1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Why use a high-precision library rather than built-in floating-point types?</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../high_precision.html" title="Using Boost.Math with High-Precision Floating-Point Libraries"> 9<link rel="prev" href="../high_precision.html" title="Using Boost.Math with High-Precision Floating-Point Libraries"> 10<link rel="next" href="use_multiprecision.html" title="Using Boost.Multiprecision"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../high_precision.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../high_precision.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="use_multiprecision.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 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> 124<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 125<td align="left"></td> 126<td align="right"><div class="copyright-footer">Copyright © 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 Råde, 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>) 133 </p> 134</div></td> 135</tr></table> 136<hr> 137<div class="spirit-nav"> 138<a accesskey="p" href="../high_precision.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../high_precision.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="use_multiprecision.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 139</div> 140</body> 141</html> 142
