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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 double is (or may be) too inaccurate.
  49           </li>
  50 <li class="listitem">
  51             Using long double (or may be) is too inaccurate.
  52           </li>
  53 <li class="listitem">
  54             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>
  55             (<a href="http://en.wikipedia.org/wiki/Darwin_(operating_system)" target="_top">Darwin</a>)
  56             is sometimes unpredictably inaccurate.
  57           </li>
  58 <li class="listitem">
  59             Loss of precision or inaccuracy caused by extreme arguments or cancellation
  60             error.
  61           </li>
  62 <li class="listitem">
  63             An accuracy as good as possible for a chosen built-in floating-point
  64             type is required.
  65           </li>
  66 <li class="listitem">
  67             As a reference value, for example, to determine the inaccuracy of a value
  68             computed with a built-in floating point type, (perhaps even using some
  69             quick'n'dirty algorithm). The accuracy of many functions and distributions
  70             in Boost.Math has been measured in this way from tables of very high
  71             precision (up to 1000 decimal digits).
  72           </li>
  73 </ul></div>
  74 <p>
  75         Many functions and distributions have differences from exact values that
  76         are only a few least significant bits - computation noise. Others, often
  77         those for which analytical solutions are not available, require approximations
  78         and iteration: these may lose several decimal digits of precision.
  79       </p>
  80 <p>
  81         Much larger loss of precision can occur for <a href="http://en.wikipedia.org/wiki/Boundary_case" target="_top">boundary</a>
  82         or <a href="http://en.wikipedia.org/wiki/Corner_case" target="_top">corner cases</a>,
  83         often caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
  84         errors</a>.
  85       </p>
  86 <p>
  87         (Some of the worst and most common examples of <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation
  88         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>:
  89         see <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a>).
  90       </p>
  91 <p>
  92         If you require a value which is as accurate as can be represented in the
  93         floating-point type, and is thus the closest representable value and has
  94         an error less than 1/2 a <a href="http://en.wikipedia.org/wiki/Least_significant_bit" target="_top">least
  95         significant bit</a> or <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ulp</a>
  96         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.
  97         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>
  98         <span class="keyword">double</span></code>) will not cause any further
  99         loss of precision. A decimal digit string will also be 'read' precisely by
 100         the compiler into a built-in floating-point type to the nearest representable
 101         value.
 102       </p>
 103 <div class="note"><table border="0" summary="Note">
 104 <tr>
 105 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
 106 <th align="left">Note</th>
 107 </tr>
 108 <tr><td align="left" valign="top"><p>
 109           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>
 110           into a built-in floating-point type is <span class="bold"><strong>not guaranteed</strong></span>
 111           by the C++ Standard to give the nearest representable value.
 112         </p></td></tr>
 113 </table></div>
 114 <p>
 115         William Kahan coined the term <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">Table-Maker's
 116         Dilemma</a> for the problem of correctly rounding functions. Using a
 117         much higher precision (50 or 100 decimal digits) is a practical way of generating
 118         (almost always) correctly rounded values.
 119       </p>
 120 </div>
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 122 <td align="left"></td>
 123 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 124       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 125       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 126       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 127         Distributed under the Boost Software License, Version 1.0. (See accompanying
 128         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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