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| 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 15. 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 15. 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 15. 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 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 53 | documentation for more details</a>. | |
| 54 | </p> | |
| 55 | <p> | |
| 56 | For small x, then <code class="computeroutput">e<sup>x</sup></code> is very close to 1, as a result calculating <code class="computeroutput">e<sup>x</sup> - 1</code> results in | |
| 57 | catastrophic cancellation errors when x is small. <code class="computeroutput"><span class="identifier">expm1</span></code> | |
| 58 | calculates <code class="computeroutput">e<sup>x</sup> - 1</code> using rational approximations (for up to 128-bit long doubles), | |
| 59 | otherwise via a series expansion when x is small (giving an accuracy of less | |
| 60 | 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 | <p> | |
| 71 | <span class="inlinemediaobject"><img src="../../../graphs/expm1.svg" align="middle"></span> | |
| 72 | </p> | |
| 73 | <h5> | |
| 74 | <a name="math_toolkit.powers.expm1.h0"></a> | |
| 75 | <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> | |
| 76 | </h5> | |
| 77 | <p> | |
| 78 | For built in floating point types <code class="computeroutput"><span class="identifier">expm1</span></code> | |
| 79 | should have approximately 1 epsilon accuracy. | |
| 80 | </p> | |
| 81 | <div class="table"> | |
| 82 | <a name="math_toolkit.powers.expm1.table_expm1"></a><p class="title"><b>Table 6.79. Error rates for expm1</b></p> | |
| 83 | <div class="table-contents"><table class="table" summary="Error rates for expm1"> | |
| 84 | <colgroup> | |
| 85 | <col> | |
| 86 | <col> | |
| 87 | <col> | |
| 88 | <col> | |
| 89 | <col> | |
| 90 | </colgroup> | |
| 91 | <thead><tr> | |
| 92 | <th> | |
| 93 | </th> | |
| 94 | <th> | |
| 95 | <p> | |
| 96 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 97 | </p> | |
| 98 | </th> | |
| 99 | <th> | |
| 100 | <p> | |
| 101 | GNU C++ version 5.1.0<br> linux<br> double | |
| 102 | </p> | |
| 103 | </th> | |
| 104 | <th> | |
| 105 | <p> | |
| 106 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 107 | </p> | |
| 108 | </th> | |
| 109 | <th> | |
| 110 | <p> | |
| 111 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 112 | </p> | |
| 113 | </th> | |
| 114 | </tr></thead> | |
| 115 | <tbody><tr> | |
| 116 | <td> | |
| 117 | <p> | |
| 118 | Random test data | |
| 119 | </p> | |
| 120 | </td> | |
| 121 | <td> | |
| 122 | <p> | |
| 123 | <span class="blue">Max = 0.996ε (Mean = 0.283ε)</span><br> <br> | |
| 124 | (<span class="emphasis"><em><math.h>:</em></span> Max = 1.31ε (Mean = 0.496ε)) | |
| 125 | </p> | |
| 126 | </td> | |
| 127 | <td> | |
| 128 | <p> | |
| 129 | <span class="blue">Max = 0.793ε (Mean = 0.126ε)</span><br> <br> | |
| 130 | (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.793ε (Mean = 0.126ε))<br> | |
| 131 | (<span class="emphasis"><em>Cephes:</em></span> Max = 1.53ε (Mean = 0.535ε)) | |
| 132 | </p> | |
| 133 | </td> | |
| 134 | <td> | |
| 135 | <p> | |
| 136 | <span class="blue">Max = 0.992ε (Mean = 0.402ε)</span><br> <br> | |
| 137 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
| 138 | = 1.26e+19ε (Mean = 4.89e+18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_expm1__tr1_cmath__Random_test_data">And | |
| 139 | other failures.</a>)</span><br> (<span class="emphasis"><em><math.h>:</em></span> | |
| 140 | Max = 0.992ε (Mean = 0.402ε)) | |
| 141 | </p> | |
| 142 | </td> | |
| 143 | <td> | |
| 144 | <p> | |
| 145 | <span class="blue">Max = 1.31ε (Mean = 0.406ε)</span><br> <br> | |
| 146 | (<span class="emphasis"><em><math.h>:</em></span> Max = 0.996ε (Mean = 0.426ε)) | |
| 147 | </p> | |
| 148 | </td> | |
| 149 | </tr></tbody> | |
| 150 | </table></div> | |
| 151 | </div> | |
| 152 | <br class="table-break"><h5> | |
| 153 | <a name="math_toolkit.powers.expm1.h1"></a> | |
| 154 | <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> | |
| 155 | </h5> | |
| 156 | <p> | |
| 157 | A mixture of spot test sanity checks, and random high precision test values | |
| 158 | calculated using NTL::RR at 1000-bit precision. | |
| 159 | </p> | |
| 160 | </div> | |
| 161 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 162 | <td align="left"></td> | |
| 163 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 164 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 165 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 166 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 167 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 168 | 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>) | |
| 169 | </p> | |
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