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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      https://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  package org.hipparchus.special;
23  
24  import org.hipparchus.CalculusFieldElement;
25  import org.hipparchus.Field;
26  import org.hipparchus.util.FastMath;
27  
28  /**
29   * This is a utility class that provides computation methods related to the
30   * error functions.
31   *
32   */
33  public class Erf {
34  
35      /**
36       * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally.
37       * This number solves {@code erf(x)=0.5} within 1ulp.
38       * More precisely, the current implementations of
39       * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br>
40       * {@code erf(X_CRIT) < 0.5},<br>
41       * {@code erf(Math.nextUp(X_CRIT) > 0.5},<br>
42       * {@code erfc(X_CRIT) = 0.5}, and<br>
43       * {@code erfc(Math.nextUp(X_CRIT) < 0.5}
44       */
45      private static final double X_CRIT = 0.4769362762044697;
46  
47      /**
48       * Default constructor.  Prohibit instantiation.
49       */
50      private Erf() {}
51  
52      /**
53       * Returns the error function.
54       *
55       * \[
56       * \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{t=0}^x e^{-t^2}dt
57       * \]
58       *
59       * <p>This implementation computes erf(x) using the
60       * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
61       * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
62       *
63       * <p>The value returned is always between -1 and 1 (inclusive).
64       * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
65       * either 1 or -1 as a double, so the appropriate extreme value is returned.
66       * </p>
67       *
68       * @param x the value.
69       * @return the error function erf(x)
70       * @throws org.hipparchus.exception.MathIllegalStateException
71       * if the algorithm fails to converge.
72       * @see Gamma#regularizedGammaP(double, double, double, int)
73       */
74      public static double erf(double x) {
75          if (FastMath.abs(x) > 40) {
76              return x > 0 ? 1 : -1;
77          }
78          final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000);
79          return x < 0 ? -ret : ret;
80      }
81  
82      /**
83       * Returns the error function.
84       *
85       * \[
86       * \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{t=0}^x e^{-t^2}dt
87       * \]
88       *
89       * <p>This implementation computes erf(x) using the
90       * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},
91       * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
92       *
93       * <p>The value returned is always between -1 and 1 (inclusive).
94       * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
95       * either 1 or -1 as a double, so the appropriate extreme value is returned.
96       * </p>
97       *
98       * @param <T> type of the field elements
99       * @param x the value.
100      * @return the error function erf(x)
101      * @throws org.hipparchus.exception.MathIllegalStateException
102      * if the algorithm fails to converge.
103      * @see Gamma#regularizedGammaP(double, double, double, int)
104      */
105     public static <T extends CalculusFieldElement<T>> T erf(T x) {
106         final Field<T> field = x.getField();
107         final T one = field.getOne();
108 
109         if (FastMath.abs(x.getReal()) > 40) {
110             return x.getReal() > 0 ? one : one.negate();
111         }
112         final T ret = Gamma.regularizedGammaP(one.multiply(0.5), x.multiply(x), 1.0e-15, 10000);
113         return x.getReal() < 0 ? ret.negate() : ret;
114     }
115 
116 
117     /**
118      * Returns the complementary error function.
119      *
120      * \[
121      * \mathrm{erfc}(x) =  \frac{2}{\sqrt{\pi}} \int_{t=x}^\infty e^{-t^2}dt = 1 - \mathrm{erf}
122      *
123      * <p>This implementation computes erfc(x) using the
124      * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function},
125      * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
126      *
127      * <p>The value returned is always between 0 and 2 (inclusive).
128      * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
129      * either 0 or 2 as a double, so the appropriate extreme value is returned.
130      * </p>
131      *
132      * @param x the value
133      * @return the complementary error function erfc(x)
134      * @throws org.hipparchus.exception.MathIllegalStateException
135      * if the algorithm fails to converge.
136      * @see Gamma#regularizedGammaQ(double, double, double, int)
137      */
138     public static double erfc(double x) {
139         if (FastMath.abs(x) > 40) {
140             return x > 0 ? 0 : 2;
141         }
142         final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
143         return x < 0 ? 2 - ret : ret;
144     }
145 
146     /**
147      * Returns the complementary error function.
148      *
149      * \[
150      * erfc(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}dt = 1 - erf(x)
151      * \]
152      *
153      * <p>This implementation computes erfc(x) using the
154      * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function}, following <a
155      * href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
156      *
157      * <p>The value returned is always between 0 and 2 (inclusive).
158      * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from either 0 or 2 as a double, so the
159      * appropriate extreme value is returned. <b>This implies that the current implementation does not allow the use of
160      * {@link org.hipparchus.dfp.Dfp Dfp} with extended precision.</b>
161      * </p>
162      *
163      * @param x the value
164      * @param <T> type of the field elements
165      *
166      * @return the complementary error function erfc(x)
167      *
168      * @throws org.hipparchus.exception.MathIllegalStateException if the algorithm fails to converge.
169      * @see Gamma#regularizedGammaQ(double, double, double, int)
170      */
171     public static <T extends CalculusFieldElement<T>> T erfc(T x) {
172         final Field<T> field = x.getField();
173         final T        zero  = field.getZero();
174         final T        one   = field.getOne();
175 
176         if (FastMath.abs(x.getReal()) > 40) {
177             return x.getReal() > 0 ? zero : one.multiply(2.);
178         }
179         final T ret = Gamma.regularizedGammaQ(one.multiply(0.5), x.multiply(x), 1.0e-15, 10000);
180         return x.getReal() < 0 ? ret.negate().add(2.) : ret;
181     }
182 
183     /**
184      * Returns the difference between erf(x1) and erf(x2).
185      * <p>
186      * The implementation uses either erf(double) or erfc(double)
187      * depending on which provides the most precise result.
188      *
189      * @param x1 the first value
190      * @param x2 the second value
191      * @return erf(x2) - erf(x1)
192      */
193     public static double erf(double x1, double x2) {
194         if(x1 > x2) {
195             return -erf(x2, x1);
196         }
197 
198         return
199         x1 < -X_CRIT ?
200             x2 < 0.0 ?
201                 erfc(-x2) - erfc(-x1) :
202                 erf(x2) - erf(x1) :
203             x2 > X_CRIT && x1 > 0.0 ?
204                 erfc(x1) - erfc(x2) :
205                 erf(x2) - erf(x1);
206     }
207 
208     /**
209      * Returns the difference between erf(x1) and erf(x2).
210      * <p>
211      * The implementation uses either erf(double) or erfc(double)
212      * depending on which provides the most precise result.
213      *
214      * @param x1 the first value
215      * @param x2 the second value
216      * @param <T> type of the field elements
217      *
218      * @return erf(x2) - erf(x1)
219      */
220     public static <T extends CalculusFieldElement<T>> T erf(T x1, T x2) {
221 
222         if (x1.getReal() > x2.getReal()) {
223             return erf(x2, x1).negate();
224         }
225 
226         return
227                 x1.getReal() < -X_CRIT ?
228                         x2.getReal() < 0.0 ?
229                                 erfc(x2.negate()).subtract(erfc(x1.negate())) :
230                                 erf(x2).subtract(erf(x1)) :
231                         x2.getReal() > X_CRIT && x1.getReal() > 0.0 ?
232                                 erfc(x1).subtract(erfc(x2)) :
233                                 erf(x2).subtract(erf(x1));
234     }
235 
236     /**
237      * Returns the inverse erf.
238      * <p>
239      * This implementation is described in the paper:
240      * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
241      * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
242      * which was published in GPU Computing Gems, volume 2, 2010.
243      * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
244      * </p>
245      * @param x the value
246      * @return t such that x = erf(t)
247      */
248     public static double erfInv(final double x) {
249 
250         // beware that the logarithm argument must be
251         // computed as (1.0 - x) * (1.0 + x),
252         // it must NOT be simplified as 1.0 - x * x as this
253         // would induce rounding errors near the boundaries +/-1
254         double w = - FastMath.log((1.0 - x) * (1.0 + x));
255         double p;
256 
257         if (w < 6.25) {
258             w -= 3.125;
259             p =  -3.6444120640178196996e-21;
260             p =   -1.685059138182016589e-19 + p * w;
261             p =   1.2858480715256400167e-18 + p * w;
262             p =    1.115787767802518096e-17 + p * w;
263             p =   -1.333171662854620906e-16 + p * w;
264             p =   2.0972767875968561637e-17 + p * w;
265             p =   6.6376381343583238325e-15 + p * w;
266             p =  -4.0545662729752068639e-14 + p * w;
267             p =  -8.1519341976054721522e-14 + p * w;
268             p =   2.6335093153082322977e-12 + p * w;
269             p =  -1.2975133253453532498e-11 + p * w;
270             p =  -5.4154120542946279317e-11 + p * w;
271             p =    1.051212273321532285e-09 + p * w;
272             p =  -4.1126339803469836976e-09 + p * w;
273             p =  -2.9070369957882005086e-08 + p * w;
274             p =   4.2347877827932403518e-07 + p * w;
275             p =  -1.3654692000834678645e-06 + p * w;
276             p =  -1.3882523362786468719e-05 + p * w;
277             p =    0.0001867342080340571352 + p * w;
278             p =  -0.00074070253416626697512 + p * w;
279             p =   -0.0060336708714301490533 + p * w;
280             p =      0.24015818242558961693 + p * w;
281             p =       1.6536545626831027356 + p * w;
282         } else if (w < 16.0) {
283             w = FastMath.sqrt(w) - 3.25;
284             p =   2.2137376921775787049e-09;
285             p =   9.0756561938885390979e-08 + p * w;
286             p =  -2.7517406297064545428e-07 + p * w;
287             p =   1.8239629214389227755e-08 + p * w;
288             p =   1.5027403968909827627e-06 + p * w;
289             p =   -4.013867526981545969e-06 + p * w;
290             p =   2.9234449089955446044e-06 + p * w;
291             p =   1.2475304481671778723e-05 + p * w;
292             p =  -4.7318229009055733981e-05 + p * w;
293             p =   6.8284851459573175448e-05 + p * w;
294             p =   2.4031110387097893999e-05 + p * w;
295             p =   -0.0003550375203628474796 + p * w;
296             p =   0.00095328937973738049703 + p * w;
297             p =   -0.0016882755560235047313 + p * w;
298             p =    0.0024914420961078508066 + p * w;
299             p =   -0.0037512085075692412107 + p * w;
300             p =     0.005370914553590063617 + p * w;
301             p =       1.0052589676941592334 + p * w;
302             p =       3.0838856104922207635 + p * w;
303         } else if (!Double.isInfinite(w)) {
304             w = FastMath.sqrt(w) - 5.0;
305             p =  -2.7109920616438573243e-11;
306             p =  -2.5556418169965252055e-10 + p * w;
307             p =   1.5076572693500548083e-09 + p * w;
308             p =  -3.7894654401267369937e-09 + p * w;
309             p =   7.6157012080783393804e-09 + p * w;
310             p =  -1.4960026627149240478e-08 + p * w;
311             p =   2.9147953450901080826e-08 + p * w;
312             p =  -6.7711997758452339498e-08 + p * w;
313             p =   2.2900482228026654717e-07 + p * w;
314             p =  -9.9298272942317002539e-07 + p * w;
315             p =   4.5260625972231537039e-06 + p * w;
316             p =  -1.9681778105531670567e-05 + p * w;
317             p =   7.5995277030017761139e-05 + p * w;
318             p =  -0.00021503011930044477347 + p * w;
319             p =  -0.00013871931833623122026 + p * w;
320             p =       1.0103004648645343977 + p * w;
321             p =       4.8499064014085844221 + p * w;
322         } else {
323             // this branch does not appears in the original code, it
324             // was added because the previous branch does not handle
325             // x = +/-1 correctly. In this case, w is positive infinity
326             // and as the first coefficient (-2.71e-11) is negative.
327             // Once the first multiplication is done, p becomes negative
328             // infinity and remains so throughout the polynomial evaluation.
329             // So the branch above incorrectly returns negative infinity
330             // instead of the correct positive infinity.
331             p = Double.POSITIVE_INFINITY;
332         }
333 
334         return p * x;
335 
336     }
337 
338     /**
339      * Returns the inverse erf.
340      * <p>
341      * This implementation is described in the paper:
342      * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
343      * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
344      * which was published in GPU Computing Gems, volume 2, 2010.
345      * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
346      * </p>
347      * @param <T> type of the filed elements
348      * @param x the value
349      * @return t such that x = erf(t)
350      */
351     public static <T extends CalculusFieldElement<T>> T erfInv(final T x) {
352         final T one = x.getField().getOne();
353 
354         // beware that the logarithm argument must be
355         // computed as (1.0 - x) * (1.0 + x),
356         // it must NOT be simplified as 1.0 - x * x as this
357         // would induce rounding errors near the boundaries +/-1
358         T w = one.subtract(x).multiply(one.add(x)).log().negate();
359         T p;
360 
361         if (w.getReal() < 6.25) {
362             w = w.subtract(3.125);
363             p = one.multiply(-3.6444120640178196996e-21);
364             p = p.multiply(w).add(-1.685059138182016589e-19);
365             p = p.multiply(w).add(1.2858480715256400167e-18);
366             p = p.multiply(w).add(1.115787767802518096e-17);
367             p = p.multiply(w).add(-1.333171662854620906e-16);
368             p = p.multiply(w).add(2.0972767875968561637e-17);
369             p = p.multiply(w).add(6.6376381343583238325e-15);
370             p = p.multiply(w).add(-4.0545662729752068639e-14);
371             p = p.multiply(w).add(-8.1519341976054721522e-14);
372             p = p.multiply(w).add(2.6335093153082322977e-12);
373             p = p.multiply(w).add(-1.2975133253453532498e-11);
374             p = p.multiply(w).add(-5.4154120542946279317e-11);
375             p = p.multiply(w).add(1.051212273321532285e-09);
376             p = p.multiply(w).add(-4.1126339803469836976e-09);
377             p = p.multiply(w).add(-2.9070369957882005086e-08);
378             p = p.multiply(w).add(4.2347877827932403518e-07);
379             p = p.multiply(w).add(-1.3654692000834678645e-06);
380             p = p.multiply(w).add(-1.3882523362786468719e-05);
381             p = p.multiply(w).add(0.0001867342080340571352);
382             p = p.multiply(w).add(-0.00074070253416626697512);
383             p = p.multiply(w).add(-0.0060336708714301490533);
384             p = p.multiply(w).add(0.24015818242558961693);
385             p = p.multiply(w).add(1.6536545626831027356);
386         }
387         else if (w.getReal() < 16.0) {
388             w = w.sqrt().subtract(3.25);
389             p = one.multiply(2.2137376921775787049e-09);
390             p = p.multiply(w).add(9.0756561938885390979e-08);
391             p = p.multiply(w).add(-2.7517406297064545428e-07);
392             p = p.multiply(w).add(1.8239629214389227755e-08);
393             p = p.multiply(w).add(1.5027403968909827627e-06);
394             p = p.multiply(w).add(-4.013867526981545969e-06);
395             p = p.multiply(w).add(2.9234449089955446044e-06);
396             p = p.multiply(w).add(1.2475304481671778723e-05);
397             p = p.multiply(w).add(-4.7318229009055733981e-05);
398             p = p.multiply(w).add(6.8284851459573175448e-05);
399             p = p.multiply(w).add(2.4031110387097893999e-05);
400             p = p.multiply(w).add(-0.0003550375203628474796);
401             p = p.multiply(w).add(0.00095328937973738049703);
402             p = p.multiply(w).add(-0.0016882755560235047313);
403             p = p.multiply(w).add(0.0024914420961078508066);
404             p = p.multiply(w).add(-0.0037512085075692412107);
405             p = p.multiply(w).add(0.005370914553590063617);
406             p = p.multiply(w).add(1.0052589676941592334);
407             p = p.multiply(w).add(3.0838856104922207635);
408         }
409         else if (!w.isInfinite()) {
410             w = w.sqrt().subtract(5.0);
411             p = one.multiply(-2.7109920616438573243e-11);
412             p = p.multiply(w).add(-2.5556418169965252055e-10);
413             p = p.multiply(w).add(1.5076572693500548083e-09);
414             p = p.multiply(w).add(-3.7894654401267369937e-09);
415             p = p.multiply(w).add(7.6157012080783393804e-09);
416             p = p.multiply(w).add(-1.4960026627149240478e-08);
417             p = p.multiply(w).add(2.9147953450901080826e-08);
418             p = p.multiply(w).add(-6.7711997758452339498e-08);
419             p = p.multiply(w).add(2.2900482228026654717e-07);
420             p = p.multiply(w).add(-9.9298272942317002539e-07);
421             p = p.multiply(w).add(4.5260625972231537039e-06);
422             p = p.multiply(w).add(-1.9681778105531670567e-05);
423             p = p.multiply(w).add(7.5995277030017761139e-05);
424             p = p.multiply(w).add(-0.00021503011930044477347);
425             p = p.multiply(w).add(-0.00013871931833623122026);
426             p = p.multiply(w).add(1.0103004648645343977);
427             p = p.multiply(w).add(4.8499064014085844221);
428         }
429         else {
430             // this branch does not appear in the original code, it
431             // was added because the previous branch does not handle
432             // x = +/-1 correctly. In this case, w is positive infinity
433             // and as the first coefficient (-2.71e-11) is negative.
434             // Once the first multiplication is done, p becomes negative
435             // infinity and remains so throughout the polynomial evaluation.
436             // So the branch above incorrectly returns negative infinity
437             // instead of the correct positive infinity.
438             p = one.multiply(Double.POSITIVE_INFINITY);
439         }
440 
441         return p.multiply(x);
442 
443     }
444 
445     /**
446      * Returns the inverse erfc.
447      * @param x the value
448      * @return t such that x = erfc(t)
449      */
450     public static double erfcInv(final double x) {
451         return erfInv(1 - x);
452     }
453 
454     /**
455      * Returns the inverse erfc.
456      * @param x the value
457      * @param <T> type of the field elements
458      * @return t such that x = erfc(t)
459      */
460     public static <T extends CalculusFieldElement<T>> T erfcInv(final T x) {
461         return erfInv(x.negate().add(1));
462     }
463 
464 }
465