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  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.runs_test"></a><a class="link" href="runs_test.html" title="Runs tests">Runs tests</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h4>
    30. 

  30. <a name="math_toolkit.runs_test.h0"></a>
    31. 

  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. 

  32.     </h4>
    33. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  42. 

  43. <span class="special">}}}</span>
    44. 

  44. </pre>
    45. 

  45. <h4>
    46. 

  46. <a name="math_toolkit.runs_test.h1"></a>
    47. 

  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. 

  48.     </h4>
    49. 

  49. <p>
    50. 

  50.       The "runs above and below median test" tries to determine if a sequence
    51. 

  51.       is random by looking at the number of consecutive values which exceed (or are
    52. 

  52.       below) the median of the sequence. For example, if we are given data {5, 2,
    53. 

  53.       0, 4, 7, 9, 10, 6, 1, 8, 3}, first we find the median (5), and transform the
    54. 

  54.       vector into a list of + and -'s, depending on whether the value is greater
    55. 

  55.       or less than the median. (Values equal to the median we simply ignore-this
    56. 

  56.       is a convention we have inherited from the wonderful <code class="computeroutput"><span class="identifier">randtests</span></code>
    57. 

  57.       package in R.) Hence our data vector is transformed into
    58. 

  58.     </p>
    59. 

  59. <p>
    60. 

  60.       {-,-,-,+,+,+,-,+,-}
    61. 

  61.     </p>
    62. 

  62. <p>
    63. 

  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. 

  64.       5 values below the median. <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35d.htm" target="_top">NIST</a>
    65. 

  65.       tells us the expected number of runs and their variance:
    66. 

  66.     </p>
    67. 

  67. <p>
    68. 

  68.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/expected_runs_above_threshold.svg" width="418" height="45"></object></span>
    69. 

  69.     </p>
    70. 

  70. <p>
    71. 

  71.       from which we derive the test statistic
    72. 

  72.     </p>
    73. 

  73. <p>
    74. 

  74.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/runs_test_statistic.svg" width="115" height="42"></object></span>
    75. 

  75.     </p>
    76. 

  76. <p>
    77. 

  77.       whose distribution we approximate as normal to extract a <span class="emphasis"><em>p</em></span>-value.
    78. 

  78.     </p>
    79. 

  79. <p>
    80. 

  80.       An example usage is as follows:
    81. 

  81.     </p>
    82. 

  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. 

  83. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
    84. 

  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. 

  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. 

  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. 

  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. 

  89. <span class="comment">// t =  -0.670820393249936919; p = 0.502334954360502017;</span>
    90. 

  90. </pre>
    91. 

  91. <p>
    92. 

  92.       We see that we have about a 50% chance of seeing this number of runs if the
    93. 

  93.       null hypothesis of randomness is true, and hence the assumption of randomness
    94. 

  94.       seems reasonable. As always, the test statistic is the first element of the
    95. 

  95.       pair, and the <span class="emphasis"><em>p</em></span>-value is the second element.
    96. 

  96.     </p>
    97. 

  97. <h4>
    98. 

  98. <a name="math_toolkit.runs_test.h2"></a>
    99. 

  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. 

  100.     </h4>
    101. 

  101. <p>
    102. 

  102.       There are two cases: Where the threshold (typically the median) has already
    103. 

  103.       been computed, and the case where the mean and sample variance must be computed
    104. 

  104.       on the fly. Computing the median is fairly expensive (requiring a call to
    105. 

  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. 

  106.       and since the order of the original data must be preserved, it must allocate.
    107. 

  107.       If you believe your data to come from a distribution where the means and median
    108. 

  108.       coincide, or if you've already computed the median in the course of some other
    109. 

  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. 

  110.       via
    111. 

  111.     </p>
    112. 

  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. 

  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. 

  114. </pre>
    115. 

  115. <p>
    116. 

  116.       The performance differences between these two cases are obvious:
    117. 

  117.     </p>
    118. 

  118. <pre class="programlisting"><span class="special">---------------------------------------------------------------------------------------</span>
    119. 

  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. 

  120. <span class="special">---------------------------------------------------------------------------------------</span>
    121. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  213. </pre>
    214. 

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  220.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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  222.         Distributed under the Boost Software License, Version 1.0. (See accompanying
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  223.         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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