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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.powers.sqrt1pm1"></a><a class="link" href="sqrt1pm1.html" title="sqrt1pm1">sqrt1pm1</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">sqrt1pm1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  30 </pre>
  31 <p>
  32         namespace boost{ namespace math{
  33       </p>
  34 <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>
  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">sqrt1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  36
  37 <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>
  38 <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">sqrt1pm1</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  39
  40 <span class="special">}}</span> <span class="comment">// namespaces</span>
  41 </pre>
  42 <p>
  43         Returns <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
  44       </p>
  45 <p>
  46         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
  47         type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code>
  48         when <span class="emphasis"><em>x</em></span> is an integer-type and T otherwise.
  49       </p>
  50 <p>
  51         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
  52         be used to control the behaviour of the function: how it handles errors,
  53         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
  54         documentation for more details</a>.
  55       </p>
  56 <p>
  57         This function is useful when you need the difference between <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code> and 1, when <span class="emphasis"><em>x</em></span> is itself
  58         close to 1.
  59       </p>
  60 <p>
  61         Implemented in terms of <code class="computeroutput"><span class="identifier">log1p</span></code>
  62         and <code class="computeroutput"><span class="identifier">expm1</span></code>.
  63       </p>
  64 <p>
  65         The following graph illustrates the behaviour of sqrt1pm1:
  66       </p>
  67 <div class="blockquote"><blockquote class="blockquote"><p>
  68           <span class="inlinemediaobject"><img src="../../../graphs/sqrt1pm1.svg" align="middle"></span>
  69
  70         </p></blockquote></div>
  71 <h5>
  72 <a name="math_toolkit.powers.sqrt1pm1.h0"></a>
  73         <span class="phrase"><a name="math_toolkit.powers.sqrt1pm1.accuracy"></a></span><a class="link" href="sqrt1pm1.html#math_toolkit.powers.sqrt1pm1.accuracy">Accuracy</a>
  74       </h5>
  75 <p>
  76         For built in floating-point types <code class="computeroutput"><span class="identifier">sqrt1pm1</span></code>
  77         should have approximately 3 epsilon accuracy.
  78       </p>
  79 <div class="table">
  80 <a name="math_toolkit.powers.sqrt1pm1.table_sqrt1pm1"></a><p class="title"><b>Table&#160;8.84.&#160;Error rates for sqrt1pm1</b></p>
  81 <div class="table-contents"><table class="table" summary="Error rates for sqrt1pm1">
  82 <colgroup>
  83 <col>
  84 <col>
  85 <col>
  86 <col>
  87 <col>
  88 </colgroup>
  89 <thead><tr>
  90 <th>
  91               </th>
  92 <th>
  93                 <p>
  94                   GNU C++ version 7.1.0<br> linux<br> double
  95                 </p>
  96               </th>
  97 <th>
  98                 <p>
  99                   GNU C++ version 7.1.0<br> linux<br> long double
 100                 </p>
 101               </th>
 102 <th>
 103                 <p>
 104                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 105                 </p>
 106               </th>
 107 <th>
 108                 <p>
 109                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 110                 </p>
 111               </th>
 112 </tr></thead>
 113 <tbody><tr>
 114 <td>
 115                 <p>
 116                   sqrt1pm1
 117                 </p>
 118               </td>
 119 <td>
 120                 <p>
 121                   <span class="blue">Max = 1.3&#949; (Mean = 0.404&#949;)</span>
 122                 </p>
 123               </td>
 124 <td>
 125                 <p>
 126                   <span class="blue">Max = 1.33&#949; (Mean = 0.404&#949;)</span>
 127                 </p>
 128               </td>
 129 <td>
 130                 <p>
 131                   <span class="blue">Max = 1.54&#949; (Mean = 0.563&#949;)</span>
 132                 </p>
 133               </td>
 134 <td>
 135                 <p>
 136                   <span class="blue">Max = 1.35&#949; (Mean = 0.497&#949;)</span>
 137                 </p>
 138               </td>
 139 </tr></tbody>
 140 </table></div>
 141 </div>
 142 <br class="table-break"><h5>
 143 <a name="math_toolkit.powers.sqrt1pm1.h1"></a>
 144         <span class="phrase"><a name="math_toolkit.powers.sqrt1pm1.testing"></a></span><a class="link" href="sqrt1pm1.html#math_toolkit.powers.sqrt1pm1.testing">Testing</a>
 145       </h5>
 146 <p>
 147         A selection of random high precision test values calculated using NTL::RR
 148         at 1000-bit precision.
 149       </p>
 150 </div>
 151 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 152 <td align="left"></td>
 153 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 154       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 155       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 156       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 157       Daryle Walker and Xiaogang Zhang<p>
 158         Distributed under the Boost Software License, Version 1.0. (See accompanying
 159         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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