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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</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">atanh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <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> |
| <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">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 14. 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">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the |
| hyperbolic tangent function</a>, at x. |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. 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 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| If x is in the range <code class="literal">]-∞;-1[</code> or in the range <code class="literal">]+1;+∞[</code> |
| then returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. |
| </p> |
| <p> |
| 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> |
| is returned, with ε   |
| denoting numeric_limits<T>::epsilon(). |
| </p> |
| <p> |
| 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> |
| is returned, with ε   |
| denoting numeric_limits<T>::epsilon(). |
| </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 type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/atanh.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.inv_hyper.atanh.h0"></a> |
| <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> |
| </h5> |
| <p> |
| Generally accuracy is to within 1 or 2 epsilon across all supported platforms. |
| </p> |
| <h5> |
| <a name="math_toolkit.inv_hyper.atanh.h1"></a> |
| <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> |
| </h5> |
| <p> |
| This function is tested using a combination of random test values designed |
| to give full function coverage computed at high precision using the "naive" |
| formula: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span> |
| </p> |
| <p> |
| along with a selection of sanity check values computed using functions.wolfram.com |
| to at least 50 decimal digits. |
| </p> |
| <h5> |
| <a name="math_toolkit.inv_hyper.atanh.h2"></a> |
| <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> |
| </h5> |
| <p> |
| For sufficiently small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/" target="_top">approximation</a>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/atanh2.svg"></span> |
| </p> |
| <p> |
| Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/" target="_top">primary |
| definition</a>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span> |
| </p> |
| <p> |
| or its equivalent form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/atanh3.svg"></span> |
| </p> |
| <p> |
| is used. |
| </p> |
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