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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.powers.sqrt1pm1"></a><a class="link" href="sqrt1pm1.html" title="sqrt1pm1">sqrt1pm1</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">sqrt1pm1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<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>
<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">sqrt1pm1</span><span class="special">(</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 15. 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">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>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
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>.
</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 is <code class="computeroutput"><span class="keyword">double</span></code>
when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 15. 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 15. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
This function is useful when you need the difference between sqrt(x) and
1, when x is itself close to 1.
</p>
<p>
Implemented in terms of <code class="computeroutput"><span class="identifier">log1p</span></code>
and <code class="computeroutput"><span class="identifier">expm1</span></code>.
</p>
<p>
The following graph illustrates the behaviour of sqrt1pm1:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/sqrt1pm1.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.powers.sqrt1pm1.h0"></a>
<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>
</h5>
<p>
For built in floating-point types <code class="computeroutput"><span class="identifier">sqrt1pm1</span></code>
should have approximately 3 epsilon accuracy.
</p>
<div class="table">
<a name="math_toolkit.powers.sqrt1pm1.table_sqrt1pm1"></a><p class="title"><b>Table 6.81. Error rates for sqrt1pm1</b></p>
<div class="table-contents"><table class="table" summary="Error rates for sqrt1pm1">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
</p>
</th>
</tr></thead>
<tbody><tr>
<td>
<p>
sqrt1pm1
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.36ε (Mean = 0.44ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.3ε (Mean = 0.404ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.33ε (Mean = 0.409ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.54ε (Mean = 0.568ε)</span>
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.powers.sqrt1pm1.h1"></a>
<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>
</h5>
<p>
A selection of random high precision test values calculated using NTL::RR
at 1000-bit precision.
</p>
</div>
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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