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<title>expm1</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.powers.expm1"></a><a class="link" href="expm1.html" title="expm1">expm1</a>
</h3></div></div></div>
<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>
</pre>
<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>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<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">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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns e<sup>x</sup> - 1.
</p>
<p>
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
type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code>
when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
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
<code class="computeroutput">e<sup>x</sup> - 1</code> results in catastrophic cancellation errors when <span class="emphasis"><em>x</em></span> is
small. <code class="computeroutput"><span class="identifier">expm1</span></code> calculates <code class="computeroutput">e<sup>x</sup> - 1</code> using
rational approximations (for up to 128-bit long doubles), otherwise via a
series expansion when x is small (giving an accuracy of less than 2ɛ).
</p>
<p>
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>
specializations of this template simply forward to the platform's native
(POSIX) implementation of this function.
</p>
<p>
The following graph illustrates the behaviour of expm1:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/expm1.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.powers.expm1.h0"></a>
<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>
</h5>
<p>
For built in floating point types <code class="computeroutput"><span class="identifier">expm1</span></code>
should have approximately 1 epsilon accuracy.
</p>
<div class="table">
<a name="math_toolkit.powers.expm1.table_expm1"></a><p class="title"><b>Table 8.82. Error rates for expm1</b></p>
<div class="table-contents"><table class="table" summary="Error rates for expm1">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody><tr>
<td>
<p>
Random test data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.992ε (Mean = 0.402ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 0.992ε (Mean = 0.402ε))<br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.992ε (Mean = 0.402ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.793ε (Mean = 0.126ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.793ε (Mean = 0.126ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.31ε (Mean = 0.428ε)</span><br> <br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.996ε (Mean = 0.426ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.31ε (Mean = 0.496ε)</span><br> <br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.31ε (Mean = 0.496ε))
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.powers.expm1.h1"></a>
<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>
</h5>
<p>
A mixture of spot test sanity checks, and random high precision test values
calculated using NTL::RR at 1000-bit precision.
</p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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>)
</p>
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