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4<title>asinh</title>
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26<div class="titlepage"><div><div><h3 class="title">
27<a name="math_toolkit.inv_hyper.asinh"></a><a class="link" href="asinh.html" title="asinh">asinh</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">asinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
30</pre>
31<pre class="programlisting"><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>
32<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">asinh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
33
34<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
35<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">asinh</span><span class="special">(</span><span class="keyword">const</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">&amp;);</span>
36</pre>
37<p>
38        Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the
39        hyperbolic sine function</a>.
40      </p>
41<p>
42        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
43        type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
44      </p>
45<div class="blockquote"><blockquote class="blockquote"><p>
46          <span class="inlinemediaobject"><img src="../../../graphs/asinh.svg" align="middle"></span>
47
48        </p></blockquote></div>
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<h5>
56<a name="math_toolkit.inv_hyper.asinh.h0"></a>
57        <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.accuracy"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.accuracy">Accuracy</a>
58      </h5>
59<p>
60        Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
61        epsilon</a> across all supported platforms.
62      </p>
63<h5>
64<a name="math_toolkit.inv_hyper.asinh.h1"></a>
65        <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.testing"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.testing">Testing</a>
66      </h5>
67<p>
68        This function is tested using a combination of random test values designed
69        to give full function coverage computed at high precision using the "naive"
70        formula:
71      </p>
72<div class="blockquote"><blockquote class="blockquote"><p>
73          <span class="inlinemediaobject"><img src="../../../equations/asinh1.svg"></span>
74
75        </p></blockquote></div>
76<p>
77        along with a selection of sanity check values computed using functions.wolfram.com
78        to at least 50 decimal digits.
79      </p>
80<h5>
81<a name="math_toolkit.inv_hyper.asinh.h2"></a>
82        <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.implementation"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.implementation">Implementation</a>
83      </h5>
84<p>
85        For sufficiently large x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/06/01/0001/" target="_top">approximation</a>:
86      </p>
87<div class="blockquote"><blockquote class="blockquote"><p>
88          <span class="inlinemediaobject"><img src="../../../equations/asinh2.svg"></span>
89
90        </p></blockquote></div>
91<p>
92        While for very small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/0001/" target="_top">approximation</a>:
93      </p>
94<div class="blockquote"><blockquote class="blockquote"><p>
95          <span class="inlinemediaobject"><img src="../../../equations/asinh3.svg"></span>
96
97        </p></blockquote></div>
98<p>
99        For 0.5 &gt; x &gt; ε the following rearrangement of the primary definition
100        is used:
101      </p>
102<div class="blockquote"><blockquote class="blockquote"><p>
103          <span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span>
104
105        </p></blockquote></div>
106<p>
107        Otherwise evaluation is via the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/" target="_top">primary
108        definition</a>:
109      </p>
110<div class="blockquote"><blockquote class="blockquote"><p>
111          <span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span>
112
113        </p></blockquote></div>
114</div>
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117<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
118      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
119      Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
120      Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
121      Daryle Walker and Xiaogang Zhang<p>
122        Distributed under the Boost Software License, Version 1.0. (See accompanying
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