/* * Licensed to the Apache Software Foundation (ASF) under one or more * contributor license agreements. See the NOTICE file distributed with * this work for additional information regarding copyright ownership. * The ASF licenses this file to You under the Apache License, Version 2.0 * (the "License"); you may not use this file except in compliance with * the License. You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ package org.apache.commons.math3.special; import org.apache.commons.math3.util.FastMath; /** This is a utility class that provides computation methods related to the error functions. */ public class Erf { /** * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally. This number * solves {@code erf(x)=0.5} within 1ulp. More precisely, the current implementations of {@link * #erf(double)} and {@link #erfc(double)} satisfy:
* {@code erf(X_CRIT) < 0.5},
* {@code erf(Math.nextUp(X_CRIT) > 0.5},
* {@code erfc(X_CRIT) = 0.5}, and
* {@code erfc(Math.nextUp(X_CRIT) < 0.5} */ private static final double X_CRIT = 0.4769362762044697; /** Default constructor. Prohibit instantiation. */ private Erf() {} /** * Returns the error function. * *

erf(x) = 2/√π ₀∫^x e^-t²dt * *

This implementation computes erf(x) using the {@link Gamma#regularizedGammaP(double, * double, double, int) regularized gamma function}, following Erf, equation (3) * *

The value returned is always between -1 and 1 (inclusive). If {@code abs(x) > 40}, then * {@code erf(x)} is indistinguishable from either 1 or -1 as a double, so the appropriate * extreme value is returned. * * @param x the value. * @return the error function erf(x) * @throws org.apache.commons.math3.exception.MaxCountExceededException if the algorithm fails * to converge. * @see Gamma#regularizedGammaP(double, double, double, int) */ public static double erf(double x) { if (FastMath.abs(x) > 40) { return x > 0 ? 1 : -1; } final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000); return x < 0 ? -ret : ret; } /** * Returns the complementary error function. * *

erfc(x) = 2/√π _x∫^∞ e^-t²dt *
* = 1 - {@link #erf(double) erf(x)} * *

This implementation computes erfc(x) using the {@link Gamma#regularizedGammaQ(double, * double, double, int) regularized gamma function}, following Erf, equation (3). * *

The value returned is always between 0 and 2 (inclusive). If {@code abs(x) > 40}, then * {@code erf(x)} is indistinguishable from either 0 or 2 as a double, so the appropriate * extreme value is returned. * * @param x the value * @return the complementary error function erfc(x) * @throws org.apache.commons.math3.exception.MaxCountExceededException if the algorithm fails * to converge. * @see Gamma#regularizedGammaQ(double, double, double, int) * @since 2.2 */ public static double erfc(double x) { if (FastMath.abs(x) > 40) { return x > 0 ? 0 : 2; } final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000); return x < 0 ? 2 - ret : ret; } /** * Returns the difference between erf(x1) and erf(x2). * *

The implementation uses either erf(double) or erfc(double) depending on which provides the * most precise result. * * @param x1 the first value * @param x2 the second value * @return erf(x2) - erf(x1) */ public static double erf(double x1, double x2) { if (x1 > x2) { return -erf(x2, x1); } return x1 < -X_CRIT ? x2 < 0.0 ? erfc(-x2) - erfc(-x1) : erf(x2) - erf(x1) : x2 > X_CRIT && x1 > 0.0 ? erfc(x1) - erfc(x2) : erf(x2) - erf(x1); } /** * Returns the inverse erf. * *

This implementation is described in the paper: Approximating the erfinv * function by Mike Giles, Oxford-Man Institute of Quantitative Finance, which was published * in GPU Computing Gems, volume 2, 2010. The source code is available here. * * @param x the value * @return t such that x = erf(t) * @since 3.2 */ public static double erfInv(final double x) { // beware that the logarithm argument must be // commputed as (1.0 - x) * (1.0 + x), // it must NOT be simplified as 1.0 - x * x as this // would induce rounding errors near the boundaries +/-1 double w = -FastMath.log((1.0 - x) * (1.0 + x)); double p; if (w < 6.25) { w -= 3.125; p = -3.6444120640178196996e-21; p = -1.685059138182016589e-19 + p * w; p = 1.2858480715256400167e-18 + p * w; p = 1.115787767802518096e-17 + p * w; p = -1.333171662854620906e-16 + p * w; p = 2.0972767875968561637e-17 + p * w; p = 6.6376381343583238325e-15 + p * w; p = -4.0545662729752068639e-14 + p * w; p = -8.1519341976054721522e-14 + p * w; p = 2.6335093153082322977e-12 + p * w; p = -1.2975133253453532498e-11 + p * w; p = -5.4154120542946279317e-11 + p * w; p = 1.051212273321532285e-09 + p * w; p = -4.1126339803469836976e-09 + p * w; p = -2.9070369957882005086e-08 + p * w; p = 4.2347877827932403518e-07 + p * w; p = -1.3654692000834678645e-06 + p * w; p = -1.3882523362786468719e-05 + p * w; p = 0.0001867342080340571352 + p * w; p = -0.00074070253416626697512 + p * w; p = -0.0060336708714301490533 + p * w; p = 0.24015818242558961693 + p * w; p = 1.6536545626831027356 + p * w; } else if (w < 16.0) { w = FastMath.sqrt(w) - 3.25; p = 2.2137376921775787049e-09; p = 9.0756561938885390979e-08 + p * w; p = -2.7517406297064545428e-07 + p * w; p = 1.8239629214389227755e-08 + p * w; p = 1.5027403968909827627e-06 + p * w; p = -4.013867526981545969e-06 + p * w; p = 2.9234449089955446044e-06 + p * w; p = 1.2475304481671778723e-05 + p * w; p = -4.7318229009055733981e-05 + p * w; p = 6.8284851459573175448e-05 + p * w; p = 2.4031110387097893999e-05 + p * w; p = -0.0003550375203628474796 + p * w; p = 0.00095328937973738049703 + p * w; p = -0.0016882755560235047313 + p * w; p = 0.0024914420961078508066 + p * w; p = -0.0037512085075692412107 + p * w; p = 0.005370914553590063617 + p * w; p = 1.0052589676941592334 + p * w; p = 3.0838856104922207635 + p * w; } else if (!Double.isInfinite(w)) { w = FastMath.sqrt(w) - 5.0; p = -2.7109920616438573243e-11; p = -2.5556418169965252055e-10 + p * w; p = 1.5076572693500548083e-09 + p * w; p = -3.7894654401267369937e-09 + p * w; p = 7.6157012080783393804e-09 + p * w; p = -1.4960026627149240478e-08 + p * w; p = 2.9147953450901080826e-08 + p * w; p = -6.7711997758452339498e-08 + p * w; p = 2.2900482228026654717e-07 + p * w; p = -9.9298272942317002539e-07 + p * w; p = 4.5260625972231537039e-06 + p * w; p = -1.9681778105531670567e-05 + p * w; p = 7.5995277030017761139e-05 + p * w; p = -0.00021503011930044477347 + p * w; p = -0.00013871931833623122026 + p * w; p = 1.0103004648645343977 + p * w; p = 4.8499064014085844221 + p * w; } else { // this branch does not appears in the original code, it // was added because the previous branch does not handle // x = +/-1 correctly. In this case, w is positive infinity // and as the first coefficient (-2.71e-11) is negative. // Once the first multiplication is done, p becomes negative // infinity and remains so throughout the polynomial evaluation. // So the branch above incorrectly returns negative infinity // instead of the correct positive infinity. p = Double.POSITIVE_INFINITY; } return p * x; } /** * Returns the inverse erfc. * * @param x the value * @return t such that x = erfc(t) * @since 3.2 */ public static double erfcInv(final double x) { return erfInv(1 - x); } }