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   4 <title>Why use a high-precision library rather than built-in floating-point types?</title>
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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <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
  28       a high-precision library rather than built-in floating-point types?</a>
  29 </h3></div></div></div>
  30 <p>
  31         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>
  32         <span class="keyword">double</span></code> if this offers higher precision
  33         than <code class="computeroutput"><span class="keyword">double</span></code>) offer enough precision,
  34         typically a dozen decimal digits.
  35       </p>
  36 <p>
  37         Some reasons why one would want to use a higher precision:
  38       </p>
  39 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  40 <li class="listitem">
  41             A much more precise result (many more digits) is just a requirement.
  42           </li>
  43 <li class="listitem">
  44             The range of the computed value exceeds the range of the type: factorials
  45             are the textbook example.
  46           </li>
  47 <li class="listitem">
  48             Using <code class="computeroutput"><span class="keyword">double</span></code> is (or may
  49             be) too inaccurate.
  50           </li>
  51 <li class="listitem">
  52             Using <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
  53             (or may be) is too inaccurate.
  54           </li>
  55 <li class="listitem">
  56             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>
  57             (<a href="http://en.wikipedia.org/wiki/Darwin_(operating_system)" target="_top">Darwin</a>)
  58             is sometimes unpredictably inaccurate.
  59           </li>
  60 <li class="listitem">
  61             Loss of precision or inaccuracy caused by extreme arguments or <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
  62             errors</a>.
  63           </li>
  64 <li class="listitem">
  65             An accuracy as good as possible for a chosen built-in floating-point
  66             type is required.
  67           </li>
  68 <li class="listitem">
  69             As a reference value, for example, to determine the inaccuracy of a value
  70             computed with a built-in floating point type, (perhaps even using some
  71             quick'n'dirty algorithm). The accuracy of many functions and distributions
  72             in Boost.Math has been measured in this way from tables of very high
  73             precision (up to 1000 decimal digits).
  74           </li>
  75 </ul></div>
  76 <p>
  77         Many functions and distributions have differences from exact values that
  78         are only a few least significant bits - computation noise. Others, often
  79         those for which analytical solutions are not available, require approximations
  80         and iteration: these may lose several decimal digits of precision.
  81       </p>
  82 <p>
  83         Much larger loss of precision can occur for <a href="http://en.wikipedia.org/wiki/Boundary_case" target="_top">boundary</a>
  84         or <a href="http://en.wikipedia.org/wiki/Corner_case" target="_top">corner cases</a>,
  85         often caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
  86         errors</a>.
  87       </p>
  88 <p>
  89         (Some of the worst and most common examples of <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
  90         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>:
  91         see <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a>).
  92       </p>
  93 <p>
  94         If you require a value which is as accurate as can be represented in the
  95         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
  96         representable value</a> correctly rounded to nearest, and has an error
  97         less than 1/2 a <a href="http://en.wikipedia.org/wiki/Least_significant_bit" target="_top">least
  98         significant bit</a> or <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ulp</a>
  99         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.
 100         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>
 101         <span class="keyword">double</span></code>) will not cause any further
 102         loss of precision. A decimal digit string will also be 'read' precisely by
 103         the compiler into a built-in floating-point type to the nearest representable
 104         value.
 105       </p>
 106 <div class="note"><table border="0" summary="Note">
 107 <tr>
 108 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
 109 <th align="left">Note</th>
 110 </tr>
 111 <tr><td align="left" valign="top"><p>
 112           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>
 113           into a built-in floating-point type is <span class="bold"><strong>not guaranteed
 114           by the C++ Standard</strong></span> to give the nearest representable value.
 115         </p></td></tr>
 116 </table></div>
 117 <p>
 118         William Kahan coined the term <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">Table-Maker's
 119         Dilemma</a> for the problem of correctly rounding functions. Using a
 120         much higher precision (50 or 100 decimal digits) is a practical way of generating
 121         (almost always) correctly rounded values.
 122       </p>
 123 </div>
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 126 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 127       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 128       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 129       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 130       Daryle Walker and Xiaogang Zhang<p>
 131         Distributed under the Boost Software License, Version 1.0. (See accompanying
 132         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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