1 | /*
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2 | * Licensed to the Apache Software Foundation (ASF) under one or more
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3 | * contributor license agreements. See the NOTICE file distributed with
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4 | * this work for additional information regarding copyright ownership.
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5 | * The ASF licenses this file to You under the Apache License, Version 2.0
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6 | * (the "License"); you may not use this file except in compliance with
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7 | * the License. You may obtain a copy of the License at
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8 | *
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9 | * http://www.apache.org/licenses/LICENSE-2.0
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10 | *
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11 | * Unless required by applicable law or agreed to in writing, software
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12 | * distributed under the License is distributed on an "AS IS" BASIS,
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13 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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14 | * See the License for the specific language governing permissions and
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15 | * limitations under the License.
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16 | */
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17 | package agents.anac.y2019.harddealer.math3.special;
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18 |
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19 | import agents.anac.y2019.harddealer.math3.util.FastMath;
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20 |
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21 | /**
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22 | * This is a utility class that provides computation methods related to the
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23 | * error functions.
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24 | *
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25 | */
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26 | public class Erf {
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27 |
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28 | /**
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29 | * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally.
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30 | * This number solves {@code erf(x)=0.5} within 1ulp.
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31 | * More precisely, the current implementations of
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32 | * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br/>
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33 | * {@code erf(X_CRIT) < 0.5},<br/>
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34 | * {@code erf(Math.nextUp(X_CRIT) > 0.5},<br/>
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35 | * {@code erfc(X_CRIT) = 0.5}, and<br/>
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36 | * {@code erfc(Math.nextUp(X_CRIT) < 0.5}
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37 | */
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38 | private static final double X_CRIT = 0.4769362762044697;
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39 |
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40 | /**
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41 | * Default constructor. Prohibit instantiation.
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42 | */
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43 | private Erf() {}
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44 |
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45 | /**
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46 | * Returns the error function.
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47 | *
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48 | * <p>erf(x) = 2/√π <sub>0</sub>∫<sup>x</sup> e<sup>-t<sup>2</sup></sup>dt </p>
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49 | *
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50 | * <p>This implementation computes erf(x) using the
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51 | * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
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52 | * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
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53 | *
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54 | * <p>The value returned is always between -1 and 1 (inclusive).
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55 | * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
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56 | * either 1 or -1 as a double, so the appropriate extreme value is returned.
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57 | * </p>
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58 | *
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59 | * @param x the value.
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60 | * @return the error function erf(x)
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61 | * @throws agents.anac.y2019.harddealer.math3.exception.MaxCountExceededException
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62 | * if the algorithm fails to converge.
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63 | * @see Gamma#regularizedGammaP(double, double, double, int)
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64 | */
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65 | public static double erf(double x) {
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66 | if (FastMath.abs(x) > 40) {
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67 | return x > 0 ? 1 : -1;
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68 | }
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69 | final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000);
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70 | return x < 0 ? -ret : ret;
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71 | }
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72 |
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73 | /**
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74 | * Returns the complementary error function.
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75 | *
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76 | * <p>erfc(x) = 2/√π <sub>x</sub>∫<sup>∞</sup> e<sup>-t<sup>2</sup></sup>dt
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77 | * <br/>
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78 | * = 1 - {@link #erf(double) erf(x)} </p>
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79 | *
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80 | * <p>This implementation computes erfc(x) using the
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81 | * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function},
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82 | * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
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83 | *
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84 | * <p>The value returned is always between 0 and 2 (inclusive).
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85 | * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
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86 | * either 0 or 2 as a double, so the appropriate extreme value is returned.
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87 | * </p>
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88 | *
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89 | * @param x the value
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90 | * @return the complementary error function erfc(x)
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91 | * @throws agents.anac.y2019.harddealer.math3.exception.MaxCountExceededException
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92 | * if the algorithm fails to converge.
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93 | * @see Gamma#regularizedGammaQ(double, double, double, int)
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94 | * @since 2.2
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95 | */
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96 | public static double erfc(double x) {
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97 | if (FastMath.abs(x) > 40) {
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98 | return x > 0 ? 0 : 2;
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99 | }
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100 | final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
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101 | return x < 0 ? 2 - ret : ret;
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102 | }
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103 |
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104 | /**
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105 | * Returns the difference between erf(x1) and erf(x2).
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106 | *
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107 | * The implementation uses either erf(double) or erfc(double)
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108 | * depending on which provides the most precise result.
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109 | *
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110 | * @param x1 the first value
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111 | * @param x2 the second value
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112 | * @return erf(x2) - erf(x1)
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113 | */
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114 | public static double erf(double x1, double x2) {
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115 | if(x1 > x2) {
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116 | return -erf(x2, x1);
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117 | }
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118 |
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119 | return
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120 | x1 < -X_CRIT ?
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121 | x2 < 0.0 ?
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122 | erfc(-x2) - erfc(-x1) :
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123 | erf(x2) - erf(x1) :
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124 | x2 > X_CRIT && x1 > 0.0 ?
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125 | erfc(x1) - erfc(x2) :
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126 | erf(x2) - erf(x1);
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127 | }
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128 |
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129 | /**
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130 | * Returns the inverse erf.
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131 | * <p>
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132 | * This implementation is described in the paper:
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133 | * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
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134 | * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
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135 | * which was published in GPU Computing Gems, volume 2, 2010.
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136 | * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
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137 | * </p>
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138 | * @param x the value
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139 | * @return t such that x = erf(t)
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140 | * @since 3.2
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141 | */
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142 | public static double erfInv(final double x) {
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143 |
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144 | // beware that the logarithm argument must be
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145 | // commputed as (1.0 - x) * (1.0 + x),
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146 | // it must NOT be simplified as 1.0 - x * x as this
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147 | // would induce rounding errors near the boundaries +/-1
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148 | double w = - FastMath.log((1.0 - x) * (1.0 + x));
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149 | double p;
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150 |
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151 | if (w < 6.25) {
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152 | w -= 3.125;
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153 | p = -3.6444120640178196996e-21;
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154 | p = -1.685059138182016589e-19 + p * w;
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155 | p = 1.2858480715256400167e-18 + p * w;
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156 | p = 1.115787767802518096e-17 + p * w;
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157 | p = -1.333171662854620906e-16 + p * w;
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158 | p = 2.0972767875968561637e-17 + p * w;
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159 | p = 6.6376381343583238325e-15 + p * w;
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160 | p = -4.0545662729752068639e-14 + p * w;
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161 | p = -8.1519341976054721522e-14 + p * w;
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162 | p = 2.6335093153082322977e-12 + p * w;
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163 | p = -1.2975133253453532498e-11 + p * w;
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164 | p = -5.4154120542946279317e-11 + p * w;
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165 | p = 1.051212273321532285e-09 + p * w;
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166 | p = -4.1126339803469836976e-09 + p * w;
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167 | p = -2.9070369957882005086e-08 + p * w;
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168 | p = 4.2347877827932403518e-07 + p * w;
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169 | p = -1.3654692000834678645e-06 + p * w;
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170 | p = -1.3882523362786468719e-05 + p * w;
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171 | p = 0.0001867342080340571352 + p * w;
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172 | p = -0.00074070253416626697512 + p * w;
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173 | p = -0.0060336708714301490533 + p * w;
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174 | p = 0.24015818242558961693 + p * w;
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175 | p = 1.6536545626831027356 + p * w;
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176 | } else if (w < 16.0) {
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177 | w = FastMath.sqrt(w) - 3.25;
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178 | p = 2.2137376921775787049e-09;
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179 | p = 9.0756561938885390979e-08 + p * w;
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180 | p = -2.7517406297064545428e-07 + p * w;
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181 | p = 1.8239629214389227755e-08 + p * w;
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182 | p = 1.5027403968909827627e-06 + p * w;
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183 | p = -4.013867526981545969e-06 + p * w;
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184 | p = 2.9234449089955446044e-06 + p * w;
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185 | p = 1.2475304481671778723e-05 + p * w;
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186 | p = -4.7318229009055733981e-05 + p * w;
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187 | p = 6.8284851459573175448e-05 + p * w;
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188 | p = 2.4031110387097893999e-05 + p * w;
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189 | p = -0.0003550375203628474796 + p * w;
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190 | p = 0.00095328937973738049703 + p * w;
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191 | p = -0.0016882755560235047313 + p * w;
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192 | p = 0.0024914420961078508066 + p * w;
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193 | p = -0.0037512085075692412107 + p * w;
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194 | p = 0.005370914553590063617 + p * w;
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195 | p = 1.0052589676941592334 + p * w;
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196 | p = 3.0838856104922207635 + p * w;
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197 | } else if (!Double.isInfinite(w)) {
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198 | w = FastMath.sqrt(w) - 5.0;
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199 | p = -2.7109920616438573243e-11;
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200 | p = -2.5556418169965252055e-10 + p * w;
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201 | p = 1.5076572693500548083e-09 + p * w;
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202 | p = -3.7894654401267369937e-09 + p * w;
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203 | p = 7.6157012080783393804e-09 + p * w;
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204 | p = -1.4960026627149240478e-08 + p * w;
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205 | p = 2.9147953450901080826e-08 + p * w;
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206 | p = -6.7711997758452339498e-08 + p * w;
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207 | p = 2.2900482228026654717e-07 + p * w;
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208 | p = -9.9298272942317002539e-07 + p * w;
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209 | p = 4.5260625972231537039e-06 + p * w;
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210 | p = -1.9681778105531670567e-05 + p * w;
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211 | p = 7.5995277030017761139e-05 + p * w;
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212 | p = -0.00021503011930044477347 + p * w;
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213 | p = -0.00013871931833623122026 + p * w;
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214 | p = 1.0103004648645343977 + p * w;
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215 | p = 4.8499064014085844221 + p * w;
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216 | } else {
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217 | // this branch does not appears in the original code, it
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218 | // was added because the previous branch does not handle
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219 | // x = +/-1 correctly. In this case, w is positive infinity
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220 | // and as the first coefficient (-2.71e-11) is negative.
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221 | // Once the first multiplication is done, p becomes negative
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222 | // infinity and remains so throughout the polynomial evaluation.
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223 | // So the branch above incorrectly returns negative infinity
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224 | // instead of the correct positive infinity.
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225 | p = Double.POSITIVE_INFINITY;
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226 | }
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227 |
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228 | return p * x;
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229 |
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230 | }
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231 |
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232 | /**
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233 | * Returns the inverse erfc.
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234 | * @param x the value
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235 | * @return t such that x = erfc(t)
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236 | * @since 3.2
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237 | */
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238 | public static double erfcInv(final double x) {
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239 | return erfInv(1 - x);
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240 | }
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241 |
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242 | }
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243 |
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