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<div class="title">bounds_binomial_proportions.hpp</div> </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00002"></a><span class="lineno"> 2</span>&#160;<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>&#160;<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>&#160;<span class="comment"> * distributed with this work for additional information</span></div>
<div class="line"><a name="l00005"></a><span class="lineno"> 5</span>&#160;<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>&#160;<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>&#160;<span class="comment"> * &quot;License&quot;); you may not use this file except in compliance</span></div>
<div class="line"><a name="l00008"></a><span class="lineno"> 8</span>&#160;<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>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00010"></a><span class="lineno"> 10</span>&#160;<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>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00012"></a><span class="lineno"> 12</span>&#160;<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>&#160;<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>&#160;<span class="comment"> * &quot;AS IS&quot; BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY</span></div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160;<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>&#160;<span class="comment"> * specific language governing permissions and limitations</span></div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span>&#160;<span class="comment"> * under the License.</span></div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span>&#160;<span class="comment"> */</span></div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160; </div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>&#160;<span class="preprocessor">#ifndef _BOUNDS_BINOMIAL_PROPORTIONS_HPP_</span></div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160;<span class="preprocessor">#define _BOUNDS_BINOMIAL_PROPORTIONS_HPP_</span></div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160; </div>
<div class="line"><a name="l00023"></a><span class="lineno"> 23</span>&#160;<span class="preprocessor">#include &lt;cmath&gt;</span></div>
<div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160;<span class="preprocessor">#include &lt;stdexcept&gt;</span></div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; </div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespacedatasketches.html">datasketches</a> {</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; </div>
<div class="line"><a name="l00081"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html"> 81</a></span>&#160;<span class="keyword">class </span><a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html">bounds_binomial_proportions</a> { <span class="comment">// confidence intervals for binomial proportions</span></div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; </div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160;<span class="keyword">public</span>:</div>
<div class="line"><a name="l00113"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html#aeaf7a3842e7bc82772b1dc33891f8c50"> 113</a></span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#aeaf7a3842e7bc82772b1dc33891f8c50">approximate_lower_bound_on_p</a>(uint64_t n, uint64_t k, <span class="keywordtype">double</span> num_std_devs) {</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; check_inputs(n, k);</div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; <span class="keywordflow">if</span> (n == 0) { <span class="keywordflow">return</span> 0.0; } <span class="comment">// the coin was never flipped, so we know nothing</span></div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == 0) { <span class="keywordflow">return</span> 0.0; }</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == 1) { <span class="keywordflow">return</span> (exact_lower_bound_on_p_k_eq_1(n, delta_of_num_stdevs(num_std_devs))); }</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == n) { <span class="keywordflow">return</span> (exact_lower_bound_on_p_k_eq_n(n, delta_of_num_stdevs(num_std_devs))); }</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>&#160; <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>&#160; <span class="keywordtype">double</span> x = abramowitz_stegun_formula_26p5p22((n - k) + 1.0, <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(k), (-1.0 * num_std_devs));</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>&#160; <span class="keywordflow">return</span> (1.0 - x); <span class="comment">// which is p</span></div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>&#160; }</div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; }</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>&#160; </div>
<div class="line"><a name="l00148"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html#a497df40100e75dd55e6f0cdccd90b60a"> 148</a></span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#a497df40100e75dd55e6f0cdccd90b60a">approximate_upper_bound_on_p</a>(uint64_t n, uint64_t k, <span class="keywordtype">double</span> num_std_devs) {</div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>&#160; check_inputs(n, k);</div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160; <span class="keywordflow">if</span> (n == 0) { <span class="keywordflow">return</span> 1.0; } <span class="comment">// the coin was never flipped, so we know nothing</span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == n) { <span class="keywordflow">return</span> 1.0; }</div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == (n - 1)) {</div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>&#160; <span class="keywordflow">return</span> (exact_upper_bound_on_p_k_eq_minusone(n, delta_of_num_stdevs(num_std_devs)));</div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>&#160; }</div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>&#160; <span class="keywordflow">else</span> <span class="keywordflow">if</span> (k == 0) {</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>&#160; <span class="keywordflow">return</span> (exact_upper_bound_on_p_k_eq_zero(n, delta_of_num_stdevs(num_std_devs)));</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; }</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; <span class="keywordtype">double</span> x = abramowitz_stegun_formula_26p5p22(<span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(n - k), k + 1.0, num_std_devs);</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160; <span class="keywordflow">return</span> (1.0 - x); <span class="comment">// which is p</span></div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160; }</div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; }</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160; </div>
<div class="line"><a name="l00170"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html#a04e345ceea6646f2789a4fc8868c470d"> 170</a></span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#a04e345ceea6646f2789a4fc8868c470d">estimate_unknown_p</a>(uint64_t n, uint64_t k) {</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>&#160; check_inputs(n, k);</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>&#160; <span class="keywordflow">if</span> (n == 0) { <span class="keywordflow">return</span> 0.5; } <span class="comment">// the coin was never flipped, so we know nothing</span></div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>&#160; <span class="keywordflow">else</span> { <span class="keywordflow">return</span> ((<span class="keywordtype">double</span>) k / (<span class="keywordtype">double</span>) n); }</div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>&#160; }</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>&#160; </div>
<div class="line"><a name="l00181"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html#ab0c083907bbd0cf5c9532f7c094ae6ef"> 181</a></span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#ab0c083907bbd0cf5c9532f7c094ae6ef">erf</a>(<span class="keywordtype">double</span> x) {</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>&#160; <span class="keywordflow">if</span> (x &lt; 0.0) { <span class="keywordflow">return</span> (-1.0 * (erf_of_nonneg(-1.0 * x))); }</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160; <span class="keywordflow">else</span> { <span class="keywordflow">return</span> (erf_of_nonneg(x)); }</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>&#160; }</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>&#160; </div>
<div class="line"><a name="l00191"></a><span class="lineno"><a class="line" href="classdatasketches_1_1bounds__binomial__proportions.html#a842aac311b818d8b074f8d1c344b903a"> 191</a></span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#a842aac311b818d8b074f8d1c344b903a">normal_cdf</a>(<span class="keywordtype">double</span> x) {</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>&#160; <span class="keywordflow">return</span> (0.5 * (1.0 + (<a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#ab0c083907bbd0cf5c9532f7c094ae6ef">erf</a>(x / (sqrt(2.0))))));</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>&#160; }</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>&#160; </div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>&#160;<span class="keyword">private</span>:</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> check_inputs(uint64_t n, uint64_t k) {</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>&#160; <span class="keywordflow">if</span> (k &gt; n) { <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">&quot;K cannot exceed N&quot;</span>); }</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>&#160; }</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>&#160; </div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>&#160; <span class="comment">//@formatter:off</span></div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>&#160; <span class="comment">// Abramowitz and Stegun formula 7.1.28, p. 88; Claims accuracy of about 7 decimal digits */</span></div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> erf_of_nonneg(<span class="keywordtype">double</span> x) {</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>&#160; <span class="comment">// The constants that appear below, formatted for easy checking against the book.</span></div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>&#160; <span class="comment">// a1 = 0.07052 30784</span></div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>&#160; <span class="comment">// a3 = 0.00927 05272</span></div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>&#160; <span class="comment">// a5 = 0.00027 65672</span></div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>&#160; <span class="comment">// a2 = 0.04228 20123</span></div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>&#160; <span class="comment">// a4 = 0.00015 20143</span></div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>&#160; <span class="comment">// a6 = 0.00004 30638</span></div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a1 = 0.0705230784;</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a3 = 0.0092705272;</div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a5 = 0.0002765672;</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a2 = 0.0422820123;</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a4 = 0.0001520143;</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>&#160; <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">double</span> a6 = 0.0000430638;</div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> x2 = x * x; <span class="comment">// x squared, x cubed, etc.</span></div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> x3 = x2 * x;</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> x4 = x2 * x2;</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> x5 = x2 * x3;</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> x6 = x3 * x3;</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> sum = ( 1.0</div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>&#160; + (a1 * x)</div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>&#160; + (a2 * x2)</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>&#160; + (a3 * x3)</div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>&#160; + (a4 * x4)</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>&#160; + (a5 * x5)</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>&#160; + (a6 * x6) );</div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> sum2 = sum * sum; <span class="comment">// raise the sum to the 16th power</span></div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> sum4 = sum2 * sum2;</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> sum8 = sum4 * sum4;</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> sum16 = sum8 * sum8;</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>&#160; <span class="keywordflow">return</span> (1.0 - (1.0 / sum16));</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>&#160; }</div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>&#160; </div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> delta_of_num_stdevs(<span class="keywordtype">double</span> kappa) {</div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span>&#160; <span class="keywordflow">return</span> (<a class="code" href="classdatasketches_1_1bounds__binomial__proportions.html#a842aac311b818d8b074f8d1c344b903a">normal_cdf</a>(-1.0 * kappa));</div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span>&#160; }</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>&#160; </div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span>&#160; <span class="comment">//@formatter:on</span></div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>&#160; <span class="comment">// Formula 26.5.22 on page 945 of Abramowitz &amp; Stegun, which is an approximation</span></div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>&#160; <span class="comment">// of the inverse of the incomplete beta function I_x(a,b) = delta</span></div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span>&#160; <span class="comment">// viewed as a scalar function of x.</span></div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span>&#160; <span class="comment">// In other words, we specify delta, and it gives us x (with a and b held constant).</span></div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span>&#160; <span class="comment">// However, delta is specified in an indirect way through yp which</span></div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span>&#160; <span class="comment">// is the number of stdDevs that leaves delta probability in the right</span></div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span>&#160; <span class="comment">// tail of a standard gaussian distribution.</span></div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span>&#160; </div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>&#160; <span class="comment">// We point out that the variable names correspond to those in the book,</span></div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>&#160; <span class="comment">// and it is worth keeping it that way so that it will always be easy to verify</span></div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>&#160; <span class="comment">// that the formula was typed in correctly.</span></div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>&#160; </div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> abramowitz_stegun_formula_26p5p22(<span class="keywordtype">double</span> a, <span class="keywordtype">double</span> b, <span class="keywordtype">double</span> yp) {</div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> b2m1 = (2.0 * b) - 1.0;</div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> a2m1 = (2.0 * a) - 1.0;</div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> lambda = ((yp * yp) - 3.0) / 6.0;</div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> htmp = (1.0 / a2m1) + (1.0 / b2m1);</div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> h = 2.0 / htmp;</div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> term1 = (yp * (sqrt(h + lambda))) / h;</div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> term2 = (1.0 / b2m1) - (1.0 / a2m1);</div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> term3 = (lambda + (5.0 / 6.0)) - (2.0 / (3.0 * h));</div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> w = term1 - (term2 * term3);</div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> xp = a / (a + (b * (exp(2.0 * w))));</div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>&#160; <span class="keywordflow">return</span> xp;</div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span>&#160; }</div>
<div class="line"><a name="l00265"></a><span class="lineno"> 265</span>&#160; </div>
<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>&#160; <span class="comment">// Formulas for some special cases.</span></div>
<div class="line"><a name="l00267"></a><span class="lineno"> 267</span>&#160; </div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> exact_upper_bound_on_p_k_eq_zero(uint64_t n, <span class="keywordtype">double</span> delta) {</div>
<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>&#160; <span class="keywordflow">return</span> (1.0 - pow(delta, (1.0 / n)));</div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>&#160; }</div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</span>&#160; </div>
<div class="line"><a name="l00272"></a><span class="lineno"> 272</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> exact_lower_bound_on_p_k_eq_n(uint64_t n, <span class="keywordtype">double</span> delta) {</div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span>&#160; <span class="keywordflow">return</span> (pow(delta, (1.0 / n)));</div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span>&#160; }</div>
<div class="line"><a name="l00275"></a><span class="lineno"> 275</span>&#160; </div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> exact_lower_bound_on_p_k_eq_1(uint64_t n, <span class="keywordtype">double</span> delta) {</div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>&#160; <span class="keywordflow">return</span> (1.0 - pow((1.0 - delta), (1.0 / n)));</div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>&#160; }</div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span>&#160; </div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>&#160; <span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> exact_upper_bound_on_p_k_eq_minusone(uint64_t n, <span class="keywordtype">double</span> delta) {</div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span>&#160; <span class="keywordflow">return</span> (pow((1.0 - delta), (1.0 / n)));</div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>&#160; }</div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>&#160; </div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span>&#160;};</div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span>&#160; </div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>&#160;}</div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span>&#160; </div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span>&#160;<span class="preprocessor">#endif </span><span class="comment">// _BOUNDS_BINOMIAL_PROPORTIONS_HPP_</span></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html">datasketches::bounds_binomial_proportions</a></div><div class="ttdoc">Confidence intervals for binomial proportions.</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:81</div></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html_a04e345ceea6646f2789a4fc8868c470d"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html#a04e345ceea6646f2789a4fc8868c470d">datasketches::bounds_binomial_proportions::estimate_unknown_p</a></div><div class="ttdeci">static double estimate_unknown_p(uint64_t n, uint64_t k)</div><div class="ttdoc">Computes an estimate of an unknown binomial proportion.</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:170</div></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html_a497df40100e75dd55e6f0cdccd90b60a"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html#a497df40100e75dd55e6f0cdccd90b60a">datasketches::bounds_binomial_proportions::approximate_upper_bound_on_p</a></div><div class="ttdeci">static double approximate_upper_bound_on_p(uint64_t n, uint64_t k, double num_std_devs)</div><div class="ttdoc">Computes upper bound of approximate Clopper-Pearson confidence interval for a binomial proportion.</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:148</div></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html_a842aac311b818d8b074f8d1c344b903a"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html#a842aac311b818d8b074f8d1c344b903a">datasketches::bounds_binomial_proportions::normal_cdf</a></div><div class="ttdeci">static double normal_cdf(double x)</div><div class="ttdoc">Computes an approximation to normal_cdf(x).</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:191</div></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html_ab0c083907bbd0cf5c9532f7c094ae6ef"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html#ab0c083907bbd0cf5c9532f7c094ae6ef">datasketches::bounds_binomial_proportions::erf</a></div><div class="ttdeci">static double erf(double x)</div><div class="ttdoc">Computes an approximation to the erf() function.</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:181</div></div>
<div class="ttc" id="aclassdatasketches_1_1bounds__binomial__proportions_html_aeaf7a3842e7bc82772b1dc33891f8c50"><div class="ttname"><a href="classdatasketches_1_1bounds__binomial__proportions.html#aeaf7a3842e7bc82772b1dc33891f8c50">datasketches::bounds_binomial_proportions::approximate_lower_bound_on_p</a></div><div class="ttdeci">static double approximate_lower_bound_on_p(uint64_t n, uint64_t k, double num_std_devs)</div><div class="ttdoc">Computes lower bound of approximate Clopper-Pearson confidence interval for a binomial proportion.</div><div class="ttdef"><b>Definition:</b> bounds_binomial_proportions.hpp:113</div></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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