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<title>sqrt1pm1</title>
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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.powers.sqrt1pm1"></a><a class="link" href="sqrt1pm1.html" title="sqrt1pm1">sqrt1pm1</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">sqrt1pm1</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">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</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>
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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">sqrt1pm1</span><span class="special">(</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 15. 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">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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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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>.
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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 is <code class="computeroutput"><span class="keyword">double</span></code>
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when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 15. 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 15. 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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This function is useful when you need the difference between sqrt(x) and
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1, when x is itself close to 1.
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</p>
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<p>
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Implemented in terms of <code class="computeroutput"><span class="identifier">log1p</span></code>
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and <code class="computeroutput"><span class="identifier">expm1</span></code>.
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</p>
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<p>
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The following graph illustrates the behaviour of sqrt1pm1:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/sqrt1pm1.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.powers.sqrt1pm1.h0"></a>
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<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>
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</h5>
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<p>
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For built in floating-point types <code class="computeroutput"><span class="identifier">sqrt1pm1</span></code>
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should have approximately 3 epsilon accuracy.
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</p>
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<div class="table">
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<a name="math_toolkit.powers.sqrt1pm1.table_sqrt1pm1"></a><p class="title"><b>Table 6.81. Error rates for sqrt1pm1</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for sqrt1pm1">
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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GNU C++ version 5.1.0<br> linux<br> double
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GNU C++ version 5.1.0<br> linux<br> long double
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
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</p>
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</th>
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</tr></thead>
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<tbody><tr>
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<td>
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<p>
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sqrt1pm1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.36ε (Mean = 0.44ε)</span>
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</p>
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</td>
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<p>
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<span class="blue">Max = 1.3ε (Mean = 0.404ε)</span>
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</p>
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</td>
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<p>
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<span class="blue">Max = 1.33ε (Mean = 0.409ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.54ε (Mean = 0.568ε)</span>
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</p>
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</td>
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</tr></tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.powers.sqrt1pm1.h1"></a>
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<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>
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</h5>
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<p>
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A selection of random high precision test values calculated using NTL::RR
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at 1000-bit precision.
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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 © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle 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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