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  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">&lt;</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">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;20.&#160;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&#160;20.&#160;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 2&#603;).
  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&#160;8.82.&#160;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 = 0.992&#949; (Mean = 0.402&#949;)</span><br> <br>
 125                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.992&#949; (Mean = 0.402&#949;))<br>
 126                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.992&#949; (Mean = 0.402&#949;))
 127                 </p>
 128               </td>
 129 <td>
 130                 <p>
 131                   <span class="blue">Max = 0.793&#949; (Mean = 0.126&#949;)</span><br> <br>
 132                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.793&#949; (Mean = 0.126&#949;))
 133                 </p>
 134               </td>
 135 <td>
 136                 <p>
 137                   <span class="blue">Max = 1.31&#949; (Mean = 0.428&#949;)</span><br> <br>
 138                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.996&#949; (Mean = 0.426&#949;))
 139                 </p>
 140               </td>
 141 <td>
 142                 <p>
 143                   <span class="blue">Max = 1.31&#949; (Mean = 0.496&#949;)</span><br> <br>
 144                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.31&#949; (Mean = 0.496&#949;))
 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 &#169; 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       R&#229;de, 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>
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