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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.powers.log1p"></a><a class="link" href="log1p.html" title="log1p">log1p</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">log1p</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">log1p</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">log1p</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 the natural logarithm of <code class="computeroutput"><span class="identifier">x</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>
There are many situations where it is desirable to compute <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code>.
However, for small <code class="computeroutput"><span class="identifier">x</span></code> then
<code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span></code> suffers from catastrophic cancellation errors
so that <code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span> <span class="special">==</span> <span class="number">1</span></code>
and <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">==</span> <span class="number">0</span></code>,
when in fact for very small x, the best approximation to <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code> would be
<code class="computeroutput"><span class="identifier">x</span></code>. <code class="computeroutput"><span class="identifier">log1p</span></code>
calculates the best approximation to <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">x</span><span class="special">)</span></code> using
a Taylor series expansion for accuracy (less than 2ɛ). Alternatively note that
there are faster methods available, for example using the equivalence:
</p>
<pre class="programlisting"><span class="identifier">log</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="special">(</span><span class="identifier">log</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="identifier">x</span><span class="special">)</span> <span class="special">/</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><span class="special">)</span>
</pre>
<p>
However, experience has shown that these methods tend to fail quite spectacularly
once the compiler's optimizations are turned on, consequently they are used
only when known not to break with a particular compiler. In contrast, the
series expansion method seems to be reasonably immune to optimizer-induced
errors.
</p>
<p>
Finally when BOOST_HAS_LOG1P is defined then the <code class="computeroutput"><span class="keyword">float</span><span class="special">/</span><span class="keyword">double</span><span class="special">/</span><span class="keyword">long</span> <span class="keyword">double</span></code>
specializations of this template simply forward to the platform's native
(POSIX) implementation of this function.
</p>
<p>
The following graph illustrates the behaviour of log1p:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.powers.log1p.h0"></a>
<span class="phrase"><a name="math_toolkit.powers.log1p.accuracy"></a></span><a class="link" href="log1p.html#math_toolkit.powers.log1p.accuracy">Accuracy</a>
</h5>
<p>
For built in floating point types <code class="computeroutput"><span class="identifier">log1p</span></code>
should have approximately 1 epsilon accuracy.
</p>
<div class="table">
<a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table 6.78. Error rates for log1p</b></p>
<div class="table-contents"><table class="table" summary="Error rates for log1p">
<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>
Random test data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.509ε (Mean = 0.057ε)</span><br> <br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.509ε (Mean = 0.057ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.846ε (Mean = 0.153ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.846ε (Mean = 0.153ε))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 0.799ε (Mean = 0.122ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.818ε (Mean = 0.227ε)</span><br> <br>
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 0.818ε (Mean = 0.227ε))<br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.227ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.53ε (Mean = 0.627ε)</span><br> <br>
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.249ε))
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.powers.log1p.h1"></a>
<span class="phrase"><a name="math_toolkit.powers.log1p.testing"></a></span><a class="link" href="log1p.html#math_toolkit.powers.log1p.testing">Testing</a>
</h5>
<p>
A mixture of spot test sanity checks, and random high precision test values
calculated using NTL::RR at 1000-bit precision.
</p>
</div>
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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