142 lines
9.9 KiB
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142 lines
9.9 KiB
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<title>atanh</title>
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<a accesskey="p" href="asinh.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../owens_t.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</a>
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</h3></div></div></div>
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<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">atanh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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">atanh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
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<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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">atanh</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">&);</span>
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</pre>
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<p>
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Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the
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hyperbolic tangent function</a>, at x.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<p>
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If x is in the range <code class="literal">]-∞;-1[</code> or in the range <code class="literal">]+1;+∞[</code>
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then returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<p>
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If x is in the range <code class="literal">[-1;-1+ε[</code>, then the result of -<a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
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is returned, with ε
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denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>.
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</p>
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<p>
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If x is in the range <code class="literal">]+1-ε;+1]</code>, then the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
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is returned, with ε
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denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>.
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</p>
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<p>
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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
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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.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/atanh.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.inv_hyper.atanh.h0"></a>
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<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.accuracy"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.accuracy">Accuracy</a>
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</h5>
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<p>
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Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
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epsilon</a> across all supported platforms.
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</p>
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<h5>
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<a name="math_toolkit.inv_hyper.atanh.h1"></a>
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<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.testing"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.testing">Testing</a>
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</h5>
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<p>
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This function is tested using a combination of random test values designed
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to give full function coverage computed at high precision using the "naive"
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formula:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span>
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</p></blockquote></div>
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<p>
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along with a selection of sanity check values computed using functions.wolfram.com
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to at least 50 decimal digits.
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</p>
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<h5>
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<a name="math_toolkit.inv_hyper.atanh.h2"></a>
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<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.implementation"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.implementation">Implementation</a>
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</h5>
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<p>
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For sufficiently small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/" target="_top">approximation</a>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/atanh2.svg"></span>
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</p></blockquote></div>
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<p>
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Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/" target="_top">primary
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definition</a>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span>
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</p></blockquote></div>
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<p>
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or its equivalent form:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/atanh3.svg"></span>
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</p></blockquote></div>
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<p>
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is used.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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<a accesskey="p" href="asinh.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../owens_t.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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