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26<div class="titlepage"><div><div><h3 class="title">
27<a name="math_toolkit.powers.log1p"></a><a class="link" href="log1p.html" title="log1p">log1p</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">log1p</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
30</pre>
31<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>
32
33<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>
34<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>
35
36<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
37<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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
38
39<span class="special">}}</span> <span class="comment">// namespaces</span>
40</pre>
41<p>
42 Returns the natural logarithm of <code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span></code>.
43 </p>
44<p>
45 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
46 type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code>
47 when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
48 </p>
49<p>
50 The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
51 be used to control the behaviour of the function: how it handles errors,
52 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
53 documentation for more details</a>.
54 </p>
55<p>
56 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>.
57 However, for small <code class="computeroutput"><span class="identifier">x</span></code> then
58 <code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span></code> suffers from catastrophic cancellation errors
59 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>
60 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>,
61 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
62 <code class="computeroutput"><span class="identifier">x</span></code>. <code class="computeroutput"><span class="identifier">log1p</span></code>
63 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
64 a Taylor series expansion for accuracy (less than 2&#603;). Alternatively note that
65 there are faster methods available, for example using the equivalence:
66 </p>
67<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>
68</pre>
69<p>
70 However, experience has shown that these methods tend to fail quite spectacularly
71 once the compiler's optimizations are turned on, consequently they are used
72 only when known not to break with a particular compiler. In contrast, the
73 series expansion method seems to be reasonably immune to optimizer-induced
74 errors.
75 </p>
76<p>
77 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>
78 specializations of this template simply forward to the platform's native
79 (POSIX) implementation of this function.
80 </p>
81<p>
82 The following graph illustrates the behaviour of log1p:
83 </p>
84<p>
85 <span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
86 </p>
87<h5>
88<a name="math_toolkit.powers.log1p.h0"></a>
89 <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>
90 </h5>
91<p>
92 For built in floating point types <code class="computeroutput"><span class="identifier">log1p</span></code>
93 should have approximately 1 epsilon accuracy.
94 </p>
95<div class="table">
96<a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table&#160;6.78.&#160;Error rates for log1p</b></p>
97<div class="table-contents"><table class="table" summary="Error rates for log1p">
98<colgroup>
99<col>
100<col>
101<col>
102<col>
103<col>
104</colgroup>
105<thead><tr>
106<th>
107 </th>
108<th>
109 <p>
110 Microsoft Visual C++ version 12.0<br> Win32<br> double
111 </p>
112 </th>
113<th>
114 <p>
115 GNU C++ version 5.1.0<br> linux<br> double
116 </p>
117 </th>
118<th>
119 <p>
120 GNU C++ version 5.1.0<br> linux<br> long double
121 </p>
122 </th>
123<th>
124 <p>
125 Sun compiler version 0x5130<br> Sun Solaris<br> long double
126 </p>
127 </th>
128</tr></thead>
129<tbody><tr>
130<td>
131 <p>
132 Random test data
133 </p>
134 </td>
135<td>
136 <p>
137 <span class="blue">Max = 0.509&#949; (Mean = 0.057&#949;)</span><br> <br>
138 (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.509&#949; (Mean = 0.057&#949;))
139 </p>
140 </td>
141<td>
142 <p>
143 <span class="blue">Max = 0.846&#949; (Mean = 0.153&#949;)</span><br> <br>
144 (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.846&#949; (Mean = 0.153&#949;))<br>
145 (<span class="emphasis"><em>Cephes:</em></span> Max = 0.799&#949; (Mean = 0.122&#949;))
146 </p>
147 </td>
148<td>
149 <p>
150 <span class="blue">Max = 0.818&#949; (Mean = 0.227&#949;)</span><br> <br>
151 (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))<br>
152 (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))
153 </p>
154 </td>
155<td>
156 <p>
157 <span class="blue">Max = 1.53&#949; (Mean = 0.627&#949;)</span><br> <br>
158 (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.249&#949;))
159 </p>
160 </td>
161</tr></tbody>
162</table></div>
163</div>
164<br class="table-break"><h5>
165<a name="math_toolkit.powers.log1p.h1"></a>
166 <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>
167 </h5>
168<p>
169 A mixture of spot test sanity checks, and random high precision test values
170 calculated using NTL::RR at 1000-bit precision.
171 </p>
172</div>
173<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
174<td align="left"></td>
175<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
176 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
177 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
178 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
179 Distributed under the Boost Software License, Version 1.0. (See accompanying
180 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>)
181 </p>
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