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 1416643 2012-12-03 19:37:14Z tn $
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