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<title>Why use a high-precision library rather than built-in floating-point types?</title>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.high_precision.why_high_precision"></a><a class="link" href="why_high_precision.html" title="Why use a high-precision library rather than built-in floating-point types?">Why use
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a high-precision library rather than built-in floating-point types?</a>
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</h3></div></div></div>
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<p>
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For nearly all applications, the built-in floating-point types, <code class="computeroutput"><span class="keyword">double</span></code> (and <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code> if this offers higher precision
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than <code class="computeroutput"><span class="keyword">double</span></code>) offer enough precision,
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typically a dozen decimal digits.
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</p>
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<p>
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Some reasons why one would want to use a higher precision:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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A much more precise result (many more digits) is just a requirement.
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</li>
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<li class="listitem">
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The range of the computed value exceeds the range of the type: factorials
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are the textbook example.
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</li>
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<li class="listitem">
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Using <code class="computeroutput"><span class="keyword">double</span></code> is (or may
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be) too inaccurate.
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</li>
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<li class="listitem">
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Using <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
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(or may be) is too inaccurate.
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</li>
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<li class="listitem">
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Using an extended-precision type implemented in software as <a href="http://en.wikipedia.org/wiki/Double-double_(arithmetic)#Double-double_arithmetic" target="_top">double-double</a>
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(<a href="http://en.wikipedia.org/wiki/Darwin_(operating_system)" target="_top">Darwin</a>)
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is sometimes unpredictably inaccurate.
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</li>
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<li class="listitem">
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Loss of precision or inaccuracy caused by extreme arguments or <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
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errors</a>.
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</li>
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<li class="listitem">
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An accuracy as good as possible for a chosen built-in floating-point
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type is required.
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</li>
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<li class="listitem">
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As a reference value, for example, to determine the inaccuracy of a value
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computed with a built-in floating point type, (perhaps even using some
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quick'n'dirty algorithm). The accuracy of many functions and distributions
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in Boost.Math has been measured in this way from tables of very high
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precision (up to 1000 decimal digits).
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</li>
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</ul></div>
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<p>
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Many functions and distributions have differences from exact values that
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are only a few least significant bits - computation noise. Others, often
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those for which analytical solutions are not available, require approximations
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and iteration: these may lose several decimal digits of precision.
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</p>
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<p>
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Much larger loss of precision can occur for <a href="http://en.wikipedia.org/wiki/Boundary_case" target="_top">boundary</a>
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or <a href="http://en.wikipedia.org/wiki/Corner_case" target="_top">corner cases</a>,
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often caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
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errors</a>.
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</p>
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<p>
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(Some of the worst and most common examples of <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
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error or loss of significance</a> can be avoided by using <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>:
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see <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a>).
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</p>
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<p>
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If you require a value which is as accurate as can be represented in the
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floating-point type, and is thus the <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic#Representable_numbers%2c_conversion_and_rounding" target="_top">closest
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representable value</a> correctly rounded to nearest, and has an error
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less than 1/2 a <a href="http://en.wikipedia.org/wiki/Least_significant_bit" target="_top">least
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significant bit</a> or <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ulp</a>
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it may be useful to use a higher-precision type, for example, <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, to generate this value.
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Conversion of this value to a built-in floating-point type ('float', <code class="computeroutput"><span class="keyword">double</span></code> or <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code>) will not cause any further
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loss of precision. A decimal digit string will also be 'read' precisely by
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the compiler into a built-in floating-point type to the nearest representable
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value.
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</p>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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In contrast, reading a value from an <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span></code>
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into a built-in floating-point type is <span class="bold"><strong>not guaranteed
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by the C++ Standard</strong></span> to give the nearest representable value.
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</p></td></tr>
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</table></div>
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<p>
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William Kahan coined the term <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">Table-Maker's
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Dilemma</a> for the problem of correctly rounding functions. Using a
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much higher precision (50 or 100 decimal digits) is a practical way of generating
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(almost always) correctly rounded values.
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</p>
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</div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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