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<a name="math_toolkit.powers.logaddexp"></a><a class="link" href="logaddexp.html" title="logaddexp and logsumexp">logaddexp and logsumexp</a>
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<h5>
<a name="math_toolkit.powers.logaddexp.h0"></a>
<span class="phrase"><a name="math_toolkit.powers.logaddexp.synopsis"></a></span><a class="link" href="logaddexp.html#math_toolkit.powers.logaddexp.synopsis">Synopsis</a>
</h5>
<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">logaddexp</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<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">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<span class="identifier">Real</span> <span class="identifier">logaddexp</span> <span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x1</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x2</span><span class="special">);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
<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">logsumexp</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<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">&lt;</span><span class="keyword">typename</span> <span class="identifier">ForwardIterator</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special">&lt;</span><span class="identifier">ForwardIterator</span><span class="special">&gt;::</span><span class="identifier">value_type</span><span class="special">&gt;</span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span> <span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ForwardContainer</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">ForwardContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">&gt;</span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">ForwardContainer</span><span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Args</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">common_type</span><span class="special">&lt;</span><span class="identifier">Args</span><span class="special">...&gt;::</span><span class="identifier">type</span><span class="special">&gt;</span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span><span class="special">(</span><span class="identifier">Args</span><span class="special">&amp;&amp;</span> <span class="special">...</span><span class="identifier">args</span><span class="special">)</span>
<span class="special">}}</span> <span class="comment">// namespace</span>
</pre>
<p>
The function <code class="computeroutput"><span class="identifier">logaddexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">)</span></code> computes log(exp(x1) + exp(x2)). The function
<code class="computeroutput"><span class="identifier">logsumexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="special">...)</span></code> is
generalized to compute log(exp(x1) + exp(x2) + ...). This function is useful
in statistics where the calculated probabilities of events may be so small
as to exceed the range of normal floating point numbers. In such cases the
logarithm of the calculated probability is stored.
</p>
<p>
The use is
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">log</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">x1</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">1e-50</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">x2</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">2.5e-50</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">x3</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">3e-50</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="identifier">x3</span><span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">probability1</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">logaddexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability2</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">logsumexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="identifier">x3</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability3</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">logsumexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability4</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">logsumexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
</pre>
<p>
Performance Data:
</p>
<pre class="programlisting"><span class="identifier">Running</span> <span class="special">./</span><span class="identifier">logaddexp_performance</span>
<span class="identifier">Run</span> <span class="identifier">on</span> <span class="identifier">Apple</span> <span class="identifier">M1</span> <span class="identifier">Pro</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">64</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">128</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">4096</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x5</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">2.23</span><span class="special">,</span> <span class="number">1.89</span><span class="special">,</span> <span class="number">1.88</span>
<span class="special">-----------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> <span class="identifier">Iterations</span>
<span class="special">-----------------------------------------------------------------------------</span>
<span class="identifier">logaddexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="number">1.05</span> <span class="identifier">ns</span> <span class="number">1.05</span> <span class="identifier">ns</span> <span class="number">597983940</span>
<span class="identifier">logaddexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="number">1.03</span> <span class="identifier">ns</span> <span class="number">1.03</span> <span class="identifier">ns</span> <span class="number">672682369</span>
</pre>
<pre class="programlisting"><span class="identifier">Running</span> <span class="special">./</span><span class="identifier">logsumexp_performance</span>
<span class="identifier">Run</span> <span class="identifier">on</span> <span class="identifier">Apple</span> <span class="identifier">M1</span> <span class="identifier">Pro</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">64</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">128</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">4096</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x5</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">1.56</span><span class="special">,</span> <span class="number">1.67</span><span class="special">,</span> <span class="number">1.81</span>
<span class="special">-----------------------------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> <span class="identifier">Iterations</span>
<span class="special">-----------------------------------------------------------------------------------------------</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">1797191</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">761</span> <span class="identifier">ns</span> <span class="number">761</span> <span class="identifier">ns</span> <span class="number">890017</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1513</span> <span class="identifier">ns</span> <span class="number">1513</span> <span class="identifier">ns</span> <span class="number">460217</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3026</span> <span class="identifier">ns</span> <span class="number">3026</span> <span class="identifier">ns</span> <span class="number">231454</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6043</span> <span class="identifier">ns</span> <span class="number">6043</span> <span class="identifier">ns</span> <span class="number">113901</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">12084</span> <span class="identifier">ns</span> <span class="number">12084</span> <span class="identifier">ns</span> <span class="number">57590</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">24240</span> <span class="identifier">ns</span> <span class="number">24240</span> <span class="identifier">ns</span> <span class="number">28835</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">48326</span> <span class="identifier">ns</span> <span class="number">48323</span> <span class="identifier">ns</span> <span class="number">14478</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">96645</span> <span class="identifier">ns</span> <span class="number">96642</span> <span class="identifier">ns</span> <span class="number">7181</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">193351</span> <span class="identifier">ns</span> <span class="number">193351</span> <span class="identifier">ns</span> <span class="number">3607</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">386537</span> <span class="identifier">ns</span> <span class="number">386538</span> <span class="identifier">ns</span> <span class="number">1810</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">774055</span> <span class="identifier">ns</span> <span class="number">774056</span> <span class="identifier">ns</span> <span class="number">894</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1548913</span> <span class="identifier">ns</span> <span class="number">1548847</span> <span class="identifier">ns</span> <span class="number">451</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3092771</span> <span class="identifier">ns</span> <span class="number">3092770</span> <span class="identifier">ns</span> <span class="number">226</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6188087</span> <span class="identifier">ns</span> <span class="number">6188089</span> <span class="identifier">ns</span> <span class="number">112</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="identifier">real_time_BigO</span> <span class="number">5.90</span> <span class="identifier">N</span> <span class="number">5.90</span> <span class="identifier">N</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="identifier">real_time_RMS</span> <span class="number">0</span> <span class="special">%</span> <span class="number">0</span> <span class="special">%</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">1806669</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">770</span> <span class="identifier">ns</span> <span class="number">770</span> <span class="identifier">ns</span> <span class="number">898340</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1534</span> <span class="identifier">ns</span> <span class="number">1534</span> <span class="identifier">ns</span> <span class="number">454768</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3063</span> <span class="identifier">ns</span> <span class="number">3063</span> <span class="identifier">ns</span> <span class="number">228057</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6126</span> <span class="identifier">ns</span> <span class="number">6126</span> <span class="identifier">ns</span> <span class="number">112667</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">12243</span> <span class="identifier">ns</span> <span class="number">12243</span> <span class="identifier">ns</span> <span class="number">56963</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">24476</span> <span class="identifier">ns</span> <span class="number">24476</span> <span class="identifier">ns</span> <span class="number">28485</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">48979</span> <span class="identifier">ns</span> <span class="number">48978</span> <span class="identifier">ns</span> <span class="number">14215</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">97929</span> <span class="identifier">ns</span> <span class="number">97929</span> <span class="identifier">ns</span> <span class="number">7070</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">195826</span> <span class="identifier">ns</span> <span class="number">195826</span> <span class="identifier">ns</span> <span class="number">3560</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">391855</span> <span class="identifier">ns</span> <span class="number">391835</span> <span class="identifier">ns</span> <span class="number">1787</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">784119</span> <span class="identifier">ns</span> <span class="number">784110</span> <span class="identifier">ns</span> <span class="number">882</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1566408</span> <span class="identifier">ns</span> <span class="number">1566386</span> <span class="identifier">ns</span> <span class="number">446</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3151649</span> <span class="identifier">ns</span> <span class="number">3150955</span> <span class="identifier">ns</span> <span class="number">223</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6300578</span> <span class="identifier">ns</span> <span class="number">6299027</span> <span class="identifier">ns</span> <span class="number">110</span>
<span class="identifier">logsumexp_performance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="identifier">real_time_BigO</span> <span class="number">6.01</span> <span class="identifier">N</span> <span class="number">6.01</span> <span class="identifier">N</span>
</pre>
<h5>
<a name="math_toolkit.powers.logaddexp.h1"></a>
<span class="phrase"><a name="math_toolkit.powers.logaddexp.references"></a></span><a class="link" href="logaddexp.html#math_toolkit.powers.logaddexp.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Higham, Nicholas J. <a href="https://nhigham.com/2021/01/05/what-is-the-log-sum-exp-function/" target="_top">What
is the Log-Sum-Exp Function?</a>
</li></ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
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