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* 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);
}
}