by a bitwise-OR combination of the bits in lhs and rhs (a bit in the result is set if at least one of the corresponding bits in lhs or rhs is set, otherwise the bit is unset)
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator ^( boost::math::big_whole const &lhs, boost::math::big_whole const &rhs ); -
- Returns: an object produced by a bitwise-XOR combination of the bits in lhs and rhs (a bit in the result is set if the corresponding bits in lhs and rhs differ, otherwise the bit is unset)
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator >>( boost::math::big_whole const &value, boost::math::big_whole const &shift ); -
- Returns: floor( value / 2shift )
- Throws: std::length_error if shift cannot be completely represented by a std::size_t object
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator <<( boost::math::big_whole const &value, boost::math::big_whole const &shift ); -
- Returns: value * 2shift
- Throws: std::length_error if shift cannot be completely represented by a std::size_t object; std::overflow_error if any of the set bits in the returned object would have a bit-place index beyond std::size_t's maximum value
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator +( boost::math::big_whole const &augend, boost::math::big_whole const &addend ); -
- Returns: the sum of augend and addend
- Note: friendly with big_whole - -
boost::math::big_whole boost::math::operator -( boost::math::big_whole const &minuend, boost::math::big_whole const &subtrahend ); -
- Returns: the difference between minuend and subtrahend
- Throws: std::range_error if the difference cannot be represented, i.e. the minuend is less than the subtrahend
- Note: friendly with big_whole - -
boost::math::big_whole & boost::math::operator &=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator |=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator ^=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator >>=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator <<=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator +=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs );
- boost::math::big_whole & boost::math::operator -=( boost::math::big_whole &lhs, boost::math::big_whole const &rhs ); -
- Effects: for operator $$=, calls lhs = lhs $$ rhs
- Returns: lhs -
- -

Numeric Limits

- -

A std::numeric_limits< boost::math::big_whole > specialization is provided, with the appropiate entries filled.

- -

Additional Routines

- -

The boost/math/big_whole.hpp header includes the core header and contains the declarations for routines that work with unlimited-length nonnegative integers but are not needed for pure computation contexts (like the input and output routines).

- -

Header Synopsis

- -
-#include <iosfwd>                         // for std::basic_ostream and std::basic_istream (declarations)
-#include <string>                         // for std::basic_string
-#include <boost/math/big_whole_core.hpp>  // for boost::math::big_whole
-
-namespace boost
-{
-namespace math
-{
-
-template < typename Ch, class Tr, class Al >
-    std::basic_string<Ch, Tr, Al>  bigwhole_to_bitstring( big_whole const &w, Ch zero_char, Ch one_char );
-template < typename Ch, class Tr, class Al >
-    void  copy_bigwhole_to_bitstring( std::basic_string<Ch, Tr, Al> &s, big_whole const &w, Ch zero_char, Ch one_char );
-
-template < typename Ch, class Tr >
-    std::basic_ostream<Ch, Tr> &  operator <<( std::basic_ostream<Ch, Tr> &os, big_whole const &w );
-
-template < typename Ch, class Tr >
-    std::basic_istream<Ch, Tr> &  operator >>( std::basic_istream<Ch, Tr> &is, big_whole &w );
-
-}
-}
-
- -

Bit-String Conversion Functions

- -
-template < typename Ch, class Tr, class Al >
-std::basic_string<Ch, Tr, Al>
-boost::math::bigwhole_to_bitstring( boost::math::big_whole const &w, Ch zero_char, Ch one_char );
-
- -

Returns: a minimal-length string for the base-2 representation of w, where zero_char and one_char are used to represent the unset and set bits, respectively

- -

Rationale: provides a base-2 representation of a big_whole object which is suitable for text processing

- -
-template < typename Ch, class Tr, class Al >
-void
-boost::math::copy_bigwhole_to_bitstring
-(
-    std::basic_string<Ch, Tr, Al> &s,
-    boost::math::big_whole const &w,
-    Ch zero_char,
-    Ch one_char
-);
-
- -

Effects: assigns to s the result of bigwhole_to_bitstring using w, zero_char, and one_char as parameters

- -

Rationale: same as bigwhole_to_bitstring, but without the need to explicitly specify the template parameters

- -

Input & Output Operators

- -
-template < typename Ch, class Tr >
-std::basic_ostream<Ch, Tr> &
-boost::math::operator <<( std::basic_ostream<Ch, Tr> &os, boost::math::big_whole const &w );
-
- -

Effects: writes characters, forming a string representation of the nonnegative integer w, to os; should use the standard numeric I/O options

- -

Returns: os

- -

Rationale: enables big_whole objects to be written by streaming

- -
-template < typename Ch, class Tr >
-std::basic_istream<Ch, Tr> &
-boost::math::operator >>( std::basic_istream<Ch, Tr> &is, boost::math::big_whole &w );
-
- -

Effects: reads characters, hopefully forming a string representation of a nonnegative integer, from is, saving any successful result in w; should use the standard numeric I/O options

- -

Returns: is

- -

Rationale: enables big_whole objects to be read by streaming

- -

References

- - - -

Credits

- -

Contributors

- -
-
Daryle Walker -
Started the library. -
- -

History

- -
-
30 Jan 2004, Daryle Walker -
Initial version -
- -
- -

Revised: 16 February 2004

- -

Copyright 2004 Daryle Walker. Use, modification, and distribution are subject to the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or a copy at <http://www.boost.org/LICENSE_1_0.txt>.)

- - diff --git a/doc/boost-build.jam b/doc/boost-build.jam new file mode 100644 index 000000000..b016488e6 --- /dev/null +++ b/doc/boost-build.jam @@ -0,0 +1,3 @@ + +boost-build ../../../../develop/boost/tools/build/v2 ; + diff --git a/doc/concepts.qbk b/doc/concepts.qbk new file mode 100644 index 000000000..02c4f056a --- /dev/null +++ b/doc/concepts.qbk @@ -0,0 +1,66 @@ +[section Using With NTL] + +The special functions and tools in this library can be used with NTL::RR and +to a lesser extent with NTL::quad_float. In order to do so you will need to +apply the following patch to NTL: libs/math/tools/ntl.diff +This patch adds trivial converting constructors to NTL::RR and NTL::quad_float, +and forces conversions to RR to proceed via `long double` rather than `double`. +The latter change is essential to accurately measure relative errors between +high precision calculations (using NTL::RR) and fixed precision calculations +(using `long double`). These occur, if for example, you are generating additional +__lanczos's. + +You will also need to include NTL via the header `` +and include that header /before/ you include any of the special function headers. +Rationale: the header `` includes some traits classes +and boilerplate code (mostly runtime substitutes for numeric_limits) that are +required by the special functions. Unfortunately, some of the stricter compilers +require these definitions to be seen before the templates that will use them are +parsed. + +Finally there is a high precision __lanczos suitable for use with NTL::RR, when +used at 1000-bit precision in libs/math/tools/ntl_rr_lanczos.hpp. The approximation +has a theoretical precision of > 90 decimal digits, and an experimental precision +of > 100 decimal digits. To use that approximation, just include that header, before +any of the special function headers (if you don't use it, you'll get a slower, but +fully generic implementation for all of the gamma-like functions). + +[endsect] + +[section Using With Boost.Interval] + +There are currently problems comparing two intervals that make it hard to use +Boost.Interval with this library. It should be just about possible to make it +work "as is", but this needs more investigation, and a simple, easy to use solution. + +[endsect] + +[section Conceptual Requirements] + +TODO. + +See boost/math/concepts/real_concept.hpp for the current concept-architype +test case. + +Basically requires: + +* The usual arithmetic operations. +* The usual comparison operations. +* The usual math functions (those in ). +* Either `std::numeric_limits` support (the easy way), or a small number +helper functions that provide the same functionality at runtime +rather than compile time (see boost/math/tools/precision.hpp). +* Promotion from the builtin floating point types to T, and the function +`real_cast` that converts from T to a narrower floating point type +(see boost/math/tools/real_cast.hpp). + +For efficient and accurate results, a __lanczos is also highly desireable. +You may be able to adapt an existing approximation from +boost/math/special_functions/lanczos.hpp: you will need change +static_cast's to lexical_cast's, and the constants to /strings/ +(in order to ensure the coefficients aren't truncated +to long double) and then specialise lanczos_traits for type T. Otherwise +you may have to hack libs/math/tools/lanczos_generator.cpp to find a suitable +approximation for your type T. + +[endsect] diff --git a/doc/equations/beta.xml b/doc/equations/beta.xml new file mode 100644 index 000000000..5b434633b --- /dev/null +++ b/doc/equations/beta.xml @@ -0,0 +1,517 @@ + +]> + + + + + + + + b + e + t + a + + + a + , + b + + + = + + + Γ + + + a + + + Γ + + + b + + + + + Γ + + + a + + + b + + + + + + + + + b + e + t + a + + + a + , + b + + + = + + + + + + a + + + g + − + 0.5 + + + a + + + b + + + g + − + 0.5 + + + + + + a + − + 0.5 + + + + + + + + b + + + g + − + 0.5 + + + a + + + b + + + g + − + 0.5 + + + + + b + + + + + e + + b + + + g + − + 0.5 + + + + + + + L + + + a + + + L + + + b + + + + + L + + + c + + + + + + + + + b + e + t + a + + + a + , + b + + + = + + + + + + a + + + g + − + 0.5 + + + a + + + b + + + g + − + 0.5 + + + + + + a + − + b + − + 0.5 + + + + + + + + + + a + + + g + − + 0.5 + + + + + b + + + g + − + 0.5 + + + + + + + + a + + + b + + + g + − + 0.5 + + + 2 + + + + + + b + + + + + e + + b + + + g + − + 0.5 + + + + + + + L + + + a + + + L + + + b + + + + + L + + + c + + + + + + + + + b + e + t + a + + + a + , + b + + + = + + + + a + + + b + + + b + + b + e + t + a + + + a + , + b + + + 1 + + + + + + + b + e + t + a + + + a + , + b + + + = + + + + + a + + + b + + + + + a + + + b + + + 1 + + + + + a + b + + + b + e + t + a + + + a + + + 1, + b + + + 1 + + + + + + + i + f + + a + ≫ + b + + a + n + d + + a + + + b + = + a + + + t + h + e + n + : + + + b + e + t + a + + + a + , + b + + + = + Γ + + + b + + + + + + + i + f + + a + = + 1 + + + t + h + e + n + : + + + b + e + t + a + + + a + , + b + + + = + + 1 + b + + + + + + + e + + l + c + − + l + a + − + l + b + + + + + + + + l + a + + + l + c + + + + + a + + + + + + + l + b + + + l + c + + + + + b + + + + + \ No newline at end of file diff --git a/doc/equations/beta1.png b/doc/equations/beta1.png new file mode 100644 index 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z); + +Returns the error function of z: + +[$../equations/erf1.png] + +[$../graphs/erf1.png] + + template + T erfc(T z); + +Returns the complement of the error function of z: + +[$../equations/erf2.png] + +[$../graphs/erf2.png] + +[h4 Accuracy] + +[table Errors In the Function erf(z) +[[Mantissa Size] [Platform and Compiler] [z < 0.5] [0.5 < z < 8] [z > 8]] +[[53] [Win32, Visual C++ 8] [Peek=0 Mean=0\n\n GSL Peek=2.0 Mean=0.3 \n\nCephes Peek=1.1 Mean=0.7] [Peek=0.9 Mean=0.09 \n\nGSL Peek=2.3 Mean=0.3 \n\nCephes Peek=1.3 Mean=0.2] [Peek=0 Mean=0\n\nGSL Peek=0 Mean=0 \n\nCephes Peek=0 Mean=0]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=0.7 Mean=0.07\n\n Native Peek=0.9 Mean=0.2] [Peek=0.9 Mean=0.2\n\n Native Peek=0.9 Mean=0.07] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=0.7 Mean=0.07\n\n Native Peek=0 Mean=0] [Peek=0.9 Mean=0.1\n\n Native Peek=0.5 Mean=0.03] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=0.8 Mean=0.1\n\n Native Peek=0.9 Mean=0.2] [Peek=0.9 Mean=0.1\n\n Native Peek=0.5 Mean=0.02] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0]] +] + +[table Errors In the Function erfc(z) +[[Mantissa Size] [Platform and Compiler] [z < 0.5] [0.5 < z < 8] [z > 8]] +[[53] [Win32, Visual C++ 8] [Peek=0.7 Mean=0.06 \n\nGSL Peek=1.0 Mean=0.4 \n\nCephes Peek=0.7 Mean=0.06] [Peek=0.99 Mean=0.3\n\nGSL Peek=2.6 Mean=0.6 \n\nCephes Peek=3.6 Mean=0.7] [Peek=1.0 Mean=0.2\n\nGSL Peek=3.9 Mean=0.4 \n\nCephes Peek=2.7 Mean=0.4]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0] [Peek=1.4 Mean=0.3\n\n Native Peek=1.3 Mean=0.3] [Peek=1.6 Mean=0.4\n\n Native Peek=1.3 Mean=0.4]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0] [Peek=1.4 Mean=0.3\n\n Native Peek=0 Mean=0] [Peek=1.5 Mean=0.4\n\n Native Peek=0 Mean=0] ] +[[113] [HPUX IA64, aCC A.06.06] [Peek=0 Mean=0\n\n Native Peek=0 Mean=0] [Peek=1.5 Mean=0.3\n\n Native Peek=0.9 Mean=0.08] [Peek=1.6 Mean=0.4\n\n Native Peek=0.9 Mean=0.1]] +] + +[h4 Testing] + +The tests for these functions come in two parts: basic sanity checks uses +spot values calculated using Mathworld's online calculator, while accuracy +checks use high-precision test values calculated at 1000-bit precision with +NTL::RR and this implementation. Note that the generic and type-specific +versions of these functions use differing implementations internally, so this +gives us reasonably independent test data. Using our test data to test other +known "good" implementations also provides an additional sanity check. + +One +should note that our tests rely on decimal to binary conversion of floating +point numbers, and assume that the result will be correctly rounded. In practice +it appears that in a few very rare cases the test data may be incorrect in the last bit: +this depends upon the compiler and standard library used, and means that the +relative errors quoted above have to treated somewhat circumspectly. Using +binary or hexadecimal coded test data would remove this issue (or at least +confirm whether it is actually an issue or not), but would make the test data +unportable, so is not used at present. + +[h4 Implementation] + +All versions of these functions first use the usual reflection formulas +to make their arguments positive: + + erf(-z) = 1 - erf(z); + + erfc(-z) = 2 - erfc(z); // prefered when -z < -0.5 + + erfc(-z) = 1 + erf(z); // prefered when -0.5 <= -z < 0 + +The generic versions of these functions are implemented in terms of +the incomplete gamma function. + +When the mantissa size is recognised (currently for 53, 64 and 113-bit reals, +plus single precision handled via promotion to double) +then a series of rational approximations are used. + +For `z <= 0.5` then a rational approximation to erf is used, based on the +observation that: + + erf(z)/z ~ 1.12.... + +Therefore erf is calculated using: + + erf(z) = z * (1.125F + R(z)); + +Where the rational approximation R(z) is optimised for absolute error: as +long as it's absolute error is small enough compared to 1.125 then any +roundoff error incured during the computation of R(z) will effectively +dissappear from the result. As a result the error for erf and erfc in this +region is very low: the last bit is incorrect in only a very small number of +cases. + +For `z > 0.5` we observe that over a small interval [a, b) then: + + erfc(z) * exp(z*z) * z ~ c + +for some constant c. + +Therefore for `z > 0.5` we calculate erfc using: + + erfc(z) = exp(-z*z) * (c + R(z)) / z; + +Again R(z) is optimised for absolute error, and the constant `c` is +the average of `erfc(z) * exp(z*z) * z` taken at the endpoints of the range. +Once again as long as the absolute error in R(z) is small +compared to `c` then `c + R(z)` will be correctly rounded, and the error +in the result will depend only on the accuracy of the exp function. In practice +in all but a very small number of cases the error is confined to the last bit +of the result. + +[endsect] + diff --git a/doc/erf_inv.qbk b/doc/erf_inv.qbk new file mode 100644 index 000000000..dd3002098 --- /dev/null +++ b/doc/erf_inv.qbk @@ -0,0 +1,70 @@ +[section The Error Function Inverses] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T erf_inv(T p); + + template + T erfc_inv(T p); + + }} // namespaces + +[h4 Description] + + template + T erf_inv(T z); + +Returns the inverse error function of z, that is a value x such that: + + p = erf(x); + +[$../graphs/erf_inv1.png] + + template + T erfc_inv(T z); + +Returns the inverse of the complement of the error function of z, that is a +value x such that: + + p = erfc(x); + +[$../graphs/erf_inv2.png] + +[h4 Accuracy] + +These functions are deliberately "detuned" to be not too accurate in order to avoid +undue iterations during their calculation. The relative error in these functions +will be similar for all types T at around 3 to 15 epsilon. + +[h4 Testing] + +There are two sets of tests: + +* Basic sanity checks attempt to "round-trip" from +/x/ to /p/ and back again. These tests have quite +generous tolerances: in general both the error functions and their +inverses change so rapidly in some places, that round tripping to more than a couple +of significant digits isn't possible. This is especially true when +/p/ is very near one: in this case there isn't enough +"information content" in the input to the inverse function to get +back where you started. +* Accuracy checks using high precision test values. These measure +the accuracy of the result, given /exact/ input values. + +[h4 Implementation] + +These functions use a rational approximation to calculate an initial +approximation to the result, then clean up the value using Halley iteration. +No more than 3 iterations are required in practice. + +[endsect] + diff --git a/doc/error.qbk b/doc/error.qbk new file mode 100644 index 000000000..e3601f379 --- /dev/null +++ b/doc/error.qbk @@ -0,0 +1,58 @@ +[section Relative Error] + +[caution __caution ] + +Given an actual value /a/ and a found value /v/ the relative error can be +calculated from: + +[$../equations/error1.png] + +Special care needs to be taken when one value is zero: we could either take the +absolute error in this case (but that's cheating as the absolute error is likely +to be very small), or we could assign a value of either 1 or infinity to the +relative error in this special case. In the test cases for the special functions +in this library, everything below a threshold is regarded as "effectively zero", +otherwise the relative error is assigned the value of 1 if only one of the terms +is zero. The threshold is currently set at `std::numeric_limits<>::min()`, in other words all +denormalised numbers are regarded as a zero. + +All the test programs calculate /quantised relative error/ whereas the graphs +in this manual are produced with the /actual error/. The difference is as +follows: in the test programs the test data is rounded to the target type +under test when +the program is compiled, the error observed will then be a whole number of +/units in the last place/ either rounded up from the actual error, or rounded +down (possibly to zero). In contrast the /true error/ is obtained by extending +the precision of the calculated value, and then comparing to the actual value: +in this case the calculated error may be some fraction of /units in the last place/. + +Note that throughout this manual and the test programs the relative error is +usually quoted in units of epsilon. However, remember that /units in the last place/ +more accurately reflect the number of contaminated digits, and that relative +error can /"wobble"/ by a factor of 2 compared to /units in the last place/. +In other words: two implementations of the same function, whose +maximum relative errors differ by a factor of 2, can actually be accurate +to the same number of binary digits. You have been warned! + +[#zero_error][h4 The Impossibility of Zero Error] + +For many of the functions in this library, it's assumed that the error is +"effectively zero" if the computation can be done with a number of guard +digits. However it should be remembered that if the result is a +/transcendental number/ +then as a point of principle we can never be sure that the result is accurate +to more than 1ulp. This is an example of /the table makers dilemma/: consider what +happens if the first guard digit is a one, and the remaining guard digits are all zero. +Do we have a tie or not? Since the only thing we can tell about a transcendental number +is that it's digits have no particular pattern, we can never tell if we have a tie, +no matter how many guard digits we have. Therefore we can never be completely sure +that the result has been rounded in the right direction. Of course transcendental +numbers that just happen to be a tie - for however many guard digits we have - are +extremely rare, and get rarer the more guard digits we have, but even so.... + +Refer to the classic text +[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic] +for more information. + +[endsect] + diff --git a/doc/fpclassify.qbk b/doc/fpclassify.qbk new file mode 100644 index 000000000..4606dc688 --- /dev/null +++ b/doc/fpclassify.qbk @@ -0,0 +1,89 @@ +[section Floating Point Classification: Infinities and NaN's] + +[h4 Synopsis] + + #define FP_ZERO /* implementation specific value */ + #define FP_NORMAL /* implementation specific value */ + #define FP_INFINITE /* implementation specific value */ + #define FP_NAN /* implementation specific value */ + #define FP_SUBNORMAL /* implementation specific value */ + + template + int fpclassify(T t); + + template + bool isfinite(T z); + + template + bool isinf(T t); + + template + bool isnan(T t); + + template + bool isnormal(T t); + +[h4 Description] + +These functions provide the same functionality as the macros with the same +name in C99, indeed if the C99 macros are available, then these functions +are implemented in terms of them, otherwise they rely on std::numeric_limits<> +to function. + +Note that the definition of these functions ['does not suppress the definition +of these names as macros by math.h] on those platforms that already provide +these as macros. That mean that the following have differing meanings: + + using namespace boost::math; + + // This might call a global macro if defined, + // might not work if the type of z is unsupported + // by the std lib: + isnan(z); + // + // This calls the boost version, works for any type + // that has numeric_limits support : + (isnan)(z); + // + // As above but with namespace qualification. + (boost::math::isnan)(z); + // + // This will cause a compiler error is isnan is a native macro: + boost::math::isnan(z); + +Detailed descriptions for each of these functions follows: + + template + int fpclassify(T t); + +Returns an integer value that classifies the value /t/: + +[table +[[FP_ZERO] [If /t/ is zero.]] +[[FP_NORMAL] [If /t/ is a non-zero, non-denormalised finite value.]] +[[FP_INFINITE] [If /t/ is plus or minus infinity.]] +[[FP_NAN] [If /t/ is a NaN.]] +[[FP_SUBNORMAL] [If /t/ is a denormalised number.]] +] + + template + bool isfinite(T z); + +Returns true only if /z/ is not an infinity or a NaN. + + template + bool isinf(T t); + +Returns true only if /z/ is plus or minus infinity. + + template + bool isnan(T t); + +Returns true only if /z/ is a NaN. + + template + bool isnormal(T t); + +Returns true only if /z/ is a normal number (not zero, infinite, NaN, or denomalised). + +[endsect] diff --git a/doc/fraction.qbk b/doc/fraction.qbk new file mode 100644 index 000000000..b84f1ecce --- /dev/null +++ b/doc/fraction.qbk @@ -0,0 +1,147 @@ +[section Continued Fraction Evaluation] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + typename detail::fraction_traits::result_type + continued_fraction_b(Gen& g, int bits); + + template + typename detail::fraction_traits::result_type + continued_fraction_b(Gen& g, int bits, boost::uintmax_t& max_terms); + + template + typename detail::fraction_traits::result_type + continued_fraction_a(Gen& g, int bits); + + template + typename detail::fraction_traits::result_type + continued_fraction_a(Gen& g, int bits, boost::uintmax_t& max_terms); + + }}} // namespaces + +[h4 Description] + +These functions all evaluate the continued fraction described by the /generator/ +type argument. The functions with an "_a" suffix evaluate the fraction: + +[$../equations/fraction2.png] + +and those with a "_b" suffix evaluate the fraction: + +[$../equations/fraction1.png] + +This latter form is somewhat more natural in that it corresponds with the usual +definition of a continued fraction, but note that the first /a/ value returned by +the generator is discarded. Further, often the first /a/ and /b/ values in a +continued fraction have different defining equations to the remaining terms, which +may make the "_a" suffixed form more appropriate. + +The generator type should be a function object which supports the following +operations: + +[table +[[Expression] [Description]] +[[Gen::result_type] [The type that is the result of invoking operator(). + This can be either an arithmetic type, or a std::pair<> of arithmetic types.]] +[[g()] [Returns an object of type Gen::result_type.\n + Each time this operator is called then the next pair of /a/ and /b/ + values is returned. Or, if result_type is an arithmetic type, + then the next /b/ value is returned and all the /a/ values + are assumed to 1.]] +] + +In all the continued fraction evaluation functions the /bits/ parameter is the +number of bits precision desired in the result, evaluation of the fraction will +continue until the last term evaluated leaves the first /bits/ bits in the result +unchanged. + +If the optional /max_terms/ parameter is specified then no more than /max_terms/ +calls to the generator will be made, and on output, +/max_terms/ will be set to actual number of +calls made. This facility is particularly useful when profiling a continued +fraction for convergence. + +[h4 Implementation] + +Internally these algorithms all use the modified Lentz algorithm: refer to +Numeric Recipes, chapter 5 for more information, also +Lentz, W.J. 1976, Applied Optics, vol. 15, pp. 668-671. + +[h4 Examples] + +The golden ratio can be computed from the simplest continued fraction of all: + +[$../equations/fraction3.png] + +We begin by defining a generator function: + + template + struct golden_ratio_fraction + { + typedef T result_type; + + result_type operator() + { + return 1; + } + }; + +The golden ration can then be computed to double precision using: + + continued_fraction_a( + golden_ratio_fraction(), + std::numeric_limits::digits); + +It's more usual though to have to define both the /a/'s and the /b/'s +when evaluating special functions by continued fractions, for example +the tan function is defined by: + +[$../equations/fraction4.png] + +So it's generator object would look like: + + template + struct tan_fraction + { + private: + T a, b; + public: + tan_fraction(T v) + : a(-v*v), b(-1) + {} + + typedef std::pair result_type; + + std::pair operator()() + { + b += 2; + return std::make_pair(a, b); + } + }; + +Notice that if the continuant is subtracted from the /b/ terms, +as is the case here, then all the /a/ terms returned by the generator +will be negative. 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+[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T beta(T a, T b, T x); + + template + T betac(T a, T b, T x); + + template + T ibeta(T a, T b, T x); + + template + T ibetac(T a, T b, T x); + + }} // namespaces + +[h4 Description] + +There are four incomplete beta functions: two are normalised versions +that return values in the range [0, 1], and two are non-normalised and +return values in the range [0, beta(a, b)]. Users interested in statistical +applications should use the normalised versions (ibeta and ibetac). + +All of these function require /a > 0/, /b > 0/ and /0 <= x <= 1/. + + template + T beta(T a, T b, T x); + +Returns the full (non-normalised) incomplete beta function of a, b and x: + +[$../equations/ibeta1.png] + + template + T betac(T a, T b, T x); + +Returns the full (non-normalised) complement of the incomplete beta function of a, b and x: + +[$../equations/ibeta2.png] + + template + T ibeta(T a, T b, T x); + +Returns the normalised incomplete beta function of a, b and x: + +[$../equations/ibeta3.png] + + template + T ibetac(T a, T zb, T x); + +Returns the normalised complement of the incomplete beta function of a, b and x: + +[$../equations/ibeta4.png] + +[h4 Accuracy] + +The following tables give peek and mean relative errors in over various domains of +a, b and x, along with comparisons to other open source implementations where available. +Note that only results for the widest floating point type on the system are given as +narrower types have __zero_error. + +[table Errors In the Function beta(a, b, x) +[[Mantissa Size] [Platform and Compiler] [0 < a,b < 10\nand\n0 < x < 1] [0 < a,b < 100\nand\n0 < x < 1][1e-5 < a,b < 1e5\nand\n0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peek=39 Mean=2.9] [Peek=91 Mean=12.7] [Peek=635 Mean=25]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peek=49.2 Mean=4.2] [Peek=180.7 Mean=30.1] [Peek~68,580 Mean=2,816]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=49.2 Mean=3.2] [Peek=67.1 Mean=13.4] [Peek~68,380 Mean=2,760]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=27.3 Mean=3.6] [Peek=49.8 Mean=9.1] [Peek~60,900 Mean=2,920]] +] + +[table Errors In the Function betac(a,b,x) +[[Mantissa Size] [Platform and Compiler] [0 < a,b < 10\nand\n0 < x < 1] [0 < a,b < 100\nand\n0 < x < 1][1e-5 < a,b < 1e5\nand\n0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peek=12.0 Mean=2.4] [Peek=91 Mean=15] [Peek=3718 Mean=113]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peek=19.8 Mean=3.8] [Peek=295.1 Mean=33.9] [Peek~104,400 Mean=5,468]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=19.8 Mean=2.6] [Peek=63.5 Mean=13.6] [Peek~104,500 Mean=5,444]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=15.6 Mean=3.5] [Peek=39.8 Mean=8.9] [Peek~89,780 Mean=4,946]] +] + +[table Errors In the Function ibeta(a,b,x) +[[Mantissa Size] [Platform and Compiler] [0 < a,b < 10\nand\n0 < x < 1] [0 < a,b < 100\nand\n0 < x < 1][1e-5 < a,b < 1e5\nand\n0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peek=42.3 Mean=2.9\n\n(GSL Peek=682 Mean=32.5)\n(Cephes Peek=42.7 Mean=7.0)] [Peek=108 Mean=16.6\n\n(GSL Peek=690 Mean=151)\n(Cephes Peek=1545 Mean=218)] [Peek=4109 Mean=203\n\n(GSL Peek~390000 Mean~18000)\n(Cephes Peek~560000 Mean~20000)]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peek=48.4 Mean=3.7] [Peek=270.7 Mean=26.8] [Peek~53,560 Mean=2,948]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=47.3 Mean=3.0] [Peek=112.9 Mean=14.3] [Peek~53,560 Mean=2,893]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=20.9 Mean=2.6] [Peek=88.1 Mean=14.3] [Peek~25,920 Mean=1,671]] +] + +[table Errors In the Function ibetac(a,b,x) +[[Mantissa Size] [Platform and Compiler] [0 < a,b < 10\nand\n0 < x < 1] [0 < a,b < 100\nand\n0 < x < 1][1e-5 < a,b < 1e5\nand\n0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peek=13.9 Mean=2.0] [Peek=56.2 Mean=14] [Peek=2670 Mean=159]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peek=20.6 Mean=3.6] [Peek=221.7 Mean=25.8] [Peek~91,790 Mean=3,047]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=15.2 Mean=2.2] [Peek=73.9 Mean=11.9] [Peek~91,790 Mean=3,029]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=9.9 Mean=2.6] [Peek=117.7 Mean=15.1] [Peek~27,860 Mean=1,195]] +] + +[h4 Testing] + +There are two sets of tests: spot tests compare values taken from Mathworld's +online calculator with this implementation: they provide a basic "sanity check" +for the implementation, with one spot-test in each implementation-domain +(see implementation notes below). + +Accuracy tests use data generated at very high precision +(with NTL's RR class set at 1000-bit precision), using the "textbook" +continued fraction representation (fraction 1 below). +Note that this continued fraction is /not/ used in the implementation, +and therefore we have test data that is fully independent of the code. + +[h4 Implementation] + +Under construction: refer to +[@http://portal.acm.org/citation.cfm?doid=131766.131776 "Algorithm 708; significant digit computation of the incomplete beta function ratios", DiDonato and Morris, ACM, 1992.] + +[endsect] + diff --git a/doc/igamma.qbk b/doc/igamma.qbk new file mode 100644 index 000000000..2dc3891ad --- /dev/null +++ b/doc/igamma.qbk @@ -0,0 +1,219 @@ +[section The Incomplete Gamma Function] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T tgamma(T a, T z); + + template + T tgamma_lower(T a, T z); + + template + T gamma_Q(T a, T z); + + template + T gamma_P(T a, T z); + + }} // namespaces + +[h4 Description] + +There are four incomplete gamma functions: two are normalised versions +that return values in the range [0, 1], and two are non-normalised and +return values in the range [0, gamma(a)]. Users interested in statistical +applications should use the normalised versions (gamma_P and gamma_Q). + +All of these function require /a > 0/ and /z >= 0/. + + template + T tgamma(T a, T z); + +Returns the full (non-normalised) upper incomplete gamma function of a and z: + +[$../equations/igamma1.png] + + template + T tgamma_lower(T a, T z); + +Returns the full (non-normalised) lower incomplete gamma function of a and z: + +[$../equations/igamma2.png] + + template + T gamma_Q(T a, T z); + +Returns the normalised upper incomplete gamma function of a and z: + +[$../equations/igamma3.png] + +This function changes rapidly from 1 to 0 around the point z == a: + +[$../graphs/gamma_q.png] + + template + T gamma_P(T a, T z); + +Returns the normalised lower incomplete gamma function of a and z: + +[$../equations/igamma4.png] + +This function changes rapidly from 0 to 1 around the point z == a: + +[$../graphs/gamma_p.png] + +[h4 Accuracy] + +The following tables give peek and mean relative errors in over various domains of +a and z, along with comparisons to other open source implementations where available. +Note that only results for the widest floating point type on the system are given as +narrower types have __zero_error. + +Note that for very large /a/ and /a ~ z/, these functions are currently quite slow. +Errors also grow as /a/ grows larger. + +[table Errors In the Function tgamma(a,z) +[[Mantissa Size] [Platform and Compiler] [0.5 < a < 100\nand\n0.01*a < z < 100*a] [1e-12 < a < 5e-2\nand\n0.01*a < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peek=5.9 Mean=1.5] [Peek=1.8 Mean=0.6]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=596 Mean=116] [Peek=3.2 Mean=0.84]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peek=40.2 Mean=2.5] [Peek=3.2 Mean=0.8]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=364 Mean=17.6] [Peek=12.7 Mean=1.8]] +] + +[table Errors In the Function tgamma_lower(a,z) +[[Mantissa Size] [Platform and Compiler] [0.5 < a < 100\nand\n0.01*a < z < 100*a] [1e-12 < a < 5e-2\nand\n0.01*a < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peek=5.5 Mean=1.4] [Peek=3.6 Mean=0.78]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=402 Mean=79] [Peek=3.4 Mean=0.8]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peek=6.8 Mean=1.4] [Peek=3.4 Mean=0.78]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=6.1 Mean=1.8] [Peek=3.7 Mean=0.89]] +] + +[table Errors In the Function gamma_Q(a,z) +[[Mantissa Size] [Platform and Compiler] [0.5 < a < 100\nand\n0.01*a < z < 100*a] [1e-12 < a < 5e-2\nand\n0.01*a < z < 100*a] [1e-6 < a < 1.7e6\nand\n1 < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peek=28.3 Mean=7.2\n\n(GSL Peek=201 Mean=13)\n(Cephes Peek=556 Mean=97)] [Peek=4.8 Mean=1.6\n\n(GSL Peek~1.3e10 Mean=1e+9)\n(Cephes Peek~3e11 Mean=4e10)] [Peek=469 Mean=33\n\n(GSL Peek=27,050 Mean=2159)\n(Cephes Peek~8e6 Mean~7e5)]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=280 Mean=33] [Peek=4.1 Mean=1.6] [Peek=11,490 Mean=732]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peek=32 Mean=9.4] [Peek=4.7 Mean=1.5] [Peek=6815 Mean=414]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=37 Mean=10] [Peek=11.2 Mean=2.0] [Peek=4,999 Mean=298]] +] + +[table Errors In the Function gamma_P(a,z) +[[Mantissa Size] [Platform and Compiler] [0.5 < a < 100\nand\n0.01*a < z < 100*a] [1e-12 < a < 5e-2\nand\n0.01*a < z < 100*a] [1e-6 < a < 1.7e6\nand\n1 < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peek=36 Mean=9.1\n\n(GSL Peek=342 Mean=46)\n(Cephes Peek=491 Mean=102)] [Peek=4.5 Mean=1.4\n\n(GSL Peek=4.8 Mean=0.76)\n(Cephes Peek=21 Mean=5.6)] [Peek=244 Mean=21\n\n(GSL Peek=1022 Mean=1054)\n(Cephes Peek~8e6 Mean~7e4)]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peek=241 Mean=36] [Peek=4.7 Mean=1.5] [Peek~30,220 Mean=1929]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peek=41 Mean=10] [Peek=4.7 Mean=1.4] [Peek~30,790 Mean=1864]] +[[113] [HPUX IA64, aCC A.06.06] [Peek=40.2 Mean=10.2] [Peek=5 Mean=1.6] [Peek=5,476 Mean=440]] +] + +[h4 Testing] + +There are two sets of tests: spot tests compare values taken from Mathworld's +online calculator with this implementation: they provide a basic "sanity check" +for the implementation. Accuracy tests use data generated at very high precision +(using NTL's RR class set at 1000-bit precision) using this implementation +with a very high precision 60-term __lanczos. +This is less than satisfactory: an independent method should really be used, +but apparently a complete lack of such methods are available. We can't even use a deliberately +naive implementation without special case handling since Legendre's continued fraction +(see below) is unstable for small a and z. + +[h4 Implementation] + +The implementation of these functions are all related since: + +[$../equations/igamma5.png] + +[$../equations/igamma6.png] + +[$../equations/igamma7.png] + +The lower incomplete gamma is computed from it's series representation: + +[$../equations/igamma8.png] + +Or by subtraction of the upper integral from either gamma(a) or 1 +when /x > max(a+1, 10)/. + +The upper integral is computed from Legendre's continued fraction representation: + +[$../equations/igamma9.png] + +When /x > 1/ or by subtraction of the lower integral from either gamma(a) or 1 +when /x < a+1/. + +For /x < 1/ computation of the upper integral is more complex as the continued +fraction representation is inaccurate in this area. However there is another +series representation for the lower integral: + +[$../equations/igamma10.png] + +That lends itself to calculation of the upper integral via rearrangement +to: + +[$../equations/igamma11.png] + +Obviously this is only accurate if the functions "powm1" and "gp1m1" are +accurately calculated. "gp1m1" is computed via the following rearrangement +of the __lanczos, note that no generic version of this function is available +and so the accuracy of the fully generic version of the upper incomplete +gamma functions is likely to limited to approximately long double precision. + +[$../equations/igamma11b.png] + +The terms in square brackets in the above are evaluated as special +functions in their own right to avoid cancellation errors. + +For the normalised incomplete gamma function's then calculation of the +leading power terms +is central to the accuracy of the function. For smallish a and x combining +the power terms with the __lanczos gives the greatest accuracy: + +[$../equations/igamma12.png] + +In the event that this causes underflow/overflow then the exponent can +be reduced by a factor of /a/ and brought inside the power term. + +When a and x are large we end up with a very large exponent with a base +near one: this will not be computed accurately via the pow function, +and taking logs simply leads to cancellation errors. The worst of the +errors can be avoided by using: + +[$../equations/igamma13.png] + +when /a-x/ is small and a and x are large. There is still a subtraction +and therefore some cancellation errors - but the terms are small so the absolute +error will be small - and it's absolute rather than relative error that +counts in the argument to the /exp/ function. Note that for sufficiently +large a and x the errors will still get you eventually, although this does +delay the inevitable much longer than other methods. Use of /log(1+x)-x/ here +is inspired by Temme (see references below). + +Finally, I note that there is an asymptotic expansion available +when a and x are large: however this requires backwards evaluation +of a series (forwards evaluation is unstable), and therefore +foreknowledge of the number of terms required (see Temme). The accuracy quoted +by it's authors is also lower than that obtained here with careful +evaluation of the leading power terms. However, the Legendre continued +fraction in particular become increasingly expensive to compute for larger +a and x, so this area needs more investigation. + +Also needing further attention is the area where a is very small and x is near 1, +and where a and x are both vanishingly small. + +[h4 References] + +* N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, +Probability in the Engineering and Informational Sciences, 8, 1994. +* W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's Ideas +and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, +Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237. +[@http://citeseer.ist.psu.edu/gautschi98incomplete.html http://citeseer.ist.psu.edu/gautschi98incomplete.html] + +[endsect] + diff --git a/doc/igamma_inv.qbk b/doc/igamma_inv.qbk new file mode 100644 index 000000000..ef6ab1ae5 --- /dev/null +++ b/doc/igamma_inv.qbk @@ -0,0 +1,81 @@ +[section The Incomplete Gamma Function Inverses] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T gamma_Q_inv(T a, T q); + + template + T gamma_P_inv(T a, T p); + + }} // namespaces + +[h4 Description] + +There are two incomplete gamma function inverses which return the inverses +of gamma_Q and gamma_P respectively. + +Both of these function require /a > 0/ and /1 >= p,q >= 0/. + + template + T gamma_Q_inv(T a, T q); + +Returns a value x such that: `q = gamma_Q(a, x);` + + template + T gamma_P_inv(T a, T p); + +Returns a value x such that: `p = gamma_P(a, x);` + +[h4 Accuracy] + +The accuracy of these functions doesn't vary much by platform or by +the type T. Given that these functions are computed by iterative methods, +they are deliberately "detuned" so as not to be too accurate: it is in +any case impossible for these function to be more accurate than the +regular forward incomplete gamma functions. + +[h4 Testing] + +There are two sets of tests: + +* Basic sanity checks attempt to "round-trip" from +/a/ and /x/ to /p/ or /q/ and back again. These tests have quite +generous tolerances: in general both the incomplete gamma and it's +inverses change so rapidly, that round tripping to more than a couple +of significant digits isn't possible. This is especially true when +/p/ or /q/ is very near one: in this case there isn't enough +"information content" in the input to the inverse function to get +back where you started. +* Accuracy checks using high precision test values. These measure +the accuracy of the result, given exact input values. + +[h4 Implementation] + +These two functions share a common implementation. + +First negative values for /p/ and /q/ are mapped to positive values +using the usual reflection formulae for the incomplete gamma function. + +Then an initial approximation is computed using the methodology described +in: + +[@http://portal.acm.org/citation.cfm?id=23109&coll=portal&dl=ACM +A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma +Function Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.] + +Finally, the last few bits are cleaned up using Halley iteration, the iteration +limit is set to 2/3 of the number of bits in T, which by experiment is +sufficient to ensure that the inverses are at least as accurate as the normal +incomplete gamma functions. + +[endsect] + diff --git a/doc/issues.qbk b/doc/issues.qbk new file mode 100644 index 000000000..3b6d0e55d --- /dev/null +++ b/doc/issues.qbk @@ -0,0 +1,43 @@ +[section Known Issues, and Todo List] + +[h4 tgamma] + +* Can the __lanczos be optimized any further? (low priority) + +[h4 lgamma] + +* Improve accuracy near 1 and 2 with either Chebyshev or rational +approximation. (High priority). + +[h4 erf/erfc] + +* Rewrite using either Chebyshev or rational +approximation. (High priority). + +[h4 Incomplete gamma] + +* Implement Temme's asymptotic expansion for large a and z, +this one is mainly an issue of performance rather than accuracy (medium priority). +* Double check behavior of the upper fraction when a is very small and z -> 1 (Medium Priority). +* Double check behavior of the upper fraction when a is very small and z -> 0 (Medium Priority). +* Double check for opportunities to chain series expansions together, passing +the result of one into the initial value of the next (Medium Priority). +* Investigate asymptotic expansions for large a small z and large z small a +(see A&S) (low priority). + + +[h4 Beta Function] + +* No known issues? + +[h4 Incomplete Beta] + +* Investigate Didonato and Morris' asymptotic expansion for large a and b +(medium priority). +* Double check for opportunities to chain series expansions together, passing +the result of one into the initial value of the next (Medium Priority). +* Investigate Didonato and Morris' asymptotic expansion for small a and b +(low priority). + +[endsect] + diff --git a/doc/lanczos.qbk b/doc/lanczos.qbk new file mode 100644 index 000000000..8cc7d9b0f --- /dev/null +++ b/doc/lanczos.qbk @@ -0,0 +1,227 @@ +[section The Lanczos Approximation] +[#lanczos] + +[caution __caution ] + +[h4 Motivation] + +['Why base gamma and gamma-like functions on the Lanczos approximation?] + +First of all I should make clear that for the gamma function +over real numbers (as opposed to +complex ones) the Lanczos approximation appears to offer no clear +advantage over more traditional methods such as Stirling's approximation. +__pugh carried out an extensive comparison of the various methods available +and discovered that they were all very similar in terms of complexity +and relative error. However, the lanczos approximation does have a couple of +properties that make it worthy of further consideration: + +* The approximation has an easy to compute truncation error, that holds for +all /z > 0/. In practice that means we can use the same approximation for all +/z > 0/, and be certain that no matter how large or small /z/ is, the truncation +error will /at worst/ be bounded by some finite value. +* The approximation has a form that is particularly amenable to analytic +manipulation, in particular ratio's of gamma or gamma-like functions +are particularly easy to compute without resorting to logarithms. + +It's the combination of these two properties that make the approximation +attractive: Stirling's approximation is highly accurate for large z, and +has some of the same analytic properties as the Lanczos approximation, but +can't easily be used across the whole range of z. + +As the simplest example, consider the ratio of two gamma functions: one could +compute the result via lgamma: + + exp(lgamma(a) - lgamma(b)); + +However, even if lgamma is uniformly accurate to 0.5ulp, the worst case +relative error in the above can easily be shown to be: + + Erel > a * log(a)/2 + b * log(b)/2 + +For small /a/ and /b/ that's not a problem, but to put the relationship another +way: ['each time a and b increase in magnitude by a factor of 10, at least one +decimal digit of precision will be lost.] In contrast, by combining like power +terms in a ratio of Lanczos approximation's, these errors can be virtually eliminated +for small /a/ and /b/, and kept under control for very large (or very small +for that matter) /a/ and /b/. Of course computing large powers is itself a +notoriously hard problem, but even so, analytic combinations of Lanczos +approximations can make the difference between obtaining a valid result, or +simply garbage. + +[h4 The Approximation] + +The Lanczos Approximation to the Gamma Function is given by: + +[$../equations/lanczos0.png] + +Where S(z) is an infinite sum, that is convergent for all z > 0, +and /g/ is an arbitrary parameter that controls the "shape" of the +terms in the sum which is given by: + +[$../equations/lanczos0a.png] + +With individual coefficients defined in closed form by: + +[$../equations/lanczos0b.png] + +However, evaluation of the sum in that form can lead to numerical instability +in the computation of the ratios of rising and falling factorials (effectively +we're multiplying by a series of numbers very close to 1, so roundoff errors +can accumulate quite rapidly). + +The Lanczos approximation is therefore often written in partial fraction form +with the leading constants absorbed by the coefficients in the sum: + +[$../equations/lanczos1.png] + +where: + +[$../equations/lanczos2.png] + +Again parameter /g/ is an arbitarily chosen constant, and /N/ is an arbitrarily chosen +number of terms to evaluate in the "Lanczos sum" part. Note that some authors +choose to define the sum from k=1 to N, and hence end up with N+1 coefficients. +This happens to confuse both the following discussion and the code (since C++ +deals with half open ranges, rather than the closed range of the sum), hence +the choice here, which is consistent with __godfrey, but not __pugh. + +[h4 Computing the Coefficients] + +The coefficients C0..CN-1 need to be computed from /N/ and /g/ +at high precision, and then stored as part of the program. +Calculation of the coefficients is performed via the method of __godfrey; +let the constants be contained in a column vector P, then: + +P = B D C F + +where B is an NxN matrix: + +[$../equations/lanczos4.png] + +D is an NxN matrix: + +[$../equations/lanczos3.png] + +C is an NxN matrix: + +[$../equations/lanczos5.png] + +and F is an N element column vector: + +[$../equations/lanczos6.png] + +Note than the matrices B, D and C contain all integer terms and depend +only on /N/, this product should be computed first, and then multiplied +by /F/ as the last step. + +[h4 Choosing the Right Parameters] + +The trick is to choose +/N/ and /g/ to give the desired level of accuracy: choosing a small value for +/g/ leads to a strictly convergent series, but one which converges only slowly. +Choosing a larger value of /g/ causes the terms in the series to be large +and\/or divergent for about the first /g-1/ terms, and to then suddenly converge +with a "crunch". + +__pugh has determined the optimal +value of /g/ for /N/ in the range /1 <= N <= 60/: however in practice choosing +these values leads to cancellation errors in the Lanczos sum as the largest +term in the series is approximately 1000 times larger than the result. +These optimal values appear not to be useful in practice unless the evaluation +can be done with a number of guard digits /and/ the coefficients are stored +at higher precision than that desired in the result. These values are best +reserved for say, computing to float precision with double precision arithmetic. + +[table Optimal choices for N and g when computing with guard digits (source: Pugh) +[[Mantissa Size] [N] [g][Max Error]] +[[24] [6] [5.581][9.51e-12]] +[[53][13][13.144565][9.2213e-23]] +] + +The alternative described by __godfrey is to perform an exhaustive +search of the /N/ and /g/ parameter space to determine the optimal combination for +a given /p/ digit floating point type. Repeating this work found a good +approximation for double precision arithmetic (close to the one __godfrey found), +but failed to find really +good approximations for 80 or 128-bit long doubles. Further it was observed +that the approximations obtained tended to optimised for the small values +of z (1 < z < 200) used to test the implementation against the factorials. +Computing ratios of gamma functions with large arguments were observed to +suffer from error resulting from the truncation of the Lancozos series. + +__pugh identified all the locations where the theoretical error of the +approximation were at a minimum, but unfortunately has published only the largest +of these minima. However, he makes the observation that the minima +coincide closely with the location where the first neglected term in the +Lanczos series (aN above) changes sign. These locations are quite easy to +locate, albeit with considerable computer time. These "sweet spots" need +only be computed once, tabulated, and then searched when required for an +approximation that delivers the required precision for some fixed precision +type. + +Unfortunately, following this path failed to find a really good approximation +for 128-bit long doubles, and those found for 64 and 80 bit reals required an +excessive number of terms. There are two competing issues here: high precision +requires a large value of /g/, but avoiding cancellation errors in the evaluation +requires a small /g/. + +At this point note that the Lanczos sum can be converted into rational form, +and doing so changes the coefficients so that /they are all positive/. That +means that the sum in rational form can be evaluated without cancellation +error, albeit with double the number of coefficients for a given N. Repeating +the search of the "sweet spots", this time evaluating the Lanczos sum in +rational form, and testing only those "sweet spots" whose theoretical error +is less than the machine epsilon for the type being tested, yielded good +approximations for all the types tested. The optimal values found were quite +close to the best cases reported by __pugh (just slightly larger /N/ and slightly +smaller /g/ for a given precision than __pugh reports), and even though converting +to rational form doubles the number of stored coefficients, it should be +noted that half of them are integers (and therefore require less storage space) +and the approximations require a smaller /N/ than would otherwise be required, +so fewer floating point operations may be required overall. + +The following table shows the optimal values for /N/ and /g/ when computing +at fixed precision. These should be taken as work in progress: there are no +values for 106-bit mantissa machines (Darwin long doubles), and further +optimisation of the values of /g/ may be possible. Errors given in the table +are estimates of the error due to truncation of the Lanczos infinite series +to /N/ terms. They are calculated from the sum of the first five neglected +terms - and are known to be rather pessimistic estimates - although it is noticeable +that the best combinations of /N/ and /g/ occurred when the estimated truncation error +almost exactly matches the machine epsilon for the type in question. + +[table Optimum value for N and g when computing at fixed precision +[[Mantissa Size][Platform/Compiler Used][N][g][Max Truncation Error]] +[[24][Win32, VC++ 7.1] [6] [1.428456135094165802001953125][9.41e-007]] +[[53][Win32, VC++ 7.1] [13] [6.024680040776729583740234375][3.23e-016]] +[[64][Suse Linux 9 IA64, gcc-3.3.3] [17] [12.2252227365970611572265625][2.34e-024]] +[[116][HP Tru64 Unix 5.1B \/ Alpha, Compaq C++ V7.1-006] [24] [20.3209821879863739013671875][4.75e-035]] +] + +Finally note that the Lanczos approximation can be written as follows +by removing a factor of exp(g) from the denominator, and then dividing +all the coefficients by exp(g): + +[$../equations/lanczos7.png] + +This form is more convenient for calculating lgamma, but for the gamma +function the division by /e/ turns a possibly exact quality into an +inexact value: this reduces accuracy in the common case that +the input is exact, and so isn't used for the gamma function. + +[h4 References] + +# [#godfrey]Paul Godfrey, [@http://my.fit.edu/~gabdo/gamma.txt "A note on the computation of the convergent +Lanczos complex Gamma approximation"]. +# [#pugh]Glendon Ralph Pugh, +[@http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf +"An Analysis of the Lanczos Gamma Approximation"], +PhD Thesis November 2004. +# Viktor T. Toth, +[@http://www.rskey.org/gamma.htm "Calculators and the Gamma Function"]. +# Mathworld, [@http://mathworld.wolfram.com/LanczosApproximation.html +The Lanczos Approximation]. + +[endsect] + diff --git a/doc/lgamma.qbk b/doc/lgamma.qbk new file mode 100644 index 000000000..ef3cbeeab --- /dev/null +++ b/doc/lgamma.qbk @@ -0,0 +1,111 @@ +[section The Log Gamma Function] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T lgamma(T z); + + template + T lgamma(T z, int* sign); + + }} // namespaces + +[h4 Description] + +The lgamma function is defined by: + +[$../equations/lgamm1.png] + +The second form of the function takes a pointer to an integer, which if non-null +is set on output to the sign of tgamma(z). + +[$../graphs/lgamma.png] + +There are effectively two versions of this function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the mantissa +of T correspond to a certain __lanczos. In practice any built in +floating point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp. + +[h4 Accuracy] + +The following table shows the peek errors (in units of epsilon) +found on various platforms +with various floating point types. Unless otherwise specified any +floating point type that is narrower than the one shown will have +__zero_error. + +[table +[[Mantissa Size] [Platform and Compiler] [Factorials and Half factorials] [Values Near Zero] [Values Near 1 or 2] [Values Near a Negative Pole]] +[[53] [Win32 Visual C++ 8] [Peek=0.88 Mean=0.14 \n\n(GSL=33) (Cephes=1.5)] [Peek=2.6 Mean=0.7 \n\n(GSL=5.2) (Cephes=1.1)] [Peek=6.8 Mean=3.0 \n\n(GSL=1168) (Cephes~500000)] [Peek=3.8 Mean=0.9 \n\n(GSL=25) (Cephes=1.6)] ] +[[64] [Linux IA32 / GCC] [Peek=25 Mean=2.6 \n\n(Native Peek=1.7 Mean=0.49)] [Peek=1.6 Mean=0.57 \n\n(Native Peek= 0.96 Mean=0.54)] [Peek=44 Mean=23 \n\n(Native Peek=0.74 Mean=0.26)] [Peek=6.0 Mean=1.8 \n\n(Native Peek=3.0 Mean=0.86)] ] +[[64] [Linux IA64 / GCC] [Peek=29 Mean=2.7 (Native=0)] [Pek=1.2 Mean=0.6 \n(Native=0)] [Peek=44 Mean=23 \n(Native=0)] [Peek=2.3 Mean=0.69 \n(Native 0)] ] +[[113] [HPUX IA64, aCC A.06.06] [Peek=218.7 Mean=17.54 \n(Native=0)] [Peek=6.1 Mean=1.3 \n(Native=0)] [Peek=238.8 Mean=105 \n(Native=0)] [Peek=1.2 Mean=0.43 \n(Native=0)] ] +] + +There's still some work to do on these error bounds, currently the larger errors +all occur in the tricky regions near 1 or 2: however for z > 4 the error +is typically less than 1 epsilon: + +[$../graphs/lgamma-errors.png] + +[h4 Testing] + +The main tests for this function involve comparisons against the logs of +the factorials which can be independently calculated to very high accuracy. + +Random tests in key problem areas are also used. + +[h4 Implementation] + +The generic version of this function is implemented by combining the series and +continued fraction representations for the incomplete gamma function: + +[$../equations/lgamm2.png] + +where /l/ is an arbitrary integration limit: choosing [^l = max(10, a)] +seems to work fairly well. For negative /z/ the logarithm version of the +reflection formula is used: + +[$../equations/lgamm3.png] + +For types of known precision the __lanczos is used, a traits class +`boost::math::lanczos::lanczos_traits` maps type T to an appropriate +approximation. The logarithmic version of the __lanczos is: + +[$../equations/lgamm4.png] + +As before the reflection formula is used for /z < 0/. + +When z is very near 1 or 2, then the logarithmic version of the __lanczos +suffers very badly from cancellation error: indeed for values sufficiently +close to 1 or 2, arbitrarily large relative errors can be obtained (even though +the absolute error is tiny). The current solution is as follows: imagine we +balance the two terms in the __lanczos by dividing the power term by it's value +at /z = 1/, and then multiplying the Lanczos coefficients by the same value. +Now each term will take the value 1 at /z = 1/ and we can rearrange the power terms +in terms of log1p. Likewise if we subtract 1 from the Lanczos sum part +(algebraically, by subtracting the value of each term at /z = 1/), we obtain +a new summation that can be also be fed into log1p. Crucially, all of the +terms tend to zero, as /z -> 1/: + +[$../equations/lgamm5.png] + +The Ck terms in the above are the same as in the __lanczos. + +A similar rearrangement can be performed at /z = 2/: + +[$../equations/lgamm6.png] + +[endsect] + diff --git a/doc/math.qbk b/doc/math.qbk new file mode 100644 index 000000000..54bc856c5 --- /dev/null +++ b/doc/math.qbk @@ -0,0 +1,105 @@ +[library Math Toolkit + [quickbook 1.3] + [copyright 2006 John Maddock] + [purpose Various Special Functions and Numerical Tools] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + + http://www.boost.org/LICENSE_1_0.txt + ) + ] + [authors [Maddock, John]] + [category math] + [last-revision $Date$] +] + +[def __effects [*Effects: ]] +[def __formula [*Formula: ]] +[def __exm1 '''ex - 1'''] +[def __ex '''ex'''] +[def __te '''2ɛ'''] +[def __lanczos [link lanczos Lanczos approximation]] +[def __zero_error [link zero_error effectively zero error]] + +[def __godfrey [link godfrey Godfrey]] +[def __pugh [link pugh Pugh]] + +[def __caution This is not an official Boost library, it is a library under + construction, the code is fully functional and robust, but + interfaces, library structure, and function names may be + changed without notice.] + +[section:intro Introduction] + +This library is work in progress, it is intended to fulfil two needs: + +# To provide a small number of high quality special functions, initially +these will be concentrated on functions used in statistical applications. +The functions currently implemented are the gamma/beta/erf functions +along with the incomplete gamma and beta functions (four variants +of each). All the implementations +are fully generic and support the use of arbitrary "real-number" +types, although they are optimised for use with types with known-about +mantissa sizes: typically `float`, `double` or `long double`. Use of these +functions with interval arithmetic (Boost.Interval) isn't quite supported +yet, but it's something I want to investigate soon. + +# To provide at least some of the tools required to implement +mathematical special functions, hopefully the presence of +these will encourage other authors to contribute more special +function implementations in the future. Currently implemented +are helpers for the evaluation of infinite series, continued +fractions and rational approximations. There are also classes +for the manipulation of polynomials, for testing a special function +against tabulated test data, and for the rapid generation of test +data and/or data for output to an external graphing application. + +[endsect] + +[section Special Functions] + +[include tgamma.qbk] +[include lgamma.qbk] +[include igamma.qbk] +[include igamma_inv.qbk] +[include erf.qbk] +[include erf_inv.qbk] +[include beta.qbk] +[include ibeta.qbk] +[include fpclassify.qbk] + +[endsect] + +[section Toolkit] + +[include series.qbk] +[include fraction.qbk] +[include rational.qbk] +[include roots.qbk] +[include polynomial.qbk] +[include relative_error.qbk] +[include test_data.qbk] + + +[endsect] + +[section Use with Non-Builtin Floating Point Types] + +[include concepts.qbk] + +[endsect] + +[section Backgrounders] + +[include lanczos.qbk] +[include error.qbk] + +[endsect] + +[section Status and Roadmap] + +[include roadmap.qbk] +[include issues.qbk] + +[endsect] diff --git a/doc/polynomial.qbk b/doc/polynomial.qbk new file mode 100644 index 000000000..5a210fc10 --- /dev/null +++ b/doc/polynomial.qbk @@ -0,0 +1,90 @@ +[section Polynomials] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + class polynomial + { + public: + // typedefs: + typedef typename std::vector::value_type value_type; + typedef typename std::vector::size_type size_type; + + // construct: + polynomial(){} + template + polynomial(const U* data, unsigned order); + template + polynomial(const U& point); + + // access: + size_type size()const; + size_type degree()const; + value_type& operator[](size_type i); + const value_type& operator[](size_type i)const; + + // operators: + template + polynomial& operator +=(const U& value); + template + polynomial& operator -=(const U& value); + template + polynomial& operator *=(const U& value); + template + polynomial& operator +=(const polynomial& value); + template + polynomial& operator -=(const polynomial& value); + template + polynomial& operator *=(const polynomial& value); + }; + + template + polynomial operator + (const polynomial& a, const polynomial& b); + template + polynomial operator - (const polynomial& a, const polynomial& b); + template + polynomial operator * (const polynomial& a, const polynomial& b); + + template + polynomial operator + (const polynomial& a, const U& b); + template + polynomial operator - (const polynomial& a, const U& b); + template + polynomial operator * (const polynomial& a, const U& b); + + template + polynomial operator + (const U& a, const polynomial& b); + template + polynomial operator - (const U& a, const polynomial& b); + template + polynomial operator * (const U& a, const polynomial& b); + + template + std::basic_ostream& operator << + (std::basic_ostream& os, const polynomial& poly); + + }}} // namespaces + +[h4 Description] + +This is a fairly trivial class for polynomial manipulation. + +Implementation is currently of the "naive" variety, with O(N^2) +multiplication for example. This class should not be used in +high performance computing environments: it is intended for the +simple manipulation of small polynomials, typically generated +for special function approximation. + +Advanced manipulations: the FFT, division, GCD, factorisation etc are +not currently provided. Submissions for these are of course welcome :-) + +[endsect] + diff --git a/doc/project-root.jam b/doc/project-root.jam new file mode 100644 index 000000000..bb13b141d --- /dev/null +++ b/doc/project-root.jam @@ -0,0 +1,7 @@ + +path-constant BOOST_ROOT : ../../../../develop/boost ; + +using lex ; +using bison ; +using testing ; + diff --git a/doc/rational.qbk b/doc/rational.qbk new file mode 100644 index 000000000..535bb87fc --- /dev/null +++ b/doc/rational.qbk @@ -0,0 +1,40 @@ +[section Rational Function Evaluation] + +[caution __caution ] + +[h4 synopsis] + +`` +#include +`` + + template + V evaluate_rational(const T* num, const U* denom, V z, unsigned count); + +[h4 Description] + +Evaluates the rational function (the ratio of two polynomials) described by +the coefficients stored in /num/ and /demom/. + +Both polynomials most have order /count-1/ with /count/ coefficients. + +Array /num/ describes the numerator, and /demon/ the denominator. + +Coefficients should be stored such that the coefficients for the x^i terms +are in num[i] and denom[i]. + +The types of the coefficients and of variable +/v/ may differ as long as /*num/ and /*denom/ are convertible to type /V/. +This allows, for example, for one or both of the coefficient tables +to be a table of integers if this is appropriate. + +[h4 Implementation] + +Evaluation is by Horners method: with the two polynomials being evaluated +in parallel to make the most of the processors floating point pipeline. +If /v/ is greater than one, then the polynomials are evaluated in reverse +order as polynomials in ['1\/v]: this avoids unnecessary numerical overflow when the +coefficients are large. + +[endsect] + diff --git a/doc/relative_error.qbk b/doc/relative_error.qbk new file mode 100644 index 000000000..f7cbac4c9 --- /dev/null +++ b/doc/relative_error.qbk @@ -0,0 +1,112 @@ +[section Relative Error and Testing] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + template + T relative_error(T a, T b); + + template + test_result test(const A& a, F1 test_func, F2 expect_func); + +[h4 Description] + + template + T relative_error(T a, T v); + +Returns the relative error between /a/ and /v/ using the usual formula: + +[$../equations/error1.png] + +In addition the value returned is zero if: + +* Both /a/ and /v/ are infinite. +* Both /a/ and /v/ are denormalised numbers or zero. + +Otherwise if only one of /a/ and /v/ is zero then the value returned is 1. + + template + test_result test(const A& a, F1 test_func, F2 expect_func); + +This function is used for testing a function against tabulated test data. + +The return type contains statistical data on the relative errors (max, mean, +variance, and the number of test cases etc), as well as the row of test data that +caused the largest relative error. Type test_result is work in progress, refer +to the header for more details. + +Parameter /a/ is a matrix of test data: and must be a standard library Sequence type, +that contains another Sequence type: +typically it will be a two dimensional instance of +[^boost::array]. Each row +of /a/ should contain all the parameters that are passed to the function +under test as well as the expected result. + +Parameter /test_func/ is the function under test, it is invoked with each row +of test data in /a/. Typically type F1 is created with Boost.Lambda: see +the example below. + +Parameter /expect_func/ is a functor that extracts the expected result +from a row of test data in /a/. Typically type F2 is created with Boost.Lambda: see +the example below. + +If the function under test returns a non-finite value when a finite result is +expected, or if a gross error is found, then a message is sent to `std::cerr`. +This is mainly a debugging/development aid (and a good place for a breakpoint). + +[h4 Example] + +Suppose we want to test the tgamma and lgamma functions, we can create a +two dimentional matrix of test data, each row is one test case, and contains +three elements: the input value, and the expected results for the tgamma and +lgamma functions respectively. + + static const boost::array, NumberOfTests> + factorials = { + /* big array of test data goes here */ + }; + +Now we can invoke the test function to test tgamma: + + using namespace boost::math::tools; + using namespace boost::lambda; + + // get a pointer to the function under test: + TestType (*funcp)(TestType) = boost::math::tgamma; + + // declare something to hold the result: + test_result result; + // + // and test tgamma against data: + // + result = test( + factorials, + bind(funcp, ret(_1[0])), // calls tgamma with factorials[row][0] + ret(_1[1]) // extracts the expected result from factorials[row][1] + ); + // + // Print out some results: + // + std::cout << "The Mean was " << result.stat.mean() << std::endl; + std::cout << "The worst error was " << (result.stat.max)() << std::endl; + std::cout << "The worst error was at row " << result.worst_case << std::endl; + // + // same again with lgamma this time: + // + funcp = boost::math::lgamma; + result = test( + factorials, + bind(funcp, ret(_1[0])), // calls tgamma with factorials[row][0] + ret(_1[2]) // extracts the expected result from factorials[row][2] + ); + // + // etc ... + // + +[endsect] + diff --git a/doc/roadmap.qbk b/doc/roadmap.qbk new file mode 100644 index 000000000..10b28d417 --- /dev/null +++ b/doc/roadmap.qbk @@ -0,0 +1,30 @@ +[section Roadmap] + +[h4 Milestone 1: Released March 31st 2006] + +* Implement gamma/beta/erf functions along with their incomplete counterparts. +* Generate high quality test data, against which future improvements can be judged. +* Provide tools for the evaluation of infinite series, continued fractions, and +rational functions. +* Provide tools for testing against tabulated test data, and collecting statistics +on error rates. +* Provide sufficient docs for people to be able to find their way around the library. + +[h4 Milestone 2] + +* Implement Newtons method for root finding. +* Implement Chebyshev approximation generation, and evaluation. +* Implement Rational function generation?? +* Implement the inverse incomplete gamma and beta functions. +* Improve erf/erfc, improve lgamma near 1 and 2. +* Find a way to integrate the statistical results generated from +the test data with Boost.Test: really needs a database of expected +results, indexed by test, floating point type, platform, and compiler. + +[h4 Milestone 3] + +* Implement the main probability distribution and density functions. +* Implement digamma (differentiate the __lanczos). +* Release?? + +[endsect] diff --git a/doc/roots.qbk b/doc/roots.qbk new file mode 100644 index 000000000..b881bb488 --- /dev/null +++ b/doc/roots.qbk @@ -0,0 +1,239 @@ +[section Root Finding] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + T bisect(F f, T min, T max, int digits); + + template + T bisect(F f, T min, T max, int digits, boost::uintmax_t& max_iter); + + template + T newton_raphson_iterate(F f, T guess, T min, T max, int digits); + + template + T newton_raphson_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + template + T halley_iterate(F f, T guess, T min, T max, int digits); + + template + T halley_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + template + T schroeder_iterate(F f, T guess, T min, T max, int digits); + + template + T schroeder_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + }}} // namespaces + +[h4 Description] + +These functions all perform the iterative root finding: `bisect` finds the root using +bisection, `newton_raphson_iterate` +performs second order [link newton Newton Raphson iteration], while `halley_iterate` and +`schroeder_iterate` perform third order [link halley Halley] and +[link schroeder Schroeder] iteration respectively. + +The functions all take the same parameters: + +[variablelist Parameters of the root finding functions +[[F f] [Type F must be a callable function object that accepts one parameter and + returns a tuple:\n\n + For a first order method (bisection) returns a single value.\n\n + For the second order iterative methods (Newton Raphson) + the tuple should have two elements containing the evaluation + of the function and it's first derivative.\n\n + For the third order methods (Halley and Schroeder) the tuple + should have three elements containing the evaluation of + the function and it's first and second derivatives.]] +[[T guess] [The initial starting value.]] +[[T min] [The minimum possible value for the result, this is used as an initial lower bracket.]] +[[T max] [The maximum possible value for the result, this is used as an initial upper bracket.]] +[[int digits] [The desired number of binary digits.]] +[[uintmax_t max_iter] [An optional maximum number of iterations to perform.]] +] + +When using these functions you should note that: + +* They may be very sensitive to the initial guess, typically they converge very rapidly +if the initial guess has two or three decimal digits correct. However convergenge +can be no better than bisection, or in some rair cases even worse than bisection if the +initial guess is a long way from the correct value and the derivatives are close to zero. +* These functions include special cases to handle zero first (and second where appropriate) +derivatives, and fall back to bisection in this case. However, it is helpful +if F is defined to return an arbitarily small value ['of the correct sign] rather +than zero. +* These functions fall back to bisection if the next computed step would take the +next value out of bounds. The bounds are updated after each step to ensure this leads +to convergenge. However, a good initial guess backed up by asymptotically-tight +bounds will improve performance no end rather than relying on bisection. +* The value of /digits/ is crucial to good performance of these functions, +if it is set too high then at best you will get one extra (unnecessary) +iteration, and at worst the last few steps will proceed by bisection. +Remember that the returned value can never be more accurate than f(x) can be +evaluated, and that if f(x) suffers from cancellation errors as it +tends to zero then the computed steps will be effectively random. The +value of /digits/ should be set so that iteration terminates before this point: +remember that for second and third order methods the number of correct +digits in the result is increasing quite +substantially with each iteration, /digits/ should be set by experiment so that the final +iteration just takes the next value into the zone where f(x) becomes inaccurate. +* Finally: you may well be able to do better than these functions by hand-coding +the heuristics used so that they are tailored to a specific function. You may also +be able to compute the ratio of derivatives used by these methods more efficiently +than computing the derivative themselves. As ever algebraic simplification can +be a big win. + +[#newton] +[h4 Newton Raphson Method] +Given an initial guess x0 the subsequent values are computed using: + +[$../equations/roots1.png] + +Out of bounds steps revert to besection of the current bounds. + +Under ideal conditions, the number of correct digits doubles with each iteration. + +[#halley] +[h4 Halley's Method] + +Given an initial guess x0 the subsequent values are computed using: + +[$../equations/roots2.png] + +Over-compensation by the second derivative (one which would proceed +in the wrong direction) causes the method to +revert to a Newton-Raphson step. + +Out of bounds steps revert to besection of the current bounds. + +Under ideal conditions, the number of correct digits trebbles with each iteration. + +[#schroeder] +[h4 Schroeder's Method] + +Given an initial guess x0 the subsequent values are computed using: + +[$../equations/roots3.png] + +Over-compensation by the second derivative (one which would proceed +in the wrong direction) causes the method to +revert to a Newton-Raphson step. Likewise a Newton step is used +whenever a Newton step would change the next value by more than 10%. + +Out of bounds steps revert to besection of the current bounds. + +Under ideal conditions, the number of correct digits trebbles with each iteration. + +[h4 Example] + +Lets suppose we want to find the cube root of a number, the equation we want to +solve along with it's derivatives are: + +[$../equations/roots4.png] + +To begin with lets solve the problem using Newton Raphson iterations, we'll +begin be defining a function object that returns the evaluation of the function +to solve, along with it's first derivative: + + template + struct cbrt_functor + { + cbrt_functor(T const& target) : a(target){} + std::tr1::tuple operator()(T const& z) + { + T sqr = z * z; + return std::tr1::make_tuple(sqr * z - a, 3 * sqr); + } + private: + T a; + }; + +Implementing the cube root is fairly trivial now, the hardest part is finding +a good approximation to begin with: in this case we'll just divide the exponent +by three: + + template + T cbrt(T z) + { + using namespace std; + int exp; + frexp(z, &exp); + T min = ldexp(0.5, exp/3); + T max = ldexp(2.0, exp/3); + T guess = ldexp(1.0, exp/3); + int digits = std::numeric_limits::digits; + return sign * tools::newton_raphson_iterate(detail::cbrt_functor(z), guess, min, max, digits); + } + +Using the test data in libs/math/test/cbrt_test.cpp this found the cube root +exact to the last digit in every case, and in no more than 6 iterations at double +precision. However, you will note that a high precision was used in this +example, exactly what was warned against earlier on in these docs! In this +particular case it's possible to compute f(x) exactly and without undue +cancellation error, so a high limit is not too much of an issue. However, +reducing the limit to `std::numeric_limits::digits * 2 / 3` gave full +precision in all but one of the test cases (and that one was out by just one bit). +The maximum number of iterations remained 6, but in most cases was reduced by one. + +Now lets adapt the functor slightly to return the second derivative as well: + + template + struct cbrt_functor + { + cbrt_functor(T const& target) : a(target){} + std::tr1::tuple operator()(T const& z) + { + T sqr = z * z; + return std::tr1::make_tuple(sqr * z - a, 3 * sqr, 6 * z); + } + private: + T a; + }; + +And then adapt the `cbrt` function to use Halley iterations: + + template + T cbrt(T z) + { + using namespace std; + int exp; + frexp(z, &exp); + T min = ldexp(0.5, exp/3); + T max = ldexp(2.0, exp/3); + T guess = ldexp(1.0, exp/3); + int digits = std::numeric_limits::digits / 2; + return sign * tools::halley_iterate(detail::cbrt_functor(z), guess, min, max, digits); + } + +Note that the iterations are set to stop at just one-half of full precision, +and yet even so not one of the test cases had a single bit wrong. +What's more the maximum number of iterations was now just 4. + +Just to complete the picture, we could have called `schroeder_iterate` in the last +example: and in fact it makes no difference to the accuracy or number of iterations +in this particular case. However, the relative performance of these two methods +may vary depending upon the nature of f(x), and the accuracy to which the initial +guess can be computed. There appear to be no generalisations that can be made +except "try them and see". + +Finally, had we called cbrt with NTL::RR set to 1000 bit precision, then full +precision can be obtained with just 7 iterations. To put that in perspective +an increase in precision by a factor of 20, has less than doubled the number of +iterations. That just goes to emphasise that most of the iterations are used +up getting the first few digits correct: after that these methods can churn out +further digits with remarkable efficiency. Or to put it another way: ['nothing beats +a really good initial guess!] + +[endsect] + diff --git a/doc/series.qbk b/doc/series.qbk new file mode 100644 index 000000000..87b35d7c4 --- /dev/null +++ b/doc/series.qbk @@ -0,0 +1,122 @@ +[section Series Evaluation] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + typename Functor::result_type sum_series(Functor& func, int bits); + + template + typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms); + + template + typename Functor::result_type sum_series(Functor& func, int bits, U init_value); + + template + typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms, U init_value); + + template + typename Functor::result_type kahan_sum_series(Functor& func, int bits); + + template + typename Functor::result_type kahan_sum_series(Functor& func, int bits, boost::uintmax_t& max_terms); + + }}} // namespaces + +[h4 Description] + +These algorithms are intended for the summation of infinite series. + +Each of the algorithms takes a nullary-function object as the first argument: +the function object will be repeatedly invoked to pull successive terms from +the series being summed. + +The second argument is the number of binary bits of +precision required, summation will stop when the next term is too small to +have any effect on the first /bits/ bits of the result. + +The optional third argument /max_terms/ sets an upper limit on the number +of terms of the series to evaluate, in addition, on exit the function will +set /max_terms/ to the actual number of terms of the series that were +evaluated: this is particularly useful for profiling the convergence +properties of a new series. + +The final optional argument /init_value/ is the initial value of the sum +to which the terms of the series should be added. This is useful in two situations: + +* Where the first value of the series has a different formula to successive +terms. In this case the first value in the series can be passed as the +last argument and the logic of the function object can then be simplified +to return subsequent terms. +* Where the series is being added (or subtracted) from some other value: +termination of the series will likely occur much more rapidly if that other +value is passed as the last argument. For example there are several functions +that can be expressed as /1 - S(z)/ where S(z) is an infinite series. In this +case pass -1 as the last argument and then negate the result of the summation +to get the result of /1 - S(z)/. + +The two /kahan_sum_series/ variants of these algorithms maintain a carry term +that corrects for roundoff error during summation. +They are inspired by the /Kahan Summation Formula/ that appears in +[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic]. +However, it should be pointed out that there are very few series that require +summation in this way. + +[h4 Example] + +Let's suppose we want to implement /log(1+x)/ via it's infinite series, + +[$../equations/log1pseries.png] + +We begin by writing a small function object to return successive terms +of the series: + + template + struct log1p_series + { + // we must define a result_type typedef: + typedef T result_type; + + log1p_series(T x) + : k(0), m_mult(-x), m_prod(-1){} + + T operator()() + { + // This is the function operator invoked by the summation + // algorithm, the first call to this operator should return + // the first term of the series, the second call the second + // term and so on. + m_prod *= m_mult; + return m_prod / ++k; + } + + private: + int k; + const T m_mult; + T m_prod; + }; + +Implementing log(1+x) is now fairly trivial: + + template + T log1p(T x) + { + // We really should add some error checking on x here! + assert(std::fabs(x) < 1); + + // construct the series functor: + log1p_series s(x); + // and add it up, with enough digits for full machine precision + // plus a couple more for luck.... ! + return tools::sum_series(s, tools::digits(x) + 2); + } + +[endsect] + diff --git a/doc/test_data.qbk b/doc/test_data.qbk new file mode 100644 index 000000000..c1458eb1a --- /dev/null +++ b/doc/test_data.qbk @@ -0,0 +1,382 @@ +[section Graphing, Profiling, and Generating Test Data for Special Functions] + +The class `test_data` and associated helper functions are designed so that in just +a few lines of code you should be able to: + +* Profile a continued fraction, or infinite series for convergence and accuracy. +* Generate csv data from a special function that can be imported into your favorite +graphing program (or spreadsheet) for further analysis. +* Generate high precision test data. + +[h4 Synopsis] + + namespace boost{ namespace math{ namespace tools{ + + enum parameter_type + { + random_in_range = 0, + periodic_in_range = 1, + power_series = 2, + dummy_param = 0x80, + }; + + template + struct parameter_info; + + template + parameter_info make_random_param(T start_range, T end_range, int n_points); + + template + parameter_info make_periodic_param(T start_range, T end_range, int n_points); + + template + parameter_info make_power_param(T basis, int start_exponent, int end_exponent); + + template + bool get_user_parameter_info(parameter_info& info, const char* param_name); + + template + class test_data + { + public: + typedef std::vector row_type; + typedef row_type value_type; + private: + typedef std::set container_type; + public: + typedef typename container_type::reference reference; + typedef typename container_type::const_reference const_reference; + typedef typename container_type::iterator iterator; + typedef typename container_type::const_iterator const_iterator; + typedef typename container_type::difference_type difference_type; + typedef typename container_type::size_type size_type; + + // creation: + test_data(){} + template + test_data(F func, const parameter_info& arg1); + + // insertion: + template + test_data& insert(F func, const parameter_info& arg1); + + template + test_data& insert(F func, const parameter_info& arg1, const parameter_info& arg2); + + template + test_data& insert(F func, const parameter_info& arg1, const parameter_info& arg2, const parameter_info& arg3); + + void clear(){ m_data.clear(); } + + // access: + iterator begin() { return m_data.begin(); } + iterator end() { return m_data.end(); } + const_iterator begin()const { return m_data.begin(); } + const_iterator end()const { return m_data.end(); } + bool operator==(const test_data& d)const{ return m_data == d.m_data; } + bool operator!=(const test_data& d)const{ return m_data != d.m_data; } + void swap(test_data& other){ m_data.swap(other.m_data); } + size_type size()const{ return m_data.size(); } + size_type max_size()const{ return m_data.max_size(); } + bool empty()const{ return m_data.empty(); } + + bool operator < (const test_data& dat)const{ return m_data < dat.m_data; } + bool operator <= (const test_data& dat)const{ return m_data <= dat.m_data; } + bool operator > (const test_data& dat)const{ return m_data > dat.m_data; } + bool operator >= (const test_data& dat)const{ return m_data >= dat.m_data; } + }; + + template + std::basic_ostream& write_csv(std::basic_ostream& os, + const test_data& data); + + template + std::basic_ostream& write_csv(std::basic_ostream& os, + const test_data& data, + const charT* separator); + + template + std::ostream& write_code(std::ostream& os, + const test_data& data, + const char* name); + + }}} // namespaces + +[h4 Description] + +This tool is best illustrated with a series of examples, if you just want the +gory details, [link test_data_reference skip onwards to the reference]. + +The functionality of test_data is split into the following parts: + +* A functor that implements the function for which data is being generated, +this is the bit you have to write. +* One of more parameters that are to be passed to the functor, these are +described in fairly abstract terms: give me N points distributed like /this/ etc. +* The class test_data, that takes the functor and descriptions of the parameters +and computes how ever many output points have been requested, these are stored +in a sorted container. +* Routines to iterate over the test_data container and output the data in either +csv format, or as C++ source code (as a table using Boost.Array). + +[h5 Example 1: Output Data for Graph Plotting] + +For example, lets say we want to graph the lgamma function between -3 and 100, +one could do this like so: + + #include + #include + + int main() + { + using namespace boost::math::tools; + + // create an object to hold the data: + test_data data; + + // insert 500 points at uniform intervals between just after -3 and 100: + double (*pf)(double) = boost::math::lgamma; + data.insert(pf, make_periodic_param(0.00001, 100.0, 500)); + + // print out in csv format: + write_csv(std::cout, data, ", "); + return 0; + } + +Which when plotted results in: + +[$../graphs/lgamma.png] + +[h5 Example 2: Creating Test Data] + +As a second example, lets suppose we want to create highly accurate test +data for a special function. Since many special functions have two or +more independent parameters, it's very hard to effectively cover all of +the possible parameter space without generating gigabytes of data at +great computational expense. A second best approach is to provide the tools +by which a user (or the library maintainer) can quickly generate more data +on demand to probe the function over a particular domain of interest. + +In this example we'll generate test data for the beta function using +NTL::RR at 1000 bit precision. Rather than call our generic +version of the beta function, we'll implement a deliberately naive version +of the beta function using lgamma, and rely on the high precision of the +data type used to get results accurate to at least 128-bit precision. In this +way our test data is independent of whatever clever tricks we may wish to +use inside the our beta function. + +To start with then, here's the function object that creates the test data: + + #include + #include + #include + #include + + #include + + using namespace boost::math::tools; + + struct beta_data_generator + { + NTL::RR operator()(NTL::RR a, NTL::RR b) + { + // + // If we throw a domain error then test_data will + // ignore this input point, we'll use this to filter + // out all cases where a < b since the beta function + // is symetrical in a and b: + // + if(a < b) + throw std::domain_error(""); + + // very naively calculate spots with lgamma: + NTL::RR g1, g2, g3; + int s1, s2, s3; + g1 = boost::math::lgamma(a, &s1); + g2 = boost::math::lgamma(b, &s2); + g3 = boost::math::lgamma(a+b, &s3); + g1 += g2 - g3; + g1 = exp(g1); + g1 *= s1 * s2 * s3; + return g1; + } + }; + +To create the data, we'll need to input the domains for a and b +for which the function will be tested: the function `get_user_parameter_info` +is designed for just that purpose. The start of main will look something like: + + // Set the precision on RR: + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the beta function:\n" + " beta(a, b)\n\n"; + + bool cont; + std::string line; + + do{ + // prompt the user for the domain of a and b to test: + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "b"); + + // create the data: + data.insert(beta_data_generator(), arg1, arg2); + + // see if the user want's any more domains tested: + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + +[caution At this point one potential stumbling block should be mentioned: +test_data<>::insert will create a matrix of test data when there are two +or more parameters, so if we have two parameters and we're asked for +a thousand points on each, that's a ['million test points in total]. +Don't say you weren't warned!] + +There's just one final step now, and that's to write the test data to file: + + std::cout << "Enter name of test data file [default=beta_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "beta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "beta_data"); + +The format of the test data looks something like: + + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1830> beta_med_data = { + SC_(0.4883005917072296142578125), SC_(0.4883005917072296142578125), SC_(3.245912809500479157065104747353807392371), + SC_(3.5808107852935791015625), SC_(0.4883005917072296142578125), SC_(1.007653173802923954909901438393379243537), + /* lots of rows skipped */ + }; + +The first two values in each row are the input parameters that were passed +to our functor and the last value is the return value from the functor. +Had our functor returned a tuple rather than a value, then we would have had +one entry for each element in the tuple in addition to the input parameters. + +The first #define serves two purposes: + +* It reduces the file sizes considerably: all those `static_cast`'s add up to a lot +of bytes otherwise (they are needed to suppress compiler warnings when `T` is +narrower than a `long double`). +* It provides a useful customisation point: for example if we were testing +a user-defined type that has more precision than a `long double` we could change +it to:\n\n +[^#define SC_(x) lexical_cast(BOOST_STRINGIZE(x))]\n\n +in order to ensure that no truncation of the values occurs prior to conversion +to `T`. Note that this isn't used by default as it's rather hard on the compiler +when the table is large. + +[h5 Example 3: Profiling a Continued Fraction for Convergence and Accuracy] + +Alternatively, lets say we want to profile a continued fraction for +convergence and error. As an example, we'll use the continued fraction +for the upper incomplete gamma function: + + template + struct upper_incomplete_gamma_fract + { + private: + T z, a; + int k; + public: + typedef std::pair result_type; + + upper_incomplete_gamma_fract(T a1, T z1) + : z(z1-a1+1), a(a1), k(0) + { + } + + result_type operator()() + { + ++k; + z += 2; + return result_type(k * (a - k), z); + } + }; + +We want to measure both the relative error, and the rate of convergence +of this fraction, so we'll write a functor that returns both as a tuple: +class test_data will unpack the tuple for us, and create one column of data +for each element in the tuple (in addition to the input parameters): + + #include + #include + #include + #include + #include + + template + struct profile_gamma_fraction + { + typedef std::tr1::tuple result_type; + + result_type operator()(T val) + { + using namespace boost::math::tools; + // estimate the true value, using arbitary precision + // arithmetic and NTL::RR: + NTL::RR rval(val); + upper_incomplete_gamma_fract f1(rval, rval); + NTL::RR true_val = continued_fraction_a(f1, 1000); + // + // Now get the aproximation at double precision, along with the number of + // iterations required: + boost::uintmax_t iters = std::numeric_limits::max(); + upper_incomplete_gamma_fract f2(val, val); + T found_val = continued_fraction_a(f2, std::numeric_limits::digits, iters); + // + // Work out the relative error, as measured in units of epsilon: + T err = real_cast(relative_error(true_val, NTL::RR(found_val)) / std::numeric_limits::epsilon()); + // + // now just return the results as a tuple: + return std::tr1::make_tuple(err, iters); + } + }; + +Feeding that functor into test_data allows rapid output of csv data, +for whatever type `T` we may be intersted in: + + int main() + { + using namespace boost::math::tools; + // create an object to hold the data: + test_data data; + // insert 500 points at uniform intervals between just after 0 and 100: + data.insert(profile_gamma_fraction(), make_periodic_param(0.01, 100.0, 100)); + // print out in csv format: + write_csv(std::cout, data, ", "); + return 0; + } + +This time there's no need to plot a graph, the first few rows are: + + a and z, Error/epsilon, Iterations required + + 0.01, 9723.14, 4726 + 1.0099, 9.54818, 87 + 2.0098, 3.84777, 40 + 3.0097, 0.728358, 25 + 4.0096, 2.39712, 21 + 5.0095, 0.233263, 16 + +So it's pretty clear that this fraction shouldn't be used for small values +of a and z. + +[h4 reference] +[#test_data_reference] + +TODO... + +[endsect] diff --git a/doc/tgamma.qbk b/doc/tgamma.qbk new file mode 100644 index 000000000..bf3691896 --- /dev/null +++ b/doc/tgamma.qbk @@ -0,0 +1,91 @@ + +[section The gamma function] + +[caution __caution ] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T tgamma(T z); + + }} // namespaces + +[h4 Description] + +Returns the "true gamma" of value z: + +[$../equations/gamm1.png] + +[$../graphs/gamma.png] + +There are effectively two versions of this function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the mantissa +of T correspond to a certain __lanczos. In practice any built in +floating point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp + +[h4 Accuracy] + +The following table shows the peek errors (in units of epsilon) +found on various platforms +with various floating point types. Unless otherwise specified any +floating point type that is narrower than the one shown will have +__zero_error. + +[table +[[Mantissa Size] [Platform and Compiler] [Factorials and Half factorials] [Values Near Zero] [Values Near 1 or 2] [Values Near a Negative Pole]] +[[53] [Win32 Visual C++ 8] [Peek=1.9 Mean=0.7 \n\n(GSL=3.9) \n(Cephes=3.0)] [Peek=2.0 Mean=1.1 \n\n(GSL=4.5) (Cephes=1)] [Peek=2.0 Mean=1.1 \n\n(GSL=7.9) (Cephes=1.0)] [Peek=2.6 Mean=1.3 \n\n(GSL=2.5) (Cephes=2.7)] ] +[[64] [Linux IA32 / GCC] [Peek=300 Mean=49.5 \n\n(Native Peek=395 Mean=89)] [Peek=3.0 Mean=1.4 \n\n(Native Peek=11 Mean=3.3)] [Peek=5.0 Mean=1.8 \n\n(Native Peek=0.92 Mean=0.2)] [Peek=157 Mean=65 \n\n(Native Peek=205 Mean=108)] ] +[[64] [Linux IA64 / GCC] [Peek=2.8 Mean=0.9 \n(Native=0.7)] [Peek=4.8 Mean=1.5 \n(Native=0)] [Peek=4.8 Mean=1.5 \n(Native=0)] [Peek=5.0 Mean=1.7 \n(Native 0)] ] +[[113] [HPUX IA64, aCC A.06.06] [Peek=2.5 Mean=1.1 \n(Native=0)] [Peek=3.5 Mean=1.7 \n(Native=0)] [Peek=3.5 Mean=1.6 \n(Native=0)] [Peek=5.2 Mean=1.92 \n(Native=0)] ] +] + +[h4 Testing] + +The gamma is relatively easy to test: factorials and half integer factorials +can be calculated exactly by other means and compared with the gamma function. +In addition some accuracy tests in known tricky areas were computed at high precision +using the generic version of this function. + +[h4 Implementation] + +The generic version of this function is implemented by combining the series and +continued fraction representations for the incomplete gamma function: + +[$../equations/gamm2.png] + +where /l/ is an arbitrary integration limit: choosing [^l = max(10, a)] +seems to work fairly well. + +For types of known precision the __lanczos is used, a traits class +`boost::math::lanczos::lanczos_traits` maps type T to an appropriate +approximation. + +For z in the range -20 < z < 1 then recursion is used to shift to z > 1 via: + +[$../equations/gamm3.png] + +For very small z, this helps to preserve the identity: + +[$../equations/gamm4.png] + +For z < -20 the reflection formula: + +[$../equations/gamm5.png] + +is used. Particular care has to be taken to evaluate the `z * sin(PI * z)` part: +a special routine is used to reduce z prior to multiplying by PI to ensure that the +result in is the range [0, PI/2]. Without this an excessive amount of error occurs +in this region: which is hard enough already as the rate of change near a negative pole +is /exceptionally/ high. + +[endsect] + diff --git a/test/Jamfile.v2 b/test/Jamfile.v2 new file mode 100644 index 000000000..c3113f7bd --- /dev/null +++ b/test/Jamfile.v2 @@ -0,0 +1,22 @@ + +# bring in the rules for testing +import testing ; + +project + : requirements + gcc:-Wno-missing-braces + ; + +run log1p_expm1_test.cpp ; +run powm1_sqrtp1m1_test.cpp ; +run test_beta.cpp ; +run test_classify.cpp ; +run test_erf.cpp ; +run test_gamma.cpp ; +run test_ibeta.cpp ; + + + + + + diff --git a/test/beta_exp_data.ipp b/test/beta_exp_data.ipp new file mode 100644 index 000000000..f29e6f0b9 --- /dev/null +++ b/test/beta_exp_data.ipp @@ -0,0 +1,364 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 351> beta_exp_data = { + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1155631.551635027016649268884796909927277), + SC_(0.216575062950141727924346923828125e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1039549.452063747329381617654200841254652), + SC_(0.216575062950141727924346923828125e-5), SC_(0.216575062950141727924346923828125e-5), SC_(923467.3524924676425690820378921903570447), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(715366.9882608199489156088500706918884474), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.216575062950141727924346923828125e-5), SC_(599284.8886895402674421924477675861806454), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.72700195232755504548549652099609375e-5), SC_(275102.4248866129549503632384850636365051), + SC_(0.14000004739500582218170166015625e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(649244.3230419389091055494874477610228104), + SC_(0.14000004739500582218170166015625e-4), SC_(0.216575062950141727924346923828125e-5), SC_(533162.2234706592346717218748633348329255), + SC_(0.14000004739500582218170166015625e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(208979.7596677320047637517666971035319141), + SC_(0.14000004739500582218170166015625e-4), SC_(0.14000004739500582218170166015625e-4), SC_(142857.0944488511634633414470885024340382), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(635967.2966761120408405544283252854891051), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(519885.1971048323697502064131013612391462), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(195702.7333019051790657557774869592595619), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(129580.0680830243894812627999923649303964), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(116303.0417171976400618567245671670493297), + SC_(0.60085076256655156612396240234375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(594458.8435535046961781833034439822035691), + SC_(0.60085076256655156612396240234375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(478376.7439822250699480739314338811953296), + SC_(0.60085076256655156612396240234375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(154194.2801792984055346510144704788292969), + SC_(0.60085076256655156612396240234375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(88071.61496041830983457563965247905141874), + SC_(0.60085076256655156612396240234375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(74794.58859459188997822531683455967562115), + SC_(0.60085076256655156612396240234375e-4), SC_(0.60085076256655156612396240234375e-4), SC_(33286.13547199056171829186221434157408122), + SC_(0.000116783194243907928466796875), SC_(0.1730655412757187150418758392333984375e-5), SC_(586378.651529572636659503110266369369984), + SC_(0.000116783194243907928466796875), SC_(0.216575062950141727924346923828125e-5), SC_(470296.5519582930697306370777452683761698), + SC_(0.000116783194243907928466796875), SC_(0.72700195232755504548549652099609375e-5), SC_(146114.0881553671010006794541334417041727), + SC_(0.000116783194243907928466796875), SC_(0.14000004739500582218170166015625e-4), SC_(79991.42293648792255405447160736354988824), + SC_(0.000116783194243907928466796875), SC_(0.17196454791701398789882659912109375e-4), SC_(66714.39657066193835073550620286643916615), + SC_(0.000116783194243907928466796875), SC_(0.60085076256655156612396240234375e-4), SC_(25205.94344806645534821918725567217595504), + SC_(0.000116783194243907928466796875), SC_(0.000116783194243907928466796875), SC_(17125.75142415007588824659456664464078893), + SC_(0.000149052008055150508880615234375), SC_(0.1730655412757187150418758392333984375e-5), SC_(584524.843186061699046596661105506295866), + SC_(0.000149052008055150508880615234375), SC_(0.216575062950141727924346923828125e-5), SC_(468442.7436147821658663815479488360288728), + SC_(0.000149052008055150508880615234375), SC_(0.72700195232755504548549652099609375e-5), SC_(144260.2798118565930535598166654066443804), + SC_(0.000149052008055150508880615234375), SC_(0.14000004739500582218170166015625e-4), SC_(78137.6145929779366207198292645775242312), + SC_(0.000149052008055150508880615234375), SC_(0.17196454791701398789882659912109375e-4), SC_(64860.58822715220034984276283119323639683), + SC_(0.000149052008055150508880615234375), SC_(0.60085076256655156612396240234375e-4), SC_(23352.13510456004391420247071206624241613), + SC_(0.000149052008055150508880615234375), SC_(0.000116783194243907928466796875), SC_(15271.94308064806187957915306088879145787), + SC_(0.000149052008055150508880615234375), SC_(0.000149052008055150508880615234375), SC_(13418.13473714855046889923959457185167481), + SC_(0.0003985252114944159984588623046875), SC_(0.1730655412757187150418758392333984375e-5), SC_(580325.0267004741284777720102301507805905), + SC_(0.0003985252114944159984588623046875), SC_(0.216575062950141727924346923828125e-5), SC_(464242.9271291948561701678823852769833447), + SC_(0.0003985252114944159984588623046875), SC_(0.72700195232755504548549652099609375e-5), SC_(140060.4633262723437444763727887594791223), + SC_(0.0003985252114944159984588623046875), SC_(0.14000004739500582218170166015625e-4), SC_(73937.79810739772240919420653098643931922), + SC_(0.0003985252114944159984588623046875), SC_(0.17196454791701398789882659912109375e-4), SC_(60660.77174157390262331984812205981539232), + SC_(0.0003985252114944159984588623046875), SC_(0.60085076256655156612396240234375e-4), SC_(19152.31861900746010982713873615733036615), + SC_(0.0003985252114944159984588623046875), SC_(0.000116783194243907928466796875), SC_(11072.12659512946959154647142760990796737), + SC_(0.0003985252114944159984588623046875), SC_(0.000149052008055150508880615234375), SC_(9218.318251649302932199820676778504299719), + SC_(0.0003985252114944159984588623046875), SC_(0.0003985252114944159984588623046875), SC_(5018.501766299587761063198864612299262498), + SC_(0.00063875340856611728668212890625), SC_(0.1730655412757187150418758392333984375e-5), SC_(579381.3241429882314787600400195790681276), + SC_(0.00063875340856611728668212890625), SC_(0.216575062950141727924346923828125e-5), SC_(463299.2245717092103071408953485735740662), + SC_(0.00063875340856611728668212890625), SC_(0.72700195232755504548549652099609375e-5), 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SC_(31.314670562744140625), SC_(0.98384749889373779296875), SC_(0.03409297661295516414885174102503442819277), + SC_(31.314670562744140625), SC_(1.1576130390167236328125), SC_(0.01721838700844971854228152401760085523744), + SC_(31.314670562744140625), SC_(3.451677799224853515625), SC_(0.1900362128922583001056640211044473004449e-4), + SC_(31.314670562744140625), SC_(7.88237094879150390625), SC_(0.2885050414250820990910991383845819731228e-8), + SC_(31.314670562744140625), SC_(15.848876953125), SC_(0.6530600415427592338710565326600917620279e-13), + SC_(31.314670562744140625), SC_(31.314670562744140625), SC_(0.8915675179813854663775491877023168025702e-19), + SC_(35.515575408935546875), SC_(0.1730655412757187150418758392333984375e-5), SC_(577811.6427939229375161086906177400275569), + SC_(35.515575408935546875), SC_(0.216575062950141727924346923828125e-5), SC_(461729.5432274267192940591041545317110485), + SC_(35.515575408935546875), SC_(0.72700195232755504548549652099609375e-5), SC_(137547.0794806155767312939551779369164394), + SC_(35.515575408935546875), SC_(0.14000004739500582218170166015625e-4), SC_(71424.41433572253924394173477030901070617), + SC_(35.515575408935546875), SC_(0.17196454791701398789882659912109375e-4), SC_(58147.38800503622156242701682984885441441), + SC_(35.515575408935546875), SC_(0.60085076256655156612396240234375e-4), SC_(16638.93535389697145567616644195839808463), + SC_(35.515575408935546875), SC_(0.000116783194243907928466796875), SC_(8558.7439531443773341365422276985698545), + SC_(35.515575408935546875), SC_(0.000149052008055150508880615234375), SC_(6704.935964257969942256683796604190388483), + SC_(35.515575408935546875), SC_(0.0003985252114944159984588623046875), SC_(2505.122219132224374561111132130667265447), + SC_(35.515575408935546875), SC_(0.00063875340856611728668212890625), SC_(1561.422298589200014741982262412434636973), + SC_(35.515575408935546875), SC_(0.0010718167759478092193603515625), SC_(928.8722467787222242808854452351889563633), + SC_(35.515575408935546875), SC_(0.00302191521041095256805419921875), SC_(326.8110378414911903385288649321631400664), + SC_(35.515575408935546875), SC_(0.00499413348734378814697265625), SC_(196.1482005962757034094399257872899036696), + SC_(35.515575408935546875), SC_(0.00928423367440700531005859375), SC_(103.6619197982581616421872226071400592126), + SC_(35.515575408935546875), SC_(0.0241700224578380584716796875), SC_(37.45772609209636964536574943912854380171), + SC_(35.515575408935546875), SC_(0.06227754056453704833984375), SC_(12.450993041772750669407167680667607419), + SC_(35.515575408935546875), SC_(0.12234418094158172607421875), SC_(4.986241033913772097760869076957541032543), + SC_(35.515575408935546875), SC_(0.249557673931121826171875), SC_(1.494248013206312554952038234048230712297), + SC_(35.515575408935546875), SC_(0.4912221431732177734375), SC_(0.3133802674518380533642801450941624241265), + SC_(35.515575408935546875), SC_(0.98384749889373779296875), SC_(0.03012063183422189552497230190183310697765), + SC_(35.515575408935546875), SC_(1.1576130390167236328125), SC_(0.01488854601694810064869001826934615844448), + SC_(35.515575408935546875), SC_(3.451677799224853515625), SC_(0.1249353030883651639444207203869412646859e-4), + SC_(35.515575408935546875), SC_(7.88237094879150390625), SC_(0.1169663515742618634451758579426537250388e-8), + SC_(35.515575408935546875), SC_(15.848876953125), SC_(0.1250851178791656557811931908536049655477e-13), + SC_(35.515575408935546875), SC_(31.314670562744140625), SC_(0.5365332615381537353158263927841062696094e-20), + SC_(35.515575408935546875), SC_(35.515575408935546875), SC_(0.2474098388705327630350038282012894724267e-21), + }; +#undef SC_ + + + + diff --git a/test/beta_med_data.ipp b/test/beta_med_data.ipp new file mode 100644 index 000000000..d31c8b714 --- /dev/null +++ b/test/beta_med_data.ipp @@ -0,0 +1,1842 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1830> beta_med_data = { + SC_(0.4883005917072296142578125), SC_(0.4883005917072296142578125), SC_(3.245912809500479157065104747353807392371), + SC_(3.5808107852935791015625), SC_(0.4883005917072296142578125), SC_(1.007653173802923954909901438393379243537), + SC_(3.5808107852935791015625), SC_(3.5808107852935791015625), SC_(0.01354763979296020361276499134253466025558), + SC_(9.76306438446044921875), SC_(0.4883005917072296142578125), SC_(0.6040068168955603700205507368871869179054), + SC_(9.76306438446044921875), SC_(3.5808107852935791015625), SC_(0.0006768556385450819292424677664459339662542), + SC_(9.76306438446044921875), SC_(9.76306438446044921875), SC_(0.1522016641487441392375569251478701938958e-5), + SC_(10.9950771331787109375), SC_(0.4883005917072296142578125), SC_(0.5691374355039684505283786467923310741394), + SC_(10.9950771331787109375), SC_(3.5808107852935791015625), SC_(0.0004623291188878679140702882645611609028529), + SC_(10.9950771331787109375), SC_(9.76306438446044921875), SC_(0.6525428759744388549684590935535556520911e-6), + SC_(10.9950771331787109375), SC_(10.9950771331787109375), SC_(0.2595645778453125149921775190075494972069e-6), + SC_(11.2553272247314453125), SC_(0.4883005917072296142578125), SC_(0.5625252582720929656922435362566677067416), + SC_(11.2553272247314453125), SC_(3.5808107852935791015625), SC_(0.0004286883639355158965650882249453525366545), + SC_(11.2553272247314453125), SC_(9.76306438446044921875), SC_(0.5507764130648082456092816216454348231704e-6), + SC_(11.2553272247314453125), SC_(10.9950771331787109375), SC_(0.215765220969841623426272969427339892725e-6), + SC_(11.2553272247314453125), SC_(11.2553272247314453125), SC_(0.1787990823319435797792105752447442768941e-6), + SC_(12.70741176605224609375), SC_(0.4883005917072296142578125), SC_(0.5294907274516666109456973690863173323156), + SC_(12.70741176605224609375), SC_(3.5808107852935791015625), SC_(0.0002889727352835592358365667915148162805071), + SC_(12.70741176605224609375), SC_(9.76306438446044921875), SC_(0.2252927431029049671249995618445578512183e-6), + SC_(12.70741176605224609375), SC_(10.9950771331787109375), SC_(0.8131970313223237520922385412367530078239e-7), + SC_(12.70741176605224609375), SC_(11.2553272247314453125), SC_(0.6627020372930043647557276756580896208093e-7), + SC_(12.70741176605224609375), SC_(12.70741176605224609375), SC_(0.2245035027874900533178284692921746500427e-7), + SC_(13.55634593963623046875), SC_(0.4883005917072296142578125), SC_(0.5127159488422374707026291329594215821195), + SC_(13.55634593963623046875), SC_(3.5808107852935791015625), SC_(0.0002338050540339516890899323768977469084643), + SC_(13.55634593963623046875), SC_(9.76306438446044921875), SC_(0.1385641428853223512568009007531028899044e-6), + SC_(13.55634593963623046875), SC_(10.9950771331787109375), SC_(0.4779208342424526648121034774140947601376e-7), + SC_(13.55634593963623046875), SC_(11.2553272247314453125), SC_(0.3858688985561532847951616400183530491078e-7), + SC_(13.55634593963623046875), SC_(12.70741176605224609375), SC_(0.1243334579233090577547671408856256594842e-7), + SC_(13.55634593963623046875), SC_(13.55634593963623046875), SC_(0.6695819170364036861651031065079190250677e-8), + SC_(14.19721508026123046875), SC_(0.4883005917072296142578125), SC_(0.5010725503977163404563348242929462964601), + SC_(14.19721508026123046875), SC_(3.5808107852935791015625), SC_(0.0002008402390975031689923237986077084533841), + SC_(14.19721508026123046875), SC_(9.76306438446044921875), SC_(0.9753123839573875998586188687154385562149e-7), + SC_(14.19721508026123046875), SC_(10.9950771331787109375), SC_(0.325396200950697967725364011405694079166e-7), + SC_(14.19721508026123046875), SC_(11.2553272247314453125), SC_(0.2609397484154104517486373559610576982077e-7), + SC_(14.19721508026123046875), SC_(12.70741176605224609375), SC_(0.8104897951369500255523549462354045969225e-8), + SC_(14.19721508026123046875), SC_(13.55634593963623046875), SC_(0.4276119047726022319466243593050360444817e-8), + SC_(14.19721508026123046875), SC_(14.19721508026123046875), SC_(0.2689992929599026714275097270923824879089e-8), + SC_(15.76973247528076171875), SC_(0.4883005917072296142578125), SC_(0.475600932703444822636860975438244705831), + SC_(15.76973247528076171875), SC_(3.5808107852935791015625), SC_(0.0001418728926781729275649568751766718949223), + SC_(15.76973247528076171875), SC_(9.76306438446044921875), SC_(0.4336326752543966160912379066984549678452e-7), + SC_(15.76973247528076171875), SC_(10.9950771331787109375), SC_(0.1338244910537290013872585434554199633933e-7), + SC_(15.76973247528076171875), SC_(11.2553272247314453125), SC_(0.1056140308679468515555449880362724923627e-7), + SC_(15.76973247528076171875), SC_(12.70741176605224609375), SC_(0.3009132692253922032267871346669237418265e-8), + SC_(15.76973247528076171875), SC_(13.55634593963623046875), SC_(0.1512669863598730427313876780451595800786e-8), + SC_(15.76973247528076171875), SC_(14.19721508026123046875), SC_(0.918353523266471209411262276890588115837e-9), + SC_(15.76973247528076171875), SC_(15.76973247528076171875), SC_(0.2882767813027467848960413793272553398391e-9), + SC_(17.126956939697265625), SC_(0.4883005917072296142578125), SC_(0.4565218812244720449676057777369567274854), + SC_(17.126956939697265625), SC_(3.5808107852935791015625), SC_(0.0001077656194412027230279434065555746466143), + SC_(17.126956939697265625), SC_(9.76306438446044921875), SC_(0.2267616977041493737061190673358616637022e-7), + SC_(17.126956939697265625), SC_(10.9950771331787109375), SC_(0.6567383488365389895737498469534509899371e-8), + SC_(17.126956939697265625), SC_(11.2553272247314453125), SC_(0.5115778604579878982282382763211068891719e-8), + SC_(17.126956939697265625), SC_(12.70741176605224609375), SC_(0.1358240483817446693334902305742392032884e-8), + SC_(17.126956939697265625), SC_(13.55634593963623046875), SC_(0.6562503597342494326172080038282840625689e-9), + SC_(17.126956939697265625), SC_(14.19721508026123046875), SC_(0.3869619725572160229261125982995702539806e-9), + SC_(17.126956939697265625), SC_(15.76973247528076171875), SC_(0.113368810788831205921628103707632404292e-9), + SC_(17.126956939697265625), SC_(17.126956939697265625), SC_(0.4211904380955363652677610570193848736124e-10), + SC_(17.394779205322265625), SC_(0.4883005917072296142578125), SC_(0.4530251392082678136392746003508150955157), + SC_(17.394779205322265625), SC_(3.5808107852935791015625), SC_(0.0001023208443499526530037973037899364285962), + SC_(17.394779205322265625), SC_(9.76306438446044921875), SC_(0.2005354902961112804072043023442896551258e-7), + SC_(17.394779205322265625), SC_(10.9950771331787109375), SC_(0.5737643064679879701819734148551932728972e-8), + SC_(17.394779205322265625), SC_(11.2553272247314453125), SC_(0.4458281433716227719378477553190576945752e-8), + SC_(17.394779205322265625), SC_(12.70741176605224609375), SC_(0.1167764940936604548122803636068842045715e-8), + SC_(17.394779205322265625), SC_(13.55634593963623046875), SC_(0.5599447115103713063560079265124538962881e-9), + SC_(17.394779205322265625), SC_(14.19721508026123046875), SC_(0.3283303102000759769499915328513568628178e-9), + SC_(17.394779205322265625), SC_(15.76973247528076171875), SC_(0.9492349452847812246661303626609556094242e-10), + SC_(17.394779205322265625), SC_(17.126956939697265625), SC_(0.348822657233525373086455562928279206484e-10), + SC_(17.394779205322265625), SC_(17.394779205322265625), SC_(0.2882805174446943738297455251918170211958e-10), + SC_(18.8463134765625), SC_(0.4883005917072296142578125), SC_(0.4353971044081953663699145675473451692464), + SC_(18.8463134765625), SC_(3.5808107852935791015625), SC_(0.7821907282155094140409267653485080042403e-4), + SC_(18.8463134765625), SC_(9.76306438446044921875), SC_(0.1057326607413087663065202876489554338397e-7), + SC_(18.8463134765625), SC_(10.9950771331787109375), SC_(0.2837956463158533935066785198610085243411e-8), + SC_(18.8463134765625), SC_(11.2553272247314453125), SC_(0.2176364981562781990024053280982715425444e-8), + SC_(18.8463134765625), SC_(12.70741176605224609375), SC_(0.5308686415267842780183531434236061793759e-9), + SC_(18.8463134765625), SC_(13.55634593963623046875), SC_(0.2445522707205327599339640190943446349336e-9), + SC_(18.8463134765625), SC_(14.19721508026123046875), SC_(0.1392224976008039575865521517870553781522e-9), + SC_(18.8463134765625), SC_(15.76973247528076171875), SC_(0.3752833192186647967829389243731752462616e-10), + SC_(18.8463134765625), SC_(17.126956939697265625), SC_(0.1301581966373817837525897476236998994169e-10), + SC_(18.8463134765625), SC_(17.394779205322265625), SC_(0.1063754790511525096235331936144872215005e-10), + SC_(18.8463134765625), 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SC_(0.4252617027499467811810299417544006679444e-59), + SC_(99.28820037841796875), SC_(96.76981353759765625), SC_(0.3484969370967557871486154959308680473194e-59), + SC_(99.28820037841796875), SC_(96.88709259033203125), SC_(0.3207141270593581779254023249982845071696e-59), + SC_(99.28820037841796875), SC_(97.0595703125), SC_(0.2838654062329006203358352082138793616487e-59), + SC_(99.28820037841796875), SC_(98.1111602783203125), SC_(0.135331897584667014131258701945417599616e-59), + SC_(99.28820037841796875), SC_(99.28820037841796875), SC_(0.5946341007968227163535653640336744451921e-60), + SC_(99.64617156982421875), SC_(0.4883005917072296142578125), SC_(0.1920415401866252505960149155087364700429), + SC_(99.64617156982421875), SC_(3.5808107852935791015625), SC_(0.2425007447493048943965527355796496259445e-6), + SC_(99.64617156982421875), SC_(9.76306438446044921875), SC_(0.4338463329689989369521219399698331619339e-14), + SC_(99.64617156982421875), SC_(10.9950771331787109375), SC_(0.2238793620730317918365780183630913591486e-15), + SC_(99.64617156982421875), SC_(11.2553272247314453125), SC_(0.1217883523395548244503225558455778824065e-15), + SC_(99.64617156982421875), SC_(12.70741176605224609375), SC_(0.4506937720795369966253108234128095796053e-17), + SC_(99.64617156982421875), SC_(13.55634593963623046875), SC_(0.7053692076383500412728910873848177350615e-18), + SC_(99.64617156982421875), SC_(14.19721508026123046875), SC_(0.1796747134889476173979330386276389059013e-18), + SC_(99.64617156982421875), SC_(15.76973247528076171875), SC_(0.6990175412575383911153375435413556806998e-20), + SC_(99.64617156982421875), SC_(17.126956939697265625), SC_(0.475219770637311911218530901770424242484e-21), + SC_(99.64617156982421875), SC_(17.394779205322265625), SC_(0.2828061146103015181288433756774729344582e-21), + SC_(99.64617156982421875), SC_(18.8463134765625), SC_(0.1805339957483893466812493111272198920639e-22), + SC_(99.64617156982421875), SC_(21.200313568115234375), SC_(0.2551145044678329055180101519714681369991e-24), + SC_(99.64617156982421875), SC_(22.111194610595703125), SC_(0.521827937109776436392706852004371656104e-25), + SC_(99.64617156982421875), SC_(27.8570384979248046875), SC_(0.4585509723284941456360070341222362036382e-29), + SC_(99.64617156982421875), SC_(29.709972381591796875), SC_(0.2797738978507146557808496231858841269347e-30), + SC_(99.64617156982421875), SC_(30.198291778564453125), SC_(0.1359628995333712367120268567405627842371e-30), + SC_(99.64617156982421875), SC_(30.8236217498779296875), SC_(0.5445278627462667801264942137517994423527e-31), + SC_(99.64617156982421875), SC_(36.1357879638671875), SC_(0.3335676798398169433467181047773937919425e-34), + SC_(99.64617156982421875), SC_(39.228778839111328125), SC_(0.5925480070878764501178860688204541780342e-36), + SC_(99.64617156982421875), SC_(39.8798675537109375), SC_(0.2596346306023602973051248853685030558771e-36), + SC_(99.64617156982421875), SC_(42.181911468505859375), SC_(0.1491806079618579894544070083102953058676e-37), + SC_(99.64617156982421875), SC_(42.21454620361328125), SC_(0.1433543971257228329446310718021257196428e-37), + SC_(99.64617156982421875), SC_(47.481121063232421875), SC_(0.2894219671371863633901114678428523677553e-40), + SC_(99.64617156982421875), SC_(48.5427093505859375), SC_(0.8715859321707892083793914056580348529658e-41), + SC_(99.64617156982421875), SC_(50.371234893798828125), SC_(0.1144304917248501740653089888841788532933e-41), + SC_(99.64617156982421875), SC_(54.69268035888671875), SC_(0.1122326874270186317133861668699838855083e-43), + SC_(99.64617156982421875), SC_(54.72658538818359375), SC_(0.108332632811810033121283078371423493224e-43), + SC_(99.64617156982421875), SC_(63.23960113525390625), SC_(0.2262074110898120193155243766072235368719e-47), + SC_(99.64617156982421875), SC_(63.979938507080078125), SC_(0.1121769230946183388130951853048359732083e-47), + SC_(99.64617156982421875), SC_(65.55123138427734375), SC_(0.2576069444251818757386717313878324944765e-48), + SC_(99.64617156982421875), SC_(65.57750701904296875), SC_(0.2513961622572047306221104339881351344355e-48), + SC_(99.64617156982421875), SC_(67.8767242431640625), SC_(0.3044620609127321948461303153395712705542e-49), + SC_(99.64617156982421875), SC_(68.13913726806640625), SC_(0.2399969117830299809884620846041547199087e-49), + SC_(99.64617156982421875), SC_(72.586639404296875), SC_(0.4654774693873818497215598980564748298819e-51), + SC_(99.64617156982421875), SC_(74.06732177734375), SC_(0.1298622741166349753386447918518841478102e-51), + SC_(99.64617156982421875), SC_(74.31581878662109375), SC_(0.1049960916780689406311743991678991020359e-51), + SC_(99.64617156982421875), SC_(75.77643585205078125), SC_(0.3039342449834348309579161767378331641473e-52), + SC_(99.64617156982421875), SC_(79.222808837890625), SC_(0.1737297384572845553919699070496733332975e-53), + SC_(99.64617156982421875), SC_(79.79488372802734375), SC_(0.1089347230301574574227379586143568671545e-53), + SC_(99.64617156982421875), SC_(79.81259918212890625), SC_(0.1073754663619106685903160281830989629804e-53), + SC_(99.64617156982421875), SC_(80.0300445556640625), SC_(0.8997668853899322225061748437651655966981e-54), + SC_(99.64617156982421875), SC_(81.47422027587890625), SC_(0.2804407307548493204640087364124890413703e-54), + SC_(99.64617156982421875), SC_(83.50250244140625), SC_(0.5586826121117789439615638077672711001932e-55), + SC_(99.64617156982421875), SC_(84.9144439697265625), SC_(0.1846564298928271058587078878704235002733e-55), + SC_(99.64617156982421875), SC_(87.8442840576171875), SC_(0.1933065218129760647098319832562443180994e-56), + SC_(99.64617156982421875), SC_(90.58013916015625), SC_(0.246368912661760433645524568855257596525e-57), + SC_(99.64617156982421875), SC_(91.3384552001953125), SC_(0.1402807696431983675171416011032570303907e-57), + SC_(99.64617156982421875), SC_(91.57439422607421875), SC_(0.1178127749000117009577583032714136533809e-57), + SC_(99.64617156982421875), SC_(93.3999786376953125), SC_(0.3085259587898341188615347001733307622013e-58), + SC_(99.64617156982421875), SC_(95.71712493896484375), SC_(0.5784821309052389104753898006604136073638e-59), + SC_(99.64617156982421875), SC_(95.75110626220703125), SC_(0.5645763032239863667263699872521759670044e-59), + SC_(99.64617156982421875), SC_(95.94964599609375), SC_(0.489819130902001275392106966836686313721e-59), + SC_(99.64617156982421875), SC_(96.48920440673828125), SC_(0.3333152772138028099176546108963123743121e-59), + SC_(99.64617156982421875), SC_(96.76981353759765625), SC_(0.2730076878660219712815291918211112895904e-59), + SC_(99.64617156982421875), SC_(96.88709259033203125), SC_(0.2511891524767099195545978116145973727058e-59), + SC_(99.64617156982421875), SC_(97.0595703125), SC_(0.222258516470169272817490714164593170175e-59), + SC_(99.64617156982421875), SC_(98.1111602783203125), SC_(0.1057582725557246879085406508118398211083e-59), + SC_(99.64617156982421875), SC_(99.28820037841796875), SC_(0.4637012277372110272033377955412817752539e-60), + SC_(99.64617156982421875), SC_(99.64617156982421875), SC_(0.3613651154665954292095783593077701467444e-60), + }; +#undef SC_ + + + diff --git a/test/beta_small_data.ipp b/test/beta_small_data.ipp new file mode 100644 index 000000000..6f6dec3ca --- /dev/null +++ b/test/beta_small_data.ipp @@ -0,0 +1,33 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 21> beta_small_data = { + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1155631.551635027016649268884796909927277), + SC_(0.216575062950141727924346923828125e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1039549.452063747329381617654200841254652), + SC_(0.216575062950141727924346923828125e-5), SC_(0.216575062950141727924346923828125e-5), SC_(923467.3524924676425690820378921903570447), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(715366.9882608199489156088500706918884474), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.216575062950141727924346923828125e-5), SC_(599284.8886895402674421924477675861806454), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.72700195232755504548549652099609375e-5), SC_(275102.4248866129549503632384850636365051), + SC_(0.14000004739500582218170166015625e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(649244.3230419389091055494874477610228104), + SC_(0.14000004739500582218170166015625e-4), SC_(0.216575062950141727924346923828125e-5), SC_(533162.2234706592346717218748633348329255), + SC_(0.14000004739500582218170166015625e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(208979.7596677320047637517666971035319141), + SC_(0.14000004739500582218170166015625e-4), SC_(0.14000004739500582218170166015625e-4), SC_(142857.0944488511634633414470885024340382), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(635967.2966761120408405544283252854891051), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(519885.1971048323697502064131013612391462), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(195702.7333019051790657557774869592595619), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(129580.0680830243894812627999923649303964), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(116303.0417171976400618567245671670493297), + SC_(0.60085076256655156612396240234375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(594458.8435535046961781833034439822035691), + SC_(0.60085076256655156612396240234375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(478376.7439822250699480739314338811953296), + SC_(0.60085076256655156612396240234375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(154194.2801792984055346510144704788292969), + SC_(0.60085076256655156612396240234375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(88071.61496041830983457563965247905141874), + SC_(0.60085076256655156612396240234375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(74794.58859459188997822531683455967562115), + SC_(0.60085076256655156612396240234375e-4), SC_(0.60085076256655156612396240234375e-4), SC_(33286.13547199056171829186221434157408122), + }; +#undef SC_ + + + diff --git a/test/big_whole_test.cpp b/test/big_whole_test.cpp deleted file mode 100644 index 1aba7ae49..000000000 --- a/test/big_whole_test.cpp +++ /dev/null @@ -1,2109 +0,0 @@ -// Boost big_whole_test.cpp test file --------------------------------------// - -// Copyright 2004 Daryle Walker. Use, modification, and distribution are -// subject to the Boost Software License, Version 1.0. (See accompanying file -// LICENSE_1_0.txt or a copy at .) - -// See for the library's home page. - -// Revision History -// 16 Feb 2004 Initial version (Daryle Walker) - -#include // for boost::math::big_whole, etc. -#include // for main, BOOST_CHECK_EQUAL, etc. - -#include // for std::copy -#include // for std::size_t -#include // for std::numeric_limits -#include // for std::set -#include // for std::range_error, std::length_error, etc. -#include // for std::valarray - - -// Control to allow extremely large memory allocations -#ifdef CONTROL_USE_BIG_MEMORY -#define PRIVATE_USE_BIG_MEMORY 1 -#else -#define PRIVATE_USE_BIG_MEMORY 0 -#endif - - -// Use internal knowledge of big_whole (i.e. cheat) to force situations -// where multiple-word representations have to be used -typedef unsigned int word_type; -typedef std::numeric_limits wlimits_type; - - -// Helper function to compare valarrays -template < typename T > -bool -equal_valarrays -( - std::valarray const & lhs, - std::valarray const & rhs -) -{ - if ( lhs.size() == rhs.size() ) - { - std::valarray s( lhs.size() ); - - s[ lhs == rhs ] = 1; - return s.sum() == s.size(); - } - else - { - return false; - } -} - -// Helper function to insert values into sets -template < class SetType, typename ValueType > -void -insert_value_range -( - SetType & s, - ValueType start, - ValueType finish -) -{ - for ( ValueType i = start ; i <= finish ; ++i ) - { - s.insert( i ); - } -} - -// Helper function to remove values from sets -template < class SetType, typename ValueType > -void -erase_value_range -( - SetType & s, - ValueType start, - ValueType finish -) -{ - for ( ValueType i = start ; i <= finish ; ++i ) - { - s.erase( i ); - } -} - -// Helper function to convert sets to valarrays -template < typename T > -std::valarray -set_to_valarray -( - std::set const & s -) -{ - std::valarray temp( s.size() ); - - std::copy( s.begin(), s.end(), &temp[0] ); - return temp; -} - - -// Unit test for the basics -void -basic_bigwhole_unit_test -( -) -{ - using boost::math::big_whole; - using std::valarray; - using std::size_t; - - typedef valarray va_bool_t; - typedef valarray va_size_t; - - // Default construction - big_whole x1; - BOOST_CHECK_EQUAL( 0u, x1.to_uintmax() ); - - // Converting assignment - x1.assign( 5u ); - BOOST_CHECK_EQUAL( 5u, x1.to_uintmax() ); - - // Converting construction - big_whole x2 = 17; - BOOST_CHECK_EQUAL( 17u, x2.to_uintmax() ); - - // Copy construction - big_whole x3( x1 ); - BOOST_CHECK_EQUAL( 5u, x3.to_uintmax() ); - - // Assignment operator - x1 = x2; - BOOST_CHECK_EQUAL( 17u, x1.to_uintmax() ); - - // Swapping - swap( x1, x3 ); - BOOST_CHECK_EQUAL( 5u, x1.to_uintmax() ); - BOOST_CHECK_EQUAL( 17u, x3.to_uintmax() ); - - // Copying assignment - x2.assign( big_whole() ); - BOOST_CHECK_EQUAL( 0u, x2.to_uintmax() ); - - // Bit-vector conversion - va_bool_t const x1_b = x1.to_bit_vector(); - bool const x1_b_check[] = { true, false, true }; - size_t const x1_b_size = sizeof( x1_b_check ) / sizeof( x1_b_check[0] ); - BOOST_CHECK( equal_valarrays(va_bool_t( x1_b_check, x1_b_size ), x1_b) ); - - BOOST_CHECK_EQUAL( 0u, x2.to_bit_vector().size() ); - - va_bool_t const x3_b = x3.to_bit_vector(); - bool const x3_b_check[] = { true, false, false, false, true }; - size_t const x3_b_size = sizeof( x3_b_check ) / sizeof( x3_b_check[0] ); - BOOST_CHECK( equal_valarrays(va_bool_t( x3_b_check, x3_b_size ), x3_b) ); - - // Bit-index conversion - va_size_t const x1_i = x1.to_bit_indices(); - size_t const x1_i_check[] = { 0, 2 }; - size_t const x1_i_size = sizeof( x1_i_check ) / sizeof( x1_i_check[0] ); - BOOST_CHECK( equal_valarrays(va_size_t( x1_i_check, x1_i_size ), x1_i) ); - BOOST_CHECK_EQUAL( x1_b_size, 1u + x1_i.max() ); - - BOOST_CHECK_EQUAL( 0u, x2.to_bit_indices().size() ); - - va_size_t const x3_i = x3.to_bit_indices(); - size_t const x3_i_check[] = { 0, 4 }; - size_t const x3_i_size = sizeof( x3_i_check ) / sizeof( x3_i_check[0] ); - BOOST_CHECK( equal_valarrays(va_size_t( x3_i_check, x3_i_size ), x3_i) ); - BOOST_CHECK_EQUAL( x3_b_size, 1u + x3_i.max() ); - - // Bit-vector construction and assignment - big_whole x4( x1_b ); - BOOST_CHECK_EQUAL( 5u, x4.to_uintmax() ); - - x4.reconfigure( x3_b ); - BOOST_CHECK_EQUAL( 17u, x4.to_uintmax() ); - - x4.reconfigure( va_bool_t() ); - BOOST_CHECK_EQUAL( 0u, x4.to_uintmax() ); - - // Bit-index construction and assignment - big_whole x5( x3_i ); - BOOST_CHECK_EQUAL( 17u, x5.to_uintmax() ); - - x5.reconfigure( x1_i ); - BOOST_CHECK_EQUAL( 5u, x5.to_uintmax() ); - - x5.reconfigure( va_size_t() ); - BOOST_CHECK_EQUAL( 0u, x5.to_uintmax() ); - - // Minimum-required bit length - BOOST_CHECK_EQUAL( x1_b_size, x1.length() ); - BOOST_CHECK_EQUAL( 0u, x2.length() ); - BOOST_CHECK_EQUAL( x3_b_size, x3.length() ); - - // Bit count - BOOST_CHECK_EQUAL( x1_i_size, x1.count() ); - BOOST_CHECK_EQUAL( 0u, x2.count() ); - BOOST_CHECK_EQUAL( x3_i_size, x3.count() ); - - BOOST_CHECK( x1.any() ); - BOOST_CHECK( !x2.any() ); - BOOST_CHECK( x3.any() ); - - BOOST_CHECK( !x1.none() ); - BOOST_CHECK( x2.none() ); - BOOST_CHECK( !x3.none() ); - - // Bit testing - BOOST_CHECK( x1.test(0) && !x1.test(1) && x1.test(2) && !x1.test(3) - && !x1.test(4) && !x1.test(5) && !x1.test(wlimits_type::digits) ); - BOOST_CHECK( !x2.test(0) && !x2.test(1) && !x2.test(2) && !x2.test(3) - && !x2.test(4) && !x2.test(5) && !x2.test(wlimits_type::digits) ); - BOOST_CHECK( x3.test(0) && !x3.test(1) && !x3.test(2) && !x3.test(3) - && x3.test(4) && !x3.test(5) && !x3.test(wlimits_type::digits) ); - - // Boolean test - BOOST_CHECK( x1 ); - BOOST_CHECK( !x2 ); - BOOST_CHECK( x3 ); -} - -// Unit test for "tests" -void -bigwhole_multi_bit_check_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - size_t const size_max = std::numeric_limits::max(); - - // Non-zero tests - big_whole const x1( 74u ); - size_t const l1 = x1.length(); - - BOOST_CHECK_EQUAL( 7u, l1 ); - - BOOST_CHECK_EQUAL( 74u, x1.tests(0, size_max).to_uintmax() ); - BOOST_CHECK_EQUAL( 74u, x1.tests(0, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 74u, x1.tests(0, l1 - 1).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x1.tests(0, 5).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x1.tests(0, 4).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x1.tests(0, 3).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x1.tests(0, 2).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x1.tests(0, 1).to_uintmax() ); - BOOST_CHECK_EQUAL( 0u, x1.tests(0, 0).to_uintmax() ); - - BOOST_CHECK_EQUAL( 37u, x1.tests(1, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 18u, x1.tests(2, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 9u, x1.tests(3, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 4u, x1.tests(4, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x1.tests(5, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 1u, x1.tests(l1 - 1, l1).to_uintmax() ); - BOOST_CHECK_EQUAL( 0u, x1.tests(l1, l1).to_uintmax() ); - - BOOST_CHECK( x1.tests(1, 1) ); - BOOST_CHECK( !x1.tests(2, 2) ); - BOOST_CHECK( x1.tests(3, 3) ); - BOOST_CHECK( !x1.tests(4, 4) ); - BOOST_CHECK( !x1.tests(5, 5) ); - BOOST_CHECK( x1.tests(l1 - 1, l1 - 1) ); - - // Zero tests - big_whole const x2; - - BOOST_CHECK( !x2.tests(0, 4) ); - BOOST_CHECK( !x2.tests(2, size_max) ); - BOOST_CHECK( !x2.tests(0, size_max) ); - BOOST_CHECK( !x2.tests(3, 3) ); -} - -// Unit test for reversing -void -bigwhole_reverse_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // Non-zero tests - big_whole const x1( 1 ); - - BOOST_CHECK_EQUAL( 1u, x1.reverse().to_uintmax() ); - BOOST_CHECK_EQUAL( 1u, x1.length() ); - - BOOST_CHECK_EQUAL( 1u, x1.reverse(0).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x1.reverse(1).to_uintmax() ); - BOOST_CHECK_EQUAL( 4u, x1.reverse(2).to_uintmax() ); - BOOST_CHECK_EQUAL( 128u, x1.reverse(7).to_uintmax() ); - - big_whole const x2( 5 ); - - BOOST_CHECK_EQUAL( 5u, x2.reverse().to_uintmax() ); - BOOST_CHECK_EQUAL( 3u, x2.length() ); - - BOOST_CHECK_EQUAL( 1u, x2.reverse(0).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x2.reverse(1).to_uintmax() ); - BOOST_CHECK_EQUAL( 5u, x2.reverse(2).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x2.reverse(3).to_uintmax() ); - BOOST_CHECK_EQUAL( 20u, x2.reverse(4).to_uintmax() ); - BOOST_CHECK_EQUAL( 160u, x2.reverse(7).to_uintmax() ); - - big_whole const x3( 74 ); - - BOOST_CHECK_EQUAL( 41u, x3.reverse().to_uintmax() ); - BOOST_CHECK_EQUAL( 7u, x3.length() ); - - BOOST_CHECK_EQUAL( 0u, x3.reverse(0).to_uintmax() ); - BOOST_CHECK_EQUAL( 1u, x3.reverse(1).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x3.reverse(2).to_uintmax() ); - BOOST_CHECK_EQUAL( 5u, x3.reverse(3).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x3.reverse(4).to_uintmax() ); - BOOST_CHECK_EQUAL( 20u, x3.reverse(5).to_uintmax() ); - BOOST_CHECK_EQUAL( 41u, x3.reverse(6).to_uintmax() ); - BOOST_CHECK_EQUAL( 82u, x3.reverse(7).to_uintmax() ); - BOOST_CHECK_EQUAL( 164u, x3.reverse(8).to_uintmax() ); - BOOST_CHECK_EQUAL( 656u, x3.reverse(10).to_uintmax() ); - - // Zero tests - big_whole const x4; - - BOOST_CHECK( !x4.length() ); - BOOST_CHECK( !x4.reverse() ); - BOOST_CHECK( !x4.reverse(0) ); - BOOST_CHECK( !x4.reverse(2 * wlimits_type::digits) ); - - // Multi-word tests - size_t const x5_i[] = { wlimits_type::digits - 1, wlimits_type::digits + 1 }; - size_t const x5_s = sizeof( x5_i ) / sizeof( x5_i[0] ); - va_size_t const x5_v( x5_i, x5_s ); - big_whole x5( x5_v ); - - BOOST_CHECK_EQUAL( 5u, x5.reverse().to_uintmax() ); - - BOOST_CHECK_EQUAL( 5u, x5.reverse(wlimits_type::digits + 1).to_uintmax() ); - BOOST_CHECK_EQUAL( 2u, x5.reverse(wlimits_type::digits).to_uintmax() ); - BOOST_CHECK_EQUAL( 1u, x5.reverse(wlimits_type::digits - 1).to_uintmax() ); - BOOST_CHECK_EQUAL( 0u, x5.reverse(wlimits_type::digits - 2).to_uintmax() ); - BOOST_CHECK_EQUAL( 0u, x5.reverse(wlimits_type::digits - 3).to_uintmax() ); - BOOST_CHECK_EQUAL( 10u, x5.reverse(wlimits_type::digits + 2).to_uintmax() ); - BOOST_CHECK_EQUAL( 20u, x5.reverse(wlimits_type::digits + 3).to_uintmax() ); -} - -// Unit test for resetting every bit -void -bigwhole_all_bit_reset_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a1; - - BOOST_CHECK( !a1 ); - - a1.reset(); - BOOST_CHECK( !a1 ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK( a2 ); - - a2.reset(); - BOOST_CHECK( !a2 ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK( a3 ); - - a3.reset(); - BOOST_CHECK( !a3 ); - - // two words - size_t const a4_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_s = sizeof( a4_i ) / sizeof( a4_i[0] ); - big_whole a4( va_size_t(a4_i, a4_s) ); - - BOOST_CHECK( a4 ); - - a4.reset(); - BOOST_CHECK( !a4 ); - - // more-than-two words - size_t const a5_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_s = sizeof( a5_i ) / sizeof( a5_i[0] ); - big_whole a5( va_size_t(a5_i, a5_s) ); - - BOOST_CHECK( a5 ); - - a5.reset(); - BOOST_CHECK( !a5 ); -} - -// Unit test for resetting single bits -void -bigwhole_single_bit_reset_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.reset( 3 ); - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.reset( 2 ); - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.reset( 3 ); - BOOST_CHECK_EQUAL( 0u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.reset( 0 ); - BOOST_CHECK_EQUAL( 24u, a3.to_uintmax() ); - - a3.reset( 1 ); - BOOST_CHECK_EQUAL( 24u, a3.to_uintmax() ); - - a3.reset( 4 ); - BOOST_CHECK_EQUAL( 8u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.reset( 5 ); - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.reset( 0 ); - BOOST_CHECK( equal_valarrays(va_size_t( a4_old_i + 1, a4_old_s - 1 ), a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - a5.reset( 4 ); - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - a5.reset( 2 * wlimits_type::digits + 5 ); - BOOST_CHECK( equal_valarrays(va_size_t( a5_old_i, a5_old_s - 1 ), a5.to_bit_indices()) ); -} - -// Unit test for resetting a group of bits -void -bigwhole_group_bit_reset_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.reset( 3, 7 ); - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.reset( 6, 9 ); - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.reset( 2, 3 ); - BOOST_CHECK_EQUAL( 0u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.reset( 3, 3 ); - BOOST_CHECK_EQUAL( 17u, a3.to_uintmax() ); - - a3.reset( 1, 2 ); - BOOST_CHECK_EQUAL( 17u, a3.to_uintmax() ); - - a3.reset( 2, 6 ); - BOOST_CHECK_EQUAL( 1u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.reset( 5, 12 ); - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.reset( 9, 2 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(va_size_t( a4_old_i, a4_old_s - 1 ), a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - a5.reset( 3, 12 ); - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - size_t const a5_new_i[] = { 1, 2 * wlimits_type::digits + 5 }; - size_t const a5_new_s = sizeof( a5_new_i ) / sizeof( a5_new_i[0] ); - - a5.reset( wlimits_type::digits - 1, 2 * wlimits_type::digits + 1 ); - BOOST_CHECK( equal_valarrays(va_size_t( a5_new_i, a5_new_s ), a5.to_bit_indices()) ); - - a5.reset( 2 * wlimits_type::digits + 1, 3 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(va_size_t( 1u, 1 ), a5.to_bit_indices()) ); - - a5.reset( wlimits_type::digits + 1, 3 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(va_size_t( 1u, 1 ), a5.to_bit_indices()) ); -} - -// Unit test for setting single bits -void -bigwhole_single_bit_set_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.set( 3 ); - BOOST_CHECK_EQUAL( 8u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.set( 2 ); - BOOST_CHECK_EQUAL( 12u, a2.to_uintmax() ); - - a2.set( 3 ); - BOOST_CHECK_EQUAL( 12u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.set( 0 ); - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.set( 1 ); - BOOST_CHECK_EQUAL( 27u, a3.to_uintmax() ); - - a3.set( 4 ); - BOOST_CHECK_EQUAL( 27u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - size_t const a4_new_i[] = { 0, 5, wlimits_type::digits + 1 }; - size_t const a4_new_s = sizeof( a4_new_i ) / sizeof( a4_new_i[0] ); - va_size_t const a4_new( a4_new_i, a4_new_s ); - - a4.set( 5 ); - BOOST_CHECK( equal_valarrays(a4_new, a4.to_bit_indices()) ); - - a4.set( 0 ); - BOOST_CHECK( equal_valarrays(a4_new, a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - size_t const a5_new_i[] = { 1, 4, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_new_s = sizeof( a5_new_i ) / sizeof( a5_new_i[0] ); - va_size_t const a5_new( a5_new_i, a5_new_s ); - - a5.set( 4 ); - BOOST_CHECK( equal_valarrays(a5_new, a5.to_bit_indices()) ); - - a5.set( 2 * wlimits_type::digits + 5 ); - BOOST_CHECK( equal_valarrays(a5_new, a5.to_bit_indices()) ); -} - -// Unit test for setting a group of bits -void -bigwhole_group_bit_set_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - typedef std::set st_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.set( 3, 7 ); - BOOST_CHECK_EQUAL( 248u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.set( 6, 9 ); - BOOST_CHECK_EQUAL( 968u, a2.to_uintmax() ); - - a2.set( 2, 3 ); - BOOST_CHECK_EQUAL( 972u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.set( 3, 3 ); - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.set( 1, 2 ); - BOOST_CHECK_EQUAL( 31u, a3.to_uintmax() ); - - a3.set( 2, 6 ); - BOOST_CHECK_EQUAL( 127u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - st_size_t a4_new( a4_old_i, a4_old_i + a4_old_s ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.set( 5, 12 ); - insert_value_range( a4_new, 5, 12 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a4_new ), a4.to_bit_indices()) ); - - a4.set( 9, 2 * wlimits_type::digits ); - insert_value_range( a4_new, 9, 2 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a4_new ), a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - st_size_t a5_new( a5_old_i, a5_old_i + a5_old_s ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - a5.set( 3, 12 ); - insert_value_range( a5_new, 3, 12 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.set( wlimits_type::digits - 1, 2 * wlimits_type::digits + 1 ); - insert_value_range( a5_new, wlimits_type::digits - 1, 2 * wlimits_type::digits + 1 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.set( 2 * wlimits_type::digits + 1, 3 * wlimits_type::digits ); - insert_value_range( a5_new, 2 * wlimits_type::digits + 1, 3 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.set( wlimits_type::digits + 1, 3 * wlimits_type::digits ); - insert_value_range( a5_new, wlimits_type::digits + 1, 3 * wlimits_type::digits ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); -} - -// Unit test for flipping single bits -void -bigwhole_single_bit_flip_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.flip( 3 ); - BOOST_CHECK_EQUAL( 8u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.flip( 2 ); - BOOST_CHECK_EQUAL( 12u, a2.to_uintmax() ); - - a2.flip( 3 ); - BOOST_CHECK_EQUAL( 4u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.flip( 0 ); - BOOST_CHECK_EQUAL( 24u, a3.to_uintmax() ); - - a3.flip( 1 ); - BOOST_CHECK_EQUAL( 26u, a3.to_uintmax() ); - - a3.flip( 4 ); - BOOST_CHECK_EQUAL( 10u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - size_t const a4_new_i[] = { 0, 5, wlimits_type::digits + 1 }; - size_t const a4_new_s = sizeof( a4_new_i ) / sizeof( a4_new_i[0] ); - va_size_t const a4_new1( a4_new_i, a4_new_s ); - va_size_t const a4_new2( a4_new_i + 1, a4_new_s - 1 ); - - a4.flip( 5 ); - BOOST_CHECK( equal_valarrays(a4_new1, a4.to_bit_indices()) ); - - a4.flip( 0 ); - BOOST_CHECK( equal_valarrays(a4_new2, a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - size_t const a5_new_i[] = { 1, 4, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_new_s = sizeof( a5_new_i ) / sizeof( a5_new_i[0] ); - va_size_t const a5_new1( a5_new_i, a5_new_s ); - va_size_t const a5_new2( a5_new_i, a5_new_s - 1 ); - - a5.flip( 4 ); - BOOST_CHECK( equal_valarrays(a5_new1, a5.to_bit_indices()) ); - - a5.flip( 2 * wlimits_type::digits + 5 ); - BOOST_CHECK( equal_valarrays(a5_new2, a5.to_bit_indices()) ); -} - -// Unit test for flipping a group of bits -void -bigwhole_group_bit_flip_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - typedef std::set st_size_t; - - // zero - big_whole a1; - - BOOST_CHECK_EQUAL( 0u, a1.to_uintmax() ); - - a1.flip( 3, 7 ); - BOOST_CHECK_EQUAL( 248u, a1.to_uintmax() ); - - a1.flip( 2, 4 ); - BOOST_CHECK_EQUAL( 228u, a1.to_uintmax() ); - - // one bit set - big_whole a2( 8 ); - - BOOST_CHECK_EQUAL( 8u, a2.to_uintmax() ); - - a2.flip( 6, 9 ); - BOOST_CHECK_EQUAL( 968u, a2.to_uintmax() ); - - a2.flip( 2, 3 ); - BOOST_CHECK_EQUAL( 964u, a2.to_uintmax() ); - - // multiple bits set - big_whole a3( 25 ); - - BOOST_CHECK_EQUAL( 25u, a3.to_uintmax() ); - - a3.flip( 3, 3 ); - BOOST_CHECK_EQUAL( 17u, a3.to_uintmax() ); - - a3.flip( 1, 2 ); - BOOST_CHECK_EQUAL( 23u, a3.to_uintmax() ); - - a3.flip( 2, 6 ); - BOOST_CHECK_EQUAL( 107u, a3.to_uintmax() ); - - // two words - size_t const a4_old_i[] = { 0, wlimits_type::digits + 1 }; - size_t const a4_old_s = sizeof( a4_old_i ) / sizeof( a4_old_i[0] ); - va_size_t const a4_old( a4_old_i, a4_old_s ); - big_whole a4( a4_old ); - st_size_t a4_new( a4_old_i, a4_old_i + a4_old_s ); - - BOOST_CHECK( equal_valarrays(a4_old, a4.to_bit_indices()) ); - - a4.flip( 5, 12 ); - insert_value_range( a4_new, 5, 12 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a4_new ), a4.to_bit_indices()) ); - - a4.flip( 9, 2 * wlimits_type::digits ); - insert_value_range( a4_new, 9, 2 * wlimits_type::digits ); - erase_value_range( a4_new, 9, 12 ); - a4_new.erase( wlimits_type::digits + 1 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a4_new ), a4.to_bit_indices()) ); - - // more-than-two words - size_t const a5_old_i[] = { 1, wlimits_type::digits + 3, wlimits_type::digits + 4, 2 * wlimits_type::digits + 5 }; - size_t const a5_old_s = sizeof( a5_old_i ) / sizeof( a5_old_i[0] ); - va_size_t const a5_old( a5_old_i, a5_old_s ); - big_whole a5( a5_old ); - st_size_t a5_new( a5_old_i, a5_old_i + a5_old_s ); - - BOOST_CHECK( equal_valarrays(a5_old, a5.to_bit_indices()) ); - - a5.flip( 3, 12 ); - insert_value_range( a5_new, 3, 12 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.flip( wlimits_type::digits - 1, 2 * wlimits_type::digits + 1 ); - insert_value_range( a5_new, wlimits_type::digits - 1, 2 * wlimits_type::digits + 1 ); - a5_new.erase( wlimits_type::digits + 3 ); - a5_new.erase( wlimits_type::digits + 4 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.flip( 2 * wlimits_type::digits + 1, 3 * wlimits_type::digits ); - insert_value_range( a5_new, 2 * wlimits_type::digits + 1, 3 * wlimits_type::digits ); - a5_new.erase( 2 * wlimits_type::digits + 1 ); - a5_new.erase( 2 * wlimits_type::digits + 5 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); - - a5.flip( wlimits_type::digits + 1, 3 * wlimits_type::digits ); - erase_value_range( a5_new, wlimits_type::digits + 1, 3 * wlimits_type::digits ); - a5_new.insert( wlimits_type::digits + 3 ); - a5_new.insert( wlimits_type::digits + 4 ); - a5_new.insert( 2 * wlimits_type::digits + 1 ); - a5_new.insert( 2 * wlimits_type::digits + 5 ); - BOOST_CHECK( equal_valarrays(set_to_valarray( a5_new ), a5.to_bit_indices()) ); -} - -// Unit test for assigning a group of bits to an arbitrary value -void -bigwhole_group_bit_assign_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - typedef std::set st_size_t; - - // no segment is zero - big_whole x( 255 ); - - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 3, 5, big_whole(2) ); - BOOST_CHECK_EQUAL( 215u, x.to_uintmax() ); - - // old middle part is zero - x.assign( 199 ); - BOOST_CHECK_EQUAL( 199u, x.to_uintmax() ); - - x.bits_assign( 3, 5, big_whole(7) ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - // new-value bits above the range are ignored - x.assign( 199 ); - BOOST_CHECK_EQUAL( 199u, x.to_uintmax() ); - - x.bits_assign( 3, 4, big_whole(14) ); - BOOST_CHECK_EQUAL( 215u, x.to_uintmax() ); - - // new middle part is zero - x.assign( 255 ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 3, 5, big_whole() ); - BOOST_CHECK_EQUAL( 199u, x.to_uintmax() ); - - // change the lowest bits - x.assign( 255 ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 0, 2, big_whole(33) ); - BOOST_CHECK_EQUAL( 249u, x.to_uintmax() ); - - // change the lowest bits to zero - x.assign( 255 ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 0, 3, big_whole() ); - BOOST_CHECK_EQUAL( 240u, x.to_uintmax() ); - - // change the highest bits - x.assign( 255 ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 5, 7, big_whole(5) ); - BOOST_CHECK_EQUAL( 191u, x.to_uintmax() ); - - // change the highest bits to zero - x.assign( 255 ); - BOOST_CHECK_EQUAL( 255u, x.to_uintmax() ); - - x.bits_assign( 4, 7, big_whole() ); - BOOST_CHECK_EQUAL( 15u, x.to_uintmax() ); - - // turn zero into a nonzero - x.reset(); - BOOST_CHECK_EQUAL( 0u, x.to_uintmax() ); - - x.bits_assign( 8, 9, big_whole(35) ); - BOOST_CHECK_EQUAL( 768u, x.to_uintmax() ); - - // keep zero a zero - x.reset(); - BOOST_CHECK_EQUAL( 0u, x.to_uintmax() ); - - x.bits_assign( 11, 45, big_whole() ); - BOOST_CHECK_EQUAL( 0u, x.to_uintmax() ); - - // multiple words - size_t const t11_old_i[] = { 0, 2 * wlimits_type::digits + 1 }; - size_t const t11_old_s = sizeof( t11_old_i ) / sizeof( t11_old_i[0] ); - va_size_t const t11_old( t11_old_i, t11_old_s ); - st_size_t t11_new( t11_old_i, t11_old_i + t11_old_s ); - - x.reconfigure( t11_old ); - BOOST_CHECK( equal_valarrays(t11_old, x.to_bit_indices()) ); - BOOST_CHECK( equal_valarrays(set_to_valarray( t11_new ), t11_old) ); - - x.bits_assign( wlimits_type::digits - 1, wlimits_type::digits + 2, big_whole(7) ); - insert_value_range( t11_new, wlimits_type::digits - 1, wlimits_type::digits + 1 ); - BOOST_CHECK( equal_valarrays(x.to_bit_indices(), set_to_valarray( t11_new )) ); -} - -// Unit test for checking if an object is even/odd -void -bigwhole_is_even_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - // zero - big_whole a; - - BOOST_CHECK_EQUAL( 0u, a.to_uintmax() ); - BOOST_CHECK( a.is_even() ); - - // non-zero odd - a.assign( 5 ); - BOOST_CHECK_EQUAL( 5u, a.to_uintmax() ); - BOOST_CHECK( !a.is_even() ); - - // non-zero even - a.assign( 18 ); - BOOST_CHECK_EQUAL( 18u, a.to_uintmax() ); - BOOST_CHECK( a.is_even() ); - - // multi-word odd - size_t const t3_i[] = { 0, wlimits_type::digits + 1 }; - size_t const t3_s = sizeof( t3_i ) / sizeof( t3_i[0] ); - va_size_t const t3( t3_i, t3_s ); - - a.reconfigure( t3 ); - BOOST_CHECK( equal_valarrays(t3, a.to_bit_indices()) ); - BOOST_CHECK( !a.is_even() ); - - // multi-word even - size_t const t4_i = 2 * wlimits_type::digits + 3; - va_size_t const t4( t4_i, 1 ); - - a.reconfigure( t4 ); - BOOST_CHECK( equal_valarrays(t4, a.to_bit_indices()) ); - BOOST_CHECK( a.is_even() ); -} - -// Unit test for comparisons -void -bigwhole_compare_unit_test -( -) -{ - using boost::math::big_whole; - - typedef std::valarray va_size_t; - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 100 ); - big_whole const c( va_size_t(2 * wlimits_type::digits + 3, 1) ); - - // "compare" directly - BOOST_CHECK( z.compare(z) == 0 ); - BOOST_CHECK( z.compare(a) < 0 ); - BOOST_CHECK( z.compare(b) < 0 ); - BOOST_CHECK( z.compare(c) < 0 ); - - BOOST_CHECK( a.compare(z) > 0 ); - BOOST_CHECK( a.compare(a) == 0 ); - BOOST_CHECK( a.compare(b) < 0 ); - BOOST_CHECK( a.compare(c) < 0 ); - - BOOST_CHECK( b.compare(z) > 0 ); - BOOST_CHECK( b.compare(a) > 0 ); - BOOST_CHECK( b.compare(b) == 0 ); - BOOST_CHECK( b.compare(c) < 0 ); - - BOOST_CHECK( c.compare(z) > 0 ); - BOOST_CHECK( c.compare(a) > 0 ); - BOOST_CHECK( c.compare(b) > 0 ); - BOOST_CHECK( c.compare(c) == 0 ); - - // use == - BOOST_CHECK( z == z ); BOOST_CHECK( !(z == a) ); BOOST_CHECK( !(z == b) ); BOOST_CHECK( !(z == c) ); - BOOST_CHECK( !(a == z) ); BOOST_CHECK( a == a ); BOOST_CHECK( !(a == b) ); BOOST_CHECK( !(a == c) ); - BOOST_CHECK( !(b == z) ); BOOST_CHECK( !(b == a) ); BOOST_CHECK( b == b ); BOOST_CHECK( !(b == c) ); - BOOST_CHECK( !(c == z) ); BOOST_CHECK( !(c == a) ); BOOST_CHECK( !(c == b) ); BOOST_CHECK( c == c ); - - // use != - BOOST_CHECK( !(z != z) ); BOOST_CHECK( z != a ); BOOST_CHECK( z != b ); BOOST_CHECK( z != c ); - BOOST_CHECK( a != z ); BOOST_CHECK( !(a != a) ); BOOST_CHECK( a != b ); BOOST_CHECK( a != c ); - BOOST_CHECK( b != z ); BOOST_CHECK( b != a ); BOOST_CHECK( !(b != b) ); BOOST_CHECK( b != c ); - BOOST_CHECK( c != z ); BOOST_CHECK( c != a ); BOOST_CHECK( c != b ); BOOST_CHECK( !(c != c) ); - - // use < - BOOST_CHECK( !(z < z) ); BOOST_CHECK( z < a ); BOOST_CHECK( z < b ); BOOST_CHECK( z < c ); - BOOST_CHECK( !(a < z) ); BOOST_CHECK( !(a < a) ); BOOST_CHECK( a < b ); BOOST_CHECK( a < c ); - BOOST_CHECK( !(b < z) ); BOOST_CHECK( !(b < a) ); BOOST_CHECK( !(b < b) ); BOOST_CHECK( b < c ); - BOOST_CHECK( !(c < z) ); BOOST_CHECK( !(c < a) ); BOOST_CHECK( !(c < b) ); BOOST_CHECK( !(c < c) ); - - // use > - BOOST_CHECK( !(z > z) ); BOOST_CHECK( !(z > a) ); BOOST_CHECK( !(z > b) ); BOOST_CHECK( !(z > c) ); - BOOST_CHECK( a > z ); BOOST_CHECK( !(a > a) ); BOOST_CHECK( !(a > b) ); BOOST_CHECK( !(a > c) ); - BOOST_CHECK( b > z ); BOOST_CHECK( b > a ); BOOST_CHECK( !(b > b) ); BOOST_CHECK( !(b > c) ); - BOOST_CHECK( c > z ); BOOST_CHECK( c > a ); BOOST_CHECK( c > b ); BOOST_CHECK( !(c > c) ); - - // use <= - BOOST_CHECK( z <= z ); BOOST_CHECK( z <= a ); BOOST_CHECK( z <= b ); BOOST_CHECK( z <= c ); - BOOST_CHECK( !(a <= z) ); BOOST_CHECK( a <= a ); BOOST_CHECK( a <= b ); BOOST_CHECK( a <= c ); - BOOST_CHECK( !(b <= z) ); BOOST_CHECK( !(b <= a) ); BOOST_CHECK( b <= b ); BOOST_CHECK( b <= c ); - BOOST_CHECK( !(c <= z) ); BOOST_CHECK( !(c <= a) ); BOOST_CHECK( !(c <= b) ); BOOST_CHECK( c <= c ); - - // use >= - BOOST_CHECK( z >= z ); BOOST_CHECK( !(z >= a) ); BOOST_CHECK( !(z >= b) ); BOOST_CHECK( !(z >= c) ); - BOOST_CHECK( a >= z ); BOOST_CHECK( a >= a ); BOOST_CHECK( !(a >= b) ); BOOST_CHECK( !(a >= c) ); - BOOST_CHECK( b >= z ); BOOST_CHECK( b >= a ); BOOST_CHECK( b >= b ); BOOST_CHECK( !(b >= c) ); - BOOST_CHECK( c >= z ); BOOST_CHECK( c >= a ); BOOST_CHECK( c >= b ); BOOST_CHECK( c >= c ); -} - -// Unit test for bitwise-and -void -bigwhole_bitwise_and_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - size_t const di[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ds = sizeof( di ) / sizeof( di[0] ); - va_size_t const dv( di, ds ); - size_t const ei[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 12 ); - big_whole const c( 100 ); - big_whole const d( dv ); - big_whole const e( ev ); - - // self-and - BOOST_CHECK_EQUAL( z, z & z ); - BOOST_CHECK_EQUAL( a, a & a ); - BOOST_CHECK_EQUAL( b, b & b ); - BOOST_CHECK_EQUAL( c, c & c ); - BOOST_CHECK_EQUAL( d, d & d ); - BOOST_CHECK_EQUAL( e, e & e ); - - // zero-and - BOOST_CHECK_EQUAL( z, z & a ); BOOST_CHECK_EQUAL( z, a & z ); - BOOST_CHECK_EQUAL( z, z & b ); BOOST_CHECK_EQUAL( z, b & z ); - BOOST_CHECK_EQUAL( z, z & c ); BOOST_CHECK_EQUAL( z, c & z ); - BOOST_CHECK_EQUAL( z, z & d ); BOOST_CHECK_EQUAL( z, d & z ); - BOOST_CHECK_EQUAL( z, z & e ); BOOST_CHECK_EQUAL( z, e & z ); - - // various combinations - word_type const ab = 4, ac = 4, ad = 0, ae = 2; - word_type const ba = ab, bc = 4, bd = 0, be = 0; - word_type const ca = ac, cb = bc, cd = 0, ce = 0; - word_type const da = ad, db = bd, dc = cd; - word_type const ea = ae, eb = be, ec = ce; - big_whole const de( va_size_t(wlimits_type::digits + 1, 1) ), ed( de ); - - BOOST_CHECK_EQUAL( ab, a & b ); BOOST_CHECK_EQUAL( ba, b & a ); - BOOST_CHECK_EQUAL( ac, a & c ); BOOST_CHECK_EQUAL( ca, c & a ); - BOOST_CHECK_EQUAL( ad, a & d ); BOOST_CHECK_EQUAL( da, d & a ); - BOOST_CHECK_EQUAL( ae, a & e ); BOOST_CHECK_EQUAL( ea, e & a ); - - BOOST_CHECK_EQUAL( bc, b & c ); BOOST_CHECK_EQUAL( cb, c & b ); - BOOST_CHECK_EQUAL( bd, b & d ); BOOST_CHECK_EQUAL( db, d & b ); - BOOST_CHECK_EQUAL( be, b & e ); BOOST_CHECK_EQUAL( eb, e & b ); - - BOOST_CHECK_EQUAL( cd, c & d ); BOOST_CHECK_EQUAL( dc, d & c ); - BOOST_CHECK_EQUAL( ce, c & e ); BOOST_CHECK_EQUAL( ec, e & c ); - - BOOST_CHECK_EQUAL( de, d & e ); BOOST_CHECK_EQUAL( ed, e & d ); -} - -// Unit test for bitwise-or -void -bigwhole_bitwise_or_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - size_t const di[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ds = sizeof( di ) / sizeof( di[0] ); - va_size_t const dv( di, ds ); - size_t const ei[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 12 ); - big_whole const c( 100 ); - big_whole const d( dv ); - big_whole const e( ev ); - - // self-or - BOOST_CHECK_EQUAL( z, z | z ); - BOOST_CHECK_EQUAL( a, a | a ); - BOOST_CHECK_EQUAL( b, b | b ); - BOOST_CHECK_EQUAL( c, c | c ); - BOOST_CHECK_EQUAL( d, d | d ); - BOOST_CHECK_EQUAL( e, e | e ); - - // zero-or - BOOST_CHECK_EQUAL( a, z | a ); BOOST_CHECK_EQUAL( a, a | z ); - BOOST_CHECK_EQUAL( b, z | b ); BOOST_CHECK_EQUAL( b, b | z ); - BOOST_CHECK_EQUAL( c, z | c ); BOOST_CHECK_EQUAL( c, c | z ); - BOOST_CHECK_EQUAL( d, z | d ); BOOST_CHECK_EQUAL( d, d | z ); - BOOST_CHECK_EQUAL( e, z | e ); BOOST_CHECK_EQUAL( e, e | z ); - - // various combinations - size_t const ed_i[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ed_s = sizeof( ed_i ) / sizeof( ed_i[0] ); - size_t const ec_i[] = { 1, 2, 5, 6, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const ec_s = sizeof( ec_i ) / sizeof( ec_i[0] ); - size_t const eb_i[] = { 1, 2, 3, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const eb_s = sizeof( eb_i ) / sizeof( eb_i[0] ); - size_t const ea_i[] = { 0, 1, 2, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const ea_s = sizeof( ea_i ) / sizeof( ea_i[0] ); - size_t const dc_i[] = { 2, 5, 6, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const dc_s = sizeof( dc_i ) / sizeof( dc_i[0] ); - size_t const db_i[] = { 2, 3, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const db_s = sizeof( db_i ) / sizeof( db_i[0] ); - size_t const da_i[] = { 0, 1, 2, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const da_s = sizeof( da_i ) / sizeof( da_i[0] ); - - va_size_t const ed_v( ed_i, ed_s ), ec_v( ec_i, ec_s ), eb_v( eb_i, eb_s ); - va_size_t const ea_v( ea_i, ea_s ), dc_v( dc_i, dc_s ), db_v( db_i, db_s ); - va_size_t const da_v( da_i, da_s ); - - big_whole const ed( ed_v ), ec( ec_v ), eb( eb_v ), ea( ea_v ); - big_whole const de( ed ), dc( dc_v ), db( db_v ), da( da_v ); - big_whole const ce( ec ), cd( dc ), be( eb ), bd( db ), ae( ea ), ad( da ); - word_type const cb = 108, ca = 103, bc = cb, ba = 15, ac = ca, ab = ba; - - BOOST_CHECK_EQUAL( ab, a | b ); BOOST_CHECK_EQUAL( ba, b | a ); - BOOST_CHECK_EQUAL( ac, a | c ); BOOST_CHECK_EQUAL( ca, c | a ); - BOOST_CHECK_EQUAL( ad, a | d ); BOOST_CHECK_EQUAL( da, d | a ); - BOOST_CHECK_EQUAL( ae, a | e ); BOOST_CHECK_EQUAL( ea, e | a ); - - BOOST_CHECK_EQUAL( bc, b | c ); BOOST_CHECK_EQUAL( cb, c | b ); - BOOST_CHECK_EQUAL( bd, b | d ); BOOST_CHECK_EQUAL( db, d | b ); - BOOST_CHECK_EQUAL( be, b | e ); BOOST_CHECK_EQUAL( eb, e | b ); - - BOOST_CHECK_EQUAL( cd, c | d ); BOOST_CHECK_EQUAL( dc, d | c ); - BOOST_CHECK_EQUAL( ce, c | e ); BOOST_CHECK_EQUAL( ec, e | c ); - - BOOST_CHECK_EQUAL( de, d | e ); BOOST_CHECK_EQUAL( ed, e | d ); -} - -// Unit test for bitwise-xor -void -bigwhole_bitwise_xor_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - size_t const di[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ds = sizeof( di ) / sizeof( di[0] ); - va_size_t const dv( di, ds ); - size_t const ei[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 12 ); - big_whole const c( 100 ); - big_whole const d( dv ); - big_whole const e( ev ); - - // self-or - BOOST_CHECK_EQUAL( z, z ^ z ); - BOOST_CHECK_EQUAL( z, a ^ a ); - BOOST_CHECK_EQUAL( z, b ^ b ); - BOOST_CHECK_EQUAL( z, c ^ c ); - BOOST_CHECK_EQUAL( z, d ^ d ); - BOOST_CHECK_EQUAL( z, e ^ e ); - - // zero-or - BOOST_CHECK_EQUAL( a, z ^ a ); BOOST_CHECK_EQUAL( a, a ^ z ); - BOOST_CHECK_EQUAL( b, z ^ b ); BOOST_CHECK_EQUAL( b, b ^ z ); - BOOST_CHECK_EQUAL( c, z ^ c ); BOOST_CHECK_EQUAL( c, c ^ z ); - BOOST_CHECK_EQUAL( d, z ^ d ); BOOST_CHECK_EQUAL( d, d ^ z ); - BOOST_CHECK_EQUAL( e, z ^ e ); BOOST_CHECK_EQUAL( e, e ^ z ); - - // various combinations - size_t const ed_i[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, 2 * wlimits_type::digits + 3 }; - size_t const ed_s = sizeof( ed_i ) / sizeof( ed_i[0] ); - size_t const ec_i[] = { 1, 2, 5, 6, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const ec_s = sizeof( ec_i ) / sizeof( ec_i[0] ); - size_t const eb_i[] = { 1, 2, 3, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const eb_s = sizeof( eb_i ) / sizeof( eb_i[0] ); - size_t const ea_i[] = { 0, 2, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const ea_s = sizeof( ea_i ) / sizeof( ea_i[0] ); - size_t const dc_i[] = { 2, 5, 6, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const dc_s = sizeof( dc_i ) / sizeof( dc_i[0] ); - size_t const db_i[] = { 2, 3, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const db_s = sizeof( db_i ) / sizeof( db_i[0] ); - size_t const da_i[] = { 0, 1, 2, wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const da_s = sizeof( da_i ) / sizeof( da_i[0] ); - - va_size_t const ed_v( ed_i, ed_s ), ec_v( ec_i, ec_s ), eb_v( eb_i, eb_s ); - va_size_t const ea_v( ea_i, ea_s ), dc_v( dc_i, dc_s ), db_v( db_i, db_s ); - va_size_t const da_v( da_i, da_s ); - - big_whole const ed( ed_v ), ec( ec_v ), eb( eb_v ), ea( ea_v ); - big_whole const de( ed ), dc( dc_v ), db( db_v ), da( da_v ); - big_whole const ce( ec ), cd( dc ), be( eb ), bd( db ), ae( ea ), ad( da ); - word_type const cb = 104, ca = 99, bc = cb, ba = 11, ac = ca, ab = ba; - - BOOST_CHECK_EQUAL( ab, a ^ b ); BOOST_CHECK_EQUAL( ba, b ^ a ); - BOOST_CHECK_EQUAL( ac, a ^ c ); BOOST_CHECK_EQUAL( ca, c ^ a ); - BOOST_CHECK_EQUAL( ad, a ^ d ); BOOST_CHECK_EQUAL( da, d ^ a ); - BOOST_CHECK_EQUAL( ae, a ^ e ); BOOST_CHECK_EQUAL( ea, e ^ a ); - - BOOST_CHECK_EQUAL( bc, b ^ c ); BOOST_CHECK_EQUAL( cb, c ^ b ); - BOOST_CHECK_EQUAL( bd, b ^ d ); BOOST_CHECK_EQUAL( db, d ^ b ); - BOOST_CHECK_EQUAL( be, b ^ e ); BOOST_CHECK_EQUAL( eb, e ^ b ); - - BOOST_CHECK_EQUAL( cd, c ^ d ); BOOST_CHECK_EQUAL( dc, d ^ c ); - BOOST_CHECK_EQUAL( ce, c ^ e ); BOOST_CHECK_EQUAL( ec, e ^ c ); - - BOOST_CHECK_EQUAL( de, d ^ e ); BOOST_CHECK_EQUAL( ed, e ^ d ); -} - -// Unit test for left-shift (value-decreasing) -void -bigwhole_left_shift_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - using std::length_error; - - typedef std::valarray va_size_t; - - typedef std::numeric_limits size_limits; - - size_t const ei[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - size_t const e1i[] = { 1, wlimits_type::digits + 3 }; - size_t const e1s = sizeof( e1i ) / sizeof( e1i[0] ); - va_size_t const e1v( e1i, e1s ); - size_t const fi[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 2 }; - size_t const fs = sizeof( fi ) / sizeof( fi[0] ); - va_size_t const fv( fi, fs ); - - big_whole const z; - big_whole const a( 1 ); - big_whole const b( 5 ); - big_whole const c( 10 ); - big_whole const d( 100 ); - big_whole const e( ev ), e1( e1v ); - big_whole const f( fv ); - - // shifting zero - BOOST_CHECK( !(z >> z) ); - BOOST_CHECK( !(z >> a) ); - BOOST_CHECK( !(z >> d) ); - - // zero-shift - BOOST_CHECK_EQUAL( z, z >> z ); - BOOST_CHECK_EQUAL( a, a >> z ); - BOOST_CHECK_EQUAL( b, b >> z ); - BOOST_CHECK_EQUAL( c, c >> z ); - BOOST_CHECK_EQUAL( d, d >> z ); - BOOST_CHECK_EQUAL( e, e >> z ); - BOOST_CHECK_EQUAL( f, f >> z ); - - // single-word shifts - BOOST_CHECK_EQUAL( z, a >> a ); - - BOOST_CHECK_EQUAL( z, b >> 3 ); - BOOST_CHECK_EQUAL( a, b >> 2 ); - BOOST_CHECK_EQUAL( 2, b >> 1 ); - - BOOST_CHECK_EQUAL( b, c >> 1 ); - BOOST_CHECK_EQUAL( 2, c >> 2 ); - BOOST_CHECK_EQUAL( a, c >> 3 ); - BOOST_CHECK_EQUAL( z, c >> 4 ); - - BOOST_CHECK_EQUAL( 50, d >> 1 ); - BOOST_CHECK_EQUAL( 25, d >> 2 ); - BOOST_CHECK_EQUAL( 12, d >> 3 ); - BOOST_CHECK_EQUAL( 6, d >> 4 ); - BOOST_CHECK_EQUAL( 3, d >> 5 ); - BOOST_CHECK_EQUAL( a, d >> 6 ); - BOOST_CHECK_EQUAL( z, d >> 7 ); - - // over shifts - BOOST_CHECK_EQUAL( z, a >> 4 ); - BOOST_CHECK_EQUAL( z, b >> 10 ); - BOOST_CHECK_EQUAL( z, c >> 25 ); - BOOST_CHECK_EQUAL( z, d >> 1000 ); - BOOST_CHECK_EQUAL( z, e >> (3 * wlimits_type::digits) ); - BOOST_CHECK_EQUAL( z, f >> (2 * wlimits_type::digits) ); - - // multi-word shifts - BOOST_CHECK_EQUAL( e1, e >> wlimits_type::digits ); - BOOST_CHECK_EQUAL( 8, e >> (2 * wlimits_type::digits) ); - BOOST_CHECK_EQUAL( 4, e1 >> (wlimits_type::digits + 1) ); - BOOST_CHECK_EQUAL( b, f >> wlimits_type::digits ); - BOOST_CHECK_EQUAL( 11, f >> (wlimits_type::digits - 1) ); - - // exceptions (hope we don't get any out-of-memory exceptions) - va_size_t const si( size_limits::digits, 1 ); - big_whole const s( si ); - big_whole const t( size_limits::max() ); // == s - 1 - big_whole const u = s | 1; // == s + 1 - - BOOST_CHECK_NO_THROW( z >> s ); - BOOST_CHECK_THROW( a >> s, length_error ); - BOOST_CHECK_THROW( f >> s, length_error ); - BOOST_CHECK_NO_THROW( z >> t ); - BOOST_CHECK_NO_THROW( a >> t ); - BOOST_CHECK_NO_THROW( f >> t ); - BOOST_CHECK_NO_THROW( z >> u ); - BOOST_CHECK_THROW( a >> u, length_error ); - BOOST_CHECK_THROW( f >> u, length_error ); -} - -// Unit test for right-shift (value-increasing) -void -bigwhole_right_shift_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - using std::length_error; - using std::overflow_error; - - typedef std::valarray va_size_t; - - typedef std::numeric_limits size_limits; - - size_t const ei[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - size_t const e1i[] = { 1, wlimits_type::digits + 3 }; - size_t const e1s = sizeof( e1i ) / sizeof( e1i[0] ); - va_size_t const e1v( e1i, e1s ); - size_t const fi[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 2 }; - size_t const fs = sizeof( fi ) / sizeof( fi[0] ); - va_size_t const fv( fi, fs ); - - big_whole const z; - big_whole const a( 1 ); - big_whole const b( 5 ); - big_whole const c( 10 ); - big_whole const d( 100 ); - big_whole const e( ev ), e1( e1v ); - big_whole const f( fv ); - - // shifting zero - BOOST_CHECK( !(z << z) ); - BOOST_CHECK( !(z << a) ); - BOOST_CHECK( !(z << d) ); - - // zero-shift - BOOST_CHECK_EQUAL( z, z << z ); - BOOST_CHECK_EQUAL( a, a << z ); - BOOST_CHECK_EQUAL( b, b << z ); - BOOST_CHECK_EQUAL( c, c << z ); - BOOST_CHECK_EQUAL( d, d << z ); - BOOST_CHECK_EQUAL( e, e << z ); - BOOST_CHECK_EQUAL( f, f << z ); - - // single-word shifts - BOOST_CHECK_EQUAL( 2, a << a ); - BOOST_CHECK_EQUAL( 32, a << b ); - - BOOST_CHECK_EQUAL( c, b << a ); - BOOST_CHECK_EQUAL( 20, b << 2 ); - BOOST_CHECK_EQUAL( 160, b << b ); - - BOOST_CHECK_EQUAL( 20, c << a ); - BOOST_CHECK_EQUAL( 320, c << b ); - BOOST_CHECK_EQUAL( 10240, c << c ); - - BOOST_CHECK_EQUAL( 200, d << a ); - BOOST_CHECK_EQUAL( 3200, d << b ); - BOOST_CHECK_EQUAL( 102400ul, d << c ); - - // multi-word shifts - BOOST_CHECK_EQUAL( e, e1 << wlimits_type::digits ); - BOOST_CHECK_EQUAL( 2 | (( a | c ) << ( wlimits_type::digits - 1 )), f ); - - // exceptions (hope we don't get any out-of-memory exceptions) - va_size_t const si( size_limits::digits, 1 ); - big_whole const s( si ); - big_whole const t( size_limits::max() ); - big_whole const u = s | 1; - - BOOST_CHECK_NO_THROW( z << s ); - BOOST_CHECK_THROW( a << s, length_error ); - BOOST_CHECK_THROW( f << s, length_error ); - - #if PRIVATE_USE_BIG_MEMORY - // This needs a lot of memory, so it shouldn't be regularly - // tested. Worse, what happens if a std::bad_alloc occurs? - BOOST_CHECK_NO_THROW( a << t ); - #endif - - BOOST_CHECK_THROW( 2 << t, overflow_error ); - BOOST_CHECK_THROW( f << t, overflow_error ); - - BOOST_CHECK_NO_THROW( z << u ); - BOOST_CHECK_THROW( a << u, length_error ); - BOOST_CHECK_THROW( f << u, length_error ); -} - -// Unit test for bitwise-and, bitwise-complementing the second value -void -bigwhole_bitwise_and_not_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - using boost::math::and_not; - - typedef std::valarray va_size_t; - - size_t const di[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ds = sizeof( di ) / sizeof( di[0] ); - va_size_t const dv( di, ds ); - size_t const ei[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 12 ); - big_whole const c( 100 ); - big_whole const d( dv ); - big_whole const e( ev ); - - // self-and - BOOST_CHECK_EQUAL( z, and_not(z, z) ); - BOOST_CHECK_EQUAL( z, and_not(a, a) ); - BOOST_CHECK_EQUAL( z, and_not(b, b) ); - BOOST_CHECK_EQUAL( z, and_not(c, c) ); - BOOST_CHECK_EQUAL( z, and_not(d, d) ); - BOOST_CHECK_EQUAL( z, and_not(e, e) ); - - // zero-and - BOOST_CHECK_EQUAL( z, and_not(z, a) ); BOOST_CHECK_EQUAL( a, and_not(a, z) ); - BOOST_CHECK_EQUAL( z, and_not(z, b) ); BOOST_CHECK_EQUAL( b, and_not(b, z) ); - BOOST_CHECK_EQUAL( z, and_not(z, c) ); BOOST_CHECK_EQUAL( c, and_not(c, z) ); - BOOST_CHECK_EQUAL( z, and_not(z, d) ); BOOST_CHECK_EQUAL( d, and_not(d, z) ); - BOOST_CHECK_EQUAL( z, and_not(z, e) ); BOOST_CHECK_EQUAL( e, and_not(e, z) ); - - // various combinations - big_whole const ab = 3, ac = 3, ad = a, ae = 5; - big_whole const ba = 8, bc = 8, bd = b, be = b; - big_whole const ca = 96, cb = 96, cd = c, ce = c; - big_whole const da = d, db = d, dc = d, de( va_size_t(di + 1, ds - 1) ); - big_whole const ea( va_size_t(ei + 1, es - 1) ), eb = e, ec = e, ed( va_size_t(ei, es - 1) ); - - BOOST_CHECK_EQUAL( ab, and_not(a, b) ); BOOST_CHECK_EQUAL( ba, and_not(b, a) ); - BOOST_CHECK_EQUAL( ac, and_not(a, c) ); BOOST_CHECK_EQUAL( ca, and_not(c, a) ); - BOOST_CHECK_EQUAL( ad, and_not(a, d) ); BOOST_CHECK_EQUAL( da, and_not(d, a) ); - BOOST_CHECK_EQUAL( ae, and_not(a, e) ); BOOST_CHECK_EQUAL( ea, and_not(e, a) ); - - BOOST_CHECK_EQUAL( bc, and_not(b, c) ); BOOST_CHECK_EQUAL( cb, and_not(c, b) ); - BOOST_CHECK_EQUAL( bd, and_not(b, d) ); BOOST_CHECK_EQUAL( db, and_not(d, b) ); - BOOST_CHECK_EQUAL( be, and_not(b, e) ); BOOST_CHECK_EQUAL( eb, and_not(e, b) ); - - BOOST_CHECK_EQUAL( cd, and_not(c, d) ); BOOST_CHECK_EQUAL( dc, and_not(d, c) ); - BOOST_CHECK_EQUAL( ce, and_not(c, e) ); BOOST_CHECK_EQUAL( ec, and_not(e, c) ); - - BOOST_CHECK_EQUAL( de, and_not(d, e) ); BOOST_CHECK_EQUAL( ed, and_not(e, d) ); -} - -// Unit test for unary + and - operators -void -bigwhole_unary_plus_minus_unit_test -( -) -{ - using boost::math::big_whole; - using std::range_error; - - big_whole const z; - big_whole const a = !z; - big_whole const b = a << ( wlimits_type::digits + 3 ); - big_whole const c = a | b; - - // plus - BOOST_CHECK_EQUAL( z, +z ); - BOOST_CHECK_EQUAL( a, +a ); - BOOST_CHECK_EQUAL( b, +b ); - BOOST_CHECK_EQUAL( c, +c ); - - // minus (hope we don't get any out-of-memory exceptions) - BOOST_CHECK_NO_THROW( -z ); - BOOST_CHECK_EQUAL( z, -z ); - - BOOST_CHECK_THROW( -a, range_error ); - BOOST_CHECK_THROW( -b, range_error ); - BOOST_CHECK_THROW( -c, range_error ); -} - -// Unit test for ++ and -- operators -void -bigwhole_double_plus_minus_unit_test -( -) -{ - using boost::math::big_whole; - using std::range_error; - using boost::math::and_not; - - typedef std::valarray va_size_t; - - big_whole const z = 0; - big_whole const aa = wlimits_type::max(); - big_whole const bb( va_size_t(wlimits_type::digits, 1) ); - - // pre-increment - big_whole a = z; - big_whole b = aa; - - BOOST_CHECK_EQUAL( 1, ++a ); - BOOST_CHECK_EQUAL( 2, ++a ); - - BOOST_CHECK_EQUAL( bb, ++b ); - BOOST_CHECK_EQUAL( bb | 1, ++b ); - - // post-increment - a.assign( 0 ); - b.assign( aa ); - - BOOST_CHECK_EQUAL( 0, a++ ); - BOOST_CHECK_EQUAL( 1, a++ ); - BOOST_CHECK_EQUAL( 2, a ); - - BOOST_CHECK_EQUAL( aa, b++ ); - BOOST_CHECK_EQUAL( bb, b++ ); - BOOST_CHECK_EQUAL( bb | 1, b ); - - // decrement errors - a.assign( 0 ); - - BOOST_CHECK_THROW( --a, range_error ); - BOOST_CHECK( !a ); - BOOST_CHECK_THROW( a--, range_error ); - BOOST_CHECK( !a ); - - // pre-decrement - a.assign( 4 ); - b.assign( bb | 1 ); - - BOOST_CHECK_EQUAL( 3, --a ); - BOOST_CHECK_EQUAL( 2, --a ); - BOOST_CHECK_EQUAL( 1, --a ); - BOOST_CHECK_EQUAL( 0, --a ); - - BOOST_CHECK_EQUAL( bb, --b ); - BOOST_CHECK_EQUAL( aa, --b ); - BOOST_CHECK_EQUAL( and_not(aa, 1), --b ); - - // post-decrement - a.assign( 4 ); - b.assign( bb | 1 ); - - BOOST_CHECK_EQUAL( 4, a-- ); - BOOST_CHECK_EQUAL( 3, a-- ); - BOOST_CHECK_EQUAL( 2, a-- ); - BOOST_CHECK_EQUAL( 1, a-- ); - BOOST_CHECK_EQUAL( 0, a ); - - BOOST_CHECK_EQUAL( bb | 1, b-- ); - BOOST_CHECK_EQUAL( bb, b-- ); - BOOST_CHECK_EQUAL( aa, b-- ); - BOOST_CHECK_EQUAL( and_not(aa, 1), b ); -} - -// Unit test for the abs and sgn functions -void -bigwhole_abs_sgn_unit_test -( -) -{ - using boost::math::big_whole; - - typedef std::valarray va_size_t; - - big_whole const z = 0; - big_whole const o = 1; - big_whole const a = wlimits_type::max(); - big_whole const b( va_size_t(wlimits_type::digits, 1) ); - big_whole const c = b | o; - big_whole const d = c << ( 2 * wlimits_type::digits + 3 ); - - BOOST_CHECK_EQUAL( z, abs(z) ); - BOOST_CHECK_EQUAL( 0, sgn(z) ); - - BOOST_CHECK_EQUAL( o, abs(o) ); - BOOST_CHECK_EQUAL( +1, sgn(o) ); - - BOOST_CHECK_EQUAL( a, abs(a) ); - BOOST_CHECK_EQUAL( +1, sgn(a) ); - - BOOST_CHECK_EQUAL( b, abs(b) ); - BOOST_CHECK_EQUAL( +1, sgn(b) ); - - BOOST_CHECK_EQUAL( c, abs(c) ); - BOOST_CHECK_EQUAL( +1, sgn(c) ); - - BOOST_CHECK_EQUAL( d, abs(d) ); - BOOST_CHECK_EQUAL( +1, sgn(d) ); -} - -// Unit test for binary + and - operators -void -bigwhole_binary_plus_minus_unit_test -( -) -{ - using boost::math::big_whole; - using std::range_error; - - typedef std::valarray va_size_t; - - int const wd = wlimits_type::digits; - big_whole const z; - big_whole const a = !z; - big_whole const b = wlimits_type::max(); - big_whole const c( va_size_t(wd, 1) ); - big_whole const d = a | c; - big_whole const e = a << ( wd + 3 ); - big_whole const f = d << ( 2 * wd + 3 ); - - // same-add - BOOST_CHECK_EQUAL( z, z + z ); - BOOST_CHECK_EQUAL( a << 1, a + a ); - BOOST_CHECK_EQUAL( b << 1, b + b ); - BOOST_CHECK_EQUAL( c << 1, c + c ); - BOOST_CHECK_EQUAL( d << 1, d + d ); - BOOST_CHECK_EQUAL( e << 1, e + e ); - BOOST_CHECK_EQUAL( f << 1, f + f ); - - // same-subtract - BOOST_CHECK_EQUAL( z, z - z ); - BOOST_CHECK_EQUAL( z, a - a ); - BOOST_CHECK_EQUAL( z, b - b ); - BOOST_CHECK_EQUAL( z, c - c ); - BOOST_CHECK_EQUAL( z, d - d ); - BOOST_CHECK_EQUAL( z, e - e ); - BOOST_CHECK_EQUAL( z, f - f ); - - // zero-add - BOOST_CHECK_EQUAL( a, z + a ); BOOST_CHECK_EQUAL( a, a + z ); - BOOST_CHECK_EQUAL( b, z + b ); BOOST_CHECK_EQUAL( b, b + z ); - BOOST_CHECK_EQUAL( c, z + c ); BOOST_CHECK_EQUAL( c, c + z ); - BOOST_CHECK_EQUAL( d, z + d ); BOOST_CHECK_EQUAL( d, d + z ); - BOOST_CHECK_EQUAL( e, z + e ); BOOST_CHECK_EQUAL( e, e + z ); - BOOST_CHECK_EQUAL( f, z + f ); BOOST_CHECK_EQUAL( f, f + z ); - - // zero-subtract - BOOST_CHECK_THROW( z - a, range_error ); BOOST_CHECK_EQUAL( a, a - z ); - BOOST_CHECK_THROW( z - b, range_error ); BOOST_CHECK_EQUAL( b, b - z ); - BOOST_CHECK_THROW( z - c, range_error ); BOOST_CHECK_EQUAL( c, c - z ); - BOOST_CHECK_THROW( z - d, range_error ); BOOST_CHECK_EQUAL( d, d - z ); - BOOST_CHECK_THROW( z - e, range_error ); BOOST_CHECK_EQUAL( e, e - z ); - BOOST_CHECK_THROW( z - f, range_error ); BOOST_CHECK_EQUAL( f, f - z ); - - // more bad subtractions - BOOST_CHECK_THROW( a - b, range_error ); BOOST_CHECK_THROW( a - c, range_error ); - BOOST_CHECK_THROW( a - d, range_error ); BOOST_CHECK_THROW( a - e, range_error ); - BOOST_CHECK_THROW( a - f, range_error ); BOOST_CHECK_THROW( b - c, range_error ); - BOOST_CHECK_THROW( b - d, range_error ); BOOST_CHECK_THROW( b - e, range_error ); - BOOST_CHECK_THROW( b - f, range_error ); BOOST_CHECK_THROW( c - d, range_error ); - BOOST_CHECK_THROW( c - e, range_error ); BOOST_CHECK_THROW( c - f, range_error ); - BOOST_CHECK_THROW( d - e, range_error ); BOOST_CHECK_THROW( d - f, range_error ); - BOOST_CHECK_THROW( e - f, range_error ); - - // additions - big_whole const ab1 = c, ac1 = d, ad1 = 3 ^ d, ae1 = a | e, af1 = a | f; - big_whole const ba1 = ab1, bc1 = b | c, bd1 = c << 1, be1 = b | e, bf1 = b | f; - big_whole const ca1 = ac1, cb1 = bc1, cd1 = 3 ^ (d << 1), ce1 = c | e, cf1 = c | f; - big_whole const da1 = ad1, db1 = bd1, dc1 = cd1, de1 = d | e, df1 = d | f; - big_whole const ea1 = ae1, eb1 = be1, ec1 = ce1, ed1 = de1, ef1 = e | f; - big_whole const fa1 = af1, fb1 = bf1, fc1 = cf1, fd1 = df1, fe1 = ef1; - - BOOST_CHECK_EQUAL( ab1, a + b ); BOOST_CHECK_EQUAL( ba1, b + a ); - BOOST_CHECK_EQUAL( ac1, a + c ); BOOST_CHECK_EQUAL( ca1, c + a ); - BOOST_CHECK_EQUAL( ad1, a + d ); BOOST_CHECK_EQUAL( da1, d + a ); - BOOST_CHECK_EQUAL( ae1, a + e ); BOOST_CHECK_EQUAL( ea1, e + a ); - BOOST_CHECK_EQUAL( af1, a + f ); BOOST_CHECK_EQUAL( fa1, f + a ); - - BOOST_CHECK_EQUAL( bc1, b + c ); BOOST_CHECK_EQUAL( cb1, c + b ); - BOOST_CHECK_EQUAL( bd1, b + d ); BOOST_CHECK_EQUAL( db1, d + b ); - BOOST_CHECK_EQUAL( be1, b + e ); BOOST_CHECK_EQUAL( eb1, e + b ); - BOOST_CHECK_EQUAL( bf1, b + f ); BOOST_CHECK_EQUAL( fb1, f + b ); - - BOOST_CHECK_EQUAL( cd1, c + d ); BOOST_CHECK_EQUAL( dc1, d + c ); - BOOST_CHECK_EQUAL( ce1, c + e ); BOOST_CHECK_EQUAL( ec1, e + c ); - BOOST_CHECK_EQUAL( cf1, c + f ); BOOST_CHECK_EQUAL( fc1, f + c ); - - BOOST_CHECK_EQUAL( de1, d + e ); BOOST_CHECK_EQUAL( ed1, e + d ); - BOOST_CHECK_EQUAL( df1, d + f ); BOOST_CHECK_EQUAL( fd1, f + d ); - - BOOST_CHECK_EQUAL( ef1, e + f ); BOOST_CHECK_EQUAL( fe1, f + e ); - - // subtractions - big_whole const six = 6, seven = 7, eight = 8, one = a, maxd = b; - - big_whole const ba2 = a ^ b; - big_whole const cb2 = a, ca2 = b; - big_whole const dc2 = a, db2 = a << 1, da2 = c; - big_whole const ed2 = b | ( six << wd ); - big_whole const ec2 = seven << wd, eb2 = ec2 | a; - big_whole const ea2 = b | ( seven << wd ); - big_whole const fe2 = ( (( (eight << wd) | seven ) << wd) | (maxd ^ seven) ) << wd; - big_whole const fd2 = ( (( (( eight << wd ) | seven) << wd ) | (maxd ^ one)) << wd ) | maxd; - big_whole const fc2 = ( (( (eight << wd) | seven ) << wd) | maxd ) << wd; - big_whole const fb2 = ( (( (( eight << wd ) | seven) << wd ) | maxd) << wd ) | one; - big_whole const fa2 = ( (( (( eight << wd ) | seven) << wd ) | maxd) << wd ) | maxd; - - BOOST_CHECK_EQUAL( ba2, b - a ); BOOST_CHECK_EQUAL( ca2, c - a ); - BOOST_CHECK_EQUAL( cb2, c - b ); BOOST_CHECK_EQUAL( da2, d - a ); - BOOST_CHECK_EQUAL( db2, d - b ); BOOST_CHECK_EQUAL( dc2, d - c ); - BOOST_CHECK_EQUAL( ea2, e - a ); BOOST_CHECK_EQUAL( eb2, e - b ); - BOOST_CHECK_EQUAL( ec2, e - c ); BOOST_CHECK_EQUAL( ed2, e - d ); - BOOST_CHECK_EQUAL( fa2, f - a ); BOOST_CHECK_EQUAL( fb2, f - b ); - BOOST_CHECK_EQUAL( fc2, f - c ); BOOST_CHECK_EQUAL( fd2, f - d ); - BOOST_CHECK_EQUAL( fe2, f - e ); -} - -// Unit test for intersects -void -bigwhole_intersects_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - size_t const di[] = { wlimits_type::digits + 1, 2 * wlimits_type::digits + 3 }; - size_t const ds = sizeof( di ) / sizeof( di[0] ); - va_size_t const dv( di, ds ); - size_t const ei[] = { 1, wlimits_type::digits - 1, wlimits_type::digits, wlimits_type::digits + 1 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 7 ); - big_whole const b( 12 ); - big_whole const c( 100 ); - big_whole const d( dv ); - big_whole const e( ev ); - - // self-intersects - BOOST_CHECK( !z.intersects(z) ); - BOOST_CHECK( a.intersects(a) ); - BOOST_CHECK( b.intersects(b) ); - BOOST_CHECK( c.intersects(c) ); - BOOST_CHECK( d.intersects(d) ); - BOOST_CHECK( e.intersects(e) ); - - // zero-intersects - BOOST_CHECK( !z.intersects(a) ); BOOST_CHECK( !a.intersects(z) ); - BOOST_CHECK( !z.intersects(b) ); BOOST_CHECK( !b.intersects(z) ); - BOOST_CHECK( !z.intersects(c) ); BOOST_CHECK( !c.intersects(z) ); - BOOST_CHECK( !z.intersects(d) ); BOOST_CHECK( !d.intersects(z) ); - BOOST_CHECK( !z.intersects(e) ); BOOST_CHECK( !e.intersects(z) ); - - // various combinations - bool const ab = true, ac = true, ad = false, ae = true; - bool const ba = ab, bc = true, bd = false, be = false; - bool const ca = ac, cb = bc, cd = false, ce = false; - bool const da = ad, db = bd, dc = cd, de = true; - bool const ea = ae, eb = be, ec = ce, ed = de; - - BOOST_CHECK_EQUAL( ab, a.intersects(b) ); BOOST_CHECK_EQUAL( ba, b.intersects(a) ); - BOOST_CHECK_EQUAL( ac, a.intersects(c) ); BOOST_CHECK_EQUAL( ca, c.intersects(a) ); - BOOST_CHECK_EQUAL( ad, a.intersects(d) ); BOOST_CHECK_EQUAL( da, d.intersects(a) ); - BOOST_CHECK_EQUAL( ae, a.intersects(e) ); BOOST_CHECK_EQUAL( ea, e.intersects(a) ); - - BOOST_CHECK_EQUAL( bc, b.intersects(c) ); BOOST_CHECK_EQUAL( cb, c.intersects(b) ); - BOOST_CHECK_EQUAL( bd, b.intersects(d) ); BOOST_CHECK_EQUAL( db, d.intersects(b) ); - BOOST_CHECK_EQUAL( be, b.intersects(e) ); BOOST_CHECK_EQUAL( eb, e.intersects(b) ); - - BOOST_CHECK_EQUAL( cd, c.intersects(d) ); BOOST_CHECK_EQUAL( dc, d.intersects(c) ); - BOOST_CHECK_EQUAL( ce, c.intersects(e) ); BOOST_CHECK_EQUAL( ec, e.intersects(c) ); - - BOOST_CHECK_EQUAL( de, d.intersects(e) ); BOOST_CHECK_EQUAL( ed, e.intersects(d) ); -} - -// Unit test for bit searching -void -bigwhole_bit_search_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - typedef std::numeric_limits size_limits; - - size_t const wd = wlimits_type::digits; - size_t const ei[] = { wd + 1, 2 * wd + 3 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - - big_whole const z; - big_whole const a( 1 ); - big_whole const b( 5 ); - big_whole const e( ev ); - - // limit violations - BOOST_CHECK_EQUAL( size_limits::min(), z.next_set_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), a.next_set_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), b.next_set_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), e.next_set_bit(size_limits::max()) ); - - BOOST_CHECK_EQUAL( size_limits::min(), z.next_reset_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), a.next_reset_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), b.next_reset_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::min(), e.next_reset_bit(size_limits::max()) ); - - BOOST_CHECK_EQUAL( size_limits::max(), z.previous_set_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), a.previous_set_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), b.previous_set_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), e.previous_set_bit(size_limits::min()) ); - - BOOST_CHECK_EQUAL( size_limits::max(), z.previous_reset_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), a.previous_reset_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), b.previous_reset_bit(size_limits::min()) ); - BOOST_CHECK_EQUAL( size_limits::max(), e.previous_reset_bit(size_limits::min()) ); - - // checking zero - BOOST_CHECK_EQUAL( 1u, z.next_reset_bit(0) ); - BOOST_CHECK_EQUAL( size_limits::min(), z.next_set_bit(0) ); - BOOST_CHECK_EQUAL( 23u, z.previous_reset_bit(24) ); - BOOST_CHECK_EQUAL( size_limits::max(), z.previous_set_bit(24) ); - - // more checks - BOOST_CHECK_EQUAL( 1u, a.next_reset_bit(0) ); - BOOST_CHECK_EQUAL( size_limits::min(), a.next_set_bit(0) ); - BOOST_CHECK_EQUAL( 23u, a.previous_reset_bit(24) ); - BOOST_CHECK_EQUAL( 0u, a.previous_set_bit(24) ); - - BOOST_CHECK_EQUAL( 1u, b.next_reset_bit(0) ); - BOOST_CHECK_EQUAL( 2u, b.next_set_bit(0) ); - BOOST_CHECK_EQUAL( 3u, b.next_reset_bit(1) ); - BOOST_CHECK_EQUAL( 2u, b.next_set_bit(1) ); - BOOST_CHECK_EQUAL( 3u, b.next_reset_bit(2) ); - BOOST_CHECK_EQUAL( size_limits::min(), b.next_set_bit(2) ); - BOOST_CHECK_EQUAL( 4u, b.next_reset_bit(3) ); - BOOST_CHECK_EQUAL( size_limits::min(), b.next_set_bit(3) ); - - BOOST_CHECK_EQUAL( 3u, b.previous_reset_bit(4) ); - BOOST_CHECK_EQUAL( 2u, b.previous_set_bit(4) ); - BOOST_CHECK_EQUAL( 1u, b.previous_reset_bit(3) ); - BOOST_CHECK_EQUAL( 2u, b.previous_set_bit(3) ); - BOOST_CHECK_EQUAL( 1u, b.previous_reset_bit(2) ); - BOOST_CHECK_EQUAL( 0u, b.previous_set_bit(2) ); - BOOST_CHECK_EQUAL( size_limits::max(), b.previous_reset_bit(1) ); - BOOST_CHECK_EQUAL( 0u, b.previous_set_bit(1) ); - - BOOST_CHECK_EQUAL( 1u, e.next_reset_bit(0) ); - BOOST_CHECK_EQUAL( wd + 1u, e.next_set_bit(0) ); - - BOOST_CHECK_EQUAL( size_limits::max(), e.previous_set_bit(wd + 1) ); - BOOST_CHECK_EQUAL( wd, e.previous_reset_bit(wd + 1) ); - BOOST_CHECK_EQUAL( wd + 2, e.next_reset_bit(wd + 1) ); - BOOST_CHECK_EQUAL( 2 * wd + 3, e.next_set_bit(wd + 1) ); - - BOOST_CHECK_EQUAL( wd + 1, e.previous_set_bit(3 * wd / 2 + 4) ); - BOOST_CHECK_EQUAL( 3 * wd / 2 + 3, e.previous_reset_bit(3 * wd / 2 + 4) ); - BOOST_CHECK_EQUAL( 3 * wd / 2 + 5, e.next_reset_bit(3 * wd / 2 + 4) ); - BOOST_CHECK_EQUAL( 2 * wd + 3, e.next_set_bit(3 * wd / 2 + 4) ); - - BOOST_CHECK_EQUAL( wd + 1, e.previous_set_bit(2 * wd + 3) ); - BOOST_CHECK_EQUAL( 2 * wd + 2, e.previous_reset_bit(2 * wd + 3) ); - BOOST_CHECK_EQUAL( 2 * wd + 4, e.next_reset_bit(2 * wd + 3) ); - BOOST_CHECK_EQUAL( size_limits::min(), e.next_set_bit(2 * wd + 3) ); - - BOOST_CHECK_EQUAL( 2 * wd + 3, e.previous_set_bit(size_limits::max()) ); - BOOST_CHECK_EQUAL( size_limits::max() - 1, e.previous_reset_bit(size_limits::max()) ); -} - -// Unit test for the scale factor -void -bigwhole_scale_unit_test -( -) -{ - using boost::math::big_whole; - using std::size_t; - - typedef std::valarray va_size_t; - - size_t const wd = wlimits_type::digits; - size_t const ei[] = { wd + 1, 2 * wd + 3 }; - size_t const es = sizeof( ei ) / sizeof( ei[0] ); - va_size_t const ev( ei, es ); - size_t const fi[] = { 1, wd - 1, wd, wd + 2 }; - size_t const fs = sizeof( fi ) / sizeof( fi[0] ); - va_size_t const fv( fi, fs ); - - big_whole const z; - big_whole const a( 1 ); - big_whole const b( 5 ); - big_whole const c( 10 ); - big_whole const d( 100 ); - big_whole const e( ev ); - big_whole const f( fv ); - - BOOST_CHECK_EQUAL( 0u, z.scale() ); - BOOST_CHECK_EQUAL( 0u, a.scale() ); - BOOST_CHECK_EQUAL( 0u, b.scale() ); - BOOST_CHECK_EQUAL( 1u, c.scale() ); - BOOST_CHECK_EQUAL( 2u, d.scale() ); - BOOST_CHECK_EQUAL( wd + 1u, e.scale() ); - BOOST_CHECK_EQUAL( 1u, (e >> wd).scale() ); - BOOST_CHECK_EQUAL( wd + 5u, (e << 4).scale() ); - BOOST_CHECK_EQUAL( 1u, f.scale() ); -} - - -// Unit test program -boost::unit_test_framework::test_suite * -init_unit_test_suite -( - int , // "argc" is unused - char * [] // "argv" is unused -) -{ - boost::unit_test_framework::test_suite * test - = BOOST_TEST_SUITE( "big_whole test" ); - - test->add( BOOST_TEST_CASE(basic_bigwhole_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_multi_bit_check_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_reverse_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_all_bit_reset_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_single_bit_reset_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_group_bit_reset_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_single_bit_set_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_group_bit_set_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_single_bit_flip_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_group_bit_flip_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_group_bit_assign_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_is_even_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_compare_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_bitwise_and_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_bitwise_or_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_bitwise_xor_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_left_shift_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_right_shift_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_bitwise_and_not_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_unary_plus_minus_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_double_plus_minus_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_abs_sgn_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_binary_plus_minus_unit_test) ); - - test->add( BOOST_TEST_CASE(bigwhole_intersects_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_bit_search_unit_test) ); - test->add( BOOST_TEST_CASE(bigwhole_scale_unit_test) ); - - return test; -} diff --git a/test/cayley_element_test.cpp b/test/cayley_element_test.cpp deleted file mode 100644 index 0259c16ea..000000000 --- a/test/cayley_element_test.cpp +++ /dev/null @@ -1,1201 +0,0 @@ -// Boost cayley_element_test.cpp test file ---------------------------------// - -// Copyright 2005 Daryle Walker. Use, modification, and distribution are -// subject to the Boost Software License, Version 1.0. (See accompanying file -// LICENSE_1_0.txt or a copy at .) - -// See for the library's home page. - -// Revision History -// 18 Dec 2005 Initial version (Daryle Walker) - -#define BOOST_TEST_MAIN "cayley element test" - -#include // for boost:...:cayley_element, etc. - -#include // for BOOST_CHECK_EQUAL -#include // for BOOST_CHECK_CLOSE - -#include // for boost::rational - -#include // for std::asin -#include // for std::size_t - - -// Types -using boost::math::cayley_element; -using boost::math::negated_cayley_element; -using boost::math::scaled_cayley_element; - -typedef boost::rational ri_type; - -typedef cayley_element ce; -typedef negated_cayley_element nce; - -typedef scaled_cayley_element ci_type; -typedef scaled_cayley_element cu_type; -typedef scaled_cayley_element cd_type; -typedef scaled_cayley_element cr_type; - - -// Testing preparations -BOOST_TEST_DONT_PRINT_LOG_VALUE( ce ); -BOOST_TEST_DONT_PRINT_LOG_VALUE( nce ); -BOOST_TEST_DONT_PRINT_LOG_VALUE( ci_type ); -BOOST_TEST_DONT_PRINT_LOG_VALUE( cu_type ); -BOOST_TEST_DONT_PRINT_LOG_VALUE( cd_type ); -BOOST_TEST_DONT_PRINT_LOG_VALUE( cr_type ); - - -// Element constructors, accessors, mutators, and Boolean conversion -BOOST_AUTO_TEST_CASE( cayley_constructor_test ) -{ - { - ce t1( 3 ); - BOOST_CHECK_EQUAL( 3u, t1.basis() ); - BOOST_CHECK( t1 ); - - t1.basis( 5 ); - BOOST_CHECK_EQUAL( 5u, t1.basis() ); - BOOST_CHECK( t1 ); - } - - { - nce t2( 7 ); - BOOST_CHECK_EQUAL( 7u, t2.basis() ); - BOOST_CHECK( not t2.negative() ); - BOOST_CHECK( t2 ); - - t2.basis( 11 ); - t2.negative( true ); - BOOST_CHECK_EQUAL( 11u, t2.basis() ); - BOOST_CHECK( t2.negative() ); - BOOST_CHECK( t2 ); - - nce const t3( 13, true ); - BOOST_CHECK_EQUAL( 13u, t3.basis() ); - BOOST_CHECK( t3.negative() ); - BOOST_CHECK( t3 ); - - ce const t4( 17 ); - nce const t5( t4 ); - BOOST_CHECK_EQUAL( t4.basis(), t5.basis() ); - BOOST_CHECK( not t5.negative() ); - BOOST_CHECK( t4 && t5 ); - } - - { - ci_type t6( 19, 2 ); - BOOST_CHECK_EQUAL( 19u, t6.basis() ); - BOOST_CHECK_EQUAL( 2, t6.scale() ); - BOOST_CHECK( t6 ); - - t6.basis( 23 ); - t6.scale( 0 ); - BOOST_CHECK_EQUAL( 23u, t6.basis() ); - BOOST_CHECK_EQUAL( 0, t6.scale() ); - BOOST_CHECK( not t6 ); - - nce const t7( 29, true ); - ci_type const t8( t7 ); - BOOST_CHECK_EQUAL( t7.basis(), t8.basis() ); - BOOST_CHECK_EQUAL( -1, t8.scale() ); - BOOST_CHECK( t7 && t8 ); - - ce const t9( 31 ); - ci_type const t10( t9 ); - BOOST_CHECK_EQUAL( t9.basis(), t10.basis() ); - BOOST_CHECK_EQUAL( 1, t10.scale() ); - BOOST_CHECK( t9 && t10 ); - - ci_type const t11( 3 ); - BOOST_CHECK_EQUAL( 0u, t11.basis() ); - BOOST_CHECK_EQUAL( 3, t11.scale() ); - BOOST_CHECK( t11 ); - - ci_type const t12; - BOOST_CHECK_EQUAL( 0u, t12.basis() ); - BOOST_CHECK_EQUAL( 0, t12.scale() ); - BOOST_CHECK( not t12 ); - - ci_type const t13( nce(ce( 37 )) ); - BOOST_CHECK_EQUAL( 37u, t13.basis() ); - BOOST_CHECK_EQUAL( 1, t13.scale() ); - BOOST_CHECK( t13 ); - - cd_type const t14( ci_type(41u, -4) ); - BOOST_CHECK_EQUAL( 41u, t14.basis() ); - BOOST_CHECK_CLOSE( -4.0, t14.scale(), 0.1 ); - BOOST_CHECK( t14 ); - } -} - -// Rung & index counting -BOOST_AUTO_TEST_CASE( cayley_rung_index_test ) -{ - BOOST_CHECK_EQUAL( 0u, ce::maximum_index_for_rung(0) ); - BOOST_CHECK_EQUAL( 0u, ce::minimum_rung_for_index(0) ); - - BOOST_CHECK_EQUAL( 1u, ce::maximum_index_for_rung(1) ); - BOOST_CHECK_EQUAL( 1u, ce::minimum_rung_for_index(1) ); - - BOOST_CHECK_EQUAL( 3u, ce::maximum_index_for_rung(2) ); - BOOST_CHECK_EQUAL( 2u, ce::minimum_rung_for_index(2) ); - BOOST_CHECK_EQUAL( 2u, ce::minimum_rung_for_index(3) ); - - BOOST_CHECK_EQUAL( 7u, ce::maximum_index_for_rung(3) ); - BOOST_CHECK_EQUAL( 3u, ce::minimum_rung_for_index(4) ); - BOOST_CHECK_EQUAL( 3u, ce::minimum_rung_for_index(5) ); - BOOST_CHECK_EQUAL( 3u, ce::minimum_rung_for_index(6) ); - BOOST_CHECK_EQUAL( 3u, ce::minimum_rung_for_index(7) ); - - BOOST_CHECK_EQUAL( 15u, ce::maximum_index_for_rung(4) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(8) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(9) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(10) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(11) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(12) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(13) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(14) ); - BOOST_CHECK_EQUAL( 4u, ce::minimum_rung_for_index(15) ); - - BOOST_CHECK_EQUAL( 31u, ce::maximum_index_for_rung(5) ); - BOOST_CHECK_EQUAL( 5u, ce(16).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, ce(17).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, ce(18).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, ce(19).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, ce(20).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, nce(21).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, nce(22).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, nce(23).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, nce(24).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, nce(25).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(26, 1u).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(27, 2u).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(28, 3u).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(29, 5u).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(30, 7u).minimum_rung() ); - BOOST_CHECK_EQUAL( 5u, cu_type(31, 11u).minimum_rung() ); -} - -// Equality operators -BOOST_AUTO_TEST_CASE( cayley_equal_op_test ) -{ - ce const t1( 0 ), t2( 1 ), t3( 0 ); - - BOOST_CHECK_EQUAL( t1, t1 ); - BOOST_CHECK_EQUAL( t2, t2 ); - BOOST_CHECK_EQUAL( t3, t3 ); - - BOOST_CHECK_EQUAL( t1, t3 ); - BOOST_CHECK_EQUAL( t3, t1 ); - - BOOST_CHECK( !(t1 == t2) ); - BOOST_CHECK( !(t2 == t1) ); - BOOST_CHECK( !(t2 == t3) ); - BOOST_CHECK( !(t3 == t2) ); - - BOOST_CHECK( !(t1 != t1) ); - BOOST_CHECK( !(t2 != t2) ); - BOOST_CHECK( !(t3 != t3) ); - BOOST_CHECK( !(t1 != t3) ); - BOOST_CHECK( !(t3 != t1) ); - BOOST_CHECK( t1 != t2 ); - BOOST_CHECK( t2 != t1 ); - BOOST_CHECK( t2 != t3 ); - BOOST_CHECK( t3 != t2 ); - - nce const t4( 2 ), t5( 3 ), t6( 2 ), t7( 2, true ), t8( 5, true ); - - BOOST_CHECK_EQUAL( t4, t4 ); BOOST_CHECK_EQUAL( t5, t5 ); - BOOST_CHECK_EQUAL( t6, t6 ); BOOST_CHECK_EQUAL( t7, t7 ); - BOOST_CHECK_EQUAL( t8, t8 ); - - BOOST_CHECK_EQUAL( t4, t6 ); BOOST_CHECK_EQUAL( t6, t4 ); - - BOOST_CHECK( !(t4 == t5) ); BOOST_CHECK( !(t4 == t7) ); - BOOST_CHECK( !(t4 == t8) ); - BOOST_CHECK( !(t5 == t4) ); BOOST_CHECK( !(t5 == t6) ); - BOOST_CHECK( !(t5 == t7) ); BOOST_CHECK( !(t5 == t8) ); - BOOST_CHECK( !(t6 == t5) ); BOOST_CHECK( !(t6 == t7) ); - BOOST_CHECK( !(t6 == t8) ); - BOOST_CHECK( !(t7 == t4) ); BOOST_CHECK( !(t7 == t5) ); - BOOST_CHECK( !(t7 == t6) ); BOOST_CHECK( !(t7 == t8) ); - BOOST_CHECK( !(t8 == t4) ); BOOST_CHECK( !(t8 == t5) ); - BOOST_CHECK( !(t8 == t6) ); BOOST_CHECK( !(t8 == t7) ); - - BOOST_CHECK( !(t4 != t4) ); BOOST_CHECK( !(t5 != t5) ); - BOOST_CHECK( !(t6 != t6) ); BOOST_CHECK( !(t7 != t7) ); - BOOST_CHECK( !(t8 != t8) ); - - BOOST_CHECK( !(t4 != t6) ); BOOST_CHECK( !(t6 != t4) ); - - BOOST_CHECK( t4 != t5 ); BOOST_CHECK( t4 != t7 ); - BOOST_CHECK( t4 != t8 ); - BOOST_CHECK( t5 != t4 ); BOOST_CHECK( t5 != t6 ); - BOOST_CHECK( t5 != t7 ); BOOST_CHECK( t5 != t8 ); - BOOST_CHECK( t6 != t5 ); BOOST_CHECK( t6 != t7 ); - BOOST_CHECK( t6 != t8 ); - BOOST_CHECK( t7 != t4 ); BOOST_CHECK( t7 != t5 ); - BOOST_CHECK( t7 != t6 ); BOOST_CHECK( t7 != t8 ); - BOOST_CHECK( t8 != t4 ); BOOST_CHECK( t8 != t5 ); - BOOST_CHECK( t8 != t6 ); BOOST_CHECK( t8 != t7 ); - - cu_type const t9, t10( 7, 0 ), t11( 11, 0 ); - cu_type const t12( 1 ), t13( 11, 1 ), t14( 11, 1 ); - - BOOST_CHECK_EQUAL( t9, t9 ); BOOST_CHECK_EQUAL( t10, t10 ); - BOOST_CHECK_EQUAL( t11, t11 ); BOOST_CHECK_EQUAL( t12, t12 ); - BOOST_CHECK_EQUAL( t13, t13 ); BOOST_CHECK_EQUAL( t14, t14 ); - - BOOST_CHECK_EQUAL( t9, t10 ); BOOST_CHECK_EQUAL( t9, t11 ); - BOOST_CHECK_EQUAL( t10, t9 ); BOOST_CHECK_EQUAL( t11, t9 ); - BOOST_CHECK_EQUAL( t10, t11 ); BOOST_CHECK_EQUAL( t11, t10 ); - BOOST_CHECK_EQUAL( t13, t14 ); BOOST_CHECK_EQUAL( t14, t13 ); - - BOOST_CHECK( !(t9 == t12) ); BOOST_CHECK( !(t9 == t13) ); - BOOST_CHECK( !(t9 == t14) ); - BOOST_CHECK( !(t10 == t12) ); BOOST_CHECK( !(t10 == t13) ); - BOOST_CHECK( !(t10 == t14) ); - BOOST_CHECK( !(t11 == t12) ); BOOST_CHECK( !(t11 == t13) ); - BOOST_CHECK( !(t11 == t14) ); - BOOST_CHECK( !(t12 == t9) ); BOOST_CHECK( !(t12 == t10) ); - BOOST_CHECK( !(t12 == t11) ); BOOST_CHECK( !(t12 == t13) ); - BOOST_CHECK( !(t12 == t14) ); - BOOST_CHECK( !(t13 == t9) ); BOOST_CHECK( !(t13 == t10) ); - BOOST_CHECK( !(t13 == t11) ); BOOST_CHECK( !(t13 == t12) ); - BOOST_CHECK( !(t14 == t9) ); BOOST_CHECK( !(t14 == t10) ); - BOOST_CHECK( !(t14 == t11) ); BOOST_CHECK( !(t14 == t12) ); - - BOOST_CHECK( !(t9 != t9) ); BOOST_CHECK( !(t10 != t10) ); - BOOST_CHECK( !(t11 != t11) ); BOOST_CHECK( !(t12 != t12) ); - BOOST_CHECK( !(t13 != t13) ); BOOST_CHECK( !(t14 != t14) ); - - BOOST_CHECK( !(t9 != t10) ); BOOST_CHECK( !(t9 != t11) ); - BOOST_CHECK( !(t10 != t9) ); BOOST_CHECK( !(t11 != t9) ); - BOOST_CHECK( !(t10 != t11) ); BOOST_CHECK( !(t11 != t10) ); - BOOST_CHECK( !(t13 != t14) ); BOOST_CHECK( !(t14 != t13) ); - - BOOST_CHECK( t9 != t12 ); BOOST_CHECK( t9 != t13 ); - BOOST_CHECK( t9 != t14 ); - BOOST_CHECK( t10 != t12 ); BOOST_CHECK( t10 != t13 ); - BOOST_CHECK( t10 != t14 ); - BOOST_CHECK( t11 != t12 ); BOOST_CHECK( t11 != t13 ); - BOOST_CHECK( t11 != t14 ); - BOOST_CHECK( t12 != t9 ); BOOST_CHECK( t12 != t10 ); - BOOST_CHECK( t12 != t11 ); BOOST_CHECK( t12 != t13 ); - BOOST_CHECK( t12 != t14 ); - BOOST_CHECK( t13 != t9 ); BOOST_CHECK( t13 != t10 ); - BOOST_CHECK( t13 != t11 ); BOOST_CHECK( t13 != t12 ); - BOOST_CHECK( t14 != t9 ); BOOST_CHECK( t14 != t10 ); - BOOST_CHECK( t14 != t11 ); BOOST_CHECK( t14 != t12 ); -} - -// Unary operators -BOOST_AUTO_TEST_CASE( cayley_unary_op_test ) -{ - ce const t1( 0 ), t2( 1 ); - - BOOST_CHECK( !(!t1) ); - BOOST_CHECK( !(!t2) ); - - nce const t2a( t2 ), t3( 2 ), t4( 3, true ); - - BOOST_CHECK( !(!t2a) ); - BOOST_CHECK( !(!t3) ); - BOOST_CHECK( !(!t4) ); - - ci_type const t2b( t2 ), t4a( t4 ); - cd_type const t5( 5, 1.0 ), t6( 7, 0.0 ); - ci_type const t7( 11, -2 ); - - BOOST_CHECK( !(!t2b) ); - BOOST_CHECK( !(!t4a) ); - BOOST_CHECK( !(!t5) ); - BOOST_CHECK( !t6 ); - BOOST_CHECK( !(!t7) ); - - BOOST_CHECK_EQUAL( +t1, t1 ); - BOOST_CHECK_EQUAL( +t2, t2 ); - - BOOST_CHECK_EQUAL( +t2a, t2a ); - BOOST_CHECK_EQUAL( +t3, t3 ); - BOOST_CHECK_EQUAL( +t4, t4 ); - - BOOST_CHECK_EQUAL( +t2b, t2b ); - BOOST_CHECK_EQUAL( +t4a, t4a ); - BOOST_CHECK_EQUAL( +t5, t5 ); - BOOST_CHECK_EQUAL( +t6, t6 ); - BOOST_CHECK_EQUAL( +t7, t7 ); - - BOOST_CHECK_EQUAL( -t1, nce(0, true) ); - BOOST_CHECK_EQUAL( -t2, nce(1, true) ); - - BOOST_CHECK_EQUAL( -t2a, nce(1, true) ); - BOOST_CHECK_EQUAL( -t3, nce(2, true) ); - BOOST_CHECK_EQUAL( -t4, nce(3) ); - - BOOST_CHECK_EQUAL( -t2b, ci_type(1, -1) ); - BOOST_CHECK_EQUAL( -t4a, ci_type(3, 1) ); - BOOST_CHECK_EQUAL( -t5, cd_type(5, -1.0) ); - BOOST_CHECK_EQUAL( -t6, t6 ); - BOOST_CHECK_EQUAL( -t7, ci_type(11, 2) ); -} - -// Unary-operator-like member functions -BOOST_AUTO_TEST_CASE( cayley_unary_self_test ) -{ - ce const t1( 0 ); - ce t2 = t1; - - t2.same_self(); - BOOST_CHECK_EQUAL( +t1, t2 ); - - nce const t3( 1 ); - nce t4 = t3; - - t4.same_self(); - BOOST_CHECK_EQUAL( +t3, t4 ); - t4.negate_self(); - BOOST_CHECK_EQUAL( -t3, t4 ); - - ci_type const t5( 2, 1 ), t6( 2, 0 ); - ci_type t7 = t5, t8 = t6; - - t7.same_self(); - BOOST_CHECK_EQUAL( +t5, t7 ); - t7.negate_self(); - BOOST_CHECK_EQUAL( -t5, t7 ); - t7.not_self(); - BOOST_CHECK_EQUAL( t6, t7 ); - t8.same_self(); - BOOST_CHECK_EQUAL( +t6, t8 ); - t8.negate_self(); - BOOST_CHECK_EQUAL( -t6, t8 ); - t8.not_self(); - BOOST_CHECK_EQUAL( t5, t8 ); -} - -// Condition functions -BOOST_AUTO_TEST_CASE( cayley_condition_function_test ) -{ - using boost::math::dot_product; - using boost::math::conj; - using boost::math::abs; - using boost::math::arg; - using boost::math::norm; - using boost::math::sgn; - using boost::math::reciprocal; - using boost::bad_rational; - using boost::math::sup; - using boost::math::l1; - - // dot product - BOOST_CHECK_EQUAL( 1, dot_product(ce( 5 ), ce( 5 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ce( 2 ), ce( 3 )) ); - - BOOST_CHECK_EQUAL( 1, dot_product(nce( 5 ), nce( 5 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(nce( 2 ), nce( 3 )) ); - BOOST_CHECK_EQUAL( -1, dot_product(nce( 5, true ), nce( 5 )) ); - BOOST_CHECK_EQUAL( -1, dot_product(nce( 5 ), nce( 5, true )) ); - BOOST_CHECK_EQUAL( 1, dot_product(nce( 5, true ), nce( 5, true )) ); - BOOST_CHECK_EQUAL( 0, dot_product(nce( 2, true ), nce( 3 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(nce( 2 ), nce( 3, true )) ); - BOOST_CHECK_EQUAL( 0, dot_product(nce( 2, true ), nce( 3, true )) ); - - BOOST_CHECK_EQUAL( 1, dot_product(ci_type( 5, 1 ), ci_type( 5, 1 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ci_type( 2, 1 ), ci_type( 3, 1 )) ); - BOOST_CHECK_EQUAL( -1, dot_product(ci_type( 5, -1 ), ci_type( 5, 1 )) ); - BOOST_CHECK_EQUAL( -1, dot_product(ci_type( 5, 1 ), ci_type( 5, -1 )) ); - BOOST_CHECK_EQUAL( 1, dot_product(ci_type( 5, -1 ), ci_type( 5, -1 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ci_type( 2, -1 ), ci_type( 3, 1 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ci_type( 2, 1 ), ci_type( 3, -1 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ci_type( 2, -1 ), ci_type( 3, -1 )) ); - BOOST_CHECK_EQUAL( 0, dot_product(ci_type( 7, 0 ), ci_type( 7, 4 )) ); - BOOST_CHECK_EQUAL( -8, dot_product(ci_type( 7, -2 ), ci_type( 7, 4 )) ); - BOOST_CHECK_EQUAL( 6, dot_product(ci_type( 11, 3 ), ci_type( 11, 2 )) ); - BOOST_CHECK_EQUAL( 10, dot_product(ci_type( 13, -2 ), ci_type( 13, -5 )) ); - - // conjugate - ce const t1( 0 ), t2( 1 ); - - BOOST_CHECK_EQUAL( conj(t1), t1 ); - BOOST_CHECK_EQUAL( conj(t2), -t2 ); - - nce const t1a( t1 ), t2a( t2 ), t3( 2, true ), t4( 0, true ); - - BOOST_CHECK_EQUAL( conj(t1a), t1a ); - BOOST_CHECK_EQUAL( conj(t2a), -t2a ); - BOOST_CHECK_EQUAL( conj(t3), -t3 ); - BOOST_CHECK_EQUAL( conj(t4), t4 ); - - ci_type const t1b( t1 ), t2b( t2 ), t3a( t3 ), t4a( t4 ); - ci_type const t5( 0, 6 ), t6( 0, -8 ), t7( 3, 9 ), t8( 5, -12 ); - ci_type const t9( 0, 0 ), t10( 7, 0 ); - - BOOST_CHECK_EQUAL( conj(t1b), t1b ); - BOOST_CHECK_EQUAL( conj(t2b), -t2b ); - BOOST_CHECK_EQUAL( conj(t3a), -t3a ); - BOOST_CHECK_EQUAL( conj(t4a), t4a ); - BOOST_CHECK_EQUAL( conj(t5), t5 ); - BOOST_CHECK_EQUAL( conj(t6), t6 ); - BOOST_CHECK_EQUAL( conj(t7), -t7 ); - BOOST_CHECK_EQUAL( conj(t8), -t8 ); - BOOST_CHECK_EQUAL( conj(t9), t9 ); - BOOST_CHECK_EQUAL( conj(t10), -t10 ); - - // absolute value (2-norm) - BOOST_CHECK_EQUAL( abs(t1), 1 ); - BOOST_CHECK_EQUAL( abs(t2), 1 ); - - BOOST_CHECK_EQUAL( abs(t1a), 1 ); - BOOST_CHECK_EQUAL( abs(t2a), 1 ); - BOOST_CHECK_EQUAL( abs(t3), 1 ); - BOOST_CHECK_EQUAL( abs(t4), 1 ); - - BOOST_CHECK_EQUAL( abs(t1b), 1 ); - BOOST_CHECK_EQUAL( abs(t2b), 1 ); - BOOST_CHECK_EQUAL( abs(t3a), 1 ); - BOOST_CHECK_EQUAL( abs(t4a), 1 ); - BOOST_CHECK_EQUAL( abs(t5), 6 ); - BOOST_CHECK_EQUAL( abs(t6), 8 ); - BOOST_CHECK_EQUAL( abs(t7), 9 ); - BOOST_CHECK_EQUAL( abs(t8), 12 ); - BOOST_CHECK_EQUAL( abs(t9), 0 ); - BOOST_CHECK_EQUAL( abs(t10), 0 ); - - // argument angle - double const pi_2 = std::asin( 1.0 ), pi = 2.0 * pi_2; - - BOOST_CHECK_CLOSE( arg(t1), 0.0, 0.1 ); - BOOST_CHECK_CLOSE( arg(t2), pi_2, 0.1 ); - - BOOST_CHECK_CLOSE( arg(t1a), 0.0, 0.1 ); - BOOST_CHECK_CLOSE( arg(t2a), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(t3), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(t4), pi, 0.1 ); - - BOOST_CHECK_CLOSE( arg(cd_type( t1b )), 0.0, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t2b )), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t3a )), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t4a )), pi, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t5 )), 0.0, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t6 )), pi, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t7 )), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t8 )), pi_2, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t9 )), 0.0, 0.1 ); - BOOST_CHECK_CLOSE( arg(cd_type( t10 )), 0.0, 0.1 ); - - // norm (quadratic, Cayley) - BOOST_CHECK_EQUAL( norm(t1), 1 ); - BOOST_CHECK_EQUAL( norm(t2), 1 ); - - BOOST_CHECK_EQUAL( norm(t1a), 1 ); - BOOST_CHECK_EQUAL( norm(t2a), 1 ); - BOOST_CHECK_EQUAL( norm(t3), 1 ); - BOOST_CHECK_EQUAL( norm(t4), 1 ); - - BOOST_CHECK_EQUAL( norm(t1b), 1 ); - BOOST_CHECK_EQUAL( norm(t2b), 1 ); - BOOST_CHECK_EQUAL( norm(t3a), 1 ); - BOOST_CHECK_EQUAL( norm(t4a), 1 ); - BOOST_CHECK_EQUAL( norm(t5), 36 ); - BOOST_CHECK_EQUAL( norm(t6), 64 ); - BOOST_CHECK_EQUAL( norm(t7), 81 ); - BOOST_CHECK_EQUAL( norm(t8), 144 ); - BOOST_CHECK_EQUAL( norm(t9), 0 ); - BOOST_CHECK_EQUAL( norm(t10), 0 ); - - // sign - BOOST_CHECK_EQUAL( sgn(t1), t1 ); - BOOST_CHECK_EQUAL( sgn(t2), t2 ); - - BOOST_CHECK_EQUAL( sgn(t1a), t1a ); - BOOST_CHECK_EQUAL( sgn(t2a), t2a ); - BOOST_CHECK_EQUAL( sgn(t3), t3 ); - BOOST_CHECK_EQUAL( sgn(t4), t4 ); - - BOOST_CHECK_EQUAL( sgn(t1b), t1b ); - BOOST_CHECK_EQUAL( sgn(t2b), t2b ); - BOOST_CHECK_EQUAL( sgn(t3a), t3a ); - BOOST_CHECK_EQUAL( sgn(t4a), t4a ); - BOOST_CHECK_EQUAL( sgn(t5), ci_type(0, 1) ); - BOOST_CHECK_EQUAL( sgn(t6), ci_type(0, -1) ); - BOOST_CHECK_EQUAL( sgn(t7), ci_type(3, 1) ); - BOOST_CHECK_EQUAL( sgn(t8), ci_type(5, -1) ); - BOOST_CHECK_EQUAL( sgn(t9), t9 ); - BOOST_CHECK_EQUAL( sgn(t10), t10 ); - - // reciprocal - BOOST_CHECK_EQUAL( reciprocal(t1), t1 ); - BOOST_CHECK_EQUAL( reciprocal(t2), -t2 ); - - BOOST_CHECK_EQUAL( reciprocal(t1a), t1a ); - BOOST_CHECK_EQUAL( reciprocal(t2a), -t2a ); - BOOST_CHECK_EQUAL( reciprocal(t3), -t3 ); - BOOST_CHECK_EQUAL( reciprocal(t4), t4 ); - - BOOST_CHECK_EQUAL( reciprocal(cr_type( t1b )), cr_type(t1b) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t2b )), cr_type(-t2b) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t3a )), cr_type(-t3a) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t4a )), cr_type(t4a) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t5 )), cr_type(0, ri_type( 1, 6 )) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t6 )), cr_type(0, ri_type( -1, - 8 )) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t7 )), cr_type(3, ri_type( -1, - 9 )) ); - BOOST_CHECK_EQUAL( reciprocal(cr_type( t8 )), cr_type(5, ri_type( 1, - 12 )) ); - BOOST_CHECK_THROW( reciprocal(cr_type( t9 )), bad_rational ); - BOOST_CHECK_THROW( reciprocal(cr_type( t10 )), bad_rational ); - - // other norms (supremum [inf-norm] and 1-norm) - BOOST_CHECK_EQUAL( sup(t1), 1 ); BOOST_CHECK_EQUAL( l1(t1), 1 ); - BOOST_CHECK_EQUAL( sup(t2), 1 ); BOOST_CHECK_EQUAL( l1(t2), 1 ); - - BOOST_CHECK_EQUAL( sup(t1a), 1 ); BOOST_CHECK_EQUAL( l1(t1a), 1 ); - BOOST_CHECK_EQUAL( sup(t2a), 1 ); BOOST_CHECK_EQUAL( l1(t2a), 1 ); - BOOST_CHECK_EQUAL( sup(t3), 1 ); BOOST_CHECK_EQUAL( l1(t3), 1 ); - BOOST_CHECK_EQUAL( sup(t4), 1 ); BOOST_CHECK_EQUAL( l1(t4), 1 ); - - BOOST_CHECK_EQUAL( sup(t1b), 1 ); BOOST_CHECK_EQUAL( l1(t1b), 1 ); - BOOST_CHECK_EQUAL( sup(t2b), 1 ); BOOST_CHECK_EQUAL( l1(t2b), 1 ); - BOOST_CHECK_EQUAL( sup(t3a), 1 ); BOOST_CHECK_EQUAL( l1(t3a), 1 ); - BOOST_CHECK_EQUAL( sup(t4a), 1 ); BOOST_CHECK_EQUAL( l1(t4a), 1 ); - BOOST_CHECK_EQUAL( sup(t5), 6 ); BOOST_CHECK_EQUAL( l1(t5), 6 ); - BOOST_CHECK_EQUAL( sup(t6), 8 ); BOOST_CHECK_EQUAL( l1(t6), 8 ); - BOOST_CHECK_EQUAL( sup(t7), 9 ); BOOST_CHECK_EQUAL( l1(t7), 9 ); - BOOST_CHECK_EQUAL( sup(t8), 12 ); BOOST_CHECK_EQUAL( l1(t8), 12 ); - BOOST_CHECK_EQUAL( sup(t9), 0 ); BOOST_CHECK_EQUAL( l1(t9), 0 ); - BOOST_CHECK_EQUAL( sup(t10), 0 ); BOOST_CHECK_EQUAL( l1(t10), 0 ); -} - -// Self-conditioning member functions -BOOST_AUTO_TEST_CASE( cayley_condition_self_test ) -{ - using boost::math::conj; - using boost::math::sgn; - using boost::math::reciprocal; - using boost::bad_rational; - - ce const c1( 0 ), c2( 1 ); - ce t1 = c1, t2 = c2; - - t1.sign_self(); BOOST_CHECK_EQUAL( sgn(c1), t1 ); - t2.sign_self(); BOOST_CHECK_EQUAL( sgn(c2), t2 ); - - nce const c3( c1 ), c4( 0, true ), c5( c2 ), c6( 1, true ); - nce t3 = c3, t4 = c4, t5 = c5, t6 = c6; - - t3.sign_self(); BOOST_CHECK_EQUAL( sgn(c3), t3 ); - t4.sign_self(); BOOST_CHECK_EQUAL( sgn(c4), t4 ); - t5.sign_self(); BOOST_CHECK_EQUAL( sgn(c5), t5 ); - t6.sign_self(); BOOST_CHECK_EQUAL( sgn(c6), t6 ); - - t3 = c3; t3.conjugate_self(); BOOST_CHECK_EQUAL( conj(c3), t3 ); - t4 = c4; t4.conjugate_self(); BOOST_CHECK_EQUAL( conj(c4), t4 ); - t5 = c5; t5.conjugate_self(); BOOST_CHECK_EQUAL( conj(c5), t5 ); - t6 = c6; t6.conjugate_self(); BOOST_CHECK_EQUAL( conj(c6), t6 ); - - t3 = c3; t3.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c3), t3 ); - t4 = c4; t4.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c4), t4 ); - t5 = c5; t5.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c5), t5 ); - t6 = c6; t6.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c6), t6 ); - - ci_type const c7( c3 ), c8( c4 ), c9( c5 ), c10( c6 ); - ci_type t7 = c7, t8 = c8, t9 = c9, t10 = c10; - - t7.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c7), t7 ); - t8.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c8), t8 ); - t9.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c9), t9 ); - t10.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c10), t10 ); - - t7 = c7; t7.sign_self(); BOOST_CHECK_EQUAL( sgn(c7), t7 ); - t8 = c8; t8.sign_self(); BOOST_CHECK_EQUAL( sgn(c8), t8 ); - t9 = c9; t9.sign_self(); BOOST_CHECK_EQUAL( sgn(c9), t9 ); - t10 = c10; t10.sign_self(); BOOST_CHECK_EQUAL( sgn(c10), t10 ); - - t7 = c7; t7.conjugate_self(); BOOST_CHECK_EQUAL( conj(c7), t7 ); - t8 = c8; t8.conjugate_self(); BOOST_CHECK_EQUAL( conj(c8), t8 ); - t9 = c9; t9.conjugate_self(); BOOST_CHECK_EQUAL( conj(c9), t9 ); - t10 = c10; t10.conjugate_self(); BOOST_CHECK_EQUAL( conj(c10), t10 ); - - cr_type const c11( 0, ri_type(4) ), c12( 0, ri_type(-3) ), c13( 0, 0 ); - cr_type const c14( 2, ri_type(-5) ), c15( 2, ri_type(7) ), c16( 2, 0 ); - cr_type t11 = c11, t12 = c12, t13 = c13; - cr_type t14 = c14, t15 = c15, t16 = c16; - - t11.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c11), t11 ); - t12.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c12), t12 ); - BOOST_CHECK_THROW( t13.reciprocate_self(), bad_rational ); - t14.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c14), t14 ); - t15.reciprocate_self(); BOOST_CHECK_EQUAL( reciprocal(c15), t15 ); - BOOST_CHECK_THROW( t16.reciprocate_self(), bad_rational ); - - t11 = c11; t11.conjugate_self(); BOOST_CHECK_EQUAL( conj(c11), t11 ); - t12 = c12; t12.conjugate_self(); BOOST_CHECK_EQUAL( conj(c12), t12 ); - t13 = c13; t13.conjugate_self(); BOOST_CHECK_EQUAL( conj(c13), t13 ); - t14 = c14; t14.conjugate_self(); BOOST_CHECK_EQUAL( conj(c14), t14 ); - t15 = c15; t15.conjugate_self(); BOOST_CHECK_EQUAL( conj(c15), t15 ); - t16 = c16; t16.conjugate_self(); BOOST_CHECK_EQUAL( conj(c16), t16 ); - - t11 = c11; t11.sign_self(); BOOST_CHECK_EQUAL( sgn(c11), t11 ); - t12 = c12; t12.sign_self(); BOOST_CHECK_EQUAL( sgn(c12), t12 ); - t13 = c13; t13.sign_self(); BOOST_CHECK_EQUAL( sgn(c13), t13 ); - t14 = c14; t14.sign_self(); BOOST_CHECK_EQUAL( sgn(c14), t14 ); - t15 = c15; t15.sign_self(); BOOST_CHECK_EQUAL( sgn(c15), t15 ); - t16 = c16; t16.sign_self(); BOOST_CHECK_EQUAL( sgn(c16), t16 ); -} - -// Scalar multiplication and division -BOOST_AUTO_TEST_CASE( cayley_scalar_muldiv_op_test ) -{ - using boost::bad_rational; - - BOOST_CHECK_EQUAL( 6 * ci_type(0, 7), ci_type(0, 42) ); - BOOST_CHECK_EQUAL( ci_type(1, 5) * -3, ci_type(1, -15) ); - - BOOST_CHECK_EQUAL( ri_type(4) / cr_type(0, 3), cr_type(0, ri_type(4, 3)) ); - BOOST_CHECK_EQUAL( ri_type(12) / cr_type(2, 3), cr_type(2, ri_type(-4)) ); - BOOST_CHECK_THROW( ri_type(4) / cr_type(0, 0), bad_rational ); - BOOST_CHECK_THROW( ri_type(12) / cr_type(2, 0), bad_rational ); - BOOST_CHECK_EQUAL( cr_type(0, 5) / ri_type(6), cr_type(0, ri_type(5, 6)) ); - BOOST_CHECK_EQUAL( cr_type(3, 7) / ri_type(6), cr_type(3, ri_type(7, 6)) ); - BOOST_CHECK_THROW( cr_type(0, 5) / ri_type(0), bad_rational ); - BOOST_CHECK_THROW( cr_type(3, 7) / ri_type(0), bad_rational ); - - BOOST_CHECK_EQUAL( 2 * nce(0), ci_type(0, 2) ); - BOOST_CHECK_EQUAL( nce(0) * 2, ci_type(0, 2) ); - BOOST_CHECK_EQUAL( 2 * nce(5), ci_type(5, 2) ); - BOOST_CHECK_EQUAL( nce(5) * 2, ci_type(5, 2) ); - BOOST_CHECK_EQUAL( 2 * nce(0, true), ci_type(0, -2) ); - BOOST_CHECK_EQUAL( nce(0, true) * 2, ci_type(0, -2) ); - BOOST_CHECK_EQUAL( 2 * nce(5, true), ci_type(5, -2) ); - BOOST_CHECK_EQUAL( nce(5, true) * 2, ci_type(5, -2) ); - - BOOST_CHECK_EQUAL( ri_type(5) / nce(0), cr_type(0, ri_type(5)) ); - BOOST_CHECK_EQUAL( ri_type(5) / nce(0, true), cr_type(0, ri_type(-5)) ); - BOOST_CHECK_EQUAL( nce(0) / ri_type(5), cr_type(0, ri_type(1, 5)) ); - BOOST_CHECK_EQUAL( nce(0, true) / ri_type(5), cr_type(0, ri_type(-1, 5)) ); - BOOST_CHECK_THROW( nce(0) / ri_type(0), bad_rational ); - BOOST_CHECK_THROW( nce(0, true) / ri_type(0), bad_rational ); - BOOST_CHECK_EQUAL( ri_type(8) / nce(7), cr_type(7, ri_type(-8)) ); - BOOST_CHECK_EQUAL( ri_type(8) / nce(7, true), cr_type(7, ri_type(8)) ); - BOOST_CHECK_EQUAL( nce(7) / ri_type(8), cr_type(7, ri_type(1, 8)) ); - BOOST_CHECK_EQUAL( nce(7, true) / ri_type(8), cr_type(7, ri_type(-1, 8)) ); - BOOST_CHECK_THROW( nce(7) / ri_type(0), bad_rational ); - BOOST_CHECK_THROW( nce(7, true) / ri_type(0), bad_rational ); - - BOOST_CHECK_EQUAL( ri_type(7) / ce(0), cr_type(0, ri_type(7)) ); - BOOST_CHECK_EQUAL( ce(0) / ri_type(7), cr_type(0, ri_type(1, 7)) ); - BOOST_CHECK_THROW( ce(0) / ri_type(0), bad_rational ); - BOOST_CHECK_EQUAL( ri_type(9) / ce(11), cr_type(11, ri_type(-9)) ); - BOOST_CHECK_EQUAL( ce(11) / ri_type(9), cr_type(11, ri_type(1, 9)) ); - BOOST_CHECK_THROW( ce(11) / ri_type(0), bad_rational ); -} - -// Element multiplication and division -BOOST_AUTO_TEST_CASE( cayley_element_muldiv_op_test ) -{ - using std::size_t; - using boost::bad_rational; - - // basic elements - size_t const top_rung = 4u, top_index = ( 1u << top_rung ); - nce const results[ top_index ][ top_index ] = - { - { +ce( 0), +ce( 1), +ce( 2), +ce( 3), +ce( 4), +ce( 5), +ce( 6), - +ce( 7), +ce( 8), +ce( 9), +ce(10), +ce(11), +ce(12), +ce(13), - +ce(14), +ce(15) }, - { +ce( 1), -ce( 0), +ce( 3), -ce( 2), +ce( 5), -ce( 4), -ce( 7), - +ce( 6), +ce( 9), -ce( 8), -ce(11), +ce(10), -ce(13), +ce(12), - +ce(15), -ce(14) }, - { +ce( 2), -ce( 3), -ce( 0), +ce( 1), +ce( 6), +ce( 7), -ce( 4), - -ce( 5), +ce(10), +ce(11), -ce( 8), -ce( 9), -ce(14), -ce(15), - +ce(12), +ce(13) }, - { +ce( 3), +ce( 2), -ce( 1), -ce( 0), +ce( 7), -ce( 6), +ce( 5), - -ce( 4), +ce(11), -ce(10), +ce( 9), -ce( 8), -ce(15), +ce(14), - -ce(13), +ce(12) }, - { +ce( 4), -ce( 5), -ce( 6), -ce( 7), -ce( 0), +ce( 1), +ce( 2), - +ce( 3), +ce(12), +ce(13), +ce(14), +ce(15), -ce( 8), -ce( 9), - -ce(10), -ce(11) }, - { +ce( 5), +ce( 4), -ce( 7), +ce( 6), -ce( 1), -ce( 0), -ce( 3), - +ce( 2), +ce(13), -ce(12), +ce(15), -ce(14), +ce( 9), -ce( 8), - +ce(11), -ce(10) }, - { +ce( 6), +ce( 7), +ce( 4), -ce( 5), -ce( 2), +ce( 3), -ce( 0), - -ce( 1), +ce(14), -ce(15), -ce(12), +ce(13), +ce(10), -ce(11), - -ce( 8), +ce( 9) }, - { +ce( 7), -ce( 6), +ce( 5), +ce( 4), -ce( 3), -ce( 2), +ce( 1), - -ce( 0), +ce(15), +ce(14), -ce(13), -ce(12), +ce(11), +ce(10), - -ce( 9), -ce( 8) }, - { +ce( 8), -ce( 9), -ce(10), -ce(11), -ce(12), -ce(13), -ce(14), - -ce(15), -ce( 0), +ce( 1), +ce( 2), +ce( 3), +ce( 4), +ce( 5), - +ce( 6), +ce( 7) }, - { +ce( 9), +ce( 8), -ce(11), +ce(10), -ce(13), +ce(12), +ce(15), - -ce(14), -ce( 1), -ce( 0), -ce( 3), +ce( 2), -ce( 5), +ce( 4), - +ce( 7), -ce( 6) }, - { +ce(10), +ce(11), +ce( 8), -ce( 9), -ce(14), -ce(15), +ce(12), - +ce(13), -ce( 2), +ce( 3), -ce( 0), -ce( 1), -ce( 6), -ce( 7), - +ce( 4), +ce( 5) }, - { +ce(11), -ce(10), +ce( 9), +ce( 8), -ce(15), +ce(14), -ce(13), - +ce(12), -ce( 3), -ce( 2), +ce( 1), -ce( 0), -ce( 7), +ce( 6), - -ce( 5), +ce( 4) }, - { +ce(12), +ce(13), +ce(14), +ce(15), +ce( 8), -ce( 9), -ce(10), - -ce(11), -ce( 4), +ce( 5), +ce( 6), +ce( 7), -ce( 0), -ce( 1), - -ce( 2), -ce( 3) }, - { +ce(13), -ce(12), +ce(15), -ce(14), +ce( 9), +ce( 8), +ce(11), - -ce(10), -ce( 5), -ce( 4), +ce( 7), -ce( 6), +ce( 1), -ce( 0), - +ce( 3), -ce( 2) }, - { +ce(14), -ce(15), -ce(12), +ce(13), +ce(10), -ce(11), +ce( 8), - +ce( 9), -ce( 6), -ce( 7), -ce( 4), +ce( 5), +ce( 2), -ce( 3), - -ce( 0), +ce( 1) }, - { +ce(15), +ce(14), -ce(13), -ce(12), +ce(11), +ce(10), -ce( 9), - +ce( 8), -ce( 7), +ce( 6), -ce( 5), -ce( 4), +ce( 3), +ce( 2), - -ce( 1), -ce( 0) } - }; - - for ( size_t i = 0 ; i < top_index ; ++i ) - for ( size_t j = 0 ; j < top_index ; ++j ) - BOOST_CHECK_MESSAGE( ce(i) * ce(j) == results[i][j], "e_" << i - << " * e_" << j << " -> " << ((ce(i) * ce(j)).negative() ? '-' - : '+') << "e_" << (ce(i) * ce(j)).basis() << " != " - << (results[i][j].negative() ? '-' : '+') << "e_" - << results[i][j].basis() << '.' ); - - // signed elements - BOOST_CHECK_EQUAL( +nce(0) * +nce(0), +nce(0) ); - BOOST_CHECK_EQUAL( -nce(0) * -nce(0), +nce(0) ); - BOOST_CHECK_EQUAL( -nce(0) * +nce(0), -nce(0) ); - BOOST_CHECK_EQUAL( +nce(0) * -nce(0), -nce(0) ); - - BOOST_CHECK_EQUAL( +nce(1) * +nce(1), -nce(0) ); - BOOST_CHECK_EQUAL( -nce(1) * -nce(1), -nce(0) ); - BOOST_CHECK_EQUAL( -nce(1) * +nce(1), +nce(0) ); - BOOST_CHECK_EQUAL( +nce(1) * -nce(1), +nce(0) ); - - BOOST_CHECK_EQUAL( +nce(0) * +nce(1), +nce(1) ); - BOOST_CHECK_EQUAL( -nce(0) * -nce(1), +nce(1) ); - BOOST_CHECK_EQUAL( -nce(0) * +nce(1), -nce(1) ); - BOOST_CHECK_EQUAL( +nce(0) * -nce(1), -nce(1) ); - BOOST_CHECK_EQUAL( +nce(1) * +nce(0), +nce(1) ); - BOOST_CHECK_EQUAL( -nce(1) * -nce(0), +nce(1) ); - BOOST_CHECK_EQUAL( -nce(1) * +nce(0), -nce(1) ); - BOOST_CHECK_EQUAL( +nce(1) * -nce(0), -nce(1) ); - - BOOST_CHECK_EQUAL( +nce(2) * +nce(5), +nce(7) ); - BOOST_CHECK_EQUAL( -nce(2) * -nce(5), +nce(7) ); - BOOST_CHECK_EQUAL( -nce(2) * +nce(5), -nce(7) ); - BOOST_CHECK_EQUAL( +nce(2) * -nce(5), -nce(7) ); - BOOST_CHECK_EQUAL( +nce(5) * +nce(2), -nce(7) ); - BOOST_CHECK_EQUAL( -nce(5) * -nce(2), -nce(7) ); - BOOST_CHECK_EQUAL( -nce(5) * +nce(2), +nce(7) ); - BOOST_CHECK_EQUAL( +nce(5) * -nce(2), +nce(7) ); - - // scaled elements - BOOST_CHECK_EQUAL( ci_type(0, 2) * ci_type(0, 3), 6 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(0, -7) * ci_type(0, -5), 35 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(0, -5) * ci_type(0, 3), -15 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(0, 11) * ci_type(0, -2), -22 * ce(0) ); - - BOOST_CHECK_EQUAL( ci_type(1, 3) * ci_type(1, 7), -21 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(1, -2) * ci_type(1, -5), -10 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(1, -11) * ci_type(1, 13), 143 * ce(0) ); - BOOST_CHECK_EQUAL( ci_type(1, 7) * ci_type(1, -3), 21 * ce(0) ); - - BOOST_CHECK_EQUAL( ci_type(0, 2) * ci_type(1, 19), 38 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(0, -3) * ci_type(1, -17), 51 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(0, -5) * ci_type(1, 13), -65 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(0, 7) * ci_type(1, -11), -77 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(1, 11) * ci_type(0, 7), 77 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(1, -13) * ci_type(0, -5), 65 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(1, -17) * ci_type(0, 3), -51 * ce(1) ); - BOOST_CHECK_EQUAL( ci_type(1, 19) * ci_type(0, -2), -38 * ce(1) ); - - BOOST_CHECK_EQUAL( ci_type(2, 23) * ci_type(5, 53), 1219 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(2, -29) * ci_type(5, -47), 1363 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(2, -31) * ci_type(5, 43), -1333 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(2, 37) * ci_type(5, -41), -1517 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(5, 41) * ci_type(2, 37), -1517 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(5, -43) * ci_type(2, -31), -1333 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(5, -47) * ci_type(2, 29), 1363 * ce(7) ); - BOOST_CHECK_EQUAL( ci_type(5, 53) * ci_type(2, -23), 1219 * ce(7) ); - - // division - BOOST_CHECK_EQUAL( ce(0) / ce(0), ce(0) ); - BOOST_CHECK_EQUAL( ce(1) / ce(0), ce(1) ); - BOOST_CHECK_EQUAL( ce(0) / ce(1), -ce(1) ); - BOOST_CHECK_EQUAL( nce(2, true) / nce(3, false), nce(1, false) ); - BOOST_CHECK_EQUAL( cr_type(14, 7) / cr_type(9, -4), cr_type(7, - ri_type( -7, 4 )) ); - - BOOST_CHECK_THROW( cr_type(9, 7) / cr_type(1, 0), bad_rational ); - - // mixed-type operations - BOOST_CHECK_EQUAL( ce(0) * nce(0, true), nce(0, true) ); - BOOST_CHECK_EQUAL( nce(1, true) * ce(1), nce(0) ); - BOOST_CHECK_EQUAL( ce(2) * ci_type(2, 5), ci_type(0, -5) ); - BOOST_CHECK_EQUAL( ci_type(3, -7) * ce(4), ci_type(7, -7) ); - BOOST_CHECK_EQUAL( nce(5) * ci_type(7, 10), ci_type(2, 10) ); - BOOST_CHECK_EQUAL( ci_type(9, -11) * nce(14, true), ci_type(7, 11) ); - - BOOST_CHECK_EQUAL( ce(0) / nce(0, true), nce(0, true) ); - BOOST_CHECK_EQUAL( nce(1, true) / ce(1), nce(0, true) ); - BOOST_CHECK_EQUAL( ce(2) / cr_type(2, 5), cr_type(0, ri_type(1, 5)) ); - BOOST_CHECK_EQUAL( cr_type(3, -7) / ce(4), cr_type(7, 7) ); - BOOST_CHECK_EQUAL( nce(5) / cr_type(7, 10), cr_type(2, ri_type(1, -10)) ); - BOOST_CHECK_EQUAL( cr_type(9, -11) / nce(14, true), cr_type(7, -11) ); - - BOOST_CHECK_THROW( ce(7) / cr_type(3, 0), bad_rational ); - BOOST_CHECK_THROW( nce(11, true) / cr_type(0, 0), bad_rational ); -} - -// Scalar multiplication and division member functions -BOOST_AUTO_TEST_CASE( cayley_scalar_muldiv_self_test ) -{ - using boost::bad_rational; - - ci_type const c1( 0, 0 ), c2( 0, 2 ), c3( 0, -3 ); - ci_type const c4( 1, 0 ), c5( 2, 5 ), c6( 3, -7 ); - ci_type t1 = c1, t2 = c2, t3 = c3, t4 = c4, t5 = c5, t6 = c6; - - t1 *= 11; BOOST_CHECK_EQUAL( t1, c1 * 11 ); - t2 *= 11; BOOST_CHECK_EQUAL( t2, c2 * 11 ); - t3 *= 11; BOOST_CHECK_EQUAL( t3, c3 * 11 ); - t4 *= 11; BOOST_CHECK_EQUAL( t4, c4 * 11 ); - t5 *= 11; BOOST_CHECK_EQUAL( t5, c5 * 11 ); - t6 *= 11; BOOST_CHECK_EQUAL( t6, c6 * 11 ); - - t1 *= 0; BOOST_CHECK_EQUAL( t1, ci_type(0) ); - t2 *= 0; BOOST_CHECK_EQUAL( t2, ci_type(0) ); - t3 *= 0; BOOST_CHECK_EQUAL( t3, ci_type(0) ); - t4 *= 0; BOOST_CHECK_EQUAL( t4, ci_type(0) ); - t5 *= 0; BOOST_CHECK_EQUAL( t5, ci_type(0) ); - t6 *= 0; BOOST_CHECK_EQUAL( t6, ci_type(0) ); - - cr_type const c1a(c1), c2a(c2), c3a(c3), c4a(c4), c5a(c5), c6a(c6); - cr_type t1a = c1a, t2a = c2a, t3a = c3a; - cr_type t4a = c4a, t5a = c5a, t6a = c6a; - - t1a /= ri_type(-13); BOOST_CHECK_EQUAL( t1a, c1a / ri_type(-13) ); - t2a /= ri_type(-13); BOOST_CHECK_EQUAL( t2a, c2a / ri_type(-13) ); - t3a /= ri_type(-13); BOOST_CHECK_EQUAL( t3a, c3a / ri_type(-13) ); - t4a /= ri_type(-13); BOOST_CHECK_EQUAL( t4a, c4a / ri_type(-13) ); - t5a /= ri_type(-13); BOOST_CHECK_EQUAL( t5a, c5a / ri_type(-13) ); - t6a /= ri_type(-13); BOOST_CHECK_EQUAL( t6a, c6a / ri_type(-13) ); - - BOOST_CHECK_THROW( t1a /= ri_type(0), bad_rational ); - BOOST_CHECK_THROW( t2a /= ri_type(0), bad_rational ); - BOOST_CHECK_THROW( t3a /= ri_type(0), bad_rational ); - BOOST_CHECK_THROW( t4a /= ri_type(0), bad_rational ); - BOOST_CHECK_THROW( t5a /= ri_type(0), bad_rational ); - BOOST_CHECK_THROW( t6a /= ri_type(0), bad_rational ); -} - -// Element multiplication and division member functions -BOOST_AUTO_TEST_CASE( cayley_element_muldiv_self_test ) -{ - // signed elements - nce const c1( 0 ), c2( 0, true ), c3( 1 ), c4( 2, true ); - nce t1 = c1, t2 = c2, t3 = c3, t4 = c4; - - t1 *= c3; BOOST_CHECK_EQUAL( t1, c1 * c3 ); - t2 *= c4; BOOST_CHECK_EQUAL( t2, c2 * c4 ); - t3 *= c1; BOOST_CHECK_EQUAL( t3, c3 * c1 ); - t4 *= c2; BOOST_CHECK_EQUAL( t4, c4 * c2 ); - - t1 = c1; t1 /= c4; BOOST_CHECK_EQUAL( t1, c1 / c4 ); - t2 = c2; t2 /= c3; BOOST_CHECK_EQUAL( t2, c2 / c3 ); - t3 = c3; t3 /= c2; BOOST_CHECK_EQUAL( t3, c3 / c2 ); - t4 = c4; t4 /= c1; BOOST_CHECK_EQUAL( t4, c4 / c1 ); - - // scaled elements - cr_type const c5( 0, 15 ), c6( 1, -4 ); - cr_type const c7( 2, ri_type(-11, 7) ), c8( 3, ri_type(2, 3) ); - cr_type t5 = c5, t6 = c6, t7 = c7, t8 = c8; - - t5 *= c7; BOOST_CHECK_EQUAL( t5, c5 * c7 ); - t6 *= c8; BOOST_CHECK_EQUAL( t6, c6 * c8 ); - t7 *= c5; BOOST_CHECK_EQUAL( t7, c7 * c5 ); - t8 *= c6; BOOST_CHECK_EQUAL( t8, c8 * c6 ); - - t5 = c5; t5 /= c8; BOOST_CHECK_EQUAL( t5, c5 / c8 ); - t6 = c6; t6 /= c7; BOOST_CHECK_EQUAL( t6, c6 / c7 ); - t7 = c7; t7 /= c6; BOOST_CHECK_EQUAL( t7, c7 / c6 ); - t8 = c8; t8 /= c5; BOOST_CHECK_EQUAL( t8, c8 / c5 ); - - BOOST_CHECK_THROW( t7 /= cr_type(5, 0), boost::bad_rational ); - - // mixed elements (via conversion) - t2 = c2; BOOST_CHECK_EQUAL( t2 /= ce(1), c2 / ce(1) ); - t3 = c3; BOOST_CHECK_EQUAL( t3 *= ce(2), c3 * ce(2) ); - t6 = c6; BOOST_CHECK_EQUAL( t6 /= ce(3), c6 / ce(3) ); - t7 = c7; BOOST_CHECK_EQUAL( t7 *= ce(5), c7 * ce(5) ); - - t5 = c5; BOOST_CHECK_EQUAL( t5 /= nce(7), c5 / nce(7) ); - t5 = c5; BOOST_CHECK_EQUAL( t5 *= nce(11, true), c5 * nce(11, true) ); - t8 = c8; BOOST_CHECK_EQUAL( t8 /= ce(13), c8 / ce(13) ); - t8 = c8; BOOST_CHECK_EQUAL( t8 *= ce(17), c8 * ce(17) ); -} - -// Shifting -BOOST_AUTO_TEST_CASE( cayley_element_shift_op_test ) -{ - // basic elements - BOOST_CHECK_EQUAL( ce(0) << 2, ce(2) ); - BOOST_CHECK_EQUAL( ce(2) << 3, ce(5) ); - - BOOST_CHECK_EQUAL( ce(7) >> 5, ce(2) ); - BOOST_CHECK_EQUAL( ce(0) >> 0, ce(0) ); - - // signed elements - BOOST_CHECK_EQUAL( nce(0) << 5, nce(5) ); - BOOST_CHECK_EQUAL( nce(3, true) << 11, nce(14, true) ); - - BOOST_CHECK_EQUAL( nce(11, true) >> 7, nce(4, true) ); - BOOST_CHECK_EQUAL( nce( 5) >> 2, nce(3) ); - - // scaled elements - BOOST_CHECK_EQUAL( ci_type(0, 2) << 5, ci_type( 5, 2) ); - BOOST_CHECK_EQUAL( ci_type(5, -1) << 7, ci_type(12, -1) ); - - BOOST_CHECK_EQUAL( ci_type(17, -20) >> 11, ci_type(6, -20) ); - BOOST_CHECK_EQUAL( ci_type( 3, 101) >> 1, ci_type(2, 101) ); -} - -// Shifting member functions -BOOST_AUTO_TEST_CASE( cayley_element_shift_self_test ) -{ - // basic elements - ce const c1( 0 ), c2( 1 ), c3( 10 ); - ce t1 = c1, t2 = c2, t3 = c3; - - t1 <<= 3; BOOST_CHECK_EQUAL( t1, c1 << 3 ); - t2 <<= 5; BOOST_CHECK_EQUAL( t2, c2 << 5 ); - t3 <<= 7; BOOST_CHECK_EQUAL( t3, c3 << 7 ); - - t1 = c1; t1 >>= 0; BOOST_CHECK_EQUAL( t1, c1 >> 0 ); - t2 = c2; t2 >>= 1; BOOST_CHECK_EQUAL( t2, c2 >> 1 ); - t3 = c3; t3 >>= 7; BOOST_CHECK_EQUAL( t3, c3 >> 7 ); - - // signed elements - nce const c4( 0, true ), c5( 3 ), c6( 11, true ); - nce t4 = c4, t5 = c5, t6 = c6; - - t4 <<= 3; BOOST_CHECK_EQUAL( t4, c4 << 3 ); - t5 <<= 5; BOOST_CHECK_EQUAL( t5, c5 << 5 ); - t6 <<= 7; BOOST_CHECK_EQUAL( t6, c6 << 7 ); - - t4 = c4; t4 >>= 0; BOOST_CHECK_EQUAL( t4, c4 >> 0 ); - t5 = c5; t5 >>= 2; BOOST_CHECK_EQUAL( t5, c5 >> 2 ); - t6 = c6; t6 >>= 5; BOOST_CHECK_EQUAL( t6, c6 >> 5 ); - - // scaled elements - cr_type const c7( 0, ri_type(2, 5) ), c8( 5, -3 ), c9( 13, 20 ); - cr_type t7 = c7, t8 = c8, t9 = c9; - - t7 <<= 3; BOOST_CHECK_EQUAL( t7, c7 << 3 ); - t8 <<= 5; BOOST_CHECK_EQUAL( t8, c8 << 5 ); - t9 <<= 7; BOOST_CHECK_EQUAL( t9, c9 << 7 ); - - t7 = c7; t7 >>= 0; BOOST_CHECK_EQUAL( t7, c7 >> 0 ); - t8 = c8; t8 >>= 3; BOOST_CHECK_EQUAL( t8, c8 >> 3 ); - t9 = c9; t9 >>= 7; BOOST_CHECK_EQUAL( t9, c9 >> 7 ); -} - -// Component functions -BOOST_AUTO_TEST_CASE( cayley_element_component_test ) -{ - ce const c1( 0 ), c2( 1 ), c3( 2 ); - nce const c4( 0 ), c5( 1, true ), c6( 3 ); - cr_type const c7( 0, 4 ), c8( 1, -3 ), c9( 5, ri_type(2, 7) ); - - // real component - BOOST_CHECK_EQUAL( real(c1), 1 ); - BOOST_CHECK_EQUAL( real(c2), 0 ); - BOOST_CHECK_EQUAL( real(c3), 0 ); - BOOST_CHECK_EQUAL( real(c4), 1 ); BOOST_CHECK_EQUAL( real(-c4), -1 ); - BOOST_CHECK_EQUAL( real(c5), 0 ); BOOST_CHECK_EQUAL( real(-c5), 0 ); - BOOST_CHECK_EQUAL( real(c6), 0 ); BOOST_CHECK_EQUAL( real(-c6), 0 ); - BOOST_CHECK_EQUAL( real(c7), 4 ); - BOOST_CHECK_EQUAL( real(c8), 0 ); - BOOST_CHECK_EQUAL( real(c9), 0 ); - - // (complex) imaginary component - BOOST_CHECK_EQUAL( imag(c1), 0 ); - BOOST_CHECK_EQUAL( imag(c2), 1 ); - BOOST_CHECK_EQUAL( imag(c3), 0 ); - BOOST_CHECK_EQUAL( imag(c4), 0 ); BOOST_CHECK_EQUAL( imag(-c4), 0 ); - BOOST_CHECK_EQUAL( imag(c5), -1 ); BOOST_CHECK_EQUAL( imag(-c5), 1 ); - BOOST_CHECK_EQUAL( imag(c6), 0 ); BOOST_CHECK_EQUAL( imag(-c6), 0 ); - BOOST_CHECK_EQUAL( imag(c7), 0 ); - BOOST_CHECK_EQUAL( imag(c8), -3 ); - BOOST_CHECK_EQUAL( imag(c9), 0 ); - - // unreal components - BOOST_CHECK_EQUAL( unreal(c9), c9 ); - BOOST_CHECK_EQUAL( unreal(c8), c8 ); - BOOST_CHECK_EQUAL( unreal(c7), cr_type() ); - BOOST_CHECK_EQUAL( unreal(c6), ci_type(c6) ); - BOOST_CHECK_EQUAL( unreal(c5), ci_type(c5) ); - BOOST_CHECK_EQUAL( unreal(c4), ci_type() ); - BOOST_CHECK_EQUAL( unreal(c3), ci_type(c3) ); - BOOST_CHECK_EQUAL( unreal(c2), ci_type(c2) ); - BOOST_CHECK_EQUAL( unreal(c1), ci_type() ); -} - -// Element to integer scalar power -BOOST_AUTO_TEST_CASE( cayley_element_integer_power_test ) -{ - ce const c1( 0 ), c2( 1 ), c3( 2 ); - nce const c1a( c1 ), c2a( c2 ), c3a( c3 ), c4 = -c1, c5 = -c2, c6 = -c3; - - BOOST_CHECK_EQUAL( pow(c1, 0), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, 1), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, 2), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, 3), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, 4), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, -1), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, -2), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, -3), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1, -4), ce(0) ); - - BOOST_CHECK_EQUAL( pow(c2, 0), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c2, 1), +ce(1) ); - BOOST_CHECK_EQUAL( pow(c2, 2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c2, 3), -ce(1) ); - BOOST_CHECK_EQUAL( pow(c2, 4), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c2, -1), -ce(1) ); - BOOST_CHECK_EQUAL( pow(c2, -2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c2, -3), +ce(1) ); - BOOST_CHECK_EQUAL( pow(c2, -4), +ce(0) ); - - BOOST_CHECK_EQUAL( pow(c3, 0), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c3, 1), +ce(2) ); - BOOST_CHECK_EQUAL( pow(c3, 2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c3, 3), -ce(2) ); - BOOST_CHECK_EQUAL( pow(c3, 4), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c3, -1), -ce(2) ); - BOOST_CHECK_EQUAL( pow(c3, -2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c3, -3), +ce(2) ); - BOOST_CHECK_EQUAL( pow(c3, -4), +ce(0) ); - - BOOST_CHECK_EQUAL( pow(c1a, 0), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, 1), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, 2), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, 3), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, 4), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, -1), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, -2), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, -3), ce(0) ); - BOOST_CHECK_EQUAL( pow(c1a, -4), ce(0) ); - - BOOST_CHECK_EQUAL( pow(c2a, 0), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c2a, 1), +ce(1) ); - BOOST_CHECK_EQUAL( pow(c2a, 2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c2a, 3), -ce(1) ); - BOOST_CHECK_EQUAL( pow(c2a, 4), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c2a, -1), -ce(1) ); - BOOST_CHECK_EQUAL( pow(c2a, -2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c2a, -3), +ce(1) ); - BOOST_CHECK_EQUAL( pow(c2a, -4), +ce(0) ); - - BOOST_CHECK_EQUAL( pow(c3a, 0), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c3a, 1), +ce(2) ); - BOOST_CHECK_EQUAL( pow(c3a, 2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c3a, 3), -ce(2) ); - BOOST_CHECK_EQUAL( pow(c3a, 4), +ce(0) ); - BOOST_CHECK_EQUAL( pow(c3a, -1), -ce(2) ); - BOOST_CHECK_EQUAL( pow(c3a, -2), -ce(0) ); - BOOST_CHECK_EQUAL( pow(c3a, -3), +ce(2) ); - BOOST_CHECK_EQUAL( pow(c3a, -4), +ce(0) ); - - BOOST_CHECK_EQUAL( pow(c4, 0), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, 1), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, 2), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, 3), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, 4), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, -1), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, -2), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, -3), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c4, -4), +nce(0) ); - - BOOST_CHECK_EQUAL( pow(c5, 0), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c5, 1), -nce(1) ); - BOOST_CHECK_EQUAL( pow(c5, 2), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c5, 3), +nce(1) ); - BOOST_CHECK_EQUAL( pow(c5, 4), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c5, -1), +nce(1) ); - BOOST_CHECK_EQUAL( pow(c5, -2), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c5, -3), -nce(1) ); - BOOST_CHECK_EQUAL( pow(c5, -4), +nce(0) ); - - BOOST_CHECK_EQUAL( pow(c6, 0), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c6, 1), -nce(2) ); - BOOST_CHECK_EQUAL( pow(c6, 2), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c6, 3), +nce(2) ); - BOOST_CHECK_EQUAL( pow(c6, 4), +nce(0) ); - BOOST_CHECK_EQUAL( pow(c6, -1), +nce(2) ); - BOOST_CHECK_EQUAL( pow(c6, -2), -nce(0) ); - BOOST_CHECK_EQUAL( pow(c6, -3), -nce(2) ); - BOOST_CHECK_EQUAL( pow(c6, -4), +nce(0) ); - - cd_type const c7( 0, 2.0 ), c8( 1, -3.0 ), c9( 2, 5.0 ); - cd_type r; - - r = pow( c7, 0 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0, 0.1 ); - r = pow( c7, 1 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 2.0, 0.1 ); - r = pow( c7, 2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 4.0, 0.1 ); - r = pow( c7, 3 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 8.0, 0.1 ); - r = pow( c7, 4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 16.0, 0.1 ); - r = pow( c7, -1 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 0.5, 0.1 ); - r = pow( c7, -2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 0.25, 0.1 ); - r = pow( c7, -3 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 0.125, 0.1 ); - r = pow( c7, -4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 0.0625, 0.1 ); - - r = pow( c8, 0 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0, 0.1 ); - r = pow( c8, 1 ); BOOST_CHECK_EQUAL( r.basis(), 1u ); - BOOST_CHECK_CLOSE( r.scale(), -3.0, 0.1 ); - r = pow( c8, 2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), -9.0, 0.1 ); - r = pow( c8, 3 ); BOOST_CHECK_EQUAL( r.basis(), 1u ); - BOOST_CHECK_CLOSE( r.scale(), 27.0, 0.1 ); - r = pow( c8, 4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 81.0, 0.1 ); - r = pow( c8, -1 ); BOOST_CHECK_EQUAL( r.basis(), 1u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0 / 3.0, 0.1 ); - r = pow( c8, -2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0 / -9.0, 0.1 ); - r = pow( c8, -3 ); BOOST_CHECK_EQUAL( r.basis(), 1u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0 / -27.0, 0.1 ); - r = pow( c8, -4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0 / 81.0, 0.1 ); - - r = pow( c9, 0 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 1.0, 0.1 ); - r = pow( c9, 1 ); BOOST_CHECK_EQUAL( r.basis(), 2u ); - BOOST_CHECK_CLOSE( r.scale(), 5.0, 0.1 ); - r = pow( c9, 2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), -25.0, 0.1 ); - r = pow( c9, 3 ); BOOST_CHECK_EQUAL( r.basis(), 2u ); - BOOST_CHECK_CLOSE( r.scale(), -125.0, 0.1 ); - r = pow( c9, 4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 625.0, 0.1 ); - r = pow( c9, -1 ); BOOST_CHECK_EQUAL( r.basis(), 2u ); - BOOST_CHECK_CLOSE( r.scale(), -0.2, 0.1 ); - r = pow( c9, -2 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), -0.04, 0.1 ); - r = pow( c9, -3 ); BOOST_CHECK_EQUAL( r.basis(), 2u ); - BOOST_CHECK_CLOSE( r.scale(), 0.008, 0.1 ); - r = pow( c9, -4 ); BOOST_CHECK_EQUAL( r.basis(), 0u ); - BOOST_CHECK_CLOSE( r.scale(), 0.0016, 0.1 ); -} diff --git a/test/cbrt_data.ipp b/test/cbrt_data.ipp new file mode 100644 index 000000000..3dc156128 --- /dev/null +++ b/test/cbrt_data.ipp @@ -0,0 +1,90 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 85> cbrt_data = { + SC_(0.266297021326439287136622624529991298914e-12), SC_(0.1888421455001568264216782004847867296461e-37), + SC_(0.5920415525016708979677559909760020673275e-12), SC_(0.2075183790343685754527643464587844863336e-36), + SC_(0.155163989296047688526414276566356420517e-11), SC_(0.3735707037930203997047439019589434984292e-35), + SC_(0.326923297461201300961874949280172586441e-11), SC_(0.3494118359373226772153200620663814420149e-34), + SC_(0.3753785910581841633870681107509881258011e-11), SC_(0.5289425440452335470097113858122815653748e-34), + SC_(0.9579165585749116473834874341264367103577e-11), SC_(0.8789881934013975442181045034737061376284e-33), + SC_(0.1858167439361402273334533674642443656921e-10), SC_(0.6415854952683870221087767100660047169015e-32), + SC_(0.5449485307451595872407779097557067871094e-10), SC_(0.1618327663648593897188312867673509349544e-30), + SC_(0.6089519166696533147842274047434329986572e-10), SC_(0.2258130336429980668979991908229127823735e-30), + SC_(0.1337744776064297980155970435589551925659e-9), SC_(0.2393975994093713145816368117219309682822e-29), + SC_(0.2554458866654840676346793770790100097656e-9), SC_(0.1666850852333084475602734478539689292864e-28), + SC_(0.9285605062636648199259070679545402526855e-9), SC_(0.8006277238962806516943344887937136945761e-27), + SC_(0.1698227447555211711005540564656257629395e-8), SC_(0.4897647988639492837638733591590200680017e-26), + SC_(0.339355921141759608872234821319580078125e-8), SC_(0.3908105631910126517922937762400154556398e-25), + SC_(0.6313728651008432279922999441623687744141e-8), SC_(0.2516852352568191806987306088580024017075e-24), + SC_(0.8383264749056706932606175541877746582031e-8), SC_(0.5891685351226332190380177353564166513441e-24), + SC_(0.1962631124285962869180366396903991699219e-7), SC_(0.7559899905537899689665020666334853594435e-23), + SC_(0.5256384838503436185419559478759765625e-7), SC_(0.1452317136611010055610488165775172825142e-21), + SC_(0.116242290459922514855861663818359375e-6), SC_(0.1570697224733838212725464129330562874032e-20), + SC_(0.1776920584006802528165280818939208984375e-6), SC_(0.5610532144067232522265416539512374653302e-20), + SC_(0.246631174150024889968335628509521484375e-6), SC_(0.1500181866107243920079950771800422025379e-19), + SC_(0.7932688959044753573834896087646484375e-6), SC_(0.4991847137949555007936240315051986249607e-18), + SC_(0.1372093493046122603118419647216796875e-5), SC_(0.2583158853420963423689975850731860849521e-17), + SC_(0.214747751670074649155139923095703125e-5), SC_(0.9903435487644105337695875806797860570175e-17), + SC_(0.527022712049074470996856689453125e-5), SC_(0.1463821071995824977534730930446568768253e-15), + SC_(0.9233162927557714283466339111328125e-5), SC_(0.7871391209588633308733754268944773271037e-15), + SC_(0.269396477960981428623199462890625e-4), SC_(0.1955130454371024476239796923077328444099e-13), + SC_(0.3208058114978484809398651123046875e-4), SC_(0.3301616917426164412576388360054394990469e-13), + SC_(0.10957030463032424449920654296875e-3), SC_(0.1315462909319166691700161404508507291713e-11), + SC_(0.126518702018074691295623779296875e-3), SC_(0.2025182580848748191763613884783980516393e-11), + SC_(0.28976381872780621051788330078125e-3), SC_(0.2432945998183714075166971697772880334043e-10), + SC_(0.687857042066752910614013671875e-3), SC_(0.325457709339121122043443511174534212782e-9), + SC_(0.145484809763729572296142578125e-2), SC_(0.3079306732417723714591019095094197606751e-8), + SC_(0.2847635187208652496337890625e-2), SC_(0.2309154822558505456380682823926992940837e-7), + SC_(0.56468211114406585693359375e-2), SC_(0.1800578620431549562186637161391699604382e-6), + SC_(0.11621631681919097900390625e-1), SC_(0.1569644571431467249504618249826407884451e-5), + SC_(0.257236398756504058837890625e-1), SC_(0.1702147780446687692243218673184664750751e-4), + SC_(0.560617186129093170166015625e-1), SC_(0.1761972888887947511388016005088640519383e-3), + SC_(0.106835305690765380859375e0), SC_(0.1219394946966688262159601656951790626948e-2), + SC_(0.2401093542575836181640625e0), SC_(0.1384290502703277530255383713042680732253e-1), + SC_(0.438671648502349853515625e0), SC_(0.8441482026963079214835649797480515710291e-1), + SC_(0.903765499591827392578125e0), SC_(0.7381885006644863290177619358266650206879e0), + SC_(0.12760250568389892578125e1), SC_(0.2077674969235041602851348126190789145085e1), + SC_(0.3411548614501953125e1), SC_(0.3970586787024071934171232101107307244092e2), + SC_(0.671881103515625e1), SC_(0.3033034012472492122469702735543251037598e3), + SC_(0.802254772186279296875e1), SC_(0.5163413756551815056187668129261680860509e3), + SC_(0.264815673828125e2), SC_(0.1857081908850696527224499732255935668945e5), + SC_(0.54742523193359375e2), SC_(0.1640493194709731775162708800053223967552e6), + SC_(0.744071502685546875e2), SC_(0.4119495333434053702710286870569689199328e6), + SC_(0.210427001953125e3), SC_(0.9317607304574811554630286991596221923828e7), + SC_(0.286463409423828125e3), SC_(0.2350755546524705072036454112094361335039e8), + SC_(0.74548876953125e3), SC_(0.4143079969757992170052602887153625488281e9), + SC_(0.15343248291015625e4), SC_(0.361203491025739912996687053237110376358e10), + SC_(0.3632982421875e4), SC_(0.47950141115752447418868541717529296875e11), + SC_(0.8027111328125e4), SC_(0.517223035506184733219444751739501953125e12), + SC_(0.128921982421875e5), SC_(0.2142796483541738858568482100963592529297e13), + SC_(0.2196087890625e5), SC_(0.10591297122360160745680332183837890625e14), + SC_(0.614975859375e5), SC_(0.232580984311524292169094085693359375e15), + SC_(0.103892109375e6), SC_(0.1121366795544843797206878662109375e16), + SC_(0.2370011875e6), SC_(0.13312253103065122768310546875e17), + SC_(0.32081496875e6), SC_(0.33018996548382043793914794921875e17), + SC_(0.533606625e6), SC_(0.151937032114340403275390625e18), + SC_(0.1836336625e7), SC_(0.6192369863782824692494140625e19), + SC_(0.38194295e7), SC_(0.55717996837103384222375e20), + SC_(0.52642505e7), SC_(0.145884664571511666937625e21), + SC_(0.1527432e8), SC_(0.3563572958789165568e22), + SC_(0.25265768e8), SC_(0.16128631219129563064832e23), + SC_(0.6509808e8), SC_(0.275870040782346842112e24), + SC_(0.114023104e9), SC_(0.1482444961322159867428864e25), + SC_(0.189604896e9), SC_(0.6816299156208797501915136e25), + SC_(0.507585472e9), SC_(0.130775849541591501155074048e27), + SC_(0.764055936e9), SC_(0.446041700029835346607865856e27), + SC_(0.2103773184e10), SC_(0.9311008970618745805030293504e28), + SC_(0.3395078656e10), SC_(0.39133574711079901758452924416e29), + SC_(0.6645239808e10), SC_(0.293448554175562916132775002112e30), + SC_(0.9947638784e10), SC_(0.984373742549644225510117474304e30), + SC_(0.19561418752e11), SC_(0.7485159350422880041966143275008e31), + SC_(0.60532621312e11), SC_(0.221803524649682352197867382243328e33), + SC_(0.78978883584e11), SC_(0.492643743012832976593364088520704e33), + SC_(0.169071362048e12), SC_(0.4832926096640938859567595590254592e34), + SC_(0.345661243392e12), SC_(0.41300191319644612608733712417292288e35), + SC_(0.994912108544e12), SC_(0.984813853841992339417900620588253184e36), + SC_(0.2023890092032e13), SC_(0.8290119158315231036253469850906656768e37), + SC_(0.4372805189632e13), SC_(0.83614267463479414022236113855701843968e38), + SC_(0.5516391612416e13), SC_(0.167866976532732026957407327191402807296e39), + }; +#undef SC_ + diff --git a/test/erf_data.ipp b/test/erf_data.ipp new file mode 100644 index 000000000..b078aca1a --- /dev/null +++ b/test/erf_data.ipp @@ -0,0 +1,505 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> erf_data = { + SC_(-7.954905033111572265625), SC_(-0.9999999999999999999999999999768236114552), SC_(1.999999999999999999999999999976823611455), + SC_(-7.925852298736572265625), SC_(-0.9999999999999999999999999999631035087875), SC_(1.999999999999999999999999999963103508787), + SC_(-7.923464298248291015625), SC_(-0.9999999999999999999999999999616689085769), SC_(1.999999999999999999999999999961668908577), + SC_(-7.870497226715087890625), SC_(-0.999999999999999999999999999910929780403), SC_(1.999999999999999999999999999910929780403), + SC_(-7.809566974639892578125), SC_(-0.9999999999999999999999999997666672740312), SC_(1.999999999999999999999999999766667274031), + SC_(-7.78337383270263671875), SC_(-0.9999999999999999999999999996477997000614), SC_(1.999999999999999999999999999647799700061), + SC_(-7.75354480743408203125), SC_(-0.9999999999999999999999999994380404842296), SC_(1.99999999999999999999999999943804048423), + SC_(-7.735670566558837890625), SC_(-0.9999999999999999999999999992571019874276), SC_(1.999999999999999999999999999257101987428), + SC_(-7.715617656707763671875), SC_(-0.9999999999999999999999999989846832415537), SC_(1.999999999999999999999999998984683241554), + SC_(-7.7047863006591796875), SC_(-0.999999999999999999999999998798456908148), SC_(1.999999999999999999999999998798456908148), + SC_(-7.53247547149658203125), SC_(-0.9999999999999999999999999830308685738632), SC_(1.999999999999999999999999983030868573863), + SC_(-7.490674495697021484375), SC_(-0.9999999999999999999999999680283814121658), SC_(1.999999999999999999999999968028381412166), + SC_(-7.4677028656005859375), SC_(-0.9999999999999999999999999547824309252063), SC_(1.999999999999999999999999954782430925206), + SC_(-7.45092296600341796875), SC_(-0.9999999999999999999999999417916071317633), SC_(1.999999999999999999999999941791607131763), + SC_(-7.448862552642822265625), SC_(-0.999999999999999999999999939960596846754), SC_(1.999999999999999999999999939960596846754), + SC_(-7.428613185882568359375), SC_(-0.9999999999999999999999999186357837637381), SC_(1.999999999999999999999999918635783763738), + SC_(-7.41693973541259765625), SC_(-0.9999999999999999999999999030902896226869), SC_(1.999999999999999999999999903090289622687), + SC_(-7.352462291717529296875), SC_(-0.9999999999999999999999997466806221320354), SC_(1.999999999999999999999999746680622132035), + SC_(-7.32713413238525390625), SC_(-0.9999999999999999999999996313499781458314), SC_(1.999999999999999999999999631349978145831), + SC_(-7.311619281768798828125), SC_(-0.9999999999999999999999995363881484604713), SC_(1.999999999999999999999999536388148460471), + SC_(-7.279047489166259765625), SC_(-0.9999999999999999999999992510468758087201), SC_(1.99999999999999999999999925104687580872), + SC_(-7.261257648468017578125), SC_(-0.999999999999999999999999027617901396665), SC_(1.999999999999999999999999027617901396665), + SC_(-7.232891559600830078125), SC_(-0.9999999999999999999999985274747006541921), SC_(1.999999999999999999999998527474700654192), + SC_(-7.20552921295166015625), SC_(-0.9999999999999999999999978059072279635353), SC_(1.999999999999999999999997805907227963535), + SC_(-7.18053722381591796875), SC_(-0.9999999999999999999999968458614521146074), SC_(1.999999999999999999999996845861452114607), + SC_(-7.14955902099609375), SC_(-0.9999999999999999999999950624010083082766), SC_(1.999999999999999999999995062401008308277), + SC_(-7.13679790496826171875), SC_(-0.9999999999999999999999940645544114571544), SC_(1.999999999999999999999994064554411457154), + SC_(-7.043527126312255859375), SC_(-0.9999999999999999999999774337000610083234), SC_(1.999999999999999999999977433700061008323), + SC_(-6.9184741973876953125), SC_(-0.9999999999999999999998683679500951423272), SC_(1.999999999999999999999868367950095142327), + SC_(-6.7863311767578125), SC_(-0.9999999999999999999991795120414840889856), SC_(1.999999999999999999999179512041484088986), + SC_(-6.784533023834228515625), SC_(-0.9999999999999999999991590255661336519112), SC_(1.999999999999999999999159025566133651911), + SC_(-6.75908756256103515625), SC_(-0.9999999999999999999988086234364717344992), SC_(1.999999999999999999998808623436471734499), + SC_(-6.7491912841796875), SC_(-0.9999999999999999999986362669470340916908), SC_(1.999999999999999999998636266947034091691), + SC_(-6.701987743377685546875), SC_(-0.999999999999999999997409009041632256952), SC_(1.999999999999999999997409009041632256952), + SC_(-6.658857822418212890625), SC_(-0.9999999999999999999953604766205454848685), SC_(1.999999999999999999995360476620545484869), + SC_(-6.649026393890380859375), SC_(-0.9999999999999999999947043111854298902328), SC_(1.999999999999999999994704311185429890233), + SC_(-6.63174724578857421875), SC_(-0.9999999999999999999933213242713558320894), SC_(1.999999999999999999993321324271355832089), + SC_(-6.562829494476318359375), SC_(-0.9999999999999999999832486398825313868824), SC_(1.999999999999999999983248639882531386882), + SC_(-6.474317073822021484375), SC_(-0.9999999999999999999461767620323245787117), SC_(1.999999999999999999946176762032324578712), + SC_(-6.456727504730224609375), SC_(-0.9999999999999999999322505846330688168028), SC_(1.999999999999999999932250584633068816803), + SC_(-6.44589138031005859375), SC_(-0.9999999999999999999219560784452381988458), SC_(1.999999999999999999921956078445238198846), + SC_(-6.439353466033935546875), SC_(-0.9999999999999999999150123383981949772306), SC_(1.999999999999999999915012338398194977231), + SC_(-6.388184070587158203125), SC_(-0.9999999999999999998348790206445813875893), SC_(1.999999999999999999834879020644581387589), + SC_(-6.293555736541748046875), SC_(-0.9999999999999999994437019971919548391819), SC_(1.999999999999999999443701997191954839182), + SC_(-6.2710094451904296875), SC_(-0.9999999999999999992589325173780715980195), SC_(1.99999999999999999925893251737807159802), + SC_(-6.242211818695068359375), SC_(-0.9999999999999999989326493643877411691996), SC_(1.9999999999999999989326493643877411692), + SC_(-6.22209262847900390625), SC_(-0.999999999999999998624109280849850203527), SC_(1.999999999999999998624109280849850203527), + SC_(-6.220756053924560546875), SC_(-0.9999999999999999986007425210281758251292), SC_(1.999999999999999998600742521028175825129), + SC_(-6.200567722320556640625), SC_(-0.9999999999999999981962301352174253788249), SC_(1.999999999999999998196230135217425378825), + SC_(-6.188919544219970703125), SC_(-0.9999999999999999979123729253091532478511), SC_(1.999999999999999997912372925309153247851), + SC_(-6.121317386627197265625), SC_(-0.9999999999999999951502004864042604960246), SC_(1.999999999999999995150200486404260496025), + SC_(-6.0960369110107421875), SC_(-0.999999999999999993368492031514234286194), SC_(1.999999999999999993368492031514234286194), + SC_(-6.08724498748779296875), SC_(-0.9999999999999999926083315666932664549236), SC_(1.999999999999999992608331566693266454924), + SC_(-6.026896953582763671875), SC_(-0.9999999999999999844955556006137776564529), SC_(1.999999999999999984495555600613777656453), + SC_(-5.9970760345458984375), SC_(-0.9999999999999999777013911828554459912853), SC_(1.999999999999999977701391182855445991285), + SC_(-5.98565387725830078125), SC_(-0.9999999999999999743831687415390670278399), SC_(1.99999999999999997438316874153906702784), + SC_(-5.96821117401123046875), SC_(-0.999999999999999968354398455085434720569), SC_(1.999999999999999968354398455085434720569), + SC_(-5.92245578765869140625), SC_(-0.9999999999999999450648430800723663224774), SC_(1.999999999999999945064843080072366322477), + SC_(-5.921500682830810546875), SC_(-0.9999999999999999444311537798288144041217), SC_(1.999999999999999944431153779828814404122), + SC_(-5.888427257537841796875), SC_(-0.9999999999999999174281958873362911444154), SC_(1.999999999999999917428195887336291144415), + SC_(-5.87206363677978515625), SC_(-0.9999999999999998996343433163177337705717), SC_(1.999999999999999899634343316317733770572), + SC_(-5.860218048095703125), SC_(-0.9999999999999998844434106367807360220097), SC_(1.99999999999999988444341063678073602201), + SC_(-5.83236789703369140625), SC_(-0.9999999999999998392204407202278816953224), SC_(1.999999999999999839220440720227881695322), + SC_(-5.822903156280517578125), SC_(-0.9999999999999998201851297606348078114632), SC_(1.999999999999999820185129760634807811463), + SC_(-5.7919788360595703125), SC_(-0.9999999999999997411398900200191738483065), SC_(1.999999999999999741139890020019173848306), + SC_(-5.782009124755859375), SC_(-0.9999999999999997089916610994263590057672), SC_(1.999999999999999708991661099426359005767), + SC_(-5.757698535919189453125), SC_(-0.9999999999999996131703510337485986984616), SC_(1.999999999999999613170351033748598698462), + SC_(-5.7298183441162109375), SC_(-0.9999999999999994646206751994613491480939), SC_(1.999999999999999464620675199461349148094), + SC_(-5.6807231903076171875), SC_(-0.9999999999999990546672380240389993944094), SC_(1.999999999999999054667238024038999394409), + SC_(-5.67137622833251953125), SC_(-0.9999999999999989471628690116228142127541), SC_(1.999999999999998947162869011622814212754), + SC_(-5.63473606109619140625), SC_(-0.9999999999999983967395492779530149899861), SC_(1.999999999999998396739549277953014989986), + SC_(-5.614176273345947265625), SC_(-0.9999999999999979723800510435190721967745), SC_(1.999999999999997972380051043519072196774), + SC_(-5.6112957000732421875), SC_(-0.9999999999999979047003987118507997439545), SC_(1.999999999999997904700398711850799743955), + SC_(-5.56195163726806640625), SC_(-0.9999999999999963321711826091909555528649), SC_(1.999999999999996332171182609190955552865), + SC_(-5.52898502349853515625), SC_(-0.999999999999994682538283828212635195642), SC_(1.999999999999994682538283828212635195642), + SC_(-5.47819042205810546875), SC_(-0.9999999999999906156035801749079852647386), SC_(1.999999999999990615603580174907985264739), + SC_(-5.4710788726806640625), SC_(-0.9999999999999898428944822493616106985002), SC_(1.9999999999999898428944822493616106985), + SC_(-5.405083179473876953125), SC_(-0.9999999999999789329565303176610969074478), SC_(1.999999999999978932956530317661096907448), + SC_(-5.402743816375732421875), SC_(-0.9999999999999783844591899263947698611185), SC_(1.999999999999978384459189926394769861119), + SC_(-5.398212432861328125), SC_(-0.9999999999999772817558127269992185563875), SC_(1.999999999999977281755812726999218556388), + SC_(-5.39404582977294921875), SC_(-0.9999999999999762190932060075070452678252), SC_(1.999999999999976219093206007507045267825), + SC_(-5.349620342254638671875), SC_(-0.9999999999999613641252395001102021837028), SC_(1.999999999999961364125239500110202183703), + SC_(-5.3191070556640625), SC_(-0.9999999999999462010476635137780577801192), SC_(1.999999999999946201047663513778057780119), + SC_(-5.2961597442626953125), SC_(-0.9999999999999310743837318684932169157806), SC_(1.999999999999931074383731868493216915781), + SC_(-5.2686710357666015625), SC_(-0.9999999999999073829182184635550018139178), SC_(1.999999999999907382918218463555001813918), + SC_(-5.261013031005859375), SC_(-0.999999999999899463963517023463646064969), SC_(1.999999999999899463963517023463646064969), + SC_(-5.218157291412353515625), SC_(-0.999999999999841220863178602997391812622), SC_(1.999999999999841220863178602997391812622), + SC_(-5.09483242034912109375), SC_(-0.9999999999994203324744342031290577923874), SC_(1.999999999999420332474434203129057792387), + SC_(-5.09044742584228515625), SC_(-0.9999999999993933525823679657367786914755), SC_(1.999999999999393352582367965736778691476), + SC_(-5.0638217926025390625), SC_(-0.9999999999992009929816947679454761069971), SC_(1.999999999999200992981694767945476106997), + SC_(-5.05747509002685546875), SC_(-0.9999999999991469518921467092111635858909), SC_(1.999999999999146951892146709211163585891), + SC_(-5.04293918609619140625), SC_(-0.9999999999990093020187848982236516240641), SC_(1.999999999999009302018784898223651624064), + SC_(-5.0100383758544921875), SC_(-0.9999999999986122073654265981749793751861), SC_(1.999999999998612207365426598174979375186), + SC_(-4.98588848114013671875), SC_(-0.9999999999982250531272305424718207921526), SC_(1.999999999998225053127230542471820792153), + SC_(-4.97671985626220703125), SC_(-0.999999999998051837006412113097549508585), SC_(1.999999999998051837006412113097549508585), + SC_(-4.88807392120361328125), SC_(-0.9999999999952475475430397811200299233621), SC_(1.999999999995247547543039781120029923362), + SC_(-4.883771419525146484375), SC_(-0.999999999995039278283384091602446410145), SC_(1.999999999995039278283384091602446410145), + SC_(-4.85447597503662109375), SC_(-0.9999999999933632419317336295299664149692), SC_(1.999999999993363241931733629529966414969), + SC_(-4.8071804046630859375), SC_(-0.9999999999894197712585954179441600183832), SC_(1.999999999989419771258595417944160018383), + SC_(-4.8020343780517578125), SC_(-0.999999999988871990172530478547610656837), SC_(1.999999999988871990172530478547610656837), + SC_(-4.67612361907958984375), SC_(-0.9999999999623486826561914023506448369931), SC_(1.999999999962348682656191402350644836993), + SC_(-4.67091083526611328125), SC_(-0.9999999999604264343616001426261420203042), SC_(1.999999999960426434361600142626142020304), + SC_(-4.63665485382080078125), SC_(-0.9999999999451799554409023277165162445309), SC_(1.999999999945179955440902327716516244531), + SC_(-4.63516998291015625), SC_(-0.9999999999444029440920085430347846425569), SC_(1.999999999944402944092008543034784642557), + SC_(-4.609210968017578125), SC_(-0.9999999999289428162110384732949528732821), SC_(1.999999999928942816211038473294952873282), + SC_(-4.579636096954345703125), SC_(-0.9999999999061772810501309067972710348531), SC_(1.999999999906177281050130906797271034853), + SC_(-4.5241947174072265625), SC_(-0.9999999998427575929141653776733036798231), SC_(1.999999999842757592914165377673303679823), + SC_(-4.4634552001953125), SC_(-0.9999999997250438347478646951934940057007), SC_(1.999999999725043834747864695193494005701), + SC_(-4.45205211639404296875), SC_(-0.9999999996948757672104213971222453301401), SC_(1.99999999969487576721042139712224533014), + SC_(-4.42528057098388671875), SC_(-0.9999999996107829436098873733936274863483), SC_(1.999999999610782943609887373393627486348), + SC_(-4.41900920867919921875), SC_(-0.9999999995880275737192502467322057148997), SC_(1.9999999995880275737192502467322057149), + 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SC_(0.209703652648431358354620963217352346183e-27), + SC_(7.84175968170166015625), SC_(0.9999999999999999999999999998595912638039), SC_(0.1404087361960659424532579514219630734144e-27), + SC_(7.88610076904296875), SC_(0.9999999999999999999999999999304798650663), SC_(0.6952013493370366965787168537834557597352e-28), + SC_(7.89655590057373046875), SC_(0.9999999999999999999999999999411317291543), SC_(0.5886827084565561878281328421149862522402e-28), + SC_(7.9050960540771484375), SC_(0.9999999999999999999999999999486179207741), SC_(0.513820792259039401977061154556515432295e-28), + SC_(7.93815517425537109375), SC_(0.9999999999999999999999999999696918627742), SC_(0.3030813722580726383815315462367670870671e-28), + SC_(7.94338130950927734375), SC_(0.9999999999999999999999999999721239170155), SC_(0.2787608298449905464917225531595312562447e-28), + }; +#undef SC_ + diff --git a/test/erf_inv_data.ipp b/test/erf_inv_data.ipp new file mode 100644 index 000000000..4215104d7 --- /dev/null +++ b/test/erf_inv_data.ipp @@ -0,0 +1,105 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> erf_inv_data = { + SC_(-0.990433037281036376953125), SC_(-1.832184533179510927322805923563700329767), + SC_(-0.936334311962127685546875), SC_(-1.311339282092737086640055105484822812599), + SC_(-0.931107819080352783203125), SC_(-1.286316461685184857889337272829270644576), + SC_(-0.928576648235321044921875), SC_(-1.274755355308702535979544636706295190547), + SC_(-0.92711746692657470703125), SC_(-1.268242390461126936128446743938528753699), + SC_(-0.907657206058502197265625), SC_(-1.190178701802009872651528128114629153367), + SC_(-0.89756715297698974609375), SC_(-1.154826929034364581020764782706068245251), + SC_(-0.80573642253875732421875), SC_(-0.917873491512746130598510739795171676962), + SC_(-0.804919183254241943359375), SC_(-0.9161942446913841771580833184012189161559), + SC_(-0.780276477336883544921875), SC_(-0.867806431357551134410815525153998291827), + SC_(-0.775070965290069580078125), SC_(-0.8580919924152284867224297625328485999768), + SC_(-0.7496345043182373046875), SC_(-0.8127924290810407931222432768905514928867), + SC_(-0.74820673465728759765625), SC_(-0.8103475955423936417307157186107256388648), + SC_(-0.74602639675140380859375), SC_(-0.806632688757205819001462030577472890805), + SC_(-0.72904598712921142578125), SC_(-0.7784313874823551106598826200232666844639), + SC_(-0.7162272930145263671875), SC_(-0.7579355487144890063429377586715787838945), + SC_(-0.701772034168243408203125), SC_(-0.7355614373173595299453301005770428516518), + SC_(-0.68477380275726318359375), SC_(-0.7101588399949052872532446197423489724803), + SC_(-0.657626628875732421875), SC_(-0.6713881533266128640126408255047881638389), + SC_(-0.652269661426544189453125), SC_(-0.6639738263692763456669198307149427734317), + SC_(-0.6262547969818115234375), SC_(-0.6289572925573171740836428308746584439674), + SC_(-0.62323606014251708984375), SC_(-0.6249936843093662471116097431474787933967), + SC_(-0.57958185672760009765625), SC_(-0.5697131213589784467617578394703976041604), + SC_(-0.576151371002197265625), SC_(-0.5655172244109430153330195150336141500139), + SC_(-0.5579319000244140625), SC_(-0.5435569422159360687847790186563654276103), + SC_(-0.446154057979583740234375), SC_(-0.4186121208546731033057205459902879301199), + SC_(-0.44300353527069091796875), SC_(-0.4152898953738801984047941692529271391195), + SC_(-0.40594112873077392578125), SC_(-0.3768620801611051992528860948080812212023), + SC_(-0.396173775196075439453125), SC_(-0.3669220210390825311547962776125822899061), + SC_(-0.38366591930389404296875), SC_(-0.3542977152760563782151668726041057557165), + SC_(-0.36689913272857666015625), SC_(-0.3375493847053488720470432821496358279516), + SC_(-0.365801036357879638671875), SC_(-0.3364591774366710656954166264654945559873), + SC_(-0.277411997318267822265625), SC_(-0.2510244067029671790889794981353227476998), + SC_(-0.236883103847503662109375), SC_(-0.213115119330839975829499967157244997714), + SC_(-0.215545952320098876953125), SC_(-0.1934073617841803428235669261097060281642), + SC_(-0.202522933483123779296875), SC_(-0.1814532246720147926398927046057793150106), + SC_(-0.18253767490386962890625), SC_(-0.1632073953550647568421821286058243218715), + SC_(-0.156477451324462890625), SC_(-0.1395756320903277910768376053314442757507), + SC_(-0.1558246612548828125), SC_(-0.1389857795955756484965030151195660030168), + SC_(-0.12251126766204833984375), SC_(-0.109002961098867662134935094105847496074), + SC_(-0.1088275909423828125), SC_(-0.09674694516640724629590870677194632943569), + SC_(-0.08402168750762939453125), SC_(-0.07460044047654119877070700345343119035515), + SC_(-0.05048263072967529296875), SC_(-0.04476895818328636312384562686715995170129), + SC_(-0.029248714447021484375), SC_(-0.0259268064334840921104659134138093242797), + SC_(-0.02486217021942138671875), SC_(-0.02203709146986755832638577823832075055744), + SC_(-0.02047121524810791015625), SC_(-0.01814413302702029459097557481591610553903), + SC_(-0.018821895122528076171875), SC_(-0.01668201759439857888105181293763417899072), + SC_(0.0073254108428955078125), SC_(0.006492067534753215749601670217642082465642), + SC_(0.09376299381256103515625), SC_(0.08328747254794857150987333986733043734817), + SC_(0.0944411754608154296875), SC_(0.08389270963798942778622198997355058545872), + SC_(0.264718532562255859375), SC_(0.2390787735821979528028028789569770109829), + SC_(0.27952671051025390625), SC_(0.2530214201700340392837551955289041822603), + SC_(0.29262602329254150390625), SC_(0.2654374523135675523971788948011709467352), + SC_(0.3109557628631591796875), SC_(0.282950508503826367238408926581528085458), + SC_(0.31148135662078857421875), SC_(0.2834552014554130441860525970673030809536), + SC_(0.32721102237701416015625), SC_(0.2986277427848421570858990348074985028421), + SC_(0.3574702739715576171875), SC_(0.3282140305634627092431945088114761850208), + SC_(0.362719058990478515625), SC_(0.3334035993712283467959295804559099468454), + SC_(0.3896572589874267578125), SC_(0.3603304982893212173104266596348905175268), + SC_(0.4120922088623046875), SC_(0.3831602323665075533579267768785894144888), + SC_(0.41872966289520263671875), SC_(0.3899906753567599452444107492361433402154), + SC_(0.45167791843414306640625), SC_(0.4244594733907945411184647153213164209335), + SC_(0.48129451274871826171875), SC_(0.4563258063707025027210352963461819167707), + SC_(0.4862649440765380859375), SC_(0.4617640058971775089811390737537561779898), + SC_(0.50937330722808837890625), SC_(0.4874174763856674076219106695373814892182), + SC_(0.5154802799224853515625), SC_(0.4943041993872143888987628020569772222018), + SC_(0.52750003337860107421875), SC_(0.5079978091910991117615000459548117088362), + SC_(0.53103363513946533203125), SC_(0.5120597685873370942783226077302881881069), + SC_(0.58441460132598876953125), SC_(0.5756584292527058478710392476034273328569), + SC_(0.5879499912261962890625), SC_(0.5800336103175463592377907341030447077804), + SC_(0.59039986133575439453125), SC_(0.5830784871670823806198622501806646778319), + SC_(0.59455978870391845703125), SC_(0.588273673825686998734497652983815773459), + SC_(0.59585726261138916015625), SC_(0.5899005483108011364541949539839185473259), + SC_(0.5962116718292236328125), SC_(0.5903454775096607218832535637355431851718), + SC_(0.6005609035491943359375), SC_(0.5958247243549040349587326482492767206448), + SC_(0.6150619983673095703125), SC_(0.6143583249050861028039832921829036722514), + SC_(0.62944734096527099609375), SC_(0.6331707263097125575937994856370309207836), + SC_(0.64380657672882080078125), SC_(0.6524069265890823819975498133014027247554), + SC_(0.6469156742095947265625), SC_(0.656635855345815020063728463464436343698), + SC_(0.67001712322235107421875), SC_(0.6888269167957872563013714303376548038671), + SC_(0.6982586383819580078125), SC_(0.7302336318927408409119676651737758401138), + SC_(0.74485766887664794921875), SC_(0.8046505193013635090578266413458426260098), + SC_(0.75686132907867431640625), SC_(0.8253191678260588578995203396384711816647), + SC_(0.81158387660980224609375), SC_(0.9300427626888758122211127950646282789481), + SC_(0.826751708984375), SC_(0.9629665092443368464606966822833571908852), + SC_(0.83147108554840087890625), SC_(0.9736479209913771931387473923084901789046), + SC_(0.84174954891204833984375), SC_(0.997713670556719074960678197806799852186), + SC_(0.8679864406585693359375), SC_(1.065050516333636716777334376076374184102), + SC_(0.90044414997100830078125), SC_(1.164612422633086435501625591693259387477), + SC_(0.91433393955230712890625), SC_(1.215315881176612875682446995412738776976), + SC_(0.91501367092132568359375), SC_(1.217962731073139868794942852653058932976), + SC_(0.918984889984130859375), SC_(1.233778505900771488542027767896521427575), + SC_(0.92977702617645263671875), SC_(1.28019542575660930623179572273596558907), + SC_(0.93538987636566162109375), SC_(1.306695301483797253764522033930388453334), + SC_(0.93773555755615234375), SC_(1.318335478463913327121670503572736587296), + SC_(0.94118559360504150390625), SC_(1.33613349872692113073358883961598631154), + SC_(0.96221935749053955078125), SC_(1.468821071545234761861756248744372345584), + SC_(0.98576259613037109375), SC_(1.733272259459038694476413373595347034928), + SC_(0.9881370067596435546875), SC_(1.77921769652839903464038407684397479173), + SC_(0.99292266368865966796875), SC_(1.904368122482929779094714951471938518496), + }; +#undef SC_ + diff --git a/test/erf_large_data.ipp b/test/erf_large_data.ipp new file mode 100644 index 000000000..fc7054fc2 --- /dev/null +++ b/test/erf_large_data.ipp @@ -0,0 +1,305 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 300> erf_large_data = { + SC_(8.2311115264892578125), SC_(0.9999999999999999999999999999997436415644), SC_(0.2563584356432915693836191701249115171878e-30), + SC_(8.3800067901611328125), SC_(0.9999999999999999999999999999999787664373), SC_(0.212335626810981756102114466368867764939e-31), + SC_(8.39224529266357421875), SC_(0.9999999999999999999999999999999827316301), SC_(0.1726836993826464997300336080711750877816e-31), + SC_(8.66370105743408203125), SC_(0.9999999999999999999999999999999998367494), SC_(0.1632506054605993373182936619108145081694e-33), + SC_(8.9759693145751953125), SC_(0.9999999999999999999999999999999999993611), SC_(0.6389109854135105643535885036658544645608e-36), + SC_(9.1102085113525390625), SC_(0.9999999999999999999999999999999999999445), SC_(0.5554662272164108694298027082501260363284e-37), + SC_(9.45745944976806640625), SC_(0.9999999999999999999999999999999999999999), SC_(0.8480477594433168680596946662186145070498e-40), + SC_(10.61029338836669921875), SC_(1), SC_(0.6786472703144874611591306359565011820753e-50), + SC_(10.8140201568603515625), SC_(1), SC_(0.8470069870191149998048954134093763207838e-52), + SC_(10.82457828521728515625), SC_(1), SC_(0.6733586817565831939437647508350297170672e-52), + SC_(10.9283580780029296875), SC_(1), SC_(0.6977670213613499261034444110870077646237e-53), + SC_(10.98818302154541015625), SC_(1), SC_(0.1870385270398381154229086866501074415355e-53), + SC_(11.31863117218017578125), SC_(1), SC_(0.1142544869245536263179923727458113771164e-56), + SC_(11.69488239288330078125), SC_(1), SC_(0.1919907987324948057617154107998359393185e-60), + SC_(11.78605365753173828125), SC_(1), SC_(0.2239752724414047638235296360419194553396e-61), + SC_(12.1997470855712890625), SC_(1), SC_(0.1061491854672859805410277580039105082623e-65), + SC_(12.4239101409912109375), SC_(1), SC_(0.4177140558891845034847722323657671701092e-68), + SC_(13.54282093048095703125), SC_(1), SC_(0.9235451601555197945993611934762413642509e-81), + SC_(14.22005176544189453125), SC_(1), SC_(0.6009313847910459672567608314935750835662e-89), + SC_(14.22926807403564453125), SC_(1), SC_(0.4620339167397254212949689945484104460402e-89), + SC_(14.359676361083984375), SC_(1), SC_(0.1100471801774583728133644248988374419641e-90), + SC_(14.41039371490478515625), SC_(1), SC_(0.2548929323782609375991749296565683531577e-91), + SC_(14.87335300445556640625), SC_(1), SC_(0.3198000452629152167848001696581299429735e-97), + SC_(14.92373943328857421875), SC_(1), SC_(0.7101988862786309879894988884459001902733e-98), + SC_(15.8191242218017578125), SC_(1), SC_(0.7438591168260150840237290305856485663443e-110), + SC_(15.96480560302734375), SC_(1), SC_(0.7187861904421355163894524608883700422307e-112), + SC_(15.99831295013427734375), SC_(1), SC_(0.2457896620902286177098546091913977024151e-112), + SC_(16.7455272674560546875), SC_(1), SC_(0.5559992295828943910250640094814395579468e-123), + SC_(17.008663177490234375), SC_(1), SC_(0.760237064211963037211163791564153667507e-127), + SC_(17.2220897674560546875), SC_(1), SC_(0.5043169273534506801418127595237229711103e-130), + SC_(17.757808685302734375), SC_(1), SC_(0.35565429074608249855471197796329792194e-138), + SC_(17.802867889404296875), SC_(1), SC_(0.7145708921803004436285314631231014067581e-139), + SC_(18.112152099609375), SC_(1), SC_(0.1053094819294519519788245447399920974245e-143), + SC_(18.2649860382080078125), SC_(1), SC_(0.4020674492293458832133643654712400773494e-146), + SC_(18.32352447509765625), SC_(1), SC_(0.4706809140073901333884996503518029500936e-147), + SC_(18.4129180908203125), SC_(1), SC_(0.1755474923764976123532737166943228663005e-148), + SC_(18.652309417724609375), SC_(1), SC_(0.2428091775485678872414439322587700295772e-152), + SC_(18.9056758880615234375), SC_(1), SC_(0.1764835785229242910502507612159515936326e-156), + SC_(19.1091136932373046875), SC_(1), SC_(0.7645206854732087495539968665185944859194e-160), + SC_(19.1576213836669921875), SC_(1), SC_(0.1191626992794708978523133508616313098621e-160), + SC_(19.31610870361328125), 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SC_(1), SC_(0.1059138463142173993761026331012035279463e-2224), + SC_(71.8696136474609375), SC_(1), SC_(0.4560525021368102193256510775118687007107e-2245), + SC_(71.89171600341796875), SC_(1), SC_(0.1900755308625425801959286367021480241643e-2246), + SC_(72.96099853515625), SC_(1), SC_(0.1012288958237199779078623773782261227323e-2313), + SC_(73.07500457763671875), SC_(1), SC_(0.5943790030417452214269807905148252301626e-2321), + SC_(73.10594940185546875), SC_(1), SC_(0.6446581210317819300626283182106697992806e-2323), + SC_(73.1313323974609375), SC_(1), SC_(0.1574358163400440570765391931676616457296e-2324), + SC_(73.20639801025390625), SC_(1), SC_(0.2666631324666853271144728061565555980879e-2329), + SC_(73.376953125), SC_(1), SC_(0.3692800481745944696527608913821150780197e-2340), + SC_(73.430145263671875), SC_(1), SC_(0.1498451636616422925337939153598716367853e-2343), + SC_(73.44467926025390625), SC_(1), SC_(0.1772057595578452829782717746213217866987e-2344), + SC_(73.60561370849609375), SC_(1), SC_(0.9327165682410597575108877219484860075423e-2355), + SC_(73.62299346923828125), SC_(1), SC_(0.7217309662507557984084259593722358671817e-2356), + SC_(73.77313995361328125), SC_(1), SC_(0.1762441226759232140980568288616069877708e-2365), + SC_(74.2175445556640625), SC_(1), SC_(0.4796668342626380739003100430828430758387e-2394), + SC_(74.27039337158203125), SC_(1), SC_(0.1873022555292045249630763229328993092174e-2397), + SC_(74.39823150634765625), SC_(1), SC_(0.1041846943079459800746159253005438318555e-2405), + SC_(74.77135467529296875), SC_(1), SC_(0.6972625788619583784396293640501867635862e-2430), + SC_(74.807342529296875), SC_(1), SC_(0.320164469944205443450167902957404361688e-2432), + SC_(75.01886749267578125), SC_(1), SC_(0.5501656655265385463409067358984172337294e-2446), + SC_(75.3089447021484375), SC_(1), SC_(0.6319468915000774061299279774718180641493e-2465), + SC_(75.34217071533203125), SC_(1), SC_(0.4232650489503930895532015931831887522867e-2467), + SC_(75.3960723876953125), SC_(1), SC_(0.1252103871082113384792176510455204354046e-2470), + SC_(75.45360565185546875), SC_(1), SC_(0.2128736422560450047027718674310538561411e-2474), + SC_(75.5235443115234375), SC_(1), SC_(0.5520057148754538352100449496414500231534e-2479), + SC_(75.7169952392578125), SC_(1), SC_(0.1082452065927623584648516750055819887681e-2491), + SC_(76.1279449462890625), SC_(1), SC_(0.8546877316832444989935075245988734481682e-2519), + SC_(76.470703125), SC_(1), SC_(0.1638058714171406332549731275799690588823e-2541), + SC_(76.938812255859375), SC_(1), SC_(0.1056702363179452330608841676726809135679e-2572), + SC_(77.5743560791015625), SC_(1), SC_(0.2358904373500824632540699420486265405702e-2615), + SC_(77.62860107421875), SC_(1), SC_(0.5201021650937707207333604556951161883551e-2619), + SC_(77.94854736328125), SC_(1), SC_(0.1249443652740786709208083871965551047597e-2640), + SC_(78.06497955322265625), SC_(1), SC_(0.1611056496774920949293225513029265281761e-2648), + SC_(78.1817626953125), SC_(1), SC_(0.1913800020341339537443259258588625217145e-2656), + SC_(78.7396087646484375), SC_(1), SC_(0.1826214635685593066183737049160168091905e-2694), + SC_(79.23296356201171875), SC_(1), SC_(0.2578910365673578199346966028090255709301e-2728), + SC_(79.28195953369140625), SC_(1), SC_(0.1091891461191934352451814384442219402849e-2731), + SC_(79.53916168212890625), SC_(1), SC_(0.1977970219864078402856631374181827871126e-2749), + SC_(79.89411163330078125), SC_(1), SC_(0.5214347553208988003055180927464712623108e-2774), + SC_(80.03131103515625), SC_(1), SC_(0.1539244869801023794218894091743149755693e-2783), + SC_(81.0540618896484375), SC_(1), SC_(0.4282424762803665933789596667679893825473e-2855), + SC_(82.27494049072265625), SC_(1), SC_(0.1058661602699748027888078315752637932386e-2941), + SC_(82.40390777587890625), SC_(1), SC_(0.6316127985700229083559360532999075433841e-2951), + SC_(82.67310333251953125), SC_(1), SC_(0.3161239679691280797883924212187842089968e-2970), + SC_(82.83135223388671875), SC_(1), SC_(0.1331877990458538379342065811181569680933e-2981), + SC_(82.8936614990234375), SC_(1), SC_(0.4360382415173864210795784221294622630074e-2986), + SC_(82.896820068359375), SC_(1), SC_(0.2582766043573584963841163098015327607626e-2986), + SC_(83.0903167724609375), SC_(1), SC_(0.290013476198442869046820654577037968453e-3000), + SC_(83.20987701416015625), SC_(1), SC_(0.6710637821460133939254155513062400075894e-3009), + SC_(83.5117340087890625), SC_(1), SC_(0.930787700439063954052855467933526662801e-3031), + SC_(84.054412841796875), SC_(1), SC_(0.2976066148553277399257428533933691648135e-3070), + SC_(84.19962310791015625), SC_(1), SC_(0.7279769443639285050812101610482378417998e-3081), + SC_(84.58744049072265625), SC_(1), SC_(0.2702912324282395041651536403393819660776e-3109), + SC_(84.58887481689453125), SC_(1), SC_(0.2120514736394887571017949592511629200089e-3109), + SC_(85.08606719970703125), SC_(1), SC_(0.4856698297979953595767807992015152776328e-3146), + SC_(85.918212890625), SC_(1), SC_(0.761735048565653930137023626638747177308e-3208), + SC_(86.31143951416015625), SC_(1), SC_(0.2931592864000316245471343335876130048243e-3237), + SC_(86.48769378662109375), SC_(1), SC_(0.1734166482219001810675374526329398742724e-3250), + SC_(86.51556396484375), SC_(1), SC_(0.1396184688760208541263525791606580135888e-3252), + SC_(86.661895751953125), SC_(1), SC_(0.137591974249259342744647335702120081949e-3263), + SC_(86.67838287353515625), SC_(1), SC_(0.7894924978682044731133080698270675959047e-3265), + SC_(86.699005126953125), SC_(1), SC_(0.2210314800408904240090566206375166498669e-3266), + SC_(86.875640869140625), SC_(1), SC_(0.1067393543048809336522125654614257792709e-3279), + SC_(87.12085723876953125), SC_(1), SC_(0.3141588749441732230579799060940491765735e-3298), + SC_(87.1272430419921875), SC_(1), SC_(0.1032456874487975042510718930756313489085e-3298), + SC_(87.350982666015625), SC_(1), SC_(0.1145259102841530225822158676931062199611e-3315), + SC_(87.4471588134765625), SC_(1), SC_(0.5719031848650119145301950377270835213481e-3323), + SC_(87.5886077880859375), SC_(1), SC_(0.100944910275613191627060617308017176992e-3333), + SC_(88.114166259765625), SC_(1), SC_(0.790369642782630853434920273428079682513e-3374), + SC_(88.35390472412109375), SC_(1), SC_(0.3336629458423611349557710086599109785783e-3392), + SC_(88.45099639892578125), SC_(1), SC_(0.1168446033182320025804519666787072055362e-3399), + SC_(88.55356597900390625), SC_(1), SC_(0.1521829146954713302263784465478849666514e-3407), + SC_(88.97168731689453125), SC_(1), SC_(0.8788241912132849825702219058327694185942e-3440), + SC_(89.04711151123046875), SC_(1), SC_(0.129507473240771950228590714535613326805e-3445), + SC_(89.05876922607421875), SC_(1), SC_(0.1623712240263443145358629211111315306735e-3446), + SC_(89.41626739501953125), SC_(1), SC_(0.3153753084460525420724595238130726065242e-3474), + SC_(89.51361846923828125), SC_(1), SC_(0.8577730193641580987437405819558340102121e-3482), + SC_(89.68304443359375), SC_(1), SC_(0.5586301300368715269248196410139332100555e-3495), + SC_(89.70983123779296875), SC_(1), SC_(0.4571434971968977143179977283713339395878e-3497), + }; +#undef SC_ + diff --git a/test/erf_small_data.ipp b/test/erf_small_data.ipp new file mode 100644 index 000000000..ee05d3acd --- /dev/null +++ b/test/erf_small_data.ipp @@ -0,0 +1,162 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 150> erf_small_data = { + SC_(0), SC_(0), SC_(1), + SC_(0.140129846432481707092372958328991613128e-44), SC_(0.1581195994027057927040695988659415347357e-44), SC_(1), + SC_(0.2802596928649634141847459166579832262561e-44), SC_(0.3162391988054115854081391977318830694715e-44), SC_(1), + SC_(0.4203895392974451212771188749869748393841e-44), SC_(0.4743587982081173781122087965978246042072e-44), SC_(1), + SC_(0.8407790785948902425542377499739496787682e-44), SC_(0.9487175964162347562244175931956492084145e-44), SC_(1), + SC_(0.1541428310757298778016102541618907744408e-43), SC_(0.1739315593429763719744765587525356882093e-43), SC_(1), + SC_(0.3222986467947079263124578041566807101945e-43), SC_(0.3636750786262233232193600773916655298922e-43), SC_(1), + SC_(0.7426881860921530475895766791436555495785e-43), SC_(0.8380338768343407013315688739894901340994e-43), SC_(1), + SC_(0.105097384824361280319279718746743709846e-42), SC_(0.1185896995520293445280521991494561510518e-42), SC_(1), + SC_(0.210194769648722560638559437493487419692e-42), SC_(0.2371793991040586890561043982989123021036e-42), SC_(1), + SC_(0.4666323886201640846176019512355420717163e-42), SC_(0.52653826601101028970455176422358531067e-42), SC_(1), + SC_(0.1223333559355565302916415926212096782608e-41), SC_(0.1380384102785621570306527598099669598243e-41), SC_(1), + SC_(0.2578389174357663410499662433253445681556e-41), SC_(0.2909400629009786585754880619133324239138e-41), SC_(1), + SC_(0.2960943655118338470861840609491592785395e-41), SC_(0.3341067135379173399836990624037344628966e-41), SC_(1), + SC_(0.7557202618103738463491673642682517695994e-41), SC_(0.8527389995787923400530473466840226968299e-41), SC_(1), + SC_(0.1465898323530191137893313517079581264932e-40), SC_(0.1654089129351705297477272073736614394871e-40), SC_(1), + SC_(0.4298903428855673810179817615616804707542e-40), SC_(0.4850793070476208308575447154009354402623e-40), SC_(1), + SC_(0.4803791265551905400833637384476161489642e-40), SC_(0.5420497987124157279688209918723341752276e-40), SC_(0.9999999999999999999999999999999999999999), + SC_(0.1055289847513733239771242274583970040145e-39), SC_(0.1190767079181896783695807335139632509788e-39), SC_(0.9999999999999999999999999999999999999999), + SC_(0.2015109230653016692500450852658398094265e-39), SC_(0.227380727529073011082233205257189905196e-39), SC_(0.9999999999999999999999999999999999999998), + SC_(0.7325049501519046019351739362618877290525e-39), SC_(0.8265433255457460713757641362399021627702e-39), SC_(0.9999999999999999999999999999999999999992), + SC_(0.1339665153968418641693291185028075750078e-38), SC_(0.1511650250621765838235665056990208282135e-38), SC_(0.9999999999999999999999999999999999999985), + SC_(0.2677046191439987831560976690835388936862e-38), SC_(0.3020723151773267572050253779518901717253e-38), SC_(0.999999999999999999999999999999999999997), + SC_(0.4980653348629733537806134066371931674319e-38), SC_(0.5620065477118294341833797281540012541512e-38), SC_(0.9999999999999999999999999999999999999944), + SC_(0.6613229117999040042200470598735932013487e-38), SC_(0.7462229963979548044442924939647721094874e-38), SC_(0.9999999999999999999999999999999999999925), + SC_(0.1548242965319156795913711730955404935546e-37), SC_(0.1747005107668316705149757196187806484869e-37), SC_(0.9999999999999999999999999999999999999825), + SC_(0.4146556501891822617936303213288858289615e-37), SC_(0.4678887971919177014040265951235884621123e-37), SC_(0.9999999999999999999999999999999999999532), + SC_(0.9169899847717825977881300331923883942907e-37), SC_(0.1034712395251710806479818721979201078743e-36), SC_(0.9999999999999999999999999999999999998965), + SC_(0.1401743180405455194869357938110469019064e-36), SC_(0.1581697802387722354144421623938376299412e-36), SC_(0.9999999999999999999999999999999999998418), + SC_(0.1945576912957096415426581064633919119024e-36), SC_(0.2195348456562787018647206880870297955178e-36), SC_(0.9999999999999999999999999999999999997805), + SC_(0.6257788193068770148365547155357311502081e-36), SC_(0.7061157829155071490944174116544753093472e-36), SC_(0.9999999999999999999999999999999999992939), + SC_(0.1082390914972224343528760935165450352846e-35), SC_(0.1221347359108108275138920730258033965505e-35), SC_(0.9999999999999999999999999999999999987787), + SC_(0.1694061057766321374547939115222498639949e-35), SC_(0.1911543205371304725231209426978241547223e-35), SC_(0.9999999999999999999999999999999999980885), + SC_(0.4157476136990655326859281227151399960547e-35), SC_(0.469120946067698479126142569921586497913e-35), SC_(0.9999999999999999999999999999999999953088), + SC_(0.7283681265086265937666854409047925692044e-35), SC_(0.8218754199287230086952619241467251109843e-35), SC_(0.9999999999999999999999999999999999917812), + SC_(0.2125163494676521664376074827660149555726e-34), SC_(0.2397990214064882285432839461962890817645e-34), SC_(0.9999999999999999999999999999999999760201), + SC_(0.2530711628583540298128114059620192234547e-34), SC_(0.2855602279620023372947687038386988804286e-34), SC_(0.999999999999999999999999999999999971444), + SC_(0.8643572969601960480494638860332170085101e-34), SC_(0.9753227668168746390533409248013332505622e-34), SC_(0.9999999999999999999999999999999999024677), + SC_(0.99805657801365854084424590887058919672e-34), SC_(0.1126186250213249491600342719979975982704e-33), SC_(0.9999999999999999999999999999999998873814), + SC_(0.2285833483419143806137872349711124495646e-33), SC_(0.2579286882139527639322447178720013355598e-33), SC_(0.9999999999999999999999999999999997420713), + SC_(0.5426235288674262982106993151912976763677e-33), SC_(0.6122850855498543097076270518307709592481e-33), SC_(0.9999999999999999999999999999999993877149), + SC_(0.1147672787261224070426617104882286355311e-32), SC_(0.1295010063788005409890233863705470341395e-32), SC_(0.9999999999999999999999999999999987049899), + SC_(0.22463880715206230259705675341401055879e-32), SC_(0.2534777501115735340046251758622995828684e-32), SC_(0.9999999999999999999999999999999974652225), + SC_(0.4454556413592197901478916560693101385393e-32), SC_(0.5026428655749137894397761320610666694693e-32), SC_(0.9999999999999999999999999999999949735713), + SC_(0.9167850888743889506336831720594191266931e-32), SC_(0.1034481194989668475298051221951978796736e-31), SC_(0.9999999999999999999999999999999896551881), + SC_(0.2029237383788489473978179376606301463249e-31), SC_(0.2289749188958332742407712486623796907403e-31), SC_(0.9999999999999999999999999999999771025081), + SC_(0.4422489809322529014977160404510081439987e-31), SC_(0.4990245367531747508971186530942781923233e-31), SC_(0.9999999999999999999999999999999500975463), + SC_(0.8427819595678069227982621602322463722706e-31), SC_(0.9509776055802459297555061290310693509558e-31), SC_(0.9999999999999999999999999999999049022394), + SC_(0.1894128825516700180187896525548839017318e-30), SC_(0.2137295506508135613325177084306249463745e-30), SC_(0.9999999999999999999999999999997862704493), + SC_(0.3460509137323572130893969054757060828136e-30), SC_(0.390476641809958306521950254784930239331e-30), SC_(0.9999999999999999999999999999996095233582), + SC_(0.7129452701155289322465925784842836259778e-30), SC_(0.8044725900776457681286359991257189545736e-30), SC_(0.9999999999999999999999999999991955274099), + SC_(0.1006606281422698106765835646474453774524e-29), SC_(0.1135833557424855221443633868642206855925e-29), SC_(0.9999999999999999999999999999988641664426), + SC_(0.2691237328241499384996035977623006944895e-29), SC_(0.3036736134897495655058606838487828118587e-29), SC_(0.9999999999999999999999999999969632638651), + SC_(0.53002073552022827485063776974422846038e-29), SC_(0.5980643560896661370936819930965482836573e-29), SC_(0.9999999999999999999999999999940193564391), + SC_(0.6328674258047448903166407618549998992304e-29), SC_(0.7141144188114391407566450273134713579864e-29), SC_(0.9999999999999999999999999999928588558119), + SC_(0.2089027321727668921272023434652731888509e-28), SC_(0.2357214909330836434674323280791160963504e-28), SC_(0.9999999999999999999999999999764278509067), + SC_(0.431842364003957104364921550167833797347e-28), SC_(0.4872819270113422778304421472654534392263e-28), SC_(0.9999999999999999999999999999512718072989), + SC_(0.5869689191576790243963724295243864109357e-28), SC_(0.6623235001100951114675477389355052037517e-28), SC_(0.999999999999999999999999999933767649989), + SC_(0.1659976352436858066871624892515432120194e-27), SC_(0.1873082733960948939621836113363260997334e-27), SC_(0.9999999999999999999999999998126917266039), + SC_(0.2259797844707783785531125574841037632032e-27), SC_(0.2549908809815603534966750948985043565705e-27), SC_(0.9999999999999999999999999997450091190184), + SC_(0.5880869455645201196495195879197812435557e-27), SC_(0.6635850578158372552104628507822996913944e-27), SC_(0.9999999999999999999999999993364149421842), + SC_(0.1210368873588132831231840717009055000898e-26), SC_(0.136575502145771107167093316323928936418e-26), SC_(0.9999999999999999999999999986342449785423), + SC_(0.2865917801972336300962667874631175760926e-26), SC_(0.3233841942353746978165316543252271930097e-26), SC_(0.9999999999999999999999999967661580576463), + SC_(0.6332274308614525819994844546119106938273e-26), SC_(0.7145206410174771386698068058159747695855e-26), SC_(0.9999999999999999999999999928547935898252), + SC_(0.1017015117562076843891927280903461796905e-25), SC_(0.1147578671278241071470420542046720634233e-25), SC_(0.9999999999999999999999999885242132872176), + SC_(0.1732407881343339865174865944566931617746e-25), SC_(0.1954812962219899413849001377507194449177e-25), SC_(0.999999999999999999999999980451870377801), + SC_(0.4851304131156321593400980501390172514877e-25), SC_(0.5474110514841189450665720204629142546924e-25), SC_(0.9999999999999999999999999452588948515881), + SC_(0.8195642344688286703600152101217105477795e-25), SC_(0.9247792082712082720339044529059539700692e-25), SC_(0.9999999999999999999999999075220791728792), + SC_(0.1869609713097047478340985963996916097412e-24), SC_(0.2109628650858126659649010477127478351642e-24), SC_(0.9999999999999999999999997890371349141873), + SC_(0.2530783866565756141965447805546793991855e-24), SC_(0.2855683791454228747266674274313813542172e-24), SC_(0.9999999999999999999999997144316208545771), + SC_(0.4209414052294289885709092151366397306124e-24), SC_(0.474981512228797723082223866616547403578e-24), SC_(0.9999999999999999999999995250184877712023), + SC_(0.1448614172287997697834930368987483835595e-23), SC_(0.1634586053169086194558576878083327447891e-23), SC_(0.9999999999999999999999983654139468309138), + SC_(0.3012998612797836509437815728233749567808e-23), SC_(0.3399804865168757552575507775525287643073e-23), SC_(0.9999999999999999999999966001951348312424), + SC_(0.415276141474016400714459480565419005095e-23), SC_(0.4685889466310888715323687240479993998499e-23), SC_(0.9999999999999999999999953141105336891113), + SC_(0.1204931390183540503963647385927016736362e-22), SC_(0.1359619478462541509417967524360939547446e-22), SC_(0.9999999999999999999999864038052153745849), + SC_(0.1993117661558407300079387840940773127252e-22), SC_(0.2248992446872631349043635494217604899453e-22), SC_(0.9999999999999999999999775100755312736865), + SC_(0.5135333031694984418963713908106404060216e-22), SC_(0.5794602809062059992419407432753169472345e-22), SC_(0.9999999999999999999999420539719093794001), + SC_(0.899483690375495720736892890190099236758e-22), SC_(0.1014958657361899781022539075381620547142e-21), SC_(0.9999999999999999999998985041342638100219), + SC_(0.1495718898928958003798630318006718936674e-21), SC_(0.1687738045382474786366114844967457859701e-21), SC_(0.9999999999999999999998312261954617525214), + SC_(0.4004143349189545415674843981449458618638e-21), SC_(0.4518191937289535418778128853745495698362e-21), SC_(0.9999999999999999999995481808062710464581), + SC_(0.6027338573124474618509081343250763374009e-21), SC_(0.6801123278944849878208640332552941589305e-21), SC_(0.9999999999999999999993198876721055150122), + SC_(0.1659584418309930229572620634728719046791e-20), SC_(0.1872640483657249799664013537449399617766e-20), SC_(0.9999999999999999999981273595163427502003), + SC_(0.2678244821868696143277384944057090132574e-20), SC_(0.3022075661378068793988187051769889525453e-20), SC_(0.999999999999999999996977924338621931206), + SC_(0.5242169890349582687359968492107897830579e-20), SC_(0.5915155294645836590589096407129433743023e-20), SC_(0.9999999999999999999940848447053541634094), + SC_(0.7847303335957885127235572164487720225878e-20), SC_(0.8854733602173995706713015709887803225844e-20), SC_(0.9999999999999999999911452663978260042933), + SC_(0.1543123850410999503479436223765675073594e-19), SC_(0.1741228805051983958166845695046398116561e-19), SC_(0.9999999999999999999825877119494801604183), + SC_(0.4775181844358503128246959748559596903306e-19), SC_(0.5388215712266861317995086503415856707862e-19), SC_(0.99999999999999999994611784287733138682), + SC_(0.6230335359741914917030257154567651767252e-19), SC_(0.7030180623951302655508677093605170542092e-19), SC_(0.9999999999999999999296981937604869734449), + SC_(0.1333737877121347004096484859664295541393e-18), SC_(0.1504962034909922627946995932175931040167e-18), SC_(0.9999999999999999998495037965090077372053), + SC_(0.272678641362033605222928850375652132243e-18), SC_(0.3076848982248274637704507517821273969355e-18), SC_(0.9999999999999999996923151017751725362295), + SC_(0.7848472665613653936552990969532928033914e-18), SC_(0.885605304939703219820034216816760452174e-18), SC_(0.99999999999999999911439469506029678018), + SC_(0.1596567770068200412154901801642381542479e-17), SC_(0.180153381060109581137526939833213468117e-17), SC_(0.9999999999999999981984661893989041886247), + SC_(0.3449535060248237706634344412748305330751e-17), SC_(0.3892383498149675248979402679995676281369e-17), SC_(0.9999999999999999961076165018503247510206), + SC_(0.4351665680924082666098176574998035448516e-17), SC_(0.4910328896519242978931639720387530850973e-17), SC_(0.9999999999999999950896711034807570210684), + SC_(0.9245716998221635790720235315021113819967e-17), SC_(0.1043267444565415210292590191714661224142e-16), SC_(0.9999999999999999895673255543458478970741), + SC_(0.2517825949792599072966925444205799067277e-16), SC_(0.2841062348118440801806623646517627970916e-16), SC_(0.9999999999999999715893765188155919819338), + SC_(0.3607757836945135819072677518803970997396e-16), SC_(0.4070918783134460417227759258926655100437e-16), SC_(0.9999999999999999592908121686553958277224), + SC_(0.6902950590186084176697045577952849271242e-16), SC_(0.7789145637455650616639975178325762794953e-16), SC_(0.9999999999999999221085436254434938336002), + SC_(0.1125254418367572150846966927417724946281e-15), SC_(0.1269713643368146509171410927223650268325e-15), SC_(0.9999999999999998730286356631853490828589), + SC_(0.4283825720300077181945663795659129391424e-15), SC_(0.4833779698254535301267982815421810779873e-15), SC_(0.9999999999999995166220301745464698732017), + SC_(0.5405621861962687634689528337617048237007e-15), SC_(0.6099591094234751315857573573713433261908e-15), SC_(0.9999999999999993900408905765248684142426), + SC_(0.1199026382331771015121724133223324315622e-14), SC_(0.1352956390621069391623702837955015493166e-14), SC_(0.9999999999999986470436093789306083762972), + SC_(0.3388160376379171188909733558602965786122e-14), SC_(0.3823129583484747580613317941389808261798e-14), SC_(0.9999999999999961768704165152524193866821), + SC_(0.4251160106447451902944578705501044169068e-14), SC_(0.4796920500102846351328633565382312006865e-14), SC_(0.9999999999999952030794998971536486713664), + SC_(0.1313415386649738336721782161475857719779e-13), SC_(0.1482030560038262349331825316596395555414e-13), SC_(0.9999999999999851796943996173765066817468), + SC_(0.1777897084111537684414372506580548360944e-13), SC_(0.2006142030951317354054567215576954023455e-13), SC_(0.9999999999999799385796904868264594543278), + SC_(0.5556494121154026410991377815662417560816e-13), SC_(0.6269832208398892455680747006005262348402e-13), SC_(0.9999999999999373016779160110754431925299), + SC_(0.9186150917630392376267423060198780149221e-13), SC_(0.1036546132124946068175722614185676168866e-12), SC_(0.9999999999998963453867875053931824277386), + SC_(0.2022372612257850033046224780264310538769e-12), SC_(0.228200312377628882307346737694658892409e-12), SC_(0.9999999999997717996876223711176926532623), + SC_(0.334987083645127414754938399710226804018e-12), SC_(0.3779924464312436744509548821977467036951e-12), SC_(0.9999999999996220075535687563255490451178), + SC_(0.9037920151860889816930466622579842805862e-12), SC_(0.1019820081323253936787897658201809797801e-11), SC_(0.9999999999989801799186767460632121023418), + SC_(0.1229327186921813641617973189568147063255e-11), SC_(0.1387147187266705574841551295246781201059e-11), SC_(0.9999999999986128528127332944251584487048), + SC_(0.1941944651959182088774014118826016783714e-11), SC_(0.2191249888923286935404641309633986461335e-11), SC_(0.9999999999978087501110767130645953586904), + SC_(0.6660515888823326235979038756340742111206e-11), SC_(0.7515587371056892484340940489030914565555e-11), SC_(0.999999999992484412628943107515659059511), + SC_(0.1305013996172332824130535300355404615402e-10), SC_(0.1472550606048923346284018897482637545021e-10), SC_(0.9999999999852744939395107665371598110252), + SC_(0.2306862860457226105381778324954211711884e-10), SC_(0.2603015993086296441348608813996585159423e-10), SC_(0.99999999997396984006913703558651391186), + SC_(0.464061636340495908825687365606427192688e-10), SC_(0.5236374826948694233006052031278254256924e-10), SC_(0.9999999999476362517305130576699394796872), + SC_(0.9020578728424766268290113657712936401367e-10), SC_(0.1017863311229943567634007843433802039614e-9), SC_(0.9999999998982136688770056432365992156566), + SC_(0.2017243178054073382554634008556604385376e-9), SC_(0.2276215177081760092443407256160138975684e-9), SC_(0.999999999772378482291823990755659274384), + SC_(0.4463814318178549456206383183598518371582e-9), SC_(0.5036875082415334987251679342248332272253e-9), SC_(0.9999999994963124917584665012748320657752), + SC_(0.7894043196898792302818037569522857666016e-9), SC_(0.8907473887532656624292837310188997818123e-9), SC_(0.9999999991092526112467343375707162689811), + SC_(0.1197530963281678850762546062469482421875e-8), SC_(0.1351268990918867631962395950793232393347e-8), SC_(0.9999999986487310090811323680376040492068), + SC_(0.3128908243610339923179708421230316162109e-8), SC_(0.3530594877843318503414910172348506997629e-8), SC_(0.9999999964694051221566814965850898276515), + SC_(0.654608101058329339139163494110107421875e-8), SC_(0.7386461438461727721842489523977686137653e-8), SC_(0.9999999926135385615382722781575104760223), + SC_(0.1037359531608217366738244891166687011719e-7), SC_(0.1170534884254671319566985248995045101925e-7), SC_(0.9999999882946511574532868043301475100495), + SC_(0.2613260008388351707253605127334594726563e-7), SC_(0.2948748151669259830348980474563029807727e-7), SC_(0.9999999705125184833074016965101952543697), + SC_(0.4653804097642932902090251445770263671875e-7), SC_(0.525125559152401230823420797240431154212e-7), SC_(0.9999999474874440847598769176579202759569), + SC_(0.8228098380413939594291150569915771484375e-7), SC_(0.9284414797271396176853241560056526014051e-7), SC_(0.9999999071558520272860382314675843994347), + SC_(0.1440129295815495424903929233551025390625e-6), SC_(0.1625011895322124534932667008476984221693e-6), SC_(0.9999998374988104677875465067332991523016), + SC_(0.373797803376874071545898914337158203125e-6), SC_(0.4217856540365096975937056642610100404098e-6), SC_(0.99999957821434596349030240629433573899), + SC_(0.72830744102247990667819976806640625e-6), SC_(0.8218069436902647194923144120911901417405e-6), SC_(0.9999991781930563097352805076855879088099), + SC_(0.10260146154905669391155242919921875e-5), SC_(0.1157733517254662264894221065154775581876e-5), SC_(0.9999988422664827453377351057789348452244), + SC_(0.2678315240700612775981426239013671875e-5), SC_(0.3022155120513748374723247514313931180799e-5), SC_(0.9999969778448794862516252767524856860688), + SC_(0.40205004552262835204601287841796875e-5), SC_(0.4536648954950918836538342339896039137139e-5), SC_(0.9999954633510450490811634616576601039609), + SC_(0.10320758519810624420642852783203125e-4), SC_(0.1164572890196433412287423508862147085112e-4), SC_(0.9999883542710980356658771257649113785291), + SC_(0.233581158681772649288177490234375e-4), SC_(0.2635681132346089850308780034445740445022e-4), SC_(0.9999736431886765391014969121996555425955), + SC_(0.4860912667936645448207855224609375e-4), SC_(0.5484952583250342934869652426491962534587e-4), SC_(0.9999451504741674965706513034757350803747), + SC_(0.0001085917465388774871826171875), SC_(0.0001225326640313436930517738593548252912385), SC_(0.9998774673359686563069482261406451747088), + SC_(0.000165569479577243328094482421875), SC_(0.0001868251497546456322442598271644064384812), SC_(0.9998131748502453543677557401728355935615), + SC_(0.0004721705918200314044952392578125), SC_(0.0005327874195307728646012144512357625925006), SC_(0.9994672125804692271353987855487642374075), + SC_(0.00095945619978010654449462890625), SC_(0.001082630055365222472802272152269070171599), SC_(0.9989173699446347775271977278477309298284), + SC_(0.0011034240014851093292236328125), SC_(0.001245080150435437151082626328085359462604), SC_(0.9987549198495645628489173736719146405374), + SC_(0.00225476245395839214324951171875), SC_(0.002544222668224971136268866790058243266433), SC_(0.9974557773317750288637311332099417567336), + SC_(0.006128217093646526336669921875), SC_(0.0069148659371010946376763193961158375309), SC_(0.9930851340628989053623236806038841624691), + SC_(0.01089772023260593414306640625), SC_(0.01229627370763712993609092392080824128196), SC_(0.987703726292362870063909076079191758718), + SC_(0.0229592286050319671630859375), SC_(0.02590216393432758991206927422057375200344), SC_(0.9740978360656724100879307257794262479966), + SC_(0.043352998793125152587890625), SC_(0.04888799071126766325637634356499925915047), SC_(0.9511120092887323367436236564350007408495), + SC_(0.06324388086795806884765625), SC_(0.07126804593959971882312270799434219648871), SC_(0.9287319540604002811768772920056578035113), + SC_(0.2158693373203277587890625), SC_(0.2398511631288111077158786048968679107576), SC_(0.7601488368711888922841213951031320892424), + SC_(0.334280669689178466796875), SC_(0.3636043520990431448970776140491250415292), SC_(0.6363956479009568551029223859508749584708), + }; +#undef SC_ + + + diff --git a/test/erfc_inv_data.ipp b/test/erfc_inv_data.ipp new file mode 100644 index 000000000..86715289c --- /dev/null +++ b/test/erfc_inv_data.ipp @@ -0,0 +1,105 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> erfc_inv_data = { + SC_(0.00956696830689907073974609375), SC_(1.832184391051582711731256541599359331735), + SC_(0.063665688037872314453125), SC_(1.311339282092737086640055105484822812599), + SC_(0.068892158567905426025390625), SC_(1.286316565305373898738127895195338854718), + SC_(0.071423359215259552001953125), SC_(1.274755321776344058215704428086211324587), + SC_(0.0728825032711029052734375), SC_(1.268242522387371561738393687518023984868), + SC_(0.09234277904033660888671875), SC_(1.190178756246535875259766567441510867604), + SC_(0.102432854473590850830078125), SC_(1.154826903977823078146497880706118113588), + SC_(0.1942635476589202880859375), SC_(0.9178735528443878579995511780412810469667), + SC_(0.19508080184459686279296875), SC_(0.9161942752629032646404767631869277618212), + SC_(0.21972350776195526123046875), SC_(0.8678064594007161661713535461829067693456), + SC_(0.224929034709930419921875), SC_(0.8580919924152284867224297625328485999768), + SC_(0.2503655254840850830078125), SC_(0.8127923779477598070926819995714374417663), + SC_(0.25179326534271240234375), SC_(0.8103475955423936417307157186107256388648), + SC_(0.2539736330509185791015625), SC_(0.8066326381314558738773191719462921998277), + SC_(0.27095401287078857421875), SC_(0.7784313874823551106598826200232666844639), + SC_(0.2837726771831512451171875), SC_(0.7579355956263440770864088550908621329508), + SC_(0.298227965831756591796875), SC_(0.7355614373173595299453301005770428516518), + SC_(0.3152261674404144287109375), SC_(0.7101588837290742217667270852502075077668), + SC_(0.342373371124267578125), SC_(0.6713881533266128640126408255047881638389), + SC_(0.347730338573455810546875), SC_(0.6639738263692763456669198307149427734317), + SC_(0.3737452030181884765625), SC_(0.6289572925573171740836428308746584439674), + SC_(0.37676393985748291015625), SC_(0.6249936843093662471116097431474787933967), + SC_(0.42041814327239990234375), SC_(0.5697131213589784467617578394703976041604), + SC_(0.4238486588001251220703125), SC_(0.5655171880456876504494070613171955224472), + SC_(0.4420680999755859375), SC_(0.5435569422159360687847790186563654276103), + SC_(0.553845942020416259765625), SC_(0.4186121208546731033057205459902879301199), + SC_(0.55699646472930908203125), SC_(0.4152898953738801984047941692529271391195), + SC_(0.59405887126922607421875), SC_(0.3768620801611051992528860948080812212023), + SC_(0.603826224803924560546875), SC_(0.3669220210390825311547962776125822899061), + SC_(0.61633408069610595703125), SC_(0.3542977152760563782151668726041057557165), + SC_(0.63310086727142333984375), SC_(0.3375493847053488720470432821496358279516), + SC_(0.634198963642120361328125), SC_(0.3364591774366710656954166264654945559873), + SC_(0.722588002681732177734375), SC_(0.2510244067029671790889794981353227476998), + SC_(0.763116896152496337890625), SC_(0.213115119330839975829499967157244997714), + SC_(0.784454047679901123046875), SC_(0.1934073617841803428235669261097060281642), + SC_(0.797477066516876220703125), SC_(0.1814532246720147926398927046057793150106), + SC_(0.81746232509613037109375), SC_(0.1632073953550647568421821286058243218715), + SC_(0.843522548675537109375), SC_(0.1395756320903277910768376053314442757507), + SC_(0.8441753387451171875), SC_(0.1389857795955756484965030151195660030168), + SC_(0.87748873233795166015625), SC_(0.109002961098867662134935094105847496074), + SC_(0.8911724090576171875), SC_(0.09674694516640724629590870677194632943569), + SC_(0.91597831249237060546875), SC_(0.07460044047654119877070700345343119035515), + SC_(0.94951736927032470703125), SC_(0.04476895818328636312384562686715995170129), + SC_(0.970751285552978515625), SC_(0.0259268064334840921104659134138093242797), + SC_(0.97513782978057861328125), SC_(0.02203709146986755832638577823832075055744), + SC_(0.97952878475189208984375), SC_(0.01814413302702029459097557481591610553903), + SC_(0.981178104877471923828125), SC_(0.01668201759439857888105181293763417899072), + SC_(1.0073254108428955078125), SC_(-0.006492067534753215749601670217642082465642), + SC_(1.09376299381256103515625), SC_(-0.08328747254794857150987333986733043734817), + SC_(1.0944411754608154296875), SC_(-0.08389270963798942778622198997355058545872), + SC_(1.264718532562255859375), SC_(-0.2390787735821979528028028789569770109829), + SC_(1.27952671051025390625), SC_(-0.2530214201700340392837551955289041822603), + SC_(1.29262602329254150390625), SC_(-0.2654374523135675523971788948011709467352), + SC_(1.3109557628631591796875), SC_(-0.282950508503826367238408926581528085458), + SC_(1.31148135662078857421875), SC_(-0.2834552014554130441860525970673030809536), + SC_(1.32721102237701416015625), SC_(-0.2986277427848421570858990348074985028421), + SC_(1.3574702739715576171875), SC_(-0.3282140305634627092431945088114761850208), + SC_(1.362719058990478515625), SC_(-0.3334035993712283467959295804559099468454), + SC_(1.3896572589874267578125), SC_(-0.3603304982893212173104266596348905175268), + SC_(1.4120922088623046875), SC_(-0.3831602323665075533579267768785894144888), + SC_(1.41872966289520263671875), SC_(-0.3899906753567599452444107492361433402154), + SC_(1.45167791843414306640625), SC_(-0.4244594733907945411184647153213164209335), + SC_(1.48129451274871826171875), SC_(-0.4563258063707025027210352963461819167707), + SC_(1.4862649440765380859375), SC_(-0.4617640058971775089811390737537561779898), + SC_(1.50937330722808837890625), SC_(-0.4874174763856674076219106695373814892182), + SC_(1.5154802799224853515625), SC_(-0.4943041993872143888987628020569772222018), + SC_(1.52750003337860107421875), SC_(-0.5079978091910991117615000459548117088362), + SC_(1.53103363513946533203125), SC_(-0.5120597685873370942783226077302881881069), + SC_(1.58441460132598876953125), SC_(-0.5756584292527058478710392476034273328569), + SC_(1.5879499912261962890625), SC_(-0.5800336103175463592377907341030447077804), + SC_(1.59039986133575439453125), SC_(-0.5830784871670823806198622501806646778319), + SC_(1.59455978870391845703125), SC_(-0.588273673825686998734497652983815773459), + SC_(1.59585726261138916015625), SC_(-0.5899005483108011364541949539839185473259), + SC_(1.5962116718292236328125), SC_(-0.5903454775096607218832535637355431851718), + SC_(1.6005609035491943359375), SC_(-0.5958247243549040349587326482492767206448), + SC_(1.6150619983673095703125), SC_(-0.6143583249050861028039832921829036722514), + SC_(1.62944734096527099609375), SC_(-0.6331707263097125575937994856370309207836), + SC_(1.64380657672882080078125), SC_(-0.6524069265890823819975498133014027247554), + SC_(1.6469156742095947265625), SC_(-0.656635855345815020063728463464436343698), + SC_(1.67001712322235107421875), SC_(-0.6888269167957872563013714303376548038671), + SC_(1.6982586383819580078125), SC_(-0.7302336318927408409119676651737758401138), + SC_(1.74485766887664794921875), SC_(-0.8046505193013635090578266413458426260098), + SC_(1.75686132907867431640625), SC_(-0.8253191678260588578995203396384711816647), + SC_(1.81158387660980224609375), SC_(-0.9300427626888758122211127950646282789481), + SC_(1.826751708984375), SC_(-0.9629665092443368464606966822833571908852), + SC_(1.83147108554840087890625), SC_(-0.9736479209913771931387473923084901789046), + SC_(1.84174954891204833984375), SC_(-0.997713670556719074960678197806799852186), + SC_(1.8679864406585693359375), SC_(-1.065050516333636716777334376076374184102), + SC_(1.90044414997100830078125), SC_(-1.164612422633086435501625591693259387477), + SC_(1.91433393955230712890625), SC_(-1.215315881176612875682446995412738776976), + SC_(1.91501367092132568359375), SC_(-1.217962731073139868794942852653058932976), + SC_(1.918984889984130859375), SC_(-1.233778505900771488542027767896521427575), + SC_(1.92977702617645263671875), SC_(-1.28019542575660930623179572273596558907), + SC_(1.93538987636566162109375), SC_(-1.306695301483797253764522033930388453334), + SC_(1.93773555755615234375), SC_(-1.318335478463913327121670503572736587296), + SC_(1.94118559360504150390625), SC_(-1.33613349872692113073358883961598631154), + SC_(1.96221935749053955078125), SC_(-1.468821071545234761861756248744372345584), + SC_(1.98576259613037109375), SC_(-1.733272259459038694476413373595347034928), + SC_(1.9881370067596435546875), SC_(-1.77921769652839903464038407684397479173), + SC_(1.99292266368865966796875), SC_(-1.904368122482929779094714951471938518496), + }; +#undef SC_ + diff --git a/test/gamma_inv_big_data.ipp b/test/gamma_inv_big_data.ipp new file mode 100644 index 000000000..76d73d37a --- /dev/null +++ b/test/gamma_inv_big_data.ipp @@ -0,0 +1,136 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 130> gamma_inv_big_data = { + SC_(464.56927490234375), SC_(0.12698681652545928955078125), SC_(440.0905015614498663381793089656310373835539712118699915057655756381750702144551901256933419148259778), SC_(489.2489328115381710888005650788332818303123263238882523231037266766796974176786885492715539575179927), + SC_(464.56927490234375), SC_(0.135477006435394287109375), SC_(440.9201626373476098986751145295281211597851207555116002836559277456581502883881279408150408804624974), SC_(488.3596775997958518494466317847702951287228545295142304855338082929544530999908165788368449119633373), + SC_(464.56927490234375), SC_(0.22103404998779296875), SC_(447.8707097559117500373730010123474103179731538915785876172851019700931527209673015311979439047788638), SC_(480.9951697728710868281134950146803822897647194680614404153327850687509247330919989892805468881282719), + SC_(464.56927490234375), SC_(0.308167040348052978515625), SC_(453.5243991344879957369314553806138325182261928645340021307574697842383156443940903587657239443012288), SC_(475.1149279136111828287528971449692848991760379145030491172523136909085503134739001275580865573262267), + SC_(464.56927490234375), SC_(0.6323592662811279296875), SC_(471.5586320968249281038445929212575977142103853999994962593917575305139999808448747718468071828960383), SC_(456.9895432265540105242329036050454287207764722373079606619854048979024366594658280592117689264157992), + SC_(464.56927490234375), SC_(0.814723670482635498046875), SC_(483.7962479352174443104604814361337114707603182758678093812861642605333000597647127303417718754975292), SC_(445.2102204107761028803290781726535172344996283727995749912027305150811353997318714622791119089755303), + SC_(464.56927490234375), SC_(0.835008561611175537109375), SC_(485.5413501990585182267303678139012739812082864500829920491986081765032489089832623030329596860813566), SC_(443.563211692283275278200948788175382734228099875239143691921387947329995022394758603528566488084557), + SC_(464.56927490234375), SC_(0.905791938304901123046875), SC_(493.1530014001506637843558286915347874415133013116095848032725512370251487272101090417362802627532), SC_(436.4721607137502194476515600944759096109462004602962052350793459815125792956858001222848555476140453), + SC_(464.56927490234375), SC_(0.9133758544921875), SC_(494.1980223559677419266737342424067342426878050583575377942941770747042395000546795305409031642061286), SC_(435.5102279299145333712817106435136572491533987684414157140918464149027877988350441152775081891128202), + SC_(464.56927490234375), SC_(0.968867778778076171875), SC_(505.5712348542740722018328982886099757924348908530379539931536263900192339710371141619690805609322114), SC_(425.2177557338328763751700654579137573643963906759571233052948938600914520447536651895743912216230868), + SC_(581.3642578125), SC_(0.12698681652545928955078125), SC_(553.9669793782038068678935676734904494531472719532817972687153514964534286624955148869921984604413705), SC_(608.9624188837483057145747405401351852945713877500076249931706854984537278152556192058495450356661075), + SC_(581.3642578125), SC_(0.135477006435394287109375), SC_(554.8986621587098884100882390919729772471677865482900127899068620712298280708065488221682075356651129), SC_(607.9711405772914939464180467890095695334450990881469482983020058157368245651758798593316096874550468), + SC_(581.3642578125), SC_(0.22103404998779296875), SC_(562.6989508379479360762737530017662823397707943257475597443912537342208024566957230695521899983347868), SC_(599.7568829483629048923260578901093044774886065740947188058513300551937073309051987160205195739811767), + SC_(581.3642578125), SC_(0.308167040348052978515625), SC_(569.0374062500558452762422305261339987947322576241495557758885978701225529419403450771958235312394718), SC_(593.191871592557187889891303913053953793280772650663480941873292317612622398375064237989516603392702), + SC_(581.3642578125), SC_(0.6323592662811279296875), SC_(589.2186806341801070520302845910583423903524143672474575926325599791162545901801711272745264959036703), SC_(572.9194442994055998344224745611620692130093494754880686659071973997039947893559220289586718667113235), + SC_(581.3642578125), SC_(0.814723670482635498046875), SC_(602.8821834284482585691264752294355355054448218400550509491940689285405704202939688051624635513663289), SC_(559.7142416966026142699409579688200021128645733168138617853078743374086688135798773103892188217158995), + SC_(581.3642578125), SC_(0.835008561611175537109375), SC_(604.828643910421477172904091611031783808541641171718978680317644118217631106164554044832916318314536), SC_(557.8658766883129854673479272570045089935130097354126524991225881637889319445059039834646698913385156), + SC_(581.3642578125), SC_(0.905791938304901123046875), SC_(613.3129282087309383781017594217145054083831628491043296373214531659991173365355850527876757575909822), SC_(549.9022051711790846699713504466950223326019400934082404934885928594446512232330430586772029938196745), + SC_(581.3642578125), SC_(0.9133758544921875), SC_(614.4770468653682831072313903473820106979961810147208338352666609984349886811227288705529284870561508), SC_(548.8211770528904363817509904880593930190623124578155064175427290011681864479910298041022961043374652), + SC_(581.3642578125), SC_(0.968867778778076171875), SC_(627.1355994680180288387606201509867766037880624816769586905572692453061589536718698854811798574447639), SC_(537.2434046094288504301880670875538722146124348492278961853627085015109570171947537038965207722816761), + SC_(1951.531005859375), SC_(0.12698681652545928955078125), SC_(1901.241568773770922566176551111324528061539415262289896946584544719985434774304519807954392335998056), SC_(2002.021320186005593269954630164748892984239050639769313631332995488067837088154452019302573500276847), + SC_(1951.531005859375), SC_(0.135477006435394287109375), SC_(1902.97354682881850141437203184951674874932885186251119744647673131837322873602175575537383500148183), SC_(2000.229742725423370235870963243869872585198799992240751333551033232397870397441633931357095980768191), + SC_(1951.531005859375), SC_(0.22103404998779296875), SC_(1917.43930352500655310596416160028640714879305407859193609469456367008395474716182237360880885823288), SC_(1985.349994080042670467264605954502004011215186704550449317111515855278342566460300427630408667542252), + SC_(1951.531005859375), SC_(0.308167040348052978515625), SC_(1929.148922991445901527665313030619405739439864321231712634684310583653565217955393471096463768241713), SC_(1973.41380900615111271612206472312245968587969819306141495965251939440367517119089187330085981855822), + SC_(1951.531005859375), SC_(0.6323592662811279296875), SC_(1966.170648608234571927835410663285873075779994557389344920315448169955531572581154381160800811608923), 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SC_(171008.2836354564209492823743402973811390337608777526702016261413773534660200796449809982051665077003), SC_(171919.9826384883939702551089975861816589350398843659033957710496659891252213316559001815815327400668), + SC_(171464.0625), SC_(0.22103404998779296875), SC_(171145.6191662746123960347552525693670128191028466757466580336222001947839794403773558314059941595646), SC_(171782.2331060803849520335453303183326526337638884076039027498474047799995742641818132862456747537524), + SC_(171464.0625), SC_(0.308167040348052978515625), SC_(171256.3361047097958387452441541471127499385772793150774116115411855101108847691601832573807619567347), SC_(171671.2895979873130236903578632050660794939876719454087826341290089368450153075267593819684360323877), + SC_(171464.0625), SC_(0.6323592662811279296875), SC_(171603.7718023434324974514302206535409055277466973477498540819203050822468613895559823545353022871572), SC_(171323.7627427548550174468619344221281973249811264358000342338320713853780519662237808191551395326248), + SC_(171464.0625), SC_(0.814723670482635498046875), SC_(171834.7812554227987093134480125253002251121865641876340406728469955856798309411540595789975061266601), SC_(171093.2116182572157852580015242631784731817460262145817086170604362162178153569504512508225251352553), + SC_(171464.0625), SC_(0.835008561611175537109375), SC_(171867.4225853696747287532499924362369112417451458992299610929008452549979384474959400833956401103073), SC_(171060.6683914212385500445147373534713684868992048983316972221307056329253829244630268336756229649048), + SC_(171464.0625), SC_(0.905791938304901123046875), SC_(172008.9390039799149245319369194588737700358279596931623900607969038096782602911984385367690545112371), SC_(170919.6726353340561334782062787174007300952740547857469982263053520346768626380151551392105674125141), + SC_(171464.0625), SC_(0.9133758544921875), SC_(172028.2607163503237074804240420988332805635207751072405443517988230279811664015979896111754298047853), SC_(170900.4340228751077815276338342892187005146619881755020781374282366220361628632145020859022759543646), + SC_(171464.0625), SC_(0.968867778778076171875), SC_(172236.905515185348613520159589244969601354143298301884427671166903159884473708788358619633391766896), SC_(170692.8701620843835232223758792946766066205588069757104160071945587119499747986058865001180462595334), + SC_(287713.625), SC_(0.12698681652545928955078125), SC_(287101.8390891334881074097768409639811422105174055074247663991980671146735409214228746030537490827894), SC_(288325.6117858325819400290984010245379181857090949256951183079858217963457877563186041586190041828876), + SC_(287713.625), SC_(0.135477006435394287109375), SC_(287123.2018824342060405509691117625160296096920082481743462259847740285682319170194185122522020998578), SC_(288304.1893914925111228677903149683935820829058722939396270910064082845368520684242652738579958079628), + SC_(287713.625), SC_(0.22103404998779296875), SC_(287301.1633297324049890437619176952336530543502592319387571077608781438650607731795664155648991042492), SC_(288125.8139425596597585955824696111531518443099937996959008431783465244705148305958926033655366242125), + SC_(287713.625), SC_(0.308167040348052978515625), SC_(287444.6161875962380911288819205876894144456711595429288695779716233686378459491093953007953541560695), SC_(287982.1345150191015439663126413778115578427703552946723363867416179375439869486600021290202033928835), + SC_(287713.625), SC_(0.6323592662811279296875), SC_(287894.6874489179014071783111281890207171400441598448189286641946463264879755117085964912831960132818), SC_(287531.9720960921756986707071354480752766558762675841713700852373467579201294942464262449709073610385), + SC_(287713.625), SC_(0.814723670482635498046875), SC_(288193.8624685657253120713231978071896495272422042999916280573686361169253355069989533169539117598378), SC_(287233.2554050650734668559921717066438674227127230795422265297442527832155003247585279503101394869551), + SC_(287713.625), SC_(0.835008561611175537109375), SC_(288236.1305748677261264763139679877739018298385456598709163321801212462074437452189204330390376852663), SC_(287191.0854018844619610439948437360860177127073148068576003551456242291546450349862447341532794138293), + SC_(287713.625), SC_(0.905791938304901123046875), SC_(288419.3697666997195573795607481350142848162288944628671823192696454698912657762168311546144078778697), SC_(287008.3668726438555694570198503384645300805316567228857999141205296735618456088406804345883798515772), + SC_(287713.625), SC_(0.9133758544921875), SC_(288444.3862532392817919544710751213857359531657572985942506273978341080423727813156579392633735805302), SC_(286983.4334860286298928127732334989709654886825090910274253943356470926729788108952795775884980653786), + SC_(287713.625), SC_(0.968867778778076171875), SC_(288714.4987040958494150791768379727138187987696180207795301950993211293148528792180512566346062773377), SC_(286714.4019734332685576845212232609186614672218264529859295616546913400761609253508670570311556417372), + SC_(811189.1875), SC_(0.12698681652545928955078125), SC_(810161.8590312371865843168933980041081328325710070299088178861493265134389880308605518391017994747884), SC_(812216.7168437188563979284647249628251019755537428376273766031544801981839282189973470746075052877665), + SC_(811189.1875), SC_(0.135477006435394287109375), SC_(810197.7498979269406216229333830071350035973327816952880714391211131397799458403161724415876017573062), SC_(812180.7663759825506891792362570130089364226324810883040952150567099852012280358577738543974186957542), + SC_(811189.1875), SC_(0.22103404998779296875), SC_(810496.7085818541638251177278868436507795786808493806841003579619404032620103009446072000163980337239), SC_(811881.3936903780003325280400757211189636198717447205229093798838652404976779035708631850160770953495), + SC_(811189.1875), SC_(0.308167040348052978515625), SC_(810737.6596810209441188397088671254688261035908114807153029865438117474573459893950971804983662488815), SC_(811640.2160215165658363785038485826735353144913795793280443540697836953543164349103546800608981662806), + SC_(811189.1875), SC_(0.6323592662811279296875), SC_(811493.4130930334681784464934025270672115225675384515035359787878148824734579705400730129452548515192), SC_(810884.3714518926483987655962044487674578474311600687306107087577719289978417856154576706566527627155), + SC_(811189.1875), SC_(0.814723670482635498046875), SC_(811995.6073944837810507278595067896837851547513348796820987055379061808847590525648652941945063796637), SC_(810382.6354791001731534931504185529321689341432235194142157415772145748880181384490815012739271443493), + SC_(811189.1875), SC_(0.835008561611175537109375), SC_(812066.5471821809268774383895112728062329337775853423876890314159367663267493240015649616128193898023), SC_(810311.7937945344017345682607983665585726031514332575059314866957462286550595648052571524209586357645), + SC_(811189.1875), SC_(0.905791938304901123046875), SC_(812374.0504188456832837387624199860361643216611730814901346962998291782800792848840022126928346732626), SC_(810004.8112205260695868832902010461963334632154865047659274031951119044731278686388069124903893504951), + SC_(811189.1875), SC_(0.9133758544921875), SC_(812416.0278043274019558628676997090748402578371572470019350929977553824684311176493949507102425008448), SC_(809962.9169349809432409688756513667346638313258479402030261893546740734029634599290918129699775355502), + SC_(811189.1875), SC_(0.968867778778076171875), SC_(812869.2110457723077173766634087729640503055024244625964488641332722990514021020625997363526699219015), SC_(809510.8146320036991512143859204546175464755726797868383946222879006660812841804439525397298317135306), + SC_(1340602.5), SC_(0.12698681652545928955078125), SC_(1339281.789217310227603638956713067406749799202177619440960783591278712272782687984485318465325634698), SC_(1341923.411657643639007333777049618448197159121016565415209623831539285788844610953752838000410332646), + SC_(1340602.5), SC_(0.135477006435394287109375), SC_(1339327.937228622487435434723558076320553619035659265800295103113312125271806499402498542109463722967), SC_(1341877.204045283265012388313242013019815575145051478003059780209276879235388989175125836420473319923), + SC_(1340602.5), SC_(0.22103404998779296875), SC_(1339712.322663835647115463627093526039417214514450079134571740611687493566900248789753047878427302874), SC_(1341492.404608383515658278630627107716714934192755525237250501053389901217595022258339024841963265735), + SC_(1340602.5), SC_(0.308167040348052978515625), SC_(1340022.109695761389373850715324652137775669433408208271739480026258057424153514144279448404087341583), SC_(1341182.391006759227701894203217959369703988360879936035471868679603927201823099528589868185221044072), + SC_(1340602.5), SC_(0.6323592662811279296875), SC_(1340993.681516583716822573407774790571084098092620222888109700458969964450847155204734529531816156073), SC_(1340210.728028324176179123620867989801125006955369657205955659737293424313505301822663996693364643671), + SC_(1340602.5), SC_(0.814723670482635498046875), SC_(1341639.21195865400971691711247042743964641649950184497692636094105194253010661428811607626202482777), SC_(1339565.655914919776901964481359375383609125899415767739147337874327439960785358915178860477236871205), + SC_(1340602.5), SC_(0.835008561611175537109375), SC_(1341730.394593718344730348431912807931874774159152781393115335369100867655446425097778085615493303823), SC_(1339474.571382988983554873353395399477444166684037486807114439003468351504601181586784778488843724471), + SC_(1340602.5), SC_(0.905791938304901123046875), SC_(1342125.631024741747482365874818097900648331829089766546145385089458404019214871024479928583367510744), SC_(1339079.85561463612134401235478017313574989598767112679109272711123705107621660725882816550403989336), + SC_(1340602.5), SC_(0.9133758544921875), SC_(1342179.58318466684205310449571855171979481228479144098285572172412554068281390306320099547841289301), SC_(1339025.98655465027920550229578505314111660672260003165765196039783266546290600690587108716340600317), + SC_(1340602.5), SC_(0.968867778778076171875), SC_(1342762.018513537047540360796759927612282441879039821617202574156172598071372824957647151677236387446), SC_(1338444.632164292546392718217459403862675869956866545071666772833839055979245806235185884461105636674), + }; +#undef SC_ + + diff --git a/test/gamma_inv_data.ipp b/test/gamma_inv_data.ipp new file mode 100644 index 000000000..77f2a6637 --- /dev/null +++ b/test/gamma_inv_data.ipp @@ -0,0 +1,206 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 200> gamma_inv_data = { + SC_(9.754039764404296875), SC_(0.12698681652545928955078125), SC_(6.349849983781954486964960115093039567468747967664820851534065876063014180392060424555352423162443804), SC_(13.35954665556050630769710564071384909890705633863167415813992116806093647564994418684947673739937248), + SC_(9.754039764404296875), SC_(0.135477006435394287109375), SC_(6.443655518427200469944002789678415067225918847146256390238048987798948404483563975737891105237977816), SC_(13.20648307403762788665455912344161337928185855631083250140430065533748994841668643195553261756810763), + SC_(9.754039764404296875), SC_(0.22103404998779296875), SC_(7.262299107203547511919478506673940833772294421637239641242141768466064388735007729524335022945474356), SC_(11.97587613793353953360795933135034836336725275703898828955465276118339699687746564051854946108002967), + SC_(9.754039764404296875), SC_(0.308167040348052978515625), SC_(7.97160002999034906812099484768140342301220827266158714468555902274184312977405873043165132107201808), SC_(11.04086573531719578182530917737022531437091511425536985032928875608312723663655612529685653444775069), + SC_(9.754039764404296875), SC_(0.6323592662811279296875), SC_(10.49594448867679616659291265911862460649618853634598426379731165830605063134254492272180891868259779), SC_(8.425658710690898403870129948430842533053798281214253294861614223449877730063968146709177371998177282), + SC_(9.754039764404296875), SC_(0.814723670482635498046875), SC_(12.4361412533700118513726558060654652709881561861912344653689388650864659162098713582998706613513665), SC_(6.942010644016112027186983623736329615032487299108703218184306594043669003923138791980905906975995029), + SC_(9.754039764404296875), SC_(0.835008561611175537109375), SC_(12.72773220471889504229650982423238107961665601639868914273535729539112172086261925240837297640844728), SC_(6.748045775833720602368139518425707341668236119625769419950098701512116894292576121300681462006500869), + SC_(9.754039764404296875), SC_(0.905791938304901123046875), SC_(14.04289410800251925225181397652529175357661325918932141591196059344562473318317362439395292198624427), SC_(5.950449506005127213772609250049267524655862647593667247023701745572192142411485108776352863016275192), + SC_(9.754039764404296875), SC_(0.9133758544921875), SC_(14.22894231389791438923272291986931994699865499051360224227851330535373836529589979766491736680448836), SC_(5.846919097207863070421132965572838346244481147780105186426552033007650921954238889468080440271030891), + SC_(9.754039764404296875), SC_(0.968867778778076171875), SC_(16.33883627400002069452382739015521709223507055939529263776286910527953871886379701300232417366169237), SC_(4.808199239905297896929793615245708425392715583004661735514849678846177677735540465249046080150658016), + SC_(12.69868183135986328125), SC_(0.12698681652545928955078125), SC_(8.784921019272498426105463510492037221421052305018408345439418701592122876845445386318018554294593597), SC_(16.81366073413268797817072489278431450377279434529634658485607365405913967176896119547533072607895572), + SC_(12.69868183135986328125), SC_(0.135477006435394287109375), SC_(8.896383236591753135866154471087553855363191438952119644843354891974586610224750070854729638914776046), SC_(16.6428583989184708274029070066937822603130962248261232385768762701676877405494336531756951743065766), + SC_(12.69868183135986328125), SC_(0.22103404998779296875), SC_(9.862617469220713982665965583371935244918334904047351290421210068334907822439623183798321618935666208), 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SC_(113.5057019266017757553572395745632780679072776470909404929295494401706417401173397780562513398354094), SC_(86.35612234649621027962836929188910335213017738233582627676662911520231216242529540127044730529945437), + SC_(99.6461334228515625), SC_(0.968867778778076171875), SC_(119.0637249422376961671341002862243780120617645758579051054825546123558214039984323476889351592427586), SC_(81.87811103947525043911330324465564757254090604246213044539733449282580273670542563644266281418367064), + }; +#undef SC_ + + diff --git a/test/gamma_inv_small_data.ipp b/test/gamma_inv_small_data.ipp new file mode 100644 index 000000000..aa813be6b --- /dev/null +++ b/test/gamma_inv_small_data.ipp @@ -0,0 +1,235 @@ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 230> gamma_inv_small_data = { + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.12698681652545928955078125), SC_(0.2239623606222809074122747811596115646210220735131141509259977248899758059576948436798908594057794725e-517862), SC_(0.4348301951174619607003912855228982264838968134589390827069898370149065135278987288014463439625604227e-34079), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.135477006435394287109375), SC_(0.9832661142546970309065948494116195914044751609094617663675341704778529043412686011701793912754530322e-501622), SC_(0.174464879621346471044494182889773112103066192989857880445657763562407515813032064473382568887549155e-36531), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.22103404998779296875), SC_(0.8367188988033804556441828789142666841098768711906267440588049611090664598162301930590308375972597743e-378782), SC_(0.258455732678645501224573885451125893770282157149004436646252270592694652815341234774299743101856032e-62682), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.308167040348052978515625), SC_(0.2205873777306860224249147129964654825654772808335654855092638386176586877302351181235652317545193535e-295387), SC_(0.8841860184607764624716199689352601821223421516721282102648409952215997935928956722103353938434066324e-92450), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.6323592662811279296875), SC_(0.3493006236342872627718268547235558974110453236408370553242260146873678814721124761086879547194341511e-115006), SC_(0.4608598583105815395151633406100700185774288566360162761619249236808047706604292396779780216945145655e-251105), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.814723670482635498046875), SC_(0.1307814977375223234061697346935750965506413900088172878810937417241710162327371948752530053786611785e-51419), SC_(0.3622247004413029118815942894731774856416774764001901351075227450895011592470141353244995696694896319e-423065), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.835008561611175537109375), SC_(0.3407200009988954922968231040073535782592944660979975072820794043769535120578823528671413904276314187e-45248), SC_(0.40372680297867266879468280760885460454290870234025769400910020388606341293084660523346882864736631e-452163), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.905791938304901123046875), SC_(0.1289547938840298804104533289571370292449761595410863583261482629787772675014213604864800473713128265e-24829), SC_(0.4473385981314550569653428208559396173592281928696895222750064028650962596615670418178431916147100131e-592788), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.9133758544921875), SC_(0.2660916718277898648353162270511365962890563186837505224306478516822433113175485818467727525519960066e-22737), SC_(0.6065564736497368801762295026989180367578487822196046971460097223432141011146248705024295896736469553e-613849), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.968867778778076171875), SC_(0.1473132778697774093465699897329346789995005439716750867355563556776005696738800270428366009192934656e-7936), SC_(0.6008673144885279892931053509531579477623416731984800692761855880946172764550286223383877167245766902e-870647), + SC_(0.216575062950141727924346923828125e-5), SC_(0.12698681652545928955078125), SC_(0.8500545195325853360129687030255002945655259106744411868936199400386357779413785394252606937409149347e-413825), SC_(0.1168004659330719336985231445704960139768804061229080049955849277439667205757483213992057220049540351e-27232), + SC_(0.216575062950141727924346923828125e-5), SC_(0.135477006435394287109375), SC_(0.7202712693841376368216354323694296792211539722228687704587193232171864702453427173668794173358037833e-400847), SC_(0.2252614264634486253201447701279415032436991420138140648287940022937659078016357382571415445200213866e-29192), + SC_(0.216575062950141727924346923828125e-5), SC_(0.22103404998779296875), SC_(0.3000327882810984063485886505392286395644270610646887344028999837107973933925695988567563167728045096e-302685), SC_(0.1498490062697145666757563645833685565995577423177516513148297134617720423214544808430852418820345691e-50089), + SC_(0.216575062950141727924346923828125e-5), SC_(0.308167040348052978515625), SC_(0.1307004433406758598014428627573055623164994782163856086219980606033295840179348304515381612579150121e-236044), SC_(0.8660761056125063203743178336674650301770491595301193399539391174485981887085468404902606561784465577e-73877), + SC_(0.216575062950141727924346923828125e-5), SC_(0.6323592662811279296875), SC_(0.1183316844216403526657583856991948601012391672622156410900987978451851712537691445331530312832797228e-91901), SC_(0.1591103740030003499118723418460479045052885251014777131066205491018123438721843692240137750119143378e-200658), + SC_(0.216575062950141727924346923828125e-5), SC_(0.814723670482635498046875), SC_(0.1674134325233505112606127764095613275142520744956844047709789044616079191515075547796473275770297974e-41089), SC_(0.3612384181299129469928622773435995792869558303198558837492185251476669534034831500723829680749590518e-338072), + SC_(0.216575062950141727924346923828125e-5), SC_(0.835008561611175537109375), SC_(0.6513834432351054603935790801928696636769937839154029815627322893900997971153735962011923847861105547e-36158), SC_(0.2131860779789139256037424176809792095632204680337931857718865949785434825481960848149849797507583928e-361324), + SC_(0.216575062950141727924346923828125e-5), SC_(0.905791938304901123046875), SC_(0.2178005192333624215784803979717837598562757410620706503281900091107825680350931337396209745019443998e-19841), SC_(0.4588085686858952503042372234625485480201105499533793863948812241019952913466033029898154440318470087e-473698), + SC_(0.216575062950141727924346923828125e-5), SC_(0.9133758544921875), SC_(0.2044546663391562477705354540370760821409485274148331480980359986548683615601729039239239614213065235e-18169), SC_(0.7629190706250225994074616748192014275571599576934851198947217095948168969106405661485602778753139218e-490528), + SC_(0.216575062950141727924346923828125e-5), SC_(0.968867778778076171875), SC_(0.4096487433832820170639333743831954761351590021205697452562807451168473548459292965564190974788844749e-6342), SC_(0.1297397706877886168379273776356177633807822967666079798581567618567115144239687805362946264983224597e-695735), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.12698681652545928955078125), SC_(0.460289138417023240211835453578728773675893958998560741266616339802513531273727174963248143404409346e-123279), SC_(0.1227830593913913689043483463822718345251662053270570059659747620207531684387617971000150489495390622e-8112), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.135477006435394287109375), SC_(0.6439991263768059081770908658752929598087302065287439367522303278837306676039083594400840449710127147e-119413), SC_(0.1936099411280065786600982265792717722767945024296565235561154800531592756609798398701501116557434413e-8696), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.22103404998779296875), SC_(0.2064880008938546016135832883384716687050715182089334885089172611730709160883330305789994119189833204e-90170), SC_(0.9636789658116208361662045675832611481515953065813422246552622845538828381107902733604814371406262203e-14922), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.308167040348052978515625), SC_(0.4672763627314931787409452529965189883928299396526058431027743531008131781333308125789010166445570443e-70318), SC_(0.5333958645484846264899444976530540348886306684971665853490471154195979393073606296468477524956455995e-22008), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.6323592662811279296875), SC_(0.1232079542911688319553284787645084489874316487664902104337227730213965329139279051931070447347538552e-27377), SC_(0.1745508475051527213009559427098382072109145551947732820968960362083588524860039683359355038443208669e-59776), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.814723670482635498046875), SC_(0.1296871584563229453031476552376127081733495432987283917208707733104304941994062901047954719016264728e-12240), SC_(0.3119684049016554066402196968135164105103042293007478934952055073327703314807997817770702753165916888e-100712), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.835008561611175537109375), SC_(0.1744156521766280393141688111513263176846896834358940986864024481897475752003777342235119371016119832e-10771), SC_(0.414282922235332770074768291902894089464647802773088955269313913489638472173573774702189380297337679e-107639), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.905791938304901123046875), SC_(0.9134084128666229552771771583922013819958821761786059137020004689307969234520514787745058008336030516e-5911), SC_(0.2085723217142930347094312090575084590094262400334108078273206186864903163220911787400224976002324296e-141115), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.9133758544921875), SC_(0.1106626349398291975361851522331692345605272541414957089479290323484012747768775482956833740133997023e-5412), SC_(0.5019947814569084821972649400070925089480757099017899755164384177383260941610254064551897764836241042e-146129), + SC_(0.72700195232755504548549652099609375e-5), SC_(0.968867778778076171875), SC_(0.2608738285704836848239817641016769732591640383513307387951906227411127886545943640386587729196529798e-1889), SC_(0.9391569555686887348572282531856270103209070385690661485800330330324704801693506326997201625650584228e-207261), + SC_(0.14000004739500582218170166015625e-4), SC_(0.12698681652545928955078125), SC_(0.339390501360481883384596846727229310838380493148059239101677398294559127652204528478268766605745691e-64017), SC_(0.8930970995970272744149034236099991596203915342210295009610487968429898591362281212215882612432334172e-4213), + SC_(0.14000004739500582218170166015625e-4), SC_(0.135477006435394287109375), SC_(0.1478165479650781320569422389970436453616410129224510514070984667774343182955852078638154528221379033e-62009), SC_(0.6166505887233514954458318978151305493474273629828733433707686152568316580969117228752011354754100409e-4516), + SC_(0.14000004739500582218170166015625e-4), SC_(0.22103404998779296875), SC_(0.2636223483866300712285073854631877501273640030610221560132162856371236675991856524728587705746297074e-46824), SC_(0.1179070268459566531464548375815239425103518998753490696351340626821057948491563316259813424943317095e-7748), + SC_(0.14000004739500582218170166015625e-4), SC_(0.308167040348052978515625), SC_(0.3085719676015113786900559879398114388323199985826467699671192420873003184123044267039751593375655573e-36515), SC_(0.1866107010075390801548946745085437000221498205178902450996305353941274930070444383308861599008651802e-11428), + SC_(0.14000004739500582218170166015625e-4), SC_(0.6323592662811279296875), SC_(0.7743073115105881135630532411055705358953657608902295030130428779685124579667759424182550282392251774e-14217), SC_(0.3891618185709909838836983643181316431906213234898591756880393231944196898745253316876972214816450983e-31041), + SC_(0.14000004739500582218170166015625e-4), SC_(0.814723670482635498046875), SC_(0.222216709392403859077345757164648641055669472167062646627925353942822049167347003774956605136793258e-6356), SC_(0.1553098038974094715628835971928945389167186664370201233578708844500900929376674522237036066734443409e-52298), + SC_(0.14000004739500582218170166015625e-4), SC_(0.835008561611175537109375), SC_(0.1762412434192703586953808931252239830607212327509661647619944439092944014510892565024867820443452495e-5593), SC_(0.1427522693565869357184324274786304328120490007393000466898389151996230157936654133010870707870098114e-55895), + SC_(0.14000004739500582218170166015625e-4), SC_(0.905791938304901123046875), SC_(0.2259490828022611962651286035615477033923299926896382236832176774513532167028895827574135582466792268e-3069), SC_(0.2241017130581831897730462424904751051004418852587147386250684104551402650352326106103864234388080914e-73279), + SC_(0.14000004739500582218170166015625e-4), SC_(0.9133758544921875), SC_(0.1005028821672110309415483560972173499716663965719086172500020531716576068606833471119991129664424692e-2810), SC_(0.6979771306818377405428798318204058016896524472057195949985191958384843690760804078528960881333370124e-75883), + SC_(0.14000004739500582218170166015625e-4), SC_(0.968867778778076171875), SC_(0.4399884537105952873524158618429125109607698051551640190711957784190261472987797258480803340986411014e-981), SC_(0.8636123050850289155805959438262889417641775752173387265269754240416817175647540561424609415147039372e-107628), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.12698681652545928955078125), SC_(0.8931367727030424252994406039858535255018561374129051577258665728233205820316949561800237503288164831e-52118), SC_(0.104535845548742628763976899233973069829736957363121321494519130075530673267495622335035426409117717e-3429), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.135477006435394287109375), SC_(0.2589093342255974756648686300693355584048765521630894862720053856422193831195464113676002513110719721e-50483), SC_(0.16198355017053579069996072009618961612118734166699913346916923061929129768370182367736118778593987e-3676), + SC_(0.17196454791701398789882659912109375e-4), SC_(0.22103404998779296875), 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SC_(0.2521961993473725269543103190526000592848006826712668650609107713979225891789595253816913570458321053), + SC_(1.1576130390167236328125), SC_(0.905791938304901123046875), SC_(2.633432043738438920475686365204867279535715696108322078844743253044475088750509042564699876424341881), SC_(0.1483234119621405878414264034510012616620586207451637965494203477538978586659149769842382367332429083), + SC_(1.1576130390167236328125), SC_(0.9133758544921875), SC_(2.72130872698585367980892827768093808349422735039784935732743888146428407234271507560375936478000293), SC_(0.1372597171858880965660262090243239004065344795529468715964073094745543966462191632895815182745602503), + SC_(1.1576130390167236328125), SC_(0.968867778778076171875), SC_(3.785791973663415980437557813576367686468274240280844818238132947111766562677797108571092977222954148), SC_(0.05460369537686154741964828809285588189958375691729530341605824976415924863104037841047769736294474202), + SC_(3.451677799224853515625), SC_(0.12698681652545928955078125), SC_(1.530624210920382284399051690330519554082768377808227128434910351585439262396314857365872406127756053), SC_(5.574698692018694700071254764468530910674337221105810057056140063183708253803233221537898874285968149), + SC_(3.451677799224853515625), SC_(0.135477006435394287109375), SC_(1.57362951537156159661431963850316197825725692404254310133476110219497865497181922972348704221144267), SC_(5.473138259331420669341549290189455776876089291573320486520460515454004459005991000402878126742323236), + SC_(3.451677799224853515625), SC_(0.22103404998779296875), SC_(1.965746189751920939352691371708867882866485644934665910275215637276934063111994360502170024656532926), SC_(4.673268932629649290920788790871645128427792301349128507557725406338670163075890517015361340870973159), + SC_(3.451677799224853515625), SC_(0.308167040348052978515625), SC_(2.327895119537809122811571079104146373091942499322663334330138868463649582849214297599497785464023609), SC_(4.087212616619640052827803625327882426668270211845906555717997902681324813869427584063421609836271661), + SC_(3.451677799224853515625), SC_(0.6323592662811279296875), SC_(3.75528141373881029632814630875677475329797026383089604597730893204313705777013022465404183283606387), SC_(2.569587565718771610879334081961858540627292872731864015723807479426770255861746498923235709828384543), + SC_(3.451677799224853515625), SC_(0.814723670482635498046875), SC_(4.96884003357348744166695303018087322225920527947015852633041450072604536715189275307836871232572009), SC_(1.808854082468155248893602071807914212511204947986080304490964939078875022325577669885755397028389777), + SC_(3.451677799224853515625), SC_(0.835008561611175537109375), SC_(5.158353930264407591141911762793226010112593356998663398064398144517377920081089377422910049379929565), SC_(1.715980010073803534303771366139081776620894782167054529829701347745407112585687815364593264795112742), + SC_(3.451677799224853515625), SC_(0.905791938304901123046875), SC_(6.033289965252111343235548988260081149382205549483973493795976997290232926915123620068323107608683628), SC_(1.352275181601738349544903018712053305322415074737749256814106096094379708637873120535301778996625571), + SC_(3.451677799224853515625), SC_(0.9133758544921875), SC_(6.159558502218759156855593347123796639713710181006289624265126514849066440847214437545387207874425825), SC_(1.307343882589167249227348096476723299076394367682521236029969747367006850476646265679909563475382611), + SC_(3.451677799224853515625), SC_(0.968867778778076171875), SC_(7.629721526111596714926608293951817133394478840350583630509809227411335042302400047948310886098339447), SC_(0.8888562589855856249970510117949218013780557896569702678778170096314482315508385771830674194073130007), + SC_(7.88237094879150390625), SC_(0.12698681652545928955078125), SC_(4.844756913723969959067287277171911999837470793758606581225223922671033928947591753701039975119122724), SC_(11.12140061565551307712696521346785564818459658303388261577657023731780253761925900663967051291410247), + SC_(7.88237094879150390625), SC_(0.135477006435394287109375), SC_(4.92589841429445857984116213454195626941969554762437568942405393109061358846779703244707367287195874), SC_(10.98108673974325765715303476569448043717025909739070272421996079460616793044380919148069611984481183), + SC_(7.88237094879150390625), SC_(0.22103404998779296875), SC_(5.638560786091699351123899548104079140326479351812437435515873530054583524774506292906280460797833098), SC_(9.856971749403947096825033526804503350139699007114203432014510548601579942970233052344921183682478769), + SC_(7.88237094879150390625), SC_(0.308167040348052978515625), SC_(6.261950304915309281829468403028535426991401847081170989871860432142896140500397036811479661870310452), SC_(9.008089510076067200136139130648726551693948927672843011870424870503873689961795526187249783039453752), + SC_(7.88237094879150390625), SC_(0.6323592662811279296875), SC_(8.515696299640880133197109983199329596272030624087086595627597017830637055606241372115218386957077541), SC_(6.663555576689267756072158572952790106439493579837933095211305263932099060610857357449696082841290954), + SC_(7.88237094879150390625), SC_(0.814723670482635498046875), SC_(10.27654630671120554490045038357659906211059116424349156516366240949281281132412336553759708614503569), SC_(5.358806007367185234198567710812204619111113995842409126062778294563755657191876758668190895570062463), + SC_(7.88237094879150390625), SC_(0.835008561611175537109375), SC_(10.54290123180201991341358069212563540909069376048726692932615624195393691868675916167585864103920169), SC_(5.189957597342458441237551570990662655102247885622660703905796425516481457633806742834164945056029809), + SC_(7.88237094879150390625), SC_(0.905791938304901123046875), SC_(11.74906177063792723279214648184444217353993726634172219771095374095196747935628306762462714050027469), SC_(4.500589730419825679581934453582185709058743448290168867874541939460911484143125315702278379652817014), + SC_(7.88237094879150390625), SC_(0.9133758544921875), SC_(11.92028503924933449320581621786613441617387709458224965565099431099659181938764039147894571145926575), SC_(4.411739446802953797688885844731597182665766011787872803903470630812608976887737596597075028566679532), + SC_(7.88237094879150390625), SC_(0.968867778778076171875), SC_(13.87112145371288914260504318784118251728461784612328590191541928786695318093130503135200110507250744), SC_(3.529662237308642030515257624627728599863992204849868392777953155280407948755004639499026009930636495), + }; +#undef SC_ + diff --git a/test/ibeta_data.ipp b/test/ibeta_data.ipp new file mode 100644 index 000000000..c56c6b6c6 --- /dev/null +++ b/test/ibeta_data.ipp @@ -0,0 +1,511 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> ibeta_data = { + SC_(0.115105740725994110107421875), SC_(27.2666988372802734375), SC_(0.913345992565155029296875), SC_(5.624704191155661855917495760118550518449), SC_(0.4314397017151442751805333868688125395477e-30), SC_(0.9999999999999999999999999999999232955748), SC_(0.7670442516666816870156810539487652961219e-31), + SC_(0.4634225666522979736328125), SC_(3.4317314624786376953125), SC_(0.24176712334156036376953125), SC_(0.9303555938707241359170267278058957882937), SC_(0.1886421251845313169553337074104181129545), SC_(0.8314186687138222136543152962911694480678), SC_(0.1685813312861777863456847037088305519322), + SC_(0.671531736850738525390625), SC_(23.0631923675537109375), SC_(0.908442795276641845703125), SC_(0.1643292146089890181565750406388380802073), SC_(0.5052411887003343404176234528376913340103e-25), SC_(0.9999999999999999999999996925432949323565), SC_(0.3074567050676435237921994231096931905187e-24), + SC_(1.190207004547119140625), SC_(72.69547271728515625), SC_(0.19963125884532928466796875), SC_(0.00559660970823449190019737902600132825008), SC_(0.9534862442725159579596623739736284684983e-9), SC_(0.9999998296314855714196326655015050202504), SC_(0.170368514428580367334498494979749619704e-6), + SC_(1.54034411907196044921875), SC_(40.498138427734375), SC_(0.3462987840175628662109375), SC_(0.002938208699935392440146870722865374716805), SC_(0.4756199818667749603935341902256155409499e-9), SC_(0.9999998381258945674369009890750742808727), SC_(0.1618741054325630990109249257191272986861e-6), + SC_(1.54871237277984619140625), SC_(81.61170196533203125), SC_(0.678767263889312744140625), SC_(0.0009678001476208569653032183047916640238319), SC_(0.559750588331056488014869497117437903989e-42), SC_(0.9999999999999999999999999999999999999994), SC_(0.5783741506002983362466276981258467292731e-39), + SC_(2.251259326934814453125), SC_(4.89832019805908203125), SC_(0.79967319965362548828125), SC_(0.02442974530836840837734643304711539009959), SC_(0.6178004958231200706879909723189065523108e-4), SC_(0.9974774927784456570206007053676991220858), SC_(0.002522507221554342979399294632300877914206), + SC_(2.8674151897430419921875), SC_(59.60579681396484375), SC_(0.575251102447509765625), SC_(0.1379388062890870134560096364710423528806e-4), SC_(0.4175062181357307894530268550925291936655e-24), SC_(0.9999999999999999999697325046252222304163), SC_(0.3026749537477776958367907987526182267324e-19), + SC_(2.922028064727783203125), SC_(26.592067718505859375), SC_(0.4733414947986602783203125), SC_(0.000115508766121651474024307828538813831237), SC_(0.3796145990560169217472146784816274010906e-9), SC_(0.9999967135537481783877208030551367243803), SC_(0.3286446251821612279196944863275619723855e-5), + SC_(3.0540943145751953125), SC_(42.39783477783203125), SC_(0.546926796436309814453125), SC_(0.2096012544527507305106729688625599112948e-4), SC_(0.1877791782437026107953033098682180353671e-16), SC_(0.9999999999991041123359027203750870846506), SC_(0.8958876640972796249129153494479842438984e-12), + SC_(3.183284282684326171875), SC_(31.6550426483154296875), SC_(0.077649272978305816650390625), SC_(0.1607872346465483285868910326819359030108e-4), SC_(0.1977878368269834064074165442158816223284e-4), SC_(0.4484060589761797174626811812282881431733), SC_(0.5515939410238202825373188187717118568267), + SC_(3.2600824832916259765625), SC_(6.254324436187744140625), SC_(0.743158161640167236328125), SC_(0.003877690538964813730515800489545485233393), SC_(0.1849121735270227071877446919569946133643e-4), SC_(0.9952540157237994764714787832333762033991), SC_(0.004745984276200523528521216766623796600881), + SC_(3.360383510589599609375), SC_(52.89206695556640625), SC_(0.81474220752716064453125), SC_(0.4293999091288321831907350544825170972342e-5), SC_(0.219834444785251736057188895022180823046e-40), SC_(0.9999999999999999999999999999999999948804), SC_(0.5119573621504776597210994477995646618901e-35), + SC_(3.4446079730987548828125), SC_(66.3605499267578125), SC_(0.096544884145259857177734375), SC_(0.1474102736157956762304488717716147895171e-5), SC_(0.8307589573110103561152668922877179107493e-7), SC_(0.9466497330301024630866270516938583276551), SC_(0.05335026696989753691337294830614167234487), + SC_(3.57116794586181640625), SC_(36.129398345947265625), SC_(0.046266771852970123291015625), SC_(0.1377307832566030389205420277524302194542e-5), SC_(0.7311615977995211503865956731396528899981e-5), SC_(0.1585130520872948273412729049865238748685), SC_(0.8414869479127051726587270950134761251315), + SC_(3.5762732028961181640625), SC_(80.64510345458984375), SC_(0.92886126041412353515625), SC_(0.5195049488417132478527686360715001708256e-6), SC_(0.2753515092058996575690963489155893940797e-94), SC_(1), SC_(0.530026729908584311094876547172139505977e-88), + SC_(3.7738864421844482421875), SC_(88.39685821533203125), SC_(0.50323975086212158203125), SC_(0.1936358080461588466938455069456644840687e-6), SC_(0.2398889407764917341676367096721572497995e-29), SC_(0.9999999999999999999999876113337095529831), SC_(0.1238866629044701687424663630369236796346e-22), + SC_(4.24311351776123046875), SC_(87.180511474609375), SC_(0.594544112682342529296875), SC_(0.4437344750551262251613273178405984994839e-7), SC_(0.1440454807297923187241336158457814894158e-36), SC_(0.9999999999999999999999999999967537910884), SC_(0.3246208911577068569804118543592340399042e-29), + SC_(4.302380084991455078125), SC_(90.84336090087890625), SC_(0.3500487506389617919921875), SC_(0.3088652285368942607072200051700648121738e-7), SC_(0.3690261013058404601783388874804478116026e-20), SC_(0.9999999999998805219664725983124099139968), SC_(0.1194780335274016875900860032189749548454e-12), + SC_(4.61713886260986328125), SC_(14.9113979339599609375), SC_(0.081217654049396514892578125), SC_(0.7777055873291569797054048770916368511548e-6), SC_(0.3050394533002200879235444490511467363605e-4), SC_(0.02486139844038035518263312395393841283781), SC_(0.9751386015596196448173668760460615871622), + SC_(4.965442657470703125), SC_(82.34554290771484375), SC_(0.3013162314891815185546875), SC_(0.6232388896113233687607109871883895899506e-8), SC_(0.1757129987256299654131721919929978230207e-16), SC_(0.9999999971806477233823653568856711213164), SC_(0.2819352276617634643114328878683629953111e-8), + SC_(5.26769924163818359375), SC_(94.65772247314453125), SC_(0.695263326168060302734375), SC_(0.1253627465271932390249428025335036505934e-8), SC_(0.3222623306430773933281977469822481222259e-51), SC_(1), SC_(0.2570638722989156911038000229513107483269e-42), + SC_(5.39501190185546875), SC_(35.276241302490234375), SC_(0.184897840023040771484375), SC_(0.1209301166284995442719813406613075674954e-6), SC_(0.2280352502477099546779864717478435018705e-7), SC_(0.8413487283667377937115292338183332209645), 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SC_(0.4363674030870381660841751091381232725007e-108), SC_(1), SC_(0.4358848338099856653448549117193567190292e-56), + SC_(92.033203125), SC_(39.246906280517578125), SC_(0.69485914707183837890625), SC_(0.3429603695399085196479912729784230311888e-35), SC_(0.4568842822110191193956928476545785495226e-35), SC_(0.4287837254261052363447145924559889366464), SC_(0.5712162745738947636552854075440110633536), + SC_(92.3379669189453125), SC_(64.45505523681640625), SC_(0.4359149932861328125), SC_(0.1865276169340167763205291709996432353182e-50), SC_(0.3113144092675435028788977675902943909337e-46), SC_(0.5991256300190186593748026830039317039068e-4), SC_(0.9999400874369980981340625197316996068296), + SC_(92.88541412353515625), SC_(24.4138278961181640625), SC_(0.47481119632720947265625), SC_(0.3517087569118679257544224321978885705116e-38), SC_(0.5033547361837899810418297479760310574297e-26), SC_(0.6987294081667341495372864749025001962669e-12), SC_(0.9999999999993012705918332658504627135251), + SC_(92.92636871337890625), SC_(21.723785400390625), SC_(0.15452297031879425048828125), SC_(0.1496355919808997710160646497363599642223e-78), SC_(0.4033357831201168472873957975134700875523e-24), SC_(0.3709950821207872136467177497502667334427e-54), SC_(1), + SC_(92.9385986328125), SC_(45.849689483642578125), SC_(0.3805077970027923583984375), SC_(0.7123009475318467090796828077561395619685e-50), SC_(0.260666357059090116628536966680822451934e-38), SC_(0.2732615576349308826639148876145050880397e-11), SC_(0.9999999999972673844236506911733608511239), + SC_(93.399322509765625), SC_(68.135955810546875), SC_(0.0497494600713253021240234375), SC_(0.6905215639710705512278514815975036528166e-125), SC_(0.6864215712687726843530898730264241411204e-48), SC_(0.1005972995129974563314022676955487947004e-76), SC_(1), + SC_(93.40106964111328125), SC_(96.49663543701171875), SC_(0.18903611600399017333984375), SC_(0.7657063231082140070082353793406337177991e-78), SC_(0.2555891770317885383063245635780211930521e-57), SC_(0.2995847993254348083418454071685611149657e-20), SC_(0.9999999999999999999970041520067456519166), + SC_(93.90015411376953125), SC_(24.007595062255859375), SC_(0.403971731662750244140625), SC_(0.9335734346720279535279023803571936252236e-44), SC_(0.7613169380702204900368864669107621241774e-26), SC_(0.1226261216568281958863292968272309752221e-17), SC_(0.9999999999999999987737387834317180411367), + SC_(94.20505523681640625), SC_(95.4943389892578125), SC_(0.3371889293193817138671875), SC_(0.9297654369865549462825173884928826911307e-63), SC_(0.2873180569011825880858826754896885391403e-57), SC_(0.3236004163085036181879114898511069822143e-5), SC_(0.9999967639958369149638181208851014889302), + SC_(94.2736968994140625), SC_(13.12218475341796875), SC_(0.709393918514251708984375), SC_(0.4136446298056877569444391910150569799517e-22), SC_(0.3591337944167607654097996956175751590464e-17), SC_(0.1151770933278035243241152703989893222014e-4), SC_(0.9999884822906672196475675884729601010678), + SC_(94.36226654052734375), SC_(75.62816619873046875), SC_(0.86930525302886962890625), SC_(0.7332140206931858197543993390917094119579e-51), SC_(0.494193161847161404627682054188893296014e-74), SC_(0.999999999999999999999993259905731481408), SC_(0.674009426851859198700510182276484925711e-23), + SC_(94.47872161865234375), SC_(4.794428348541259765625), SC_(0.3019829094409942626953125), SC_(0.2037612994500144366883878655315176352494e-51), SC_(0.5445271727836805564325394307425243332438e-8), SC_(0.3741985884898362402462482734954103002849e-43), SC_(1), + SC_(94.5174102783203125), SC_(2.595887660980224609375), SC_(0.9688708782196044921875), SC_(0.337761821443508199381120070196992983155e-5), SC_(0.7003104354886139691687572520659797605122e-5), SC_(0.3253740952886229454250383398376993551864), SC_(0.6746259047113770545749616601623006448136), + SC_(95.02220916748046875), SC_(49.05890655517578125), SC_(0.095445640385150909423828125), SC_(0.1014296591921831167485134416924426771213e-100), SC_(0.3255049410331931492209518274123644776711e-40), SC_(0.3116071260553918747274711046692546109596e-60), SC_(1), + SC_(95.16304779052734375), SC_(87.8050537109375), SC_(0.68924558162689208984375), SC_(0.3591386390171570330217034323842412396658e-55), SC_(0.3500715414450358091351879835688815561025e-61), SC_(0.9999990252477395079575524827693321958401), SC_(0.974752260492042447517230667804159929911e-6), + SC_(95.61345672607421875), SC_(65.196868896484375), SC_(0.3377856314182281494140625), SC_(0.4333859722014505980501156640750924994333e-58), SC_(0.2842452980133411144067320657040665393916e-47), SC_(0.1524690030842662316266754200421873949762e-10), SC_(0.9999999999847530996915733768373324579958), + SC_(95.71669769287109375), SC_(10.986175537109375), SC_(0.0333652384579181671142578125), SC_(0.3372800085083612728245814732509161788213e-143), SC_(0.3482559491895011198520376372913856125954e-15), SC_(0.9684831208004220966663378958012853318006e-128), SC_(1), + SC_(95.7506866455078125), SC_(99.6461334228515625), SC_(0.01787211932241916656494140625), SC_(0.7925981877666200622765224435998813190314e-170), SC_(0.5647605029015430138123291624587681412693e-59), SC_(0.140342354625460930114931308563781999924e-110), SC_(1), + SC_(95.7693939208984375), SC_(72.4906005859375), SC_(0.872441589832305908203125), SC_(0.4388142633049923644259492345034689476008e-50), SC_(0.6107180919399320674461648364002632421015e-72), SC_(0.9999999999999999999998608253780676540823), SC_(0.1391746219323459176926850046732349646854e-21), + SC_(95.92913818359375), SC_(58.095668792724609375), SC_(0.1387105882167816162109375), SC_(0.1152963776753726723714207279128040302867e-87), SC_(0.1964676344164222838212765957522296319386e-44), SC_(0.5868466733355003420468381699190122888759e-43), SC_(1), + SC_(95.94924163818359375), SC_(50.36627197265625), SC_(0.043119497597217559814453125), SC_(0.1205105244843612170118997464853132481334e-133), SC_(0.5379464386572334970803452281919284791023e-41), SC_(0.2240195599866172144586170468512280290734e-92), SC_(1), + SC_(95.974395751953125), SC_(57.37546539306640625), SC_(0.12707412242889404296875), SC_(0.5509848073830600725513709899102562059558e-91), SC_(0.3907686547756237468365485949936283442849e-44), SC_(0.1410002569677527400672535793471909277884e-46), SC_(1), + SC_(96.18981170654296875), SC_(97.99256134033203125), SC_(0.240151941776275634765625), SC_(0.1047338624529782694403256402509401725883e-72), SC_(0.1274994550921882776529206680489908409938e-58), SC_(0.8214455691379210750681642067899947422341e-14), SC_(0.9999999999999917855443086207892493183579), + SC_(96.3088531494140625), SC_(60.5928192138671875), SC_(0.488668859004974365234375), SC_(0.1173424000075725323832188274859866785982e-48), SC_(0.1455980145613272932867048313858796718915e-45), SC_(0.0008052850602336237442086193512349341441502), SC_(0.9991947149397663762557913806487650658558), + SC_(96.48885345458984375), SC_(96.76949310302734375), SC_(0.0185489989817142486572265625), SC_(0.1435982258016881740011473579074188068313e-169), SC_(0.2405150906117843685469869117960864428404e-58), SC_(0.5970445573141612300096758741661990216964e-111), SC_(1), + SC_(96.86492919921875), SC_(33.23390960693359375), SC_(0.822622716426849365234375), SC_(0.3896126830362454261608108680217826494112e-32), SC_(0.575732196536069935996623203779523859796e-34), SC_(0.985438141759546703124794958382314271643), SC_(0.01456185824045329687520504161768572835698), + SC_(97.0592803955078125), SC_(98.11096954345703125), SC_(0.0319296605885028839111328125), SC_(0.2998869565906142596550640980837126732485e-148), SC_(0.6376859616143078939902779400501204314318e-59), SC_(0.4702737313386163666185665116771431706157e-89), SC_(1), + SC_(97.297454833984375), SC_(14.22172069549560546875), SC_(0.62565600872039794921875), SC_(0.4586358283116612515366441141888728005777e-27), SC_(0.2345703889146973155066568361856950228888e-18), SC_(0.1955216212655529321825058592196298245196e-8), SC_(0.9999999980447837873444706781749414078037), + SC_(97.86806488037109375), SC_(23.2400360107421875), SC_(0.580998599529266357421875), SC_(0.4847838580994435860838596547982930354105e-33), SC_(0.1112477977342289130957389321924258614326e-25), SC_(0.4357693786731819507126485869030923714513e-7), SC_(0.9999999564230621326818049287351413096908), + SC_(97.9748382568359375), SC_(35.46381378173828125), SC_(0.43892610073089599609375), SC_(0.2873257135274528993245007467042681059185e-45), SC_(0.1374390079964617837539348395028823804413e-33), SC_(0.2090568883720763685344749440226251580624e-11), SC_(0.9999999999979094311162792363146552505598), + SC_(98.16379547119140625), SC_(49.7869415283203125), SC_(0.7558143138885498046875), SC_(0.3971958028052333965608830276024843615237e-41), SC_(0.2499072385121668210487453408177231300208e-43), SC_(0.9937475495928450856212412081950470642903), SC_(0.006252450407154914378758791804952935709679), + SC_(98.2663421630859375), SC_(89.943603515625), SC_(0.79222810268402099609375), SC_(0.9703020130843336605117450383730907257821e-57), SC_(0.9342902223591938348587142832364306200003e-73), SC_(0.9999999999999999037114001866973304919699), SC_(0.962885998133026695080301263760365136479e-16), + SC_(98.3052520751953125), SC_(55.438030242919921875), SC_(0.711244642734527587890625), SC_(0.9148851548605620864233078020635181477222e-44), SC_(0.2676495491351328940765645110959640891094e-45), SC_(0.9715765392732393467357708912202385044325), SC_(0.02842346072676065326422910877976149556749), + SC_(98.4063720703125), SC_(29.338825225830078125), SC_(0.679551780223846435546875), SC_(0.7234863966571889292957198162735201822678e-32), SC_(0.6644618653733999798962457728260649872896e-30), SC_(0.01077102753452599131081375931128823317142), SC_(0.9892289724654740086891862406887117668286), + SC_(98.79819488525390625), SC_(76.92431640625), SC_(0.498594343662261962890625), SC_(0.8547721404370727048220658289647343126557e-54), SC_(0.1805898574966365262303134358305105727825e-52), SC_(0.04519314170159976658173488989109380979713), SC_(0.9548068582984002334182651101089061902029), + SC_(99.61347198486328125), SC_(44.7584381103515625), SC_(0.2373598515987396240234375), SC_(0.4979063210905853961279025180666656173251e-69), SC_(0.6862752454273526015268742998992848087559e-39), SC_(0.7255198616125700349403582088841519836769e-30), SC_(0.9999999999999999999999999999992744801384), + SC_(99.90804290771484375), SC_(67.421295166015625), SC_(0.73282539844512939453125), SC_(0.4022690295295743797802135225234299639637e-49), SC_(0.3062978751814600527884142222488817508074e-53), SC_(0.9999238632526556439070749307531342152962), SC_(0.7613674734435609292506924686578470377178e-4), + }; +#undef SC_ + + diff --git a/test/ibeta_large_data.ipp b/test/ibeta_large_data.ipp new file mode 100644 index 000000000..16464b84a --- /dev/null +++ b/test/ibeta_large_data.ipp @@ -0,0 +1,1222 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1210> ibeta_large_data = { + SC_(0.104760829344741068780422210693359375e-4), SC_(39078.1875), SC_(0.913384497165679931640625), SC_(95444.37547576888548779405522478045372688), SC_(0.4076397251031275963346153642645211144346e-41521), SC_(1), SC_(0.4270966445860582673748568606264586002504e-41526), + SC_(0.1127331415773369371891021728515625e-4), SC_(0.0226620174944400787353515625), SC_(0.1355634629726409912109375), SC_(88703.20318198098901713372585734808194292), SC_(45.9460483635769831377505786833842050448), SC_(0.9994822930837981780736860587426162687504), SC_(0.0005177069162018219263139412573837312496096), + SC_(0.113778432933031581342220306396484375e-4), SC_(0.03654421865940093994140625), SC_(0.9688708782196044921875), SC_(87893.29210911967129223056449206866305889), SC_(24.13233514927218107101077084629346084654), SC_(0.999725511349976252733137998478545702411), SC_(0.0002744886500237472668620015214542975889712), + SC_(0.1142846667789854109287261962890625e-4), SC_(0.00244517601095139980316162109375), SC_(0.1355634629726409912109375), SC_(87498.94901523223439182946063290895919618), SC_(410.8174698800966982396187764410683048418), SC_(0.9953268278792474174353255397883295739773), SC_(0.004673172120752582564674460211670426022665), + SC_(0.1184685606858693063259124755859375e-4), SC_(0.015964560210704803466796875), SC_(0.3082362115383148193359375), SC_(84409.76586511709213323237371074024560834), SC_(63.42758275377907514602214367744158071269), SC_(0.9992491395179357013663738064942314292026), SC_(0.000750860482064298633626193505768570797446), + SC_(0.1215800511999987065792083740234375e-4), SC_(24110.49609375), SC_(0.1355634629726409912109375), SC_(82239.66859351932879196086451784571067159), SC_(0.1228963536503215267489197519054078145969e-1528), SC_(1), SC_(0.1494368298804234679967895435418141236754e-1533), + SC_(0.130341722979210317134857177734375e-4), SC_(26168.341796875), SC_(0.12707412242889404296875), SC_(76710.65536325700740870385917451316226344), SC_(0.8987481009726700678494101665597031705645e-1548), SC_(1), SC_(0.1171607903382812021950729983554623721701e-1552), + SC_(0.13885271982871927320957183837890625e-4), SC_(0.04976274073123931884765625), SC_(0.632396042346954345703125), SC_(72019.23463150248088958543324182892626471), SC_(19.53662131349699149968802706905083676178), SC_(0.9997288040735046086759949196286322002492), SC_(0.0002711959264953913240050803713677997508421), + SC_(0.139016165121574886143207550048828125e-4), SC_(0.3188882328686304390430450439453125e-4), SC_(0.81474220752716064453125), SC_(71935.56143195648321209894597982190925135), SC_(31357.46843093207518473308985368320533006), SC_(0.6964222225589077802406713530999497187926), SC_(0.3035777774410922197593286469000502812074), + 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SC_(0.5772928801419627986284807700357329751852e-18560), SC_(4716.320526853965142300050830950109310733), SC_(0.1224032329556379061882981898965492987967e-18563), SC_(1), + SC_(433260.8125), SC_(3.3808853626251220703125), SC_(0.905801355838775634765625), SC_(0.9743394652123491454497722494064540082877e-18624), SC_(0.2560546738053348250776027384729315592872e-18), SC_(0.3805200860942258123591032689689538525655e-18605), SC_(1), + SC_(434088.21875), SC_(21.81499481201171875), SC_(0.12707412242889404296875), SC_(0.7666944874274611518715379495155788957394e-388925), SC_(0.3007553193509600488237813376679700633905e-103), SC_(0.254923001555554620514995644832550525135e-388821), SC_(1), + SC_(434689.125), SC_(39291.4765625), SC_(0.81474220752716064453125), SC_(0.4229593173885190819792612716082535703903e-67453), SC_(0.2997707408973057615935360098045888677795e-58830), SC_(0.141094262943231928276825070136084907e-8622), SC_(1), + SC_(435121.25), SC_(397.58978271484375), SC_(0.632396042346954345703125), SC_(0.1391846018183584493348497097868590409154e-86771), SC_(0.1011089115991303029245123638141426629899e-1381), SC_(0.13765809523317592290575564449364874034e-85389), SC_(1), + SC_(437285.3125), SC_(0.2788266647257842123508453369140625e-4), SC_(0.3082362115383148193359375), SC_(0.7080016520655375267035486305854570379383e-223509), SC_(35851.012645549836380276487988456969042), SC_(0.1974844222854668306991988406185103636153e-223513), SC_(1), + SC_(442472.9375), SC_(49476.66015625), SC_(0.1355634629726409912109375), SC_(0.6336326725813869671779066098442524839183e-387139), SC_(0.2249187802897843416031847629628799944729e-69724), SC_(0.2817162140773738373578099290063302965899e-317414), SC_(1), + SC_(450819.5), SC_(499216.46875), SC_(0.221111953258514404296875), SC_(0.3800121390355127178063074898342430570494e-349644), SC_(0.9678656246284879041515322169506269463219e-285456), SC_(0.3926290275898362964767484676866590121196e-64188), SC_(1), + SC_(457779.125), SC_(0.326781928539276123046875), SC_(0.81474220752716064453125), SC_(0.5550829230252479526732424261273786286927e-40738), SC_(0.03865177027724154668769750896774259956337), SC_(0.1436112548128448741327568829062085238803e-40736), SC_(1), + SC_(458356.625), SC_(0.00327513157390058040618896484375), SC_(0.81474220752716064453125), SC_(0.3934256588938313890657511884630295473806e-40789), SC_(292.0201861612453957241730382306629407247), SC_(0.1347255010229302168172554227989064367004e-40791), SC_(1), + SC_(462545.71875), SC_(0.01835658587515354156494140625), SC_(0.81474220752716064453125), SC_(0.6842410988092162996870266776720414780451e-41162), SC_(42.43584359872765638163172784469089833824), SC_(0.1612413094174311960166824477001882166453e-41163), SC_(1), + SC_(466997.53125), SC_(0.356361567974090576171875), SC_(0.9688708782196044921875), SC_(0.3241995530691347719937107476224957921527e-6418), SC_(0.02384677265659975718672488537621955689915), SC_(0.1359511233397070742246772924547520956389e-6416), SC_(1), + SC_(468932.8125), SC_(0.182366857188753783702850341796875e-4), SC_(0.632396042346954345703125), SC_(0.1103527708644207448237778040457649555939e-93327), SC_(54820.89133734763217172708014125126167222), SC_(0.2012969292771077366711511442513115417448e-93332), SC_(1), + SC_(470942.5), SC_(0.326362073421478271484375), SC_(0.913384497165679931640625), SC_(0.1471321546322475787061574604050565023955e-18534), SC_(0.03855775001051170472627007361165177373414), SC_(0.3815890569136841848282960464759255989168e-18533), SC_(1), + SC_(473791.3125), SC_(0.00037331905332393944263458251953125), SC_(0.9688708782196044921875), SC_(0.5426122640062687858436002133933513536523e-6511), SC_(2665.063154701061606205642057038907522539), SC_(0.2036020283606124332820904603012161247452e-6514), SC_(1), + SC_(474410.90625), SC_(1.53685867786407470703125), SC_(0.3082362115383148193359375), SC_(0.117766994152457958795877225944368333002e-242484), SC_(0.1678625980035251962785258430723784518394e-8), SC_(0.7015678033887262640021676404777871768517e-242476), SC_(1), + SC_(481669.8125), SC_(360.35894775390625), SC_(0.8350250720977783203125), SC_(0.1292404819014634354658936369356745477441e-38001), SC_(0.1184119790891173687044477395438058404198e-1283), SC_(0.1091447697231683701613307519286097336645e-36717), SC_(1), + SC_(485412.125), SC_(483725.21875), SC_(0.81474220752716064453125), SC_(0.8593009078995238625969168132860357389775e-291741), SC_(0.4682031506788712249402837988058297965792e-397392), SC_(1), SC_(0.544865187939051001712434628531605526675e-105651), + SC_(489093.59375), SC_(101.60869598388671875), SC_(0.221111953258514404296875), SC_(0.2679049247465053509778519007374647739724e-320562), SC_(0.1239584973345346670544577370734255574289e-418), SC_(0.2161246953675901063139086740469757800594e-320143), SC_(1), + SC_(489183.34375), SC_(118808.328125), SC_(0.905801355838775634765625), SC_(0.1458359410123570793275225181353576621217e-130434), SC_(0.259453747115697820698795192063284012701e-142915), SC_(1), SC_(0.1779079596666185278653733063531830995884e-12480), + SC_(491651.625), SC_(32.955524444580078125), SC_(0.3082362115383148193359375), SC_(0.1033057257566332368173504705063859426151e-251301), SC_(0.603837473523874333668117011509794982175e-152), SC_(0.1710820051524158477748585452714475118872e-251149), SC_(1), + SC_(499024.09375), SC_(0.3617647588253021240234375), SC_(0.221111953258514404296875), SC_(0.1198688160746637782814955712359261486421e-327059), SC_(0.0213597393543653960374826366383814420494), SC_(0.5611904437877215617130419280865220458283e-327058), SC_(1), + }; +#undef SC_ + + + diff --git a/test/ibeta_small_data.ipp b/test/ibeta_small_data.ipp new file mode 100644 index 000000000..be84f0aea --- /dev/null +++ b/test/ibeta_small_data.ipp @@ -0,0 +1,512 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> ibeta_small_data = { + SC_(0.011510574258863925933837890625), SC_(2.726669788360595703125), SC_(0.913345992565155029296875), SC_(85.50804647765112454526520766832090120908), SC_(0.0004969774007283958673037601318581948437309), SC_(0.9999941879795790272650838125485136788379), SC_(0.5812020420972734916187451486321162137375e-5), + SC_(0.0463422574102878570556640625), SC_(0.34317314624786376953125), SC_(0.24176712334156036376953125), SC_(20.36367089471426800179696837574809289039), SC_(3.630773740238788693031723031598949317136), SC_(0.8486827348798151491124959025982318049325), SC_(0.1513172651201848508875040974017681950675), + SC_(0.06715317070484161376953125), SC_(2.30631923675537109375), SC_(0.908442795276641845703125), SC_(13.78727207173260438424514878253152024011), SC_(0.001859217475218142037637565734833320618317), SC_(0.9998651679038930266805148691555636600556), SC_(0.0001348320961069733194851308444363399443688), + SC_(0.1190207004547119140625), SC_(7.26954746246337890625), SC_(0.19963125884532928466796875), SC_(6.225047194692518378537141946282961080975), SC_(0.08420413075357450486624447742592649170405), SC_(0.9866538632858121421788318729611834919717), SC_(0.01334613671418785782116812703881650802825), + SC_(0.1540344059467315673828125), SC_(4.049813747406005859375), SC_(0.3462987840175628662109375), SC_(4.872033861103043731550905432482222659763), SC_(0.08561968850485947325255754361164029251743), SC_(0.9827297959310547093553196782752749843992), SC_(0.01727020406894529064468032172472501560077), + SC_(0.15487124025821685791015625), SC_(8.16117000579833984375), SC_(0.678767263889312744140625), SC_(4.38038087876606402301557100951561214911), SC_(0.1540537895441550779731794373253219559374e-4), SC_(0.999996483108386751271915947803000461492), SC_(0.3516891613248728084052196999538507955892e-5), + SC_(0.2251259386539459228515625), SC_(0.4898320138454437255859375), SC_(0.79967319965362548828125), SC_(4.764013990849157580013322088552892632018), SC_(0.9820281524501242644370508756885101403883), SC_(0.8290948573018540323461920074910160298235), SC_(0.1709051426981459676538079925089839701765), + SC_(0.2867415249347686767578125), SC_(5.96057987213134765625), SC_(0.575251102447509765625), SC_(1.911839728052784613470170990261300207059), SC_(0.001412230939458232329324385634994224021816), SC_(0.9992618688130328170087283879882410324961), SC_(0.0007381311869671829912716120117589675039269), + SC_(0.2922028005123138427734375), SC_(2.659206867218017578125), SC_(0.4733414947986602783203125), SC_(2.306780982820498015018593188560459962841), SC_(0.09785807660771959252183084474688989446611), SC_(0.9593044635018992897241131636398162350677), SC_(0.04069553649810071027588683636018376493234), + SC_(0.30540943145751953125), SC_(4.23978328704833984375), SC_(0.546926796436309814453125), SC_(1.925766552541500211935789109529683925689), SC_(0.01136621113828240461645649770280082638802), SC_(0.9941324563027414190664847981329029657831), SC_(0.005867543697258580933515201867097034216903), + SC_(0.318328440189361572265625), SC_(3.1655042171478271484375), SC_(0.077649272978305816650390625), SC_(1.337499870967964297403057650377070193563), SC_(0.6794195418585711764434318012650276929832), SC_(0.6631399660602084538462156083737590843824), SC_(0.3368600339397915461537843916262409156176), + SC_(0.326008260250091552734375), SC_(0.625432431697845458984375), SC_(0.743158161640167236328125), SC_(3.078454179457483361692247221422692039972), SC_(0.7360974885092882497469458885357603771633), SC_(0.8070290947450602569822924525981547293294), SC_(0.1929709052549397430177075474018452706706), + SC_(0.3360383510589599609375), SC_(5.28920650482177734375), SC_(0.81474220752716064453125), SC_(1.550409503660757234080944256429985207784), SC_(0.2836427788609597588510100729645041742761e-4), SC_(0.9999817056339550018928818220116909157891), SC_(0.1829436604499810711817798830908421094105e-4), + SC_(0.344460785388946533203125), SC_(6.63605499267578125), SC_(0.096544884145259857177734375), SC_(1.138099220831291561221229605806786314538), SC_(0.2341007981779031558780678617792900307434), SC_(0.8293974676177770014617748040947131716032), SC_(0.1706025323822229985382251959052868283968), + SC_(0.3571167886257171630859375), SC_(3.6129398345947265625), SC_(0.046266771852970123291015625), SC_(0.9053135817270937856051949826437502245961), SC_(0.7221758562362922538389449085733137575244), SC_(0.5562638752728180895224241888333329380732), SC_(0.4437361247271819104775758111666670619268), + SC_(0.357627332210540771484375), SC_(8.064510345458984375), SC_(0.92886126041412353515625), SC_(1.197177044295353240338824861021333896969), SC_(0.7152754878374974083055684082879234042397e-10), SC_(0.9999999999402531571079558956505173243843), SC_(0.597468428920441043494826756157346017527e-10), + SC_(0.377388656139373779296875), SC_(8.8396854400634765625), SC_(0.50323975086212158203125), SC_(1.048244561726343473675201855489355128708), SC_(0.0003378999437661089518583588179790252882915), SC_(0.999677755487891791262427317664954237941), SC_(0.0003222445121082087375726823350457620590389), + SC_(0.424311339855194091796875), SC_(8.71805095672607421875), SC_(0.594544112682342529296875), SC_(0.8450634150576302527396320143587314116559), SC_(0.5692485563137910015348490929019764830298e-4), SC_(0.9999326428995458545641785392037209582211), SC_(0.6735710045414543582146079627904177891161e-4), + SC_(0.4302380084991455078125), SC_(9.08433628082275390625), SC_(0.3500487506389617919921875), SC_(0.8041671576172428143402874467426430937693), SC_(0.003657242502533209987243300892131746946366), SC_(0.995472725877070621917637160657188675184), SC_(0.004527274122929378082362839342811324816028), + SC_(0.4617138803005218505859375), SC_(1.49113976955413818359375), SC_(0.081217654049396514892578125), SC_(0.6708426431384324314892486199862307536228), SC_(1.06148098502757264044386168615001572695), SC_(0.3872501836441769856742304061740122471689), SC_(0.6127498163558230143257695938259877528311), + SC_(0.4965442717075347900390625), SC_(8.234554290771484375), SC_(0.3013162314891815185546875), SC_(0.6255445999459755311064947246119255484269), SC_(0.01047121638037352884799746710410909796172), SC_(0.9835362327294694289867309219284499434766), SC_(0.01646376727053057101326907807155005652337), + SC_(0.526769936084747314453125), SC_(9.4657726287841796875), SC_(0.695263326168060302734375), SC_(0.5223867123087661970619213259031225900887), SC_(0.1604184643861934563272584858102025314868e-5), SC_(0.9999969291337627338160603638685334704082), SC_(0.3070866237266183939636131466529591820232e-5), + SC_(0.539501190185546875), SC_(3.5276241302490234375), SC_(0.184897840023040771484375), SC_(0.6337086405553985369329386662133114583969), SC_(0.2298885748176341425212030686754049306522), SC_(0.7338011624802040649313396945278096189402), SC_(0.2661988375197959350686603054721903810598), + SC_(0.596188604831695556640625), SC_(9.52472019195556640625), SC_(0.587086021900177001953125), SC_(0.3957027656575689988900171488472459553927), SC_(0.2784518837222712883966491998692226153708e-4), SC_(0.9999296360008322244237915105214427770265), SC_(0.7036399916777557620848947855722297349452e-4), + SC_(0.597795426845550537109375), SC_(4.3524570465087890625), SC_(0.3404516875743865966796875), SC_(0.5857283108172326731547619033650850279516), SC_(0.05176933170314495414753269319008084049281), SC_(0.9187929048671107105645293523464974421304), SC_(0.08120709513288928943547064765350255786955), + SC_(0.604711830615997314453125), SC_(2.7766406536102294921875), SC_(0.65411365032196044921875), SC_(0.8104703643388840470345296228912560465778), SC_(0.02126375318260348616345995572262831556693), SC_(0.9744344343526894636219031995625457203905), SC_(0.02556556564731053637809680043745427960951), + 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SC_(0.9999996392922183063089445295584381333051), SC_(0.3607077816936910554704415618666949469689e-6), + SC_(9.20332050323486328125), SC_(3.924690723419189453125), SC_(0.69485914707183837890625), SC_(0.0002310503112810029066611893416788379084469), SC_(0.0002841260239293677938812757004432580527586), SC_(0.4484878195863833882626599955886481131739), SC_(0.5515121804136166117373400044113518868261), + SC_(9.2337970733642578125), SC_(6.445505619049072265625), SC_(0.4359149932861328125), SC_(0.3518617297258378498751803839224728042655e-5), SC_(0.2846752000482794851986542503386822247384e-4), SC_(0.1100044454892298999779387975978826443182), SC_(0.8899955545107701000220612024021173556818), + SC_(9.2885417938232421875), SC_(2.441382884979248046875), SC_(0.47481119632720947265625), SC_(0.4761255483176591056175849284001070081479e-4), SC_(0.004586185824940369681133540819136408293863), SC_(0.01027505966586902534866379616980543048538), SC_(0.9897249403341309746513362038301945695146), + SC_(9.292636871337890625), SC_(2.1723785400390625), SC_(0.15452297031879425048828125), SC_(0.2624075056699678414562124647032627161364e-8), SC_(0.007521320523051118214918717491848671514352), SC_(0.3488847647374800274842030525271354542582e-6), SC_(0.9999996511152352625199725157969474728645), + SC_(9.2938594818115234375), SC_(4.584969043731689453125), SC_(0.3805077970027923583984375), SC_(0.3033084082238698622351283936762757320358e-5), SC_(0.0002160623300405638004025369269425911334952), SC_(0.01384366758374349929295072479433910689838), SC_(0.9861563324162565007070492752056608931016), + SC_(9.33993244171142578125), SC_(6.81359577178955078125), SC_(0.0497494600713253021240234375), SC_(0.5518070299877559210026646747715538340941e-13), SC_(0.2146633044365963053975520213612198005479e-4), SC_(0.2570569897903910302588882280913495968099e-8), SC_(0.9999999974294301020960896974111177190865), + SC_(9.340106964111328125), SC_(9.6496639251708984375), SC_(0.18903611600399017333984375), SC_(0.3760693208748964009075562773732041031133e-8), SC_(0.2241360584188258329348724623108553890864e-5), SC_(0.001675051252959087450179950156748128851185), SC_(0.9983249487470409125498200498432518711488), + SC_(9.39001560211181640625), SC_(2.4007594585418701171875), SC_(0.403971731662750244140625), SC_(0.1134959668917263176160589414159781875943e-4), SC_(0.004844797940116370573271103607794275185958), SC_(0.002337160599663038722280937028362435559564), SC_(0.9976628394003369612777190629716375644404), + SC_(9.420505523681640625), SC_(9.54943370819091796875), SC_(0.3371889293193817138671875), SC_(0.1805691602626601023406871171319958471924e-6), SC_(0.2091846165246050032401702854147298202011e-5), SC_(0.07946133712253384615168111931846197902557), SC_(0.9205386628774661538483188806815380209744), + SC_(9.4273700714111328125), SC_(1.3122184276580810546875), SC_(0.709393918514251708984375), SC_(0.003017013444722590640311000913969701534262), SC_(0.04315436163162597522990240737111380539958), SC_(0.06534380749400867203205742252815333005062), SC_(0.9346561925059913279679425774718466699494), + SC_(9.43622684478759765625), SC_(7.562816619873046875), SC_(0.86930525302886962890625), SC_(0.1050377778686616769098759960271509592204e-4), SC_(0.9810847461408665570279905884259728037155e-8), SC_(0.9990668412278015532617620281113608196916), SC_(0.0009331587721984467382379718886391803084328), + SC_(9.447872161865234375), SC_(0.4794428348541259765625), SC_(0.3019829094409942626953125), SC_(0.1527301820162022411297571755969196094484e-5), SC_(0.6378057092170519230917855314560001309605), SC_(0.2394613501875548352256068326033443059895e-5), SC_(0.9999976053864981244516477439316739665569), + SC_(9.45174121856689453125), SC_(0.259588778018951416015625), SC_(0.9688708782196044921875), SC_(0.4793292977141484299090680804809562384231), SC_(1.485529678525234390368678419714687359757), SC_(0.2439509926720312131315430238386720081336), SC_(0.7560490073279687868684569761613279918664), + SC_(9.50222110748291015625), SC_(4.90589046478271484375), SC_(0.095445640385150909423828125), SC_(0.1494828487479906048651881175052573767788e-10), SC_(0.0001378928182363681434821788452535606188609), SC_(0.1084050891519629129496674016152519495829e-6), SC_(0.9999998915949108480370870503325983847481), + SC_(9.51630496978759765625), SC_(8.78050518035888671875), SC_(0.68924558162689208984375), SC_(0.3487950539831946130214243133355575362614e-5), SC_(0.2597595086647305428948201602139996658804e-6), SC_(0.9306884723462189452892185573366006247018), SC_(0.06931152765378105471078144266339937529823), + SC_(9.5613460540771484375), SC_(6.51968669891357421875), SC_(0.3377856314182281494140625), SC_(0.4444839214175533505542773743694858127194e-6), SC_(0.2448798812291126478463498053849728750408e-4), SC_(0.01782751107179748780824836030020235500629), SC_(0.9821724889282025121917516396997976449937), + SC_(9.57166957855224609375), SC_(1.0986175537109375), SC_(0.0333652384579181671142578125), SC_(0.7641149432459472271709082287842355172746e-15), SC_(0.07915132376380966467712577074419741010585), SC_(0.9653849195574867821285790668820816696202e-14), SC_(0.9999999999999903461508044251321787142093), + SC_(9.57506847381591796875), SC_(9.9646129608154296875), SC_(0.01787211932241916656494140625), SC_(0.1658996384153681417135737467814313678843e-17), SC_(0.1513424851355072886884114448435894461463e-5), SC_(0.1096186825970546076551374980438478385039e-11), SC_(0.9999999999989038131740294539234486250196), + SC_(9.57693958282470703125), SC_(7.249060153961181640625), SC_(0.872441589832305908203125), SC_(0.1265918745248280535982657188870855265927e-4), SC_(0.1650447533802566668031298868233442384058e-7), SC_(0.9986979428474589734503870049545106253976), SC_(0.001302057152541026549612995045489374602406), + SC_(9.59291362762451171875), SC_(5.809566974639892578125), SC_(0.1387105882167816162109375), SC_(0.3226608546341382422172814363970325173078e-9), SC_(0.4951645101829454492779247772056654797778e-4), SC_(0.6516193012187428032807345415266845925099e-5), SC_(0.9999934838069878125719671926545847331541), + SC_(9.59492397308349609375), SC_(5.036627292633056640625), SC_(0.043119497597217559814453125), SC_(0.70463813192480350035637684220091276433e-14), SC_(0.0001140624644588322714943677495101932553999), SC_(0.6177651300315304735247529704567042174447e-10), SC_(0.9999999999382234869968469526475247029543), + SC_(9.59743976593017578125), SC_(5.737546443939208984375), SC_(0.12707412242889404296875), SC_(0.1472749033768804794567481194100647733112e-9), SC_(0.5323124639940113790246026061688731976865e-4), SC_(0.276669260771152772688627996297166358751e-5), SC_(0.9999972333073922884722731137200370283364), + SC_(9.61898136138916015625), SC_(9.79925632476806640625), SC_(0.240151941776275634765625), SC_(0.1353843923794317977412052106059477829317e-7), SC_(0.1632783113888411433729758988565696586839e-5), SC_(0.008223447729414564260765034318619123062768), SC_(0.9917765522705854357392349656813808769372), + SC_(9.63088512420654296875), SC_(6.059281826019287109375), SC_(0.488668859004974365234375), SC_(0.581055997045127220418244921269735439886e-5), SC_(0.3187439986268775277133352875803521062324e-4), SC_(0.1541877713596934710853644077040271007484), SC_(0.8458122286403065289146355922959728992516), + SC_(9.6488857269287109375), SC_(9.67694950103759765625), SC_(0.0185489989817142486572265625), SC_(0.1749481595836606883382303122964370689815e-17), SC_(0.1758002864820450185505025032329301561348e-5), SC_(0.995152869681782569163934165768269529895e-12), SC_(0.9999999999990048471303182174308360658342), + SC_(9.686492919921875), SC_(3.323390960693359375), SC_(0.822622716426849365234375), SC_(0.0007256304807823739090101393096545824752395), SC_(0.0002824860673401657300814152223119963953949), SC_(0.7197882845329221918507242489433549020479), SC_(0.2802117154670778081492757510566450979521), + SC_(9.70592784881591796875), SC_(9.81109714508056640625), SC_(0.0319296605885028839111328125), SC_(0.2412648809123635167598641163183182877987e-15), SC_(0.153247481975863672328821013292784593199e-5), SC_(0.1574348092142740770669069066716607378887e-9), SC_(0.9999999998425651907857259229330930933283), + SC_(9.7297458648681640625), SC_(1.422172069549560546875), SC_(0.62565600872039794921875), SC_(0.0007515285211317483408827317273891061880333), SC_(0.03308064211512736093153684853289103287133), SC_(0.02221342902327139455287559010633701337676), SC_(0.9777865709767286054471244098936629866232), + SC_(9.7868061065673828125), SC_(2.324003696441650390625), SC_(0.580998599529266357421875), SC_(0.0001869421048606273257218647038338269425064), SC_(0.004902502132333050539451707571566560474365), SC_(0.03673133964106605423746931249168049356184), SC_(0.9632686603589339457625306875083195064382), + SC_(9.7974834442138671875), SC_(3.546381473541259765625), SC_(0.43892610073089599609375), SC_(0.8846383934988994958104111659144072787642e-5), SC_(0.0006942444162306305791078085627402992944978), SC_(0.01258213581077315616174240184063815608328), SC_(0.9874178641892268438382575981593618439167), + SC_(9.816379547119140625), SC_(4.97869396209716796875), SC_(0.7558143138885498046875), SC_(0.8471586497414725426535462974925383298225e-4), SC_(0.2600787706776670771042095868990896095329e-4), SC_(0.7651102050188879474911089577484436075733), SC_(0.2348897949811120525088910422515563924267), + SC_(9.82663440704345703125), SC_(8.9943599700927734375), SC_(0.79222810268402099609375), SC_(0.2564909668495030018187514949876471506381e-5), SC_(0.1328272157019951695723866392922830834089e-7), SC_(0.9948480487253848974511529374575346247921), SC_(0.005151951274615102548847062542465375207862), + SC_(9.83052539825439453125), SC_(5.54380321502685546875), SC_(0.711244642734527587890625), SC_(0.4142099063694028106257790388260333526588e-4), SC_(0.1706128131291109079313412403482492307802e-4), SC_(0.7082657573983930918019463517369821910095), SC_(0.2917342426016069081980536482630178089905), + SC_(9.84063720703125), SC_(2.9338824748992919921875), SC_(0.679551780223846435546875), SC_(0.0003620677940926280979183654032292271566579), SC_(0.001398063860838349825529157603027032578726), SC_(0.2057049500122911434857054575556857430709), SC_(0.7942950499877088565142945424443142569291), + SC_(9.8798198699951171875), SC_(7.69243144989013671875), SC_(0.498594343662261962890625), SC_(0.2102276377997196459781769309658523199559e-5), SC_(0.5090672027747954852585150635916233053816e-5), SC_(0.2922690751289229878585770869253518082993), SC_(0.7077309248710770121414229130746481917007), + SC_(9.9613475799560546875), SC_(4.475843906402587890625), SC_(0.2373598515987396240234375), SC_(0.2596148971864002645551220232138375643592e-7), SC_(0.0001912089032647389537199491735362492125718), SC_(0.0001357570950881480228485230092737049890207), SC_(0.999864242904911851977151476990726295011), + SC_(9.9908046722412109375), SC_(6.74212932586669921875), SC_(0.73282539844512939453125), SC_(0.1400972426630799368521044086815964034595e-4), SC_(0.1993917002921952527694349144803786583093e-5), SC_(0.8754085417575787330833090864097798601476), SC_(0.1245914582424212669166909135902201398524), + }; +#undef SC_ + + + diff --git a/test/igamma_big_data.ipp b/test/igamma_big_data.ipp new file mode 100644 index 000000000..cf494d516 --- /dev/null +++ b/test/igamma_big_data.ipp @@ -0,0 +1,298 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 287> igamma_big_data = { + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655441178896580822765827178955078125e-7), SC_(-1), SC_(0.2993117912029903240422582053899047098779533956541385552546168739467202263345519375546082579259835689e-4), SC_(577797.90391619920649973095027584836128926282155274216943338008435801586707772349755359609709258809), SC_(0.9999700688208797009675957741794610095290122046604345861444745383126053279773665448062442806862583006), + SC_(0.1730655412757187150418758392333984375e-5), SC_(0.8653277063785935752093791961669921875e-6), SC_(13.38277582367555562294055416847373915500263659346494217875993623821221196887112446988546507684731302), SC_(0.2316099655383339755405731669847576162961123204586203558223557251327214945205980959748823352959805302e-4), 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SC_(0.8234759069615652019245476397172569750885541471334500479714655987220876801411179354025002722103325368e-21057953), SC_(0.1026762455478787231848021279669200901989201648818956533640028599511278074166334792418412467632444988), SC_(1), + SC_(788352.3125), SC_(1), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), SC_(0.4666440397038730095030131088923330001395921365337505093937538015695853474965515882892883491065022498e-6), SC_(0.1433908471829622508376220400714432285652733465995812102127318484064704211691023381454450301792300023e-4306319), + SC_(788352.3125), SC_(394176.15625), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), SC_(0.2006257520633973004844236201403980302780110902909388684784270381180202010418543872474966575049005566), SC_(0.6164848172783960800240480360461358265727377175617248578986815365384881763390067655148178076072402911e-66132), + SC_(788352.3125), SC_(788350.3125), SC_(0.162961216861866486196123635890099179907765436181771479621640418745608625572387287597608545912315394), SC_(0.5007488568506734531584407973208886646226494394047062415392632213315407911442369980288546641886893887), SC_(0.1624738083656850062172671442384129180894308808036804291026031361346630666772952933350920757512800554), SC_(0.4992511431493265468415592026791113353773505605952937584607367786684592088557630019711453358113106113), + SC_(788352.3125), SC_(788352.3125), SC_(0.1626687717822932899593035312904247710866671194963517534486901004447793852560311314241268483442598936), SC_(0.4998502286855928443434206403315169955579486533591805378314370596112904256380207126657714482687295139), SC_(0.1627662534452582024540872488380873269105291974891001552755534544354923069936514495085737733193355558), SC_(0.5001497713144071556565793596684830044420513466408194621685629403887095743619792873342285517312704861), + SC_(788352.3125), SC_(788354.3125), SC_(0.1623763274446328985015387393773169322760017933096679464665287143383813538555079409462603516012733209), SC_(0.4989516028002693214791004285569604766689865869988938477436476252444986636812350914588652370815612146), SC_(0.1630586977829185939118520407511951657211945236757839622577148405418903383941746399864402700623221286), SC_(0.5010483971997306785208995714430395233310134130011061522563523747555013363187649085411347629184387854), + SC_(788352.3125), SC_(1576704.625), SC_(0.62966881391456604663265189232296651965212021691993178179931323658375951801607008550560163062833006), SC_(0.1934852628337369167484390536774609470572595166386051921886173923840391906771189648683653522231175771e-105062), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), + SC_(788352.3125), SC_(78835232), SC_(0.4164740077948943840500786922849307561907212018086545639004814350805550398839532201156819761152062039e-28012316), SC_(0.1279745496059271390694582975257723346837265207507390129414161314216004095755396957400977650588924393e-32318629), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), + SC_(1736170), SC_(1), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), SC_(0.2118914928047443977144200433110895948573532788053230331876929560838004923241637943455403176192920957e-6), SC_(0.1678638954845296242754981240024346426980228514280855063671268705695352948287648087179192945942541815e-10078987), + SC_(1736170), SC_(868085), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), SC_(0.1448940961583454163407547513975813077281954384418820421315934004081116003090269523192520567562712675), SC_(0.1147874654706537747553860317681134464810452807999399860509102977323900833593912850990760752412482674e-145637), + SC_(1736170), SC_(1736168), SC_(0.6317777290022475594523330311129906002358832226132013915031845078518775760353610272181933223363237277), SC_(0.500504618033028326360191641057343755999405997498206839871815176619438983969420893139030447209664989), SC_(0.6305037889687944450359199539867683474985226650229812801702613966546458595016893892268912941388886983), SC_(0.499495381966971673639808358942656244000594002501793160128184823380561016030579106860969552790335011), + SC_(1736170), SC_(1736170), SC_(0.6310133650036994505936467834743236622047002119530026486269595591827934240774463130132409395432514033), SC_(0.49989907641044580817879562048675209725080620507540160516376663057508236554638837205467803575236936), SC_(0.6312681529673425538946062016254352855297056756831800230464863453237300114596041034318436769319610226), SC_(0.50010092358955419182120437951324790274919379492459839483623336942491763445361162794532196424763064), + SC_(1736170), SC_(1736172), SC_(0.63024900188566778658967343804876836928228054774250423301619921925975649543556103050219224039778483), SC_(0.4992935354854227744062593568501916692180042004614849112072827608836331168589189268904245367712978671), SC_(0.632032516085374217898579547050990578452125339893678438657246685246766940101489385942892376077427596), SC_(0.5007064645145772255937406431498083307819957995385150887927172391163668831410810731095754632287021329), + SC_(1736170), SC_(3472340), SC_(0.6014445720839096678691921265326588051367345601002845203175453094832083718879547519600219908634900144), SC_(0.4764741965410820548147263974496875110171884751124591798801059384427379413001614590138374358389181743e-231373), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), + SC_(1736170), SC_(173616992), SC_(0.1028718049106474382663595835099645962497203244264180111334446299544683261400046764833876858973945412e-61095576), SC_(0.8149672117199407490039815813486797316083480325648451121983304934903758539219857770713382104685664004e-71174558), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), + }; +#undef SC_ + + diff --git a/test/igamma_med_data.ipp b/test/igamma_med_data.ipp new file mode 100644 index 000000000..fdf974408 --- /dev/null +++ b/test/igamma_med_data.ipp @@ -0,0 +1,712 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 700> igamma_med_data = { + SC_(0.9759566783905029296875), SC_(0.009759566746652126312255859375), SC_(1.003339192007827076679495082852814844336347745360235976200689788561799007730110137330271807878843422), SC_(0.9890348932311204142405476428315735322132825375129739181325170152149560168983641202727073881868421137), SC_(0.01112369385656880593053787042266942213006008594885039559843996330162391632861423778173725952365124566), SC_(0.01096510676887958575945235716842646778671746248702608186748298478504398310163587972729261181315788632), + SC_(0.9759566783905029296875), SC_(0.48797833919525146484375), SC_(0.6107075742899761923771199703161020169732161418698525715736270035568544567698769445946517198087313906), SC_(0.60200090392622800556746880021527940973985765617427715855663599678855269649431562996389062378697249), SC_(0.4037553115744196902329129829593822494931916894392338002255027483065684672888474305173573475937632775), SC_(0.39799909607377199443253119978472059026014234382572284144336400321144730350568437003610937621302751), + SC_(0.9759566783905029296875), SC_(0.8783609867095947265625), SC_(0.4103188243195825736649265764483608742565650301514019080978258727846258380934756822777647401521337079), SC_(0.4044690348331089562847173039266064850324957624072768224521011340506874965505832795400716939018472498), SC_(0.6041440615448133089451063768271233922098428011576844637013038790787970859652486928342443272503609601), SC_(0.5955309651668910437152826960733935149675042375927231775478988659493125034494167204599283060981527502), + SC_(0.9759566783905029296875), SC_(0.9759566783905029296875), SC_(0.3716156455565117999679764359314206771522167659503088545148165288456984447337491300360711644068227913), SC_(0.3663176354055262891120913532650947355579995910412343114483952548046833846145392929954730440121592797), SC_(0.6428472403078840826420565173440635893141910653587775172843132230177244793249752450759379029956718767), SC_(0.6336823645944737108879086467349052644420004089587656885516047451953166153854607070045269559878407203), + SC_(0.9759566783905029296875), SC_(1.0735523700714111328125), SC_(0.3365957734341682416974116339033939077306129158759127039900907555371173447101305942124236590906355356), SC_(0.331797030846884313930450730877113627495191936019305203409885944607670085443529202208539000516429768), SC_(0.6778671124302276409126213193720903587357949154331736678090389963263055793485937808995854083118591324), SC_(0.668202969153115686069549269122886372504808063980694796590114055392329914556470797791460999483570232), + SC_(0.9759566783905029296875), SC_(1.951913356781005859375), SC_(0.1385088659605976262491193813236600847409759615506046462442726458845512900567187865481680178692283672), SC_(0.1365341875889111868905471971345765310104098007339706625166294395122010561441478434485743781967094833), SC_(0.8759540199037982563609135719518241817254318697584817255548571059788716340020055885638410495332663008), SC_(0.8634658124110888131094528028654234689895901992660293374833705604877989438558521565514256218032905167), + SC_(0.9759566783905029296875), SC_(100), SC_(0.3329367866516218882556933190525174260789494242751242200806022302568879706613898466339952656223149765e-43), SC_(0.3281902091153740166486606492940994602463196281189732879787170311504671582868285770684776274764428056e-43), SC_(1.014462885864395882610032953275484266466407798015407706636940926294091018806981767406982208969380557), SC_(0.9999999999999999999999999999999999999999999671809790884625983351339350705900539753680371881026712021), + SC_(3.6673679351806640625), SC_(0.03667367994785308837890625), SC_(4.015456203482594363703108922457754241931144753022213644581757557362565251633751067204031319947614119), SC_(0.9999996416031942390887720081336876066391718689312749653540593938756701333363441103622007068271772676), SC_(0.1439127192779586379672567684497393093482122996236962325320321035788002657163475408231449064922934748e-5), SC_(0.3583968057609112279918663123933608281310687250346459406061243298666636558896377992931728227323669815e-6), + SC_(3.6673679351806640625), SC_(1.83368396759033203125), SC_(3.385019930725903437101102788992704693233898009050769147408163022452742444496498683915469724406386969), SC_(0.8429972949548684405745767660490086691148623810631514321669785188982553992208514843501333648638513432), SC_(0.6304377118838837061883858060327340460903402260944407341359198552308585951399095467639698269902920728), SC_(0.1570027050451315594254232339509913308851376189368485678330214811017446007791485156498666351361486568), + SC_(3.6673679351806640625), SC_(3.300631046295166015625), SC_(2.042636694164828921905502083385700066735693044660836778705857972635214071130121556552436672447216837), SC_(0.5086933734500178903640699120946947210401396612791668913147219471061294630383430188409151403350782137), SC_(1.972820948444958221383986511639738672588545190484373102838224905048386968506286674127002878949462205), SC_(0.4913066265499821096359300879053052789598603387208331086852780528938705369616569811590848596649217863), + SC_(3.6673679351806640625), SC_(3.6673679351806640625), SC_(1.728735745904876644611421612335214154871948437794169498245863240719631120462386158739239016328353788), SC_(0.4305202295151868380055049423689706093538316338471304831830580882615064594948400767660055044416881203), SC_(2.286721896704910498678066982690224584452289797351040383298219636963969919174022071940200535068325254), SC_(0.5694797704848131619944950576310293906461683661528695168169419117384935405051599232339944955583118797), + SC_(3.6673679351806640625), SC_(4.034104824066162109375), SC_(1.444516814327011000070882393866730351220066002187575174421871857951121331631687572841352290352819824), SC_(0.3597390242642850341727041436017742040531479500883038068894793884921274228917627066755248387784064007), SC_(2.570940828282776143218606201158708388104172232957634707122211019732479708004720657838087261043859218), SC_(0.6402609757357149658272958563982257959468520499116961931105206115078725771082372933244751612215935993), + SC_(3.6673679351806640625), SC_(7.334735870361328125), SC_(0.1928842173031202025225624838692485746218833743579218885223983296392167688850553059225165015834992419), SC_(0.04803542571495237241980779345714029329653710992944984439518803096715759866423134446016679398725604201), SC_(3.822573425306666940766926111156190164702354860787287993021684548044384270751352884171158251598726148), SC_(0.951964574285047627580192206542859706703462890070550155604811969032842401335768655539833206012743958), + SC_(3.6673679351806640625), SC_(366.736785888671875), SC_(0.372800134674871004133305800038544316286141292939638700513746479143277577659533359615901901430435418e-152), SC_(0.9284125692646457240171069953349802726912792879264744141937156428280981577209798205498168718568776692e-153), SC_(4.015457642609787143289488595025438739324238235145209881544082877683601039636408230679439551396679042), SC_(1), + SC_(3.927384853363037109375), SC_(0.0392738468945026397705078125), SC_(5.481066062312759609088467038496268309673263041819793219043840830040297205342994832270601061056033285), SC_(0.9999998644970030817287440986146656923084586349422512633027737770225692386615800459657372422641664819), SC_(0.7427009783886154185672763382162962257956018203370314581397997422675096103629829713614292219536073734e-6), SC_(0.1355029969182712559013853343076915413650577487366972262229774307613384199540342627577358335181356689e-6), + SC_(3.927384853363037109375), SC_(1.9636924266815185546875), SC_(4.681812913714047542283856210321072659033064415174669810637868258672116849915694109286206104312772615), SC_(0.8541791370671521729901850166624917295981362963337091768339429521680602021206356329487839388666490246), 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SC_(0.2129921818936519596147962439174955247123397346404678611936355708343225692418480262403174155802927073e155), SC_(0.592035600286549645870405814185993904133267672331248255377119793948359424847024228821680434117092343), + SC_(99.29168701171875), SC_(198.5833740234375), SC_(0.827233542896868997011855796813781815677230400991992891565280334025260219053460972781573951373944216e140), SC_(0.229938818782866162381957384742067701211139315001140649983601537152353780603212112398035629790115604e-14), SC_(0.3597624564984971602315425958176462786603342308131862864219592650234045339793184808775710578693209953e155), SC_(0.9999999999999977006118121713383761804261525793229878886068499885935001639846284764621939678788760196), + SC_(99.29168701171875), SC_(9929.1689453125), SC_(0.4834473233080081830333768520751663665093597177735535847660483721488578903585873065526523207939684243e-3919), SC_(0.1343795925826480955658960684191085655544382968324780717244341336895034665163230020619295313734211649e-4073), SC_(0.3597624564984979874650854926866432905161310445950019636523602570162960992596525061377901113302937769e155), SC_(1), + SC_(99.40981292724609375), SC_(0.994098126888275146484375), SC_(0.6189775403998407011012861882761402733097287676441233200787688479850177921372028259997245685694001147e155), SC_(1), SC_(0.00208738650682762943074724179328328460912087074928959890241928233000629188305710213804783747970955094), SC_(0.3372313808800301725169188483181072887659421985588169372890476556137167360016957658877023326983095951e-157), + SC_(99.40981292724609375), SC_(49.704906463623046875), SC_(0.6189775401772136033366649694131709554114237751328974114695223637149822231246713201702283730746409055e155), SC_(0.9999999996403308953329536449926347585079614442743745335075024035068298900441595001383387493915938425), SC_(0.2226270977646212188629693178983049925112259086092464842700355690125315058294961954947592092115827498e146), SC_(0.3596691046670463550073652414920385557256254664924975964931701099558404998616612506084061575015153272e-9), + SC_(99.40981292724609375), SC_(97.40981292724609375), SC_(0.3508911365196629713996826314769176339475002271815756817668854924735257948695597303600379465355294383e155), SC_(0.5668883176164969556694640046035019373021062926401136236228527719055084133507101313260179116199909852), SC_(0.2680864038801777297016035567992226393622285404625476383118833555114919972676430956396866220338706763e155), SC_(0.4331116823835030443305359953964980626978937073598863763771472280944915866492898686739820883800090148), + SC_(99.40981292724609375), SC_(99.40981292724609375), SC_(0.3012327110073060776126306088558402484398597921259981479632252878490593671505499604951644728373842357e155), SC_(0.4866617790569895164132848585800606033635106037347255456377686770273865390342095779746840038072262761), SC_(0.317744829392534623488655579420300024869868975518125172115543560135958424986652865504560095732015879e155), SC_(0.5133382209430104835867151414199393966364893962652744543622313229726134609657904220253159961927737239), + SC_(99.40981292724609375), SC_(101.40981292724609375), SC_(0.2525536615874421367012932901172485860763539011272499932099422653784982044920184787105406746658177969e155), SC_(0.4080174886867464334217982943121703482051615723869957786212134749545282348319943206732155228552149625), SC_(0.3664238788123985643999928981588916872333748665168733268688265826065195876451843472891838939035823178e155), SC_(0.5919825113132535665782017056878296517948384276130042213787865250454717651680056793267844771447850375), + SC_(99.40981292724609375), SC_(198.8196258544921875), SC_(0.1371819884134727259345696989796477150536656151060715801081402111868202863481570951651604961881278798e141), SC_(0.221626762620267814217739611891735278077275735293365054216933274355798919338201866535892704439999524e-14), SC_(0.6189775403998393292814020535488809276127389711669727834226177872692167107350909577968610869984484631e155), SC_(0.9999999999999977837323737973218578226038810826472192272426470663494578306672564420108066179813346411), + SC_(99.40981292724609375), SC_(9940.9814453125), SC_(0.1194537579446871083493681851309120071876520743662587465911793406448948030112600502063890936876032829e-3923), SC_(0.1929856095707828218066317814300956090965211451849739733633670381338185644181153034168965828385051012e-4078), SC_(0.6189775403998407011012861882761402733097287676441233200787688479850177921372028259997245685694001147e155), SC_(1), + SC_(99.6479034423828125), SC_(0.996479034423828125), SC_(0.1848618324328595326256540103756933748634832912163837099714564642354930615866313317180770173409626267e156), SC_(1), SC_(0.002632953621713545975680045402254573025958139457581454220024431257252884916204475441566275959699470982), SC_(0.1424281901278791824660099798410885257699748667114052276410912168756372599058364270087640660099318927e-157), + SC_(99.6479034423828125), SC_(49.82395172119140625), SC_(0.1848618323694318986984968527056010536947721607403919764188529025226665743038734891777793577541211654e156), SC_(0.9999999996568916736763733473241955787224548174526161914240157740209114383969351598000863202596909474), SC_(0.6342763392715715767009232116871113047599173355260356171282648728275784254029765958684146128176885942e146), SC_(0.3431083263236266526758044212775451825473838085759842259790885616030648401999136797403090526350547118e-9), + SC_(99.6479034423828125), SC_(97.6479034423828125), SC_(0.104781143510080106941519063115430082168881879110811778245289071826214589716454123584899890589776219e156), SC_(0.5668078809514984818732949250616079106146259355202233070637680474174180977553905187864349475271238675), SC_(0.8008068892277942568413494726026329269460141210557193172616739240927847187017720813317712675118640764e155), SC_(0.4331921190485015181267050749383920893853740644797766929362319525825819022446094812135650524728761325), + SC_(99.6479034423828125), SC_(99.6479034423828125), SC_(0.8996813603114861425627742217275710021870970554377666868921695606836039683900496541598731785012627279e155), SC_(0.4866777249101670793371012779886767047310287873694062698229024013112932567958949261113829499504654837), SC_(0.9489369640171091836937658820293627464477358567260704128223950816713266474762636630208969949083635387e155), SC_(0.5133222750898329206628987220113232952689712126305937301770975986887067432041050738886170500495345163), + SC_(99.6479034423828125), SC_(101.6479034423828125), SC_(0.7544659013596046014875316445704815685260451593376435295742314736115164671432240751181442400434537482e155), SC_(0.4081242144094947811955420069383153567301560395689175929907159931951304506349246950413365552505599538), SC_(0.1094152422968990724769008459186452180108787752826193570140333168743414148723089242062625933366172518e156), SC_(0.5918757855905052188044579930616846432698439604310824070092840068048695493650753049586634447494400462), + SC_(99.6479034423828125), SC_(199.295806884765625), SC_(0.3804006384186390812353329092798841093741132631690706197243637647942568433128668478754826060017677696e141), SC_(0.2057756506101917034700047476092601152101401495116289370474041581162582830156798739943260060771748412e-14), SC_(0.1848618324328591522250155917366121395305740113322743358581932951648733372228665374612337044741147512e156), SC_(0.9999999999999979422434938980829652999525239073988478985985048837106295259584188374171698432012600567), + SC_(99.6479034423828125), SC_(9964.7900390625), SC_(0.6187187441149196707967788811980241646983120637234991013972314808151116439559127711068191156963192852e-3933), SC_(0.3346925300762848381679977351230198692364721942446516591127645420968259313810798749854292537402422257e-4088), SC_(0.1848618324328595326256540103756933748634832912163837099714564642354930615866313317180770173409626267e156), SC_(1), + }; +#undef SC_ + + + diff --git a/test/igamma_small_data.ipp b/test/igamma_small_data.ipp new file mode 100644 index 000000000..17407be8c --- /dev/null +++ b/test/igamma_small_data.ipp @@ -0,0 +1,264 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 252> igamma_small_data = { + SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.165048164480104120332981665342231281101703643798828125e-13), SC_(31.15790848492937617968754572910879626038987427321665501824062445965678033809696860327421033588997389), SC_(0.5142555520027874508595595722328287870710653491133268223717400674935981376413784737279829921742056631e-10), SC_(605883754914.8750136902267543620760601500284208480879474455026885251060577270869941538181411625849707), SC_(0.9999999999485744447997212549140440427767171212928934650886673177628259932506401862358621522444918976), + SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.825240808847993445596102901617996394634246826171875e-12), 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SC_(2.832141251899738927184198830918552560705390731127443324742929027769956627452021708549562866974237375), SC_(0.1491682522274481432092518761039267269452211637367753155145843941704919377471471128043007783037985871), SC_(16.15407873532219650535129629222863492136799506800665709466281568880458765136323436187097046887212565), SC_(0.85083174777255185679074812389607327305477883626322468448541560582950806225285288719409849230013501), + SC_(0.05124260485172271728515625), SC_(0.0461183451116085052490234375), SC_(2.354535041475650840188296544626824459348071689176520405240621833882890783690260790374595349243249817), SC_(0.1240128389463671496190159389825520574179898719077050768343103933526658807883438828524452225211835791), SC_(16.63168494574628459234719857852036302272531410995758001416512288269165349512499528004593798660311321), SC_(0.8759871610536328503809840610174479425820101280922949231656896066473341192116561171462170724905049019), + SC_(0.05124260485172271728515625), SC_(0.05124260485172271728515625), SC_(2.268581955935316913153430290814781029572894726877321715668417204937011887527880456166238463358814554), SC_(0.119485708975357549703407111440249731302572941365773848592259365032731048905891169977364697529085816), SC_(16.71763803128661851938206483233240645250049107225677870373732751163753239128737561425429487248754847), SC_(0.8805142910246424502965928885597502686974270586342261514077406349672689510941088300213449330467672867), + SC_(0.05124260485172271728515625), SC_(0.0563668645918369293212890625), SC_(2.190826928493114755175474738392958609573869639135473828240129903375769251681938201165488344589073467), SC_(0.1153903689079543131568507907786239460633868239209515305709967719710419887432512190607714956691512904), SC_(16.79539305872882067736002038475422887249951615999862659116561481319877502713331786925504499125728956), SC_(0.8846096310920456868431492092213760539366131760790484694290032280289580112567487809379809556858218957), + SC_(0.05124260485172271728515625), SC_(0.1024852097034454345703125), SC_(1.705832980948029655675174791358196762256920326164848611959849532446454449849713723349992966594024533), SC_(0.08984584514959194964111225566819986886942747214355191893796722859826575799935451129133496979377311521), SC_(17.2803870062739057768603203317889907198164654729692518074458951841280898289655423470705403692523385), SC_(0.9101541548504080503588877443318001311305725278564480810620327714017342420006454887076845744989248092), + SC_(0.05124260485172271728515625), SC_(100), SC_(0.4666332762470370840214357140870669083484141793110941527015291021554978221892371620740912810024635374e-45), SC_(0.2457747126921997105292850810706506718048727997191850109419737442424277356633748861407006331354595514e-46), SC_(18.98621998722193543253549512314718748207338579866746714315870763255310856472818916227141762310922602), SC_(0.9999999999999999999999999999999999999999999999754225287307800289470714918929349328195127200280814989), + }; +#undef SC_ + + + diff --git a/test/log1p_expm1_test.cpp b/test/log1p_expm1_test.cpp new file mode 100644 index 000000000..9c3e23d6d --- /dev/null +++ b/test/log1p_expm1_test.cpp @@ -0,0 +1,593 @@ +// (C) Copyright John Maddock 2005. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include + +#include + +#ifdef BOOST_NO_STDC_NAMESPACE +namespace std{ using ::sqrt; using ::exp; } +#endif + +// +// These test values were generated at 256-bit precision with NTL, +// we'll use these as a comparison for our own algorithms: +// +long double data[][3] = { + { 0.7071067811865475244008443621048490392848L, 0.5347999967395703705239932642507040249904L, 1.028114981647472451108126112746351175174L, }, + { -0.7071067811865475244008443621048490392848L, -1.227947177299515679941225385708880593066L, -0.5069313086047602121540426193848225247642L, }, + { 0.5L, 0.405465108108164381978013115464349136572L, 0.6487212707001281468486507878141635716538L, }, + { -0.5L, -0.6931471805599453094172321214581765680755L, -0.3934693402873665763962004650088195465581L, }, + { 0.3535533905932737622004221810524245196424L, 0.3027332756136080026565008168738916532429L, 0.4241190194809816004138284781974755046316L, }, + { -0.3535533905932737622004221810524245196424L, -0.4362646682381306258028444378052416278323L, -0.2978114986734404037618125202537819364955L, }, + { 0.25L, 0.2231435513142097557662950903098345033746L, 0.2840254166877414840734205680624364583363L, }, + { -0.25L, -0.2876820724517809274392190059938274315035L, -0.2211992169285951317548297330216793527032L, }, + { 0.1767766952966368811002110905262122598212L, 0.1627790866561079067978577664745092478057L, 0.1933645794479495947956947190066751660075L, }, + { -0.1767766952966368811002110905262122598212L, -0.1945277849706882079548540492230348777333L, -0.1620331144212442112773768997051282549751L, }, + { 0.125L, 0.1177830356563834545387941094705217050685L, 0.1331484530668263168290072278117938725655L, }, + { -0.125L, -0.1335313926245226231463436209313499745894L, -0.117503097415404597135107856770949263778L, }, + { 0.0883883476483184405501055452631061299106L, 0.08469802192359444824992033285217108168932L, 0.09241227540153062617063135584671815396877L, }, + { -0.0883883476483184405501055452631061299106L, -0.09254119938462034112310437534311466334391L, -0.08459468781377732889540834505145255847266L, }, + { 0.0625L, 0.0606246218164348425806061320404202632862L, 0.06449445891785942956339059464288967310073L, }, + { -0.0625L, -0.06453852113757117167292391568399292812891L, -0.06058693718652421388028917537769491547532L, }, + { 0.0441941738241592202750527726315530649553L, 0.0432454624194302551610854640871481968471L, 0.04518528280947901327000081260336002430674L, }, + { -0.0441941738241592202750527726315530649553L, -0.04520049725523360571871295632901500898476L, -0.04323183989734343213613354147207468639883L, }, + { 0.03125L, 0.0307716586667536883710282075967721640917L, 0.0317434074991026709387478152815071441945L, }, + { -0.03125L, -0.03174869831458030115699628274852562992756L, -0.03076676552365591815189080675364721639527L, }, + { 0.02209708691207961013752638631577653247765L, 0.02185648424813694627588486704954425304883L, 0.02234303578078870447841893477675632033383L, }, + { -0.02209708691207961013752638631577653247765L, -0.02234488474624582074086983060944116403229L, -0.02185473466225040063461756538971286083712L, }, + { 0.015625L, 0.01550418653596525415085404604244683587787L, 0.01574770858668574745853507208235174890672L, }, + { -0.015625L, -0.01574835696813916860754951146082826952093L, -0.01550356299459159401301117030297963029214L, }, + { 0.01104854345603980506876319315788826623883L, 0.01098795417351052684893633616419361769653L, 0.01110980401773807767277088699717081730887L, }, + { -0.01104854345603980506876319315788826623883L, -0.01111003193719750909051920695450559324905L, -0.0109877324634695920927755509825266486829L, }, + { 0.0078125L, 0.007782140442054948947462900061136763678126L, 0.007843097206447977693453559760123579193392L, }, + { -0.0078125L, -0.007843177461025892873184042490943581654592L, -0.00778206173975648789406277388638216992911L, }, + { 0.005524271728019902534381596578944133119413L, 0.005509068902949034570040757911345564446908L, 0.005539558653829430569890102084787436958606L, }, + { -0.005524271728019902534381596578944133119413L, -0.00553958694674479599776921231398085523465L, -0.005509040998094137069364339639027839483272L, }, + { 0.00390625L, 0.003898640415657323013937343095842907010724L, 0.003913889338347573443609603903460281898814L, }, + { -0.00390625L, -0.003913899321136329092317783643572664842706L, -0.003898630529882509937135338054830955713597L, }, + { 0.002762135864009951267190798289472066559706L, 0.002758328176699244507992500461404507081665L, 0.002765954075939611809020032526494603236975L, }, + { -0.002762135864009951267190798289472066559706L, -0.002765957600334472995309858879390478447378L, -0.002758324676557722165983108723111282889863L, }, + { 0.001953125L, 0.00195122013126174943967404953184153850035L, 0.001955033591002812046518898047477215867356L, }, + { -0.001953125L, -0.001955034835803350557627492241866812137664L, -0.001951218892524527289957340917326404744062L, }, + { 0.001381067932004975633595399144736033279853L, 0.001380115134839422362840066413196608682497L, 0.001382022045502678201141870910789095246896L, }, + { -0.001381067932004975633595399144736033279853L, -0.001382022485291226579353866252268409285565L, -0.001380114696566700781276188886360328730002L, }, + { 0.0009765625L, 0.0009760859730554588959608249080171866726118L, 0.0009770394924165352428452926116065064658516L, }, + { -0.0009765625L, -0.000977039647826612785968075151753465835865L, -0.0009760858180243377652882103896705696807979L, }, + { 0.0006905339660024878167976995723680166399266L, 0.0006902956571239922163465924178424595489941L, 0.0006907724394697735895620855250217767118257L, }, + { -0.0006905339660024878167976995723680166399266L, -0.0006907724943958822152082937884673220890625L, -0.0006902956022926226579739822473659270196051L, }, + { 0.00048828125L, 0.0004881620795013511885370496926454098503177L, 0.0004884004786944731261736238071633537881055L, }, + { -0.00048828125L, -0.0004884004981088744649849635598969109834608L, -0.0004881620601106346120642438969970786650047L, }, + { 0.0003452669830012439083988497861840083199633L, 0.0003452073920725940347605594851408457654352L, 0.0003453265945064500836804861194113777618491L, }, + { -0.0003452669830012439083988497861840083199633L, -0.0003453266013692502439328491689895266497932L, -0.0003452073852157150645328468804957180492811L, }, + { 0.000244140625L, 0.0002441108275273627091604790858234536994612L, 0.0002441704297478549370052339241357737575196L, }, + { -0.000244140625L, -0.0002441704321739144566954654183814336430665L, -0.0002441108251027834869064123317332192226443L, }, + { 0.0001726334915006219541994248930920041599816L, 0.0001726185920541657488047052564217192486562L, 0.0001726483935193326386701093624723297495011L, }, + { -0.0001726334915006219541994248930920041599816L, -0.0001726483943769975333358785791023362275292L, -0.0001726185911968709284892274915373687749247L, }, + { 0.0001220703125L, 0.0001220628625256773716230553671622032006662L, 0.0001220777633837710765035196704053169651876L, }, + { -0.0001220703125L, -0.0001220777636869822415828707903119755525239L, -0.0001220628622225587251301833935672790920603L, }, + { 0.8631674575031097709971244654600207999082e-4L, 0.8631302067436859538170634567406786112601e-4L, 0.8632047114779673053115205017424336944456e-4L, }, + { -0.8631674575031097709971244654600207999082e-4L, -0.8632047125499327478558483802377432851254e-4L, -0.8631302056719518077375662982669623892259e-4L, }, + { 0.6103515625e-4L, 0.6103329368063852491315878964896399244278e-4L, 0.610370189330454217791205985471093033817e-4L, }, + { -0.6103515625e-4L, -0.6103701897094392572114242980681797646601e-4L, -0.6103329364274580338274063706796778431152e-4L, }, + { 0.4315837287515548854985622327300103999541e-4L, 0.4315744157937625036775176424278410274829e-4L, 0.4315930421112837090401086653281340081162e-4L, }, + { -0.4315837287515548854985622327300103999541e-4L, -0.4315930422452721604826114516606004744788e-4L, -0.4315744156597885082640026911399617719788e-4L, }, + { 0.30517578125e-4L, 0.3051711247318637856906951416899468124305e-4L, 0.3051804379102429545128481254124317032922e-4L, }, + { -0.30517578125e-4L, -0.3051804379576142772845440263529078774196e-4L, -0.3051711246844960769262435293040988383306e-4L, }, + { 0.2157918643757774427492811163650051999771e-4L, 0.2157895361028356677465179910857365631679e-4L, 0.2157941926989617246887680913732626816569e-4L, }, + { -0.2157918643757774427492811163650051999771e-4L, -0.2157941927157098293412689381708235824685e-4L, -0.2157895360860884665958278826510182396057e-4L, }, + { 0.152587890625e-4L, 0.1525867264836239740575732513488881988266e-4L, 0.1525890547841394814004262248066173018701e-4L, }, + { -0.152587890625e-4L, -0.152589054790060783804405477297578319824e-4L, -0.1525867264777028975290466417266610381266e-4L, }, + { 0.1078959321878887213746405581825025999885e-4L, 0.1078953501154664660426995857578938351957e-4L, 0.1078965142665913183071896911252005879737e-4L, }, + { -0.1078959321878887213746405581825025999885e-4L, -0.1078965142686848031534799026109936625941e-4L, -0.1078953501133730376652725297740371278837e-4L, }, + { 0.762939453125e-5L, 0.7629365427567572155885296849132278805619e-5L, 0.7629423635154471743184330338220061308624e-5L, }, + { -0.762939453125e-5L, -0.7629423635228487317358417985971365713551e-5L, -0.7629365427493557993432788023738794879446e-5L, }, + { 0.5394796609394436068732027909125129999427e-5L, 0.5394782057531543905412357533557238337288e-5L, 0.5394811161335832678521641682790705055058e-5L, }, + { -0.5394796609394436068732027909125129999427e-5L, -0.5394811161362001062632444450174708967681e-5L, -0.5394782057505375874231949463613865956897e-5L, }, + { 0.3814697265625e-5L, 0.3814689989685889480711784976833932293351e-5L, 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0.4156000133564589919546802616566101790699e-38L, 0.4156000133564589919546802616566101790716e-38L, }, + { -0.4156000133564589919546802616566101790707e-38L, -0.4156000133564589919546802616566101790716e-38L, -0.4156000133564589919546802616566101790699e-38L, }, + { 0.2938735877055718769921841343055614194547e-38L, 0.2938735877055718769921841343055614194542e-38L, 0.2938735877055718769921841343055614194551e-38L, }, + { -0.2938735877055718769921841343055614194547e-38L, -0.2938735877055718769921841343055614194551e-38L, -0.2938735877055718769921841343055614194542e-38L, }, +}; + +template +void test(T) +{ + static const T two = 2; + // + // Fudge factor is 3 epsilon, should be less than that, but some + // platforms have poor long double support (some of the test values + // actually test the accuracy of std::log and std::exp, and it's + // usually these, rather than our series expansion that fails otherwise). + // + static const T factor = std::pow(two, 1-std::numeric_limits::digits) * 300; + for(unsigned i = 0; i < sizeof(data)/sizeof(data[0]); ++i) + { + T input_value = static_cast(data[i][0]); + T expected_log1p = static_cast(data[i][1]); + T expected_expm1 = static_cast(data[i][2]); + BOOST_CHECK_CLOSE(boost::math::log1p(input_value), expected_log1p, factor); + BOOST_CHECK_CLOSE(boost::math::expm1(input_value), expected_expm1, factor); + BOOST_CHECK_CLOSE(boost::math::expm1(-input_value), -boost::math::expm1(input_value)/static_cast(std::exp(input_value)), 2*factor); + } + // + // C99 Appendix F special cases: + static const T zero = 0; + static const T m_one = -1; + BOOST_CHECK_EQUAL(boost::math::log1p(zero), zero); + BOOST_CHECK_EQUAL(boost::math::log1p(-zero), zero); + BOOST_CHECK_EQUAL(boost::math::expm1(zero), zero); + if(std::numeric_limits::has_infinity) + { + BOOST_CHECK_EQUAL(boost::math::log1p(m_one), -std::numeric_limits::infinity()); + BOOST_CHECK_EQUAL(boost::math::expm1(-std::numeric_limits::infinity()), m_one); + BOOST_CHECK_EQUAL(boost::math::expm1(std::numeric_limits::infinity()), std::numeric_limits::infinity()); + } +} + + +int test_main(int, char* []) +{ + std::cout << "Running float tests" << std::endl; + test(float(0)); + std::cout << "Running double tests" << std::endl; + test(double(0)); + // + // The long double version of these tests fails on some platforms + // due to poor std lib support (not enough digits returned from + // std::log and std::exp): + // +#if !defined(__CYGWIN__) && !defined(__FreeBSD__) && !(defined(__GNUC__) && defined(__sun)) + std::cout << "Running long double tests" << std::endl; + test((long double)(0)); +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} diff --git a/test/modulo_test.cpp b/test/modulo_test.cpp deleted file mode 100644 index 6d8b5f8b2..000000000 --- a/test/modulo_test.cpp +++ /dev/null @@ -1,206 +0,0 @@ -// Boost modular arithmetic test program file ------------------------------// - -// Copyright 2002 Daryle Walker. Use, modification, and distribution are -// subject to the Boost Software License, Version 1.0. (See accompanying file -// LICENSE_1_0.txt or a copy at .) - -// See for the library's home page. - -// Revision History -// 20 Sep 2002 Initial version (Daryle Walker) - - -#include // for BOOST_NO_MEMBER_TEMPLATES -#include // for boost::exit_success -#include // for boost::math::modulo, etc. -#include // for main, BOOST_TEST - -#include // for std::setw -#include // for std::ios_base -#include // for std::cout -#include // for std::ostream, std::endl -#include // for std::ostringstream, etc. -#include // for std::string - - -// Control macro -#ifndef CONTROL_BACKWARD_CONVERSION_TEST -#define CONTROL_BACKWARD_CONVERSION_TEST 0 -#endif - -// Macros to compact code -#define PRIVATE_PRINT_UNARY_TABLE( Modulus, Op, Stream ) do { \ - typedef ::boost::math::modulo< (Modulus) > modulo_type; \ - (Stream) << "Unary Operation mod-" << (Modulus) \ - << " (" << #Op << " Value):\n"; \ - for ( unsigned i = 0u ; i < (Modulus) ; ++i ) \ - (Stream) << ' ' << ::std::setw(6) << i; \ - (Stream) << '\n'; \ - for ( unsigned j = 0u ; j < (Modulus) ; ++j ) \ - (Stream) << ' ' << ::std::setw(6) << ( Op modulo_type(j) ); \ - (Stream) << '\n' << ::std::endl; } while (false) - -#define PRIVATE_PRINT_ALL_UNARY_TABLES( Modulus, Stream ) do { \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), +, (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), -, (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), ~, (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), (bool), (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), !, (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), ++, (Stream) ); \ - PRIVATE_PRINT_UNARY_TABLE( (Modulus), --, (Stream) ); } while (false) - -#define PRIVATE_PRINT_BINARY_TABLE( Modulus, Op, Stream ) do { \ - typedef ::boost::math::modulo< (Modulus) > modulo_type; \ - (Stream) << "Binary Operation mod-" << (Modulus) \ - << " (Left " << #Op << " Top):\n"; \ - (Stream) << ::std::setw(6) << ' '; \ - for ( unsigned i = 0u ; i < (Modulus) ; ++i ) \ - (Stream) << ' ' << ::std::setw(6) << i; \ - (Stream) << '\n'; \ - for ( unsigned j = 0u ; j < (Modulus) ; ++j ) { \ - (Stream) << ::std::setw(6) << j; \ - for ( unsigned k = 0u ; k < (Modulus) ; ++k ) { \ - ::std::ostringstream oss; \ - ::std::string answer; \ - oss.copyfmt( (Stream) ); \ - try { oss << ( modulo_type(j) Op modulo_type(k) ); \ - answer = oss.str(); } catch (...) { answer.assign( "-" ); } \ - (Stream) << ' ' << ::std::setw(6) << answer; \ - } (Stream) << '\n'; \ - } (Stream) << ::std::endl; } while (false) - -#define PRIVATE_PRINT_ALL_BINARY_TABLES( Modulus, Stream ) do { \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), +, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), -, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), *, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), /, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), &, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), |, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), ^, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), &&, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), ||, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), ==, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), !=, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), >=, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), <=, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), >, (Stream) ); \ - PRIVATE_PRINT_BINARY_TABLE( (Modulus), <, (Stream) ); } while (false) - -#define PRIVATE_PRINT_BINARY_FUNC_TABLE( Modulus, Func, Stream ) do { \ - typedef ::boost::math::modulo< (Modulus) > modulo_type; \ - (Stream) << "Binary Operation mod-" << (Modulus) \ - << " " << #Func << "(Left, Top):\n"; \ - (Stream) << ::std::setw(6) << ' '; \ - for ( unsigned i = 0u ; i < (Modulus) ; ++i ) \ - (Stream) << ' ' << ::std::setw(6) << i; \ - (Stream) << '\n'; \ - for ( unsigned j = 0u ; j < (Modulus) ; ++j ) { \ - (Stream) << ::std::setw(6) << j; \ - for ( unsigned k = 0u ; k < (Modulus) ; ++k ) { \ - ::std::ostringstream oss; \ - ::std::string answer; \ - oss.copyfmt( (Stream) ); \ - try { oss << (Func)( modulo_type(j), k ); \ - answer = oss.str(); } catch (...) { answer.assign( "-" ); } \ - (Stream) << ' ' << ::std::setw(6) << answer; \ - } (Stream) << '\n'; \ - } (Stream) << ::std::endl; } while (false) - -#define PRIVATE_PRINT_ALL_TABLES( Modulus, Stream ) do { \ - PRIVATE_PRINT_ALL_UNARY_TABLES( (Modulus), (Stream) ); \ - PRIVATE_PRINT_ALL_BINARY_TABLES( (Modulus), (Stream) ); } while (false) - - -// Main testing function -int -test_main -( - int , // "argc" is unused - char * [] // "argv" is unused -) -{ - using boost::math::modulo; - using std::cout; - using boost::math::pow; - using std::endl; - - typedef modulo<6> modulo6_t; - typedef modulo<3> modulo3_t; - - // Try out most of the operations - cout.setf( std::ios_base::boolalpha ); - PRIVATE_PRINT_ALL_TABLES( 7, cout ); - PRIVATE_PRINT_BINARY_FUNC_TABLE( 7, pow, cout ); - - // Test input - std::istringstream as( "[3]" ), bs( "[2]" ), cs( "-4" ); - modulo6_t a, b, c; - - cout << "Doing input tests." << endl; - BOOST_TEST( as >> a ); - BOOST_TEST( residue(a) == 3 ); - BOOST_TEST( bs >> b ); - BOOST_TEST( residue(b) == 2 ); - BOOST_TEST( !(cs >> c) ); - BOOST_TEST( (cs.clear(), cs.str( "[ty" ), !( cs >> c )) ); - BOOST_TEST( (cs.clear(), cs.str( "[5?" ), !( cs >> c )) ); - BOOST_TEST( (cs.clear(), cs.str( "[1]r0]]" ), ( cs >> c )) ); - BOOST_TEST( residue(c) == 1 ); - - // Test post-(in/de)crement - cout << "Doing post-(in/de)crement tests." << endl; - BOOST_TEST( (c++, residue( c ) == 2) ); - BOOST_TEST( (c--, residue( c ) == 1) ); - - // Test swap - cout << "Doing swap test." << endl; - swap( a, b ); - BOOST_TEST( (residue( a ) == 2) && (residue( b ) == 3) ); - - #ifndef BOOST_NO_MEMBER_TEMPLATES - // Test cross-conversions - cout << "Doing cross-conversion tests." << endl; - BOOST_TEST( residue(modulo3_t( modulo6_t(0) )) == 0 ); - BOOST_TEST( residue(modulo3_t( modulo6_t(1) )) == 1 ); - BOOST_TEST( residue(modulo3_t( modulo6_t(2) )) == 2 ); - BOOST_TEST( residue(modulo3_t( modulo6_t(3) )) == 0 ); - BOOST_TEST( residue(modulo3_t( modulo6_t(4) )) == 1 ); - BOOST_TEST( residue(modulo3_t( modulo6_t(5) )) == 2 ); - - #if CONTROL_BACKWARD_CONVERSION_TEST - // Should cause an error #if compiled - BOOST_TEST( residue(modulo6_t( modulo3_t(5) )) == 2 ); - #endif - #endif - - // Test quasi-traditional notation - cout << "Doing cross-comparison tests." << endl; - BOOST_TEST( 17 == modulo<5>(2) ); - BOOST_TEST( 5 != modulo<2>(0) ); - BOOST_TEST( modulo<5>(3) != 17 ); - BOOST_TEST( modulo<2>(1) == 5 ); - - // Test assignment - cout << "Doing assignment tests." << endl; - BOOST_TEST( (c.assign( 4 ), residue( c ) == 4) ); - BOOST_TEST( (c.assign( 17 ), residue( c ) == 5) ); - - // Test inversion status - cout << "Doing invertibilty tests." << endl; - BOOST_TEST( !modulo6_t(0).is_invertible() ); - BOOST_TEST( modulo6_t(1).is_invertible() ); - BOOST_TEST( !modulo6_t(2).is_invertible() ); - BOOST_TEST( !modulo6_t(3).is_invertible() ); - BOOST_TEST( !modulo6_t(4).is_invertible() ); - BOOST_TEST( modulo6_t(5).is_invertible() ); - BOOST_TEST( !modulo3_t(0).is_invertible() ); - BOOST_TEST( modulo3_t(1).is_invertible() ); - BOOST_TEST( modulo3_t(2).is_invertible() ); - - // Test Chinese remainder theorem - cout << "Doing Chinese Remainder Theorem tests." << endl; - BOOST_TEST( boost::math::chinese_remainder(modulo<7>( 3 ), modulo<13>( 7 )) - == modulo<91>(59) ); - - return boost::exit_success; -} diff --git a/test/powm1_sqrtp1m1_test.cpp b/test/powm1_sqrtp1m1_test.cpp new file mode 100644 index 000000000..2082c9bf4 --- /dev/null +++ b/test/powm1_sqrtp1m1_test.cpp @@ -0,0 +1,1603 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +template +void test_powm1_sqrtp1m1(T, const char* name) +{ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 141> sqrtp1m1_data = { + SC_(-0.990433037281036376953125), SC_(-0.902189148255607021082179865003660033379), + SC_(-0.928576648235321044921875), SC_(-0.7327485233629588435419837213946140663952), + SC_(-0.804919183254241943359375), SC_(-0.5583204592175928547330219789723812512248), + SC_(-0.780276477336883544921875), SC_(-0.5312532425038905348288090678272629719499), + SC_(-0.775070965290069580078125), SC_(-0.525733160857803152349794525146692520785), + SC_(-0.74602639675140380859375), SC_(-0.4960420620244223284705423670972348730775), + SC_(-0.72904598712921142578125), SC_(-0.4794675678972648545670296583048773096015), + SC_(-0.7162272930145263671875), SC_(-0.4672967927771847619430106324762494277417), + SC_(-0.68477380275726318359375), SC_(-0.4385499156267435234127335149836228483716), + SC_(-0.62323606014251708984375), SC_(-0.3861890031471553130754715740017444288273), + SC_(-0.576151371002197265625), SC_(-0.3489634196162225368137418186622322770468), + SC_(-0.5579319000244140625), SC_(-0.3351179804088653673227401220715512851566), + SC_(-0.44300353527069091796875), SC_(-0.2536780421766293214167684432407843582845), + SC_(-0.40594112873077392578125), SC_(-0.2292478535422518369418742953348121057781), + SC_(-0.38366591930389404296875), SC_(-0.2149305249240001453212315877592832894955), + SC_(-0.277411997318267822265625), SC_(-0.1499482352928545459530860365223688354153), + SC_(-0.202522933483123779296875), SC_(-0.1069842854031759981174720343292165991554), + SC_(-0.156477451324462890625), SC_(-0.08156516362044655434273010203540784115386), + SC_(-0.029248714447021484375), SC_(-0.01473288619127324715091701268111937769847), + SC_(0.1431564604442703013402649929484277542953e-29), SC_(0.7157823022213515067013249647418825993242e-30), + SC_(0.1791466932348087634896446282571611213266e-29), SC_(0.8957334661740438174482231412854044374117e-30), + SC_(0.6013619202535540063110633226832922483532e-29), SC_(0.3006809601267770031555316613411940789777e-29), + SC_(0.115805324961653822428570241697281798758e-28), SC_(0.5790266248082691121428512084847326346288e-29), + SC_(0.1422457400834001098175711728787848259007e-28), SC_(0.711228700417000549087855864391394898182e-29), + SC_(0.4970121018327539153628705477876439795096e-28), SC_(0.2485060509163769576814352738907342268877e-28), + SC_(0.9660079415057497591758174164417478444323e-28), SC_(0.483003970752874879587908708209209280428e-28), + SC_(0.1232929313253182131376331095427391968754e-27), SC_(0.6164646566265910656881655476946945507336e-28), + SC_(0.3296523285617759312781860549364832953326e-27), SC_(0.1648261642808879656390930274546578154505e-27), + SC_(0.528364435768055252017009628713605422886e-27), SC_(0.26418221788402762600850481432190658932e-27), + SC_(0.886586057273120049620324386849842094685e-27), SC_(0.4432930286365600248101621933266666927236e-27), + SC_(0.2499669674831043259218157022821422146034e-26), SC_(0.1249834837415521629609078510629667512608e-26), + SC_(0.4131050397232622964314362671638736040881e-26), SC_(0.2065525198616311482157181333686170847381e-26), + SC_(0.7679738097881433551381658732998641759182e-26), SC_(0.3839869048940716775690829359127023723085e-26), + SC_(0.199929739820949207249437007767740538737e-25), SC_(0.9996486991047460362471850338422150855758e-26), + SC_(0.5151477415246978459754129800826163591626e-25), SC_(0.2575738707623489229877064867240932346063e-25), + SC_(0.101200734533556026342258477595279955025e-24), SC_(0.5060036726677801317112923751744139374608e-25), + SC_(0.2064292695896540981798546456623054911033e-24), SC_(0.1032146347948270490899273175045223276369e-24), + SC_(0.4063294332896333395257434433879773416284e-24), SC_(0.2031647166448166697628717010560376261299e-24), + SC_(0.8138195767936862452966745688936976428456e-24), SC_(0.406909788396843122648337201659060874841e-24), + SC_(0.9575550627132253801929510132578249716542e-24), SC_(0.478777531356612690096475392014950219861e-24), + SC_(0.2855160956298500804375620841706273850616e-23), SC_(0.1427580478149250402187809401860126128887e-23), + SC_(0.65201444297915461398563707001320281266e-23), SC_(0.3260072214895773069928180036030590895584e-23), + SC_(0.1310988374636350038320977491775043421995e-22), SC_(0.6554941873181750191604865975243736714241e-23), + SC_(0.2590288837798696209228010176465529547374e-22), SC_(0.1295144418899348104613996701237435743037e-22), + SC_(0.2937779542193655202274099291941187976629e-22), SC_(0.1468889771096827601137038857784795823862e-22), + SC_(0.7863513178004503049754083414326234074965e-22), SC_(0.3931756589002251524876964413613741224383e-22), + SC_(0.1903818607087388763706780167350761726053e-21), SC_(0.9519093035436943818533447771092722109619e-22), + SC_(0.3812242142377350870566942975497647799754e-21), SC_(0.1906121071188675435283289822871922426703e-21), + SC_(0.5493133580141330277178034419485741501887e-21), SC_(0.2746566790070665138588640028286254797086e-21), + SC_(0.9672153634284186955666772243312215295852e-21), SC_(0.4836076817142093477832216739707042687854e-21), + SC_(0.1702169477623814384559878647986894129041e-20), SC_(0.8510847388119071922795771513771277983779e-21), + SC_(0.4817114569977399785676754474621208412799e-20), SC_(0.2408557284988699892835476663213068137742e-20), + SC_(0.7538352992756463183303278501219690799218e-20), SC_(0.3769176496378231591644535904879420358976e-20), + SC_(0.2596305715949999708394617609422128090557e-19), SC_(0.1298152857974999854188882800497720744566e-19), + SC_(0.4444587480324321591032923385589104015025e-19), SC_(0.2222293740162160795491768745456732380306e-19), + SC_(0.9715574921498573937069095571295029856174e-19), SC_(0.4857787460749286968416557290578449901693e-19), + SC_(0.2036598542733453787268262970278076551267e-18), SC_(0.1018299271366726893582284814835737930738e-18), + SC_(0.4248971931658660264162106698360155121463e-18), SC_(0.2124485965829330131855381318229788476899e-18), + SC_(0.6521097487613458963613731825259556273977e-18), SC_(0.3260548743806729481275307007092796054895e-18), + SC_(0.1436126164096190058281493628911107407475e-17), SC_(0.718063082048095028882939519555349077068e-18), + SC_(0.3118908901459261162419055180006211003274e-17), SC_(0.1559454450729630579993578498052878596544e-17), + SC_(0.3593346613595175715618300349429858897565e-17), SC_(0.1796673306797587856195132689035439820049e-17), + SC_(0.9445874854124767215374919304693435151421e-17), SC_(0.4722937427062383596534390682373393588305e-17), + SC_(0.2566182432094081539023303073498993853718e-16), SC_(0.1283091216047040761280036193264127770924e-16), + SC_(0.3363765695149349330660137891158001366421e-16), SC_(0.1681882847574674651186419380749519252198e-16), + SC_(0.1073581901339262605326457800103412409953e-15), SC_(0.53679095066963128825600266401137964529e-16), + SC_(0.186668406231853462907965823802669547149e-15), SC_(0.9333420311592672709834617625880158673234e-16), + SC_(0.3727540802657755688795382376099496468669e-15), SC_(0.1863770401328877670715685744569461355954e-15), + SC_(0.6211646767866855090717281839829411183018e-15), SC_(0.3105823383933427063051696310530511222847e-15), + SC_(0.1561186859754253464932505224282976996619e-14), SC_(0.7805934298771264278032012284733416682125e-15), + SC_(0.3092010764722992466335682593125966377556e-14), SC_(0.1546005382361495038101520151206746652194e-14), + SC_(0.6192850577371690132255643845837767003104e-14), SC_(0.3096425288685840272203037716324248888074e-14), + SC_(0.1047879028014987723427253740737796761096e-13), SC_(0.5239395140074924891505551783318071383455e-14), + SC_(0.1978473638988408750405412206418986897916e-13), SC_(0.989236819494199482255280888212449451182e-14), + SC_(0.4041816252346730475863978426787070930004e-13), SC_(0.2020908126173344817583717046094650765545e-13), + SC_(0.9410302262901834580155480125540634617209e-13), SC_(0.4705151131450806597841891098742774312111e-13), + SC_(0.1334530223958893535574077304772799834609e-12), SC_(0.6672651119794245056505554066901540498883e-13), + SC_(0.266297021326439287136622624529991298914e-12), SC_(0.1331485106632107793053653966876952769505e-12), + SC_(0.5920415525016708979677559909760020673275e-12), SC_(0.29602077625079163483389193486561174812e-12), + SC_(0.155163989296047688526414276566356420517e-11), SC_(0.7758199464799374943373933162927267207972e-12), + SC_(0.326923297461201300961874949280172586441e-11), SC_(0.1634616487304670519279090616362400067267e-11), + SC_(0.3753785910581841633870681107509881258011e-11), SC_(0.1876892955289159453352533516401529801122e-11), + SC_(0.9579165585749116473834874341264367103577e-11), SC_(0.4789582792863088185252592080679746767978e-11), + SC_(0.1858167439361402273334533674642443656921e-10), SC_(0.9290837196763851538764282981715353513818e-11), + SC_(0.5449485307451595872407779097557067871094e-10), SC_(0.2724742653688676823559737485578304836506e-10), + SC_(0.6089519166696533147842274047434329986572e-10), SC_(0.3044759583301913769320592805847923970217e-10), + SC_(0.1337744776064297980155970435589551925659e-9), SC_(0.6688723880097794765058899684143827289977e-10), + SC_(0.2554458866654840676346793770790100097656e-9), SC_(0.1277229433245854586915920539477320784002e-9), + SC_(0.9285605062636648199259070679545402526855e-9), SC_(0.4642802530240543332889135775318577029962e-9), + SC_(0.1698227447555211711005540564656257629395e-8), SC_(0.8491137234171087978551371134983090509552e-9), + SC_(0.339355921141759608872234821319580078125e-8), SC_(0.1696779604269267531629088042873986404561e-8), + SC_(0.6313728651008432279922999441623687744141e-8), SC_(0.3156864320521319970871232105329124668304e-8), + SC_(0.8383264749056706932606175541877746582031e-8), SC_(0.4191632365743462521529019590756957227221e-8), + SC_(0.1962631124285962869180366396903991699219e-7), SC_(0.9813155573280803193195787399755096477409e-8), + SC_(0.5256384838503436185419559478759765625e-7), SC_(0.2628192384714742037346966666026188341227e-7), + SC_(0.116242290459922514855861663818359375e-6), SC_(0.5812114354092759417537613189484537667262e-7), + SC_(0.1776920584006802528165280818939208984375e-6), SC_(0.8884602525353202473263890111106254261705e-7), + SC_(0.246631174150024889968335628509521484375e-6), SC_(0.1233155794716463747702014149912298889881e-6), + SC_(0.7932688959044753573834896087646484375e-6), SC_(0.3966343692928262265327186740527956086072e-6), + SC_(0.1372093493046122603118419647216796875e-5), SC_(0.6860465111931535414103361330991297079332e-6), + SC_(0.214747751670074649155139923095703125e-5), SC_(0.1073738181893531617762304794194699568343e-5), + SC_(0.527022712049074470996856689453125e-5), SC_(0.2635110088342783512028120843144783486949e-5), + SC_(0.9233162927557714283466339111328125e-5), SC_(0.461657080741584719962950298693491037714e-5), + SC_(0.269396477960981428623199462890625e-4), SC_(0.1346973318119308516416495011442959614762e-4), + SC_(0.3208058114978484809398651123046875e-4), SC_(0.1604016193149502975581143111396377659995e-4), + SC_(0.00010957030463032424449920654296875), SC_(0.5478365169091582645735757270110530879622e-4), + SC_(0.000126518702018074691295623779296875), SC_(0.6325735026285620681070344590408428362373e-4), + SC_(0.00028976381872780621051788330078125), SC_(0.0001448714155003885621472683658275189243597), + SC_(0.000687857042066752910614013671875), SC_(0.0003438693979519525361574726383087724411954), + SC_(0.00145484809763729572296142578125), SC_(0.0007271596682270997885618336103424497281234), + SC_(0.0073254108428955078125), SC_(0.003656022172385267560713803761753901548099), + SC_(0.09376299381256103515625), SC_(0.04583124537975104196281645139037475336575), + SC_(0.0944411754608154296875), SC_(0.04615542605332570615786127417302399776704), + SC_(0.264718532562255859375), SC_(0.1245970534205822199491885900676517027912), + SC_(0.27952671051025390625), SC_(0.1311616641799057655376985599143742875242), + SC_(0.31148135662078857421875), SC_(0.1451992650280511659331021017070743457453), + SC_(0.3574702739715576171875), SC_(0.1651052630434546351309425398611544399279), + SC_(0.362719058990478515625), SC_(0.1673555837834839069815597770182209283592), + SC_(0.45167791843414306640625), SC_(0.2048559741455171397876451754340651729329), + SC_(0.58441460132598876953125), SC_(0.2587353182166570039327003862804562417757), + SC_(0.59585726261138916015625), SC_(0.263272441958340655449252054139087309968), + SC_(0.5962116718292236328125), SC_(0.2634127084326893190571263246997038824839), + SC_(0.6005609035491943359375), SC_(0.265132761234643838648662687260938123223), + SC_(0.62944734096527099609375), SC_(0.2764980771490691893425893413189776581002), + SC_(0.67001712322235107421875), SC_(0.2922914234886615056408835783570718269519), + SC_(0.6982586383819580078125), SC_(0.3031725282486421443925178739003813209589), + SC_(0.75686132907867431640625), SC_(0.3254664571684469135448954017993676523574), + SC_(0.81158387660980224609375), SC_(0.3459509190939327287182180832834475919551), + SC_(0.826751708984375), SC_(0.3515737896927326105907938446356122609441), + SC_(0.83147108554840087890625), SC_(0.3533185454830658156956304096558121255029), + SC_(0.8679864406585693359375), SC_(0.3667430046129994134441282712127036765511), + SC_(0.91433393955230712890625), SC_(0.3835945719582406373102286771719657103388), + SC_(0.91501367092132568359375), SC_(0.3838401898056457601350544781643172326518), + SC_(0.918984889984130859375), SC_(0.3852743013512272831269535677278325535267), + SC_(0.92977702617645263671875), SC_(0.3891641465919182859360893492188078577616), + SC_(0.93538987636566162109375), SC_(0.3911829054317989358513931994433199352401), + SC_(0.93773555755615234375), SC_(0.3920257029078710024155036194055376074007), + SC_(0.94118559360504150390625), SC_(0.3932643660142325996680401116017235829116), + SC_(0.96221935749053955078125), SC_(0.4007924034240546775728864480529865192659), + SC_(0.98576259613037109375), SC_(0.4091708896121758495875515950932191750449), + SC_(0.99292266368865966796875), SC_(0.4117091285702801135545007937655927942821), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1400> powm1_data = { + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1431564604442703013402649929484277542953e-29), SC_(-0.4876113153308343652049349438365788782568e-28), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1791466932348087634896446282571611213266e-29), SC_(-0.6101991796549119337733033929476086235147e-28), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6013619202535540063110633226832922483532e-29), SC_(-0.2048324441766037485142714404837079817647e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.115805324961653822428570241697281798758e-28), SC_(-0.3944494481885382670819636026287935618847e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1422457400834001098175711728787848259007e-28), SC_(-0.4845092719324132218645425927278781231393e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4970121018327539153628705477876439795096e-28), SC_(-0.1692894082166527210362866376720838911671e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9660079415057497591758174164417478444323e-28), SC_(-0.3290360780895537432240111567751307160722e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1232929313253182131376331095427391968754e-27), SC_(-0.4199533030361316903341770129910084958631e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3296523285617759312781860549364832953326e-27), SC_(-0.1122842832471787971733666733700489713877e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.528364435768055252017009628713605422886e-27), SC_(-0.1799684601724219422633144864504456459061e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.886586057273120049620324386849842094685e-27), SC_(-0.301983851933262353667369307173490332305e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2499669674831043259218157022821422146034e-26), SC_(-0.851423131205077246153257242575159396693e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4131050397232622964314362671638736040881e-26), SC_(-0.1407094665264327834604146533729082291379e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7679738097881433551381658732998641759182e-26), SC_(-0.261582829282236901328884539002832986633e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.199929739820949207249437007767740538737e-25), SC_(-0.6809891995464351323188379271234568515929e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.5151477415246978459754129800826163591626e-25), SC_(-0.1754666656712664684348626881941041760604e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.101200734533556026342258477595279955025e-24), SC_(-0.34470413088736644015953540038270992073e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2064292695896540981798546456623054911033e-24), SC_(-0.7031275246329503001381414764473488822476e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4063294332896333395257434433879773416284e-24), SC_(-0.1384015983694437753942599100031834846519e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.8138195767936862452966745688936976428456e-24), SC_(-0.2771985511871706510319651593329877641942e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9575550627132253801929510132578249716542e-24), SC_(-0.3261569070527988349552625022548697814845e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2855160956298500804375620841706273850616e-23), SC_(-0.9725085302202836634236273890970136659468e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.65201444297915461398563707001320281266e-23), SC_(-0.2220854156138726384270360939328805807496e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1310988374636350038320977491775043421995e-22), SC_(-0.4465413322989495361133815710858355990476e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2590288837798696209228010176465529547374e-22), SC_(-0.8822893101478278733146155237640398711321e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2937779542193655202274099291941187976629e-22), SC_(-0.1000649598540977920935091326953457942208e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7863513178004503049754083414326234074965e-22), SC_(-0.2678424705352926224139081344679444879469e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1903818607087388763706780167350761726053e-21), SC_(-0.6484677619663470968358777401078973025876e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3812242142377350870566942975497647799754e-21), SC_(-0.1298504028134944266344521819289603363198e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.5493133580141330277178034419485741501887e-21), SC_(-0.1871039617763820057590094637996686252126e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9672153634284186955666772243312215295852e-21), SC_(-0.3294473432116758998094638207256180094583e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1702169477623814384559878647986894129041e-20), SC_(-0.5797831933845967032001409153808505427486e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4817114569977399785676754474621208412799e-20), SC_(-0.1640777904312887822627372613029956055284e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7538352992756463183303278501219690799218e-20), SC_(-0.2567670510166787584153650238170661635392e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2596305715949999708394617609422128090557e-19), SC_(-0.8843387446340099351857154955251648734277e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4444587480324321591032923385589104015025e-19), SC_(-0.1513889866135372642811595930728137815273e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9715574921498573937069095571295029856174e-19), SC_(-0.3309263341636914772803140564669851761043e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2036598542733453787268262970278076551267e-18), SC_(-0.6936945012060519151079315126759585887524e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4248971931658660264162106698360155121463e-18), SC_(-0.1447260421199383628860906433743869048343e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6521097487613458963613731825259556273977e-18), SC_(-0.2221178781221441480704197986591689852107e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1436126164096190058281493628911107407475e-17), SC_(-0.4891650475869226303643170206367209825168e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3118908901459261162419055180006211003274e-17), SC_(-0.1062344840825147041561189485134686674784e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3593346613595175715618300349429858897565e-17), SC_(-0.1223945090048411651974452688903537147646e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9445874854124767215374919304693435151421e-17), SC_(-0.3217399653341720695630524182811530497325e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2566182432094081539023303073498993853718e-16), SC_(-0.8740783246589107366187111286071499874759e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3363765695149349330660137891158001366421e-16), SC_(-0.114574655589157486912055633512820155344e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1073581901339262605326457800103412409953e-15), SC_(-0.3656773025840533327467103325113205753957e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.186668406231853462907965823802669547149e-15), SC_(-0.6358192065586750730853832514556047648021e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3727540802657755688795382376099496468669e-15), SC_(-0.1269653544166010514919541179082417894329e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6211646767866855090717281839829411183018e-15), SC_(-0.2115775453968538439661164062909478793222e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1561186859754253464932505224282976996619e-14), SC_(-0.5317625036268033276607060773206517931366e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3092010764722992466335682593125966377556e-14), SC_(-0.1053182951942663933832876385506518241085e-12), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6192850577371690132255643845837767003104e-14), SC_(-0.2109373203492102058861790697529436159834e-12), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1047879028014987723427253740737796761096e-13), SC_(-0.3569225374615943825992302718383009611711e-12), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1978473638988408750405412206418986897916e-13), SC_(-0.673896330253118868079779235488630600479e-12), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4041816252346730475863978426787070930004e-13), SC_(-0.1376700243226680691549667180741589712713e-11), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9410302262901834580155480125540634617209e-13), SC_(-0.3205283121576780981640095118048982901968e-11), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1334530223958893535574077304772799834609e-12), SC_(-0.4545600218337299481258164553284154943185e-11), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.266297021326439287136622624529991298914e-12), SC_(-0.9070456229087110845090862250063546365371e-11), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.5920415525016708979677559909760020673275e-12), SC_(-0.2016577940297158071169597347971748413736e-10), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.155163989296047688526414276566356420517e-11), SC_(-0.5285106706035087367329172863247919698404e-10), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.326923297461201300961874949280172586441e-11), SC_(-0.1113547363379608255313626394695583529731e-9), + 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SC_(0.0002488605167115786508690708436538612267867), + SC_(9.68867778778076171875), SC_(0.000126518702018074691295623779296875), SC_(0.0002873599340757661525877158215704810459439), + SC_(9.68867778778076171875), SC_(0.00028976381872780621051788330078125), SC_(0.0006582580089946178862548800972604646079047), + SC_(9.68867778778076171875), SC_(0.000687857042066752910614013671875), SC_(0.001563315133764469862768389959281037577749), + SC_(9.68867778778076171875), SC_(0.00145484809763729572296142578125), SC_(0.003309362765364723055307278727456538237784), + SC_(9.68867778778076171875), SC_(0.002847635187208652496337890625), SC_(0.006487815096079959006202372484557638978379), + SC_(9.68867778778076171875), SC_(0.0056468211114406585693359375), SC_(0.01290626953766988799660445158177424847916), + SC_(9.68867778778076171875), SC_(0.011621631681919097900390625), SC_(0.02674359636941844621991721651837741451166), + SC_(9.68867778778076171875), SC_(0.0257236398756504058837890625), SC_(0.06015731222731434671073651753326632370858), + SC_(9.68867778778076171875), SC_(0.0560617186129093170166015625), SC_(0.1357733748610437010694097215927829841061), + SC_(9.68867778778076171875), SC_(0.106835305690765380859375), SC_(0.2745822634838372037418174975910269667509), + SC_(9.68867778778076171875), SC_(0.2401093542575836181640625), SC_(0.7250883224868308476529921117348408126984), + SC_(9.68867778778076171875), SC_(0.438671648502349853515625), SC_(1.707985158316040663526167707818201486792), + SC_(9.68867778778076171875), SC_(0.903765499591827392578125), SC_(6.786671267376742594913276377075578069747), + }; +#undef SC_ + + + using namespace boost::lambda; + using namespace std; + + typedef T (*func_t)(const T&); + func_t f = &boost::math::sqrtp1m1; + T eps = ldexp(T(1.0), 1-boost::math::tools::digits(T())); + + boost::math::tools::test_result result = boost::math::tools::test( + sqrtp1m1_data, + bind(f, ret(_1[0])), + ret(_1[1])); + + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n" + "Test results for type " << name << std::endl << std::endl; + std::cout << "sqrtp1m1 max error=" << ((result.stat.max)()/eps) << " at point " << sqrtp1m1_data[result.worst_case][0] << std::endl << std::endl; + + typedef T (*func2_t)(T, T); + func2_t f2 = &boost::math::powm1; + result = boost::math::tools::test( + powm1_data, + bind(f2, ret(_1[0]), ret(_1[1])), + ret(_1[2])); + std::cout << "powm1 max error=" << ((result.stat.max)()/eps) << " at point " << powm1_data[result.worst_case][0] << std::endl << std::endl; +} + +int main() +{ + test_powm1_sqrtp1m1(1.0F, "float"); + test_powm1_sqrtp1m1(1.0, "double"); + test_powm1_sqrtp1m1(1.0L, "long double"); + test_powm1_sqrtp1m1(boost::math::concepts::real_concept(), "real_concept"); + return 0; +} + + diff --git a/test/test_beta.cpp b/test/test_beta.cpp new file mode 100644 index 000000000..3e7e1bfda --- /dev/null +++ b/test/test_beta.cpp @@ -0,0 +1,123 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include "test_beta_hooks.hpp" + + +template +void print_test_result(const boost::math::tools::test_result& result, + T worst, const char* name, const char* test) +{ + using namespace std; + T eps = pow(T(2), 1-boost::math::tools::digits(worst)); + std::cout << setprecision(4); + std::cout << test << "(" << name << ") Max = " << (result.stat.max)()/eps + << " RMS Mean=" << result.stat.rms()/eps << " worst case at point: " << worst << std::endl; +} + +template +void do_test_beta(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); + pg funcp = boost::math::beta; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test beta against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case][0], type_name, "boost::math::beta"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::beta; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case][0], type_name, "other::beta"); + } +#endif + std::cout << std::endl; +} +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and beta(a, b): + // +# include "beta_small_data.ipp" + + do_test_beta(beta_small_data, name, "Beta Function: Small Values"); + +# include "beta_med_data.ipp" + + do_test_beta(beta_med_data, name, "Beta Function: Medium Values"); + +# include "beta_exp_data.ipp" + + do_test_beta(beta_exp_data, name, "Beta Function: Divergent Values"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 10 decimal places expressed as a persentage: + // + T tolerance = std::pow(10.0, -8); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(1), static_cast(1)), static_cast(1), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(1), static_cast(4)), static_cast(0.25), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), static_cast(1)), static_cast(0.25), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(1e-50), static_cast(4)), static_cast(1e50), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), static_cast(1e-50)), static_cast(1e50), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), static_cast(20)), static_cast(0.00002823263692828910220214568040654997176736), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(0.0125), static_cast(0.000023)), static_cast(43558.24045647538375006349016083320744662), tolerance); +} + +int test_main(int, char* []) +{ + //test_spots(0.0F); + test_spots(0.0); + test_spots(0.0L); + test_spots(boost::math::concepts::real_concept(0.1)); +#ifndef __HP_aCC + test_beta(0.1F, "float"); + test_beta(0.1, "double"); +#endif + test_beta(0.1L, "long double"); + test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); + return 0; +} + + diff --git a/test/test_beta_hooks.hpp b/test/test_beta_hooks.hpp new file mode 100644 index 000000000..ff2909eb3 --- /dev/null +++ b/test/test_beta_hooks.hpp @@ -0,0 +1,81 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_BETA_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_BETA_OTHER_HOOKS_HPP + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double beta(double, double); + float betaf(float, float); + long double betal(long double, long double); + + double incbet(double, double, double); + float incbetf(float, float, float); + long double incbetl(long double, long double, long double); +} +inline float beta(float a, float b) +{ return betaf(a, b); } +inline long double beta(long double a, long double b) +{ +#ifdef BOOST_MSVC + return beta((double)a, (double)b); +#else + return betal(a, b); +#endif +} +inline float ibeta(float a, float b, float x) +{ return incbetf(a, b, x); } +inline double ibeta(double a, double b, double x) +{ return incbet(a, b, x); } +inline long double ibeta(long double a, long double b, long double x) +{ +#ifdef BOOST_MSVC + return incbet((double)a, (double)b, (double)x); +#else + return incbetl(a, b); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#include +#include +#include + +namespace other{ +inline float beta(float a, float b) +{ return (float)gsl_sf_beta(a, b); } +inline double beta(double a, double b) +{ return gsl_sf_beta(a, b); } +inline long double beta(long double a, long double b) +{ return gsl_sf_beta(a, b); } + +inline float ibeta(float a, float b, float x) +{ return (float)gsl_sf_beta_inc(a, b, x); } +inline double ibeta(double a, double b, double x) +{ return gsl_sf_beta_inc(a, b, x); } +inline long double ibeta(long double a, long double b, long double x) +{ + return gsl_sf_beta_inc((double)a, (double)b, (double)x); +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept beta(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept ibeta(boost::math::concepts::real_concept, boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + diff --git a/test/test_cbrt.cpp b/test/test_cbrt.cpp new file mode 100644 index 000000000..709292401 --- /dev/null +++ b/test/test_cbrt.cpp @@ -0,0 +1,97 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +template +void print_test_result(const boost::math::tools::test_result& result, + T worst, const char* name, const char* test) +{ + using namespace std; + T eps = pow(T(2), 1-boost::math::tools::digits(worst)); + std::cout << setprecision(4); + std::cout << test << "(" << name << ") Max = " << (result.stat.max)()/eps + << " RMS Mean=" << result.stat.rms()/eps << " worst case at point: " << worst << std::endl; +} + +template +void do_test_cbrt(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::cbrt; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test erf against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[0])); + result += boost::math::tools::test( + data, + boost::lambda::bind(funcp, + -boost::lambda::ret(boost::lambda::_1[1])), + -boost::lambda::ret(boost::lambda::_1[0])); + print_test_result(result, data[result.worst_case][1], type_name, "boost::math::cbrt"); + std::cout << std::endl; +} +template +void test_cbrt(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "cbrt_data.ipp" + + do_test_cbrt(cbrt_data, name, "cbrt Function"); + +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 10 decimal places expressed as a persentage: + // + T tolerance = std::pow(10.0, -8); + //BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0.125)), static_cast(0.859683795198666182606970553478), tolerance); +} + +int test_main(int, char* []) +{ + //test_spots(0.0F); + test_spots(0.0); + test_spots(0.0L); + test_spots(boost::math::concepts::real_concept(0.1)); + + test_cbrt(0.1F, "float"); + test_cbrt(0.1, "double"); + test_cbrt(0.1L, "long double"); + test_cbrt(boost::math::concepts::real_concept(0.1), "real_concept"); + return 0; +} + + diff --git a/test/test_classify.cpp b/test/test_classify.cpp new file mode 100644 index 000000000..dfc96eabc --- /dev/null +++ b/test/test_classify.cpp @@ -0,0 +1,136 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +template +void test_classify(T t, const char* type) +{ + std::cout << "Testing type " << type << std::endl; + t = 2; + T u = 2; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + t = (std::numeric_limits::max)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + t = (std::numeric_limits::min)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + if(std::numeric_limits::has_denorm) + { + t /= 2; + if(t != 0) + { + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_SUBNORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_SUBNORMAL); + } + t = std::numeric_limits::denorm_min(); + if(t != 0) + { + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_SUBNORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_SUBNORMAL); + } + } + else + { + std::cout << "Denormalised forms not tested" << std::endl; + } + t = 0; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_ZERO); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_ZERO); + t /= -u; // create minus zero if it exists + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_ZERO); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_ZERO); + // inifinity: + if(std::numeric_limits::has_infinity) + { + // At least one std::numeric_limits::infinity)() returns zero + // (Compaq true64 cxx), hence the check. + t = (std::numeric_limits::infinity)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); + } + else + { + std::cout << "Infinity not tested" << std::endl; + } +#if !defined(__BORLANDC__) && !(defined(__DECCXX) && !defined(_IEEE_FP)) + // divide by zero on Borland triggers a C++ exception :-( + // divide by zero on Compaq CXX triggers a C style signal :-( + t = 2; + u = 0; + t /= u; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); + t = -2; + t /= u; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); +#else + std::cout << "Infinities from divide by zero not tested" << std::endl; +#endif +#ifndef __BORLANDC__ + // NaN's: + // Note that Borland throws an exception if we even try to obtain a Nan + // by calling std::numeric_limits::quiet_NaN() !!!!!!! + if(std::numeric_limits::has_quiet_NaN) + { + t = std::numeric_limits::quiet_NaN(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NAN); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NAN); + } + else + { + std::cout << "Quite NaN's not tested" << std::endl; + } + if(std::numeric_limits::has_signaling_NaN) + { + t = std::numeric_limits::signaling_NaN(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NAN); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NAN); + } + else + { + std::cout << "Signaling NaN's not tested" << std::endl; + } +#endif +} + +int test_main(int, char* [] ) +{ + // start by printing some information: +#ifdef isnan + std::cout << "Platform has isnan macro." << std::endl; +#endif +#ifdef fpclassify + std::cout << "Platform has fpclassify macro." << std::endl; +#endif +#ifdef BOOST_HAS_FPCLASSIFY + std::cout << "Platform has FP_NORMAL macro." << std::endl; +#endif + std::cout << "FP_ZERO: " << (int)FP_ZERO << std::endl; + std::cout << "FP_NORMAL: " << (int)FP_NORMAL << std::endl; + std::cout << "FP_INFINITE: " << (int)FP_INFINITE << std::endl; + std::cout << "FP_NAN: " << (int)FP_NAN << std::endl; + std::cout << "FP_SUBNORMAL: " << (int)FP_SUBNORMAL << std::endl; + + // then run the tests: + test_classify(float(0), "float"); + test_classify(double(0), "double"); + test_classify((long double)(0), "long double"); + return 0; +} + + + + + + + diff --git a/test/test_erf.cpp b/test/test_erf.cpp new file mode 100644 index 000000000..fe5de1977 --- /dev/null +++ b/test/test_erf.cpp @@ -0,0 +1,230 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include "test_erf_hooks.hpp" + + +template +void print_test_result(const boost::math::tools::test_result& result, + T worst, const char* name, const char* test) +{ + using namespace std; + T eps = pow(T(2), 1-boost::math::tools::digits(worst)); + std::cout << setprecision(4); + std::cout << test << "(" << name << ") Max = " << (result.stat.max)()/eps + << " RMS Mean=" << result.stat.rms()/eps << " worst case at point: " << worst << std::endl; +} + +template +void do_test_erf(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::erf; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test erf against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case][0], type_name, "boost::math::erf"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::erf; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case][0], type_name, "other::erf"); + } +#endif + // + // test erfc against data: + // + funcp = boost::math::erfc; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case][0], type_name, "boost::math::erfc"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::erfc; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case][0], type_name, "other::erfc"); + } +#endif + std::cout << std::endl; +} + +template +void do_test_erf_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::erf; + + boost::math::tools::test_result result; + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + // + // test erf_inv against data: + // + funcp = boost::math::erf_inv; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case][0], type_name, "boost::math::erf"); + std::cout << std::endl; +} + +template +void do_test_erfc_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::erf; + + boost::math::tools::test_result result; + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + // + // test erfc_inv against data: + // + funcp = boost::math::erfc_inv; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case][0], type_name, "boost::math::erfc_inv"); + std::cout << std::endl; +} + +template +void test_erf(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "erf_small_data.ipp" + + do_test_erf(erf_small_data, name, "Erf Function: Small Values"); + +# include "erf_data.ipp" + + do_test_erf(erf_data, name, "Erf Function: Medium Values"); + +# include "erf_large_data.ipp" + + do_test_erf(erf_large_data, name, "Erf Function: Large Values"); + +# include "erf_inv_data.ipp" + + do_test_erf_inv(erf_inv_data, name, "Inverse Erf Function"); + +# include "erfc_inv_data.ipp" + + do_test_erfc_inv(erfc_inv_data, name, "Inverse Erfc Function"); + +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 10 decimal places expressed as a persentage: + // + T tolerance = std::pow(10.0, -8); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0.125)), static_cast(0.859683795198666182606970553478), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0.5)), static_cast(0.479500122186953462317253346108), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(1)), static_cast(0.157299207050285130658779364917), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(5)), static_cast(1.53745979442803485018834348538e-12), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(-0.125)), static_cast(1.14031620480133381739302944652), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(-0.5)), static_cast(1.52049987781304653768274665389), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0)), static_cast(1), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0.125)), static_cast(0.140316204801333817393029446522), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0.5)), static_cast(0.520499877813046537682746653892), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(1)), static_cast(0.842700792949714869341220635083), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(5)), static_cast(0.99999999999846254020557196515), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(-0.125)), static_cast(-0.140316204801333817393029446522), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(-0.5)), static_cast(-0.520499877813046537682746653892), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0)), static_cast(0), tolerance); + + tolerance = boost::math::tools::epsilon(tolerance) * 100 * 200; // 200 eps %. +#if defined(__CYGWIN__) + // some platforms long double is only reliably accurate to double precision: + if(sizeof(T) == sizeof(long double)) + tolerance = boost::math::tools::epsilon(double(0)) * 100 * 200; // 200 eps %. +#endif + + for(T i = -0.95; i < 1; i += 0.125) + { + T inv = boost::math::erf_inv(i); + T b = boost::math::erf(inv); + BOOST_CHECK_CLOSE(b, i, tolerance); + } + for(T j = 0.125; j < 2; j += 0.125) + { + T inv = boost::math::erfc_inv(j); + T b = boost::math::erfc(inv); + BOOST_CHECK_CLOSE(b, j, tolerance); + } +} + +int test_main(int, char* []) +{ + //test_spots(0.0F); + test_spots(0.0, "double"); + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); + + test_erf(0.1F, "float"); + test_erf(0.1, "double"); + test_erf(0.1L, "long double"); + test_erf(boost::math::concepts::real_concept(0.1), "real_concept"); + return 0; +} + + diff --git a/test/test_erf_hooks.hpp b/test/test_erf_hooks.hpp new file mode 100644 index 000000000..e1c42af50 --- /dev/null +++ b/test/test_erf_hooks.hpp @@ -0,0 +1,105 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP + +#ifdef TEST_NATIVE +namespace other{ +inline float erf(float a) +{ + return ::erff(a); +} +inline float erfc(float a) +{ + return ::erfcf(a); +} +inline double erf(double a) +{ + return ::erf(a); +} +inline double erfc(double a) +{ + return ::erfc(a); +} +inline long double erf(long double a) +{ + return ::erfl(a); +} +inline long double erfc(long double a) +{ + return ::erfcl(a); +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double erf(double); + float erff(float); + long double erfl(long double); +} +inline float erf(float a) +{ return erff(a); } +inline long double erf(long double a) +{ +#ifdef BOOST_MSVC + return erf((double)a); +#else + return erfl(a); +#endif +} +extern "C" { + double erfc(double); + float erfcf(float); + long double erfcl(long double); +} +inline float erfc(float a) +{ return erfcf(a); } +inline long double erfc(long double a) +{ +#ifdef BOOST_MSVC + return erfc((double)a); +#else + return erfcl(a); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#include + +namespace other{ +inline float erf(float a) +{ return (float)gsl_sf_erf(a); } +inline double erf(double a) +{ return gsl_sf_erf(a); } +inline long double erf(long double a) +{ return gsl_sf_erf(a); } +inline float erfc(float a) +{ return (float)gsl_sf_erfc(a); } +inline double erfc(double a) +{ return gsl_sf_erfc(a); } +inline long double erfc(long double a) +{ return gsl_sf_erfc(a); } +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept erf(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept erfc(boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + diff --git a/test/test_gamma.cpp b/test/test_gamma.cpp new file mode 100644 index 000000000..add442156 --- /dev/null +++ b/test/test_gamma.cpp @@ -0,0 +1,464 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include "test_gamma_hooks.hpp" + + +template +void print_test_result(const boost::math::tools::test_result& result, + const Seq& worst, int row, const char* name, const char* test) +{ + using namespace std; + T eps = pow(T(2), 1-boost::math::tools::digits(worst[0])); + std::cout << setprecision(4); + std::cout << test << "(" << name << ") Max = " << (result.stat.max)()/eps + << " RMS Mean=" << result.stat.rms()/eps + << "\n worst case at row: " + << row << "\n { "; + for(unsigned i = 0; i < worst.size(); ++i) + { + if(i) + std::cout << ", "; + std::cout << worst[i]; + } + std::cout << " }" << std::endl; +} + +template +void do_test_gamma(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::tgamma; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::tgamma"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::tgamma; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "other::tgamma"); + } +#endif + // + // test lgamma against data: + // + funcp = boost::math::lgamma; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::lgamma"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::lgamma; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "other::lgamma"); + } +#endif + + std::cout << std::endl; +} + +template +void do_test_gammap1m1(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + pg funcp = boost::math::tgammap1m1; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgammap1m1 against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, boost::lambda::ret(boost::lambda::_1[0])), + boost::lambda::ret(boost::lambda::_1[1])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::tgammap1m1"); + std::cout << std::endl; +} + +template +void do_test_gamma_2(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); + pg funcp = boost::math::tgamma; + + using namespace boost::lambda; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma(T, T) against data: + // + if(data[0][2] > 0) + { + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[2])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::tgamma"); + // + // test tgamma_lower(T, T) against data: + // + funcp = boost::math::tgamma_lower; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[4])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::tgamma_lower"); + } + // + // test gamma_Q(T, T) against data: + // + funcp = boost::math::gamma_Q; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[3])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::gamma_Q"); +#if defined(TEST_CEPHES) || defined(TEST_GSL) + // + // test other gamma_Q(T, T) against data: + // + if(boost::is_floating_point::value) + { + funcp = other::gamma_Q; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[3])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "other::gamma_Q"); + } +#endif + // + // test gamma_P(T, T) against data: + // + funcp = boost::math::gamma_P; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[5])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::gamma_P"); +#if defined(TEST_CEPHES) || defined(TEST_GSL) + // + // test other gamma_P(T, T) against data: + // + if(boost::is_floating_point::value) + { + funcp = other::gamma_P; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[5])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "other::gamma_P"); + } +#endif + // + // test gamma_P_inv(T, T) against data: + // + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::tools::digits(value_type(0))/2)) * 100; + if(boost::math::tools::digits(value_type(0)) < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete gamma is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_P_inv(data[i][0], data[i][5]), value_type(0)); + else if((1 - data[i][5] > 0.001) && (fabs(data[i][5]) >= boost::math::tools::min_value(data[i][5]))) + { + value_type inv = boost::math::gamma_P_inv(data[i][0], data[i][5]); + BOOST_CHECK_CLOSE(data[i][1], inv, precision); + } + else if(1 == data[i][5]) + BOOST_CHECK_EQUAL(boost::math::gamma_P_inv(data[i][0], data[i][5]), boost::math::tools::max_value(data[i][5])); + else + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_P_inv(data[i][0], data[i][5]); + BOOST_CHECK_CLOSE(data[i][1], inv, 100000); + } + + if(data[i][3] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_Q_inv(data[i][0], data[i][3]), boost::math::tools::max_value(data[i][3])); + else if((1 - data[i][3] > 0.001) && (fabs(data[i][3]) >= boost::math::tools::min_value(data[i][3]))) + { + value_type inv = boost::math::gamma_Q_inv(data[i][0], data[i][3]); + BOOST_CHECK_CLOSE(data[i][1], inv, precision); + } + else if(1 == data[i][3]) + BOOST_CHECK_EQUAL(boost::math::gamma_Q_inv(data[i][0], data[i][3]), value_type(0)); + else + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_Q_inv(data[i][0], data[i][3]); + BOOST_CHECK_CLOSE(data[i][1], inv, 100); + } + } + std::cout << std::endl; +} + +template +void do_test_gamma_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); + pg funcp = boost::math::gamma_P_inv; + + using namespace boost::lambda; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test gamma_P_inv(T, T) against data: + // + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[2])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::gamma_P_inv"); + // + // test gamma_Q_inv(T, T) against data: + // + funcp = boost::math::gamma_Q_inv; + result = boost::math::tools::test( + data, + bind(funcp, ret(_1[0]), ret(_1[1])), + ret(_1[3])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::gamma_Q_inv"); +} + +template +void test_gamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value, gamma and lgamma: + // + // gamma and lgamma at integer and half integer values: + // boost::array, N> factorials; + // + // gamma and lgamma for z near 0: + // boost::array, N> near_0; + // + // gamma and lgamma for z near 1: + // boost::array, N> near_1; + // + // gamma and lgamma for z near 2: + // boost::array, N> near_2; + // + // gamma and lgamma for z near -10: + // boost::array, N> near_m10; + // + // gamma and lgamma for z near -55: + // boost::array, N> near_m55; + // + // The last two cases are chosen more or less at random, + // except that one is even and the other odd, and both are + // at negative poles. The data near zero also tests near + // a pole, the data near 1 and 2 are to probe lgamma as + // the result -> 0. + // +# include "test_gamma_data.ipp" + + do_test_gamma(factorials, name, "factorials"); + do_test_gamma(near_0, name, "near 0"); + do_test_gamma(near_1, name, "near 1"); + do_test_gamma(near_2, name, "near 2"); + do_test_gamma(near_m10, name, "near -10"); + do_test_gamma(near_m55, name, "near -55"); + + // + // And now tgammap1m1 which computes gamma(1+dz)-1: + // + do_test_gammap1m1(gammap1m1_data, name, "tgammap1m1(dz)"); + + // + // Now the data for the incomplete gamma function, each + // row has the following entries: + // Parameter a, parameter z, + // Expected tgamma(a, z), Expected gamma_Q(a, z) + // Expected tgamma_lower(a, z), Expected gamma_P(a, z) + // +# include "igamma_med_data.ipp" + + do_test_gamma_2(igamma_med_data, name, "tgamma(a, z) medium values"); + +# include "igamma_small_data.ipp" + + do_test_gamma_2(igamma_small_data, name, "tgamma(a, z) small values"); + +# include "igamma_big_data.ipp" + + do_test_gamma_2(igamma_big_data, name, "tgamma(a, z) large values"); + +# include "gamma_inv_data.ipp" + + do_test_gamma_inv(gamma_inv_data, name, "incomplete gamma inverse(a, z) medium values"); + +# include "gamma_inv_big_data.ipp" + + do_test_gamma_inv(gamma_inv_big_data, name, "incomplete gamma inverse(a, z) large values"); + +# include "gamma_inv_small_data.ipp" + + do_test_gamma_inv(gamma_inv_small_data, name, "incomplete gamma inverse(a, z) small values"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 10 decimal places expressed as a persentage: + // + T tolerance = std::pow(10.0, -8); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(3.5)), static_cast(3.3233509704478425512), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.1)), static_cast(9.5135076986687318363), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-0.1)), static_cast(-10.686287021193193549), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-3.2)), static_cast(0.68905641200597974292), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-52.001)), static_cast(-1.2349010280547181826e-65), tolerance); + + int sign = 1; + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(3.5), &sign), static_cast(1.2009736023470742248208170321516), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(0.1), &sign), static_cast(2.2527126517342059598704913469682), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-0.1), &sign), static_cast(2.3689613327287886552092298040686), tolerance); + BOOST_CHECK(sign == -1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-3.2), &sign), static_cast(-0.37243213612996769536089336752751), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-52.001), &sign), static_cast(-149.4570402169712972001218882919), tolerance); + BOOST_CHECK(sign == -1); + + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(1)), static_cast(23.912163676143750904), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(5)), static_cast(10.571838841565097875), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(10)), static_cast(0.70206451384706574415), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(100)), static_cast(3.8734332808745531497e-36), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(0.5)), static_cast(0.562418231594407124279494957302), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(0.9)), static_cast(0.318532103604121098738593603904), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(5)), static_cast(0.00277460326041280931949083572726), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(100)), static_cast(3.70174786040827892025356644813e-45), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(1)), static_cast(0.087836323856249096291), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(5)), static_cast(13.428161158434902125), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(10)), static_cast(23.297935486152934256), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(100)), static_cast(24.000000000000000000), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::gamma_Q(static_cast(5), static_cast(1)), static_cast(0.996340153172656287654543544187), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_Q(static_cast(5), static_cast(5)), static_cast(0.440493285065212411442581665663), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_Q(static_cast(5), static_cast(10)), static_cast(0.0292526880769610726727661331928), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_Q(static_cast(5), static_cast(100)), static_cast(1.61393053369773047904057392250e-37), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::gamma_P(static_cast(5), static_cast(1)), static_cast(0.00365984682734371234545645581271), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_P(static_cast(5), static_cast(5)), static_cast(0.559506714934787588557418334337), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_P(static_cast(5), static_cast(10)), static_cast(0.970747311923038927327233866807), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_P(static_cast(5), static_cast(100)), static_cast(1.00000000000000000000000000000), tolerance); + + //BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(1), static_cast(1.5)), static_cast(0.35798614547512708694), tolerance); + //BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(3), static_cast(6)), static_cast(12.724961860971380343), tolerance); + //BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(7), static_cast(9)), static_cast(2.8326711932871945905), tolerance); + + tolerance = boost::math::tools::epsilon(tolerance) * 100 * 200; // 200 eps %. +#if defined(__CYGWIN__) + // some platforms long double is only reliably accurate to double precision: + if(sizeof(T) == sizeof(long double)) + tolerance = boost::math::tools::epsilon(double(0)) * 100 * 200; // 200 eps %. +#endif +#if 0 + // this is way too slow... + for(T a = 0.5; a < 500; a += 2.5) + { + for(T p = 0.05; p < 1; p += 0.04) + { + T x = boost::math::gamma_P_inv(a, p); + T p2 = boost::math::gamma_P(a, x); + BOOST_CHECK_CLOSE(p2, p, tolerance); + + x = boost::math::gamma_Q_inv(a, p); + p2 = boost::math::gamma_Q(a, x); + BOOST_CHECK_CLOSE(p2, p, tolerance); + } + } +#endif +} + +int test_main(int, char* []) +{ + //test_spots(0.0F); + test_spots(0.0); + test_spots(0.0L); + test_spots(boost::math::concepts::real_concept(0.1)); + + test_gamma(0.1F, "float"); + test_gamma(0.1, "double"); + test_gamma(0.1L, "long double"); + test_gamma(boost::math::concepts::real_concept(0.1), "real_concept"); + return 0; +} + + diff --git a/test/test_gamma_data.ipp b/test/test_gamma_data.ipp new file mode 100644 index 000000000..911827912 --- /dev/null +++ b/test/test_gamma_data.ipp @@ -0,0 +1,582 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 198> factorials = { + SC_(1), SC_(1), SC_(0), + SC_(2), SC_(1), SC_(0), + SC_(3), SC_(2), SC_(0.6931471805599453094172321214581765680755), + SC_(4), SC_(6), SC_(1.791759469228055000812477358380702272723), + SC_(5), SC_(24), SC_(3.178053830347945619646941601297055408874), + SC_(6), SC_(120), SC_(4.7874917427820459942477009345232430484), + SC_(7), SC_(720), SC_(6.579251212010100995060178292903945321123), + SC_(8), SC_(5040), SC_(8.52516136106541430016553103634712505076), + SC_(9), SC_(40320), SC_(10.60460290274525022841722740072165475499), + SC_(10), SC_(362880), SC_(12.80182748008146961120771787456670616428), + SC_(11), SC_(3628800), SC_(15.10441257307551529522570932925107037188), + SC_(12), SC_(39916800), SC_(17.5023078458738858392876529072161996717), + SC_(13), SC_(479001600), SC_(19.9872144956618861495173623870550785125), + SC_(14), SC_(6227020800.0), SC_(22.55216385312342288557084982862039711731), + SC_(15), SC_(87178291200.0), SC_(25.19122118273868150009343469352175341502), + SC_(16), SC_(1307674368000.0), SC_(27.89927138384089156608943926367046675919), + SC_(17), SC_(20922789888000.0), SC_(30.6718601060806728037583677495031730315), + SC_(18), SC_(355687428096000.0), SC_(33.50507345013688888400790236737629956708), + SC_(19), SC_(6402373705728000.0), SC_(36.39544520803305357621562496267952754445), + SC_(20), SC_(121645100408832000.0), SC_(39.33988418719949403622465239456738108169), + SC_(21), SC_(0.243290200817664e19), SC_(42.33561646075348502965987597070992185737), + SC_(22), SC_(0.5109094217170944e20), SC_(45.38013889847690802616047395107562729165), + SC_(23), SC_(0.112400072777760768e22), SC_(48.47118135183522387963964965049893315955), + SC_(24), SC_(0.2585201673888497664e23), SC_(51.60667556776437357044640248230912927799), + SC_(25), SC_(0.62044840173323943936e24), SC_(54.78472939811231919009334408360618468687), + SC_(26), SC_(0.15511210043330985984e26), SC_(58.00360522298051993929486275005855996592), + SC_(27), SC_(0.403291461126605635584e27), SC_(61.2617017610020019847655823130820551388), + SC_(28), SC_(0.10888869450418352160768e29), SC_(64.55753862700633105895131802384963225274), + SC_(29), SC_(0.304888344611713860501504e30), SC_(67.88974313718153498289113501020916511853), + SC_(30), SC_(0.8841761993739701954543616e31), SC_(71.25703896716800901007440704257107672402), + SC_(31), SC_(0.26525285981219105863630848e33), SC_(74.65823634883016438548764373417796663627), + SC_(32), SC_(0.822283865417792281772556288e34), SC_(78.09222355331531063141680805872032384672), + SC_(33), SC_(0.26313083693369353016721801216e36), SC_(81.5579594561150371785029686660112066871), + SC_(34), SC_(0.868331761881188649551819440128e37), SC_(85.05446701758151741396015748089886169157), + SC_(35), SC_(0.29523279903960414084761860964352e39), SC_(88.58082754219767880362692422023016479523), + SC_(36), SC_(0.103331479663861449296666513375232e41), SC_(92.13617560368709248333303629689953216439), + SC_(37), SC_(0.3719933267899012174679994481508352e42), SC_(95.71969454214320248495799101366093670984), + SC_(38), SC_(0.137637530912263450463159795815809024e44), SC_(99.33061245478742692932608668469238387374), + SC_(39), SC_(0.5230226174666011117600072241000742912e45), SC_(102.9681986145138126987523462380384139791), + SC_(40), SC_(0.203978820811974433586402817399028973568e47), SC_(106.6317602606434591262010789165262582885), + SC_(41), SC_(0.815915283247897734345611269596115894272e48), SC_(110.3206397147573954290535346141269756323), + SC_(42), SC_(0.3345252661316380710817006205344075166515e50), SC_(114.0342117814617032329202979871643832206), + SC_(43), SC_(0.1405006117752879898543142606244511569936e52), SC_(117.771881399745071538838128088988265223), + SC_(44), SC_(0.6041526306337383563735513206851399750726e53), SC_(121.5330815154386339623109706023341122586), + SC_(45), SC_(0.265827157478844876804362581101461589032e55), SC_(125.3172711493568951252073784232155946945), + SC_(46), SC_(0.1196222208654801945619631614956577150644e57), SC_(129.1239336391272148825986282302868337433), + SC_(47), SC_(0.5502622159812088949850305428800254892962e58), SC_(132.9525750356163098828226131835552064299), + SC_(48), SC_(0.2586232415111681806429643551536119799692e60), SC_(136.8027226373263684696435638533273801388), + SC_(49), SC_(0.1241391559253607267086228904737337503852e62), SC_(140.6739236482342593987077375760826121157), + SC_(50), SC_(0.6082818640342675608722521633212953768876e63), SC_(144.565743946344886008918443062968971575), + SC_(51), SC_(0.3041409320171337804361260816606476884438e65), SC_(148.4777669517730320675371938508795234221), + SC_(52), SC_(0.1551118753287382280224243016469303211063e67), SC_(152.4095925844973578391819737056751756623), + SC_(53), SC_(0.8065817517094387857166063685640376697529e68), SC_(156.3608363030787851940699253901568474033), + SC_(54), SC_(0.427488328406002556429801375338939964969e70), SC_(160.3311282166309070282143945291859051737), + SC_(55), SC_(0.2308436973392413804720927426830275810833e72), SC_(164.3201122631951814118173623614116588557), + SC_(56), SC_(0.1269640335365827592596510084756651695958e74), SC_(168.327445448427652330480065272602975795), + SC_(57), SC_(0.7109985878048634518540456474637249497365e75), SC_(172.3527971391628015638371143804206852289), + SC_(58), SC_(0.4052691950487721675568060190543232213498e77), SC_(176.3958484069973517152413870492310644708), + SC_(59), SC_(0.2350561331282878571829474910515074683829e79), SC_(180.4562914175437710518418912030511526443), + SC_(60), SC_(0.1386831185456898357379390197203894063459e81), SC_(184.5338288614494905024579415767708502684), + SC_(61), SC_(0.8320987112741390144276341183223364380754e82), SC_(188.6281734236715911872884103898359167487), + SC_(62), SC_(0.507580213877224798800856812176625227226e84), 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SC_(230.4390435657769523213935127204501618205), + SC_(72), SC_(0.8504785885678623175211676442399260102886e102), SC_(234.7017234428182677427229672529631959172), + SC_(73), SC_(0.6123445837688608686152407038527467274078e104), SC_(238.9783895618343230537651540911827770308), + SC_(74), SC_(0.4470115461512684340891257138125051110077e106), SC_(243.2688490029827141828572629486213196017), + SC_(75), SC_(0.3307885441519386412259530282212537821457e108), SC_(247.5729140961868839366425907411109433336), + SC_(76), SC_(0.2480914081139539809194647711659403366093e110), SC_(251.8904022097231943772393546444858443173), + SC_(77), SC_(0.188549470166605025498793226086114655823e112), SC_(256.2211355500095254560828463192900509907), + SC_(78), SC_(0.1451830920282858696340707840863082849837e114), SC_(260.5649409718632093052501426406983600202), + SC_(79), SC_(0.1132428117820629783145752115873204622873e116), SC_(264.9216497985528010421161074406443808977), + SC_(80), SC_(0.8946182130782975286851441715398316520698e117), SC_(269.2910976510198225362890529821257918199), + SC_(81), SC_(0.7156945704626380229481153372318653216558e119), SC_(273.6731242856937041485587408011846857317), + SC_(82), SC_(0.5797126020747367985879734231578109105412e121), SC_(278.0675734403661429141397217488747885503), + SC_(83), SC_(0.4753643337012841748421382069894049466438e123), SC_(282.4742926876303960274237172433703727067), + SC_(84), SC_(0.3945523969720658651189747118012061057144e125), SC_(286.8931332954269939508991894666617431598), + SC_(85), SC_(0.3314240134565353266999387579130131288001e127), SC_(291.3239500942703075662342516899438017302), + SC_(86), SC_(0.2817104114380550276949479442260611594801e129), SC_(295.7666013507606240210845456410431159053), + SC_(87), SC_(0.2422709538367273238176552320344125971528e131), SC_(300.220948647014131753974620275847139509), + SC_(88), SC_(0.210775729837952771721360051869938959523e133), SC_(304.6868567656687154725531375451315768191), + SC_(89), SC_(0.1854826422573984391147968456455462843802e135), SC_(309.1641935801469219448667774874712358232), + SC_(90), SC_(0.1650795516090846108121691926245361930984e137), SC_(313.6528299498790617831845930281410850426), + SC_(91), SC_(0.1485715964481761497309522733620825737886e139), SC_(318.1526396202093268499930749566705006595), + SC_(92), SC_(0.1352001527678402962551665687594951421476e141), SC_(322.6634991267261768911519151416789989939), + SC_(93), SC_(0.1243841405464130725547532432587355307758e143), SC_(327.1852877037752172007931322164055482485), + SC_(94), SC_(0.1156772507081641574759205162306240436215e145), SC_(331.7178871969284731381175417778704311636), + SC_(95), SC_(0.1087366156656743080273652852567866010042e147), SC_(336.2611819791984770343557245691007814406), + SC_(96), SC_(0.103299784882390592625997020993947270954e149), SC_(340.8150588707990178689655113342148226173), + SC_(97), SC_(0.9916779348709496892095714015418938011582e150), SC_(345.3794070622668541074469171784282311623), + SC_(98), SC_(0.9619275968248211985332842594956369871234e152), SC_(349.9541180407702369295636388001321928762), + SC_(99), SC_(0.942689044888324774562618574305724247381e154), SC_(354.5390855194408088491915764084767289035), + SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519), + SC_(2.5), SC_(1.329340388179137020473625612505858887098), SC_(0.2846828704729191596324946696827019243201), + SC_(3.5), SC_(3.323350970447842551184064031264647217745), SC_(1.20097360234707422481602188145071299577), + SC_(4.5), SC_(11.63172839656744892914422410942626526211), SC_(2.453736570842442220504142503435716157332), + SC_(5.5), SC_(52.34277778455352018114900849241819367949), SC_(3.957813967618716293877400855822590998551), + SC_(6.5), SC_(287.8852778150443609963195467083000652372), SC_(5.662562059857141528522112312329543730298), + SC_(7.5), SC_(1871.254305797788346476077053603950424042), 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SC_(0.9186498057706952351408903494872271390691e52), SC_(119.6495745463449012688534009037863717517), + SC_(44.5), SC_(0.3996126655102524272862873020269438054951e54), SC_(123.4223354844395396780146860516126324938), + SC_(45.5), SC_(0.1778276361520623301423978494019899934453e56), SC_(127.2178246736117342069152694708243051451), + SC_(46.5), SC_(0.8091157444918836021479102147790544701761e57), SC_(131.0355369995686389386568775343746269115), + SC_(47.5), SC_(0.3762388211887258749987782498722603286319e59), SC_(134.8749893121619495665640549743813332585), + SC_(48.5), SC_(0.1787134400646447906244196686893236561001e61), SC_(138.7357190232025450917566096180371978672), + SC_(49.5), SC_(0.8667601843135272345284353931432197320857e62), SC_(142.6172828211459826044560991182829830129), + SC_(50.5), SC_(0.4290462912351959810915755196058937673824e64), SC_(146.519255490720627221891301048634987154), + SC_(51.5), SC_(0.2166683770737739704512456374009763525281e66), SC_(150.4412288270019413633582671940897997428), + SC_(52.5), SC_(0.111584214192993594782391503261502821552e68), SC_(154.3828106346716318247096373876850639954), + SC_(53.5), SC_(0.5858171245132163726075553921228898131479e69), SC_(158.3436238042692098863937625798187805011), + SC_(54.5), SC_(0.3134121616145707593450421347857460500341e71), SC_(162.3233054581711707502809292753838809346), + SC_(55.5), SC_(0.1708096280799410638430479634582315972686e73), SC_(166.3215061598403691412410136061349060176), + SC_(56.5), SC_(0.9479934358436729043289161971931853648408e74), SC_(170.337889180592757967587122392630702318), + SC_(57.5), SC_(0.535616291251675190945837651414149731135e76), SC_(174.372129818745153226752021764788547422), + SC_(58.5), SC_(0.3079793674697132347938566495631360954026e78), SC_(178.4239147665484579827423018083667546119), + SC_(59.5), SC_(0.1801679299697822423544061399944346158105e80), SC_(182.4929415207862687921690476023189480579), + SC_(60.5), SC_(0.1071999183320204342008716532966885964073e82), SC_(186.5789178333378528681067028421770777551), + SC_(61.5), SC_(0.648559505908723626915273502444966008264e83), SC_(190.6815611983746486468133578766491597866), + SC_(62.5), SC_(0.3988640961338650305528932040036540950824e85), SC_(194.8005983731871208326581343651509165116), + SC_(63.5), SC_(0.2492900600836656440955582525022838094265e87), SC_(198.9357649299294766470431802433713028621), + SC_(64.5), SC_(0.1582991881531276840006794903389502189858e89), SC_(203.0868048358281226106733889296294186889), + SC_(65.5), SC_(0.1021029763587673561804382712686228912459e91), SC_(207.253470059629849416124244558439614861), + SC_(66.5), SC_(0.6687744951499261829818706768094799376603e92), SC_(211.435520202271055650856436448150964186), + SC_(67.5), SC_(0.4447350392747009116829440000783041585441e94), SC_(215.6327221499328641065535845020238208848), + SC_(68.5), SC_(0.3001961515104231153859872000528553070173e96), SC_(219.8448497478113482459228474245594090702), + SC_(69.5), SC_(0.2056343637846398340394012320362058853068e98), SC_(224.0716834930795278518208054310810978927), + SC_(70.5), SC_(0.1429158828303246846573838562651630902882e100), SC_(228.3130102456502742995923582284984651071), + SC_(71.5), SC_(0.1007556973953789026834556186669399786532e102), SC_(232.5686229554684972683913220137349879525), + SC_(72.5), SC_(0.7204032363769591541867076734686208473705e103), SC_(236.8383204051684592390895209118072592891), + SC_(73.5), SC_(0.5222923463732953867853630632647501143436e105), SC_(241.121906967029088331456320155937181966), + SC_(74.5), SC_(0.3838848745843721092872418514995913340426e107), SC_(245.4191923732478793236450387582878905619), + SC_(75.5), SC_(0.2859942315653572214189951793671955438617e109), SC_(249.7299914986333931552202349119338344817), + SC_(76.5), SC_(0.2159256448318447021713413604222326356156e111), SC_(254.0541241548883721745992390899601342039), + SC_(77.5), SC_(0.1651831182963611971610761407230079662459e113), SC_(258.3914148957208623282220320602201355807), + SC_(78.5), SC_(0.1280169166796799277998340090603311738406e115), SC_(262.7416928320801636393347235965305038626), + SC_(79.5), SC_(0.1004932795935487433228696971123599714649e117), SC_(267.1047914568685263873419367114758025432), + SC_(80.5), SC_(0.7989215727687125094168140920432617731457e118), SC_(271.4805484785288126034644189659692094502), + SC_(81.5), SC_(0.6431318660788135700805353440948257273823e120), SC_(275.8688056629533302899592924197644087302), + SC_(82.5), SC_(0.5241524708542330596156363054372829678165e122), SC_(280.2694086832001473146069926644269820387), + SC_(83.5), SC_(0.4324257884547422741828999519857584484486e124), SC_(284.6822069765407826152477086910826481146), + SC_(84.5), SC_(0.3610755333597097989427214599081083044546e126), SC_(289.1070536083975924130902273463692509124), + SC_(85.5), SC_(0.3051088256889547801065996336223515172642e128), SC_(293.5438051427607205757799701080417115539), + SC_(86.5), SC_(0.2608680459640563369911426867471105472609e130), SC_(297.9923215187034351091622558923164399324), + SC_(87.5), SC_(0.2256508597589087314973384240362506233806e132), SC_(302.4524659326412687466785241592992548335), + SC_(88.5), SC_(0.1974445022890451400601711210317192954581e134), SC_(306.9241047260048374915681634477366332741), + SC_(89.5), SC_(0.1747383845258049489532514421130715764804e136), SC_(311.4071072780187213241622269369206800347), + SC_(90.5), SC_(0.1563908541505954293131600406911990609499e138), SC_(315.9013459032995310109227992454839056422), + SC_(91.5), SC_(0.1415337230062888635284098368255351501597e140), SC_(320.4066957540054114483446541626587380014), + SC_(92.5), SC_(0.1295033565507543101284950006953646623961e142), SC_(324.9230347262868870790740563815487018843), + SC_(93.5), SC_(0.1197906048094477368688578756432123127164e144), SC_(329.4502433708052665886256792643481601196), + SC_(94.5), SC_(0.1120042154968336339723821137264035123898e146), SC_(333.988204807099907903519925338728239387), + SC_(95.5), SC_(0.1058439836445077841039010974714513192084e148), SC_(338.5368046415996049733937816714808196625), + SC_(96.5), SC_(0.101081004380504933819225548085236009844e150), SC_(343.095930889086289536826499502225010241), + SC_(97.5), SC_(0.9754316922718726113555265390225274949948e151), SC_(347.6654738974312297792641984341498924371), + SC_(98.5), SC_(0.95104589996507579607163837554696430762e153), SC_(352.245326275435031271896458324405747818), + SC_(99.5), SC_(0.9367802114655996591305637999137598430057e155), SC_(356.835382823613074469259023532110402225), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 41> near_1 = { + SC_(0.5), SC_(1.772453850905516027298167483341145182798), SC_(0.5723649429247000870717136756765293558236), + SC_(0.625), SC_(1.434518848090556775636019739456423136632), SC_(0.3608294954889401811849576858227794878574), + SC_(0.75), SC_(1.225416702465177645129098303362890526851), SC_(0.2032809514312953714814329718624296997597), + SC_(0.875), SC_(1.089652357422896951252376755102892971148), SC_(0.08585870722533432350236558376948770226972), + SC_(0.875), SC_(1.089652357422896951252376755102892971148), SC_(0.08585870722533432350236558376948770226972), + SC_(0.9375), SC_(1.040177011186767171459762817361921128614), SC_(0.03939090173458230065822754634095384450336), + SC_(0.96875), SC_(1.019032525056673950564730047928136024882), SC_(0.01885367233441289053559206570480080890174), + SC_(0.984375), SC_(1.009263984715686303151364227987264939567), SC_(0.009221337197578781045045446027854805411837), + SC_(0.9921875), SC_(1.004570300975031369541819081985749775618), SC_(0.004559888861804558865096599455042458443847), + SC_(0.99609375), SC_(1.002269894807266338070518646683408822875), SC_(0.002267322487909119869224853055660925945503), + SC_(0.998046875), SC_(1.001131154070271719475148653701632936175), SC_(0.001130514797538261731446855551493110644591), + SC_(0.9990234375), SC_(1.000564631256105134179583334797007222469), SC_(0.0005644719118551233842574575277837954032273), + SC_(0.99951171875), SC_(1.000282079501403060312266009234616105409), SC_(0.0002820397244605020216499093258679218227703), + SC_(0.999755859375), SC_(1.000140980758729162527664293184201592009), SC_(0.0001409708218759223705137152436852900410971), + SC_(0.9998779296875), SC_(1.000070475636328154358952919083132410835), SC_(0.7047315303717008615499045661742960649129e-4), + SC_(0.99993896484375), SC_(1.000035234133024267207862236519063784064), SC_(0.3523351231678223923789189421658142521323e-4), + SC_(0.999969482421875), SC_(1.00001761614530457531538182039987833925), SC_(0.1761599014210985977615986870219441253152e-4), + SC_(0.9999847412109375), SC_(1.000008807842360071243230645907272632701), SC_(0.8807803571255486980463673710461571970668e-5), + SC_(0.99999237060546875), SC_(1.000004403863608190592851185888812529877), SC_(0.4403853911211722516241836921789602523957e-5), + SC_(0.999996185302734375), SC_(1.000002201917411285169225005738085227502), SC_(0.2201914987068584780432891571768087250967e-5), + SC_(1), SC_(1), SC_(0), + SC_(1.000003814697265625), SC_(0.9999977981113740328314320876032760067576), SC_(-0.2201891050127487637714181831108975522931e-5), + SC_(1.00000762939453125), SC_(0.9999955962515330814147665236857616584616), SC_(-0.4403758163447332570221840098229107074209e-5), + SC_(1.0000152587890625), SC_(0.9999911926182050168670695717547199129391), SC_(-0.8807420580197905194061157631901679899078e-5), + SC_(1.000030517578125), SC_(0.99998238569695577840308912119540189507), SC_(-0.1761445817787918059338915122874759423839e-4), + SC_(1.00006103515625), SC_(0.9999647732360171681023427372223702590454), SC_(-0.352273844598538899122262467277208094941e-4), + SC_(1.0001220703125), SC_(0.9999295538398379138630109146622273453912), SC_(-0.7044864160936656733821449407577769386531e-4), + SC_(1.000244140625), SC_(0.9998591371459403420587898072239427065107), SC_(-0.0001408727761632663509703058417841195184664), + SC_(1.00048828125), SC_(0.9997183921173586652300583206642029951561), SC_(-0.0002816475415868068318239205463917072406589), + SC_(1.0009765625), SC_(0.999437255220281084345213592031181905165), SC_(-0.0005629031799912046317021499216851450277288), + SC_(1.001953125), SC_(0.998876391856702293840181665944397250951), SC_(-0.001124239864176365593088235002674045499535), + SC_(1.00390625), SC_(0.9977602892435009045255150050890440579513), SC_(-0.002242222659961150144765481909230529929808), + SC_(1.0078125), SC_(0.995550440714294209465143243929765941358), SC_(-0.004459488037952299086670078922352109198758), + SC_(1.015625), SC_(0.9912190698420517341764818662191450397148), SC_(-0.008819709705733069204889229698627066380001), + SC_(1.03125), SC_(0.98290109928362691478263486825456935047), SC_(-0.01724677500176806740289126202224623179737), + SC_(1.0625), SC_(0.9675800675995248847599762987154317516646), SC_(-0.03295710029357781908319883575047418315706), + SC_(1.125), SC_(0.9417426998497014880874037301518917030763), SC_(-0.06002318412603958293140584320743011427822), + SC_(1.125), SC_(0.9417426998497014880874037301518917030763), SC_(-0.06002318412603958293140584320743011427822), + SC_(1.25), SC_(0.9064024770554770779826712889669180007488), SC_(-0.09827183642181316146385380269663584022562), + SC_(1.375), SC_(0.8889135691562253407424275640662446912078), SC_(-0.1177552707410787744513620333179885042465), + SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 41> near_2 = { + SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519), + SC_(1.625), SC_(0.8965742800565979847725123371602644603951), SC_(-0.1091741337567953724659793453255625768435), + SC_(1.75), SC_(0.9190625268488832338468237275221678951384), SC_(-0.08440112102048555595778603413139773174384), + SC_(1.875), SC_(0.9534458127450348323458296607150313497544), SC_(-0.0476726853991882996439780371618622723197), + SC_(1.875), SC_(0.9534458127450348323458296607150313497544), SC_(-0.0476726853991882996439780371618622723197), + SC_(1.9375), SC_(0.9751659479875942232435276412768010580753), SC_(-0.02514761940298887101469636934303908362555), + SC_(1.96875), SC_(0.9871877586486528896095822339303817741048), SC_(-0.01289502598016741062140421704372482102582), + SC_(1.984375), SC_(0.9934942349545037046646241619249639248858), SC_(-0.006527019770560387562504065432973464109098), + SC_(1.9921875), SC_(0.9967220954986639369672736204077361054958), SC_(-0.003283288599221334008087443035901123210745), + SC_(1.99609375), SC_(0.9983547780306754539374306832198017571606), SC_(-0.001646576833227209223092930587911738897203), + SC_(1.998046875), SC_(0.9991758197849782200230487539873719343464), SC_(-0.0008245200382650888261806366903737014930725), + SC_(1.9990234375), SC_(0.9995875173583940940094860854466195201034), SC_(-0.0004125677359714894017106176239696704326377), + SC_(1.99951171875), SC_(0.9997936605172715158492229105972945155141), SC_(-0.0002063607736483724433350542340289891606905), + SC_(1.999755859375), SC_(0.9998968057145986134157190626438734177924), SC_(-0.0001031996102979920861817501746961436019694), + SC_(1.9998779296875), SC_(0.9999483967208452041447977734631271456482), SC_(-0.5160461064981215542788033369454594603258e-4), + SC_(1.99993896484375), SC_(0.9999741968262534527384281628962293685498), SC_(-0.2580350665416168648325053559023655125278e-4), + SC_(1.999969482421875), SC_(0.9999870980295774847216527132886600812439), SC_(-0.1290205365365156795229453393309637521044e-4), + SC_(1.9999847412109375), SC_(0.9999935489189005625751513729787156979985), SC_(-0.6451101907750591399976874019296260011728e-5), + SC_(1.99999237060546875), SC_(0.9999967744354781276641509840822795646932), SC_(-0.3225569724016764801116581064181763189594e-5), + SC_(1.999996185302734375), SC_(0.9999983872117460118412633921783404380531), SC_(-0.1612789554532533173009818453505550062943e-5), + SC_(2), SC_(1), SC_(0), + SC_(2.000003814697265625), SC_(1.00000161280024011931074434129623713986), SC_(0.161279893955840184299760314572495677042e-5), + SC_(2.00000762939453125), SC_(1.000003225612466396944257297966490462185), SC_(0.322560726412023958566345675090317173141e-5), + SC_(2.0000152587890625), SC_(1.000006451272877535864519325623917830418), SC_(0.6451252068164492211696167502987139983584e-5), + SC_(2.000030517578125), SC_(1.000012902737534909133631207655399313341), SC_(0.1290265429530719797568036294024708700466e-4), + SC_(2.00006103515625), SC_(1.000025806242196124228325546227327679105), SC_(0.2580590922078463500093254292124318294867e-4), + SC_(2.0001220703125), SC_(1.000051615552953128452105520486771074315), SC_(0.5161422091631080428484087308642550680092e-4), + SC_(2.000244140625), SC_(1.000103243380595112650112753954221989398), SC_(0.0001032380513640963581901732440393341809948), + SC_(2.00048828125), SC_(1.000206535863509719265815185078589813025), SC_(0.0002065145379145443567131291462537026096587), + SC_(2.0009765625), SC_(1.000413268164832140091644464679649856244), SC_(0.000413182793064254264258674986332041644883), + SC_(2.001953125), SC_(1.000827322309547415507838270760694901832), SC_(0.000826980267085383846585814529167493000815), + SC_(2.00390625), SC_(1.001657790373358329933817798077673136303), SC_(0.001656417755696172869171861186612377080916), + SC_(2.0078125), SC_(1.003328178532374632976589675522967237775), SC_(0.003322652404102649860792821138784654479368), + SC_(2.015625), SC_(1.00670686780833379252298939537881918096), SC_(0.006684476830232184945964816343819769497868), + SC_(2.03125), SC_(1.013616758636240255869592207887524642672), SC_(0.01352488366498562096813694557452593229433), + SC_(2.0625), SC_(1.028053821824495190057474817385146236144), SC_(0.02766752152285702349740729628994608012915), + SC_(2.125), SC_(1.059460537330914174098329196420878165961), SC_(0.05775985153034387160738826626309159079026), + SC_(2.125), SC_(1.059460537330914174098329196420878165961), SC_(0.05775985153034387160738826626309159079026), + SC_(2.25), SC_(1.133003096319346347478339111208647500936), SC_(0.124871714892396594302441287613198663149), + SC_(2.375), SC_(1.222256157589809843520837900591086450411), SC_(0.2006984603774558413588851802726110913487), + SC_(2.5), SC_(1.329340388179137020473625612505858887098), SC_(0.2846828704729191596324946696827019243201), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_0 = { + SC_(-0.5), SC_(-3.544907701811032054596334966682290365595), SC_(1.265512123484645396488945797134705923899), + SC_(-0.375), SC_(-3.825383594908151401696052638550461697686), SC_(1.341658748500666418041408813274783487436), + SC_(-0.25), SC_(-4.901666809860710580516393213451562107405), SC_(1.589575312551185990315897214778782835911), + SC_(-0.125), SC_(-8.717218859383175610019014040823143769183), SC_(2.165300248905170251754061948144017406496), + SC_(-0.125), SC_(-8.717218859383175610019014040823143769183), SC_(2.165300248905170251754061948144017406496), + SC_(-0.0625), SC_(-16.64283217898827474335620507779073805782), SC_(2.811979623974363538327156032173660116805), + SC_(-0.03125), SC_(-32.60904080181356641807136153370035279623), SC_(3.484589575134139437621752672995683649279), + SC_(-0.015625), SC_(-64.59289502180392340168731059118495613226), SC_(4.168104420557250637548438174776914213865), + SC_(-0.0078125), SC_(-128.5849985248040153013528424941759712791), SC_(4.856590152781421724785721449662278434972), + SC_(-0.00390625), SC_(-256.581093070660182546052773550952658656), SC_(5.54744476696747159520708182472107347055), + SC_(-0.001953125), SC_(-512.5791508839791203712761106952360633215), SC_(6.239455139837046046486535948675082223324), + SC_(-0.0009765625), SC_(-1024.578182406251657399893334832135395808), SC_(6.932036277511308217556578672109549476158), + SC_(-0.00048828125), SC_(-2048.577698818873467519520786912493783878), SC_(7.624901025883858905611203245365810170653), + SC_(-0.000244140625), SC_(-4096.57745718775464971331294488248972087), SC_(8.317907137541219635377299172741804106947), + SC_(-0.0001220703125), SC_(-8192.577336412800240508542313129020709561), SC_(9.010983820432326192510172569412912814588), + SC_(-0.6103515625e-4), SC_(-16384.5772760354695939336148831283410381), SC_(9.704095761351551114080487592308688534482), + SC_(-0.30517578125e-4), SC_(-32768.57724584934032393443149086321342054), SC_(10.39722532438932175111825798174135071555), + SC_(-0.152587890625e-4), SC_(-65536.57723075690962899636361017901925666), SC_(11.09036369676269620616269440700453555078), + SC_(-0.762939453125e-5), SC_(-131072.577223210852757386190636818435916), SC_(11.78350647337298147181546230662592344689), + SC_(-0.3814697265625e-5), SC_(-262144.5772194378639394013199042046138782), SC_(12.47665145199400263809495861913874999345), + SC_(0.3814697265625e-5), SC_(262143.4227881080344625629331726731855155), SC_(12.47664704818796544202254047206534711638), + SC_(0.762939453125e-5), SC_(131071.4227918809440471962777925401520979), SC_(11.78349766576090681276037584294890342818), + SC_(0.152587890625e-4), SC_(65535.42279942668398540027145451732421438), SC_(11.09034608153854475277051988217319318753), + SC_(0.30517578125e-4), SC_(32767.42281451784694671242432333092929765), SC_(10.39719009394100176207788843272141977354), + SC_(0.6103515625e-4), SC_(16383.4228446989052821887834066513143242), SC_(9.704025300454774477951337474167744232248), + SC_(0.0001220703125), SC_(8191.422905055952190365785412912966413445), SC_(9.010842898637679655856679364462219607288), + SC_(0.000244140625), SC_(4095.423025749771641072803050389269325868), SC_(8.317625293943180446655815151656334697388), + SC_(0.00048828125), SC_(2047.42326705635054639115944072028773408), SC_(7.62433733861781159675772941549355054159), + SC_(0.0009765625), SC_(1023.423749345567830369498718239930270889), SC_(6.930908902419461889540619064660080535727), + SC_(0.001953125), SC_(511.4247126306315744461730129635313924869), SC_(6.23720038517533141916200085812091506718), + SC_(0.00390625), SC_(255.4266340463362315585318413027952788355), SC_(5.542935221819601325193091489756182014674), + SC_(0.0078125), SC_(127.4304564114296588115383352230100404938), SC_(4.84757077588166486683395477128488386733), + SC_(0.015625), SC_(63.43802046989131098729483943802528254174), SC_(4.150063373653938787298503499050432342073), + SC_(0.03125), SC_(31.45283517707606127304431578414621921504), SC_(3.44848912779795847968326934526863660858), + SC_(0.0625), SC_(15.48128108159239815615962077944690802663), SC_(2.739631621946203418585729650082232089145), + SC_(0.125), SC_(7.53394159879761190469922984121513362461), SC_(2.019418357553796345320290521167099589948), + SC_(0.125), SC_(7.53394159879761190469922984121513362461), SC_(2.019418357553796345320290521167099589948), + SC_(0.25), SC_(3.625609908221908311930685155867672002995), SC_(1.288022524698077457370610440219717295925), + SC_(0.375), SC_(2.370436184416600908646473504176652509887), SC_(0.8630739822706474624050890941340154953325), + SC_(0.5), SC_(1.772453850905516027298167483341145182798), SC_(0.5723649429247000870717136756765293558236), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_m10 = { + SC_(-10.5), SC_(-0.2640121820547716316246385325311240439682e-6), SC_(-15.14727059071784114610117639552631963436), + SC_(-10.375), SC_(-0.3853824777091100016167565620752110328498e-6), SC_(-14.76902954720701012688042720516103787762), + SC_(-10.25), SC_(-0.67808180432946731304891004492754985848e-6), SC_(-14.20399790093109065161116876070387206737), + SC_(-10.125), SC_(-0.1684831262052517456216882327889895477394e-5), SC_(-13.293845140389538484236785089823191703), + SC_(-10.125), SC_(-0.1684831262052517456216882327889895477394e-5), SC_(-13.293845140389538484236785089823191703), + SC_(-10.0625), SC_(-0.3830328107020316635889325503807551568979e-5), SC_(-12.47256009081165579039929355805785844157), + SC_(-10.03125), SC_(-0.820629952953019778657392524711140975814e-5), SC_(-11.71060846332720387792243958913415617938), + SC_(-10.015625), SC_(-0.1700699874643038383461557487368349913225e-4), SC_(-10.98188560766303889037266091827081352261), + SC_(-10.0078125), SC_(-0.3463458256516313677243695342138690766685e-4), SC_(-10.27065787896126424410583159193311350064), + SC_(-10.00390625), SC_(-0.6990332874758209033426446699035926999779e-4), SC_(-9.568397288290682386853008683681967849367), + SC_(-10.001953125), SC_(-0.0001404477363900451708521851118608466504012), SC_(-8.870675121382310094805966387146254501189), + SC_(-10.0009765625), SC_(-0.000281540041401883411812666338203440274148), SC_(-8.175235877509399161097817609681419866606), + SC_(-10.00048828125), SC_(-0.000563726404385624063397951573975953521758), SC_(-7.480941522771659111292091779019988346536), + SC_(-10.000244140625), SC_(-0.001128100008866690679009902807702968873353), SC_(-6.787220469493554911614571257238006425123), + SC_(-10.0001220703125), SC_(-0.002256847657595184957233851276605048793085), SC_(-6.093786281167310238545736320487058532071), + SC_(-10.00006103515625), SC_(-0.004514343175062893955072428860155326550553), SC_(-5.400495578872419834979145899538225218653), + SC_(-10.000030517578125), SC_(-0.009029334320035575647839997474459714791226), SC_(-4.707276632982054173483611347195057396393), + SC_(-10.0000152587890625), SC_(-0.01805931666500754906434806474181437991942), SC_(-4.01409356864116185854622778461065066607), + SC_(-10.00000762939453125), SC_(-0.03611928138246679574769960801636554264536), SC_(-3.320928445911808853642423587159533484381), + SC_(-10.000003814697265625), SC_(-0.07223921083114343778444263107651750106133), SC_(-2.627772294197426150094657771000510514181), + SC_(-9.999996185302734375), SC_(0.07224050699108014022628046448675851453374), SC_(-2.627754351749084271591553302449947161499), + SC_(-9.99999237060546875), SC_(0.03612057754240364069840471083444586728855), SC_(-3.320892561015125097640948165422843317137), + SC_(-9.9999847412109375), SC_(0.01806061282494496405052242015364798732676), SC_(-4.014021798847794354581145064057511795502), + SC_(-9.999969482421875), SC_(0.009030630479975270775894162659645956519741), SC_(-4.707133093395319229856390889602684453056), + SC_(-9.99993896484375), SC_(0.004515639335011709650690623512369371764253), SC_(-5.400208499698950462148264857327856480671), + SC_(-9.9998779296875), SC_(0.002258143817580482923824809786660261248415), SC_(-6.093212122820375608272453375481778776963), + SC_(-9.999755859375), SC_(0.00112939616899791774095825334343857186485), SC_(-6.786072152799718574175843462405206935614), + SC_(-9.99951171875), SC_(0.0005650225651005676902373042742306456894427), SC_(-7.478644889384249821277500306107773873963), + SC_(-9.9990234375), SC_(0.000282836204451696233602080064105049188933), SC_(-8.170642610736687659976646968506313698216), + SC_(-9.998046875), SC_(0.0001417439087793817388175538505551912206884), SC_(-8.861488587853743723990903404527520549863), + SC_(-9.99609375), SC_(0.7119953849576511774290270246050426493188e-4), SC_(-9.550024221368402701862865053477154735263), + SC_(-9.9921875), SC_(0.3593094176075641243360143967272818899195e-4), SC_(-10.23391174619552949434163600960412224542), + SC_(-9.984375), SC_(0.1830395592410737084371847753712605958898e-4), SC_(-10.90839335076217169965796414369624068766), + SC_(-9.96875), SC_(0.9505651717966543211040666528035462899479e-5), SC_(-11.56362401857045907442600014919001566007), + SC_(-9.9375), SC_(0.5139309871979484321498376816774659227047e-5), SC_(-12.17859175366355833461560769108992933228), + SC_(-9.875), SC_(0.3033137834984411836791652786842133042328e-5), SC_(-12.70591288519170921633064985576606088605), + SC_(-9.875), SC_(0.3033137834984411836791652786842133042328e-5), SC_(-12.70591288519170921633064985576606088605), + SC_(-9.75), SC_(0.2197547155462853887868709703817446704567e-5), SC_(-13.02816874893114553892881831326149507953), + SC_(-9.625), SC_(0.2248214426276410648574460028763198846825e-5), SC_(-13.00537424512744790544056216033820032846), + SC_(-9.5), SC_(0.2772127911575102132058704591576802461667e-5), SC_(-12.79589533355436345901781053661879076815), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_m55 = { + SC_(-55.5), SC_(0.3313939247684676728377268347296671102738e-73), SC_(-169.1931592947433577934436950412776436064), + SC_(-55.375), SC_(0.5931885913251829148011431845907438457576e-73), SC_(-168.6109546898941580979033836431457031234), + SC_(-55.25), SC_(0.128134265213561204650151776242601279892e-72), SC_(-167.8408033134133338381597995376176917559), + SC_(-55.125), SC_(0.3913191103476296924523858848738104144085e-72), SC_(-166.7243586084319013982262850893209212539), + SC_(-55.125), SC_(0.3913191103476296924523858848738104144085e-72), SC_(-166.7243586084319013982262850893209212539), + SC_(-55.0625), SC_(0.986732607487198018754292852511389106856e-72), SC_(-165.7994828862238820223463651561184383051), + SC_(-55.03125), SC_(0.2226660343675615375290464186909601183619e-71), SC_(-164.9856238363615162450940197678605306484), + SC_(-55.015625), SC_(0.4736069454740001055763353712089973133951e-71), SC_(-164.2309191328226253115354307486462442346), + SC_(-55.0078125), SC_(0.977114118293834733521912343859491871413e-71), SC_(-163.5066934315334134525852432438180103697), + SC_(-55.00390625), SC_(0.1984981336813056540008261636773673389822e-70), SC_(-162.7979320905915813518737104001603528406), + SC_(-55.001953125), SC_(0.4001152702583918680236220298606474660766e-70), SC_(-162.0969591073259992663458061954608696748), + SC_(-55.0009765625), SC_(0.8033716579133006533597549451686567408424e-70), SC_(-161.399894344940666691934457343256688548), + SC_(-55.00048828125), SC_(0.1609895558223411179719291838965309144853e-69), SC_(-160.7047872033597077904481004857770657217), + SC_(-55.000244140625), SC_(0.3222948938373906907742658942729222148632e-69), SC_(-160.0106597497627978468127607946271352327), + SC_(-55.0001220703125), SC_(0.6449058492709206027088103024110200975353e-69), SC_(-159.3170223595527956784727838139907674684), + SC_(-55.00006103515625), SC_(0.1290127899946570906748117205135768082314e-68), SC_(-158.6236300558849309281660984198839415596), + SC_(-55.000030517578125), SC_(0.2580572071228903781598785159467838581806e-68), SC_(-157.9303603092003033255947568117785799832), + SC_(-55.0000152587890625), SC_(0.5161460448765772204325064962419091742529e-68), SC_(-157.2371518444353362397006108889220356827), + SC_(-55.00000762939453125), SC_(0.1032323722132728230665906965174285853426e-67), SC_(-156.5439740214872098856520824742320518588), + SC_(-55.000003814697265625), SC_(0.2064679077519460715554797358664728667652e-67), SC_(-155.8508115196617565149201605014267824236), + SC_(-54.999996185302734375), SC_(-0.2064742345776389872444192676691131506575e-67), SC_(-155.8507808769879053144034254273344043224), + SC_(-54.99999237060546875), SC_(-0.1032386990389669331883069223703195417382e-67), SC_(-156.543912736139507484654652677755511734), + SC_(-54.9999847412109375), SC_(-0.5162093131335660989607568848213731056733e-68), SC_(-157.2370292737399314379940741096346901828), + SC_(-54.999969482421875), SC_(-0.258120475380070365932807313377307528986e-68), SC_(-157.9301151678094937244882657625299630878), + SC_(-54.99993896484375), SC_(-0.129076058252601515431668371755594450098e-68), SC_(-158.6231397731033117444057763960015760334), + SC_(-54.9998779296875), SC_(-0.6455385318809423304687659289501264852653e-69), SC_(-159.3160417939895574585734203633457962732), + SC_(-54.999755859375), SC_(-0.3229275765697223526481784197301709161285e-69), SC_(-160.0086986186363225879842786767213113941), + SC_(-54.99951171875), SC_(-0.1616222390439127298761439614470246404862e-69), SC_(-160.70086494110676672055309472267163668), + SC_(-54.9990234375), SC_(-0.8096985096986489455731873769112931053315e-70), SC_(-161.3920498204348601342401201789602276343), + SC_(-54.998046875), SC_(-0.4064422003228156070372890110113661523823e-70), SC_(-162.0812700583149908077227373306519861132), + SC_(-54.99609375), SC_(-0.2048253768707794232109022149495560465895e-70), SC_(-162.7665539925744016887591545718340828031), + SC_(-54.9921875), SC_(-0.1040399076590405899054222579682253865197e-70), SC_(-163.4439372355377521596244089121137210432), + SC_(-54.984375), SC_(-0.5369420317824801296090020359957584504561e-71), SC_(-164.1054067411408869986598886088213583004), + SC_(-54.96875), SC_(-0.2862019916619447982849309567142574291595e-71), SC_(-164.7345990554747140245090340482576614581), + SC_(-54.9375), SC_(-0.1630184748946170270451320476784206632433e-71), SC_(-165.2974333442636798880161449262335724888), + SC_(-54.875), SC_(-0.1068084665867092473960298264876009169786e-71), SC_(-165.7202596830189416811721153572453840111), + SC_(-54.875), SC_(-0.1068084665867092473960298264876009169786e-71), SC_(-165.7202596830189416811721153572453840111), + SC_(-54.75), SC_(-0.9545836185625177195542164628822374598687e-72), SC_(-165.8326067306542059861208673933569437798), + SC_(-54.625), SC_(-0.1206186475649396145903288691554707530168e-71), SC_(-165.5986629859610336595561650158366255984), + SC_(-54.5), SC_(-0.183923628246499558424938393274965246202e-71), SC_(-165.1767762739909689670975862547818473059), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 141> gammap1m1_data = { + SC_(-0.4952165186405181884765625), SC_(0.7559827693907095754807809442951050489732), + SC_(-0.4642883241176605224609375), SC_(0.6574328869566978138139138799311066062094), + SC_(-0.4024595916271209716796875), SC_(0.4948624198600628575485791492257182331098), + SC_(-0.3901382386684417724609375), SC_(0.466994695902624837582482771934607569504), + SC_(-0.3875354826450347900390625), SC_(0.4612760821854033810460931227828015343156), + SC_(-0.373013198375701904296875), SC_(0.4303940312894863907637421435297095229576), + SC_(-0.364522993564605712890625), SC_(0.4131121158463471061555369932726425070959), + SC_(-0.35811364650726318359375), SC_(0.4004260060282522370607617707987066040664), + SC_(-0.342386901378631591796875), SC_(0.3705491389143732767395411311340356537958), + SC_(-0.311618030071258544921875), SC_(0.3168394199636867612664810682360364987851), + SC_(-0.2880756855010986328125), SC_(0.2795634828731819295069040561405863199118), + SC_(-0.27896595001220703125), SC_(0.2659494215075372040950689557832739194382), + SC_(-0.221501767635345458984375), SC_(0.1892800230490205183297725744929501015544), + SC_(-0.202970564365386962890625), SC_(0.1675837744862475402389911769714649852773), + SC_(-0.191832959651947021484375), SC_(0.1551782986717939911912971517092154248689), + SC_(-0.1387059986591339111328125), SC_(0.1019358076187795495751950704491293377337), + SC_(-0.1012614667415618896484375), SC_(0.06964849559574637322009665674759450635903), + SC_(-0.0782387256622314453125), SC_(0.05168944861403749320872745190411594799105), + SC_(-0.0146243572235107421875), SC_(0.008655823217365420051945171708362679166147), + SC_(0.1431564604442703013402649929484277542953e-29), SC_(-0.8263215150028947028222714725368343067077e-30), + SC_(0.1791466932348087634896446282571611213266e-29), SC_(-0.1034062776504410791508686126702507453693e-29), + SC_(0.6013619202535540063110633226832922483532e-29), SC_(-0.3471155206456177563387123499681270341265e-29), + SC_(0.115805324961653822428570241697281798758e-28), SC_(-0.6684464764687909153739175517973613571261e-29), + SC_(0.1422457400834001098175711728787848259007e-28), SC_(-0.8210646944165041873250036406993552111264e-29), + SC_(0.4970121018327539153628705477876439795096e-28), SC_(-0.2868831708235014101261168518349749234889e-28), + SC_(0.9660079415057497591758174164417478444323e-28), SC_(-0.5575949162564024099347963251548294827165e-28), + SC_(0.1232929313253182131376331095427391968754e-27), SC_(-0.7116661133260258147780879126862696392479e-28), + SC_(0.3296523285617759312781860549364832953326e-27), SC_(-0.1902804880171240659658521971091414147824e-27), + SC_(0.528364435768055252017009628713605422886e-27), SC_(-0.3049802291021812635024061181080113889506e-27), + SC_(0.886586057273120049620324386849842094685e-27), SC_(-0.5117513605413324831805933701656815521236e-27), + SC_(0.2499669674831043259218157022821422146034e-26), SC_(-0.1442848493391799075204112945868205575921e-26), + SC_(0.4131050397232622964314362671638736040881e-26), SC_(-0.2384507001780369908735157814349257650306e-26), + SC_(0.7679738097881433551381658732998641759182e-26), SC_(-0.4432865132438264916724504940164877771849e-26), + SC_(0.199929739820949207249437007767740538737e-25), SC_(-0.1154025777043396680338534232061371984153e-25), + SC_(0.5151477415246978459754129800826163591626e-25), SC_(-0.2973513461467014566158126478567428007026e-25), + SC_(0.101200734533556026342258477595279955025e-24), SC_(-0.5841464927231005972586520769367701801272e-25), + SC_(0.2064292695896540981798546456623054911033e-24), SC_(-0.1191542081013299677372796818685763621433e-24), + SC_(0.4063294332896333395257434433879773416284e-24), SC_(-0.2345397140053387487331153502526122099635e-24), + SC_(0.8138195767936862452966745688936976428456e-24), SC_(-0.4697494081288516881709792363579127898501e-24), + SC_(0.9575550627132253801929510132578249716542e-24), SC_(-0.5527157822038433842858279196451987644551e-24), + SC_(0.2855160956298500804375620841706273850616e-23), SC_(-0.1648043629790735548426012109835882111878e-23), + SC_(0.65201444297915461398563707001320281266e-23), SC_(-0.3763529502296153146061378052316774680657e-23), + SC_(0.1310988374636350038320977491775043421995e-22), SC_(-0.7567230263439006455136418784154172627278e-23), + SC_(0.2590288837798696209228010176465529547374e-22), SC_(-0.1495155293796993236481538654450094828034e-22), + SC_(0.2937779542193655202274099291941187976629e-22), SC_(-0.1695732371781431498672748630538267989951e-22), + SC_(0.7863513178004503049754083414326234074965e-22), SC_(-0.4538942987503834952636621253829968594981e-22), + SC_(0.1903818607087388763706780167350761726053e-21), SC_(-0.109891392314185724619218859435147914583e-21), + SC_(0.3812242142377350870566942975497647799754e-21), SC_(-0.2200485882977986685255989911406099731464e-21), + SC_(0.5493133580141330277178034419485741501887e-21), SC_(-0.3170722751854215599980143372792630827825e-21), + SC_(0.9672153634284186955666772243312215295852e-21), SC_(-0.5582918591043124472447977064584587087382e-21), + SC_(0.1702169477623814384559878647986894129041e-20), SC_(-0.9825188868017248805929057833465618943933e-21), + SC_(0.4817114569977399785676754474621208412799e-20), SC_(-0.2780513989416366360400948471972900830387e-20), + SC_(0.7538352992756463183303278501219690799218e-20), SC_(-0.4351255434976382024407146526549699565796e-20), + SC_(0.2596305715949999708394617609422128090557e-19), SC_(-0.1498628330119729391523576917959646915902e-19), + SC_(0.4444587480324321591032923385589104015025e-19), SC_(-0.2565485517668431887674899070788880246089e-19), + SC_(0.9715574921498573937069095571295029856174e-19), SC_(-0.5607982038213457280620337061143959968827e-19), + SC_(0.2036598542733453787268262970278076551267e-18), SC_(-0.1175556581981383412558675525031327289634e-18), + SC_(0.4248971931658660264162106698360155121463e-18), SC_(-0.2452573158680304024128136998709195441154e-18), + SC_(0.6521097487613458963613731825259556273977e-18), SC_(-0.3764079622200518159115220576707372289309e-18), + SC_(0.1436126164096190058281493628911107407475e-17), SC_(-0.8289545186912702312307596583951073368993e-18), + SC_(0.3118908901459261162419055180006211003274e-17), SC_(-0.1800283075323116855097674712567357424907e-17), + SC_(0.3593346613595175715618300349429858897565e-17), SC_(-0.2074135954788010816821644850649799519675e-17), + SC_(0.9445874854124767215374919304693435151421e-17), SC_(-0.5452306934500297136990208691880314661067e-17), + SC_(0.2566182432094081539023303073498993853718e-16), SC_(-0.1481240698799817909729203530750507585117e-16), + SC_(0.3363765695149349330660137891158001366421e-16), SC_(-0.1941618252298598450712101459398214835952e-16), + SC_(0.1073581901339262605326457800103412409953e-15), SC_(-0.6196882910077942026172170471644452240087e-16), + SC_(0.186668406231853462907965823802669547149e-15), SC_(-0.1077479282192287023344331697438668504475e-15), + SC_(0.3727540802657755688795382376099496468669e-15), SC_(-0.2151594942853688563255514533064804448857e-15), + SC_(0.6211646767866855090717281839829411183018e-15), SC_(-0.3585459819247720488263755672803206966864e-15), + SC_(0.1561186859754253464932505224282976996619e-14), SC_(-0.9011415112885851355703262394946230181441e-15), + SC_(0.3092010764722992466335682593125966377556e-14), SC_(-0.1784757049442269726369719145429686196432e-14), + SC_(0.6192850577371690132255643845837767003104e-14), SC_(-0.3574610363653403859138387348420733335946e-14), + SC_(0.1047879028014987723427253740737796761096e-13), SC_(-0.6048521898920322580919537460083385958307e-14), + SC_(0.1978473638988408750405412206418986897916e-13), SC_(-0.1142005977018810929368580238123776497134e-13), + SC_(0.4041816252346730475863978426787070930004e-13), SC_(-0.2332999655507978182935820166341591465938e-13), + SC_(0.9410302262901834580155480125540634617209e-13), SC_(-0.5431773877604405885692410233977439758357e-13), + SC_(0.1334530223958893535574077304772799834609e-12), SC_(-0.7703117505534481432167486795079037095016e-13), + SC_(0.266297021326439287136622624529991298914e-12), SC_(-0.1537108122261681912683838906901126667239e-12), + SC_(0.5920415525016708979677559909760020673275e-12), SC_(-0.3417356583762410657228951913532598329392e-12), + SC_(0.155163989296047688526414276566356420517e-11), SC_(-0.8956308525005437046951559037146364454014e-12), + SC_(0.326923297461201300961874949280172586441e-11), SC_(-0.1887052485148118271327652592003779278951e-11), + SC_(0.3753785910581841633870681107509881258011e-11), SC_(-0.2166744030260566997416579153839836535144e-11), + SC_(0.9579165585749116473834874341264367103577e-11), SC_(-0.5529244432689301568675461523208969657637e-11), + SC_(0.1858167439361402273334533674642443656921e-10), SC_(-0.1072563353975220567027440177997243696045e-10), + SC_(0.5449485307451595872407779097557067871094e-10), SC_(-0.3145528284818088265306644806429650505592e-10), + SC_(0.6089519166696533147842274047434329986572e-10), SC_(-0.3514965854368603547185748339618264962e-10), + SC_(0.1337744776064297980155970435589551925659e-9), SC_(-0.7721672402075083279577276691763777798079e-10), + SC_(0.2554458866654840676346793770790100097656e-9), SC_(-0.1474473672534405166880467009401481006846e-9), + SC_(0.9285605062636648199259070679545402526855e-9), SC_(-0.5359796691714968347006887450221011388011e-9), + SC_(0.1698227447555211711005540564656257629395e-8), SC_(-0.980243482442200345890191122898173278899e-9), + SC_(0.339355921141759608872234821319580078125e-8), SC_(-0.1958815525210918994679361737180445367869e-8), + SC_(0.6313728651008432279922999441623687744141e-8), SC_(-0.3644383041872783822741274272616254061398e-8), + SC_(0.8383264749056706932606175541877746582031e-8), SC_(-0.4838951666662356901644608730186546413827e-8), + SC_(0.1962631124285962869180366396903991699219e-7), SC_(-0.1132861391263510827635198807950784606364e-7), + SC_(0.5256384838503436185419559478759765625e-7), SC_(-0.3034067396263077516870264512967026960836e-7), + SC_(0.116242290459922514855861663818359375e-6), SC_(-0.6709685761311096545819618213723265238395e-7), + SC_(0.1776920584006802528165280818939208984375e-6), SC_(-0.1025666084085592538025029643343934584008e-6), + SC_(0.246631174150024889968335628509521484375e-6), SC_(-0.142359317011220184093241753035251343271e-6), + SC_(0.7932688959044753573834896087646484375e-6), SC_(-0.4578866108069128537168607745624614794722e-6), + SC_(0.1372093493046122603118419647216796875e-5), SC_(-0.791991995861101927407880386227298092424e-6), + SC_(0.214747751670074649155139923095703125e-5), SC_(-0.1239553101482841564081262476002391564028e-5), + SC_(0.527022712049074470996856689453125e-5), SC_(-0.3042030180348038757030022358659056016131e-5), + SC_(0.9233162927557714283466339111328125e-5), SC_(-0.5329441960781667799420300524501852822265e-5), + SC_(0.269396477960981428623199462890625e-4), SC_(-0.1554926893050895772896453957379474045684e-4), + SC_(0.3208058114978484809398651123046875e-4), SC_(-0.1851639610824637542072201121307897455015e-4), + SC_(0.00010957030463032424449920654296875), SC_(-0.6323382317252073303439803607481865268353e-4), + SC_(0.000126518702018074691295623779296875), SC_(-0.7301274674392621832822527575061291232133e-4), + SC_(0.00028976381872780621051788330078125), SC_(-0.0001671731931845284705043154051351730706298), + SC_(0.000687857042066752910614013671875), SC_(-0.0003965741858362509385654340724912705823236), + SC_(0.00145484809763729572296142578125), SC_(-0.0008376704829300962209037156359622989708524), + SC_(0.00366270542144775390625), SC_(-0.002100946766981816464155333982357752904999), + SC_(0.046881496906280517578125), SC_(-0.02497588947336944943591732868959193811385), + SC_(0.04722058773040771484375), SC_(-0.02514197077286474061941968460870171431946), + SC_(0.1323592662811279296875), SC_(-0.06091072639354085529687250076608036167983), + SC_(0.139763355255126953125), SC_(-0.06350237679012056526225278765134891026831), + SC_(0.155740678310394287109375), SC_(-0.06883441875617310404657299803607175885806), + SC_(0.17873513698577880859375), SC_(-0.07590347542832235360535014668688262042636), + SC_(0.1813595294952392578125), SC_(-0.07666619009216679212771233211297326572625), + SC_(0.225838959217071533203125), SC_(-0.08828121583826801836069461556215827733624), + SC_(0.292207300662994384765625), SC_(-0.1013142147509511845410398110728244632327), + SC_(0.297928631305694580078125), SC_(-0.1022125377703106514269725748156778633457), + SC_(0.29810583591461181640625), SC_(-0.1022398124529602122123098193784073635017), + SC_(0.30028045177459716796875), SC_(-0.1025718475377491669113360177547935208456), + SC_(0.314723670482635498046875), SC_(-0.1046527148786850527567817015912636515928), + SC_(0.335008561611175537109375), SC_(-0.1072166100764849221291653907100739024018), + SC_(0.34912931919097900390625), SC_(-0.1087597918806720999316678995289547437333), + SC_(0.378430664539337158203125), SC_(-0.1113472284261879798379642546535625292991), + SC_(0.405791938304901123046875), SC_(-0.1130363541503776146472634379575969350382), + SC_(0.4133758544921875), SC_(-0.1133834162443148976453781329005327033269), + SC_(0.415735542774200439453125), SC_(-0.1134808289122708542397946052703635909288), + SC_(0.43399322032928466796875), SC_(-0.1140666251072475325460987089801937766512), + SC_(0.457166969776153564453125), SC_(-0.1143882508090213200510081514983929178479), + SC_(0.457506835460662841796875), SC_(-0.1143895043011533114503086722896019260013), + SC_(0.4594924449920654296875), SC_(-0.1143948425572877686750521180317509748309), + SC_(0.464888513088226318359375), SC_(-0.1143922664386800836288584840918904877633), + SC_(0.467694938182830810546875), SC_(-0.1143810844126184461958335384166685988825), + SC_(0.468867778778076171875), SC_(-0.114374421494214380845931849419009841586), + SC_(0.470592796802520751953125), SC_(-0.1143624939837521390307936225841266176506), + SC_(0.481109678745269775390625), SC_(-0.1142351916017016455044428590981136786372), + SC_(0.492881298065185546875), SC_(-0.1139822218283491394475161618135268104636), + SC_(0.496461331844329833984375), SC_(-0.1138823102618022293565213085801050842487), + }; +#undef SC_ + + diff --git a/test/test_gamma_hooks.hpp b/test/test_gamma_hooks.hpp new file mode 100644 index 000000000..2accb1753 --- /dev/null +++ b/test/test_gamma_hooks.hpp @@ -0,0 +1,176 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_GAMMA_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_GAMMA_OTHER_HOOKS_HPP + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double gamma(double); + float gammaf(float); + long double gammal(long double); + double lgam(double); + float lgamf(float); + long double lgaml(long double); + float igamf(float, float); + double igam(double, double); + long double igaml(long double, long double); + float igamcf(float, float); + double igamc(double, double); + long double igamcl(long double, long double); +} +inline float tgamma(float x) +{ return gammaf(x); } +inline double tgamma(double x) +{ return gamma(x); } +inline long double tgamma(long double x) +{ +#ifdef BOOST_MSVC + return gamma((double)x); +#else + return gammal(x); +#endif +} +inline float lgamma(float x) +{ return lgamf(x); } +inline double lgamma(double x) +{ return lgam(x); } +inline long double lgamma(long double x) +{ +#ifdef BOOST_MSVC + return lgam((double)x); +#else + return lgaml(x); +#endif +} +inline float gamma_Q(float x, float y) +{ return igamcf(x, y); } +inline double gamma_Q(double x, double y) +{ return igamc(x, y); } +inline long double gamma_Q(long double x, long double y) +{ +#ifdef BOOST_MSVC + return igamc((double)x, (double)y); +#else + return igamcl(x, y); +#endif +} +inline float gamma_P(float x, float y) +{ return igamf(x, y); } +inline double gamma_P(double x, double y) +{ return igam(x, y); } +inline long double gamma_P(long double x, long double y) +{ +#ifdef BOOST_MSVC + return igam((double)x, (double)y); +#else + return igaml(x, y); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_NATIVE +#include +namespace other{ +#if defined(__FreeBSD__) +// no float version: +inline float tgamma(float x) +{ return ::tgamma(x); } +#else +inline float tgamma(float x) +{ return ::tgammaf(x); } +#endif +inline double tgamma(double x) +{ return ::tgamma(x); } +inline long double tgamma(long double x) +{ +#if defined(__CYGWIN__) || defined(__FreeBSD__) + // no long double versions: + return ::tgamma(x); +#else + return ::tgammal(x); +#endif +} +#if defined(__FreeBSD__) +inline float lgamma(float x) +{ return ::lgamma(x); } +#else +inline float lgamma(float x) +{ return ::lgammaf(x); } +#endif +inline double lgamma(double x) +{ return ::lgamma(x); } +inline long double lgamma(long double x) +{ +#if defined(__CYGWIN__) || defined(__FreeBSD__) + // no long double versions: + return ::lgamma(x); +#else + return ::lgammal(x); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#define TEST_OTHER +#include + +namespace other{ +float tgamma(float z) +{ + return (float)gsl_sf_gamma(z); +} +double tgamma(double z) +{ + return gsl_sf_gamma(z); +} +long double tgamma(long double z) +{ + return gsl_sf_gamma(z); +} +float lgamma(float z) +{ + return (float)gsl_sf_lngamma(z); +} +double lgamma(double z) +{ + return gsl_sf_lngamma(z); +} +long double lgamma(long double z) +{ + return gsl_sf_lngamma(z); +} +inline float gamma_Q(float x, float y) +{ return (float)gsl_sf_gamma_inc_Q(x, y); } +inline double gamma_Q(double x, double y) +{ return gsl_sf_gamma_inc_Q(x, y); } +inline long double gamma_Q(long double x, long double y) +{ return gsl_sf_gamma_inc_Q(x, y); } +inline float gamma_P(float x, float y) +{ return (float)gsl_sf_gamma_inc_P(x, y); } +inline double gamma_P(double x, double y) +{ return gsl_sf_gamma_inc_P(x, y); } +inline long double gamma_P(long double x, long double y) +{ return gsl_sf_gamma_inc_P(x, y); } +} +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept tgamma(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept lgamma(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept gamma_Q(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept gamma_P(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } +} +#endif + +#endif + + diff --git a/test/test_ibeta.cpp b/test/test_ibeta.cpp new file mode 100644 index 000000000..bdcc80357 --- /dev/null +++ b/test/test_ibeta.cpp @@ -0,0 +1,310 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include "test_beta_hooks.hpp" + +template +void print_test_result(const boost::math::tools::test_result& result, + const Seq& worst, int row, const char* name, const char* test) +{ + using namespace std; + T eps = pow(T(2), 1-boost::math::tools::digits(worst[0])); + std::cout << setprecision(4); + std::cout << test << "(" << name << ") Max = " << (result.stat.max)()/eps + << " RMS Mean=" << result.stat.rms()/eps + << "\n worst case at row: " + << row << "\n { "; + for(unsigned i = 0; i < worst.size(); ++i) + { + if(i) + std::cout << ", "; + std::cout << worst[i]; + } + std::cout << " }" << std::endl; +} + + +template +void do_test_beta(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type, value_type); + pg funcp = boost::math::beta; + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test beta against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[3])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::beta"); + + funcp = boost::math::betac; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[4])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::betac"); + + funcp = boost::math::ibeta; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[5])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::ibeta"); + + funcp = boost::math::ibetac; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[6])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "boost::math::ibetac"); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::ibeta; + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[5])); + print_test_result(result, data[result.worst_case], result.worst_case, type_name, "other::ibeta"); + } +#endif + std::cout << std::endl; +} + +template +void test_inverses(const T& data) +{ + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::tools::digits(value_type(0))/2)) * 100; + if(boost::math::tools::digits(value_type(0)) < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete beta is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + BOOST_CHECK_EQUAL(boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]), value_type(0)); + else if((1 - data[i][5] > 0.001) && (fabs(data[i][5]) >= boost::math::tools::min_value(data[i][5]))) + { + value_type inv = boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + } + else if(1 == data[i][5]) + BOOST_CHECK_EQUAL(boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]), value_type(1)); + + if(data[i][6] == 0) + BOOST_CHECK_EQUAL(boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]), value_type(1)); + else if((1 - data[i][6] > 0.001) && (fabs(data[i][6]) >= boost::math::tools::min_value(data[i][6]))) + { + value_type inv = boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + } + else if(data[i][6] == 1) + BOOST_CHECK_EQUAL(boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]), value_type(0)); + } +} + +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // +# include "ibeta_small_data.ipp" + + do_test_beta(ibeta_small_data, name, "Incomplete Beta Function: Small Values"); + test_inverses(ibeta_small_data); + +# include "ibeta_data.ipp" + + do_test_beta(ibeta_data, name, "Incomplete Beta Function: Medium Values"); + test_inverses(ibeta_data); + +# include "ibeta_large_data.ipp" + + do_test_beta(ibeta_large_data, name, "Incomplete Beta Function: Large and Diverse Values"); + test_inverses(ibeta_large_data); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 10 decimal places expressed as a persentage, + // One check per domain of the implementation: + // + T tolerance = std::pow(10.0, -8); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.015964560210704803), + static_cast(1.1846856068586931e-005), + static_cast(0.69176378846168518)), + static_cast(0.0007508604820642986204162462167319506309750), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(42.434902191162109), + static_cast(0.30012050271034241), + static_cast(0.91574394702911377)), + static_cast(0.002844243156314242058287766324242276991912), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(9.7131776809692383), + static_cast(99.406852722167969), + static_cast(0.083912998437881470)), + static_cast(0.4612716118626361034813232775095335302198), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(72.695472717285156), + static_cast(1.1902070045471191), + static_cast(0.80036874115467072)), + static_cast(1.703685144285803673344984949797496197040e-7), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4.9854421615600586), + static_cast(1.0665277242660522), + static_cast(0.75997146964073181)), + static_cast(0.2755954254731642667260071858810487404614), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(6.8127136230468750), + static_cast(1.0562920570373535), + static_cast(0.17416560649871826)), + static_cast(7.702362015088558153029455563361002570531e-6), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.48983201384544373), + static_cast(0.22512593865394592), + static_cast(0.20032680034637451)), + static_cast(0.170905142698145967653807992508983970176), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4.0498137474060059), + static_cast(0.15403440594673157), + static_cast(0.65370121598243713)), + static_cast(0.0172702040689452906446803217247250156007), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(7.2695474624633789), + static_cast(0.11902070045471191), + static_cast(0.80036874115467072)), + static_cast(0.013346136714187857821168127038816508028), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(2.7266697883605957), + static_cast(0.011510574258863926), + static_cast(0.086654007434844971)), + static_cast(5.812020420972734916187451486321162137375e-6), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.34317314624786377), + static_cast(0.046342257410287857), + static_cast(0.75823287665843964)), + static_cast(0.151317265120184850887504097401768195067), tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.34317314624786377), + static_cast(0.046342257410287857), + static_cast(0)), + static_cast(0), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibetac( + static_cast(0.34317314624786377), + static_cast(0.046342257410287857), + static_cast(0)), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.34317314624786377), + static_cast(0.046342257410287857), + static_cast(1)), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibetac( + static_cast(0.34317314624786377), + static_cast(0.046342257410287857), + static_cast(1)), + static_cast(0), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(1), + static_cast(0.046342257410287857), + static_cast(0.32)), + static_cast(0.0177137046180187568703202426065033413304), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.046342257410287857), + static_cast(1), + static_cast(0.32)), + static_cast(0.948565954109602496577407403168592262389), tolerance); +} + +int test_main(int, char* []) +{ +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + //test_spots(0.0F); + test_spots(0.0); + //test_spots(0.0L); + //test_spots(boost::math::concepts::real_concept(0.1)); + +#ifndef __HP_aCC + //test_beta(0.1F, "float"); +#endif + test_beta(0.1, "double"); + //test_beta(0.1L, "long double"); + //test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); + return 0; +} + + diff --git a/test/test_roots.cpp b/test/test_roots.cpp new file mode 100644 index 000000000..bbb285d0b --- /dev/null +++ b/test/test_roots.cpp @@ -0,0 +1,297 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +// +// Implement various versions of inverse of the incomplete beta +// using different root finding algorithms, and deliberately "bad" +// starting conditions: that way we get all the pathological cases +// we could ever wish for!!! +// + +template +struct ibeta_roots_1 // for first order algorithms +{ + ibeta_roots_1(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + T operator()(const T& x) + { + return boost::math::detail::ibeta_imp(a, b, x, L(), invert, true) - target; + } +private: + T a, b, target; + bool invert; +}; + +template +struct ibeta_roots_2 // for second order algorithms +{ + ibeta_roots_2(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + std::tr1::tuple operator()(const T& x) + { + T f = boost::math::detail::ibeta_imp(a, b, x, L(), invert, true) - target; + T f1 = invert ? + -boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true) + : boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true); + T y = 1 - x; + if(y == 0) + y = boost::math::tools::min_value(x) * 8; + f1 /= y * x; + + // make sure we don't have a zero derivative: + if(f1 == 0) + f1 = (invert ? -1 : 1) * boost::math::tools::min_value(f1) * 64; + + return std::tr1::make_tuple(f, f1); + } +private: + T a, b, target; + bool invert; +}; + +template +struct ibeta_roots_3 // for third order algorithms +{ + ibeta_roots_3(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + std::tr1::tuple operator()(const T& x) + { + T f = boost::math::detail::ibeta_imp(a, b, x, L(), invert, true) - target; + T f1 = invert ? + -boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true) + : boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true); + T y = 1 - x; + if(y == 0) + y = boost::math::tools::min_value(x) * 8; + f1 /= y * x; + T f2 = f1 * (-y * a + (b - 2) * x + 1) / (y * x); + if(invert) + f2 = -f2; + + // make sure we don't have a zero derivative: + if(f1 == 0) + f1 = (invert ? -1 : 1) * boost::math::tools::min_value(f1) * 64; + + return std::tr1::make_tuple(f, f1, f2); + } +private: + T a, b, target; + bool invert; +}; + +double inverse_ibeta_bisect(double a, double b, double z) +{ + typedef boost::math::lanczos::lanczos13m53 L; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::bisect(ibeta_roots_1(a, b, z, invert), min, max, precision); +} + +double inverse_ibeta_newton(double a, double b, double z) +{ + typedef boost::math::lanczos::lanczos13m53 L; + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::newton_raphson_iterate(ibeta_roots_2(a, b, z, invert), guess, min, max, precision); +} + +double inverse_ibeta_halley(double a, double b, double z) +{ + typedef boost::math::lanczos::lanczos13m53 L; + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::halley_iterate(ibeta_roots_3(a, b, z, invert), guess, min, max, precision); +} + +double inverse_ibeta_schroeder(double a, double b, double z) +{ + typedef boost::math::lanczos::lanczos13m53 L; + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::schroeder_iterate(ibeta_roots_3(a, b, z, invert), guess, min, max, precision); +} + + +template +void test_inverses(const T& data) +{ + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::tools::digits(value_type(0))/2)) * 100; + if(boost::math::tools::digits(value_type(0)) < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete beta is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + { + BOOST_CHECK_EQUAL(inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]), value_type(0)); + } + else if((1 - data[i][5] > 0.001) && (fabs(data[i][5]) >= boost::math::tools::min_value(data[i][5]))) + { + value_type inv = inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + inv = inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + inv = inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + inv = inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + } + else if(1 == data[i][5]) + { + BOOST_CHECK_EQUAL(inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]), value_type(1)); + } + + } +} + +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // +# include "ibeta_small_data.ipp" + + test_inverses(ibeta_small_data); + +# include "ibeta_data.ipp" + + test_inverses(ibeta_data); + +# include "ibeta_large_data.ipp" + + test_inverses(ibeta_large_data); +} + +int test_main(int, char* []) +{ + test_beta(0.1, "double"); + return 0; +} + + diff --git a/tools/beta_data.cpp b/tools/beta_data.cpp new file mode 100644 index 000000000..b73f276b8 --- /dev/null +++ b/tools/beta_data.cpp @@ -0,0 +1,72 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +#include + +using namespace boost::math::tools; + +struct beta_data_generator +{ + NTL::RR operator()(NTL::RR a, NTL::RR b) + { + if(a < b) + throw std::domain_error(""); + // very naively calculate spots: + NTL::RR g1, g2, g3; + int s1, s2, s3; + g1 = boost::math::lgamma(a, &s1); + g2 = boost::math::lgamma(b, &s2); + g3 = boost::math::lgamma(a+b, &s3); + g1 += g2 - g3; + g1 = exp(g1); + g1 *= s1 * s2 * s3; + return g1; + } +}; + + +int main() +{ + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the beta function:\n" + " beta(a, b)\n\n"; + + bool cont; + std::string line; + + do{ + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "b"); + data.insert(beta_data_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=beta_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "beta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "beta_data"); + + return 0; +} + diff --git a/tools/cbrt_data.cpp b/tools/cbrt_data.cpp new file mode 100644 index 000000000..0f106bd93 --- /dev/null +++ b/tools/cbrt_data.cpp @@ -0,0 +1,68 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include + +#include + +using namespace boost::math::tools; +using namespace std; + +struct cube_data_generator +{ + NTL::RR operator()(NTL::RR z) + { + NTL::RR result = z*z*z; + // if result is out of range of a float, + // don't include in test data as it messes up our results: + if(result > (std::numeric_limits::max)()) + throw std::domain_error(""); + if(result < (std::numeric_limits::min)()) + throw std::domain_error(""); + return result; + } +}; + +int main(int argc, char* argv[]) +{ + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the cbrt function:\n\n"; + + bool cont; + std::string line; + + do{ + if(0 == get_user_parameter_info(arg1, "z")) + return 1; + data.insert(cube_data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=cbrt_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "cbrt_data.ipp"; + std::ofstream ofs(line.c_str()); + ofs << std::scientific; + write_code(ofs, data, "cbrt_data"); + + return 0; +} + + + + diff --git a/tools/erf_data.cpp b/tools/erf_data.cpp new file mode 100644 index 000000000..df4c7e463 --- /dev/null +++ b/tools/erf_data.cpp @@ -0,0 +1,202 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include // for inverses +#include +#include +#include + +#include + +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; +using namespace std; + +struct erf_data_generator +{ + std::tr1::tuple operator()(NTL::RR z) + { + // very naively calculate spots using the gamma function at high precision: + int sign = 1; + if(z < 0) + { + sign = -1; + z = -z; + } + NTL::RR g1, g2; + g1 = boost::math::tgamma_lower(NTL::RR(0.5), z * z); + g1 /= sqrt(boost::math::constants::pi()); + g1 *= sign; + + if(z < 0.5) + { + g2 = 1 - (sign * g1); + } + else + { + g2 = boost::math::tgamma(NTL::RR(0.5), z * z); + g2 /= sqrt(boost::math::constants::pi()); + } + if(sign < 1) + g2 = 2 - g2; + return std::tr1::make_tuple(g1, g2); + } +}; + +double double_factorial(int N) +{ + double result = 1; + while(N > 2) + { + N -= 2; + result *= N; + } + return result; +} + +void asymptotic_limit(int Bits) +{ + // + // The following block of code estimates how large z has + // to be before we can use the asymptotic expansion for + // erf/erfc and still get convergence: the series becomes + // divergent eventually so we have to be careful! + // + double result = (std::numeric_limits::max)(); + int terms = 0; + for(int n = 1; n < 15; ++n) + { + double lim = (Bits-n) * log(2.0) - log(sqrt(3.14)) + log(double_factorial(2*n+1)); + double x = 1; + while(x*x + (2*n+1)*log(x) <= lim) + x += 0.1; + if(x < result) + { + result = x; + terms = n; + } + } + + std::cout << "Erf asymptotic limit for " + << Bits << " bit numbers is " + << result << " after approximately " + << terms << " terms." << std::endl; + + result = (std::numeric_limits::max)(); + terms = 0; + for(int n = 1; n < 30; ++n) + { + double x = pow(double_factorial(2*n+1)/pow(2.0, n-Bits), 1 / (2.0*n)); + if(x < result) + { + result = x; + terms = n; + } + } + + std::cout << "Erfc asymptotic limit for " + << Bits << " bit numbers is " + << result << " after approximately " + << terms << " terms." << std::endl; +} + +NTL::RR erfc_inv(NTL::RR r) +{ + std::cout << r << std::endl; + return boost::math::erfc_inv(r); +} + + +int main(int argc, char* argv[]) +{ + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + if(argc >= 2) + { + if(strcmp(argv[1], "--limits") == 0) + { + asymptotic_limit(24); + asymptotic_limit(53); + asymptotic_limit(64); + asymptotic_limit(106); + asymptotic_limit(113); + return 0; + } + else if(strcmp(argv[1], "--erf_inv") == 0) + { + NTL::RR (*f)(NTL::RR); + f = boost::math::erf_inv; + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse erf function:\n"; + std::cout << "Enter the number of data points: "; + int points; + std::cin >> points; + data.insert(f, make_random_param(NTL::RR(-1), NTL::RR(1), points)); + } + else if(strcmp(argv[1], "--erfc_inv") == 0) + { + NTL::RR (*f)(NTL::RR); + f = /*boost::math::*/ erfc_inv; + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse erfc function:\n"; + std::cout << "Enter the number of data points: "; + int points; + std::cin >> points; + data.insert(f, make_random_param(NTL::RR(0), NTL::RR(2), points)); + } + } + else + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the erf and erfc functions:\n" + " erf(z) and erfc(z)\n\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "a")) + return 1; + data.insert(erf_data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + std::cout << "Enter name of test data file [default=erf_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "erf_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "erf_data"); + + return 0; +} + +/* Output for asymptotic limits: + +Erf asymptotic limit for 24 bit numbers is 2.8 after approximately 6 terms. +Erfc asymptotic limit for 24 bit numbers is 4.12064 after approximately 17 terms. +Erf asymptotic limit for 53 bit numbers is 4.3 after approximately 11 terms. +Erfc asymptotic limit for 53 bit numbers is 6.19035 after approximately 29 terms. +Erf asymptotic limit for 64 bit numbers is 4.8 after approximately 12 terms. +Erfc asymptotic limit for 64 bit numbers is 7.06004 after approximately 29 terms. +Erf asymptotic limit for 106 bit numbers is 6.5 after approximately 14 terms. +Erfc asymptotic limit for 106 bit numbers is 11.6626 after approximately 29 terms. +Erf asymptotic limit for 113 bit numbers is 6.8 after approximately 14 terms. +Erfc asymptotic limit for 113 bit numbers is 12.6802 after approximately 29 terms. +*/ + diff --git a/tools/ibeta_data.cpp b/tools/ibeta_data.cpp new file mode 100644 index 000000000..1865a9cca --- /dev/null +++ b/tools/ibeta_data.cpp @@ -0,0 +1,294 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include + +#include + +// speed up beta function computation with a 90-decimal digit approximation: +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +template +struct ibeta_fraction1_t +{ + typedef std::pair result_type; + + ibeta_fraction1_t(T a_, T b_, T x_) : a(a_), b(b_), x(x_), k(1) {} + + result_type operator()() + { + T aN; + if(k & 1) + { + int m = (k - 1) / 2; + aN = -(a + m) * (a + b + m) * x; + aN /= a + 2*m; + aN /= a + 2*m + 1; + } + else + { + int m = k / 2; + aN = m * (b - m) *x; + aN /= a + 2*m - 1; + aN /= a + 2*m; + } + ++k; + return std::make_pair(aN, T(1)); + } + +private: + T a, b, x; + int k; +}; + +// +// This function caches previous calls to beta +// just so we can speed things up a bit: +// +template +T get_beta(T a, T b) +{ + static std::map, T> m; + + if(a < b) + std::swap(a, b); + + std::pair p(a, b); + std::map, T>::const_iterator i = m.find(p); + if(i != m.end()) + return i->second; + + T r = beta(a, b); + p.first = a; + p.second = b; + m[p] = r; + + return r; +} + +// +// compute the continued fraction: +// +template +T get_ibeta_fraction1(T a, T b, T x) +{ + ibeta_fraction1_t f(a, b, x); + T fract = boost::math::tools::continued_fraction_a(f, boost::math::tools::digits(x)); + T denom = (a * (fract + 1)); + T num = pow(x, a) * pow(1 - x, b); + if(num == 0) + return 0; + else if(denom == 0) + return -1; + return num / denom; +} +// +// calculate the incomplete beta from the fraction: +// +template +std::pair ibeta_fraction1(T a, T b, T x) +{ + T bet = get_beta(a, b); + if(x > ((a+1)/(a+b+2))) + { + T fract = get_ibeta_fraction1(b, a, 1-x); + if(fract/bet > 0.75) + { + fract = get_ibeta_fraction1(a, b, x); + return std::make_pair(fract, bet - fract); + } + return std::make_pair(bet - fract, fract); + } + T fract = get_ibeta_fraction1(a, b, x); + if(fract/bet > 0.75) + { + fract = get_ibeta_fraction1(b, a, 1-x); + return std::make_pair(bet - fract, fract); + } + return std::make_pair(fract, bet - fract); + +} +// +// calculate the regularised incomplete beta from the fraction: +// +template +std::pair ibeta_fraction1_regular(T a, T b, T x) +{ + T bet = get_beta(a, b); + if(x > ((a+1)/(a+b+2))) + { + T fract = get_ibeta_fraction1(b, a, 1-x); + if(fract == 0) + bet = 1; // normalise so we don't get 0/0 + else if(bet == 0) + return std::make_pair(T(-1), T(-1)); // Yikes!! + if(fract / bet > 0.75) + { + fract = get_ibeta_fraction1(a, b, x); + return std::make_pair(fract / bet, 1 - (fract / bet)); + } + return std::make_pair(1 - (fract / bet), fract / bet); + } + T fract = get_ibeta_fraction1(a, b, x); + if(fract / bet > 0.75) + { + fract = get_ibeta_fraction1(b, a, 1-x); + return std::make_pair(1 - (fract / bet), fract / bet); + } + return std::make_pair(fract / bet, 1 - (fract / bet)); +} + +// +// we absolutely must trunctate the input values to float +// precision: we have to be certain that the input values +// can be represented exactly in whatever width floating +// point type we are testing, otherwise the output will +// necessarily be off. +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(NTL::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +std::tr1::mt19937 rnd; +std::tr1::uniform_real ur_a(1.0F, 5.0F); +std::tr1::variate_generator > gen(rnd, ur_a); +std::tr1::uniform_real ur_a2(0.0F, 100.0F); +std::tr1::variate_generator > gen2(rnd, ur_a2); + +struct beta_data_generator +{ + std::tr1::tuple operator()(NTL::RR ap, NTL::RR bp, NTL::RR x_) + { + float a = truncate_to_float(real_cast(gen() * pow(NTL::RR(10), ap))); + float b = truncate_to_float(real_cast(gen() * pow(NTL::RR(10), bp))); + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::endl; + std::pair ib_full = ibeta_fraction1(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + std::pair ib_reg = ibeta_fraction1_regular(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +// medium sized values: +struct beta_data_generator_medium +{ + std::tr1::tuple operator()(NTL::RR x_) + { + NTL::RR a = gen2(); + NTL::RR b = gen2(); + NTL::RR x = x_; + a = ConvPrec(a, 22); + b = ConvPrec(b, 22); + x = ConvPrec(x, 22); + std::cout << a << " " << b << " " << x << std::endl; + //NTL::RR exp_beta = boost::math::beta(a, b, x); + std::pair ib_full = ibeta_fraction1(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + /*exp_beta = boost::math::tools::relative_error(ib_full.first, exp_beta); + if(exp_beta > 1e-40) + { + std::cout << exp_beta << std::endl; + }*/ + std::pair ib_reg = ibeta_fraction1_regular(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +struct beta_data_generator_small +{ + std::tr1::tuple operator()(NTL::RR x_) + { + float a = truncate_to_float(gen2()/10); + float b = truncate_to_float(gen2()/10); + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::endl; + std::pair ib_full = ibeta_fraction1(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + std::pair ib_reg = ibeta_fraction1_regular(NTL::RR(a), NTL::RR(b), NTL::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + + +int main(int argc, char* argv[]) +{ + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2, arg3, arg4; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the incomplete beta functions:\n" + " beta(a, b, x) and ibeta(a, b, x)\n\n" + "This is not an interactive program be prepared for a long wait!!!\n\n"; + + arg1 = make_periodic_param(NTL::RR(-5), NTL::RR(6), 11); + arg2 = make_periodic_param(NTL::RR(-5), NTL::RR(6), 11); + arg3 = make_random_param(NTL::RR(0.0001), NTL::RR(1), 10); + arg4 = make_random_param(NTL::RR(0.0001), NTL::RR(1), 100 /*500*/); + + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3.type |= dummy_param; + arg4.type |= dummy_param; + + // comment out all but one of the following when running + // or this program will take forever to complete! + data.insert(beta_data_generator(), arg1, arg2, arg3); + data.insert(beta_data_generator_medium(), arg4); + data.insert(beta_data_generator_small(), arg4); + + test_data::const_iterator i, j; + i = data.begin(); + j = data.end(); + while(i != j) + { + NTL::RR v1 = beta((*i)[0], (*i)[1], (*i)[2]); + NTL::RR v2 = relative_error(v1, (*i)[3]); + std::string s = boost::lexical_cast((*i)[3]); + NTL::RR v3 = boost::lexical_cast(s); + NTL::RR v4 = relative_error(v3, (*i)[3]); + if(v2 > 1e-40) + { + std::cout << v2 << std::endl; + } + if(v4 > 1e-60) + { + std::cout << v4 << std::endl; + } + ++ i; + } + + std::cout << "Enter name of test data file [default=ibeta_data.ipp]"; + std::string line; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ibeta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "ibeta_data"); + + return 0; +} + diff --git a/tools/igamma_data.cpp b/tools/igamma_data.cpp new file mode 100644 index 000000000..09e314d00 --- /dev/null +++ b/tools/igamma_data.cpp @@ -0,0 +1,155 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include + +#include +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; + +// +// Force trunctation to float precision of input values: +// we must ensure that the input values are exactly representable +// in whatever type we are testing, or the output values will all +// be thrown off: +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(NTL::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +// +// Our generator takes two arguments, but the second is interpreted +// as an instruction not a value, the second argument won't be placed +// in the output matrix by class test_data, so we add our algorithmically +// derived second argument to the output. +// +struct igamma_data_generator +{ + std::tr1::tuple operator()(NTL::RR a, NTL::RR x) + { + // very naively calculate spots: + NTL::RR z; + switch((int)real_cast(x)) + { + case 1: + z = truncate_to_float((std::min)(NTL::RR(1), a/100)); + break; + case 2: + z = truncate_to_float(a / 2); + break; + case 3: + z = truncate_to_float((std::max)(0.9*a, a - 2)); + break; + case 4: + z = a; + break; + case 5: + z = truncate_to_float((std::min)(1.1*a, a + 2)); + break; + case 6: + z = truncate_to_float(a * 2); + break; + case 7: + z = truncate_to_float((std::max)(NTL::RR(100), a*100)); + break; + default: + BOOST_ASSERT(0 == "Can't get here!!"); + } + + //NTL::RR g = boost::math::tgamma(a); + NTL::RR lg = boost::math::tgamma_lower(a, z); + NTL::RR ug = boost::math::tgamma(a, z); + NTL::RR rlg = boost::math::gamma_P(a, z); + NTL::RR rug = boost::math::gamma_Q(a, z); + + return std::tr1::make_tuple(z, ug, rug, lg, rlg); + } +}; + +struct gamma_inverse_generator +{ + std::tr1::tuple operator()(const NTL::RR a, const NTL::RR p) + { + NTL::RR x1 = boost::math::gamma_P_inv(a, p); + NTL::RR x2 = boost::math::gamma_Q_inv(a, p); + std::cout << "Inverse for " << a << " " << p << std::endl; + return std::tr1::make_tuple(x1, x2); + } +}; + + +int main(int argc, char const* argv[]) +{ + NTL::RR::SetPrecision(1000); + NTL::RR::SetOutputPrecision(100); + + bool cont; + std::string line; + + parameter_info arg1, arg2; + test_data data; + + if((argc >= 2) && (std::strcmp(argv[1], "-inverse") == 0)) + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete gamma function:\n" + " gamma_P_inv(a, p) and gamma_Q_inv(a, q)\n\n"; + do{ + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "p"); + data.insert(gamma_inverse_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else + { + arg2 = make_periodic_param(NTL::RR(1), NTL::RR(8), 7); + arg2.type |= boost::math::tools::dummy_param; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the incomplete gamma function:\n" + " gamma(a, z)\n\n"; + + do{ + get_user_parameter_info(arg1, "a"); + data.insert(igamma_data_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + std::cout << "Enter name of test data file [default=igamma_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "igamma_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "igamma_data"); + + return 0; +} + diff --git a/tools/lanczos_generator.cpp b/tools/lanczos_generator.cpp new file mode 100644 index 000000000..56c1c855b --- /dev/null +++ b/tools/lanczos_generator.cpp @@ -0,0 +1,4351 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +// +// this is a sort of recursive include, since this file +// is used to create the contents of gamma.hpp. +// However, we will be using the "generic" version of gamma +// which doesn't use a lanczos approximation. +// +#include + +#include +#include +#include + +using boost::numeric::ublas::matrix; +using boost::numeric::ublas::vector; + +struct lanczos_spot_data +{ + int N; + double g; + double err; +}; + +lanczos_spot_data sweet_spots[] = { +// a few selected spots from Pugh's thesis: +3, 2.03209, 6.4e-7, +4, 3.655180, 8.5e-8, +9, 8.406094, 6.9e-15, +10, 9.656578, 2.1e-16, +11, 10.900511, 6.1e-18, +12, 12.066012, 1.1e-19, +13, 13.144565, 5.2e-21, +14, 13.726821, 4.0e-22, +15, 14.977863, 1.2e-23, +20, 20.298892, 7.8e-31, +21, 21.508926, 2.1e-32, +22, 22.618910, 1.8e-34, +23, 23.118012, 5.2e-35, + +// some more we've found, these are all the points where the first +// negleted term from the Lanczos series changes sign, there is one +// point just above that point, and one just below: + +3, 0.58613894134759903, 0.00036580426080686315, +3, 0.58613894879817963, 0.00036580416320668711, +3, 1.249951496720314, 4.2305536798953929e-005, +3, 1.2499515041708946, 4.2305534015367749e-005, +3, 1.8384982049465179, 1.164778449938813e-005, +3, 1.8384982123970985, 1.1647786529958048e-005, +3, 2.3276706635951996, 5.3732582230468998e-006, +3, 2.3276706710457802, 5.3732600294516588e-006, +3, 2.6038404405117035, 7.4060771171390379e-007, +3, 2.6038404479622841, 7.406058543248471e-007, +4, 0.62704569846391678, 0.00012800533557793425, +4, 0.62704570591449738, 0.00012800531551557324, +4, 1.3344274088740349, 9.0166140418663663e-006, +4, 1.3344274163246155, 9.0166116338646694e-006, +4, 1.9880444705486298, 1.4955557778725745e-006, +4, 1.9880444779992104, 1.4955557079994109e-006, +4, 2.5835030898451805, 4.5877676859600527e-007, +4, 2.5835030972957611, 4.5877667876325684e-007, +4, 3.1086832508444786, 2.1441725587707906e-007, +4, 3.1086832582950592, 2.1441723688775765e-007, +4, 3.5476611852645874, 1.0813760990515544e-007, +4, 3.547661192715168, 1.0813766234070327e-007, +4, 3.6492579728364944, 1.047642251124206e-007, +4, 3.649257980287075, 1.0476415796074403e-007, +5, 0.65435918420553207, 5.5274429026631278e-005, +5, 0.65435919165611267, 5.527441185903385e-005, +5, 1.389187254011631, 2.6307833416554052e-006, +5, 1.3891872614622116, 2.6307826816540255e-006, +5, 2.0801975876092911, 2.9014694764756466e-007, +5, 2.0801975950598717, 2.9014696142909125e-007, +5, 2.7272208333015442, 5.9176544536206063e-008, +5, 2.7272208407521248, 5.9176528019009275e-008, +5, 3.3269794136285782, 1.9379184595472579e-008, +5, 3.3269794210791588, 1.9379189057179851e-008, +5, 3.8722873777151108, 9.0123694745032944e-009, +5, 3.8722873851656914, 9.0123661007867372e-009, +5, 4.3516943231225014, 4.9816097608842378e-009, +5, 4.351694330573082, 4.9816087871575021e-009, +6, 0.67425807565450668, 2.7371903086326379e-005, +6, 0.67425808310508728, 2.7371893712684938e-005, +6, 1.4284561350941658, 9.4138428977455083e-007, +6, 1.4284561425447464, 9.4138417896969612e-007, +6, 2.1447814926505089, 7.4109049995920916e-008, +6, 2.1447815001010895, 7.4109044064461768e-008, +6, 2.8244904428720474, 1.0708333024058614e-008, +6, 2.824490450322628, 1.0708333661155651e-008, +6, 3.4669468700885773, 2.4940116660002578e-009, +6, 3.4669468775391579, 2.4940118560592828e-009, +6, 4.0696377158164978, 8.5267622527507512e-010, +6, 4.0696377232670784, 8.5267624048985822e-010, +6, 4.6278460025787354, 3.9323405231792876e-010, +6, 4.6278460100293159, 3.9323393550021793e-010, +6, 5.1345816180109978, 2.1943462473419721e-010, +6, 5.1345816254615784, 2.1943458105218707e-010, +6, 5.5917193591594696, 1.242916896389048e-010, +6, 5.5917193666100502, 1.2429175679293363e-010, +7, 0.68959406018257141, 1.4934177419305624e-005, +7, 0.68959406763315201, 1.4934171696393048e-005, +7, 1.4584139510989189, 3.8891482410396904e-007, +7, 1.4584139585494995, 3.8891475436265016e-007, +7, 2.1934122517704964, 2.2977989848570384e-008, +7, 2.193412259221077, 2.2977982819591052e-008, +7, 2.8964230641722679, 2.4742239281355561e-009, +7, 2.8964230716228485, 2.4742234478616864e-009, +7, 3.5678729563951492, 4.2816138965285379e-010, +7, 3.5678729638457298, 4.2816128067858218e-010, +7, 4.2069913893938065, 1.0931003472338519e-010, +7, 4.2069913968443871, 1.0931004659777642e-010, +7, 4.8118086308240891, 3.8518213120095777e-011, +7, 4.8118086382746696, 3.8518212172324096e-011, +7, 5.3790060579776764, 1.7613602961109952e-011, +7, 5.379006065428257, 1.7613603275796266e-011, +7, 5.9039439857006073, 9.7098281187573355e-012, +7, 5.9039439931511879, 9.7098271522398387e-012, +7, 6.3838147521018982, 5.7878801570305524e-012, +7, 6.3838147595524788, 5.7878771973838702e-012, +7, 6.7833591029047966, 2.829092368392454e-012, +7, 6.7833591103553772, 2.8290901754246269e-012, +8, 0.70188850909471512, 8.7610752024780369e-006, +8, 0.70188851654529572, 8.7610704217796693e-006, +8, 1.4822541922330856, 1.7881701068328982e-007, +8, 1.4822541996836662, 1.7881694190606741e-007, +8, 2.231784276664257, 8.2272386690167009e-009, +8, 2.2317842841148376, 8.2272367814664617e-009, +8, 2.9525627419352531, 6.8591536474980462e-010, +8, 2.9525627493858337, 6.8591515664364383e-010, +8, 3.6455128863453865, 9.1549235855884194e-011, +8, 3.6455128937959671, 9.1549217088498416e-011, +8, 4.3106606528162956, 1.8015466799942756e-011, +8, 4.3106606602668762, 1.8015467001952356e-011, +8, 4.9472371935844421, 4.918149362493385e-012, +8, 4.9472372010350227, 4.9181495178774722e-012, +8, 5.553666315972805, 1.7720036810370511e-012, +8, 5.5536663234233856, 1.7720035779693502e-012, +8, 6.1274929493665695, 8.0403304333886761e-013, +8, 6.1274929568171501, 8.0403274526493467e-013, +8, 6.6654317826032639, 4.3551924163930176e-013, +8, 6.6654317900538445, 4.3551918580885179e-013, +8, 7.1646013185381889, 2.6142608803934048e-013, +8, 7.1646013259887695, 2.6142610866873453e-013, +8, 7.6312204822897911, 1.5767278184484503e-013, +8, 7.6312204897403717, 1.576728292094547e-013, +8, 7.8779716044664383, 6.0579161144149095e-014, +8, 7.8779716119170189, 6.0579145495868576e-014, +9, 0.7120363786816597, 5.4383680639900953e-006, +9, 0.7120363861322403, 5.4383647430909934e-006, +9, 1.5018193200230598, 8.9331552269793478e-008, +9, 1.5018193274736404, 8.9331529217486569e-008, +9, 2.2630845010280609, 3.2922940247769628e-009, +9, 2.2630845084786415, 3.2922936681951672e-009, +9, 2.9980199560523033, 2.1887825488935961e-010, +9, 2.9980199635028839, 2.1887818854849753e-010, +9, 3.7078000977635384, 2.3213138957857667e-011, +9, 3.707800105214119, 2.3213132917173771e-011, +9, 4.3928612694144249, 3.6226509291513855e-012, +9, 4.3928612768650055, 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29.468719460070133, 1.3798854862844653e-60, +48, 29.468719467520714, 1.3798850951568444e-60, +48, 30.124579109251499, 3.8565307240880389e-61, +48, 30.12457911670208, 3.8565299743759719e-61, +48, 30.775601848959923, 1.134342736876812e-61, +48, 30.775601856410503, 1.1343429076941469e-61, +48, 31.42175617814064, 3.5066060548219019e-62, +48, 31.421756185591221, 3.5066056252289294e-62, +48, 32.063007235527039, 1.1377415805787006e-62, +48, 32.063007242977619, 1.1377415420726448e-62, +48, 32.699316792190075, 3.8694437991207317e-63, +48, 32.699316799640656, 3.8694430678757391e-63, +48, 33.330643333494663, 1.3776788233453704e-63, +48, 33.330643340945244, 1.3776791661638305e-63, +48, 33.956942074000835, 5.1285952783952753e-64, +48, 33.956942081451416, 5.1285947602355625e-64, +48, 34.578165091574192, 1.9937127927134763e-64, +48, 34.578165099024773, 1.9937126593845751e-64, +48, 35.194261386990547, 8.0836531292607215e-65, +48, 35.194261394441128, 8.0836543290621667e-65, +48, 35.805177040398121, 3.4143034915050112e-65, +48, 35.805177047848701, 3.4143040521491458e-65, +48, 36.410855405032635, 1.5004193222265229e-65, +48, 36.410855412483215, 1.5004189889736636e-65, +48, 37.011237367987633, 6.8526723104762059e-66, +48, 37.011237375438213, 6.8526715926507402e-66, +}; + +// +// A bunch of helper functions used for calculating the coefficients: +// +NTL::RR factorial(unsigned n) +{ + static boost::array, 201> result; + static bool init = false; + if(!init) + { + unsigned k = 1; + NTL::RR fact = 1; + do{ + result[k-1][0] = fact; + fact *= k++; + } + while(k < result.size()); + init = true; + } + //static const int array_size = result.size(); + if(n < result.size()) + return result[n][0]; + + unsigned i = (unsigned)result.size()-1; + NTL::RR r = result[i][0]; + while(i < n) + r *= ++i; + return r; +} + +template +T binomial(int n, int k, T) +{ + T result; + if(k < 0) + { + result = 0; + } + else if(k > n) + { + result = 1; + } + else + { + result = factorial(n); + result /= factorial(k); + result /= factorial(n-k); + } + return result; +} +// +// Functions for creating the matrices that generate the coefficents. +// See http://my.fit.edu/~gabdo/gamma.txt and http://www.rskey.org/gamma.htm +// +template +matrix make_B(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if(i == 0) + result(i, j) = 1; + else if(j >= i) + { + T r = binomial(i+j-1, j-i, T()); + if((j-i) %2) + r = -r; + result(i, j) = r; + } + else + result(i, j) = 0; + } + } + return result; +} + +template +matrix make_C(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if((i==0) && (j == 0)) + result(i, j) = 0.5; + else if(j > i) + result(i, j) = 0; + else + { + T r = 0; + for(unsigned k = 0; k <= i; ++k) + { + r += binomial(2*i, 2*k, T()) * binomial(k, k+j-i, T()); + } + if((i-j)%2) + r = -r; + result(i, j) = r; + } + } + } + return result; +} + +template +matrix make_D(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if(i != j) + result(i, j) = 0; + else if(i == 0) + result(i, j) = 1; + else if(i == 1) + result(i, j) = -1; + else + { + T r = result(i-1, i-1) * (2 * (2*i - 1)); + r /= (i-1); + result(i, j) = r; + } + } + } + return result; +} + +template +matrix make_F(unsigned n, T g) +{ + using namespace std; + matrix result(n, 1); + for(unsigned i = 0; i < n; ++i) + { + T r = factorial(2 * i); + r /= factorial(i); + r *= exp(double(i) + g + 0.5); + r /= ldexp(1.0, (2*i)-1); + r /= pow(g + i + 0.5, i); + r /= sqrt(g + i + 0.5); + result(i, 0) = r; + } + return result; +} + +template +struct lanczos_info +{ + int n; // number of coefficients + T r; // the arbitrary parameter + std::vector c; // the coefficients + T err; // error found +}; +// +// Create the coefficients, caching previous matrix values as appropriate: +// +template +lanczos_info generate_lanczos(unsigned n, T g) +{ + static unsigned last_n = 0; + static matrix last_prefix; + + matrix B, C, D; + matrix F, P, E, T1, T2; + + if(n == last_n) + { + F = make_F(n, g); + P = prod(last_prefix, F); + } + else + { + B = make_B(n, g); + C = make_C(n, g); + D = make_D(n, g); + F = make_F(n, g); + + T1 = prod(D, B); + T2 = prod(T1, C); + P = prod(T2, F); + + last_n = n; + last_prefix = T2; + } + + lanczos_info result; + result.n = n; + result.r = g; + for(unsigned i = 0; i < n; ++i) + result.c.push_back(P(i, 0)); + + return result; +} +// +// Returns the factorials and half factorials: +// +template +std::vector > const & get_test_data() +{ + // returns factorials and half factorials in the range [0, 100]: + static std::vector > data; + if(data.empty()) + { + T fact = 1; + int k = 1; + std::vector item; + do + { + data.push_back(item); + data.back().push_back(k-1); + data.back().push_back(fact); + data.back().push_back(log(fact)); + fact = fact * T(k++); + }while(k < 100); + + fact = 0.5; + T srpi = sqrt(boost::math::constants::pi()); + T mul = 1.5; + do{ + data.push_back(item); + data.back().push_back(mul-1); + data.back().push_back(fact*srpi); + data.back().push_back(log(fact*srpi)); + fact *= mul; + mul += 1; + }while(mul < 100); + } + + return data; +} + +struct sort_on_0 +{ + template + bool operator ()(const T& a, const T& b) + { + return ((a[0]) < (b[0])); + } +}; +// +// Get test data for points near 0, 1, 2 etc. +template +std::vector > const & get_test_data_near_x(T x) +{ + // + // this *must* be called with a *very* high precision + // type T in order to stand any chance of long double + // precision results: + // + using namespace std; + using namespace boost::math; + static std::vector > data; + static T last_x(-10000); + if(last_x != x) + data.clear(); + if(!data.size()) + { + last_x = x; + std::vector item; + for(int i = -4; i > -20; --i) + { + double val = ldexp(2.0, i); + data.push_back(item); + data.back().push_back(val+x); + data.back().push_back(tgamma(T(val)+x)); + data.back().push_back(lgamma(T(val)+x)); + data.push_back(item); + data.back().push_back(-val+x); + data.back().push_back(tgamma(T(-val)+x)); + data.back().push_back(lgamma(T(-val)+x)); + } + T v = x-0.5; + T interval = T(1)/8; + while(v <= x+0.5) + { + if((floor(v) != v) || (v > 0)) + { + data.push_back(item); + data.back().push_back(v); + data.back().push_back(tgamma(v)); + data.back().push_back(lgamma(v)); + } + v += interval; + } + // + // sort data: + // + std::sort(data.begin(), data.end(), sort_on_0()); + } + return data; +} + +template +std::vector > const & get_test_data_near_1() +{ + return get_test_data_near_x(T(1)); +} + +template +std::vector > const & get_test_data_near_2() +{ + return get_test_data_near_x(T(2)); +} + +// +// Converts Lanczos approximation into rational form via +// polynomial arithmetic: +// +template +struct lanczos_rational +{ + std::vector num, denom; + int N; + T g; + + T gamma(T z) + { + using namespace std; + BOOST_ASSERT(num.size() == denom.size()); + BOOST_ASSERT(num.size() == N); + T zgh = z + g - T(0.5); + T prefix = pow(zgh, T(z - 0.5)) / exp(zgh); + T s1, s2; + if(z < 1) + { + s1 = num[N-1]; + s2 = denom[N-1]; + for(unsigned i = N-2; i >= 0; --i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += num[i]; + } + s1 /= s2; + return prefix * s1; + } + else + { + z = 1/z; + s1 = num[0]; + s2 = denom[0]; + for(int i = 1; i < N; ++i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += denom[i]; + } + s1 /= s2; + return prefix * s1; + } + } + T factorial(T z) + { + return this->gamma(z+1); + } + template + lanczos_rational(const lanczos_info& info) + { + U l_denom_coef[2] = { 0, 1 }; + boost::math::tools::polynomial l_denom(l_denom_coef, 1), l_num(info.c[1]); + for(unsigned i = 2; i < info.c.size(); ++i) + { + l_denom_coef[0] = i - 1; + boost::math::tools::polynomial bot(l_denom_coef, 1), top(info.c[i]); + l_num *= bot; + top *= l_denom; + l_denom *= bot; + l_num += top; + } + l_num += l_denom * info.c[0]; + for(unsigned i = 0; i < info.c.size(); ++i) + { + num.push_back(boost::math::tools::real_cast(l_num[i])); + denom.push_back(boost::math::tools::real_cast(l_denom[i])); + } + //std::cout << l_num << std::endl; + //std::cout << l_denom << std::endl; + BOOST_ASSERT(num.size() == l_num.degree()+1); + BOOST_ASSERT(denom.size() == l_denom.degree()+1); + g = boost::math::tools::real_cast(info.r); + N = info.n; + /* + for(unsigned i = 0; i < num.size(); ++i) + std::cout << num[i] << " "; + std::cout << std::endl; + for(unsigned i = 0; i < denom.size(); ++i) + std::cout << denom[i] << " "; + std::cout << std::endl; + */ + } +}; +// +// Code to test an approximation against the factorials and +// half factorials, returns the max error found: +// +template +T get_max_error(const lanczos_info& dat, R) +{ + T max_error = 0; + std::vector > const & tests = get_test_data(); + lanczos_rational rational(dat); + + for(unsigned i = 0; i < tests.size(); ++i) + { + R input = boost::math::tools::real_cast(tests[i][0]); + if(std::numeric_limits::is_specialized && (boost::math::tools::real_cast(tests[i][1]) > (std::numeric_limits::max)())) + continue; + + R gamr = rational.factorial(input); + if(std::numeric_limits::is_specialized && (gamr > (std::numeric_limits::max)())) + continue; + T gam = gamr; + T exp = tests[i][1]; + T err = boost::math::tools::relative_error(gam, exp); + if(err > max_error) + max_error = err; + } + + return max_error; +} +// +// This is what prints the "best" approximation out as code: +// +template +void print_code(const lanczos_info& l, const char* name) +{ + using namespace std; + lanczos_info l2(l); + T factor = exp(l.r); + for(unsigned k = 0; k < l2.c.size(); ++k) + l2.c[k] /= factor; + + lanczos_rational rat(l); + T max_term = 0; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + if(rat.denom[i] > max_term) + max_term = rat.denom[i]; + } + const char* denom_type; + const char* cast_type; + const char* suffix_type; + if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint16_t"; + cast_type = "static_cast"; + suffix_type = "u"; + } + else if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint32_t"; + cast_type = "static_cast"; + suffix_type = "u"; + } +#ifdef BOOST_HAS_LONG_LONG + else if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint64_t"; + cast_type = ""; + suffix_type = "uLL"; + } +#endif + else + { + denom_type = "T"; + cast_type = "static_cast"; + suffix_type = "L"; + } + + std::cout << + "//\n" + "// Lanczos Coefficients for N=" << l.n << " G=" << l.r << "\n" + "// Max experimental error (with "; + if(std::strlen(name) == 0) + std::cout << "arbitary"; + else + std::cout << name; + std::cout << " precision arithmetic) " << l.err << + "\n// Generated with compiler: " << BOOST_COMPILER << " on " << BOOST_PLATFORM << " at " << __DATE__ << "\n" + "//\n" + "struct lanczos" << l.n << name << "\n" + "{\n" + " template \n" + " static T lanczos_sum(const T& z)\n" + " {\n" + " static const T num[" << rat.num.size() << "] = {\n"; + + for(unsigned i = 0; i < rat.num.size(); ++i) + { + std::cout << " static_cast(" << rat.num[i] << "L),\n"; + } + std::cout << + " };\n" + " static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n"; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << "),\n"; + } + std::cout << + " };\n" + " return boost::math::tools::evaluate_rational(num, denom, z, " << rat.denom.size() << ");\n" + " }\n\n" + " template \n" + " static T lanczos_sum_expG_scaled(const T& z)\n" + " {\n" + " static const T num[" << rat.num.size() << "] = {\n"; + + for(unsigned i = 0; i < rat.num.size(); ++i) + { + std::cout << " static_cast(" << (rat.num[i]/factor) << "L),\n"; + } + std::cout << + " };\n" + " static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n"; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << "),\n"; + } + std::cout << + " };\n" + " return boost::math::tools::evaluate_rational(num, denom, z, " << rat.denom.size() << ");\n" + " }\n\n"; + + std::cout << + "\n template\n" + " static T lanczos_sum_near_1(const T& dz)\n" + " {\n" + " static const T d[" << l2.n-1 << "] = {\n"; + + factor = sqrt((l.r + 0.5)/boost::math::constants::e()) / exp(l.r); + + for(int i = 1; i < l.n; ++i) + { + std::cout << " static_cast(" << l.c[i]*factor << "L),\n"; + } + std::cout << + " };\n" + " T result = 0;\n" + " for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n" + " {\n" + " result += (-d[k-1]*dz)/(k*dz + k*k);\n" + " }\n" + " return result;\n" + " }\n"; + + std::cout << + "\n template\n" + " static T lanczos_sum_near_2(const T& z)\n" + " {\n" + " static const T d[" << l2.n-1 << "] = {\n"; + + factor = pow(boost::math::constants::e()/(l.r+1.5), T(1.5)) * exp(l.r); + //pow((l.r + 1.5)/boost::math::constants::e(), 1.5) / exp(l.r); + + for(int i = 1; i < l.n; ++i) + { + std::cout << " static_cast(" << l.c[i]/factor << "L),\n"; + } + std::cout << + " };\n" + " T result = 0;\n" + " T dz = z - 2;\n" + " for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n" + " {\n" + " result += (-d[k-1]*dz)/(z + k*z + k*k - 1);\n" + " }\n" + " return result;\n" + " }\n\n" + " static double g(){ return " << l.r << "; }\n" + "};\n\n"; +} +// +// Print out the test values: +// +void print_test_values(const std::vector >& v, const char* name, int offset = 1) +{ + std::cout << "#define SC_(x) static_cast(BOOST_JOIN(x, L))\n"; + std::cout << + " static const boost::array, " << v.size() << "> " << name << " = {\n"; + for(unsigned i = 0; i < v.size(); ++i) + { + std::cout << " SC_(" << (v[i][0] + offset) << "), SC_(" << v[i][1] << "), SC_(" << v[i][2] << "),\n"; + } + std::cout << " };\n#undef SC_\n\n"; +} +// +// Get the error for a specific approximation, and print out it's code: +// +void calculate_lanczos_spot(int n, NTL::RR r, const char* suffix = "") +{ + lanczos_info info = generate_lanczos(n, r); + // note error is calculated at high precision: + info.err = get_max_error(info, r); + print_code(info, suffix); +} +// +// Scan the sweet-spots for the best approximation: +// +template +void find_best_lanczos(const char* name, T eps, int max_scan = 100) +{ + using namespace std; + + lanczos_info best; + best.err = 100; // best case had better be better than this! + for(int i = 0; i < sizeof(sweet_spots)/sizeof(sweet_spots[0]); ++i) + { + if((sweet_spots[i].err < eps*10) && (sweet_spots[i].N < max_scan)) + { + lanczos_info info = generate_lanczos(sweet_spots[i].N, NTL::RR(sweet_spots[i].g)); + NTL::RR err = get_max_error(info, eps); + if(err/eps < 1000) + { + std::cout << sweet_spots[i].N << " " << sweet_spots[i].g << " " << err/eps << std::endl; + } + if(err < best.err) + { + best = info; + best.err = err; + } + } + } + std::cout << std::endl; + + print_code(best, name); +} +int main(int argc, char*argv[]) +{ + bool test_double(false), test_long(false), test_float(false), test_quad(false), spots(false), test_data(false); + + if(argc < 2) + { + std::cout << + "Useage:\n" + " -float test type float for the best approximation\n" + " -double test type double for the best approximation\n" + " -long-double test type long double for the best approximation\n" + " -spots print out the best cases found in previous runs\n" + " -data print out the test data\n" << std::flush; + return 0; + } + + for(int i = 1; i < argc; ++i) + { + if(std::strcmp(argv[i], "-float") == 0) + test_float = true; + else if(std::strcmp(argv[i], "-double") == 0) + test_double = true; + else if(std::strcmp(argv[i], "-long-double") == 0) + test_long = true; + else if(std::strcmp(argv[i], "-quad-float") == 0) + test_quad = true; + else if(std::strcmp(argv[i], "-spots") == 0) + spots = true; + else if(std::strcmp(argv[i], "-data") == 0) + test_data = true; + } + + NTL::RR::SetPrecision(1000); + boost::math::tools::setprecision(std::cout, NTL::RR(0), 40); + + if(spots) + { + // these are optimal N and R from Pugh: + calculate_lanczos_spot(31, 32.080670); + calculate_lanczos_spot(6, 5.581); + calculate_lanczos_spot(11, 10.900511); + calculate_lanczos_spot(13, 13.144565); + calculate_lanczos_spot(22, 22.61891); + // these are the best cases we've already determined + // from previous runs: + calculate_lanczos_spot(6, 1.428456135094165802001953125); + calculate_lanczos_spot(13, 6.024680040776729583740234375); + calculate_lanczos_spot(17, 12.2252227365970611572265625); + boost::math::tools::setprecision(std::cout, NTL::RR(0), 90); + calculate_lanczos_spot(24, 20.3209821879863739013671875); + } + if(test_float) + { + find_best_lanczos("float", std::numeric_limits::epsilon()); + } + if(test_double) + { + find_best_lanczos("double", std::numeric_limits::epsilon()); + } + + if(test_long) + { + find_best_lanczos("long_double", std::numeric_limits::epsilon()); + } + if(test_quad) + { + find_best_lanczos("quad_float", pow(NTL::quad_float(2), NTL::quad_float(-105))); + } + if(test_data) + { + std::cout << "Test Data follows:\n\n"; + + NTL::RR::SetPrecision(1000); + + std::vector > const & tests = get_test_data(); + print_test_values(tests, "factorials"); + print_test_values(get_test_data_near_1(), "near_1", 0); + print_test_values(get_test_data_near_2(), "near_2", 0); + print_test_values(get_test_data_near_x(NTL::RR(0)), "near_0", 0); + print_test_values(get_test_data_near_x(NTL::RR(-10)), "near_m10", 0); + print_test_values(get_test_data_near_x(NTL::RR(-55)), "near_m55", 0); + } + return 0; +} + + + diff --git a/tools/ntl.diff b/tools/ntl.diff new file mode 100644 index 000000000..4c6baff39 --- /dev/null +++ b/tools/ntl.diff @@ -0,0 +1,135 @@ +diff -b -r -u WinNTL-5_4/include/NTL/RR.h WinNTL-5_4_patch/include/NTL/RR.h +--- WinNTL-5_4/include/NTL/RR.h 2005-03-25 07:59:12.000000000 +0000 ++++ WinNTL-5_4_patch/include/NTL/RR.h 2006-03-23 17:06:36.742860800 +0000 +@@ -17,6 +17,13 @@ + + RR() { e = 0; } + ++inline RR(long double t) ++: e(0) ++{ ++ *this = t; ++} ++RR(quad_float const& t); ++ + inline RR(INIT_VAL_TYPE, const ZZ& a); + inline RR(INIT_VAL_TYPE, int a); + inline RR(INIT_VAL_TYPE, long a); +@@ -30,7 +37,7 @@ + inline RR(INIT_VAL_TYPE, const RR& a); + + +-inline RR& operator=(double a); ++inline RR& operator=(long double a); + + RR(RR& z, INIT_TRANS_TYPE) : x(z.x, INIT_TRANS), e(z.e) { } + +@@ -346,8 +353,9 @@ + void conv(RR& z, unsigned long a); + inline void conv(RR& z, unsigned int a) { conv(z, (unsigned long)(a)); } + void conv(RR& z, const char *s); +-void conv(RR& z, double a); +-inline void conv(RR& z, float a) { conv(z, double(a)); } ++void conv(RR& z, long double a); ++inline void conv(RR& z, float a) { conv(z, (long double)(a)); } ++inline void conv(RR& z, double a) { conv(z, (long double)(a)); } + void conv(RR& z, const xdouble& a); + void conv(RR& z, const quad_float& a); + +@@ -381,7 +389,7 @@ + inline RR to_RR(const quad_float& a) { return RR(INIT_VAL, a); } + inline RR to_RR(const char *a) { return RR(INIT_VAL, a); } + +-inline RR& RR::operator=(double a) { conv(*this, a); return *this; } ++inline RR& RR::operator=(long double a) { conv(*this, a); return *this; } + + void conv(ZZ& z, const RR& a); + void conv(long& z, const RR& a); +@@ -488,6 +496,11 @@ + inline RR cos(const RR& a) + { RR z; cos(z, a); NTL_OPT_RETURN(RR, z); } + ++inline RR::RR(quad_float const& t) ++: e(0) ++{ ++ conv(*this, t); ++} + + + +diff -b -r -u WinNTL-5_4/include/NTL/quad_float.h WinNTL-5_4_patch/include/NTL/quad_float.h +--- WinNTL-5_4/include/NTL/quad_float.h 2005-03-25 07:59:14.000000000 +0000 ++++ WinNTL-5_4_patch/include/NTL/quad_float.h 2006-03-23 12:25:57.453315200 +0000 +@@ -67,7 +67,7 @@ + static void SetOutputPrecision(long p); + static long OutputPrecision() { return oprec; } + +- quad_float(double x, double y) : hi(x), lo(y) { } // internal use only ++ quad_float(double x, double y = 0) : hi(x), lo(y) { } // internal use only + + ~quad_float() {} + +diff -b -r -u WinNTL-5_4/src/RR.cpp WinNTL-5_4_patch/src/RR.cpp +--- WinNTL-5_4/src/RR.cpp 2005-03-25 07:59:08.000000000 +0000 ++++ WinNTL-5_4_patch/src/RR.cpp 2006-03-25 13:31:14.338248000 +0000 +@@ -570,7 +570,7 @@ + xcopy(z, t); + } + +-void RoundPrec(RR& x, const RR& a, const RR& b, long p) ++void RoundPrec(RR& x, const RR& a, long p) + { + if (p < 1 || NTL_OVERFLOW(p, 1, 0)) + Error("RoundPrec: bad precsion"); +@@ -657,7 +657,7 @@ + RR::prec = old_p; + } + +- ++/* + void conv(RR& z, double a) + { + if (a == 0) { +@@ -687,6 +687,41 @@ + + xcopy(z, t); + } ++*/ ++void conv(RR& z, long double a) ++{ ++ if (a == 0) { ++ clear(z); ++ return; ++ } ++ ++ if (a == 1) { ++ set(z); ++ return; ++ } ++ ++ //if (!IsFinite(&a)) ++ //Error("RR: conversion of a non-finite double"); ++ ++ int e; ++ long double f, term; ++ RR t; ++ clear(z); ++ ++ f = frexp(a, &e); ++ ++ while(f) ++ { ++ // extract 30 bits from f: ++ f = ldexp(f, 30); ++ term = floor(f); ++ e -= 30; ++ conv(t.x, (int)term); ++ t.e = e; ++ z += t; ++ f -= term; ++ } ++} + + void ConvPrec(RR& x, double a, long p) + { + diff --git a/tools/ntl_rr_lanczos.hpp b/tools/ntl_rr_lanczos.hpp new file mode 100644 index 000000000..65824760a --- /dev/null +++ b/tools/ntl_rr_lanczos.hpp @@ -0,0 +1,903 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_LARGE_LANCZOS_HPP +#define BOOST_LARGE_LANCZOS_HPP + +#include +#include +#include + +struct lanczos13UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[13] = { + boost::lexical_cast("44012138428004.60895436261759919070125699"), + boost::lexical_cast("41590453358593.20051581730723108131357995"), + boost::lexical_cast("18013842787117.99677796276038389462742949"), + boost::lexical_cast("4728736263475.388896889723995205703970787"), + boost::lexical_cast("837910083628.4046470415724300225777912264"), + boost::lexical_cast("105583707273.4299344907359855510105321192"), + boost::lexical_cast("9701363618.494999493386608345339104922694"), + boost::lexical_cast("654914397.5482052641016767125048538245644"), + boost::lexical_cast("32238322.94213356530668889463945849409184"), + boost::lexical_cast("1128514.219497091438040721811544858643121"), + boost::lexical_cast("26665.79378459858944762533958798805525125"), + boost::lexical_cast("381.8801248632926870394389468349331394196"), + boost::lexical_cast("2.506628274631000502415763426076722427007"), + }; + static const boost::uint32_t denom[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 13); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[13] = { + boost::lexical_cast("86091529.53418537217994842267760536134841"), + boost::lexical_cast("81354505.17858011242874285785316135398567"), + boost::lexical_cast("35236626.38815461910817650960734605416521"), + boost::lexical_cast("9249814.988024471294683815872977672237195"), + boost::lexical_cast("1639024.216687146960253839656643518985826"), + boost::lexical_cast("206530.8157641225032631778026076868855623"), + boost::lexical_cast("18976.70193530288915698282139308582105936"), + boost::lexical_cast("1281.068909912559479885759622791374106059"), + boost::lexical_cast("63.06093343420234536146194868906771599354"), + boost::lexical_cast("2.207470909792527638222674678171050209691"), + boost::lexical_cast("0.05216058694613505427476207805814960742102"), + boost::lexical_cast("0.0007469903808915448316510079585999893674101"), + boost::lexical_cast("0.4903180573459871862552197089738373164184e-5"), + }; + static const boost::uint32_t denom[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 13); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[12] = { + boost::lexical_cast("4.832115561461656947793029596285626840312"), + boost::lexical_cast("-19.86441536140337740383120735104359034688"), + boost::lexical_cast("33.9927422807443239927197864963170585331"), + boost::lexical_cast("-31.41520692249765980987427413991250886138"), + boost::lexical_cast("17.0270866009599345679868972409543597821"), + boost::lexical_cast("-5.5077216950865501362506920516723682167"), + boost::lexical_cast("1.037811741948214855286817963800439373362"), + boost::lexical_cast("-0.106640468537356182313660880481398642811"), + boost::lexical_cast("0.005276450526660653288757565778182586742831"), + boost::lexical_cast("-0.0001000935625597121545867453746252064770029"), + boost::lexical_cast("0.462590910138598083940803704521211569234e-6"), + boost::lexical_cast("-0.1735307814426389420248044907765671743012e-9"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& z) + { + static const T d[12] = { + boost::lexical_cast("26.96979819614830698367887026728396466395"), + boost::lexical_cast("-110.8705424709385114023884328797900204863"), + boost::lexical_cast("189.7258846119231466417015694690434770085"), + boost::lexical_cast("-175.3397202971107486383321670769397356553"), + boost::lexical_cast("95.03437648691551457087250340903980824948"), + boost::lexical_cast("-30.7406022781665264273675797983497141978"), + boost::lexical_cast("5.792405601630517993355102578874590410552"), + boost::lexical_cast("-0.5951993240669148697377539518639997795831"), + boost::lexical_cast("0.02944979359164017509944724739946255067671"), + boost::lexical_cast("-0.0005586586555377030921194246330399163602684"), + boost::lexical_cast("0.2581888478270733025288922038673392636029e-5"), + boost::lexical_cast("-0.9685385411006641478305219367315965391289e-9"), + }; + T result = 0; + T dz = z - 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 13.1445650000000000545696821063756942749; } +}; + + +// +// Lanczos Coefficients for N=22 G=22.61891 +// Max experimental error (with arbitary precision arithmetic) 2.9524e-38 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006 +// +struct lanczos22UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[22] = { + boost::lexical_cast("46198410803245094237463011094.12173081986"), + boost::lexical_cast("43735859291852324413622037436.321513777"), + boost::lexical_cast("19716607234435171720534556386.97481377748"), + boost::lexical_cast("5629401471315018442177955161.245623932129"), + boost::lexical_cast("1142024910634417138386281569.245580222392"), + boost::lexical_cast("175048529315951173131586747.695329230778"), + boost::lexical_cast("21044290245653709191654675.41581372963167"), + boost::lexical_cast("2033001410561031998451380.335553678782601"), + boost::lexical_cast("160394318862140953773928.8736211601848891"), + boost::lexical_cast("10444944438396359705707.48957290388740896"), + boost::lexical_cast("565075825801617290121.1466393747967538948"), + boost::lexical_cast("25475874292116227538.99448534450411942597"), + boost::lexical_cast("957135055846602154.6720835535232270205725"), + boost::lexical_cast("29874506304047462.23662392445173880821515"), + boost::lexical_cast("769651310384737.2749087590725764959689181"), + boost::lexical_cast("16193289100889.15989633624378404096011797"), + boost::lexical_cast("273781151680.6807433264462376754578933261"), + boost::lexical_cast("3630485900.32917021712188739762161583295"), + boost::lexical_cast("36374352.05577334277856865691538582936484"), + boost::lexical_cast("258945.7742115532455441786924971194951043"), + boost::lexical_cast("1167.501919472435718934219997431551246996"), + boost::lexical_cast("2.50662827463100050241576528481104525333"), + }; + static const boost::uint64_t denom[22] = { + (0uLL), + (2432902008176640000uLL), + (8752948036761600000uLL), + (13803759753640704000uLL), + (12870931245150988800uLL), + (8037811822645051776uLL), + (3599979517947607200uLL), + (1206647803780373360uLL), + (311333643161390640uLL), + (63030812099294896uLL), + (10142299865511450uLL), + (1307535010540395uLL), + (135585182899530uLL), + (11310276995381uLL), + (756111184500uLL), + (40171771630uLL), + (1672280820uLL), + (53327946uLL), + (1256850uLL), + (20615uLL), + (210uLL), + (1uLL), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 22); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[22] = { + boost::lexical_cast("6939996264376682180.277485395074954356211"), + boost::lexical_cast("6570067992110214451.87201438870245659384"), + boost::lexical_cast("2961859037444440551.986724631496417064121"), + boost::lexical_cast("845657339772791245.3541226499766163431651"), + boost::lexical_cast("171556737035449095.2475716923888737881837"), + boost::lexical_cast("26296059072490867.7822441885603400926007"), + boost::lexical_cast("3161305619652108.433798300149816829198706"), + boost::lexical_cast("305400596026022.4774396904484542582526472"), + boost::lexical_cast("24094681058862.55120507202622377623528108"), + boost::lexical_cast("1569055604375.919477574824168939428328839"), + boost::lexical_cast("84886558909.02047889339710230696942513159"), + boost::lexical_cast("3827024985.166751989686050643579753162298"), + boost::lexical_cast("143782298.9273215199098728674282885500522"), + boost::lexical_cast("4487794.24541641841336786238909171265944"), + boost::lexical_cast("115618.2025760830513505888216285273541959"), + boost::lexical_cast("2432.580773108508276957461757328744780439"), + boost::lexical_cast("41.12782532742893597168530008461874360191"), + boost::lexical_cast("0.5453771709477689805460179187388702295792"), + boost::lexical_cast("0.005464211062612080347167337964166505282809"), + boost::lexical_cast("0.388992321263586767037090706042788910953e-4"), + boost::lexical_cast("0.1753839324538447655939518484052327068859e-6"), + boost::lexical_cast("0.3765495513732730583386223384116545391759e-9"), + }; + static const boost::uint64_t denom[22] = { + (0uLL), + (2432902008176640000uLL), + (8752948036761600000uLL), + (13803759753640704000uLL), + (12870931245150988800uLL), + (8037811822645051776uLL), + (3599979517947607200uLL), + (1206647803780373360uLL), + (311333643161390640uLL), + (63030812099294896uLL), + (10142299865511450uLL), + (1307535010540395uLL), + (135585182899530uLL), + (11310276995381uLL), + (756111184500uLL), + (40171771630uLL), + (1672280820uLL), + (53327946uLL), + (1256850uLL), + (20615uLL), + (210uLL), + (1uLL), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 22); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[21] = { + boost::lexical_cast("8.318998691953337183034781139546384476554"), + boost::lexical_cast("-63.15415991415959158214140353299240638675"), + boost::lexical_cast("217.3108224383632868591462242669081540163"), + boost::lexical_cast("-448.5134281386108366899784093610397354889"), + boost::lexical_cast("619.2903759363285456927248474593012711346"), + boost::lexical_cast("-604.1630177420625418522025080080444177046"), + boost::lexical_cast("428.8166750424646119935047118287362193314"), + boost::lexical_cast("-224.6988753721310913866347429589434550302"), + boost::lexical_cast("87.32181627555510833499451817622786940961"), + boost::lexical_cast("-25.07866854821128965662498003029199058098"), + boost::lexical_cast("5.264398125689025351448861011657789005392"), + boost::lexical_cast("-0.792518936256495243383586076579921559914"), + boost::lexical_cast("0.08317448364744713773350272460937904691566"), + boost::lexical_cast("-0.005845345166274053157781068150827567998882"), + boost::lexical_cast("0.0002599412126352082483326238522490030412391"), + boost::lexical_cast("-0.6748102079670763884917431338234783496303e-5"), + boost::lexical_cast("0.908824383434109002762325095643458603605e-7"), + boost::lexical_cast("-0.5299325929309389890892469299969669579725e-9"), + boost::lexical_cast("0.994306085859549890267983602248532869362e-12"), + boost::lexical_cast("-0.3499893692975262747371544905820891835298e-15"), + boost::lexical_cast("0.7260746353663365145454867069182884694961e-20"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& z) + { + static const T d[21] = { + boost::lexical_cast("75.39272007105208086018421070699575462226"), + boost::lexical_cast("-572.3481967049935412452681346759966390319"), + boost::lexical_cast("1969.426202741555335078065370698955484358"), + boost::lexical_cast("-4064.74968778032030891520063865996757519"), + boost::lexical_cast("5612.452614138013929794736248384309574814"), + boost::lexical_cast("-5475.357667500026172903620177988213902339"), + boost::lexical_cast("3886.243614216111328329547926490398103492"), + boost::lexical_cast("-2036.382026072125407192448069428134470564"), + boost::lexical_cast("791.3727954936062108045551843636692287652"), + boost::lexical_cast("-227.2808432388436552794021219198885223122"), + boost::lexical_cast("47.70974355562144229897637024320739257284"), + boost::lexical_cast("-7.182373807798293545187073539819697141572"), + boost::lexical_cast("0.7537866989631514559601547530490976100468"), + boost::lexical_cast("-0.05297470142240154822658739758236594717787"), + boost::lexical_cast("0.00235577330936380542539812701472320434133"), + boost::lexical_cast("-0.6115613067659273118098229498679502138802e-4"), + boost::lexical_cast("0.8236417010170941915758315020695551724181e-6"), + boost::lexical_cast("-0.4802628430993048190311242611330072198089e-8"), + boost::lexical_cast("0.9011113376981524418952720279739624707342e-11"), + boost::lexical_cast("-0.3171854152689711198382455703658589996796e-14"), + boost::lexical_cast("0.6580207998808093935798753964580596673177e-19"), + }; + T result = 0; + T dz = z - 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 22.61890999999999962710717227309942245483; } +}; + +// +// Lanczos Coefficients for N=31 G=32.08067 +// Max experimental error (with arbitary precision arithmetic) 0.162e-52 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at May 9 2006 +// +struct lanczos31UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[31] = { + boost::lexical_cast("0.2579646553333513328235723061836959833277e46"), + boost::lexical_cast("0.2444796504337453845497419271639377138264e46"), + boost::lexical_cast("0.1119885499016017172212179730662673475329e46"), + boost::lexical_cast("0.3301983829072723658949204487793889113715e45"), + boost::lexical_cast("0.7041171040503851585152895336505379417066e44"), + boost::lexical_cast("0.1156687509001223855125097826246939403504e44"), + boost::lexical_cast("1522559363393940883866575697565974893306000"), + boost::lexical_cast("164914363507650839510801418717701057005700"), + boost::lexical_cast("14978522943127593263654178827041568394060"), + boost::lexical_cast("1156707153701375383907746879648168666774"), + boost::lexical_cast("76739431129980851159755403434593664173.2"), + boost::lexical_cast("4407916278928188620282281495575981079.306"), + boost::lexical_cast("220487883931812802092792125175269667.3004"), + boost::lexical_cast("9644828280794966468052381443992828.433924"), + boost::lexical_cast("369996467042247229310044531282837.6549068"), + boost::lexical_cast("12468380890717344610932904378961.13494291"), + boost::lexical_cast("369289245210898235894444657859.0529720075"), + boost::lexical_cast("9607992460262594951559461829.34885209022"), + boost::lexical_cast("219225935074853412540086410.981421315799"), + boost::lexical_cast("4374309943598658046326340.720767382079549"), + boost::lexical_cast("76008779092264509404014.10530947173485581"), + boost::lexical_cast("1143503533822162444712.335663112617754987"), + boost::lexical_cast("14779233719977576920.37884890049671578409"), + boost::lexical_cast("162409028440678302.9992838032166348069916"), + boost::lexical_cast("1496561553388385.733407609544964535634135"), + boost::lexical_cast("11347624460661.81008311053190661436107043"), + boost::lexical_cast("68944915931.32004991941950530448472223832"), + boost::lexical_cast("322701221.6391432296123937035480931903651"), + boost::lexical_cast("1092364.213992634267819050120261755371294"), + boost::lexical_cast("2380.151399852411512711176940867823024864"), + boost::lexical_cast("2.506628274631000502415765284811045253007"), + }; + static const T denom[31] = { + boost::lexical_cast("0"), + boost::lexical_cast("0.8841761993739701954543616e31"), + boost::lexical_cast("0.3502799997985980526649278464e32"), + boost::lexical_cast("0.622621928420356134910574592e32"), + boost::lexical_cast("66951000306085302338993639424000"), + boost::lexical_cast("49361465831621147825759587123200"), + boost::lexical_cast("26751280755793398822580822142976"), + boost::lexical_cast("11139316913434780466101123891200"), + boost::lexical_cast("3674201658710345201899117607040"), + boost::lexical_cast("981347603630155088295475765440"), + boost::lexical_cast("215760462268683520394805979744"), + boost::lexical_cast("39539238727270799376544542000"), + boost::lexical_cast("6097272817323042122728617800"), + boost::lexical_cast("796974693974455191377937300"), + boost::lexical_cast("88776380550648116217781890"), + boost::lexical_cast("8459574446076318147830625"), + boost::lexical_cast("691254538651580660999025"), + boost::lexical_cast("48487623689430693038025"), + boost::lexical_cast("2918939500751087661105"), + boost::lexical_cast("150566737512021319125"), + boost::lexical_cast("6634460278534540725"), + boost::lexical_cast("248526574856284725"), + boost::lexical_cast("7860403394108265"), + boost::lexical_cast("207912996295875"), + boost::lexical_cast("4539323721075"), + boost::lexical_cast("80328850875"), + boost::lexical_cast("1122686019"), + boost::lexical_cast("11921175"), + boost::lexical_cast("90335"), + boost::lexical_cast("435"), + boost::lexical_cast("1"), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 31); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[31] = { + boost::lexical_cast("30137154810677525966583148469478.52374216"), + boost::lexical_cast("28561746428637727032849890123131.36314653"), + boost::lexical_cast("13083250730789213354063781611435.74046294"), + boost::lexical_cast("3857598154697777600846539129354.783647"), + boost::lexical_cast("822596651552555685068015316144.0952185852"), + boost::lexical_cast("135131964033213842052904200372.039133532"), + boost::lexical_cast("17787555889683709693655685146.19771358863"), + boost::lexical_cast("1926639793777927562221423874.149673297196"), + boost::lexical_cast("174989113988888477076973808.6991839697774"), + boost::lexical_cast("13513425905835560387095425.01158383184045"), + boost::lexical_cast("896521313378762433091075.1446749283094845"), + boost::lexical_cast("51496223433749515758124.71524415105430686"), + boost::lexical_cast("2575886794780078381228.37205955912263407"), + boost::lexical_cast("112677328855422964200.4155776009524490958"), + boost::lexical_cast("4322545967487943330.625233358130724324796"), + boost::lexical_cast("145663957202380774.0362027607207590519723"), + boost::lexical_cast("4314283729473470.686566233465428332496534"), + boost::lexical_cast("112246988185485.8877916434026906290603878"), + boost::lexical_cast("2561143864972.040563435178307062626388193"), + boost::lexical_cast("51103611767.9626550674442537989885239605"), + boost::lexical_cast("887985348.0369447209508500133077232094491"), + boost::lexical_cast("13359172.3954672607019822025834072685839"), + boost::lexical_cast("172660.8841147568768783928167105965064459"), + boost::lexical_cast("1897.370795407433013556725714874693719617"), + boost::lexical_cast("17.48383210090980598861217644749573257178"), + boost::lexical_cast("0.1325705316732132940835251054350153028901"), + boost::lexical_cast("0.0008054605783673449641889260501816356090452"), + boost::lexical_cast("0.377001130700104515644336869896819162464e-5"), + boost::lexical_cast("0.1276172868883867038813825443204454996531e-7"), + boost::lexical_cast("0.2780651912081116274907381023821492811093e-10"), + boost::lexical_cast("0.2928410648650955854121639682890739211234e-13"), + }; + static const T denom[31] = { + boost::lexical_cast("0"), + boost::lexical_cast("0.8841761993739701954543616e31"), + boost::lexical_cast("0.3502799997985980526649278464e32"), + boost::lexical_cast("0.622621928420356134910574592e32"), + boost::lexical_cast("66951000306085302338993639424000"), + boost::lexical_cast("49361465831621147825759587123200"), + boost::lexical_cast("26751280755793398822580822142976"), + boost::lexical_cast("11139316913434780466101123891200"), + boost::lexical_cast("3674201658710345201899117607040"), + boost::lexical_cast("981347603630155088295475765440"), + boost::lexical_cast("215760462268683520394805979744"), + boost::lexical_cast("39539238727270799376544542000"), + boost::lexical_cast("6097272817323042122728617800"), + boost::lexical_cast("796974693974455191377937300"), + boost::lexical_cast("88776380550648116217781890"), + boost::lexical_cast("8459574446076318147830625"), + boost::lexical_cast("691254538651580660999025"), + boost::lexical_cast("48487623689430693038025"), + boost::lexical_cast("2918939500751087661105"), + boost::lexical_cast("150566737512021319125"), + boost::lexical_cast("6634460278534540725"), + boost::lexical_cast("248526574856284725"), + boost::lexical_cast("7860403394108265"), + boost::lexical_cast("207912996295875"), + boost::lexical_cast("4539323721075"), + boost::lexical_cast("80328850875"), + boost::lexical_cast("1122686019"), + boost::lexical_cast("11921175"), + boost::lexical_cast("90335"), + boost::lexical_cast("435"), + boost::lexical_cast("1"), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 31); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[30] = { + boost::lexical_cast("11.80038544942943603508206880307972596807"), + boost::lexical_cast("-130.6355975335626214564236363322099481079"), + boost::lexical_cast("676.2177719145993049893392276809256538927"), + boost::lexical_cast("-2174.724497783850503069990936574060452057"), + boost::lexical_cast("4869.877180638131076410069103742986502022"), + boost::lexical_cast("-8065.744271864238179992762265472478229172"), + boost::lexical_cast("10245.03825618572106228191509520638651539"), + boost::lexical_cast("-10212.83902362683215459850403668669647192"), + boost::lexical_cast("8110.289185383288952562767679576754140336"), + boost::lexical_cast("-5179.310892558291062401828964000776095156"), + boost::lexical_cast("2673.987492589052370230989109591011091273"), + boost::lexical_cast("-1118.342574651205183051884250033505609141"), + boost::lexical_cast("378.5812742511620662650096436471920295596"), + boost::lexical_cast("-103.3725999812126067084828735543906768961"), + boost::lexical_cast("22.62913974335996321848099677797888917792"), + boost::lexical_cast("-3.936414819950859548507275533569588041446"), + boost::lexical_cast("0.5376818198843817355682124535902641644854"), + boost::lexical_cast("-0.0567827903603478957483409124122554243201"), + boost::lexical_cast("0.004545544993648879420352693271088478106482"), + boost::lexical_cast("-0.0002689795568951033950042375135970897959935"), + boost::lexical_cast("0.1139493459006846530734617710847103572122e-4"), + boost::lexical_cast("-0.3316581197839213921885210451302820192794e-6"), + boost::lexical_cast("0.6285613334898374028443777562554713906213e-8"), + boost::lexical_cast("-0.7222145115734409070310317999856424167091e-10"), + boost::lexical_cast("0.4562976983547274766890241815002584238219e-12"), + boost::lexical_cast("-0.1380593023819058919640038942493212141072e-14"), + boost::lexical_cast("0.1629663871586410129307496385264268190679e-17"), + boost::lexical_cast("-0.5429994291916548849493889660077076739993e-21"), + boost::lexical_cast("0.2922682842441892106795386303084661338957e-25"), + boost::lexical_cast("-0.8456967065309046044689041041336866118459e-31"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& z) + { + static const T d[30] = { + boost::lexical_cast("147.9979641587472136175636384176549713358"), + boost::lexical_cast("-1638.404318611773924210055619836375434296"), + boost::lexical_cast("8480.981744216135641122944743711911653273"), + boost::lexical_cast("-27274.93942104458448200467097634494071176"), + boost::lexical_cast("61076.98019918759324489193232276937262854"), + boost::lexical_cast("-101158.8762737154296509560513952101409264"), + boost::lexical_cast("128491.1252383947174824913796141607174379"), + boost::lexical_cast("-128087.2892038336581928787480535905496026"), + boost::lexical_cast("101717.5492545853663296795562084430123258"), + boost::lexical_cast("-64957.8330410311808907869707511362206858"), + boost::lexical_cast("33536.59139229792478811870738772305570317"), + boost::lexical_cast("-14026.01847115365926835980820243003785821"), + boost::lexical_cast("4748.087094096186515212209389240715050212"), + boost::lexical_cast("-1296.477510211815971152205100242259733245"), + boost::lexical_cast("283.8099337545793198947620951499958085157"), + boost::lexical_cast("-49.36969067101255103452092297769364837753"), + boost::lexical_cast("6.743492833270653628580811118017061581404"), + boost::lexical_cast("-0.7121578704864048548351804794951487823626"), + boost::lexical_cast("0.0570092738016915476694118877057948681298"), + boost::lexical_cast("-0.003373485297696102660302960722607722438643"), + boost::lexical_cast("0.0001429128843527532859999752593761934089751"), + boost::lexical_cast("-0.41595867130858508233493767243236888636e-5"), + boost::lexical_cast("0.7883284669307241040059778207492255409785e-7"), + boost::lexical_cast("-0.905786322462384670803148223703187214379e-9"), + boost::lexical_cast("0.5722790216999820323272452464661250331451e-11"), + boost::lexical_cast("-0.1731510870832349779315841757234562309727e-13"), + boost::lexical_cast("0.2043890314358438601429048378015983874378e-16"), + boost::lexical_cast("-0.6810185176079344204740000170500311171065e-20"), + boost::lexical_cast("0.3665567641131713928114853776588342403919e-24"), + boost::lexical_cast("-0.1060655106553177007425710511436497259484e-29"), + }; + T result = 0; + T dz = z - 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 32.08066999999999779902282170951366424561; } +}; + +// +// Lanczos Coefficients for N=61 G=63.192152 +// Max experimental error (with 1000-bit precision arithmetic) 3.740e-113 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 12 2006 +// +struct lanczos61UDT +{ + template + static T lanczos_sum(const T& z) + { + using namespace boost; + static const T d[61] = { + boost::lexical_cast("2.50662827463100050241576528481104525300698674060993831662992357634229365460784197494659584"), + boost::lexical_cast("13349415823254323512107320481.3495396037261649201426994438803767191136434970492309775123879"), + boost::lexical_cast("-300542621510568204264185787475.230003734889859348050696493467253861933279360152095861484548"), + boost::lexical_cast("3273919938390136737194044982676.40271056035622723775417608127544182097346526115858803376474"), + boost::lexical_cast("-22989594065095806099337396006399.5874206181563663855129141706748733174902067950115092492439"), + boost::lexical_cast("116970582893952893160414263796102.542775878583510989850142808618916073286745084692189044738"), + boost::lexical_cast("-459561969036479455224850813196807.283291532483532558959779434457349912822256480548436066098"), + boost::lexical_cast("1450959909778264914956547227964788.89797179379520834974601372820249784303794436366366810477"), + boost::lexical_cast("-3782846865486775046285288437885921.41537699732805465141128848354901016102326190612528503251"), + boost::lexical_cast("8305043213936355459145388670886540.09976337905520168067329932809302445437208115570138102767"), + boost::lexical_cast("-15580988484396722546934484726970745.4927787160273626078250810989811865283255762028143642311"), + boost::lexical_cast("25262722284076250779006793435537600.0822901485517345545978818780090308947301031347345640449"), + boost::lexical_cast("-35714428027687018805443603728757116.5304655170478705341887572982734901197345415291580897698"), + boost::lexical_cast("44334726194692443174715432419157343.2294160783772787096321009453791271387235388689346602833"), + boost::lexical_cast("-48599573547617297831555162417695106.187829304963846482633791012658974681648157963911491985"), + boost::lexical_cast("47258466493366798944386359199482189.0753349196625125615316002614813737880755896979754845101"), + boost::lexical_cast("-40913448215392412059728312039233342.142914753896559359297977982314043378636755884088383226"), + boost::lexical_cast("31626312914486892948769164616982902.7262756989418188077611392594232674722318027323102462687"), + boost::lexical_cast("-21878079174441332123064991795834438.4699982361692990285700077902601657354101259411789722708"), + boost::lexical_cast("13567268503974326527361474986354265.3136632133935430378937191911532112778452274286122946396"), + boost::lexical_cast("-7551494211746723529747611556474669.62996644923557605747803028485900789337467673523741066527"), + boost::lexical_cast("3775516572689476384052312341432597.70584966904950490541958869730702790312581801585742038997"), + boost::lexical_cast("-1696271471453637244930364711513292.79902955514107737992185368006225264329876265486853482449"), + boost::lexical_cast("684857608019352767999083000986166.20765273693720041519286231015176745354062413008561259139"), + boost::lexical_cast("-248397566275708464679881624417990.410438107634139924805871051723444048539177890346227250473"), + boost::lexical_cast("80880368999557992138783568858556.1512378233079327986518410244522800950609595592170022878937"), + boost::lexical_cast("-23618197945394013802495450485616.9025005749893350650829964098117490779655546610665927669729"), + boost::lexical_cast("6176884636893816103087134481332.06708966653493024119556843727320635285468825056891248447124"), + boost::lexical_cast("-1444348683723439589948246285262.64080678953468490544615312565485170860503207005915261691108"), + boost::lexical_cast("301342031656979076702313946827.961658905182634508217626783081241074250132289461989777865387"), + boost::lexical_cast("-55959656587719766738301589651.3940625826610668990368881615587469329021742236397809951765678"), + boost::lexical_cast("9223339169004064297247180402.36227016155682738556103138196079389248843082157924368301293963"), + boost::lexical_cast("-1344882881571942601385730283.42710150182526891377514071881534880944872423492272147871101373"), + boost::lexical_cast("172841913316760599352601139.54409257740173055624405575900164401527761357324625574736896079"), + boost::lexical_cast("-19496120443876233531343952.3802212016691702737346568192063937387825469602063310488794471653"), + boost::lexical_cast("1920907372583710284097959.44121420322495784420169085871802458519363819782779653621724219067"), + boost::lexical_cast("-164429314798240461613359.399597503536657962383155875723527581699785846599051112719962464604"), + boost::lexical_cast("12154026644351189572525.1452249886865981747374191977801688548318519692423556934568426042152"), + boost::lexical_cast("-770443988366210815096.519382051977221101156336663806705367929328924137169970381042234329058"), + boost::lexical_cast("41558909851418707920.4696085656889424895313728719601503526476333404973280596225722152966128"), + boost::lexical_cast("-1890879946549708819.24562220042687554209318172044783707920086716716717574156283898330017796"), + boost::lexical_cast("71844996557297623.9583461685535340440524052492427928388171299145330229958643439878608673403"), + boost::lexical_cast("-2253785109518255.55600197759875781765803818232939130127735487613049577235879610065545755637"), + boost::lexical_cast("57616883849355.997562563968344493719962252675875692642406455612671495250543228005045106721"), + boost::lexical_cast("-1182495730353.08218118278997948852215670614084013289033222774171548915344541249351599628436"), + boost::lexical_cast("19148649358.6196967288062261380599423925174178776792840639099120170800869284300966978300613"), + boost::lexical_cast("-239779605.891370259668403359614360511661030470269478602533200704639655585967442891496784613"), + boost::lexical_cast("2267583.00284368310957842936892685032434505866445291643236133553754152047677944820353796872"), + boost::lexical_cast("-15749.490806784673108773558070497383604733010677027764233749920147549999361110299643477893"), + boost::lexical_cast("77.7059495149052727171505425584459982871343274332635726864135949842508025564999785370162956"), + boost::lexical_cast("-0.261619987273930331397625130282851629108569607193781378836014468617185550622160348688297247"), + boost::lexical_cast("0.000572252321659691600529444769356185993188551770817110673186068921175991328434642504616377475"), + boost::lexical_cast("-0.765167220661540041663007112207436426423746402583423562585653954743978584117929356523307954e-6"), + boost::lexical_cast("0.579179571056209077507916813937971472839851499147582627425979879366849876944438724610663401e-9"), + boost::lexical_cast("-0.224804733043915149719206760378355636826808754704148660354494460792713189958510735070096991e-12"), + boost::lexical_cast("0.392711975389579343321746945135488409914483448652884894759297584020979857734289645857714768e-16"), + boost::lexical_cast("-0.258603588346412049542768766878162221817684639789901440429511261589010049357907537684380983e-20"), + boost::lexical_cast("0.499992460848751668441190360024540741752242879565548017176883304716370989218399797418478685e-25"), + boost::lexical_cast("-0.196211614533318174187346267877390498735734213905206562766348625767919122511096089367364025e-30"), + boost::lexical_cast("0.874722648949676363732094858062907290148733370978226751462386623191111439121706262759209573e-37"), + boost::lexical_cast("-0.163907874717737848669759890242660846846105433791283903651924563157080252845636658802930428e-44"), + }; + T result = d[0]; + for(int k = 1; k < sizeof(d)/sizeof(d[0]); ++k) + { + result += d[k]/(z+(k-1)); + } + return result; + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + using namespace boost; + static const T d[61] = { + boost::lexical_cast("0.901751806425638853077358552989167785490911341809902155556127108480303870921448984935411583e-27"), + boost::lexical_cast("4.80241125306810017699523302110401965428995345115391817406006361151407344955277298373661032"), + boost::lexical_cast("-108.119283021710869401330097315436214587270846871451487282117128515476478251641970487922552"), + boost::lexical_cast("1177.78262074811362219818923738088833932279000985161077740440010901595132448469513438139561"), + boost::lexical_cast("-8270.43570321334374279057432173814835581983913167617217749736484999327758232081395983082867"), + boost::lexical_cast("42079.807161997077661752306902088979258826568702655699995911391774839958572703348502730394"), + boost::lexical_cast("-165326.003834443330215001219988296482004968548294447320869281647211603153902436231468280089"), + boost::lexical_cast("521978.361504895300685499370463597042533432134369277742485307843747923127933979566742421213"), + boost::lexical_cast("-1360867.51629992863544553419296636395576666570468519805449755596254912681418267100657262281"), + boost::lexical_cast("2987713.73338656161102517003716335104823650191612448011720936412226357385029800040631754755"), + boost::lexical_cast("-5605212.64915921452169919008770165304171481697939254152852673009005154549681704553438450709"), + boost::lexical_cast("9088186.58332916818449459635132673652700922052988327069535513580836143146727832380184335474"), + boost::lexical_cast("-12848155.5543636394746355365819800465364996596310467415907815393346205151090486190421959769"), + boost::lexical_cast("15949281.2867656960575878805158849857756293807220033618942867694361569866468996967761600898"), + boost::lexical_cast("-17483546.9948295433308250581770557182576171673272450149400973735206019559576269174369907171"), + boost::lexical_cast("17001087.8599749419660906448951424280111036786456594738278573653160553115681287326064596964"), + boost::lexical_cast("-14718487.0643665950346574802384331125115747311674609017568623694516187494204567579595827859"), + boost::lexical_cast("11377468.7255609717716845971105161298889777425898291183866813269222281486121330837791392732"), + boost::lexical_cast("-7870571.64253038587947746661946939286858490057774448573157856145556080330153403858747655263"), + boost::lexical_cast("4880783.08440908743641723492059912671377140680710345996273343885045364205895751515063844239"), + boost::lexical_cast("-2716626.7992639625103140035635916455652302249897918893040695025407382316653674141983775542"), + boost::lexical_cast("1358230.46602865696544327299659410214201327791319846880787515156343361311278133805428800255"), + boost::lexical_cast("-610228.440751458395860905749312275043435828322076830117123636938979942213530882048883969802"), + boost::lexical_cast("246375.416501158654327780901087115642493055617468601787093268312234390446439555559050129729"), + boost::lexical_cast("-89360.2599028475206119333931211015869139511677735549267100272095432140508089207221096740632"), + boost::lexical_cast("29096.4637987498328341260960356772198979319790332957125131055960448759586930781530063775634"), + boost::lexical_cast("-8496.57401431514433694413130585404918350686834939156759654375188338156288564260152505382438"), + boost::lexical_cast("2222.11523574301594407443285016240908726891841242444092960094015874546135316534057865883047"), + boost::lexical_cast("-519.599993280949289705514822058693289933461756514489674453254304308040888101533569480646682"), + boost::lexical_cast("108.406868361306987817730701109400305482972790224573776407166683184990131682003417239181112"), + boost::lexical_cast("-20.1313142142558596796857948064047373605439974799116521459977609253211918146595346493447238"), + boost::lexical_cast("3.31806787671783168020012913552384112429614503798293169239082032849759155847394955909684383"), + boost::lexical_cast("-0.483817477111537877685595212919784447924875428848331771524426361483392903320495411973587861"), + boost::lexical_cast("0.0621793463102927384924303843912913542297042029136293808338022462765755471002366206711862637"), + boost::lexical_cast("-0.00701366932085103924241526535768453911099671087892444015581511551813219720807206445462785293"), + boost::lexical_cast("0.000691040514756294308758606917671220770856269030526647010461217455799229645004351524024364997"), + boost::lexical_cast("-0.591529398871361458428147660822525365922497109038495896497692806150033516658042357799869656e-4"), + boost::lexical_cast("0.437237367535177689875119370170434437030440227275583289093139147244747901678407875809020739e-5"), + boost::lexical_cast("-0.277164853397051135996651958345647824709602266382721185838782221179129726200661453504250697e-6"), + boost::lexical_cast("0.149506899012035980148891401548317536032574502641368034781671941165064546410613201579653674e-7"), + boost::lexical_cast("-0.68023824066463262779882895193964639544061678698791279217407325790147925675797085217462974e-9"), + boost::lexical_cast("0.258460163734186329938721529982859244969655253624066115559707985878606277800329299821882688e-10"), + boost::lexical_cast("-0.810792256024669306744649981276512583535251727474303382740940985102669076169168931092026491e-12"), + boost::lexical_cast("0.207274966207031327521921078048021807442500113231320959236850963529304158700096495799022922e-13"), + boost::lexical_cast("-0.425399199286327802950259994834798737777721414442095221716122926637623478450472871269742436e-15"), + boost::lexical_cast("0.688866766744198529064607574117697940084548375790020728788313274612845280173338912495478431e-17"), + boost::lexical_cast("-0.862599751805643281578607291655858333628582704771553874199104377131082877406179933909898885e-19"), + boost::lexical_cast("0.815756005678735657200275584442908437977926312650210429668619446123450972547018343768177988e-21"), + boost::lexical_cast("-0.566583084099007858124915716926967268295318152203932871370429534546567151650626184750291695e-23"), + boost::lexical_cast("0.279544761599725082805446124351997692260093135930731230328454667675190245860598623539891708e-25"), + boost::lexical_cast("-0.941169851584987983984201821679114408126982142904386301937192011680047611188837432096199601e-28"), + boost::lexical_cast("0.205866011331040736302780507155525142187875191518455173304638008169488993406425201915370746e-30"), + boost::lexical_cast("-0.27526655245712584371295491216289353976964567057707464008951584303679019796521332324352501e-33"), + boost::lexical_cast("0.208358067979444305082929004102609366169534624348056112144990933897581971394396210379638792e-36"), + boost::lexical_cast("-0.808728107661779323263133007119729988596844663194254976820030366188579170595441991680169012e-40"), + boost::lexical_cast("0.141276924383478964519776436955079978012672985961918248012931336621229652792338950573694356e-43"), + boost::lexical_cast("-0.930318449287651389310440021745842417218125582685428432576258687100661462527604238849332053e-48"), + boost::lexical_cast("0.179870748819321661641184169834635133045197146966203370650788171790610563029431722308057539e-52"), + boost::lexical_cast("-0.705865243912790337263229413370018392321238639017433365017168104310561824133229343750737083e-58"), + boost::lexical_cast("0.3146787805734405996448268100558028857930560442377698646099945108125281507396722995748398e-64"), + boost::lexical_cast("-0.589653534231618730406843260601322236697428143603814281282790370329151249078338470962782338e-72"), + }; + T result = d[0]; + for(int k = 1; k < sizeof(d)/sizeof(d[0]); ++k) + { + result += d[k]/(z+(k-1)); + } + return result; + } + + template + static T lanczos_sum_near_1(const T& dz) + { + using namespace boost; + static const T d[60] = { + boost::lexical_cast("23.2463658527729692390378860713647146932236940604550445351214987229819352880524561852919518"), + boost::lexical_cast("-523.358012551815715084547614110229469295755088686612838322817729744722233637819564673967396"), + boost::lexical_cast("5701.12892340421080714956066268650092612647620400476183901625272640935853188559347587495571"), + boost::lexical_cast("-40033.5506451901904954336453419007623117537868026994808919201793803506999271787018654246699"), + boost::lexical_cast("203689.884259074923009439144410340256983393397995558814367995938668111650624499963153485034"), + boost::lexical_cast("-800270.648969745331278757692597096167418585957703057412758177038340791380011708874081291202"), + boost::lexical_cast("2526668.23380061659863999395867315313385499515711742092815402701950519696944982260718031476"), + boost::lexical_cast("-6587362.57559198722630391278043503867973853468105110382293763174847657538179665571836023631"), + boost::lexical_cast("14462211.3454541602975917764900442754186801975533106565506542322063393991552960595701762805"), + boost::lexical_cast("-27132375.1879227404375395522940895789625516798992585980380939378508607160857820002128106898"), + boost::lexical_cast("43991923.8735251977856804364757478459275087361742168436524951824945035673768875988985478116"), + boost::lexical_cast("-62192284.0030124679010201921841372967696262036115679150017649233887633598058364494608060812"), + boost::lexical_cast("77203473.0770033513405070667417251568915937590689150831268228886281254637715669678358204991"), + boost::lexical_cast("-84630180.2217173903516348977915150565994784278120192219937728967986198118628659866582594874"), + boost::lexical_cast("82294807.2253549409847505891112074804416229757832871133388349982640444405470371147991706317"), + boost::lexical_cast("-71245738.2484649177928765605893043553453557808557887270209768163561363857395639001251515788"), + boost::lexical_cast("55073334.3180266913441333534260714059077572215147571872597651029894142803987981342430068594"), + boost::lexical_cast("-38097984.1648990787690036742690550656061009857688125101191167768314179751258568264424911474"), + boost::lexical_cast("23625729.5032184580395130592017474282828236643586203914515183078852982915252442161768527976"), + boost::lexical_cast("-13149998.4348054726172055622442157883429575511528431835657668083088902165366619827169829685"), + boost::lexical_cast("6574597.77221556423683199818131482663205682902023554728024972161230111356285973623550338976"), + boost::lexical_cast("-2953848.1483469149918109110050192571921018042012905892107136410603990336401921230407043408"), + boost::lexical_cast("1192595.29584357246380113611351829515963605337523874715861849584306265512819543347806085356"), + boost::lexical_cast("-432553.812019608638388918135375154289816441900572658692369491476137741687213006403648722272"), + boost::lexical_cast("140843.215385933866391177743292449477205328233960902455943995092958295858485718885800427129"), + boost::lexical_cast("-41128.186992679630058614841985110676526199977321524879849001760603476646382839182691529968"), + boost::lexical_cast("10756.2849191854701631989789887757784944313743544315113894758328432005767448056040879337769"), + boost::lexical_cast("-2515.15559672041299884426826962296210458052543246529646213159169885444118227871246315458787"), + boost::lexical_cast("524.750087004805200600237632074908875763734578390662349666321453103782638818305404274166951"), + boost::lexical_cast("-97.4468596421732493988298219295878130651986131393383646674144877163795143930682205035917965"), + boost::lexical_cast("16.0613108128210806736384551896802799172445298357754547684100294231532127326987175444453058"), + boost::lexical_cast("-2.34194813526540240672426202485306862230641838409943369059203455578340880900483887447559874"), + boost::lexical_cast("0.300982934748016059399829007219431333744032924923669397318820178988611410275964499475465815"), + boost::lexical_cast("-0.033950095985367909789000959795708551814461844488183964432565731809399824963680858993718525"), + boost::lexical_cast("0.00334502394288921146242772614150438404658527112198421937945605441697314216921393987758378122"), + boost::lexical_cast("-0.000286333429067523984607730553301991502191011265745476190940771685897687956262049750683013485"), + boost::lexical_cast("0.211647426149364947402896718485536530479491688838087899435991994237067890628274492042231115e-4"), + boost::lexical_cast("-0.134163345121302758109675190598169832775248626443483098532368562186356128620805552609175683e-5"), + boost::lexical_cast("0.723697303042715085329782938306424498336642078597508935450663080894255773653328980495047891e-7"), + boost::lexical_cast("-0.329273487343139063533251321553223583999676337945788660475231347828772272134656322947906888e-8"), + boost::lexical_cast("0.12510922551028971731767784013117088894558604838820475961392154031378891971216135267744134e-9"), + boost::lexical_cast("-0.392468958215589939603666430583400537413757786057185505426804034745840192914621891690369903e-11"), + boost::lexical_cast("0.100332717101049934370760667782927946803279422001380028508200697081188326364078428184546051e-12"), + boost::lexical_cast("-0.205917088291197705194762747639836655808855850989058813560983717575008725393428497910009756e-14"), + boost::lexical_cast("0.333450178247893143608439314203175490705783992567107481617660357577257627854979230819461489e-16"), + boost::lexical_cast("-0.417546693906616047110563550428133589051498362676394888715581845170969319500638944065594319e-18"), + boost::lexical_cast("0.394871691642184410859178529844325390739857256666676534513661579365702353214518478078730801e-20"), + boost::lexical_cast("-0.274258012587811199757875927548699264063511843669070634471054184977334027224611843434000922e-22"), + boost::lexical_cast("0.135315354265459854889496635967343027244391821142592599244505313738163473730636430399785048e-24"), + boost::lexical_cast("-0.455579032003288910408487905303223613382276173706542364543918076752861628464036586507967767e-27"), + boost::lexical_cast("0.99650703862462739161520123768147312466695159780582506041370833824093136783202694548427718e-30"), + boost::lexical_cast("-0.1332444609228706921659395801935919548447859029572115502899861345555006827214220195650058e-32"), + boost::lexical_cast("0.100856999148765307000182397631280249632761913433008379786888200467467364474581430670889392e-35"), + boost::lexical_cast("-0.39146979455613683472384690509165395074425354524713697428673406058157887065953366609738731e-39"), + boost::lexical_cast("0.683859606707931248105140296850112494069265272540298100341919970496564103098283709868586478e-43"), + boost::lexical_cast("-0.450326344248604222735147147805963966503893913752040066400766411031387063854141246972061792e-47"), + boost::lexical_cast("0.870675378039492904184581895322153006461319724931909799151743284769901586333730037761678891e-52"), + boost::lexical_cast("-0.341678395249272265744518787745356400350877656459401143889000625280131819505857966769964401e-57"), + boost::lexical_cast("0.152322191370871666358069530949353871960316638394428595988162174042653299702098929238880862e-63"), + boost::lexical_cast("-0.285425405297633795767452984791738825078111150078605004958179057245980222485147999495352632e-71"), + }; + T result = 0; + for(int k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& z) + { + using namespace boost; + static const T d[60] = { + boost::lexical_cast("557.56438192770795764344217888434355281097193198928944200046501607026919782564033547346298"), + boost::lexical_cast("-12552.748616427645475141433405567201788681683808077269330800392600825597799119572762385222"), + boost::lexical_cast("136741.650054039199076788077149441364242294724343897779563222338447737802381279007988884806"), + boost::lexical_cast("-960205.223613240309942047656967301131022760634321049075674684679438471862998829007639437133"), + boost::lexical_cast("4885504.47588736223774859617054275229642041717942140469884121916073195308537421162982679289"), + boost::lexical_cast("-19194501.738192166918904824982935279260356575935661514109550613809352009246483412530545583"), + boost::lexical_cast("60602169.8633537742937457094837494059849674261357199218329545854990149896822944801504450843"), + boost::lexical_cast("-157997975.522506767297528502540724511908584668874487506510120462561270100749019845014382885"), + boost::lexical_cast("346876323.86092543685419723290495817330608574729543216092477261152247521712190505658568876"), + boost::lexical_cast("-650770365.471136883718747607976242475416651908858429752332176373467422603953536408709972919"), + boost::lexical_cast("1055146856.05909309330903130910708372830487315684258450293308627289334336871273080305128138"), + boost::lexical_cast("-1491682726.25614447429071368736790697283307005456720192465860871846879804207692411263924608"), + boost::lexical_cast("1851726287.94866167094858600116562210167031458934987154557042242638980748286188183300900268"), + boost::lexical_cast("-2029855953.68334371445800569238095379629407314338521720558391277508374519526853827142679839"), + boost::lexical_cast("1973842002.53354868177824629525448788555435194808657489238517523691040148611221295436287925"), + boost::lexical_cast("-1708829941.98209573247426625323314413060108441455314880934710595647408841619484540679859098"), + boost::lexical_cast("1320934627.12433688699625456833930317624783222321555050330381730035733198249283009357314954"), + boost::lexical_cast("-913780636.858542526294419197161614811332299249415125108737474024007693329922089123296358727"), + boost::lexical_cast("566663423.929632170286007468016419798879660054391183200464733820209439185545886930103546787"), + boost::lexical_cast("-315402880.436816230388857961460509181823167373029384218959199936902955049832392362044305869"), + boost::lexical_cast("157691811.550465734461741500275930418786875005567018699867961482249002625886064187146134966"), + boost::lexical_cast("-70848085.5705405970640618473551954585013808128304384354476488268600720054598122945113512731"), + boost::lexical_cast("28604413.4050137708444142264980840059788755325900041515286382001704951527733220637586013815"), + boost::lexical_cast("-10374808.7067303054787164054055989420809074792801592763124972441820101840292558840131568633"), + boost::lexical_cast("3378126.32016207486657791623723515804931231041318964254116390764473281291389374196880720069"), + boost::lexical_cast("-986460.090390653140964189383080344920103075349535669020623874668558777188889544398718979744"), + boost::lexical_cast("257989.631187387317948158483575125380011593209850756066176921901006833159795100137743395985"), + boost::lexical_cast("-60326.0391159227288325790327830741260824763549807922845004854215660451399733578621565837087"), + boost::lexical_cast("12586.1375399649496159880821645216260841794563919652590583420570326276086323953958907053394"), + boost::lexical_cast("-2337.26417330316922535871922886167444795452055677161063205953141105726549966801856628447293"), + boost::lexical_cast("385.230745012608736644117458716226876976056390433401632749144285378123105481506733917763829"), + boost::lexical_cast("-56.1716559403731491675970177460841997333796694857076749852739159067307309470690838101179615"), + boost::lexical_cast("7.21907953468550196348585224042498727840087634483369357697580053424523903859773769748599575"), + boost::lexical_cast("-0.814293485887386870805786409956942772883474224091975496298369877683530509729332902182019049"), + boost::lexical_cast("0.0802304419995150047616460464220884371214157889148846405799324851793571580868840034085001373"), + boost::lexical_cast("-0.00686771095380619535195996193943858680694970000948742557733102777115482017857981277171196115"), + boost::lexical_cast("0.000507636621977556438232617777542864427109623356049335590894564220687567763620803789858345916"), + boost::lexical_cast("-0.32179095465362720747836116655088181481893063531138957363431280817392443948706754917605911e-4"), + boost::lexical_cast("0.173578890579848508947329833426585354230744194615295570820295052665075101971588563893718407e-5"), + boost::lexical_cast("-0.789762880006288893888161070734302768702358633525134582027140158619195373770299678322596835e-7"), + boost::lexical_cast("0.300074637200885066788470310738617992259402710843493097610337134266720909870967550606601658e-8"), + boost::lexical_cast("-0.941337297619721713151110244242536308266701344868601679868536153775533330272973088246835359e-10"), + boost::lexical_cast("0.24064815013182536657310186836079333949814111498828401548170442715552017773994482539471456e-11"), + boost::lexical_cast("-0.493892399304811910466345686492277504628763169549384435563232052965821874553923373100791477e-13"), + boost::lexical_cast("0.799780678476644196901221989475355609743387528732994566453160178199295104357319723742820593e-15"), + boost::lexical_cast("-0.100148627870893347527249092785257443532967736956154251497569191947184705954310833302770086e-16"), + boost::lexical_cast("0.947100256812658897084619699699028861352615460106539259289295071616221848196411749449858071e-19"), + boost::lexical_cast("-0.657808193528898116367845405906343884364280888644748907819280236995018351085371701094007759e-21"), + boost::lexical_cast("0.324554050057463845012469010247790763753999056976705084126950591088538742509983426730851614e-23"), + boost::lexical_cast("-0.10927068902162908990029309141242256163212535730975970310918370355165185052827948996110107e-25"), + boost::lexical_cast("0.239012340507870646690121104637467232366271566488184795459093215303237974655782634371712486e-28"), + boost::lexical_cast("-0.31958700972990573259359660326375143524864710953063781494908314884519046349402409667329667e-31"), + boost::lexical_cast("0.241905641292988284384362036555782113275737930713192053073501265726048089991747342247551645e-34"), + boost::lexical_cast("-0.93894080230619233745797029179332447129464915420290457418429337322820997038069119047864035e-38"), + boost::lexical_cast("0.164023814025085488413251990798690797467351995518990067783355251949198292596815470576539877e-41"), + boost::lexical_cast("-0.108010831192689925518484618970761942019888832176355541674171850211917230280206410356465451e-45"), + boost::lexical_cast("0.208831600642796805563854019033577205240227465154130766898180386564934443551840379116390645e-50"), + boost::lexical_cast("-0.819516067465171848863933747691434138146789031214932473898084756489529673230665363014007306e-56"), + boost::lexical_cast("0.365344970579318347488211604761724311582675708113250505307342682118101409913523622073678179e-62"), + boost::lexical_cast("-0.684593199208628857931267904308244537968349564351534581234005234847904343404822808648361291e-70"), + }; + T result = 0; + T dz = z - 2; + for(int k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 63.19215200000000010049916454590857028961181640625; } +}; + +namespace boost{ namespace math{ namespace lanczos{ + +template<> +struct lanczos_traits +{ + typedef NTL::RR value_type; +#ifdef L13 + typedef lanczos13UDT evaluation_type; +#elif defined(L22) + typedef lanczos22UDT evaluation_type; +#elif defined(L31) + typedef lanczos31UDT evaluation_type; +#else + typedef lanczos61UDT evaluation_type; +#endif +}; + +}}} + +#endif + +