1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>expm1</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="../powers.html" title="Basic Functions"> 9<link rel="prev" href="log1p.html" title="log1p"> 10<link rel="next" href="cbrt.html" title="cbrt"> 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="log1p.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.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="cbrt.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.powers.expm1"></a><a class="link" href="expm1.html" title="expm1">expm1</a> 28</h3></div></div></div> 29<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">expm1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 30</pre> 31<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 32 33<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 34<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 35 36<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 37<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 38 39<span class="special">}}</span> <span class="comment">// namespaces</span> 40</pre> 41<p> 42 Returns e<sup>x</sup> - 1. 43 </p> 44<p> 45 The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 46 type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code> 47 when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise. 48 </p> 49<p> 50 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 51 be used to control the behaviour of the function: how it handles errors, 52 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 53 documentation for more details</a>. 54 </p> 55<p> 56 For small <span class="emphasis"><em>x</em></span>, then <code class="computeroutput">e<sup>x</sup></code> is very close to 1, as a result calculating 57 <code class="computeroutput">e<sup>x</sup> - 1</code> results in catastrophic cancellation errors when <span class="emphasis"><em>x</em></span> is 58 small. <code class="computeroutput"><span class="identifier">expm1</span></code> calculates <code class="computeroutput">e<sup>x</sup> - 1</code> using 59 rational approximations (for up to 128-bit long doubles), otherwise via a 60 series expansion when x is small (giving an accuracy of less than 2ɛ). 61 </p> 62<p> 63 Finally when BOOST_HAS_EXPM1 is defined then the <code class="computeroutput"><span class="keyword">float</span><span class="special">/</span><span class="keyword">double</span><span class="special">/</span><span class="keyword">long</span> <span class="keyword">double</span></code> 64 specializations of this template simply forward to the platform's native 65 (POSIX) implementation of this function. 66 </p> 67<p> 68 The following graph illustrates the behaviour of expm1: 69 </p> 70<div class="blockquote"><blockquote class="blockquote"><p> 71 <span class="inlinemediaobject"><img src="../../../graphs/expm1.svg" align="middle"></span> 72 73 </p></blockquote></div> 74<h5> 75<a name="math_toolkit.powers.expm1.h0"></a> 76 <span class="phrase"><a name="math_toolkit.powers.expm1.accuracy"></a></span><a class="link" href="expm1.html#math_toolkit.powers.expm1.accuracy">Accuracy</a> 77 </h5> 78<p> 79 For built in floating point types <code class="computeroutput"><span class="identifier">expm1</span></code> 80 should have approximately 1 epsilon accuracy. 81 </p> 82<div class="table"> 83<a name="math_toolkit.powers.expm1.table_expm1"></a><p class="title"><b>Table 8.82. Error rates for expm1</b></p> 84<div class="table-contents"><table class="table" summary="Error rates for expm1"> 85<colgroup> 86<col> 87<col> 88<col> 89<col> 90<col> 91</colgroup> 92<thead><tr> 93<th> 94 </th> 95<th> 96 <p> 97 GNU C++ version 7.1.0<br> linux<br> long double 98 </p> 99 </th> 100<th> 101 <p> 102 GNU C++ version 7.1.0<br> linux<br> double 103 </p> 104 </th> 105<th> 106 <p> 107 Sun compiler version 0x5150<br> Sun Solaris<br> long double 108 </p> 109 </th> 110<th> 111 <p> 112 Microsoft Visual C++ version 14.1<br> Win32<br> double 113 </p> 114 </th> 115</tr></thead> 116<tbody><tr> 117<td> 118 <p> 119 Random test data 120 </p> 121 </td> 122<td> 123 <p> 124 <span class="blue">Max = 0.992ε (Mean = 0.402ε)</span><br> <br> 125 (<span class="emphasis"><em><cmath>:</em></span> Max = 0.992ε (Mean = 0.402ε))<br> 126 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.992ε (Mean = 0.402ε)) 127 </p> 128 </td> 129<td> 130 <p> 131 <span class="blue">Max = 0.793ε (Mean = 0.126ε)</span><br> <br> 132 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.793ε (Mean = 0.126ε)) 133 </p> 134 </td> 135<td> 136 <p> 137 <span class="blue">Max = 1.31ε (Mean = 0.428ε)</span><br> <br> 138 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.996ε (Mean = 0.426ε)) 139 </p> 140 </td> 141<td> 142 <p> 143 <span class="blue">Max = 1.31ε (Mean = 0.496ε)</span><br> <br> 144 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.31ε (Mean = 0.496ε)) 145 </p> 146 </td> 147</tr></tbody> 148</table></div> 149</div> 150<br class="table-break"><h5> 151<a name="math_toolkit.powers.expm1.h1"></a> 152 <span class="phrase"><a name="math_toolkit.powers.expm1.testing"></a></span><a class="link" href="expm1.html#math_toolkit.powers.expm1.testing">Testing</a> 153 </h5> 154<p> 155 A mixture of spot test sanity checks, and random high precision test values 156 calculated using NTL::RR at 1000-bit precision. 157 </p> 158</div> 159<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 160<td align="left"></td> 161<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 162 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 163 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 164 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 165 Daryle Walker and Xiaogang Zhang<p> 166 Distributed under the Boost Software License, Version 1.0. (See accompanying 167 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>) 168 </p> 169</div></td> 170</tr></table> 171<hr> 172<div class="spirit-nav"> 173<a accesskey="p" href="log1p.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.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="cbrt.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 174</div> 175</body> 176</html> 177
