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 */
017 package org.apache.commons.math3.special;
018
019 import org.apache.commons.math3.util.FastMath;
020
021 /**
022 * This is a utility class that provides computation methods related to the
023 * error functions.
024 *
025 * @version $Id: Erf.java 1456905 2013-03-15 11:37:35Z luc $
026 */
027 public class Erf {
028
029 /**
030 * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally.
031 * This number solves {@code erf(x)=0.5} within 1ulp.
032 * More precisely, the current implementations of
033 * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br/>
034 * {@code erf(X_CRIT) < 0.5},<br/>
035 * {@code erf(Math.nextUp(X_CRIT) > 0.5},<br/>
036 * {@code erfc(X_CRIT) = 0.5}, and<br/>
037 * {@code erfc(Math.nextUp(X_CRIT) < 0.5}
038 */
039 private static final double X_CRIT = 0.4769362762044697;
040
041 /**
042 * Default constructor. Prohibit instantiation.
043 */
044 private Erf() {}
045
046 /**
047 * Returns the error function.
048 *
049 * <p>erf(x) = 2/√π <sub>0</sub>∫<sup>x</sup> e<sup>-t<sup>2</sup></sup>dt </p>
050 *
051 * <p>This implementation computes erf(x) using the
052 * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
053 * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
054 *
055 * <p>The value returned is always between -1 and 1 (inclusive).
056 * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
057 * either 1 or -1 as a double, so the appropriate extreme value is returned.
058 * </p>
059 *
060 * @param x the value.
061 * @return the error function erf(x)
062 * @throws org.apache.commons.math3.exception.MaxCountExceededException
063 * if the algorithm fails to converge.
064 * @see Gamma#regularizedGammaP(double, double, double, int)
065 */
066 public static double erf(double x) {
067 if (FastMath.abs(x) > 40) {
068 return x > 0 ? 1 : -1;
069 }
070 final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000);
071 return x < 0 ? -ret : ret;
072 }
073
074 /**
075 * Returns the complementary error function.
076 *
077 * <p>erfc(x) = 2/√π <sub>x</sub>∫<sup>∞</sup> e<sup>-t<sup>2</sup></sup>dt
078 * <br/>
079 * = 1 - {@link #erf(double) erf(x)} </p>
080 *
081 * <p>This implementation computes erfc(x) using the
082 * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function},
083 * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
084 *
085 * <p>The value returned is always between 0 and 2 (inclusive).
086 * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
087 * either 0 or 2 as a double, so the appropriate extreme value is returned.
088 * </p>
089 *
090 * @param x the value
091 * @return the complementary error function erfc(x)
092 * @throws org.apache.commons.math3.exception.MaxCountExceededException
093 * if the algorithm fails to converge.
094 * @see Gamma#regularizedGammaQ(double, double, double, int)
095 * @since 2.2
096 */
097 public static double erfc(double x) {
098 if (FastMath.abs(x) > 40) {
099 return x > 0 ? 0 : 2;
100 }
101 final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
102 return x < 0 ? 2 - ret : ret;
103 }
104
105 /**
106 * Returns the difference between erf(x1) and erf(x2).
107 *
108 * The implementation uses either erf(double) or erfc(double)
109 * depending on which provides the most precise result.
110 *
111 * @param x1 the first value
112 * @param x2 the second value
113 * @return erf(x2) - erf(x1)
114 */
115 public static double erf(double x1, double x2) {
116 if(x1 > x2) {
117 return -erf(x2, x1);
118 }
119
120 return
121 x1 < -X_CRIT ?
122 x2 < 0.0 ?
123 erfc(-x2) - erfc(-x1) :
124 erf(x2) - erf(x1) :
125 x2 > X_CRIT && x1 > 0.0 ?
126 erfc(x1) - erfc(x2) :
127 erf(x2) - erf(x1);
128 }
129
130 /**
131 * Returns the inverse erf.
132 * <p>
133 * This implementation is described in the paper:
134 * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
135 * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
136 * which was published in GPU Computing Gems, volume 2, 2010.
137 * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
138 * </p>
139 * @param x the value
140 * @return t such that x = erf(t)
141 * @since 3.2
142 */
143 public static double erfInv(final double x) {
144
145 // beware that the logarithm argument must be
146 // commputed as (1.0 - x) * (1.0 + x),
147 // it must NOT be simplified as 1.0 - x * x as this
148 // would induce rounding errors near the boundaries +/-1
149 double w = - FastMath.log((1.0 - x) * (1.0 + x));
150 double p;
151
152 if (w < 6.25) {
153 w = w - 3.125;
154 p = -3.6444120640178196996e-21;
155 p = -1.685059138182016589e-19 + p * w;
156 p = 1.2858480715256400167e-18 + p * w;
157 p = 1.115787767802518096e-17 + p * w;
158 p = -1.333171662854620906e-16 + p * w;
159 p = 2.0972767875968561637e-17 + p * w;
160 p = 6.6376381343583238325e-15 + p * w;
161 p = -4.0545662729752068639e-14 + p * w;
162 p = -8.1519341976054721522e-14 + p * w;
163 p = 2.6335093153082322977e-12 + p * w;
164 p = -1.2975133253453532498e-11 + p * w;
165 p = -5.4154120542946279317e-11 + p * w;
166 p = 1.051212273321532285e-09 + p * w;
167 p = -4.1126339803469836976e-09 + p * w;
168 p = -2.9070369957882005086e-08 + p * w;
169 p = 4.2347877827932403518e-07 + p * w;
170 p = -1.3654692000834678645e-06 + p * w;
171 p = -1.3882523362786468719e-05 + p * w;
172 p = 0.0001867342080340571352 + p * w;
173 p = -0.00074070253416626697512 + p * w;
174 p = -0.0060336708714301490533 + p * w;
175 p = 0.24015818242558961693 + p * w;
176 p = 1.6536545626831027356 + p * w;
177 } else if (w < 16.0) {
178 w = FastMath.sqrt(w) - 3.25;
179 p = 2.2137376921775787049e-09;
180 p = 9.0756561938885390979e-08 + p * w;
181 p = -2.7517406297064545428e-07 + p * w;
182 p = 1.8239629214389227755e-08 + p * w;
183 p = 1.5027403968909827627e-06 + p * w;
184 p = -4.013867526981545969e-06 + p * w;
185 p = 2.9234449089955446044e-06 + p * w;
186 p = 1.2475304481671778723e-05 + p * w;
187 p = -4.7318229009055733981e-05 + p * w;
188 p = 6.8284851459573175448e-05 + p * w;
189 p = 2.4031110387097893999e-05 + p * w;
190 p = -0.0003550375203628474796 + p * w;
191 p = 0.00095328937973738049703 + p * w;
192 p = -0.0016882755560235047313 + p * w;
193 p = 0.0024914420961078508066 + p * w;
194 p = -0.0037512085075692412107 + p * w;
195 p = 0.005370914553590063617 + p * w;
196 p = 1.0052589676941592334 + p * w;
197 p = 3.0838856104922207635 + p * w;
198 } else if (!Double.isInfinite(w)) {
199 w = FastMath.sqrt(w) - 5.0;
200 p = -2.7109920616438573243e-11;
201 p = -2.5556418169965252055e-10 + p * w;
202 p = 1.5076572693500548083e-09 + p * w;
203 p = -3.7894654401267369937e-09 + p * w;
204 p = 7.6157012080783393804e-09 + p * w;
205 p = -1.4960026627149240478e-08 + p * w;
206 p = 2.9147953450901080826e-08 + p * w;
207 p = -6.7711997758452339498e-08 + p * w;
208 p = 2.2900482228026654717e-07 + p * w;
209 p = -9.9298272942317002539e-07 + p * w;
210 p = 4.5260625972231537039e-06 + p * w;
211 p = -1.9681778105531670567e-05 + p * w;
212 p = 7.5995277030017761139e-05 + p * w;
213 p = -0.00021503011930044477347 + p * w;
214 p = -0.00013871931833623122026 + p * w;
215 p = 1.0103004648645343977 + p * w;
216 p = 4.8499064014085844221 + p * w;
217 } else {
218 // this branch does not appears in the original code, it
219 // was added because the previous branch does not handle
220 // x = +/-1 correctly. In this case, w is positive infinity
221 // and as the first coefficient (-2.71e-11) is negative.
222 // Once the first multiplication is done, p becomes negative
223 // infinity and remains so throughout the polynomial evaluation.
224 // So the branch above incorrectly returns negative infinity
225 // instead of the correct positive infinity.
226 p = Double.POSITIVE_INFINITY;
227 }
228
229 return p * x;
230
231 }
232
233 /**
234 * Returns the inverse erfc.
235 * @param x the value
236 * @return t such that x = erfc(t)
237 * @since 3.2
238 */
239 public static double erfcInv(final double x) {
240 return erfInv(1 - x);
241 }
242
243 }
244