/*
* 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.
*
* @version $Id: Erf.java 1416643 2012-12-03 19:37:14Z tn $
*/
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.
*
* <p>erf(x) = 2/√π <sub>0</sub>∫<sup>x</sup> e<sup>-t<sup>2</sup></sup>dt </p>
*
* <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.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.
*
* <p>erfc(x) = 2/√π <sub>x</sub>∫<sup>∞</sup> e<sup>-t<sup>2</sup></sup>dt
* <br/>
* = 1 - {@link #erf(double) erf(x)} </p>
*
* <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.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);
}
}