001/* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017package org.apache.commons.numbers.gamma; 018 019/** 020 * Inverse of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a>. 021 * <p> 022 * This implementation is described in the paper: 023 * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating 024 * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance, 025 * which was published in GPU Computing Gems, volume 2, 2010. 026 * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>. 027 * </p> 028 */ 029public final class InverseErf { 030 /** Private constructor. */ 031 private InverseErf() { 032 // intentionally empty. 033 } 034 035 /** 036 * Returns the inverse error function. 037 * 038 * @param x Value. 039 * @return t such that {@code x =} {@link Erf#value(double) Erf.value(t)}. 040 */ 041 public static double value(final double x) { 042 // Beware that the logarithm argument must be 043 // computed as (1 - x) * (1 + x), 044 // it must NOT be simplified as 1 - x * x as this 045 // would induce rounding errors near the boundaries +/-1 046 double w = -Math.log((1 - x) * (1 + x)); 047 double p; 048 049 if (w < 6.25) { 050 w -= 3.125; 051 p = -3.6444120640178196996e-21; 052 p = -1.685059138182016589e-19 + p * w; 053 p = 1.2858480715256400167e-18 + p * w; 054 p = 1.115787767802518096e-17 + p * w; 055 p = -1.333171662854620906e-16 + p * w; 056 p = 2.0972767875968561637e-17 + p * w; 057 p = 6.6376381343583238325e-15 + p * w; 058 p = -4.0545662729752068639e-14 + p * w; 059 p = -8.1519341976054721522e-14 + p * w; 060 p = 2.6335093153082322977e-12 + p * w; 061 p = -1.2975133253453532498e-11 + p * w; 062 p = -5.4154120542946279317e-11 + p * w; 063 p = 1.051212273321532285e-09 + p * w; 064 p = -4.1126339803469836976e-09 + p * w; 065 p = -2.9070369957882005086e-08 + p * w; 066 p = 4.2347877827932403518e-07 + p * w; 067 p = -1.3654692000834678645e-06 + p * w; 068 p = -1.3882523362786468719e-05 + p * w; 069 p = 0.0001867342080340571352 + p * w; 070 p = -0.00074070253416626697512 + p * w; 071 p = -0.0060336708714301490533 + p * w; 072 p = 0.24015818242558961693 + p * w; 073 p = 1.6536545626831027356 + p * w; 074 } else if (w < 16.0) { 075 w = Math.sqrt(w) - 3.25; 076 p = 2.2137376921775787049e-09; 077 p = 9.0756561938885390979e-08 + p * w; 078 p = -2.7517406297064545428e-07 + p * w; 079 p = 1.8239629214389227755e-08 + p * w; 080 p = 1.5027403968909827627e-06 + p * w; 081 p = -4.013867526981545969e-06 + p * w; 082 p = 2.9234449089955446044e-06 + p * w; 083 p = 1.2475304481671778723e-05 + p * w; 084 p = -4.7318229009055733981e-05 + p * w; 085 p = 6.8284851459573175448e-05 + p * w; 086 p = 2.4031110387097893999e-05 + p * w; 087 p = -0.0003550375203628474796 + p * w; 088 p = 0.00095328937973738049703 + p * w; 089 p = -0.0016882755560235047313 + p * w; 090 p = 0.0024914420961078508066 + p * w; 091 p = -0.0037512085075692412107 + p * w; 092 p = 0.005370914553590063617 + p * w; 093 p = 1.0052589676941592334 + p * w; 094 p = 3.0838856104922207635 + p * w; 095 } else if (w < Double.POSITIVE_INFINITY) { 096 w = Math.sqrt(w) - 5; 097 p = -2.7109920616438573243e-11; 098 p = -2.5556418169965252055e-10 + p * w; 099 p = 1.5076572693500548083e-09 + p * w; 100 p = -3.7894654401267369937e-09 + p * w; 101 p = 7.6157012080783393804e-09 + p * w; 102 p = -1.4960026627149240478e-08 + p * w; 103 p = 2.9147953450901080826e-08 + p * w; 104 p = -6.7711997758452339498e-08 + p * w; 105 p = 2.2900482228026654717e-07 + p * w; 106 p = -9.9298272942317002539e-07 + p * w; 107 p = 4.5260625972231537039e-06 + p * w; 108 p = -1.9681778105531670567e-05 + p * w; 109 p = 7.5995277030017761139e-05 + p * w; 110 p = -0.00021503011930044477347 + p * w; 111 p = -0.00013871931833623122026 + p * w; 112 p = 1.0103004648645343977 + p * w; 113 p = 4.8499064014085844221 + p * w; 114 } else if (w == Double.POSITIVE_INFINITY) { 115 // this branch does not appears in the original code, it 116 // was added because the previous branch does not handle 117 // x = +/-1 correctly. In this case, w is positive infinity 118 // and as the first coefficient (-2.71e-11) is negative. 119 // Once the first multiplication is done, p becomes negative 120 // infinity and remains so throughout the polynomial evaluation. 121 // So the branch above incorrectly returns negative infinity 122 // instead of the correct positive infinity. 123 p = Double.POSITIVE_INFINITY; 124 } else { 125 // this branch does not appears in the original code, it 126 // occurs when the input is NaN or not in the range [-1, 1]. 127 return Double.NaN; 128 } 129 130 return p * x; 131 } 132} 133