From 5b95219e094cfd8806ecad8398b28fa451ff45d4 Mon Sep 17 00:00:00 2001
From: John Maddock <john@johnmaddock.co.uk>
Date: Wed, 4 Oct 2006 09:57:05 +0000
Subject: [PATCH] Fixed digamma 128-bit error rates with a more accurate value
 for the root. Updated png's: hopefully they'll all render OK now. Tweeked
 docs to reflect recent advances.

[SVN r3237]
---
 doc/digamma.qbk                               |   9 +++++++--
 doc/equations/chf.png                         | Bin 873 -> 922 bytes
 doc/equations/dist_tutorial1.png              | Bin 2218 -> 1883 bytes
 doc/equations/dist_tutorial2.png              | Bin 796 -> 817 bytes
 doc/equations/dist_tutorial3.png              | Bin 1359 -> 1436 bytes
 doc/equations/dist_tutorial4.png              | Bin 609 -> 630 bytes
 doc/equations/hazard.png                      | Bin 776 -> 826 bytes
 doc/equations/students_t_dist.png             | Bin 1192 -> 465 bytes
 doc/erf.qbk                                   |   9 ---------
 .../boost/math/special_functions/digamma.hpp  |   4 ++--
 minimax/f.cpp                                 |  19 ++++++++++++++++--
 test/test_digamma.cpp                         |  11 +---------
 12 files changed, 27 insertions(+), 25 deletions(-)

diff --git a/doc/digamma.qbk b/doc/digamma.qbk
index 1ab430f6e..6d92d6fa9 100644
--- a/doc/digamma.qbk
+++ b/doc/digamma.qbk
@@ -40,9 +40,14 @@ than the one shown will have __zero_error.
 [[53]            [Win32 Visual C++ 8]  [Peak=0.98 Mean=0.36] [Peak=0.99 Mean=0.5]    [Peak=0.95 Mean=0.5] [Peak=214 Mean=16] ]
 [[64]            [Linux IA32 / GCC]  [Peak=1.4 Mean=0.4]  [Peak=1.3 Mean=0.45]    [Peak=0.98 Mean=0.35]       [Peak=180 Mean=13]   ]
 [[64]            [Linux IA64 / GCC]  [Peak=0.92 Mean=0.4]   [Peak=1.3 Mean=0.45]    [Peak=0.98 Mean=0.4]    [Peak=180 Mean=13]           ]
-[[113]           [HPUX IA64, aCC A.06.06]    [Peak=0.9 Mean=0.4] [Peak=200 Mean=14]    [Peak=0.99 Mean=0.4] [Peak=200 Mean=14]  ]
+[[113]           [HPUX IA64, aCC A.06.06]    [Peak=0.9 Mean=0.4] [Peak=1.1 Mean=0.5]    [Peak=0.99 Mean=0.4] [Peak=64 Mean=6]  ]
 ]
 
+As shown above, error rates for positive arguments are generally very low.
+For negative arguments there are an infinite number of irrational roots:
+relative errors very close to these can be arbitrarily large, although
+absolute error will remain very low.
+
 [h4 Testing]
 
 There are two sets of tests: spot values are computed using
@@ -119,7 +124,7 @@ rational approximation.  Note that since X0 is irrational, we need twice as many
 digits in X0 as in x in order to avoid cancellation error during the subtraction
 (this assumes that /x/ is an exact value, if it's not then all bets are off).  That
 means that even when x is the value of the root rounded to the nearest 
-representable value, the result of digamma(x) ['[*will not be zero*]].
+representable value, the result of digamma(x) ['[*will not be zero]].
 
 
 [endsect][/section:tgamma The Gamma Function]
diff --git a/doc/equations/chf.png b/doc/equations/chf.png
index e13dcf00d23916e0e105314af892bfd2ca31713a..2923d3137390378a2e1f693dfbd19acc4e40cdd4 100644
GIT binary patch
delta 859
zcmaFKHj90NBnKM<1H;y`<(nodn%6Vyc)B=-RNQ)V*SF}gf<W8Dji;y2`mJBock_wV
zyp)Ag78wccXIuDut=G;+Cp=y*v(GqCP;EEAp5einJ2M#=6qzry-hTUSW8K~@x=W@V
z*m?AgcH93~6B$7go6Xg>s-!ebDPpUZP-b4>&c@Kf!@wcgAbS6CdiZPxg_Hv+O5f`L
zR&BlM<HXi5fl;D6HS*ZhY3Tw+37fVA^@G(4Uh;AR={vA{=P&W&oRSP+9>YD|6!)D!
zKH09HF{k$Ztj#s?Z%!o(E5;wc)_BRP>Au;m*&BU&mxo>7*7dRa?1EUc>gc+@maOw1
z-2Eh%3dL5opS!^IX~xv28>N|V)iZ8gw?65y&C_2o@>gTDit;NTU--Iocl0!=1!V~w
z+e&`#R}Q#v&wTNZ7k0B9tEaBMwE1HXQ|T2(4Z-y@(!2IPS6<c-kQ)0xxB6I|U%t+>
z&4%YicBTF8SsD8_T=)F0o~M7WG1i0>x`|H+H%q$z{kMXMyk_C<TK!}0hL0;Xzuyt9
z-}Lrq_5St8k9uj{+ID8Imq^gdx5jO|_V3jEWMpx3vr)g8hv_BGZ)^3v+zYQM^Y(Op
zjFGcExAyIu@TuENYSxCHQclg99kwaoI{s^pN>I|CnG<^F?S8*h#W>8%>A=3Fdlz1c
zJTt|pD9PsQynaUBfA3#EHGV$-T<Mj_sI4<wyK2|hf3AOe<cHw*8(!azUHf1><?!?;
z-&KF^jNGv;Q}y>Y;guJDl=Zr<*;%+l{PE83H8M*xvz>F(|Hg^Vn|v>q-^XLF?M*x9
zQ)@C#`DCtL_l;*|ct(+G`iGgdx0f6=mFk-ipZs^4^y7`+|NQjnpM1=H^^~pGZg6ZV
z+m>{HN88SA`BD9?^;fl)pKiJ_F)T}X<Hx_}_tn-EiwV9~OHuBW6qLUG|IO-!omFje
zD^(i|H+4pCke4h75=>?VCl>}q=75fmhG6!K*2X#~HXp?}P;z<Edp1p+JE^g_bH+Uh
zMP`MqEYMWLA=&V<!ho?r&MA7=@tm)ICm48?J(AXc+mK)SL06G^!qv-B#U(4&nx?k!
zNVMHK9O1=xZ|*9^4GLlw@#}S;dN@0=F{nS-&%n&^D=Yd)g}_xu1_lNOPgg&ebxsLQ
E0NGQXH~;_u

delta 810
zcmbQm{*rBiBnJ}%1A~=_!PAL~=Jia!JzX3_DsH`*^SZA^fyeFYkALgm_!(Ir;kd%o
z6mah6#+z@HvR<m|ZvSf0$I4KZbeM(VgwO)9*SmJjKOwY$+fMSP`((CHMk$;PI*eIt
z8}IW8M>3Tx@l<|)h=r43uPTFrH$w{Nh0sHmk2nN2v7DY2Q#X0$2_c3FstzshT@Dx2
zU*<g_q~pHf%QS`t9YzsBolPtZDVzy<8(pR`fR!~wMjli8`qlkX>wCRm8LQVHc=gO3
zvo1$?=`<hSAa!3`Kj7V~=uWkMdBdNqzQUWsXPZe^ec!b|%{AhvQa<;#jotR=8#Zmf
zKkeddz3R5K>LpqGeO`rqydthqxKh&QQ)K=3SuA@mUo*RV?eppDs}J_v_sQ-`n^!zB
zev(w{?UPsE9b(<bzQcYV|C5lisZNOx%l_@Zv(}XH<kQ>F^eVqSITR7RS9Qke)xO#7
zOa2%?+p^K9>1E54Z`*Ed&^fQ#w14*$qm`EDr?cNa^S@$F{@dgHzb^cmT^9beXIkEL
zLF<`YoR3x1pFg@=%XMMX>en}?hUsnmaa1};_~NGxXWeFm%N_KSduL^OZS&!^tL+s|
ziyM2|-+8rX@5F-wWjk1ESMF?BCVbAr@XzXRa<g=Q+^PD!_jjX=Q{9b)ABDx-PDJlr
zmF!q=(CZVl=F?4|xzD2o4xTfMs<!T)Ju7-wC}Wj=tjJ^yr>)iXI#n@y9K*lshdW57
z&zm1)d~fTK_VmBA@9}+j#;PLA*8b06Mp4D#9$C3Ji%kSy3eBpWkbPmtu9&y=8@<+E
z+O+qx+TxsYh04l4-pI85fjLr-94~9GU$u44f}31ICNp#Erzsuj`L<4N`l{76w={%(
zpWT%A{r1Olac;t{3w!Ht<a5?96Y`b@rxAt|LeCDKap2s<;xLUt)S&DP(<T<>tSu&2
z+jSb(2}Ls59FO>2^!1I2ZqGWOYJJyyP=aHKWK!^E&}dbZmyrA4A^GrvWD4g6kIB=E
z3s3O}xJ_fY_3wh9wd&1dHJqDT7%#|+Pb+aS-o%n2c>Gqx_rsBP40k_@s~=$asagCr
U>c?b31_lNOPgg&ebxsLQ0QC!er~m)}

diff --git a/doc/equations/dist_tutorial1.png b/doc/equations/dist_tutorial1.png
index 80d54afe8882e4755290cd423ca77a810e2ff43e..172eaab0b275f585c1c1731b55299109867b75a4 100644
GIT binary patch
delta 1879
zcmZ1_c$;s6WIY=L1H*E@ZMh5#3@qu6zK#qG>ra@ocD-a^V2~_vjVKAuPb(=;EJ|f?
zOvz75Rq)JBOiv9;O-!jQJeg|4z`*v_)5S5Q;?|qHuk+>@2(&#^JLUWJU;UR&m)5Py
z6*Nj{&OP>{a_K6|2}hUzv@tgO@%Z=e-|P%q9yA|dP%wC5ZBTD#SGRZnKV!GLef#SF
zuIJNu=q;>}I>XyJMrxJSf);<@2HV3m=l{K8U}{dYIq1-NhBrx{y?F}zhV%*f{E4-9
zFWz6t+{jqgTQb+eH+K7UpSQUOw!WJ1OIk+xWcT3-Ntrp(s||O$ZT})w)bcRvMDSuh
z&5epiENr5!%eDtz-FJOzy_LY^&mNx^-6#%V7mx@szHrg}=G8ULA69JGt5P}9Xu;Yg
zX<6lO?tL$p$XsnDAkNg>=#q2ANLY%gnbB!_V2{bv%ZdgpZaKUH5|5Y!B=o9Hzt39D
z%XGMz(IKJZbmV)^T!To#8F@^~E=-Mz20bzg25*nFUSMEZ7Wu;|=GT_bIrV4k)~!);
zl;C1we$@2w!!EWKp~Wj6hCJCMD521NVjiD<&jAJngBhAn3_lA?Fen%>aPSCd*=+jd
zoX}A3BoHW|G+oz-FK+ggj{7cNhoY9L8B_$^xRhnKt%`-I`Q7FsZb$E{ily_F&#w)?
zxoq|on{Jczjw_EJY+f->`j_HK>xCQY*ZAwl*riYYe5Th$or9+<_UELTb>1m`c8_%v
z0>is(u0)7i&fwfTW37OudrZ)CyZM}<izEv{&f|H!^ohjvdt0{_N!k`GE&V=e{Ys5H
zllEAAesv*alZE<`Tf5VK&a|C$QF&@^$diN*{t;CYXW}YKpSW&4*MCK0_J^(m35Tm+
z){DByJT2H}_dAdAj@IGmk|6C87B-vdTy3+Z<!pU3TD@4<8hJj;y1jXOAol&mmyy#~
zq)M07czjya_WV-z_M9)VJIi=KCmdL@g}IUOn27ko$pW0Bysy93umAAkZ|RnH&Fl7$
zPYOtI@H!+g2tdP+fr*)cg^i(+(IJ7Mp&k^x3>-WlS&*P(TyhmZhg$mI5ZlI;ItFpk
z41bPGzXi!MSABI-c^G)*+nui0hJN>JGWA@<vme>!-RGDnz?bxC&OSwhxv{6de?Q%P
z$=xc*bo-j6rF$0hy1koT`qS&wt<Cu@9_G8J1hRh&+gapSsI&I*z4y_r?u+ZcoesNM
z<bVCmJ@fS{>+7%o`};8b-1*}5(i`)_U(fheo4NZ*-2aw3%gbryUwQ>R&nc$JD$ki>
z;jOnK^}fIizl9es_1YF$PcJNb;<)qqs^?Q`U+<chYmmMF&Q<dw1M{<DE0x2i&)vQ!
zZ_CV*^?jC`AIN^4yD<LU3uz|d1&i;;KUb>{D7LanFU!nTh(5OCUtRt+-L;QHn|IvO
z|J8P&r6jYMJtVK)eO9dZ>k6~-fX{NOsd{&7%$)C^IB3%Ia{ZYY|BU{dr)<1dl(^p)
zm@(V>XXot){jYX>I}{|nuSz{Q{l)ZKwaO=@EmXYD*1b+i(%M;ZJi+H~{d%A3QlHq#
zMVHqftN&O2)qDNETUyS36=h5Fz1~01eAT8Hv_5vlMd!mA-!JI@eW&j7%0l~`!m7>R
zyjyx31%J1)UtZ{(UH0ehd(W4a9ntq!UwfXPYf-anM~dxlmxO0?Pj1zZ`ra$R>9DzA
z0(){B@9k~nZA~kL`GjV0GdU$R)b8`Tvg-S%>-XMEni(9ZSK0gYV8zFxf5OWd{3;A=
z+RW~We{h*3{4tJC?fX>*f1ZicZXc>Na5yTEsc#cyzxYGR`?doN&o%F{8~>8t%kbr~
zwz|vBsz*gp+xvGgJ$GIb^LW<53)}8pw0hlj%gCnh65GVn)2tKypC6neCHw5-4W+e{
zdJ7V6or$*dv+e$pC6iu%?2I(ux0;1PM>buR2;Q8N`eV)Amou7Aw8^Z^G4>Cdzm!*1
zXO_RmA~~;bOPDXqy=JVwu~EloMkQ<J`+!YP&u{;vr201O*17Q0*Dhb4k;x-C<IutE
z&R-UpmCsUdY&}<{r5Ww{X=~`ud40jG1y&|2Gc~r}I~O0cZPNMmU$f5MR8z0d-}2gI
zhuimiQ+js@NR${pSuy*~x+!Z@=WcUyQxrPmWS=B27QQ%V@41Rum(o-!PqTHT{z~uM
zEb7GMoFFHX<eIgg<GG}ugn&fDfith4F$Zq!VrgJ!ZaBaI$zNFW95l<#vk{QVtB>E>
z<8~!}&t!4F?fC(2N}t8wH!JG#ZC`)Jr~S6@j6;t32}jIR+HZ4Dl(Rqmea6vh`QIDM
zbF$8r8c*e&nH)3edxTF~=StZ;`D&*}2OkI~UcBT`6qZ|5#JBSP`uehx{!5#ts>Nr^
z6!^z*_{_Me;ncB3UBXy4FMjdfPVPod(}Vyeo?|E63@05Fnr;#{kw>Eawz-6Mm~y4T
zkyPdOM+LusJ$$!iTZ$;BzgTMfZD9!;jUz=nY&m!u^!|t?F#NBulV>yFPG(?WVDNPH
Kb6Mw<&;$TXU15m;

literal 2218
zcmeAS@N?(olHy`uVBq!ia0y~yV3c5BVCdmsW?*2LaKSB}fq{Xuz$3Dlfr0M`2s2LA
z=96Y%U|>mi^mSxl*f=lI$MHJ@1A{`cN02WALzOB6LqjtI!_WT=3=J<B7=kz$7)2u(
z7_6o<Fo@@0kCLCjz`)E9;1lA?00vA<Ow7#8EG#UntgLKoZ0zjp92^{+oSa--T-@B;
zJUl$Syu5sTeEj_U0s;bpf`URqLc+qrA|fK9qM~AAV&dZB5)u-Ul9EzVQqt1WGBPr<
zva)h=a`N)>3JMB}ii%1~O3KR0Dk>_fs;X*gYU=9h8X6j!nwnZ#TH4y$IyySKy1IIL
zdiwhM1_lO(hK5E)M#jd*CMG7Prlw|QX6EMR78VwkmX=mlR@T<mHa0f4wzhV5cJ}u6
z4h{~Ej*d=FPR`EGE-o&vuC8uwZtm{x9v&W^o}OM_Uf$l`K0ZFazP^5be*XUc0RaJl
zfq_9mLBYYnAt525p`l@6Vd3H75fKrQk&#hRQPI)SF)=Z*v9WP+aq;o-2?+^_iHS)`
zNy*8{DJdzbsi|pcY3b?d85tRwnVDHxS=rgyIXOAGxw(0HdHMPI1qB6#g@r{$Ma9L%
zB_$=LrKM$MW##4N6%`egm6cUhRn^tiH8nN0wY7D1b@lc24Gj&Ajg3uBP0h{CEiEmr
zt*vcsZSC#t9UUE=ot<4>UEST?Jv}|Wy}f;Xef|CY6DCZUIC0{nNs}f|o;+pBl&Mpv
zPMbDu`t<2DX3Us5bLOmBvu4kpJ!j6GxpU{vn>TO%{P_zOELga3;i5&07B61BWXY1H
zOP4NNwru(G<ttXKSh;fLs#U92uU@@o&6>4q*RET)ZvFc88#Zj%xN+mAO`A4v-n?bY
zmaSX2ZriqP`}XZScI?=>bLXyIyLRv1y=Tv!y?giW+qZB3{{06I95{IJ;Gsi@4j(>z
z<j9euM~@ymcI^1^<0np>IC=8qsZ*y;pFVx&%$c)i&z?JX?)>@l7cN}5c=6(;OP4NR
zzI^4%m8(~;Ub}Yf`t|EKZrr$e^X9Evw{G9Qedo@dyLa#2yLa#Y{re9dJb3u<;iE^7
z9zTBk<jIq#PoF+}_U!rd=PzEoc=__>t5>gHzkdDZ&6~Gx-@beI?*04sA3l8e`0?YX
zPoF-2{`}?3m#<&He*5<A`}glZe*F0P^XIQ$zkdJz{pZi0zkmP!`}gnv|Npv43(XlA
z7??_e{DS{8Jl$^K#lXPI=IP=XQgQ1|?CniU6nK`z{{KI{T0X7h1JfcEA&Z9(LvnQ`
za*LN5bFj##Cl_&?W6#Opab{S!^TE3rxeuBIR_t2fAFSlWaUtfy*8av{P45~QcUJHj
zu>{pPZa-&kP_u`PiTTj=wML6Yj<bqfU3jxcR`9+>vxng#g)PSO4>KvT9JDaKU1+#-
z`Jt~&JTH=0>|1bs#gT%`Jm<tc(rW%E9PT;v$b08M?R5F&+%Cq6GxU@yjb97594iq#
zX??C`)61TQV+AMXc2^}FJF--6M##RC=3G+OXM8_?heQ0X2rpa1&Z;7l8%74P+F$b$
zg7>N#_E@lZF}=8zw0QTdw^2O1%v7aQFU2#d9qX8MnT?4}*XO+Et#b{TaTz;~Sr|Ao
z1Sfcw^lILDxI`nY__n$p*CyXXU+>(A%P>~kx;&%sdUwL#`L6d@EVA>hT$*C;+<C0v
z#xX^<2bb?`3bSn6R(8vMDR2FWUZV|b^p?0i4hVk!yOi}uBzyU>pif7vW_J{OhepoV
z_IxP5?M#A8Fjvprj63%?zTp47rR3q3e+{1O$KEUJYfY_sbJ(Phfv0%cW)0>~Y@27g
zF~~4TEcw0Y2TPP@ee$}FCl`Mk7<30c+|p-s|MP>{ewp5rV>_4JxVMct_e(s_j%M}A
zAHK5q-1M6gV3xc-`?81fB->uYPs`@079_|>upetIM5c}xW_<9v`~8A5=L`mO)$fMy
zl38MUnp8i^{_Z()CU%eAqp7;*BYfnSwR}J7Gb>Bd<DOzveemN}ktv&$wT+X~a<`@!
z>RjwJ_~`V^=DMxs{7m~*#_DN7jw(jijkJGkvgn99yW!&dTK===G^cE1Ql2zP-1z2K
z{u_&YRrRL`7{8Rf^zY5aMZ5oK{eEAdm8KV<uBxAQEp*!Zo5|(<Wd|=Y{i$sYm^ew)
zIQ#a+s&5e;((@`)nIirkUpKk<XLHY#Dnoshio)I86^B*ld_VcX#=&*+W1Y7fC0ON)
zU!+~TxBWq~`I$7=CvOj*`o?{5X>$)((77XPik|qJc3bI`r}saNQ#TLWG%uTtjiu%M
z{KY{kOFlUH-0_g<R@@LZt)iZ%aN-KP)o0$vyx%!xLA;Ay_SU`Gf*vX6Yo;5$)Hw6g
zHuzEB`UyMFeBKmP_nlug!AbtQZ`a4QXFp`ERgwSkvrpt>lxO7C`-V$4N4;DW^7`+N
z{lAm#-lf`BC-48ea$$quwYGhpQ7#7>Dr469WPUgH?^$HXVz*F>Rlv(ZF3ab5uC)Bd
z&v*Z`+$yR2C1-GazJoc#(hsK{j>-$Hdc6Dj-L1Fwesa3EW=7oVC%d<+GN!zkd7P{L
z^pE=Y%ub^Fjpya0$~U!hhwfYH=@!qTp!xW$+=Xl*yJe}WJD;DOBzd(i>Rq(+1_qzf
zh+Ci1ZhTR6aebE-^(3i(iD;rrGRqsz=jW2XA60mCWb%nRWw*S=tIW(r^!;C(+&IS<
zr<h+hXIg|y2y0Z}A>PuoDap?#%yC<BXHr7PG{4!WeHy){uD^TsLpZ<A3$J9Jx5rI*
zy&p3fUu(#}X=#w>#n&@?67%C_PY&$RgmC%ImtIP2P~G7A>^9>uiJAGwjz=heR&5l#
zQM%Rn+3DFr^K1EQ-u_D7=D4X~mhSV}!7(Cx5`$CrEnE@e{Y?I5O;nec=Atbg#j8&I
zIlR@C!{V)c<b#)op7U!v`}pnJG39CUv=e_)IVS!R4L5XqCcSRztw}%2R=+-$vM>C)
b_~ZW!PSs9HIwG}BpeBN+tDnm{r-UW|PdpYi

diff --git a/doc/equations/dist_tutorial2.png b/doc/equations/dist_tutorial2.png
index 8b2bf93486089cfb3cacebc6bf95900d609d77ec..3bf4ff04891fad162f7f3790c04468fe07ea3470 100644
GIT binary patch
delta 805
zcmbQkwvlauWIY=L1B0Y?%`XN929|V3Uq=Rp^(V|(yIz8XOI#yLg7ec#$`gxH85~pc
zlTsBta}(23gHjVyDhp4h+AuIM9rko_45_&FW^Qa>vjI=bbkXUze%pVE$obtMvXt}Z
z*5t$HLbr0ShFz{m>UY+-T_|JE@a55TMFtKJhTGfro_kzh!}C7>jfbjiOL^vhZu!Hr
zd9H}I@IN~K!Oh|92ZcqVdjtcL4mk5qoc(s;{f@uKCs*We`FK55ao*8~4)f0+Uw^t%
zaIKEn?C+xGJ|(Wd_}RbBd8u3XxW+&CfxGu5o_m^xOXIp5q)s)Lu+9AYQ+ZGJ{bu%8
zCzw9CIrtvDaKdg{Wqf@tACp#ziDh!+@9Wm@Cvm(yB(k}A`_?5Biu>%Dg7kFbT&A2_
z5X<z*qPRg&xVOT=F!-s_gtS|_l72340vu8l0-9w%UP*WQ5X)P}-*{Qb=w?|GThz*|
z2Ya}?#V1IpU3l7fg73%jprto7HK#m!smj9;t~TpZx_0x{0w;&bY<tw}e{8+&7G5`9
zNZm_fUS@|d6H|qP{=R2fnJ&9(JPVbWIV(G^9Jn6m^te-)Nyy}aDdUB>uk+7#3aJS!
zn6e_M@xrYJ{ZlVD%XcjPU-3A!%I^DruXUR(0uQyM|GjIesBkdvIp>@mYtlD$>TNx(
zn-;dZRsLa1LyOghYY(ELrsi3SyVY|VKk#0&F8OMO8VgGsOK$^1tVG1~j{+B*m}}eL
zw>QjTkBqXJk)f`%VH3m5-<l6(EM|I1`-?wYX>*4+MPc!w({BZIY{NIFCa0x63UPDf
z(B3w`>J^J-VuO&H>E_+}4oPp496jw)7Mz@W@haQ5sgoVM_?e3CEZSbTk42uv<mQh0
zWnS?${K02cnKJD-JQnEoH;4V%TAmpA_0!6SEe#^+|4w{Oj;gs4u<gd-i0FfPPY<u;
zQFip6d$I3*a231V8fMLuwNtkX%0E2ZX7rTrP3L`4M$0dzvmPCEu@+jvx^UA%pM`(7
zUU(b%(20%7*}*{~gM~#vz`=omk@0h^5EIk)EIGX$TFgBU;u{zlTyIbLKkcjddIkmt
N22WQ%mvv4FO#pyrW%U36

delta 784
zcmdnUHivD3WIYoD1H&}W*J~LV7&r?&B8wRq_>O=u<5X=vX$A%cmUKs7M+SzC^Add=
zzcVl}C?tCX`7$t6sWLD$G&3;#{LjG9@PdIMh=YMqG=hP_YB~dhc>eV$`3VdROp`oa
z978H@y@~PUJFLLt6Y%eUYE_4nB1g2rgVdT8yOwNy{jr{{%wq3%b`L(w6$}lz%Rc*m
zFHW5l^Xo!Y)1R^}X&#TZEGm^be(Z~lztB_59)C|))y`nH^OKE(Cf85$nvrnCvuKsh
z>-YcXFPrdt?uW}JmoI3gxJ(L9T<5Ips<q?o-dQ2<PM=!!p0`|HrE@~J-UZGG4u+}6
zd;a)MS9yAROK9gS)%uHJs%^8BO4jR6+EeE(o#0<N?b(M<%I2O$67z5TSu6F`%`m8v
z;fA-s?bVY{JdS2ePRd@eOonaCkG96^t<t$ECsIpqu<#~)FgRDT(y{yes>)+r3t4U$
zg<hJ~I&ngQu9<*9;ZzB|n+s$OEq(Sza~K`>sy^!XL+F>k;q;D0&Y2(TV;)ZoEt~Gk
zBegaD-8Fwvo#T9p69eaEC$b-yaJXgpitC}gp*Aw<&wCa>-P<1$JkR>}xpOyJR@zpc
zx$#RfG2KdK+2f{(voe;Kc-)9QpSVyc__f-4q2o6fdad!vC{9k_`$FLOxjPyTKh-7%
zEPpafF)^Kc13&ApM`^*P^A+1`^$*rRU(3y1$=ASX%*UYC!!Tn*%<BcLkwu3NSFGK#
zs=AY54QI2d_nR*+9a9W0A6&7<JWBPIT%+mR#QO_;cOTw4z18nj`uyM>rlwnu@82un
zda-fBi3GvRh1aefvJ4c8<2wKHg=_gu4}DJk78kGYSwH7G#;Gx!zRDE%f#>{l)p+TE
zkM%`ej<2pJfBPV*^ygNk=nX}cW83bVZuK~QI*)nt-)ACWa{X*+ljRoke)^yh*S2Sk
z|G#H(=Oh$!pQr9>)O<HbVmGf(=k0G936mK2Gt^z?^*Pk}GgVMDbdggK-;905EDsYI
scvucNs4)ars*Bq2G#oscb}65M<*ExGLzP<qD9tl?y85}Sb4q9e0O3zuH~;_u

diff --git a/doc/equations/dist_tutorial3.png b/doc/equations/dist_tutorial3.png
index 205ba53a5956fffca8e5bd09c269fa534ca4420f..d6d5d801483396a9172ec8f6a77601c26ca2c260 100644
GIT binary patch
delta 1429
zcmX@lHHUkGWIY=L14FQ`1PcQL153K2uOkD)`V;1?T`w6J7$i$vBT9nv(@M${i&7aJ
zQ}UBi6+Ckj(^G>|6H_V+Po~;1FtF-*x;TbZ+<J5OcHUzP0hSAAmF!Lb|2LRFb#ARe
zLWWB8q{^3R^O<;jjQU)~y3c0a`}6hl=g<5OtTlqx0@eq7>J#f~|JMEa-S1W}Xsx6D
zCA~4feqVQeAos-!T9bsUHz-EM9&fy%ICbg6o7|>x{YTc8D^+ekr`ziHhO_maa-nqK
zyw$0-b@i*O>Lw@5duZXLA|E*Kc*ezB?}e>-UQ47~*KGGYa_mh1EBAfhKmOjk|7~Vs
z)wJ|wUyY=NmngsCTu@qnX!=H*Bl}i_@0XISKA-5W&3}I1is^TS{nj7JO{rX0QJ$o)
z@MCd?rI*6RQ-_b~WF#uxY{+=3_|4VIW$ULiuRNB?sk?tGzg)9e_wb5OXFj|=B6s9i
zjA*jR{+jED&kI;fPd1p@du^ut`WJHZZwR}V%J$z-%ygHu6%w$XB6+ey_Edd@S(3im
zsd-M*7dVL@bQ8E%9BH&>+b!wI1{;ILCdxXbtbee$-;aOq@4GYC?pf_pn-qV8>ka46
zyPw12IFgeadK)dn)J`9DSF*k%bnPNz`cuWSf3+8$uUD%`;#c_amgmH0rL!+sXM<dP
zl=c6+zQgyLc1O$IX65+t?RS695svEm!fiI?`~?;(Lrfgc#?15CH}6I>M^66t2b({*
zm33u2%?#6LU-!Ff+4s3Uk*`l5Yv}WRV>;=IXiMF)eVeV=Hmdt&$v)V@R-HVlPvVr4
z<J0uQgU1*;-Ylw?<Z~$#xan9DzxJJyb;lcpUF*Zrlhl70m8zDl$W2~UJC)OcJLZ<&
zh5Blbs_omIo;N>Bn!h!=e}<w@%K9k|Ns9K{4u1aK#ZcvbPs8jON3g}F{#kZPMl-a^
zq-T7JSa&K~@LWNG#reEnhVIJw`-}Gedi6$|Z(n3rsgQxpp0?hEfOL@!u@8*rSD5dT
z`*~(}SV~)~=kbI8lfLiz_N)26?*D&x&+k1Q+{$cLueRX)o@&R8Wk)wi%0xK&CWzm=
z*>F+m{3*us_cknyue`80<J-H>O6PW@JZ<RBym7+)zUvg<Gpman-U$E6NLxR5CbQYH
zh5mP|&tJU0HKpg@P0I-j%=*%9XKcN`LwkLV+p~t=msf6BbR1{gXfE@p@WH89MZJF|
z7Ht<?cWB*_6EYqF_0p<o^ShG1d{!<jXloZcpPsDtD~j>SukXu+tY5r2!sqa2LxrEq
zn}DzvNB9z3^c-XNEOa+5`tqy)(Sb#mR6^!gnMgA$Y<ey+N!$H5kBoq|cTgL%8QTqs
z9riuD6HL_Dm8uuJ*;!@uFeg2aEp&JjDP3+Y{c%r(QPD#d_9J$CHr9XVlac3KXvMy`
zLgCxFi*GYL;;+7&^Yp;OUe+|l0)gw-g{u!AyLbEOcH5ku=1Lihg4%_O)iM@xg%>Q0
z*wgK{UoCbmi`c|0Xx$>Zi)E{W)AJ8)how&+e%Sl*>+}U|U#DN3bXn9QA$b8$$KOX6
zUR%A|BO_paI;4#`N&N2V0*5!Q^^2=YHkL(hG?V^3Ga@ZVvE$R692tS#7Z>yBXsOCR
zyI+|l^XAP$@tdXhuFX}^VLx*2SV-ed35BvB)<SMyHYlH-GM71N|HLmncB22iG8T(h
z7!^Cb5&!#k&FuA2_U~UkUweRYBfHQ8^`|pZ^^FQ0-tc;q{toBk>)7j<y!)^Ay|?l8
z)f<|AB_`XOzgc~If1K&3i~n8cv)i9O_|3LjHi?D($UnzTyJo$Ad-UMU@4qhHKhSIY
z>)Gmz?Sd&!98&THTDR|iEx1(4^}oNxkCYUXuNIl#5=*nMHaE|EYV-N#5x&FLPu3i8
yd}D5ttg_$svD|e3SN4;cqiYuO{1Z-K*k=@J#d(GAe>npK1B0ilpUXO@geCyQ>bR-^

delta 1351
zcmbQkeV%KAWIYoD1H&Ev{$K_M2F?PH$YKTtz9S&aI8~cZnt_3VCEd~2k%3|3yhI<z
z?+gqK3dtTpz6=aistgPb%?u1b|1&T&ykKAm;$UDDjbLD~n$Ex=o_{?`egXpn%TZ4k
z$B>F!Z|2^f^jLw%W%88&|EKTeU+I~e%g($@;ljnsK5F&CcP@D7>Rl|p|DC-p^me!2
zfmu0=xi{VX{Ill&1@X89AFEw89iRRh2XgK<+nLj9XufVXdz)cIy@%8B7hjeIc&~bK
zt@i$F`Q5Yien`mfdQ+b!D|IS2)b0R#!u`vSl9x<AU(6YD@7iotp|ry@t_CeMdiiut
zv*CBXXF9U?*2~<}5v*tL$~EPbD(P8RqH#V%ch`=stjT9YeIBlN(zV>CyJ*r*BMY9X
zG2W}Tr%KI8@VV_UZ~2aQ2P(F?hHT>X%nLlQSyv+2X_@5hO^am%SC%Q<_j^&TqO+wb
zJ=)-ymfM|uMLScHZe4Tmbq?9&+ot<v+Icy#o|n`2a<1e|{-nosKq6;DeSC}I1=eF{
zCagaFvgy;@c`SU+KQc2OEHZd)XuQ<miPw#qwtmBjvd@)^_t)vp$?bSqk<fFzLM9{7
zqCj%LbBC9N-7y^l-aQA}`e%0PJ->X5dkbswvXeR4cIrl%31|L3s9kO_|Eh6!s>%6n
z2|dD@eCgZHFMfSZY&T2b9*?utS$y@UGi}d5%{?*8In+&hn!z%&O3q8y3ObpSSFwHC
zko>1Mq59soUp-%EY_t6PclEE5{H*LNnvykV*5vgYS6n=r@32c`-p#%s1GnHUX%-=^
zeTEYixJ_F6Bee{8Lo5y0nhkf}?J1RFO*W92eB-Ren#%_yc#fr%8ryLnG)NP?wBxOG
zz0FFUuFe{t70gW$4!iHGUMMQx-_bF7_H4s4MVYjekA{~y9oVjxY|GE%x}&c3@TA)>
ziMuVm8)N1e3fHD*p4niL|9+}s&IQ-)>sv)%KX$&sA-`gU8pj)<gxqNzhHFpXeE$0E
z*Z+AUlaw4zEDHbiYToyi!q&T78XQ8hxSxOh{<yt<y{eSsjRJ#Ll`C1pA32<;j$gT3
zta9O{)x{n1Ob@4CKIjwJZKKxm#!9V7_tOjkvGY^8jNgRoembou_o8l*;|-(g=?}J^
zNOtm&ohr72<t4wOX7aTiy4FkXzPVZYB<hKw;+i!zt+M6O-|sX=YYA{^SlP{wZJM{H
zbP?y`I6v8A2kI*%nkQ-}G9MM_Y1A`d+r60k)b*9S<&D>Wm8|`&fB*UJoHE~aEv*}R
z?9#fNo+K)!r1-Q|XOy{x9Qt6W8&`Yv_m>|MPxtl3ci8B(yh;Diz>vJE%Cb)J(*}V_
zza-8tdVW~(#iA3^Q*FBE@i!jm)cUKFoXhb>NaK6O2ast`_SOq){`t;PuGBjJ!uOMF
zey)4(YZ=&TdYSd<>*acufu#n1A(hXiHrxNt{uwy`m-R6wCk3rUgJY?E%UT=yIb|m?
zIe8qExWi+Y#>#Ho#puKlDe-yU;zN#7()~>i3;KkrR-E0Y>aH{;KbnWhPK|4Jx+F&#
zXWQB9o{NQc3b7^s>u0JzH_Kt~&J2N|H$pZ!eplo9mLCm@*5J@Ke)eJRU$I4wpJtl=
zmlkl+lYP9{#5ep)yYq=1>yM~<Ov#V#FXjwN>b#cBcT`Yc-!bmn*PllNUu&dtv<e;V
z5?J9X?P_&GQRLnMbBFS06JD8f*uIx$X`FEAxCqxaF7^4(e}1Uq@U@%z#^ILc>8;#N
zP8?cm#qC;Zmzs*sz8m5hsCYMcO2iwFaJxy@xO^p<o+ocG`)$r}ZPB!sSJ?aZGB7YO
Nc)I$ztaD0e0su1QW`h6#

diff --git a/doc/equations/dist_tutorial4.png b/doc/equations/dist_tutorial4.png
index 4a94b80a1fa4aa808d283e051266fe8023a5160e..3e89ffb00596267f757f9281be33fe33c6016c84 100644
GIT binary patch
delta 616
zcmaFJ@{MJJWIY=L1A{cntCb843@qu6zK#qG>ra@ocD-a^V2~_vjVKAuPb(=;EJ|f?
zOvz75Rq)JBOiv9;O-!jQJeg|4z`(@e>EaktaqCS?G#|4f#~UgAtH0~NMBH}Y)Oqt&
z*8=TB6TU3=jb?R|m{4~6`(MV!B|-^ICnA-665O}tRxdqoUmy8^Gr)q~g{#TfVS=5)
z8;y_5A|1>2eo<s(QEGKGsF9y^g)zXv!DMfU+@&vpPegXQs&LQ0Y3Q=PMXK6Yc4js2
zBMuQMzui*|9NC(a*c1dNiZnga$WL?SaOv_~vh~`YM+{A`&a8gFblvUMiN)pi2V*^F
z1@B^w(QSy(Jh~(_hP(d3`&;6iBC~qmH5Q+_r0+4OY0mlbZE3ezb3Oc&9ZnXk@KF=+
zeYL^uq@~az&zG}`JXe?*e7(nS{%eATYwlvxRF$r8MnQUyDmwIROL(FJ?0&ICvAg)!
zADOv4Qdpw+qDvKH#P2(f7VfVm-*)bmn=ya+f-?8wzKJmm=k^I%u;s0*{}xu|zk!F%
z@9>$VnRi!BKB+i6t|qDV()X<Xm<7o)>nA+7oqQ?7_onry+RV9uflpV-yF_lDdciGw
z`JVQR*=}=8-7aier}A~Dzx?7f^QzkXl}-<==6>3larNXCVOuS!o_*p{ng^v6`X;CL
z8h8b}ciSH7I<|B6i?gekcBE$9TK+Kc28TVHQJCKM9V<-kUffc}X%@9UwZ7Ev<rBs&
z3bX2u9ow;Bv&Tg1T-BtdxhLGJRf;cOyb>}`Z{n@6|LW3SA{QQYgr3m<caP(fdOjl?
a!{iDh?%y2VI~f=l7(8A5T-G@yGywnr@C|bS

delta 595
zcmeyy@{nbMWIYoD1H+WG+J6iT44efXk;M!Qd`Cc-ajG_-Gy?+zOS+@4BLl<6d5J!b
z-x(Mf6p}rHd>I(3R2di=ni&{={%2rlc)`FB#KFKQ8o|I|HJyP$JpX!>`~(ID#_OIg
zjv*Dd-poGOcUXbP<?yQi|4-j^Ub@1oNm5!yEyJQTbXvWV{o$Uu+ur=%mwBN_NiO8^
zVu6&@D$V2BHZPAVb00hSK-{PJk7n7;!%I0HO+30_$|EbuOP|~Iw8LADI@>&16Jyi&
zbYK45e@B9ylOt>^xU}U`KYBcW$Yp2|d85hZP7%u)@t%%(^{Iz--PcC!zT#n{WZu8s
zWNUce(}M-&CfUq;>us2xA8vCj%A2s{WX6gq4qWOV7hIYeIlJ(K1K%ejo3yvbk1xJ#
zExE1tLP_9<4cUKhTRf`Do1q=^WIj*u`SgcUm#_8TI`=85@lNu2>-m0ig3c|UJ$hYp
zzuWG;)M@kR|A~U^W~F@p-@fAc{8wh&eE&P&Jos|!^$!_s7I&Cs-~Do^H!rd8oqxRX
zpvRNddjvo2kem4ZW5L=97ujStU0Ev~HQ)UPyWRQ4ui`T*E3~C&A1^fx$cf1}jZON@
zvcABsYKqO7#m{EtsK(DfZmQQPeXJm_!dXgxq0XN(HWG4iU(Jie9vW^pI(~j#E$h`O
zJ+qD<`!$CvKhs>7b4K(dLx&q|X5W3{4mAJVB(=}^oBNVK25j{j#{C8Voo$Y_KL`oD
z{r8AzLcr%463b>zS9!RdBWdX;CIhp-|C)3E@H6bud-%jxthbGUfq}u()z4*}Q$iB}
D$q5fL

diff --git a/doc/equations/hazard.png b/doc/equations/hazard.png
index 7a5978611c9374bad55fb93c3298cc054f7da0e7..b65aebd46b1b476e91e0d02d3ab100c8e609c896 100644
GIT binary patch
delta 761
zcmeBR+r>6Pl7o$bfkE{i&)JEJ7WHR5T^vIyZoQd%F|SpDr)~L7@3(*TzeS|)HdJhJ
zYGe}Krysegc*;wyznL3feEnIokAY#$P62TSiG#hI40GHWu78!TudB<twe_9z6Q&It
zH~jhi^8DwYKW)G6{Ia)%=j}chJ%-r<MZe+_EB_tq;9x6cWVoc%AbV+!LC^%Ho`UK-
z^~=n%IB!4g%bhEysK+4Osw&3j$#dxK(!X1o77Chp7b@y9%{cOS$3o`gf}U?4ToYca
ztd=Jz&M<@P5mSMOm)Waz0*{y;oVZw=8L-DT@3NK-$iB}U9Z$Zje7o>P#;W3Ar?6k=
zq8g6uc3pjCoqMC{^EV&=&$3O53d@_9<G()8rT(;`+nnobL#_Q6rt}x4J)e8e&FA&G
zvQ4&o-Q8BRzJK?&MNa0h_s-8N7i)i8Bdw5r!>iiY%`NDDOpy!oAJ@LkedeF$>@$@$
zDivCp<?K22an|7w>6JP=m#zv3ekU_6biL2>Fw1IRPK$3W#f6=xCu}{vFZ))@!bQt?
zcDZdVHvY7`zAwyFu;<#c8^ULfy6&2=)5BFrcxA82qp9!yUzWVhy7ai(jbL%diQh`k
z>^!pC%yeh%y0a_i7A)KROv&?2=Lx_0#TU;N_hdXyTyk%v@V7bK-&&&bxYf^fc5HJp
zxY`+YdcjWZN!#7u&f)&XVqDlMYx(rX$0u<SlXPqyc3g22nz4hYewSI!ywG=l(heS+
zmvHZRSw+MZ=}RVWI?sl=uby?~QqjrdX8wylf7tRlcy8<0sMO`Lg?_8=OKdp1Due&&
zIqOE-uVtUU{jaV}KCt(8>$V&08`QrXzNOUQ8?@ohr<r$*W(9iN35YjL=kU(E=MtrL
zIcj#w#wW~$j1dBrUzW{Wy0vy%n&^kMEA<~S8FXHLw!ETEk0B*^<24PALdJ^9uBTsn
z8TXtMJ#gb<xa{#)e=n_6+7oj75%Xt3aRv#nF%4ObfomO|Vi*>9nfOKT;9%JILpXsU
Zr+n+Yr&5V885kHCJYD@<);T3K0RRumciR8}

delta 711
zcmdnR*1<MGl7oqXfnn92*|R1pTGY?-ba4!+xb<ev%}K2aJTB{}{r`XZT(hazRE8!U
zCyT!(%{yE!sWk4(dtUjEfr0f#B`d=Lr41U}a*O}(`*6efuYe9ycHoOY?P*6#O-f|n
zYi{7sX%sMYsh;DswS%RbL18ho+vcqSbCqu?&vjKO%2vO)-#|vTM^a+4lQ^S^@Vm15
z=?t3EX5TwI|8}}2ylZ;?t;pPK$Gv!`=2z8AHgGURusHbbib&0r7HiC0r9AiO&bRx|
z*j|41R6vJmak%@vRECRnpS6@?1h(+}jdn6PfAX!yY*S|!xy7G_x6Md>d_Mo`w||bx
zAA?%U?>wIJR&2-SHCYpHeVdc&?N*X?dw*BvTF?63&-)75r@vZ!Yeu=$Hs2-Uto|oo
zuM+xxqjauC>_#cus<_Rk-u8X@a`&$8w`a*!!8gB37ak0#T=0F>(H^C+94)mAciBuo
zwTW-ME7A9(=A~%T3*9Fk=87*aT=$>p8nwb~-`5#adbw9`TYpIDZ`9t8?#Axrmp7+n
zuqU)m+;?(vYG1wN?X_w1+}}kD)UB62bUXK+;ktJR73VIkyv^&rN4fmX@9F2>KYO2e
z+~T#{v7GH)e_Q>lErst(izQE6_jtpWMWR|#;Tvjo#dhZD#@x{@KDvJHmvWU7b6=J4
z;}a)kUJ1?Rab0ya$GlCD{qf;9g16g^HoI~OcqYYjuXhVs{4FrIr=IikHv3a<w{_=i
z`5kh>Zn5#rbLly^>nsb;-(I>!oNN8JiSggNozptDta)q9Br7{(-z1*a>GLNsR+Oec
zJ)|AM@@(0J@EO;B20L!x;E{?h+I(}uxzAsd+^RnDbRRG=eYoMlZf=Gj!biU-uyr>a
zOl><ipR39Bfpq!$P2a1$Pqw@+H~AMLP^81))f#?82}Ew-Xh>JG<@j*&1%t+s`3%eq
ZS8Dri%ce~1VPIfj@O1TaS?83{1OQYtSRDWW

diff --git a/doc/equations/students_t_dist.png b/doc/equations/students_t_dist.png
index 5d4657d8c4e667f2ea3a082205331f38ad2552f4..3a0a239ae6d2cb6b241c882642729a27075e0270 100644
GIT binary patch
delta 450
zcmZ3%d69X7WIY=L149w3eLMpL153K2uOkD)`V;1?T`w6J7$i$vBT9nv(@M${i&7aJ
zQ}UBi6+Ckj(^G>|6H_V+Po~;1Ffb;2x;TbZ+<G(nB3H8kkL%kP{%`-we~Zkk*2r0u
z*{Wjt@s7A7i|Teu&&AV9W5q?{`JE;>PZfx9UOMIa_aBN+>Nl64y;sV*<(R?dbd!vw
zZugX}tDc*@FQ1aDeB<begA2SGztxHL_-t74B1(Nhg*NBhy?I;HHQclkL#2182l7QV
zD~l?#GvAt!d39q}!qi5G<>$6C^lkYc_5HW$yBC!Pml!g6SPU*+_c)!&78KJV6t#GL
zP*2R0aJ6~TnPNhcmD=^8K0P<NbM~$b=U;F=scT=x>#Jd6L3iprF0+bkaGPXkm3mmY
zP4v!F4cVR-J-%PGd?jY4ST|2RmR7ZJ>h*}3+vPhN8A=O+(z&A?wVBo~2$x+eF*D&c
zW0a$|)#Tpl!^Lm?Gf#N1zm>dtF7u(d^3>T<FVz)J|I~K#lj6!Scb3SsZt#5>wR+8x
zn}J5f-^4{zmK3w^{`;6qnK|LwuluL=R~wp~U$~utnL%rwwA#cwDgPN57#KWV{an^L
HB{Ts5Gts^4

delta 1183
zcmcb}yn=IrWIZzj1B2`B^SKNR44efXk;M!Qd`Cc-ajG_-Gy?+zOS+@4BLl<6d5J!b
z-x(Mf6p}rHd>I(3R2di=ni&{={%2rlc)`FB#KFKQ8o|I|HJyP$JpX!>`~(IDW`+Qt
z5LX5;U}9ooW@ct#VPR!uWn*JwXJ_Z&;Naxstmoq5;^yY&;o;%s<>lk!<LBoW5D*X)
z6ciE?5*8K~5fKp;6%`W`6BiekkdTm+l$4T^l9ra1k&%&=m6el|lb4rQP*6}*R8&$@
zQdU-0QBhG<RaH|{Q&(5l(9qD-)YQ__($?12(b3V>)z#C})7RHGFfcGQG&C|YGB!3g
zsW&k(H8nLeGcz|gx3I9Vw6wIcva+_ewz09XwY9agv$MCicW`iUbaZrba&mTdc5!iW
zb#--fb8~lh_wexW^z`)d^78if_VMxY_4W1h^Yi!j4+sbd3=9kk3JMMm4habf4Gj$o
z3kweqkBEqfjEszmii(boj){qhjg5_qtB;G1k55QQNK8yjN=iyjPEJWlNli^nOG`^n
zPtVB6$jr>l%F4>l&d$ln$<58p%gf8p&o3w_C@d^2Dk>^2E-oo4DJ?B6D=RB6FR!Sm
zsI084s;a84uCA%6sjaQ8tE;Q8uWx8*Xl!h3YHDh3Zf<F5X>Dz7Yinz7Z|~^nsPF9T
z?CR?3?(XjC>FMq5?d$97@9&>5VZy|T6DLiYG<ovmDO09QojP^ev}x0)PoFVk#>|;B
zXU&>5d-m)(bLPyQJ9pl^dGqJbU$9`o!i5VLEn2jA@!};*mMmSmblI|H%a<=-v0}x_
zl`B`RTD5xh>NRWDtX;cy-MV$_*RS8OVZ+9a8#is*w7Gus<}F*cY~8wb+qP}nw{PFE
zW5>>&J9q8cwR`vOJ$v@--Me?+zJ2@m?>}(hz`=tD4;?ym`0(K)M~)mldi2<_W5<sl
zKXKy3$&)8fojP^;^yxEa&YV4a_T0I1=g*(NaN)wmix)3lx^(&S<ttaNT)ler+O=!f
zuV24$<HpUKH*ej#b^G@1J9qBfy<30x-o1PG@85s$;K9R(4<9{x^!V}PCr_R{efsp-
zvuDqrKY#J!#mkp3U%h(u`t|EKZ{ECp`}W<tckkc7|M20%$B!RBefsqI^XD&LzI^@q
z_1m{^-@kwV@#DwOpFe;7`t|$w?>~S3{Qdj)-@kwV|NqxbT4>I|z`#@z<QM#(;puh*
zF9rri7EhOY$B>F!cV>9<wHoj^e_rr?|EB-Te12{5@tly*I6*7vk%e%=&Fd5TGF0W9
zj(tgZQqh^&^h`C0UH{Xv+vg;s>uME_Cf(X~p!3MOeal@mO0J!i%3;&}x93v1QRfZc
z8G99$wk7<p*eW>xfn#%^@#3)ND!y}DmQHEv?%-Z>iF@8V;hOrdXD61%zf{>4uibt;
z?q1GCe}Pg%j;CuJ?<7U;Ia(ODQOVHLEoo2S?F7!G^~px7Di`K>>@`?hE@Yb#_58MV
zXP>NIXHQ-J&0x8W>TNye1+=pg?@Jdv+H>U96&e1OmAXMeI;9$m4?fCrJnUQlb3wC^
sh?;!jvAili$EeWV-jn}J@V_@_2>xT+qVchI1E{3%boFyt=akR{0M@l8?f?J)

diff --git a/doc/erf.qbk b/doc/erf.qbk
index ccf337f81..82b4dda35 100644
--- a/doc/erf.qbk
+++ b/doc/erf.qbk
@@ -70,15 +70,6 @@ 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
diff --git a/include/boost/math/special_functions/digamma.hpp b/include/boost/math/special_functions/digamma.hpp
index 0a925890e..a22f80c72 100644
--- a/include/boost/math/special_functions/digamma.hpp
+++ b/include/boost/math/special_functions/digamma.hpp
@@ -158,8 +158,8 @@ T digamma_imp_1_2(T x, const mpl::int_<0>*)
    static const T root1 = 1569415565.0 / 1073741824uL;
    static const T root2 = (381566830.0 / 1073741824uL) / 1073741824uL;
    static const T root3 = ((111616537.0 / 1073741824uL) / 1073741824uL) / 1073741824uL;
-   static const T root4 = (((503992063.0 / 1073741824uL) / 1073741824uL) / 1073741824uL) / 1073741824uL;
-   static const T root5 = 0.75231638452626400509999138382223723380394592598054e-36L;
+   static const T root4 = (((503992070.0 / 1073741824uL) / 1073741824uL) / 1073741824uL) / 1073741824uL;
+   static const T root5 = 0.52112228569249997894452490385577338504019838794544e-36L;
 
    static const T P[] = {    
       0.25479851061131551526977464225335883769L,
diff --git a/minimax/f.cpp b/minimax/f.cpp
index ed59131ce..862fd97c6 100644
--- a/minimax/f.cpp
+++ b/minimax/f.cpp
@@ -122,6 +122,10 @@ NTL::RR f(const NTL::RR& x, int variant)
       }
    case 10:
       {
+         //
+         // Digamma over the interval [1,2], set x-offset to 1
+         // and optimise for absolute error over [0,1].
+         //
          int current_precision = NTL::RR::precision();
          if(current_precision < 1000)
             NTL::RR::SetPrecision(1000);
@@ -129,11 +133,16 @@ NTL::RR f(const NTL::RR& x, int variant)
          // This value for the root of digamma is calculated using our
          // differentiated lanczos approximation.  It agrees with Cody
          // to ~ 25 digits and to Morris to 35 digits.  See:
-         // TOLMS ALGORITHM 708 (Didonato and Morris).
+         // TOMS ALGORITHM 708 (Didonato and Morris).
          // and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher.
          //
          //NTL::RR root = boost::lexical_cast<NTL::RR>("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871");
-
+         //
+         // Actually better to calculate the root on the fly, it appears to be more
+         // accurate: convergence is easier with the 1000-bit value, the approximation
+         // produced agrees with functions.mathworld.com values to 35 digits even quite
+         // near the root.
+         //
          static boost::math::tools::eps_tolerance<NTL::RR> tol(1000);
          static boost::uintmax_t max_iter = 1000;
          static const NTL::RR root = boost::math::tools::bracket_and_solve_root(&boost::math::digamma, NTL::RR(1.4), NTL::RR(1.5), true, tol, max_iter).first;
@@ -142,6 +151,12 @@ NTL::RR f(const NTL::RR& x, int variant)
          double lim = 1e-65;
          if(fabs(x2 - root) < lim)
          {
+            //
+            // This is a problem area:
+            // x2-root suffers cancellation error, so does digamma.
+            // That gets compounded again when Remez calculates the error
+            // function.  This cludge seems to stop the worst of the problems:
+            //
             static const NTL::RR a = boost::math::digamma(root - lim) / -lim;
             static const NTL::RR b = boost::math::digamma(root + lim) / lim;
             NTL::RR fract = (x2 - root + lim) / (2*lim);
diff --git a/test/test_digamma.cpp b/test/test_digamma.cpp
index 3f7dd4578..ecf244de0 100644
--- a/test/test_digamma.cpp
+++ b/test/test_digamma.cpp
@@ -42,14 +42,6 @@ void expected_results()
    // Define the max and mean errors expected for
    // various compilers and platforms.
    //
-   add_expected_result(
-      ".*aCC.*",                     // compiler
-      ".*",                          // stdlib
-      ".*",                          // platform
-      ".*",                          // test type(s)
-      ".*Root.*",                    // test data group
-      ".*", 220, 40);                // test function
-
    add_expected_result(
       ".*",                     // compiler
       ".*",                          // stdlib
@@ -137,12 +129,11 @@ void test_spots(T, const char* t)
    // Special tolerance (200eps) for when we're very near the root,
    // and T has more than 64-bits in it's mantissa:
    //
-   T tolerance2 = boost::math::tools::digits<T>() > 64 ? boost::math::tools::epsilon<T>() * 20000 : tolerance;
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(0.125)), static_cast<T>(-8.3884926632958548678027429230863430000514460424495L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(0.5)), static_cast<T>(-1.9635100260214234794409763329987555671931596046604L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(1)), static_cast<T>(-0.57721566490153286060651209008240243104215933593992L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(1.5)), static_cast<T>(0.036489973978576520559023667001244432806840395339566L), tolerance);
-   BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(1.5) - static_cast<T>(1)/32), static_cast<T>(0.00686541147073577672813890866512415766586241385896200579891429L), tolerance2);
+   BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(1.5) - static_cast<T>(1)/32), static_cast<T>(0.00686541147073577672813890866512415766586241385896200579891429L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(2)), static_cast<T>(0.42278433509846713939348790991759756895784066406008L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(8)), static_cast<T>(2.0156414779556099965363450527747404261006978069172L), tolerance);
    BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast<T>(12)), static_cast<T>(2.4426616799758120167383652547949424463027180089374L), tolerance);