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| <div class="title">binomial_bounds.hpp</div> </div> |
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| <div class="contents"> |
| <div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">/*</span></div> |
| <div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment"> * Licensed to the Apache Software Foundation (ASF) under one</span></div> |
| <div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment"> * or more contributor license agreements. See the NOTICE file</span></div> |
| <div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment"> * distributed with this work for additional information</span></div> |
| <div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment"> * regarding copyright ownership. The ASF licenses this file</span></div> |
| <div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment"> * to you under the Apache License, Version 2.0 (the</span></div> |
| <div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment"> * "License"); you may not use this file except in compliance</span></div> |
| <div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment"> * with the License. You may obtain a copy of the License at</span></div> |
| <div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="comment"> * http://www.apache.org/licenses/LICENSE-2.0</span></div> |
| <div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment"> * Unless required by applicable law or agreed to in writing,</span></div> |
| <div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="comment"> * software distributed under the License is distributed on an</span></div> |
| <div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="comment"> * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY</span></div> |
| <div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="comment"> * KIND, either express or implied. See the License for the</span></div> |
| <div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment"> * specific language governing permissions and limitations</span></div> |
| <div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment"> * under the License.</span></div> |
| <div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00019"></a><span class="lineno"> 19</span>  </div> |
| <div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="preprocessor">#ifndef BINOMIAL_BOUNDS_HPP_</span></div> |
| <div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="preprocessor">#define BINOMIAL_BOUNDS_HPP_</span></div> |
| <div class="line"><a name="l00022"></a><span class="lineno"> 22</span>  </div> |
| <div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="preprocessor">#include <algorithm></span></div> |
| <div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="preprocessor">#include <cmath></span></div> |
| <div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="preprocessor">#include <stdexcept></span></div> |
| <div class="line"><a name="l00026"></a><span class="lineno"> 26</span>  </div> |
| <div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment">/*</span></div> |
| <div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment"> * This class enables the estimation of error bounds given a sample set size, the sampling</span></div> |
| <div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="comment"> * probability theta, the number of standard deviations and a simple noDataSeen flag. This can</span></div> |
| <div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="comment"> * be used to estimate error bounds for fixed threshold sampling as well as the error bounds</span></div> |
| <div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="comment"> * calculations for sketches.</span></div> |
| <div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="comment"> * </span></div> |
| <div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="comment"> * author Alexander Saydakov</span></div> |
| <div class="line"><a name="l00034"></a><span class="lineno"> 34</span> <span class="comment"> * author Lee Rhodes</span></div> |
| <div class="line"><a name="l00035"></a><span class="lineno"> 35</span> <span class="comment"> * author Kevin Lang</span></div> |
| <div class="line"><a name="l00036"></a><span class="lineno"> 36</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  </div> |
| <div class="line"><a name="l00038"></a><span class="lineno"><a class="line" href="namespacedatasketches.html"> 38</a></span> <span class="keyword">namespace </span><a class="code" href="namespacedatasketches.html">datasketches</a> {</div> |
| <div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  </div> |
| <div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="keyword">static</span> constexpr <span class="keywordtype">double</span> delta_of_num_std_devs[] = {</div> |
| <div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  0.5000000000000000000, <span class="comment">// not actually using this value</span></div> |
| <div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  0.1586553191586026479,</div> |
| <div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  0.0227502618904135701,</div> |
| <div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  0.0013498126861731796</div> |
| <div class="line"><a name="l00045"></a><span class="lineno"> 45</span> };</div> |
| <div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  </div> |
| <div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="keyword">static</span> constexpr <span class="keywordtype">double</span> lb_equiv_table[] = {</div> |
| <div class="line"><a name="l00048"></a><span class="lineno"> 48</span>  1.0, 2.0, 3.0, <span class="comment">// fake values for k = 0</span></div> |
| <div class="line"><a name="l00049"></a><span class="lineno"> 49</span>  0.78733703534118149, 3.14426768537558132, 13.56789685109913535, <span class="comment">// k = 1</span></div> |
| <div class="line"><a name="l00050"></a><span class="lineno"> 50</span>  0.94091379266077979, 2.64699271711145911, 6.29302733018320737, <span class="comment">// k = 2</span></div> |
| <div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  0.96869128474958188, 2.46531676590527127, 4.97375283467403051, <span class="comment">// k = 3</span></div> |
| <div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  0.97933572521046131, 2.37418810664669877, 4.44899975481712318, <span class="comment">// k = 4</span></div> |
| <div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  0.98479165917274258, 2.31863116255024693, 4.16712379778553554, <span class="comment">// k = 5</span></div> |
| <div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  0.98806033915698777, 2.28075536565225434, 3.99010556144099837, <span class="comment">// k = 6</span></div> |
| <div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  0.99021896790580399, 2.25302005857281529, 3.86784477136922078, <span class="comment">// k = 7</span></div> |
| <div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  0.99174267079089873, 2.23168103978522936, 3.77784896945266269, <span class="comment">// k = 8</span></div> |
| <div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  0.99287147837287648, 2.21465899260871879, 3.70851932988722410, <span class="comment">// k = 9</span></div> |
| <div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  0.99373900046805375, 2.20070155496262032, 3.65326029076638292, <span class="comment">// k = 10</span></div> |
| <div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  0.99442519013851438, 2.18900651202670815, 3.60803817612955413, <span class="comment">// k = 11</span></div> |
| <div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  0.99498066823221620, 2.17903457780744247, 3.57024330407946877, <span class="comment">// k = 12</span></div> |
| <div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  0.99543899410224412, 2.17040883161922693, 3.53810982030634591, <span class="comment">// k = 13</span></div> |
| <div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  0.99582322541263579, 2.16285726913676513, 3.51039837124298515, <span class="comment">// k = 14</span></div> |
| <div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  0.99614973311747690, 2.15617827879603396, 3.48621230377099778, <span class="comment">// k = 15</span></div> |
| <div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  0.99643042892560629, 2.15021897666090922, 3.46488605693562590, <span class="comment">// k = 16</span></div> |
| <div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  0.99667418783778317, 2.14486114872480016, 3.44591466064832730, <span class="comment">// k = 17</span></div> |
| <div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  0.99688774875812669, 2.14001181420209718, 3.42890765690452781, <span class="comment">// k = 18</span></div> |
| <div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  0.99707632299691795, 2.13559675336844634, 3.41355809420343803, <span class="comment">// k = 19</span></div> |
| <div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  0.99724399084971083, 2.13155592217421486, 3.39962113251016262, <span class="comment">// k = 20</span></div> |
| <div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  0.99739400151915447, 2.12784018863251845, 3.38689892877548004, <span class="comment">// k = 21</span></div> |
| <div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  0.99752896842633731, 2.12440890875851096, 3.37522975271599535, <span class="comment">// k = 22</span></div> |
| <div class="line"><a name="l00071"></a><span class="lineno"> 71</span>  0.99765101725122918, 2.12122815311133195, 3.36448003577621080, <span class="comment">// k = 23</span></div> |
| <div class="line"><a name="l00072"></a><span class="lineno"> 72</span>  0.99776189496810730, 2.11826934724291505, 3.35453840911279144, <span class="comment">// k = 24</span></div> |
| <div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  0.99786304821586214, 2.11550823850916458, 3.34531123809287578, <span class="comment">// k = 25</span></div> |
| <div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  0.99795568665180667, 2.11292409529477254, 3.33671916527694634, <span class="comment">// k = 26</span></div> |
| <div class="line"><a name="l00075"></a><span class="lineno"> 75</span>  0.99804083063483517, 2.11049908609763293, 3.32869446834217797, <span class="comment">// k = 27</span></div> |
| <div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  0.99811933910984862, 2.10821776918189130, 3.32117898316676019, <span class="comment">// k = 28</span></div> |
| <div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  0.99819195457286014, 2.10606671027090897, 3.31412243534683171, <span class="comment">// k = 29</span></div> |
| <div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  0.99825930555178388, 2.10403415237001923, 3.30748113008135647, <span class="comment">// k = 30</span></div> |
| <div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  0.99832193858154028, 2.10210975877822648, 3.30121691946897045, <span class="comment">// k = 31</span></div> |
| <div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  0.99838032666573895, 2.10028440670842542, 3.29529629751144171, <span class="comment">// k = 32</span></div> |
| <div class="line"><a name="l00081"></a><span class="lineno"> 81</span>  0.99843488390555990, 2.09855000145353188, 3.28968974413223236, <span class="comment">// k = 33</span></div> |
| <div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  0.99848596721417948, 2.09689934193824001, 3.28437111460505093, <span class="comment">// k = 34</span></div> |
| <div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  0.99853390005924325, 2.09532599155502908, 3.27931717312372939, <span class="comment">// k = 35</span></div> |
| <div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  0.99857895741078551, 2.09382418262592296, 3.27450718840060517, <span class="comment">// k = 36</span></div> |
| <div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  0.99862138880970974, 2.09238872751677718, 3.26992261182860489, <span class="comment">// k = 37</span></div> |
| <div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  0.99866141580770318, 2.09101494715108061, 3.26554677962434425, <span class="comment">// k = 38</span></div> |
| <div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  0.99869923565267982, 2.08969860402822860, 3.26136468165239535, <span class="comment">// k = 39</span></div> |
| <div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  0.99873502010169091, 2.08843585627218431, 3.25736275677081721, <span class="comment">// k = 40</span></div> |
| <div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  0.99876893292508839, 2.08722321436752623, 3.25352872241415980, <span class="comment">// k = 41</span></div> |
| <div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  0.99880111078502409, 2.08605749165553789, 3.24985141664350863, <span class="comment">// k = 42</span></div> |
| <div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  0.99883168573342118, 2.08493577529222307, 3.24632068399498053, <span class="comment">// k = 43</span></div> |
| <div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  0.99886077231613513, 2.08385540129560809, 3.24292724848112357, <span class="comment">// k = 44</span></div> |
| <div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  0.99888847451828155, 2.08281392374021834, 3.23966263299664092, <span class="comment">// k = 45</span></div> |
| <div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  0.99891488795844907, 2.08180908991394631, 3.23651906111521726, <span class="comment">// k = 46</span></div> |
| <div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  0.99894010085196783, 2.08083882998420222, 3.23348939240611344, <span class="comment">// k = 47</span></div> |
| <div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  0.99896419358239541, 2.07990122528650545, 3.23056705515594444, <span class="comment">// k = 48</span></div> |
| <div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  0.99898723510594323, 2.07899450946285924, 3.22774598963252402, <span class="comment">// k = 49</span></div> |
| <div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  0.99900929266780736, 2.07811704477046533, 3.22502059972006805, <span class="comment">// k = 50</span></div> |
| <div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  0.99903043086155208, 2.07726730587160091, 3.22238570890294795, <span class="comment">// k = 51</span></div> |
| <div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  0.99905070073845081, 2.07644388314946582, 3.21983651940365689, <span class="comment">// k = 52</span></div> |
| <div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  0.99907015770423868, 2.07564546080757850, 3.21736857351049821, <span class="comment">// k = 53</span></div> |
| <div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  0.99908884779227947, 2.07487081196367740, 3.21497773796417619, <span class="comment">// k = 54</span></div> |
| <div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  0.99910681586905525, 2.07411879634256024, 3.21266015316183484, <span class="comment">// k = 55</span></div> |
| <div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  0.99912410177549305, 2.07338834403498140, 3.21041222805715165, <span class="comment">// k = 56</span></div> |
| <div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  0.99914074347179849, 2.07267845454973099, 3.20823061166797174, <span class="comment">// k = 57</span></div> |
| <div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  0.99915677607464204, 2.07198819052374006, 3.20611216970604573, <span class="comment">// k = 58</span></div> |
| <div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  0.99917223149395795, 2.07131667846186929, 3.20405396962596001, <span class="comment">// k = 59</span></div> |
| <div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  0.99918714153457699, 2.07066309019154460, 3.20205326110445299, <span class="comment">// k = 60</span></div> |
| <div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  0.99920153247185794, 2.07002665203046377, 3.20010746990493544, <span class="comment">// k = 61</span></div> |
| <div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  0.99921543193525508, 2.06940663431663552, 3.19821417453343315, <span class="comment">// k = 62</span></div> |
| <div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  0.99922886570365677, 2.06880235245998279, 3.19637109973109546, <span class="comment">// k = 63</span></div> |
| <div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  0.99924185357357942, 2.06821315729285971, 3.19457610621114441, <span class="comment">// k = 64</span></div> |
| <div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  0.99925441845175555, 2.06763843812092318, 3.19282717869864996, <span class="comment">// k = 65</span></div> |
| <div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  0.99926658263325407, 2.06707761824370095, 3.19112241228646099, <span class="comment">// k = 66</span></div> |
| <div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  0.99927836173816331, 2.06653015295219689, 3.18946001739936946, <span class="comment">// k = 67</span></div> |
| <div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  0.99928977431994781, 2.06599552505539918, 3.18783829446098821, <span class="comment">// k = 68</span></div> |
| <div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  0.99930083753795884, 2.06547324585920933, 3.18625564538041317, <span class="comment">// k = 69</span></div> |
| <div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  0.99931156864562354, 2.06496285191821016, 3.18471055124089730, <span class="comment">// k = 70</span></div> |
| <div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  0.99932197985521043, 2.06446390392778767, 3.18320157510865442, <span class="comment">// k = 71</span></div> |
| <div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  0.99933208559809827, 2.06397598606787369, 3.18172735837393361, <span class="comment">// k = 72</span></div> |
| <div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  0.99934190032416836, 2.06349869971447220, 3.18028661102792398, <span class="comment">// k = 73</span></div> |
| <div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  0.99935143390791836, 2.06303166975550312, 3.17887810481605015, <span class="comment">// k = 74</span></div> |
| <div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  0.99936070171270330, 2.06257453607466346, 3.17750067581857820, <span class="comment">// k = 75</span></div> |
| <div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  0.99936971103502970, 2.06212696042919674, 3.17615321728274580, <span class="comment">// k = 76</span></div> |
| <div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  0.99937847392385493, 2.06168861430600714, 3.17483467831510779, <span class="comment">// k = 77</span></div> |
| <div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  0.99938700168914352, 2.06125918927764928, 3.17354405480557489, <span class="comment">// k = 78</span></div> |
| <div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  0.99939530099953799, 2.06083838987589729, 3.17228039269048168, <span class="comment">// k = 79</span></div> |
| <div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  0.99940338278830154, 2.06042593411496000, 3.17104278166036124, <span class="comment">// k = 80</span></div> |
| <div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  0.99941125463777780, 2.06002155276328835, 3.16983035274597569, <span class="comment">// k = 81</span></div> |
| <div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  0.99941892470027938, 2.05962498741951094, 3.16864227952240185, <span class="comment">// k = 82</span></div> |
| <div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  0.99942640059737187, 2.05923599161263837, 3.16747776846497686, <span class="comment">// k = 83</span></div> |
| <div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  0.99943368842187397, 2.05885433061945378, 3.16633606416374391, <span class="comment">// k = 84</span></div> |
| <div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  0.99944079790603269, 2.05847977868873500, 3.16521644518826406, <span class="comment">// k = 85</span></div> |
| <div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  0.99944773295734990, 2.05811212058944193, 3.16411821883858124, <span class="comment">// k = 86</span></div> |
| <div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  0.99945450059186669, 2.05775114781260982, 3.16304072400711789, <span class="comment">// k = 87</span></div> |
| <div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  0.99946110646314423, 2.05739666442039493, 3.16198332650733960, <span class="comment">// k = 88</span></div> |
| <div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  0.99946755770463369, 2.05704847678819647, 3.16094541781455973, <span class="comment">// k = 89</span></div> |
| <div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  0.99947385746861528, 2.05670640500335367, 3.15992641851471490, <span class="comment">// k = 90</span></div> |
| <div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  0.99948001256305474, 2.05637027420314666, 3.15892576988736096, <span class="comment">// k = 91</span></div> |
| <div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  0.99948602689656241, 2.05603991286400856, 3.15794293484717059, <span class="comment">// k = 92</span></div> |
| <div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  0.99949190674294641, 2.05571516158917689, 3.15697740043813724, <span class="comment">// k = 93</span></div> |
| <div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  0.99949765436329585, 2.05539586490317561, 3.15602867309343083, <span class="comment">// k = 94</span></div> |
| <div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  0.99950327557880314, 2.05508187237845164, 3.15509627710042651, <span class="comment">// k = 95</span></div> |
| <div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  0.99950877461972709, 2.05477304104951486, 3.15417975753007340, <span class="comment">// k = 96</span></div> |
| <div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  0.99951415481862682, 2.05446923022574879, 3.15327867462917766, <span class="comment">// k = 97</span></div> |
| <div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  0.99951942042375208, 2.05417030908833453, 3.15239260700215596, <span class="comment">// k = 98</span></div> |
| <div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  0.99952457390890004, 2.05387614661762541, 3.15152114915238712, <span class="comment">// k = 99</span></div> |
| <div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  0.99952962005008317, 2.05358662050909402, 3.15066390921020911, <span class="comment">// k = 100</span></div> |
| <div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  0.99953456216121594, 2.05330161104427589, 3.14982051097524618, <span class="comment">// k = 101</span></div> |
| <div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  0.99953940176368405, 2.05302100378725072, 3.14899059183684926, <span class="comment">// k = 102</span></div> |
| <div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  0.99954414373920031, 2.05274468493067275, 3.14817379948561893, <span class="comment">// k = 103</span></div> |
| <div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  0.99954879047621148, 2.05247255013657082, 3.14736979964868624, <span class="comment">// k = 104</span></div> |
| <div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  0.99955334485656522, 2.05220449388099269, 3.14657826610371671, <span class="comment">// k = 105</span></div> |
| <div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  0.99955780993869325, 2.05194041831310869, 3.14579888316276879, <span class="comment">// k = 106</span></div> |
| <div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  0.99956218652590678, 2.05168022402710903, 3.14503134811607765, <span class="comment">// k = 107</span></div> |
| <div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  0.99956647932785359, 2.05142381889103831, 3.14427536967733090, <span class="comment">// k = 108</span></div> |
| <div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  0.99957069025060719, 2.05117111251445294, 3.14353066260227365, <span class="comment">// k = 109</span></div> |
| <div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  0.99957482032178291, 2.05092201793428330, 3.14279695558593630, <span class="comment">// k = 110</span></div> |
| <div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  0.99957887261450651, 2.05067645094720774, 3.14207398336887422, <span class="comment">// k = 111</span></div> |
| <div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  0.99958284988383639, 2.05043432833224415, 3.14136149076028914, <span class="comment">// k = 112</span></div> |
| <div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  0.99958675435604505, 2.05019557189746138, 3.14065923143530767, <span class="comment">// k = 113</span></div> |
| <div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  0.99959058650074439, 2.04996010556124020, 3.13996696426707445, <span class="comment">// k = 114</span></div> |
| <div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  0.99959434898201494, 2.04972785368377686, 3.13928445867830419, <span class="comment">// k = 115</span></div> |
| <div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  0.99959804437042976, 2.04949874512311681, 3.13861149103462367, <span class="comment">// k = 116</span></div> |
| <div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  0.99960167394553423, 2.04927271043337100, 3.13794784369528656, <span class="comment">// k = 117</span></div> |
| <div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  0.99960523957651048, 2.04904968140490951, 3.13729330661277572, <span class="comment">// k = 118</span></div> |
| <div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  0.99960874253329735, 2.04882959397491504, 3.13664767767019725, <span class="comment">// k = 119</span></div> |
| <div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  0.99961218434327748, 2.04861238220240693, 3.13601075688413289 <span class="comment">// k = 120</span></div> |
| <div class="line"><a name="l00169"></a><span class="lineno"> 169</span> };</div> |
| <div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  </div> |
| <div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="keyword">static</span> constexpr <span class="keywordtype">double</span> ub_equiv_table[] = {</div> |
| <div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  1.0, 2.0, 3.0, <span class="comment">// fake values for k = 0</span></div> |
| <div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  0.99067760836669549, 1.75460517119302040, 2.48055626001627161, <span class="comment">// k = 1</span></div> |
| <div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  0.99270518097577565, 1.78855957509907171, 2.53863835259832626, <span class="comment">// k = 2</span></div> |
| <div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  0.99402032633599902, 1.81047286499563143, 2.57811676180597260, <span class="comment">// k = 3</span></div> |
| <div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  0.99492607629539975, 1.82625928017762362, 2.60759550546498531, <span class="comment">// k = 4</span></div> |
| <div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  0.99558653966013821, 1.83839160339161367, 2.63086812358551470, <span class="comment">// k = 5</span></div> |
| <div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  0.99608981951632813, 1.84812399034444752, 2.64993712523727254, <span class="comment">// k = 6</span></div> |
| <div class="line"><a name="l00179"></a><span class="lineno"> 179</span>  0.99648648035983456, 1.85617372053235385, 2.66598485907860550, <span class="comment">// k = 7</span></div> |
| <div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  0.99680750790483330, 1.86298655802610824, 2.67976541374471822, <span class="comment">// k = 8</span></div> |
| <div class="line"><a name="l00181"></a><span class="lineno"> 181</span>  0.99707292880049181, 1.86885682585270274, 2.69178781407745760, <span class="comment">// k = 9</span></div> |
| <div class="line"><a name="l00182"></a><span class="lineno"> 182</span>  0.99729614928489241, 1.87398826101983218, 2.70241106542158604, <span class="comment">// k = 10</span></div> |
| <div class="line"><a name="l00183"></a><span class="lineno"> 183</span>  0.99748667952445658, 1.87852708449801753, 2.71189717290596377, <span class="comment">// k = 11</span></div> |
| <div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  0.99765127712748836, 1.88258159501103250, 2.72044290303773550, <span class="comment">// k = 12</span></div> |
| <div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  0.99779498340305395, 1.88623391878036273, 2.72819957382063194, <span class="comment">// k = 13</span></div> |
| <div class="line"><a name="l00186"></a><span class="lineno"> 186</span>  0.99792160418357412, 1.88954778748873764, 2.73528576807902368, <span class="comment">// k = 14</span></div> |
| <div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  0.99803398604944960, 1.89257337682371940, 2.74179612106766513, <span class="comment">// k = 15</span></div> |
| <div class="line"><a name="l00188"></a><span class="lineno"> 188</span>  0.99813449883217231, 1.89535099316557876, 2.74780718300419835, <span class="comment">// k = 16</span></div> |
| <div class="line"><a name="l00189"></a><span class="lineno"> 189</span>  0.99822494122659577, 1.89791339232732525, 2.75338173141955167, <span class="comment">// k = 17</span></div> |
| <div class="line"><a name="l00190"></a><span class="lineno"> 190</span>  0.99830679915913834, 1.90028752122407241, 2.75857186416826039, <span class="comment">// k = 18</span></div> |
| <div class="line"><a name="l00191"></a><span class="lineno"> 191</span>  0.99838117410831728, 1.90249575897183831, 2.76342117562634826, <span class="comment">// k = 19</span></div> |
| <div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  0.99844913407071090, 1.90455689090418900, 2.76796659454200267, <span class="comment">// k = 20</span></div> |
| <div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  0.99851147736424650, 1.90648682834171268, 2.77223944710058845, <span class="comment">// k = 21</span></div> |
| <div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  0.99856879856019987, 1.90829917277082473, 2.77626682032629901, <span class="comment">// k = 22</span></div> |
| <div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  0.99862183849734265, 1.91000561415842185, 2.78007199816156003, <span class="comment">// k = 23</span></div> |
| <div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  0.99867096266018507, 1.91161621560812023, 2.78367524259661536, <span class="comment">// k = 24</span></div> |
| <div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  0.99871656986212543, 1.91313978579765376, 2.78709435016625662, <span class="comment">// k = 25</span></div> |
| <div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  0.99875907577771272, 1.91458400425526065, 2.79034488416175463, <span class="comment">// k = 26</span></div> |
| <div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  0.99879885565047744, 1.91595563175945927, 2.79344064132371273, <span class="comment">// k = 27</span></div> |
| <div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  0.99883610756373287, 1.91726064301425936, 2.79639384757751941, <span class="comment">// k = 28</span></div> |
| <div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  0.99887095169674467, 1.91850441099725799, 2.79921543574803877, <span class="comment">// k = 29</span></div> |
| <div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  0.99890379414739527, 1.91969155477030995, 2.80191513182441554, <span class="comment">// k = 30</span></div> |
| <div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  0.99893466279047516, 1.92082633358913313, 2.80450167352080371, <span class="comment">// k = 31</span></div> |
| <div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  0.99896392088177777, 1.92191254955568525, 2.80698295731653502, <span class="comment">// k = 32</span></div> |
| <div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  0.99899147889385631, 1.92295362479495680, 2.80936614404217266, <span class="comment">// k = 33</span></div> |
| <div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  0.99901764688726757, 1.92395267400968351, 2.81165765979318394, <span class="comment">// k = 34</span></div> |
| <div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  0.99904238606342233, 1.92491244978191389, 2.81386337393604435, <span class="comment">// k = 35</span></div> |
| <div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  0.99906590152386343, 1.92583552644848055, 2.81598868034527072, <span class="comment">// k = 36</span></div> |
| <div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  0.99908829040739988, 1.92672418013918900, 2.81803841726804194, <span class="comment">// k = 37</span></div> |
| <div class="line"><a name="l00210"></a><span class="lineno"> 210</span>  0.99910959420023460, 1.92758051694144683, 2.82001709302821268, <span class="comment">// k = 38</span></div> |
| <div class="line"><a name="l00211"></a><span class="lineno"> 211</span>  0.99912996403594434, 1.92840654943159961, 2.82192875763732332, <span class="comment">// k = 39</span></div> |
| <div class="line"><a name="l00212"></a><span class="lineno"> 212</span>  0.99914930224576892, 1.92920397044028391, 2.82377730628954282, <span class="comment">// k = 40</span></div> |
| <div class="line"><a name="l00213"></a><span class="lineno"> 213</span>  0.99916781270195543, 1.92997447498220254, 2.82556612075063640, <span class="comment">// k = 41</span></div> |
| <div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  0.99918553179077207, 1.93071949211818605, 2.82729843191989971, <span class="comment">// k = 42</span></div> |
| <div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  0.99920250730914972, 1.93144048613876862, 2.82897728689417249, <span class="comment">// k = 43</span></div> |
| <div class="line"><a name="l00216"></a><span class="lineno"> 216</span>  0.99921873345181211, 1.93213870990595638, 2.83060537017752267, <span class="comment">// k = 44</span></div> |
| <div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  0.99923435180002684, 1.93281536508689555, 2.83218527795750674, <span class="comment">// k = 45</span></div> |
| <div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  0.99924930425362390, 1.93347145882316340, 2.83371938965598247, <span class="comment">// k = 46</span></div> |
| <div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  0.99926370394567243, 1.93410820221384938, 2.83520990872793277, <span class="comment">// k = 47</span></div> |
| <div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  0.99927750755296074, 1.93472643138986200, 2.83665891945119597, <span class="comment">// k = 48</span></div> |
| <div class="line"><a name="l00221"></a><span class="lineno"> 221</span>  0.99929082941537217, 1.93532697329771963, 2.83806833931606661, <span class="comment">// k = 49</span></div> |
| <div class="line"><a name="l00222"></a><span class="lineno"> 222</span>  0.99930366295501472, 1.93591074716263734, 2.83943997143404658, <span class="comment">// k = 50</span></div> |
| <div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  0.99931598804721489, 1.93647857274021362, 2.84077557836653227, <span class="comment">// k = 51</span></div> |
| <div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  0.99932789059798210, 1.93703110239354714, 2.84207662106302905, <span class="comment">// k = 52</span></div> |
| <div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  0.99933946180485123, 1.93756904936378760, 2.84334468086129277, <span class="comment">// k = 53</span></div> |
| <div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  0.99935053819703512, 1.93809302131219852, 2.84458116874117195, <span class="comment">// k = 54</span></div> |
| <div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  0.99936126637970801, 1.93860365411038060, 2.84578731838604426, <span class="comment">// k = 55</span></div> |
| <div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  0.99937166229284458, 1.93910149816429112, 2.84696443486512862, <span class="comment">// k = 56</span></div> |
| <div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  0.99938169190727422, 1.93958709548454067, 2.84811369085281285, <span class="comment">// k = 57</span></div> |
| <div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  0.99939136927613959, 1.94006085573701625, 2.84923617230361970, <span class="comment">// k = 58</span></div> |
| <div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  0.99940074328745254, 1.94052339623206649, 2.85033291216254270, <span class="comment">// k = 59</span></div> |
| <div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  0.99940993070470086, 1.94097508636855309, 2.85140492437699322, <span class="comment">// k = 60</span></div> |
| <div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  0.99941868577388959, 1.94141633372043998, 2.85245314430358121, <span class="comment">// k = 61</span></div> |
| <div class="line"><a name="l00234"></a><span class="lineno"> 234</span>  0.99942734443487780, 1.94184757038001976, 2.85347839582286156, <span class="comment">// k = 62</span></div> |
| <div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  0.99943556385736088, 1.94226915100517772, 2.85448160365493209, <span class="comment">// k = 63</span></div> |
| <div class="line"><a name="l00236"></a><span class="lineno"> 236</span>  0.99944374522542034, 1.94268143723749631, 2.85546346373061510, <span class="comment">// k = 64</span></div> |
| <div class="line"><a name="l00237"></a><span class="lineno"> 237</span>  0.99945159955424856, 1.94308482059116727, 2.85642486111805738, <span class="comment">// k = 65</span></div> |
| <div class="line"><a name="l00238"></a><span class="lineno"> 238</span>  0.99945915301904620, 1.94347956957849988, 2.85736639994965458, <span class="comment">// k = 66</span></div> |
| <div class="line"><a name="l00239"></a><span class="lineno"> 239</span>  0.99946660663832176, 1.94386600964031686, 2.85828887832701639, <span class="comment">// k = 67</span></div> |
| <div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  0.99947383703224091, 1.94424436597356021, 2.85919278275500233, <span class="comment">// k = 68</span></div> |
| <div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  0.99948075442870277, 1.94461502153473020, 2.86007887186090670, <span class="comment">// k = 69</span></div> |
| <div class="line"><a name="l00242"></a><span class="lineno"> 242</span>  0.99948766082269458, 1.94497821937304138, 2.86094774077355396, <span class="comment">// k = 70</span></div> |
| <div class="line"><a name="l00243"></a><span class="lineno"> 243</span>  0.99949422748713346, 1.94533411296001191, 2.86179981848076181, <span class="comment">// k = 71</span></div> |
| <div class="line"><a name="l00244"></a><span class="lineno"> 244</span>  0.99950070756119658, 1.94568300035135167, 2.86263579405672886, <span class="comment">// k = 72</span></div> |
| <div class="line"><a name="l00245"></a><span class="lineno"> 245</span>  0.99950704321753392, 1.94602523449961495, 2.86345610449197352, <span class="comment">// k = 73</span></div> |
| <div class="line"><a name="l00246"></a><span class="lineno"> 246</span>  0.99951320334216121, 1.94636083782822311, 2.86426125541271404, <span class="comment">// k = 74</span></div> |
| <div class="line"><a name="l00247"></a><span class="lineno"> 247</span>  0.99951920293474927, 1.94669011080745236, 2.86505169255406145, <span class="comment">// k = 75</span></div> |
| <div class="line"><a name="l00248"></a><span class="lineno"> 248</span>  0.99952501670378524, 1.94701327348536779, 2.86582788270862920, <span class="comment">// k = 76</span></div> |
| <div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  0.99953071209267819, 1.94733044372333097, 2.86659027602854621, <span class="comment">// k = 77</span></div> |
| <div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  0.99953632734991515, 1.94764180764266825, 2.86733927778843167, <span class="comment">// k = 78</span></div> |
| <div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  0.99954171164873173, 1.94794766430732125, 2.86807526143834934, <span class="comment">// k = 79</span></div> |
| <div class="line"><a name="l00252"></a><span class="lineno"> 252</span>  0.99954699274462655, 1.94824807472994621, 2.86879864789403882, <span class="comment">// k = 80</span></div> |
| <div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  0.99955216611081710, 1.94854317889829076, 2.86950970901679625, <span class="comment">// k = 81</span></div> |
| <div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  0.99955730019613043, 1.94883320227168610, 2.87020887436986527, <span class="comment">// k = 82</span></div> |
| <div class="line"><a name="l00255"></a><span class="lineno"> 255</span>  0.99956213770650493, 1.94911826561721568, 2.87089648477021342, <span class="comment">// k = 83</span></div> |
| <div class="line"><a name="l00256"></a><span class="lineno"> 256</span>  0.99956704264963037, 1.94939848545763539, 2.87157281693902178, <span class="comment">// k = 84</span></div> |
| <div class="line"><a name="l00257"></a><span class="lineno"> 257</span>  0.99957166306481327, 1.94967401618316671, 2.87223821840905202, <span class="comment">// k = 85</span></div> |
| <div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  0.99957632713136491, 1.94994497791333288, 2.87289293193450135, <span class="comment">// k = 86</span></div> |
| <div class="line"><a name="l00259"></a><span class="lineno"> 259</span>  0.99958087233392234, 1.95021155752212394, 2.87353731228213860, <span class="comment">// k = 87</span></div> |
| <div class="line"><a name="l00260"></a><span class="lineno"> 260</span>  0.99958532555996271, 1.95047376805584349, 2.87417154907075201, <span class="comment">// k = 88</span></div> |
| <div class="line"><a name="l00261"></a><span class="lineno"> 261</span>  0.99958956246481989, 1.95073180380688882, 2.87479599765507032, <span class="comment">// k = 89</span></div> |
| <div class="line"><a name="l00262"></a><span class="lineno"> 262</span>  0.99959389351869277, 1.95098572880579013, 2.87541081987382086, <span class="comment">// k = 90</span></div> |
| <div class="line"><a name="l00263"></a><span class="lineno"> 263</span>  0.99959807862052230, 1.95123574036898617, 2.87601637401948551, <span class="comment">// k = 91</span></div> |
| <div class="line"><a name="l00264"></a><span class="lineno"> 264</span>  0.99960214057801977, 1.95148186921983324, 2.87661283691068093, <span class="comment">// k = 92</span></div> |
| <div class="line"><a name="l00265"></a><span class="lineno"> 265</span>  0.99960607527256684, 1.95172415829728152, 2.87720042968334155, <span class="comment">// k = 93</span></div> |
| <div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  0.99960996433179616, 1.95196280898670693, 2.87777936649376898, <span class="comment">// k = 94</span></div> |
| <div class="line"><a name="l00267"></a><span class="lineno"> 267</span>  0.99961379137860717, 1.95219787713926962, 2.87834989933620022, <span class="comment">// k = 95</span></div> |
| <div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  0.99961756088146103, 1.95242944583677058, 2.87891216133900230, <span class="comment">// k = 96</span></div> |
| <div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  0.99962125605327401, 1.95265762420910960, 2.87946647367488140, <span class="comment">// k = 97</span></div> |
| <div class="line"><a name="l00270"></a><span class="lineno"> 270</span>  0.99962486179100551, 1.95288245314810638, 2.88001290210658567, <span class="comment">// k = 98</span></div> |
| <div class="line"><a name="l00271"></a><span class="lineno"> 271</span>  0.99962843240297161, 1.95310404286672679, 2.88055166523392359, <span class="comment">// k = 99</span></div> |
| <div class="line"><a name="l00272"></a><span class="lineno"> 272</span>  0.99963187276145504, 1.95332251980147475, 2.88108300006589957, <span class="comment">// k = 100</span></div> |
| <div class="line"><a name="l00273"></a><span class="lineno"> 273</span>  0.99963525453173929, 1.95353785898848287, 2.88160703591438505, <span class="comment">// k = 101</span></div> |
| <div class="line"><a name="l00274"></a><span class="lineno"> 274</span>  0.99963855412988778, 1.95375019354571577, 2.88212393551896184, <span class="comment">// k = 102</span></div> |
| <div class="line"><a name="l00275"></a><span class="lineno"> 275</span>  0.99964190254169694, 1.95395953472205974, 2.88263389761985422, <span class="comment">// k = 103</span></div> |
| <div class="line"><a name="l00276"></a><span class="lineno"> 276</span>  0.99964506565942202, 1.95416607430155409, 2.88313700661564098, <span class="comment">// k = 104</span></div> |
| <div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  0.99964834424233118, 1.95436972855640079, 2.88363350163803034, <span class="comment">// k = 105</span></div> |
| <div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  0.99965136548857458, 1.95457068540693513, 2.88412349413960101, <span class="comment">// k = 106</span></div> |
| <div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  0.99965436594726498, 1.95476896383092935, 2.88460710620208260, <span class="comment">// k = 107</span></div> |
| <div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  0.99965736463468602, 1.95496457504532373, 2.88508450078833789, <span class="comment">// k = 108</span></div> |
| <div class="line"><a name="l00281"></a><span class="lineno"> 281</span>  0.99966034130443404, 1.95515761150707590, 2.88555580586194083, <span class="comment">// k = 109</span></div> |
| <div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  0.99966326130828520, 1.95534810382198998, 2.88602118761679094, <span class="comment">// k = 110</span></div> |
| <div class="line"><a name="l00283"></a><span class="lineno"> 283</span>  0.99966601446035952, 1.95553622237747504, 2.88648066384146773, <span class="comment">// k = 111</span></div> |
| <div class="line"><a name="l00284"></a><span class="lineno"> 284</span>  0.99966887679593697, 1.95572186728168163, 2.88693444915907094, <span class="comment">// k = 112</span></div> |
| <div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  0.99967161286551232, 1.95590523410490391, 2.88738271495714116, <span class="comment">// k = 113</span></div> |
| <div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  0.99967435412270333, 1.95608626483223702, 2.88782540459769166, <span class="comment">// k = 114</span></div> |
| <div class="line"><a name="l00287"></a><span class="lineno"> 287</span>  0.99967701261934394, 1.95626497627117146, 2.88826277189363623, <span class="comment">// k = 115</span></div> |
| <div class="line"><a name="l00288"></a><span class="lineno"> 288</span>  0.99967963265157778, 1.95644153684824573, 2.88869486674335008, <span class="comment">// k = 116</span></div> |
| <div class="line"><a name="l00289"></a><span class="lineno"> 289</span>  0.99968216317182623, 1.95661589936000269, 2.88912184353694101, <span class="comment">// k = 117</span></div> |
| <div class="line"><a name="l00290"></a><span class="lineno"> 290</span>  0.99968479674396349, 1.95678821614791332, 2.88954376359643561, <span class="comment">// k = 118</span></div> |
| <div class="line"><a name="l00291"></a><span class="lineno"> 291</span>  0.99968729031337489, 1.95695842061650183, 2.88996069422501023, <span class="comment">// k = 119</span></div> |
| <div class="line"><a name="l00292"></a><span class="lineno"> 292</span>  0.99968963358631413, 1.95712651709766305, 2.89037285320668502 <span class="comment">// k = 120</span></div> |
| <div class="line"><a name="l00293"></a><span class="lineno"> 293</span> };</div> |
| <div class="line"><a name="l00294"></a><span class="lineno"> 294</span>  </div> |
| <div class="line"><a name="l00295"></a><span class="lineno"> 295</span> <span class="keyword">class </span>binomial_bounds {</div> |
| <div class="line"><a name="l00296"></a><span class="lineno"> 296</span>  </div> |
| <div class="line"><a name="l00297"></a><span class="lineno"> 297</span> <span class="keyword">public</span>:</div> |
| <div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> get_lower_bound(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">unsigned</span> num_std_devs) {</div> |
| <div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  check_theta(theta);</div> |
| <div class="line"><a name="l00300"></a><span class="lineno"> 300</span>  check_num_std_devs(num_std_devs);</div> |
| <div class="line"><a name="l00301"></a><span class="lineno"> 301</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> estimate = num_samples / theta;</div> |
| <div class="line"><a name="l00302"></a><span class="lineno"> 302</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> lb = compute_approx_binomial_lower_bound(num_samples, theta, num_std_devs);</div> |
| <div class="line"><a name="l00303"></a><span class="lineno"> 303</span>  <span class="keywordflow">return</span> std::min(estimate, std::max(<span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(num_samples), lb));</div> |
| <div class="line"><a name="l00304"></a><span class="lineno"> 304</span>  }</div> |
| <div class="line"><a name="l00305"></a><span class="lineno"> 305</span>  </div> |
| <div class="line"><a name="l00306"></a><span class="lineno"> 306</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> get_upper_bound(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">unsigned</span> num_std_devs) {</div> |
| <div class="line"><a name="l00307"></a><span class="lineno"> 307</span>  check_theta(theta);</div> |
| <div class="line"><a name="l00308"></a><span class="lineno"> 308</span>  check_num_std_devs(num_std_devs);</div> |
| <div class="line"><a name="l00309"></a><span class="lineno"> 309</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> estimate = num_samples / theta;</div> |
| <div class="line"><a name="l00310"></a><span class="lineno"> 310</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> ub = compute_approx_binomial_upper_bound(num_samples, theta, num_std_devs);</div> |
| <div class="line"><a name="l00311"></a><span class="lineno"> 311</span>  <span class="keywordflow">return</span> std::max(estimate, ub);</div> |
| <div class="line"><a name="l00312"></a><span class="lineno"> 312</span>  }</div> |
| <div class="line"><a name="l00313"></a><span class="lineno"> 313</span>  </div> |
| <div class="line"><a name="l00314"></a><span class="lineno"> 314</span> <span class="keyword">private</span>:</div> |
| <div class="line"><a name="l00315"></a><span class="lineno"> 315</span>  <span class="comment">// our "classic" bounds, but now with continuity correction</span></div> |
| <div class="line"><a name="l00316"></a><span class="lineno"> 316</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> cont_classic_lb(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">double</span> num_std_devs) {</div> |
| <div class="line"><a name="l00317"></a><span class="lineno"> 317</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> n_hat = (num_samples - 0.5) / theta;</div> |
| <div class="line"><a name="l00318"></a><span class="lineno"> 318</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> b = num_std_devs * std::sqrt((1.0 - theta) / theta);</div> |
| <div class="line"><a name="l00319"></a><span class="lineno"> 319</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> d = 0.5 * b * std::sqrt((b * b) + (4.0 * n_hat));</div> |
| <div class="line"><a name="l00320"></a><span class="lineno"> 320</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> center = n_hat + (0.5 * (b * b));</div> |
| <div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  <span class="keywordflow">return</span> (center - d);</div> |
| <div class="line"><a name="l00322"></a><span class="lineno"> 322</span>  }</div> |
| <div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  </div> |
| <div class="line"><a name="l00324"></a><span class="lineno"> 324</span>  <span class="comment">// our "classic" bounds, but now with continuity correction</span></div> |
| <div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> cont_classic_ub(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">double</span> num_std_devs) {</div> |
| <div class="line"><a name="l00326"></a><span class="lineno"> 326</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> n_hat = (num_samples + 0.5) / theta;</div> |
| <div class="line"><a name="l00327"></a><span class="lineno"> 327</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> b = num_std_devs * std::sqrt((1.0 - theta) / theta);</div> |
| <div class="line"><a name="l00328"></a><span class="lineno"> 328</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> d = 0.5 * b * std::sqrt((b * b) + (4.0 * n_hat));</div> |
| <div class="line"><a name="l00329"></a><span class="lineno"> 329</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> center = n_hat + (0.5 * (b * b));</div> |
| <div class="line"><a name="l00330"></a><span class="lineno"> 330</span>  <span class="keywordflow">return</span> (center + d);</div> |
| <div class="line"><a name="l00331"></a><span class="lineno"> 331</span>  }</div> |
| <div class="line"><a name="l00332"></a><span class="lineno"> 332</span>  </div> |
| <div class="line"><a name="l00333"></a><span class="lineno"> 333</span>  <span class="comment">// This is a special purpose calculator for NStar, using a computational</span></div> |
| <div class="line"><a name="l00334"></a><span class="lineno"> 334</span>  <span class="comment">// strategy inspired by its Bayesian definition. It is only appropriate</span></div> |
| <div class="line"><a name="l00335"></a><span class="lineno"> 335</span>  <span class="comment">// for a very limited set of inputs. However, the procedure compute_approx_binomial_lower_bound()</span></div> |
| <div class="line"><a name="l00336"></a><span class="lineno"> 336</span>  <span class="comment">// below does in fact only call it for suitably limited inputs.</span></div> |
| <div class="line"><a name="l00337"></a><span class="lineno"> 337</span>  <span class="comment">// Outside of this limited range, two different bad things will happen.</span></div> |
| <div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  <span class="comment">// First, because we are not using logarithms, the values of intermediate</span></div> |
| <div class="line"><a name="l00339"></a><span class="lineno"> 339</span>  <span class="comment">// quantities will exceed the dynamic range of doubles. Second, even if that</span></div> |
| <div class="line"><a name="l00340"></a><span class="lineno"> 340</span>  <span class="comment">// problem were fixed, the running time of this procedure is essentially linear</span></div> |
| <div class="line"><a name="l00341"></a><span class="lineno"> 341</span>  <span class="comment">// in est = (numSamples / p), and that can be Very, Very Big.</span></div> |
| <div class="line"><a name="l00342"></a><span class="lineno"> 342</span>  <span class="keyword">static</span> <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> special_n_star(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> p, <span class="keywordtype">double</span> delta) {</div> |
| <div class="line"><a name="l00343"></a><span class="lineno"> 343</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> q = 1.0 - p;</div> |
| <div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  <span class="comment">// Use a different algorithm if the following is true; this one will be too slow, or worse.</span></div> |
| <div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  <span class="keywordflow">if</span> ((num_samples / p) >= 500.0) <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">"out of range"</span>);</div> |
| <div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  <span class="keywordtype">double</span> cur_term = std::pow(p, num_samples); <span class="comment">// curTerm = posteriorProbability (k, k, p)</span></div> |
| <div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  <span class="keywordflow">if</span> (cur_term <= 1e-100) <span class="keywordflow">throw</span> std::logic_error(<span class="stringliteral">"out of range"</span>); <span class="comment">// sanity check for non-use of logarithms</span></div> |
| <div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  <span class="keywordtype">double</span> tot = cur_term;</div> |
| <div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> m = num_samples;</div> |
| <div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  <span class="keywordflow">while</span> (tot <= delta) { <span class="comment">// this test can fail even the first time</span></div> |
| <div class="line"><a name="l00351"></a><span class="lineno"> 351</span>  cur_term = (cur_term * q * (m)) / ((m + 1) - num_samples);</div> |
| <div class="line"><a name="l00352"></a><span class="lineno"> 352</span>  tot += cur_term;</div> |
| <div class="line"><a name="l00353"></a><span class="lineno"> 353</span>  m += 1;</div> |
| <div class="line"><a name="l00354"></a><span class="lineno"> 354</span>  }</div> |
| <div class="line"><a name="l00355"></a><span class="lineno"> 355</span>  <span class="comment">// we have reached a state where tot > delta, so back up one</span></div> |
| <div class="line"><a name="l00356"></a><span class="lineno"> 356</span>  <span class="keywordflow">return</span> (m - 1);</div> |
| <div class="line"><a name="l00357"></a><span class="lineno"> 357</span>  }</div> |
| <div class="line"><a name="l00358"></a><span class="lineno"> 358</span>  </div> |
| <div class="line"><a name="l00359"></a><span class="lineno"> 359</span>  <span class="comment">// The following procedure has very limited applicability.</span></div> |
| <div class="line"><a name="l00360"></a><span class="lineno"> 360</span>  <span class="comment">// The above remarks about special_n_star() also apply here.</span></div> |
| <div class="line"><a name="l00361"></a><span class="lineno"> 361</span>  <span class="keyword">static</span> <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> special_n_prime_b(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> p, <span class="keywordtype">double</span> delta) {</div> |
| <div class="line"><a name="l00362"></a><span class="lineno"> 362</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> q = 1.0 - p;</div> |
| <div class="line"><a name="l00363"></a><span class="lineno"> 363</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> one_minus_delta = 1.0 - delta;</div> |
| <div class="line"><a name="l00364"></a><span class="lineno"> 364</span>  <span class="keywordtype">double</span> cur_term = std::pow(p, num_samples); <span class="comment">// curTerm = posteriorProbability (k, k, p)</span></div> |
| <div class="line"><a name="l00365"></a><span class="lineno"> 365</span>  <span class="keywordflow">if</span> (cur_term <= 1e-100) <span class="keywordflow">throw</span> std::logic_error(<span class="stringliteral">"out of range"</span>); <span class="comment">// sanity check for non-use of logarithms</span></div> |
| <div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  <span class="keywordtype">double</span> tot = cur_term;</div> |
| <div class="line"><a name="l00367"></a><span class="lineno"> 367</span>  <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> m = num_samples;</div> |
| <div class="line"><a name="l00368"></a><span class="lineno"> 368</span>  <span class="keywordflow">while</span> (tot < one_minus_delta) {</div> |
| <div class="line"><a name="l00369"></a><span class="lineno"> 369</span>  cur_term = (cur_term * q * (m)) / ((m + 1) - num_samples);</div> |
| <div class="line"><a name="l00370"></a><span class="lineno"> 370</span>  tot += cur_term;</div> |
| <div class="line"><a name="l00371"></a><span class="lineno"> 371</span>  m += 1;</div> |
| <div class="line"><a name="l00372"></a><span class="lineno"> 372</span>  }</div> |
| <div class="line"><a name="l00373"></a><span class="lineno"> 373</span>  <span class="keywordflow">return</span> m; <span class="comment">// no need to back up</span></div> |
| <div class="line"><a name="l00374"></a><span class="lineno"> 374</span>  }</div> |
| <div class="line"><a name="l00375"></a><span class="lineno"> 375</span>  </div> |
| <div class="line"><a name="l00376"></a><span class="lineno"> 376</span>  <span class="keyword">static</span> <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> special_n_prime_f(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> p, <span class="keywordtype">double</span> delta) {</div> |
| <div class="line"><a name="l00377"></a><span class="lineno"> 377</span>  <span class="comment">// Use a different algorithm if the following is true; this one will be too slow, or worse.</span></div> |
| <div class="line"><a name="l00378"></a><span class="lineno"> 378</span>  <span class="keywordflow">if</span> ((num_samples / p) >= 500.0) <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">"out of range"</span>); <span class="comment">//A super-small delta could also make it slow.</span></div> |
| <div class="line"><a name="l00379"></a><span class="lineno"> 379</span>  <span class="keywordflow">return</span> special_n_prime_b(num_samples + 1, p, delta);</div> |
| <div class="line"><a name="l00380"></a><span class="lineno"> 380</span>  }</div> |
| <div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  </div> |
| <div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  <span class="comment">// The following computes an approximation to the lower bound of a Frequentist</span></div> |
| <div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  <span class="comment">// confidence interval based on the tails of the Binomial distribution.</span></div> |
| <div class="line"><a name="l00384"></a><span class="lineno"> 384</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> compute_approx_binomial_lower_bound(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">unsigned</span> num_std_devs) {</div> |
| <div class="line"><a name="l00385"></a><span class="lineno"> 385</span>  <span class="keywordflow">if</span> (theta == 1) <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(num_samples);</div> |
| <div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  <span class="keywordflow">if</span> (num_samples == 0) <span class="keywordflow">return</span> 0;</div> |
| <div class="line"><a name="l00387"></a><span class="lineno"> 387</span>  <span class="keywordflow">if</span> (num_samples == 1) {</div> |
| <div class="line"><a name="l00388"></a><span class="lineno"> 388</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> delta = delta_of_num_std_devs[num_std_devs];</div> |
| <div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_lb = std::log(1 - delta) / std::log(1 - theta);</div> |
| <div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  <span class="keywordflow">return</span> std::floor(raw_lb); <span class="comment">// round down</span></div> |
| <div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  }</div> |
| <div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  <span class="keywordflow">if</span> (num_samples > 120) {</div> |
| <div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  <span class="comment">// plenty of samples, so gaussian approximation to binomial distribution isn't too bad</span></div> |
| <div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_lb = cont_classic_lb(num_samples, theta, num_std_devs);</div> |
| <div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  <span class="keywordflow">return</span> (raw_lb - 0.5); <span class="comment">// fake round down</span></div> |
| <div class="line"><a name="l00396"></a><span class="lineno"> 396</span>  }</div> |
| <div class="line"><a name="l00397"></a><span class="lineno"> 397</span>  <span class="comment">// at this point we know 2 <= num_samples <= 120</span></div> |
| <div class="line"><a name="l00398"></a><span class="lineno"> 398</span>  <span class="keywordflow">if</span> (theta > (1 - 1e-5)) { <span class="comment">// empirically-determined threshold</span></div> |
| <div class="line"><a name="l00399"></a><span class="lineno"> 399</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(num_samples);</div> |
| <div class="line"><a name="l00400"></a><span class="lineno"> 400</span>  }</div> |
| <div class="line"><a name="l00401"></a><span class="lineno"> 401</span>  <span class="keywordflow">if</span> (theta < (num_samples / 360.0)) { <span class="comment">// empirically-determined threshold</span></div> |
| <div class="line"><a name="l00402"></a><span class="lineno"> 402</span>  <span class="comment">// here we use the Gaussian approximation, but with a modified num_std_devs</span></div> |
| <div class="line"><a name="l00403"></a><span class="lineno"> 403</span>  <span class="keyword">const</span> <span class="keywordtype">unsigned</span> index = 3 * <span class="keyword">static_cast<</span><span class="keywordtype">unsigned</span><span class="keyword">></span>(num_samples) + (num_std_devs - 1);</div> |
| <div class="line"><a name="l00404"></a><span class="lineno"> 404</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_lb = cont_classic_lb(num_samples, theta, lb_equiv_table[index]);</div> |
| <div class="line"><a name="l00405"></a><span class="lineno"> 405</span>  <span class="keywordflow">return</span> raw_lb - 0.5; <span class="comment">// fake round down</span></div> |
| <div class="line"><a name="l00406"></a><span class="lineno"> 406</span>  }</div> |
| <div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="comment">// This is the most difficult range to approximate; we will compute an "exact" LB.</span></div> |
| <div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  <span class="comment">// We know that est <= 360, so specialNStar() shouldn't be ridiculously slow.</span></div> |
| <div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> delta = delta_of_num_std_devs[num_std_devs];</div> |
| <div class="line"><a name="l00410"></a><span class="lineno"> 410</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(special_n_star(num_samples, theta, delta)); <span class="comment">// no need to round</span></div> |
| <div class="line"><a name="l00411"></a><span class="lineno"> 411</span>  }</div> |
| <div class="line"><a name="l00412"></a><span class="lineno"> 412</span>  </div> |
| <div class="line"><a name="l00413"></a><span class="lineno"> 413</span>  <span class="comment">// The following computes an approximation to the upper bound of a Frequentist</span></div> |
| <div class="line"><a name="l00414"></a><span class="lineno"> 414</span>  <span class="comment">// confidence interval based on the tails of the Binomial distribution.</span></div> |
| <div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  <span class="keyword">static</span> <span class="keywordtype">double</span> compute_approx_binomial_upper_bound(<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> num_samples, <span class="keywordtype">double</span> theta, <span class="keywordtype">unsigned</span> num_std_devs) {</div> |
| <div class="line"><a name="l00416"></a><span class="lineno"> 416</span>  <span class="keywordflow">if</span> (theta == 1) <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(num_samples);</div> |
| <div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  <span class="keywordflow">if</span> (num_samples == 0) {</div> |
| <div class="line"><a name="l00418"></a><span class="lineno"> 418</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> delta = delta_of_num_std_devs[num_std_devs];</div> |
| <div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_ub = std::log(delta) / std::log(1 - theta);</div> |
| <div class="line"><a name="l00420"></a><span class="lineno"> 420</span>  <span class="keywordflow">return</span> std::ceil(raw_ub); <span class="comment">// round up</span></div> |
| <div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  }</div> |
| <div class="line"><a name="l00422"></a><span class="lineno"> 422</span>  <span class="keywordflow">if</span> (num_samples > 120) {</div> |
| <div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  <span class="comment">// plenty of samples, so gaussian approximation to binomial distribution isn't too bad</span></div> |
| <div class="line"><a name="l00424"></a><span class="lineno"> 424</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_ub = cont_classic_ub(num_samples, theta, num_std_devs);</div> |
| <div class="line"><a name="l00425"></a><span class="lineno"> 425</span>  <span class="keywordflow">return</span> (raw_ub + 0.5); <span class="comment">// fake round up</span></div> |
| <div class="line"><a name="l00426"></a><span class="lineno"> 426</span>  }</div> |
| <div class="line"><a name="l00427"></a><span class="lineno"> 427</span>  <span class="comment">// at this point we know 2 <= num_samples <= 120</span></div> |
| <div class="line"><a name="l00428"></a><span class="lineno"> 428</span>  <span class="keywordflow">if</span> (theta > (1 - 1e-5)) { <span class="comment">// empirically-determined threshold</span></div> |
| <div class="line"><a name="l00429"></a><span class="lineno"> 429</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(num_samples + 1);</div> |
| <div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  }</div> |
| <div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  <span class="keywordflow">if</span> (theta < (num_samples / 360.0)) { <span class="comment">// empirically-determined threshold</span></div> |
| <div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  <span class="comment">// here we use the Gaussian approximation, but with a modified num_std_devs</span></div> |
| <div class="line"><a name="l00433"></a><span class="lineno"> 433</span>  <span class="keyword">const</span> <span class="keywordtype">unsigned</span> index = 3 * <span class="keyword">static_cast<</span><span class="keywordtype">unsigned</span><span class="keyword">></span>(num_samples) + (num_std_devs - 1);</div> |
| <div class="line"><a name="l00434"></a><span class="lineno"> 434</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> raw_ub = cont_classic_ub(num_samples, theta, ub_equiv_table[index]);</div> |
| <div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  <span class="keywordflow">return</span> raw_ub + 0.5; <span class="comment">// fake round up</span></div> |
| <div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  }</div> |
| <div class="line"><a name="l00437"></a><span class="lineno"> 437</span>  <span class="comment">// This is the most difficult range to approximate; we will compute an "exact" UB.</span></div> |
| <div class="line"><a name="l00438"></a><span class="lineno"> 438</span>  <span class="comment">// We know that est <= 360, so specialNPrimeF() shouldn't be ridiculously slow.</span></div> |
| <div class="line"><a name="l00439"></a><span class="lineno"> 439</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> delta = delta_of_num_std_devs[num_std_devs];</div> |
| <div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(special_n_prime_f(num_samples, theta, delta)); <span class="comment">// no need to round</span></div> |
| <div class="line"><a name="l00441"></a><span class="lineno"> 441</span>  }</div> |
| <div class="line"><a name="l00442"></a><span class="lineno"> 442</span>  </div> |
| <div class="line"><a name="l00443"></a><span class="lineno"> 443</span>  <span class="keyword">static</span> <span class="keywordtype">void</span> check_theta(<span class="keywordtype">double</span> theta) {</div> |
| <div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <span class="keywordflow">if</span> (theta < 0 || theta > 1) {</div> |
| <div class="line"><a name="l00445"></a><span class="lineno"> 445</span>  <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">"theta must be in [0, 1]"</span>);</div> |
| <div class="line"><a name="l00446"></a><span class="lineno"> 446</span>  }</div> |
| <div class="line"><a name="l00447"></a><span class="lineno"> 447</span>  }</div> |
| <div class="line"><a name="l00448"></a><span class="lineno"> 448</span>  </div> |
| <div class="line"><a name="l00449"></a><span class="lineno"> 449</span>  <span class="keyword">static</span> <span class="keywordtype">void</span> check_num_std_devs(<span class="keywordtype">unsigned</span> num_std_devs) {</div> |
| <div class="line"><a name="l00450"></a><span class="lineno"> 450</span>  <span class="keywordflow">if</span> (num_std_devs < 1 || num_std_devs > 3) {</div> |
| <div class="line"><a name="l00451"></a><span class="lineno"> 451</span>  <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">"num_std_devs must be 1, 2 or 3"</span>);</div> |
| <div class="line"><a name="l00452"></a><span class="lineno"> 452</span>  }</div> |
| <div class="line"><a name="l00453"></a><span class="lineno"> 453</span>  }</div> |
| <div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  </div> |
| <div class="line"><a name="l00455"></a><span class="lineno"> 455</span> };</div> |
| <div class="line"><a name="l00456"></a><span class="lineno"> 456</span>  </div> |
| <div class="line"><a name="l00457"></a><span class="lineno"> 457</span> } <span class="comment">/* namespace datasketches */</span></div> |
| <div class="line"><a name="l00458"></a><span class="lineno"> 458</span>  </div> |
| <div class="line"><a name="l00459"></a><span class="lineno"> 459</span> <span class="preprocessor"># endif</span></div> |
| <div class="ttc" id="anamespacedatasketches_html"><div class="ttname"><a href="namespacedatasketches.html">datasketches</a></div><div class="ttdoc">DataSketches namespace.</div><div class="ttdef"><b>Definition:</b> binomial_bounds.hpp:38</div></div> |
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