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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.runs_test"></a><a class="link" href="runs_test.html" title="Runs tests">Runs tests</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.runs_test.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.runs_test.synopsis"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.synopsis">Synopsis</a>
  32     </h4>
  33 <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">statistics</span><span class="special">/</span><span class="identifier">runs_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34
  35 <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span> <span class="special">{</span>
  36
  37 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  38 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">runs_above_and_below_threshold</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">threshold</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  41 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">runs_above_and_below_median</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  42
  43 <span class="special">}}}</span>
  44 </pre>
  45 <h4>
  46 <a name="math_toolkit.runs_test.h1"></a>
  47       <span class="phrase"><a name="math_toolkit.runs_test.background"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.background">Background</a>
  48     </h4>
  49 <p>
  50       The "runs above and below median test" tries to determine if a sequence
  51       is random by looking at the number of consecutive values which exceed (or are
  52       below) the median of the sequence. For example, if we are given data {5, 2,
  53       0, 4, 7, 9, 10, 6, 1, 8, 3}, first we find the median (5), and transform the
  54       vector into a list of + and -'s, depending on whether the value is greater
  55       or less than the median. (Values equal to the median we simply ignore-this
  56       is a convention we have inherited from the wonderful <code class="computeroutput"><span class="identifier">randtests</span></code>
  57       package in R.) Hence our data vector is transformed into
  58     </p>
  59 <p>
  60       {-,-,-,+,+,+,-,+,-}
  61     </p>
  62 <p>
  63       which is 5 runs, with <span class="emphasis"><em>n</em></span><sub>a</sub> = 5 values above and <span class="emphasis"><em>n</em></span><sub>b</sub> =
  64       5 values below the median. <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35d.htm" target="_top">NIST</a>
  65       tells us the expected number of runs and their variance:
  66     </p>
  67 <p>
  68       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/expected_runs_above_threshold.svg" width="418" height="45"></object></span>
  69     </p>
  70 <p>
  71       from which we derive the test statistic
  72     </p>
  73 <p>
  74       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/runs_test_statistic.svg" width="115" height="42"></object></span>
  75     </p>
  76 <p>
  77       whose distribution we approximate as normal to extract a <span class="emphasis"><em>p</em></span>-value.
  78     </p>
  79 <p>
  80       An example usage is as follows:
  81     </p>
  82 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
  83 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
  84 <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">statistics</span><span class="special">/</span><span class="identifier">runs_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  85
  86 <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
  87 <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">v</span><span class="special">{</span><span class="number">5</span><span class="special">,</span> <span class="number">2</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="number">7</span><span class="special">,</span> <span class="number">9</span><span class="special">,</span> <span class="number">10</span><span class="special">,</span> <span class="number">6</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">3</span><span class="special">};</span>
  88 <span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</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">statistics</span><span class="special">::</span><span class="identifier">runs_above_and_below_median</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
  89 <span class="comment">// t =  -0.670820393249936919; p = 0.502334954360502017;</span>
  90 </pre>
  91 <p>
  92       We see that we have about a 50% chance of seeing this number of runs if the
  93       null hypothesis of randomness is true, and hence the assumption of randomness
  94       seems reasonable. As always, the test statistic is the first element of the
  95       pair, and the <span class="emphasis"><em>p</em></span>-value is the second element.
  96     </p>
  97 <h4>
  98 <a name="math_toolkit.runs_test.h2"></a>
  99       <span class="phrase"><a name="math_toolkit.runs_test.performance"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.performance">Performance</a>
 100     </h4>
 101 <p>
 102       There are two cases: Where the threshold (typically the median) has already
 103       been computed, and the case where the mean and sample variance must be computed
 104       on the fly. Computing the median is fairly expensive (requiring a call to
 105       <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">median</span></code>),
 106       and since the order of the original data must be preserved, it must allocate.
 107       If you believe your data to come from a distribution where the means and median
 108       coincide, or if you've already computed the median in the course of some other
 109       analysis, then you can get away with a call to <code class="computeroutput"><span class="identifier">runs_above_and_below_threshold</span></code>
 110       via
 111     </p>
 112 <pre class="programlisting"><span class="identifier">Real</span> <span class="identifier">threshold</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>
 113 <span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</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">statistics</span><span class="special">::</span><span class="identifier">runs_above_and_below_threshold</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">threshold</span><span class="special">);</span>
 114 </pre>
 115 <p>
 116       The performance differences between these two cases are obvious:
 117     </p>
 118 <pre class="programlisting"><span class="special">---------------------------------------------------------------------------------------</span>
 119 <span class="identifier">Benchmark</span>                                            <span class="identifier">Time</span>   <span class="identifier">Bytes</span><span class="special">/</span><span class="identifier">second</span>
 120 <span class="special">---------------------------------------------------------------------------------------</span>
 121 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>                   <span class="number">260</span> <span class="identifier">ns</span> <span class="number">118.421</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 122 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>                  <span class="number">318</span> <span class="identifier">ns</span> <span class="number">192.797</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 123 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>                  <span class="number">417</span> <span class="identifier">ns</span> <span class="number">303.509</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 124 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>                  <span class="number">625</span> <span class="identifier">ns</span> <span class="number">390.578</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 125 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>                 <span class="number">743</span> <span class="identifier">ns</span> <span class="number">657.827</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 126 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>                <span class="number">1308</span> <span class="identifier">ns</span> <span class="number">767.181</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 127 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>                <span class="number">1896</span> <span class="identifier">ns</span> <span class="number">1034.31</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 128 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>               <span class="number">6582</span> <span class="identifier">ns</span> <span class="number">594.458</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 129 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>              <span class="number">26067</span> <span class="identifier">ns</span> <span class="number">300.001</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 130 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>              <span class="number">62023</span> <span class="identifier">ns</span> <span class="number">252.125</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 131 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>             <span class="number">124976</span> <span class="identifier">ns</span> <span class="number">250.256</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 132 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>            <span class="number">242171</span> <span class="identifier">ns</span> <span class="number">258.29</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 133 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>            <span class="number">528683</span> <span class="identifier">ns</span> <span class="number">236.714</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 134 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>            <span class="number">965354</span> <span class="identifier">ns</span> <span class="number">259.185</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 135 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>          <span class="number">2514693</span> <span class="identifier">ns</span> <span class="number">199.068</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 136 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>          <span class="number">4223084</span> <span class="identifier">ns</span> <span class="number">237.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 137 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>          <span class="number">8638963</span> <span class="identifier">ns</span> <span class="number">231.755</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 138 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>        <span class="number">16215682</span> <span class="identifier">ns</span> <span class="number">246.995</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 139 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>        <span class="number">39180496</span> <span class="identifier">ns</span> <span class="number">204.443</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 140 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>        <span class="number">82495779</span> <span class="identifier">ns</span> <span class="number">194.307</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 141 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>       <span class="number">142675936</span> <span class="identifier">ns</span> <span class="number">224.547</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 142 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>      <span class="number">287638068</span> <span class="identifier">ns</span> <span class="number">223.088</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 143 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>              <span class="number">17.25</span> <span class="identifier">N</span>
 144 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>                  <span class="number">191</span> <span class="identifier">ns</span> <span class="number">320.129</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 145 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>                 <span class="number">233</span> <span class="identifier">ns</span> <span class="number">523.526</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 146 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>                 <span class="number">334</span> <span class="identifier">ns</span> <span class="number">730.8</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 147 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>                 <span class="number">456</span> <span class="identifier">ns</span> <span class="number">1070.93</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 148 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>                <span class="number">688</span> <span class="identifier">ns</span> <span class="number">1.38789</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 149 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>               <span class="number">1257</span> <span class="identifier">ns</span> <span class="number">1.51807</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 150 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>               <span class="number">2663</span> <span class="identifier">ns</span> <span class="number">1.43406</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 151 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>              <span class="number">4100</span> <span class="identifier">ns</span> <span class="number">1.86266</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 152 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>             <span class="number">23493</span> <span class="identifier">ns</span> <span class="number">665.851</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 153 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>             <span class="number">57968</span> <span class="identifier">ns</span> <span class="number">539.551</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 154 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>            <span class="number">142272</span> <span class="identifier">ns</span> <span class="number">439.746</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 155 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>           <span class="number">260948</span> <span class="identifier">ns</span> <span class="number">479.639</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 156 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>           <span class="number">551577</span> <span class="identifier">ns</span> <span class="number">453.623</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 157 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>          <span class="number">1056583</span> <span class="identifier">ns</span> <span class="number">473.654</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 158 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>         <span class="number">2123956</span> <span class="identifier">ns</span> <span class="number">471.35</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 159 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>         <span class="number">5028745</span> <span class="identifier">ns</span> <span class="number">398.111</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 160 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>        <span class="number">10399212</span> <span class="identifier">ns</span> <span class="number">384.981</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 161 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>       <span class="number">23089767</span> <span class="identifier">ns</span> <span class="number">348.496</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 162 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>       <span class="number">37626884</span> <span class="identifier">ns</span> <span class="number">425.962</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 163 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>       <span class="number">79281747</span> <span class="identifier">ns</span> <span class="number">404.088</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 164 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>      <span class="number">172055781</span> <span class="identifier">ns</span> <span class="number">373.391</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 165 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>     <span class="number">391377449</span> <span class="identifier">ns</span> <span class="number">332.01</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 166 <span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>             <span class="number">22.52</span> <span class="identifier">N</span>
 167 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">41.6</span> <span class="identifier">ns</span> <span class="number">739.55</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 168 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">58.4</span> <span class="identifier">ns</span> <span class="number">1050.48</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 169 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">66.5</span> <span class="identifier">ns</span> <span class="number">1.79606</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 170 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>               <span class="number">115</span> <span class="identifier">ns</span> <span class="number">2.0762</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 171 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>              <span class="number">198</span> <span class="identifier">ns</span> <span class="number">2.41515</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 172 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>              <span class="number">365</span> <span class="identifier">ns</span> <span class="number">2.61328</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 173 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>              <span class="number">720</span> <span class="identifier">ns</span> <span class="number">2.65053</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 174 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>            <span class="number">1424</span> <span class="identifier">ns</span> <span class="number">2.68123</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 175 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>            <span class="number">3009</span> <span class="identifier">ns</span> <span class="number">2.5379</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 176 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>           <span class="number">16748</span> <span class="identifier">ns</span> <span class="number">933.699</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 177 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>           <span class="number">40190</span> <span class="identifier">ns</span> <span class="number">778.105</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 178 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>          <span class="number">86500</span> <span class="identifier">ns</span> <span class="number">723.067</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 179 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>         <span class="number">176692</span> <span class="identifier">ns</span> <span class="number">708.108</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 180 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>         <span class="number">356863</span> <span class="identifier">ns</span> <span class="number">701.198</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 181 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>        <span class="number">714807</span> <span class="identifier">ns</span> <span class="number">700.08</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 182 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>       <span class="number">1429078</span> <span class="identifier">ns</span> <span class="number">700.415</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 183 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>       <span class="number">2877227</span> <span class="identifier">ns</span> <span class="number">695.785</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 184 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>      <span class="number">5795662</span> <span class="identifier">ns</span> <span class="number">691.222</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 185 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>     <span class="number">11562715</span> <span class="identifier">ns</span> <span class="number">692.427</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 186 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>     <span class="number">23364846</span> <span class="identifier">ns</span> <span class="number">686.464</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 187 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>     <span class="number">46442540</span> <span class="identifier">ns</span> <span class="number">689.871</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 188 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>    <span class="number">92284501</span> <span class="identifier">ns</span> <span class="number">694.006</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 189 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>            <span class="number">5.51</span> <span class="identifier">N</span>
 190 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>              <span class="number">45.1</span> <span class="identifier">ns</span> <span class="number">1.32169</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 191 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>             <span class="number">53.6</span> <span class="identifier">ns</span> <span class="number">2.22712</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 192 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>             <span class="number">71.4</span> <span class="identifier">ns</span> <span class="number">3.34079</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 193 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">112</span> <span class="identifier">ns</span> <span class="number">4.24946</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 194 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">196</span> <span class="identifier">ns</span> <span class="number">4.87317</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 195 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">378</span> <span class="identifier">ns</span> <span class="number">5.04476</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 196 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>             <span class="number">702</span> <span class="identifier">ns</span> <span class="number">5.44134</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 197 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">1417</span> <span class="identifier">ns</span> <span class="number">5.3865</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 198 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>           <span class="number">3031</span> <span class="identifier">ns</span> <span class="number">5.03872</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 199 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">16813</span> <span class="identifier">ns</span> <span class="number">1.81669</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 200 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">41182</span> <span class="identifier">ns</span> <span class="number">1.48565</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 201 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">86939</span> <span class="identifier">ns</span> <span class="number">1.40536</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 202 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>        <span class="number">177255</span> <span class="identifier">ns</span> <span class="number">1.37892</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 203 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">356391</span> <span class="identifier">ns</span> <span class="number">1.3713</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 204 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">718417</span> <span class="identifier">ns</span> <span class="number">1.36057</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 205 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>      <span class="number">1442288</span> <span class="identifier">ns</span> <span class="number">1.35583</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 206 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">2942259</span> <span class="identifier">ns</span> <span class="number">1.33217</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 207 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">5870235</span> <span class="identifier">ns</span> <span class="number">1.33244</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 208 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">11743081</span> <span class="identifier">ns</span> <span class="number">1.33192</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 209 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">23521002</span> <span class="identifier">ns</span> <span class="number">1.32976</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 210 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">46917407</span> <span class="identifier">ns</span> <span class="number">1.33339</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 211 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>   <span class="number">93823876</span> <span class="identifier">ns</span> <span class="number">1.33305</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 212 <span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">5.59</span> <span class="identifier">N</span>          <span class="number">5.59</span> <span class="identifier">N</span>
 213 </pre>
 214 </div>
 215 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 216 <td align="left"></td>
 217 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 218       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 219       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 220       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 221       Daryle Walker and Xiaogang Zhang<p>
 222         Distributed under the Boost Software License, Version 1.0. (See accompanying
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