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.newInstance(0.5), x.square(), 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.newInstance(2.);
178 }
179 final T ret = Gamma.regularizedGammaQ(one.newInstance(0.5), x.square(), 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.newInstance(-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.newInstance(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.newInstance(-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