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 */
017package org.apache.commons.math3.special;
018
019import 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 1456905 2013-03-15 11:37:35Z luc $
026 */
027public 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/&radic;&pi; <sub>0</sub>&int;<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/&radic;&pi; <sub>x</sub>&int;<sup>&infin;</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    /**
131     * Returns the inverse erf.
132     * <p>
133     * This implementation is described in the paper:
134     * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
135     * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
136     * which was published in GPU Computing Gems, volume 2, 2010.
137     * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
138     * </p>
139     * @param x the value
140     * @return t such that x = erf(t)
141     * @since 3.2
142     */
143    public static double erfInv(final double x) {
144
145        // beware that the logarithm argument must be
146        // commputed as (1.0 - x) * (1.0 + x),
147        // it must NOT be simplified as 1.0 - x * x as this
148        // would induce rounding errors near the boundaries +/-1
149        double w = - FastMath.log((1.0 - x) * (1.0 + x));
150        double p;
151
152        if (w < 6.25) {
153            w = w - 3.125;
154            p =  -3.6444120640178196996e-21;
155            p =   -1.685059138182016589e-19 + p * w;
156            p =   1.2858480715256400167e-18 + p * w;
157            p =    1.115787767802518096e-17 + p * w;
158            p =   -1.333171662854620906e-16 + p * w;
159            p =   2.0972767875968561637e-17 + p * w;
160            p =   6.6376381343583238325e-15 + p * w;
161            p =  -4.0545662729752068639e-14 + p * w;
162            p =  -8.1519341976054721522e-14 + p * w;
163            p =   2.6335093153082322977e-12 + p * w;
164            p =  -1.2975133253453532498e-11 + p * w;
165            p =  -5.4154120542946279317e-11 + p * w;
166            p =    1.051212273321532285e-09 + p * w;
167            p =  -4.1126339803469836976e-09 + p * w;
168            p =  -2.9070369957882005086e-08 + p * w;
169            p =   4.2347877827932403518e-07 + p * w;
170            p =  -1.3654692000834678645e-06 + p * w;
171            p =  -1.3882523362786468719e-05 + p * w;
172            p =    0.0001867342080340571352 + p * w;
173            p =  -0.00074070253416626697512 + p * w;
174            p =   -0.0060336708714301490533 + p * w;
175            p =      0.24015818242558961693 + p * w;
176            p =       1.6536545626831027356 + p * w;
177        } else if (w < 16.0) {
178            w = FastMath.sqrt(w) - 3.25;
179            p =   2.2137376921775787049e-09;
180            p =   9.0756561938885390979e-08 + p * w;
181            p =  -2.7517406297064545428e-07 + p * w;
182            p =   1.8239629214389227755e-08 + p * w;
183            p =   1.5027403968909827627e-06 + p * w;
184            p =   -4.013867526981545969e-06 + p * w;
185            p =   2.9234449089955446044e-06 + p * w;
186            p =   1.2475304481671778723e-05 + p * w;
187            p =  -4.7318229009055733981e-05 + p * w;
188            p =   6.8284851459573175448e-05 + p * w;
189            p =   2.4031110387097893999e-05 + p * w;
190            p =   -0.0003550375203628474796 + p * w;
191            p =   0.00095328937973738049703 + p * w;
192            p =   -0.0016882755560235047313 + p * w;
193            p =    0.0024914420961078508066 + p * w;
194            p =   -0.0037512085075692412107 + p * w;
195            p =     0.005370914553590063617 + p * w;
196            p =       1.0052589676941592334 + p * w;
197            p =       3.0838856104922207635 + p * w;
198        } else if (!Double.isInfinite(w)) {
199            w = FastMath.sqrt(w) - 5.0;
200            p =  -2.7109920616438573243e-11;
201            p =  -2.5556418169965252055e-10 + p * w;
202            p =   1.5076572693500548083e-09 + p * w;
203            p =  -3.7894654401267369937e-09 + p * w;
204            p =   7.6157012080783393804e-09 + p * w;
205            p =  -1.4960026627149240478e-08 + p * w;
206            p =   2.9147953450901080826e-08 + p * w;
207            p =  -6.7711997758452339498e-08 + p * w;
208            p =   2.2900482228026654717e-07 + p * w;
209            p =  -9.9298272942317002539e-07 + p * w;
210            p =   4.5260625972231537039e-06 + p * w;
211            p =  -1.9681778105531670567e-05 + p * w;
212            p =   7.5995277030017761139e-05 + p * w;
213            p =  -0.00021503011930044477347 + p * w;
214            p =  -0.00013871931833623122026 + p * w;
215            p =       1.0103004648645343977 + p * w;
216            p =       4.8499064014085844221 + p * w;
217        } else {
218            // this branch does not appears in the original code, it
219            // was added because the previous branch does not handle
220            // x = +/-1 correctly. In this case, w is positive infinity
221            // and as the first coefficient (-2.71e-11) is negative.
222            // Once the first multiplication is done, p becomes negative
223            // infinity and remains so throughout the polynomial evaluation.
224            // So the branch above incorrectly returns negative infinity
225            // instead of the correct positive infinity.
226            p = Double.POSITIVE_INFINITY;
227        }
228
229        return p * x;
230
231    }
232
233    /**
234     * Returns the inverse erfc.
235     * @param x the value
236     * @return t such that x = erfc(t)
237     * @since 3.2
238     */
239    public static double erfcInv(final double x) {
240        return erfInv(1 - x);
241    }
242
243}
244