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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&#160;20.&#160;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&#160;20.&#160;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&#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 <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; &#949; 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 evalution 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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 116 <td align="left"></td>
 117 <td align="right"><div class="copyright-footer">Copyright &#169; 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       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 121       Daryle Walker and Xiaogang Zhang<p>
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