Erf.java
/*
* 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
*
* https://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.
*/
/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/
package org.hipparchus.special;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.Field;
import org.hipparchus.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:<br>
* {@code erf(X_CRIT) < 0.5},<br>
* {@code erf(Math.nextUp(X_CRIT) > 0.5},<br>
* {@code erfc(X_CRIT) = 0.5}, and<br>
* {@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.
*
* \[
* \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{t=0}^x e^{-t^2}dt
* \]
*
* <p>This implementation computes erf(x) using the
* {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
* following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
*
* <p>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.
* </p>
*
* @param x the value.
* @return the error function erf(x)
* @throws org.hipparchus.exception.MathIllegalStateException
* 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 error function.
*
* \[
* \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{t=0}^x e^{-t^2}dt
* \]
*
* <p>This implementation computes erf(x) using the
* {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
* following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
*
* <p>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.
* </p>
*
* @param <T> type of the field elements
* @param x the value.
* @return the error function erf(x)
* @throws org.hipparchus.exception.MathIllegalStateException
* if the algorithm fails to converge.
* @see Gamma#regularizedGammaP(double, double, double, int)
*/
public static <T extends CalculusFieldElement<T>> T erf(T x) {
final Field<T> field = x.getField();
final T one = field.getOne();
if (FastMath.abs(x.getReal()) > 40) {
return x.getReal() > 0 ? one : one.negate();
}
final T ret = Gamma.regularizedGammaP(one.newInstance(0.5), x.square(), 1.0e-15, 10000);
return x.getReal() < 0 ? ret.negate() : ret;
}
/**
* Returns the complementary error function.
*
* \[
* \mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_{t=x}^\infty e^{-t^2}dt = 1 - \mathrm{erf}
*
* <p>This implementation computes erfc(x) using the
* {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function},
* following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
*
* <p>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.
* </p>
*
* @param x the value
* @return the complementary error function erfc(x)
* @throws org.hipparchus.exception.MathIllegalStateException
* if the algorithm fails to converge.
* @see Gamma#regularizedGammaQ(double, double, double, int)
*/
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 complementary error function.
*
* \[
* erfc(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}dt = 1 - erf(x)
* \]
*
* <p>This implementation computes erfc(x) using the
* {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function}, following <a
* href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
*
* <p>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. <b>This implies that the current implementation does not allow the use of
* {@link org.hipparchus.dfp.Dfp Dfp} with extended precision.</b>
* </p>
*
* @param x the value
* @param <T> type of the field elements
*
* @return the complementary error function erfc(x)
*
* @throws org.hipparchus.exception.MathIllegalStateException if the algorithm fails to converge.
* @see Gamma#regularizedGammaQ(double, double, double, int)
*/
public static <T extends CalculusFieldElement<T>> T erfc(T x) {
final Field<T> field = x.getField();
final T zero = field.getZero();
final T one = field.getOne();
if (FastMath.abs(x.getReal()) > 40) {
return x.getReal() > 0 ? zero : one.newInstance(2.);
}
final T ret = Gamma.regularizedGammaQ(one.newInstance(0.5), x.square(), 1.0e-15, 10000);
return x.getReal() < 0 ? ret.negate().add(2.) : ret;
}
/**
* Returns the difference between erf(x1) and erf(x2).
* <p>
* 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 difference between erf(x1) and erf(x2).
* <p>
* 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
* @param <T> type of the field elements
*
* @return erf(x2) - erf(x1)
*/
public static <T extends CalculusFieldElement<T>> T erf(T x1, T x2) {
if (x1.getReal() > x2.getReal()) {
return erf(x2, x1).negate();
}
return
x1.getReal() < -X_CRIT ?
x2.getReal() < 0.0 ?
erfc(x2.negate()).subtract(erfc(x1.negate())) :
erf(x2).subtract(erf(x1)) :
x2.getReal() > X_CRIT && x1.getReal() > 0.0 ?
erfc(x1).subtract(erfc(x2)) :
erf(x2).subtract(erf(x1));
}
/**
* Returns the inverse erf.
* <p>
* This implementation is described in the paper:
* <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
* the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
* which was published in GPU Computing Gems, volume 2, 2010.
* The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
* </p>
* @param x the value
* @return t such that x = erf(t)
*/
public static double erfInv(final double x) {
// beware that the logarithm argument must be
// computed 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 erf.
* <p>
* This implementation is described in the paper:
* <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
* the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
* which was published in GPU Computing Gems, volume 2, 2010.
* The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
* </p>
* @param <T> type of the filed elements
* @param x the value
* @return t such that x = erf(t)
*/
public static <T extends CalculusFieldElement<T>> T erfInv(final T x) {
final T one = x.getField().getOne();
// beware that the logarithm argument must be
// computed 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
T w = one.subtract(x).multiply(one.add(x)).log().negate();
T p;
if (w.getReal() < 6.25) {
w = w.subtract(3.125);
p = one.newInstance(-3.6444120640178196996e-21);
p = p.multiply(w).add(-1.685059138182016589e-19);
p = p.multiply(w).add(1.2858480715256400167e-18);
p = p.multiply(w).add(1.115787767802518096e-17);
p = p.multiply(w).add(-1.333171662854620906e-16);
p = p.multiply(w).add(2.0972767875968561637e-17);
p = p.multiply(w).add(6.6376381343583238325e-15);
p = p.multiply(w).add(-4.0545662729752068639e-14);
p = p.multiply(w).add(-8.1519341976054721522e-14);
p = p.multiply(w).add(2.6335093153082322977e-12);
p = p.multiply(w).add(-1.2975133253453532498e-11);
p = p.multiply(w).add(-5.4154120542946279317e-11);
p = p.multiply(w).add(1.051212273321532285e-09);
p = p.multiply(w).add(-4.1126339803469836976e-09);
p = p.multiply(w).add(-2.9070369957882005086e-08);
p = p.multiply(w).add(4.2347877827932403518e-07);
p = p.multiply(w).add(-1.3654692000834678645e-06);
p = p.multiply(w).add(-1.3882523362786468719e-05);
p = p.multiply(w).add(0.0001867342080340571352);
p = p.multiply(w).add(-0.00074070253416626697512);
p = p.multiply(w).add(-0.0060336708714301490533);
p = p.multiply(w).add(0.24015818242558961693);
p = p.multiply(w).add(1.6536545626831027356);
}
else if (w.getReal() < 16.0) {
w = w.sqrt().subtract(3.25);
p = one.newInstance(2.2137376921775787049e-09);
p = p.multiply(w).add(9.0756561938885390979e-08);
p = p.multiply(w).add(-2.7517406297064545428e-07);
p = p.multiply(w).add(1.8239629214389227755e-08);
p = p.multiply(w).add(1.5027403968909827627e-06);
p = p.multiply(w).add(-4.013867526981545969e-06);
p = p.multiply(w).add(2.9234449089955446044e-06);
p = p.multiply(w).add(1.2475304481671778723e-05);
p = p.multiply(w).add(-4.7318229009055733981e-05);
p = p.multiply(w).add(6.8284851459573175448e-05);
p = p.multiply(w).add(2.4031110387097893999e-05);
p = p.multiply(w).add(-0.0003550375203628474796);
p = p.multiply(w).add(0.00095328937973738049703);
p = p.multiply(w).add(-0.0016882755560235047313);
p = p.multiply(w).add(0.0024914420961078508066);
p = p.multiply(w).add(-0.0037512085075692412107);
p = p.multiply(w).add(0.005370914553590063617);
p = p.multiply(w).add(1.0052589676941592334);
p = p.multiply(w).add(3.0838856104922207635);
}
else if (!w.isInfinite()) {
w = w.sqrt().subtract(5.0);
p = one.newInstance(-2.7109920616438573243e-11);
p = p.multiply(w).add(-2.5556418169965252055e-10);
p = p.multiply(w).add(1.5076572693500548083e-09);
p = p.multiply(w).add(-3.7894654401267369937e-09);
p = p.multiply(w).add(7.6157012080783393804e-09);
p = p.multiply(w).add(-1.4960026627149240478e-08);
p = p.multiply(w).add(2.9147953450901080826e-08);
p = p.multiply(w).add(-6.7711997758452339498e-08);
p = p.multiply(w).add(2.2900482228026654717e-07);
p = p.multiply(w).add(-9.9298272942317002539e-07);
p = p.multiply(w).add(4.5260625972231537039e-06);
p = p.multiply(w).add(-1.9681778105531670567e-05);
p = p.multiply(w).add(7.5995277030017761139e-05);
p = p.multiply(w).add(-0.00021503011930044477347);
p = p.multiply(w).add(-0.00013871931833623122026);
p = p.multiply(w).add(1.0103004648645343977);
p = p.multiply(w).add(4.8499064014085844221);
}
else {
// this branch does not appear 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 = one.multiply(Double.POSITIVE_INFINITY);
}
return p.multiply(x);
}
/**
* Returns the inverse erfc.
* @param x the value
* @return t such that x = erfc(t)
*/
public static double erfcInv(final double x) {
return erfInv(1 - x);
}
/**
* Returns the inverse erfc.
* @param x the value
* @param <T> type of the field elements
* @return t such that x = erfc(t)
*/
public static <T extends CalculusFieldElement<T>> T erfcInv(final T x) {
return erfInv(x.negate().add(1));
}
}