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 * http://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 // (C) Copyright John Maddock 2006.
19 // Use, modification and distribution are subject to the
20 // Boost Software License, Version 1.0. (See accompanying file
21 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
22
23 package org.apache.commons.numbers.gamma;
24
25 /**
26 * Implementation of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> and
27 * its inverse.
28 *
29 * <p>This code has been adapted from the <a href="https://www.boost.org/">Boost</a>
30 * {@code c++} implementation {@code <boost/math/special_functions/erf.hpp>}.
31 * The erf/erfc functions and their inverses are copyright John Maddock 2006 and subject to
32 * the Boost Software License.
33 *
34 * <p>Additions made to support the erfcx function are original work under the Apache software
35 * license.
36 *
37 * @see
38 * <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/sf_erf/error_function.html">
39 * Boost C++ Error functions</a>
40 */
41 final class BoostErf {
42 /**
43 * The multiplier used to split the double value into high and low parts. From
44 * Dekker (1971): "The constant should be chosen equal to 2^(p - p/2) + 1,
45 * where p is the number of binary digits in the mantissa". Here p is 53
46 * and the multiplier is {@code 2^27 + 1}.
47 */
48 private static final double MULTIPLIER = 1.0 + 0x1.0p27;
49 /** 1 / sqrt(pi). Used for the scaled complementary error function erfcx. */
50 private static final double ONE_OVER_ROOT_PI = 0.5641895835477562869480794515607725858;
51 /** Threshold for the scaled complementary error function erfcx
52 * where the approximation {@code (1 / sqrt(pi)) / x} can be used. */
53 private static final double ERFCX_APPROX = 6.71e7;
54 /** Threshold for the erf implementation for |x| where the computation
55 * uses {@code erf(x)}; otherwise {@code erfc(x)} is computed. The final result is
56 * achieved by suitable application of symmetry. */
57 private static final double COMPUTE_ERF = 0.5;
58 /** Threshold for the scaled complementary error function erfcx for negative x
59 * where {@code 2 * exp(x*x)} will overflow. Value is 26.62873571375149. */
60 private static final double ERFCX_NEG_X_MAX = Math.sqrt(Math.log(Double.MAX_VALUE / 2));
61 /** Threshold for the scaled complementary error function erfcx for x
62 * where {@code exp(x*x) == 1; x <= t}. Value is (1 + 5/16) * 2^-27 = 9.778887033462524E-9.
63 * <p>Note: This is used for performance. If set to 0 then the result is computed
64 * using expm1(x*x) with the same final result. */
65 private static final double EXP_XX_1 = 0x1.5p-27;
66
67 /** Private constructor. */
68 private BoostErf() {
69 // intentionally empty.
70 }
71
72 // Code ported from Boost 1.77.0
73 //
74 // boost/math/special_functions/erf.hpp
75 // boost/math/special_functions/detail/erf_inv.hpp
76 //
77 // Original code comments, including measured deviations, are preserved.
78 //
79 // Changes to the Boost implementation:
80 // - Update method names to replace underscores with camel case
81 // - Explicitly inline the polynomial function evaluation
82 // using Horner's method (https://en.wikipedia.org/wiki/Horner%27s_method)
83 // - Support odd function for f(0.0) = -f(-0.0)
84 // - Support the scaled complementary error function erfcx
85 // Erf:
86 // - Change extended precision z*z to compute the square round-off
87 // using Dekker's method
88 // - Change extended precision exp(-z*z) to compute using a
89 // round-off addition to the standard exp result (see NUMBERS-177)
90 // - Change the erf threshold for z when erf(z)=1 from
91 // z > 5.8f to z > 5.930664
92 // - Change the erfc threshold for z when erfc(z)=0 from
93 // z < 28 to z < 27.3
94 // - Change rational function approximation for z > 4 to a function
95 // suitable for erfcx (see NUMBERS-177)
96 // Inverse erf:
97 // - Change inverse erf edge case detection to include NaN
98 // - Change edge case detection for integer z
99 //
100 // Note:
101 // Constants using the 'f' suffix are machine
102 // representable as a float, e.g.
103 // assert 0.0891314744949340820313f == 0.0891314744949340820313;
104 // The values are unchanged from the Boost reference.
105
106 /**
107 * Returns the complementary error function.
108 *
109 * @param x the value.
110 * @return the complementary error function.
111 */
112 static double erfc(double x) {
113 return erfImp(x, true, false);
114 }
115
116 /**
117 * Returns the error function.
118 *
119 * @param x the value.
120 * @return the error function.
121 */
122 static double erf(double x) {
123 return erfImp(x, false, false);
124 }
125
126 /**
127 * 53-bit implementation for the error function.
128 *
129 * <p>Note: The {@code scaled} flag only applies when
130 * {@code z >= 0.5} and {@code invert == true}.
131 * This functionality is used to compute erfcx(z) for positive z.
132 *
133 * @param z Point to evaluate
134 * @param invert true to invert the result (for the complementary error function)
135 * @param scaled true to compute the scaled complementary error function
136 * @return the error function result
137 */
138 private static double erfImp(double z, boolean invert, boolean scaled) {
139 if (Double.isNaN(z)) {
140 return Double.NaN;
141 }
142
143 if (z < 0) {
144 // Here the scaled flag is ignored.
145 if (!invert) {
146 return -erfImp(-z, invert, false);
147 } else if (z < -0.5) {
148 return 2 - erfImp(-z, invert, false);
149 } else {
150 return 1 + erfImp(-z, false, false);
151 }
152 }
153
154 double result;
155
156 //
157 // Big bunch of selection statements now to pick
158 // which implementation to use,
159 // try to put most likely options first:
160 //
161 if (z < COMPUTE_ERF) {
162 //
163 // We're going to calculate erf:
164 //
165 // Here the scaled flag is ignored.
166 if (z < 1e-10) {
167 if (z == 0) {
168 result = z;
169 } else {
170 final double c = 0.003379167095512573896158903121545171688;
171 result = z * 1.125f + z * c;
172 }
173 } else {
174 // Maximum Deviation Found: 1.561e-17
175 // Expected Error Term: 1.561e-17
176 // Maximum Relative Change in Control Points: 1.155e-04
177 // Max Error found at double precision = 2.961182e-17
178
179 final double Y = 1.044948577880859375f;
180 final double zz = z * z;
181 double P;
182 P = -0.000322780120964605683831;
183 P = -0.00772758345802133288487 + P * zz;
184 P = -0.0509990735146777432841 + P * zz;
185 P = -0.338165134459360935041 + P * zz;
186 P = 0.0834305892146531832907 + P * zz;
187 double Q;
188 Q = 0.000370900071787748000569;
189 Q = 0.00858571925074406212772 + Q * zz;
190 Q = 0.0875222600142252549554 + Q * zz;
191 Q = 0.455004033050794024546 + Q * zz;
192 Q = 1.0 + Q * zz;
193 result = z * (Y + P / Q);
194 }
195 // Note: Boost threshold of 5.8f has been raised to approximately 5.93 (6073 / 1024);
196 // threshold of 28 has been lowered to approximately 27.3 (6989/256) where exp(-z*z) = 0.
197 } else if (scaled || (invert ? (z < 27.300781f) : (z < 5.9306640625f))) {
198 //
199 // We'll be calculating erfc:
200 //
201 // Here the scaled flag is used.
202 invert = !invert;
203 if (z < 1.5f) {
204 // Maximum Deviation Found: 3.702e-17
205 // Expected Error Term: 3.702e-17
206 // Maximum Relative Change in Control Points: 2.845e-04
207 // Max Error found at double precision = 4.841816e-17
208 final double Y = 0.405935764312744140625f;
209 final double zm = z - 0.5;
210 double P;
211 P = 0.00180424538297014223957;
212 P = 0.0195049001251218801359 + P * zm;
213 P = 0.0888900368967884466578 + P * zm;
214 P = 0.191003695796775433986 + P * zm;
215 P = 0.178114665841120341155 + P * zm;
216 P = -0.098090592216281240205 + P * zm;
217 double Q;
218 Q = 0.337511472483094676155e-5;
219 Q = 0.0113385233577001411017 + Q * zm;
220 Q = 0.12385097467900864233 + Q * zm;
221 Q = 0.578052804889902404909 + Q * zm;
222 Q = 1.42628004845511324508 + Q * zm;
223 Q = 1.84759070983002217845 + Q * zm;
224 Q = 1.0 + Q * zm;
225 result = Y + P / Q;
226 if (scaled) {
227 result /= z;
228 } else {
229 result *= expmxx(z) / z;
230 }
231 } else if (z < 2.5f) {
232 // Max Error found at double precision = 6.599585e-18
233 // Maximum Deviation Found: 3.909e-18
234 // Expected Error Term: 3.909e-18
235 // Maximum Relative Change in Control Points: 9.886e-05
236 final double Y = 0.50672817230224609375f;
237 final double zm = z - 1.5;
238 double P;
239 P = 0.000235839115596880717416;
240 P = 0.00323962406290842133584 + P * zm;
241 P = 0.0175679436311802092299 + P * zm;
242 P = 0.04394818964209516296 + P * zm;
243 P = 0.0386540375035707201728 + P * zm;
244 P = -0.0243500476207698441272 + P * zm;
245 double Q;
246 Q = 0.00410369723978904575884;
247 Q = 0.0563921837420478160373 + Q * zm;
248 Q = 0.325732924782444448493 + Q * zm;
249 Q = 0.982403709157920235114 + Q * zm;
250 Q = 1.53991494948552447182 + Q * zm;
251 Q = 1.0 + Q * zm;
252 result = Y + P / Q;
253 if (scaled) {
254 result /= z;
255 } else {
256 result *= expmxx(z) / z;
257 }
258 // Lowered Boost threshold from 4.5 to 4.0 as this is the limit
259 // for the Cody erfc approximation
260 } else if (z < 4.0f) {
261 // Maximum Deviation Found: 1.512e-17
262 // Expected Error Term: 1.512e-17
263 // Maximum Relative Change in Control Points: 2.222e-04
264 // Max Error found at double precision = 2.062515e-17
265 final double Y = 0.5405750274658203125f;
266 final double zm = z - 3.5;
267 double P;
268 P = 0.113212406648847561139e-4;
269 P = 0.000250269961544794627958 + P * zm;
270 P = 0.00212825620914618649141 + P * zm;
271 P = 0.00840807615555585383007 + P * zm;
272 P = 0.0137384425896355332126 + P * zm;
273 P = 0.00295276716530971662634 + P * zm;
274 double Q;
275 Q = 0.000479411269521714493907;
276 Q = 0.0105982906484876531489 + Q * zm;
277 Q = 0.0958492726301061423444 + Q * zm;
278 Q = 0.442597659481563127003 + Q * zm;
279 Q = 1.04217814166938418171 + Q * zm;
280 Q = 1.0 + Q * zm;
281 result = Y + P / Q;
282 if (scaled) {
283 result /= z;
284 } else {
285 result *= expmxx(z) / z;
286 }
287 } else {
288 // Rational function approximation for erfc(x > 4.0)
289 //
290 // This approximation is not the Boost implementation.
291 // The Boost function is suitable for [4.5 < z < 28].
292 //
293 // This function is suitable for erfcx(z) as it asymptotes
294 // to (1 / sqrt(pi)) / z at large z.
295 //
296 // Taken from "Rational Chebyshev approximations for the error function"
297 // by W. J. Cody, Math. Comp., 1969, PP. 631-638.
298 //
299 // See NUMBERS-177.
300
301 final double izz = 1 / (z * z);
302 double p;
303 p = 1.63153871373020978498e-2;
304 p = 3.05326634961232344035e-1 + p * izz;
305 p = 3.60344899949804439429e-1 + p * izz;
306 p = 1.25781726111229246204e-1 + p * izz;
307 p = 1.60837851487422766278e-2 + p * izz;
308 p = 6.58749161529837803157e-4 + p * izz;
309 double q;
310 q = 1;
311 q = 2.56852019228982242072e00 + q * izz;
312 q = 1.87295284992346047209e00 + q * izz;
313 q = 5.27905102951428412248e-1 + q * izz;
314 q = 6.05183413124413191178e-2 + q * izz;
315 q = 2.33520497626869185443e-3 + q * izz;
316
317 result = izz * p / q;
318 result = (ONE_OVER_ROOT_PI - result) / z;
319
320 if (!scaled) {
321 // exp(-z*z) can be sub-normal so
322 // multiply by any sub-normal after divide by z
323 result *= expmxx(z);
324 }
325 }
326 } else {
327 //
328 // Any value of z larger than 27.3 will underflow to zero:
329 //
330 result = 0;
331 invert = !invert;
332 }
333
334 if (invert) {
335 // Note: If 0.5 <= z < 28 and the scaled flag is true then
336 // invert will have been flipped to false and the
337 // the result is unchanged as erfcx(z)
338 result = 1 - result;
339 }
340
341 return result;
342 }
343
344 /**
345 * Returns the scaled complementary error function.
346 * <pre>
347 * erfcx(x) = exp(x^2) * erfc(x)
348 * </pre>
349 *
350 * @param x the value.
351 * @return the scaled complementary error function.
352 */
353 static double erfcx(double x) {
354 if (Double.isNaN(x)) {
355 return Double.NaN;
356 }
357
358 // For |z| < 0.5 erfc is computed using erf
359 final double ax = Math.abs(x);
360 if (ax < COMPUTE_ERF) {
361 // Use the erf(x) result.
362 // (1 - erf(x)) * exp(x*x)
363
364 final double erfx = erf(x);
365 if (ax < EXP_XX_1) {
366 // No exponential required
367 return 1 - erfx;
368 }
369
370 // exp(x*x) - exp(x*x) * erf(x)
371 // Avoid use of exp(x*x) with expm1:
372 // exp(x*x) - 1 - (erf(x) * (exp(x*x) - 1)) - erf(x) + 1
373
374 // Sum small to large: |erf(x)| > expm1(x*x)
375 // -erf(x) * expm1(x*x) + expm1(x*x) - erf(x) + 1
376 // Negative x: erf(x) < 0, summed terms are positive, no cancellation occurs.
377 // Positive x: erf(x) > 0 so cancellation can occur.
378 // When terms are ordered by absolute magnitude the magnitude of the next term
379 // is above the round-off from adding the previous term to the sum. Thus
380 // cancellation is negligible compared to errors in the largest computed term (erf(x)).
381
382 final double em1 = Math.expm1(x * x);
383 return -erfx * em1 + em1 - erfx + 1;
384 }
385
386 // Handle negative arguments
387 if (x < 0) {
388 // erfcx(x) = 2*exp(x*x) - erfcx(-x)
389
390 if (x < -ERFCX_NEG_X_MAX) {
391 // Overflow
392 return Double.POSITIVE_INFINITY;
393 }
394
395 final double e = expxx(x);
396 return e - erfImp(-x, true, true) + e;
397 }
398
399 // Approximation for large positive x
400 if (x > ERFCX_APPROX) {
401 return ONE_OVER_ROOT_PI / x;
402 }
403
404 // Compute erfc scaled
405 return erfImp(x, true, true);
406 }
407
408 /**
409 * Returns the inverse complementary error function.
410 *
411 * @param z Value (in {@code [0, 2]}).
412 * @return t such that {@code z = erfc(t)}
413 */
414 static double erfcInv(double z) {
415 //
416 // Begin by testing for domain errors, and other special cases:
417 //
418 if (z < 0 || z > 2 || Double.isNaN(z)) {
419 // Argument outside range [0,2] in inverse erfc function
420 return Double.NaN;
421 }
422 // Domain bounds must be detected as the implementation computes NaN.
423 // (log(q=0) creates infinity and the rational number is
424 // infinity / infinity)
425 if (z == (int) z) {
426 // z return
427 // 2 -inf
428 // 1 0
429 // 0 inf
430 return z == 1 ? 0 : (1 - z) * Double.POSITIVE_INFINITY;
431 }
432
433 //
434 // Normalise the input, so it's in the range [0,1], we will
435 // negate the result if z is outside that range. This is a simple
436 // application of the erfc reflection formula: erfc(-z) = 2 - erfc(z)
437 //
438 double p;
439 double q;
440 double s;
441 if (z > 1) {
442 q = 2 - z;
443 p = 1 - q;
444 s = -1;
445 } else {
446 p = 1 - z;
447 q = z;
448 s = 1;
449 }
450
451 //
452 // And get the result, negating where required:
453 //
454 return s * erfInvImp(p, q);
455 }
456
457 /**
458 * Returns the inverse error function.
459 *
460 * @param z Value (in {@code [-1, 1]}).
461 * @return t such that {@code z = erf(t)}
462 */
463 static double erfInv(double z) {
464 //
465 // Begin by testing for domain errors, and other special cases:
466 //
467 if (z < -1 || z > 1 || Double.isNaN(z)) {
468 // Argument outside range [-1, 1] in inverse erf function
469 return Double.NaN;
470 }
471 // Domain bounds must be detected as the implementation computes NaN.
472 // (log(q=0) creates infinity and the rational number is
473 // infinity / infinity)
474 if (z == (int) z) {
475 // z return
476 // -1 -inf
477 // -0 -0
478 // 0 0
479 // 1 inf
480 return z == 0 ? z : z * Double.POSITIVE_INFINITY;
481 }
482
483 //
484 // Normalise the input, so it's in the range [0,1], we will
485 // negate the result if z is outside that range. This is a simple
486 // application of the erf reflection formula: erf(-z) = -erf(z)
487 //
488 double p;
489 double q;
490 double s;
491 if (z < 0) {
492 p = -z;
493 q = 1 - p;
494 s = -1;
495 } else {
496 p = z;
497 q = 1 - z;
498 s = 1;
499 }
500 //
501 // And get the result, negating where required:
502 //
503 return s * erfInvImp(p, q);
504 }
505
506 /**
507 * Common implementation for inverse erf and erfc functions.
508 *
509 * @param p P-value
510 * @param q Q-value (1-p)
511 * @return the inverse
512 */
513 private static double erfInvImp(double p, double q) {
514 double result = 0;
515
516 if (p <= 0.5) {
517 //
518 // Evaluate inverse erf using the rational approximation:
519 //
520 // x = p(p+10)(Y+R(p))
521 //
522 // Where Y is a constant, and R(p) is optimised for a low
523 // absolute error compared to |Y|.
524 //
525 // double: Max error found: 2.001849e-18
526 // long double: Max error found: 1.017064e-20
527 // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
528 //
529 final float Y = 0.0891314744949340820313f;
530 double P;
531 P = -0.00538772965071242932965;
532 P = 0.00822687874676915743155 + P * p;
533 P = 0.0219878681111168899165 + P * p;
534 P = -0.0365637971411762664006 + P * p;
535 P = -0.0126926147662974029034 + P * p;
536 P = 0.0334806625409744615033 + P * p;
537 P = -0.00836874819741736770379 + P * p;
538 P = -0.000508781949658280665617 + P * p;
539 double Q;
540 Q = 0.000886216390456424707504;
541 Q = -0.00233393759374190016776 + Q * p;
542 Q = 0.0795283687341571680018 + Q * p;
543 Q = -0.0527396382340099713954 + Q * p;
544 Q = -0.71228902341542847553 + Q * p;
545 Q = 0.662328840472002992063 + Q * p;
546 Q = 1.56221558398423026363 + Q * p;
547 Q = -1.56574558234175846809 + Q * p;
548 Q = -0.970005043303290640362 + Q * p;
549 Q = 1.0 + Q * p;
550 final double g = p * (p + 10);
551 final double r = P / Q;
552 result = g * Y + g * r;
553 } else if (q >= 0.25) {
554 //
555 // Rational approximation for 0.5 > q >= 0.25
556 //
557 // x = sqrt(-2*log(q)) / (Y + R(q))
558 //
559 // Where Y is a constant, and R(q) is optimised for a low
560 // absolute error compared to Y.
561 //
562 // double : Max error found: 7.403372e-17
563 // long double : Max error found: 6.084616e-20
564 // Maximum Deviation Found (error term) 4.811e-20
565 //
566 final float Y = 2.249481201171875f;
567 final double xs = q - 0.25f;
568 double P;
569 P = -3.67192254707729348546;
570 P = 21.1294655448340526258 + P * xs;
571 P = 17.445385985570866523 + P * xs;
572 P = -44.6382324441786960818 + P * xs;
573 P = -18.8510648058714251895 + P * xs;
574 P = 17.6447298408374015486 + P * xs;
575 P = 8.37050328343119927838 + P * xs;
576 P = 0.105264680699391713268 + P * xs;
577 P = -0.202433508355938759655 + P * xs;
578 double Q;
579 Q = 1.72114765761200282724;
580 Q = -22.6436933413139721736 + Q * xs;
581 Q = 10.8268667355460159008 + Q * xs;
582 Q = 48.5609213108739935468 + Q * xs;
583 Q = -20.1432634680485188801 + Q * xs;
584 Q = -28.6608180499800029974 + Q * xs;
585 Q = 3.9713437953343869095 + Q * xs;
586 Q = 6.24264124854247537712 + Q * xs;
587 Q = 1.0 + Q * xs;
588 final double g = Math.sqrt(-2 * Math.log(q));
589 final double r = P / Q;
590 result = g / (Y + r);
591 } else {
592 //
593 // For q < 0.25 we have a series of rational approximations all
594 // of the general form:
595 //
596 // let: x = sqrt(-log(q))
597 //
598 // Then the result is given by:
599 //
600 // x(Y+R(x-B))
601 //
602 // where Y is a constant, B is the lowest value of x for which
603 // the approximation is valid, and R(x-B) is optimised for a low
604 // absolute error compared to Y.
605 //
606 // Note that almost all code will really go through the first
607 // or maybe second approximation. After than we're dealing with very
608 // small input values indeed.
609 //
610 // Limit for a double: Math.sqrt(-Math.log(Double.MIN_VALUE)) = 27.28...
611 // Branches for x >= 44 (supporting 80 and 128 bit long double) have been removed.
612 final double x = Math.sqrt(-Math.log(q));
613 if (x < 3) {
614 // Max error found: 1.089051e-20
615 final float Y = 0.807220458984375f;
616 final double xs = x - 1.125f;
617 double P;
618 P = -0.681149956853776992068e-9;
619 P = 0.285225331782217055858e-7 + P * xs;
620 P = -0.679465575181126350155e-6 + P * xs;
621 P = 0.00214558995388805277169 + P * xs;
622 P = 0.0290157910005329060432 + P * xs;
623 P = 0.142869534408157156766 + P * xs;
624 P = 0.337785538912035898924 + P * xs;
625 P = 0.387079738972604337464 + P * xs;
626 P = 0.117030156341995252019 + P * xs;
627 P = -0.163794047193317060787 + P * xs;
628 P = -0.131102781679951906451 + P * xs;
629 double Q;
630 Q = 0.01105924229346489121;
631 Q = 0.152264338295331783612 + Q * xs;
632 Q = 0.848854343457902036425 + Q * xs;
633 Q = 2.59301921623620271374 + Q * xs;
634 Q = 4.77846592945843778382 + Q * xs;
635 Q = 5.38168345707006855425 + Q * xs;
636 Q = 3.46625407242567245975 + Q * xs;
637 Q = 1.0 + Q * xs;
638 final double R = P / Q;
639 result = Y * x + R * x;
640 } else if (x < 6) {
641 // Max error found: 8.389174e-21
642 final float Y = 0.93995571136474609375f;
643 final double xs = x - 3;
644 double P;
645 P = 0.266339227425782031962e-11;
646 P = -0.230404776911882601748e-9 + P * xs;
647 P = 0.460469890584317994083e-5 + P * xs;
648 P = 0.000157544617424960554631 + P * xs;
649 P = 0.00187123492819559223345 + P * xs;
650 P = 0.00950804701325919603619 + P * xs;
651 P = 0.0185573306514231072324 + P * xs;
652 P = -0.00222426529213447927281 + P * xs;
653 P = -0.0350353787183177984712 + P * xs;
654 double Q;
655 Q = 0.764675292302794483503e-4;
656 Q = 0.00263861676657015992959 + Q * xs;
657 Q = 0.0341589143670947727934 + Q * xs;
658 Q = 0.220091105764131249824 + Q * xs;
659 Q = 0.762059164553623404043 + Q * xs;
660 Q = 1.3653349817554063097 + Q * xs;
661 Q = 1.0 + Q * xs;
662 final double R = P / Q;
663 result = Y * x + R * x;
664 } else if (x < 18) {
665 // Max error found: 1.481312e-19
666 final float Y = 0.98362827301025390625f;
667 final double xs = x - 6;
668 double P;
669 P = 0.99055709973310326855e-16;
670 P = -0.281128735628831791805e-13 + P * xs;
671 P = 0.462596163522878599135e-8 + P * xs;
672 P = 0.449696789927706453732e-6 + P * xs;
673 P = 0.149624783758342370182e-4 + P * xs;
674 P = 0.000209386317487588078668 + P * xs;
675 P = 0.00105628862152492910091 + P * xs;
676 P = -0.00112951438745580278863 + P * xs;
677 P = -0.0167431005076633737133 + P * xs;
678 double Q;
679 Q = 0.282243172016108031869e-6;
680 Q = 0.275335474764726041141e-4 + Q * xs;
681 Q = 0.000964011807005165528527 + Q * xs;
682 Q = 0.0160746087093676504695 + Q * xs;
683 Q = 0.138151865749083321638 + Q * xs;
684 Q = 0.591429344886417493481 + Q * xs;
685 Q = 1.0 + Q * xs;
686 final double R = P / Q;
687 result = Y * x + R * x;
688 } else {
689 // x < 44
690 // Max error found: 5.697761e-20
691 final float Y = 0.99714565277099609375f;
692 final double xs = x - 18;
693 double P;
694 P = -0.116765012397184275695e-17;
695 P = 0.145596286718675035587e-11 + P * xs;
696 P = 0.411632831190944208473e-9 + P * xs;
697 P = 0.396341011304801168516e-7 + P * xs;
698 P = 0.162397777342510920873e-5 + P * xs;
699 P = 0.254723037413027451751e-4 + P * xs;
700 P = -0.779190719229053954292e-5 + P * xs;
701 P = -0.0024978212791898131227 + P * xs;
702 double Q;
703 Q = 0.509761276599778486139e-9;
704 Q = 0.144437756628144157666e-6 + Q * xs;
705 Q = 0.145007359818232637924e-4 + Q * xs;
706 Q = 0.000690538265622684595676 + Q * xs;
707 Q = 0.0169410838120975906478 + Q * xs;
708 Q = 0.207123112214422517181 + Q * xs;
709 Q = 1.0 + Q * xs;
710 final double R = P / Q;
711 result = Y * x + R * x;
712 }
713 }
714 return result;
715 }
716
717 /**
718 * Compute {@code exp(x*x)} with high accuracy. This is performed using
719 * information in the round-off from {@code x*x}.
720 *
721 * <p>This is accurate at large x to 1 ulp.
722 *
723 * <p>At small x the accuracy cannot be improved over using exp(x*x).
724 * This occurs at {@code x <= 1}.
725 *
726 * <p>Warning: This has no checks for overflow. The method is never called
727 * when {@code x*x > log(MAX_VALUE/2)}.
728 *
729 * @param x Value
730 * @return exp(x*x)
731 */
732 static double expxx(double x) {
733 // Note: If exp(a) overflows this can create NaN if the
734 // round-off b is negative or zero:
735 // exp(a) * exp1m(b) + exp(a)
736 // inf * 0 + inf or inf * -b + inf
737 final double a = x * x;
738 final double b = squareLowUnscaled(x, a);
739 return expxx(a, b);
740 }
741
742 /**
743 * Compute {@code exp(-x*x)} with high accuracy. This is performed using
744 * information in the round-off from {@code x*x}.
745 *
746 * <p>This is accurate at large x to 1 ulp until exp(-x*x) is close to
747 * sub-normal. For very small exp(-x*x) the adjustment is sub-normal and
748 * bits can be lost in the adjustment for a max observed error of {@code < 2} ulp.
749 *
750 * <p>At small x the accuracy cannot be improved over using exp(-x*x).
751 * This occurs at {@code x <= 1}.
752 *
753 * @param x Value
754 * @return exp(-x*x)
755 */
756 static double expmxx(double x) {
757 final double a = x * x;
758 final double b = squareLowUnscaled(x, a);
759 return expxx(-a, -b);
760 }
761
762 /**
763 * Compute {@code exp(a+b)} with high accuracy assuming {@code a+b = a}.
764 *
765 * <p>This is accurate at large positive a to 1 ulp. If a is negative and exp(a) is
766 * close to sub-normal a bit of precision may be lost when adjusting result
767 * as the adjustment is sub-normal (max observed error {@code < 2} ulp).
768 * For the use case of multiplication of a number less than 1 by exp(-x*x), a = -x*x,
769 * the result will be sub-normal and the rounding error is lost.
770 *
771 * <p>At small |a| the accuracy cannot be improved over using exp(a) as the
772 * round-off is too small to create terms that can adjust the standard result by
773 * more than 0.5 ulp. This occurs at {@code |a| <= 1}.
774 *
775 * @param a High bits of a split number
776 * @param b Low bits of a split number
777 * @return exp(a+b)
778 */
779 private static double expxx(double a, double b) {
780 // exp(a+b) = exp(a) * exp(b)
781 // = exp(a) * (exp(b) - 1) + exp(a)
782 // Assuming:
783 // 1. -746 < a < 710 for no under/overflow of exp(a)
784 // 2. a+b = a
785 // As b -> 0 then exp(b) -> 1; expm1(b) -> b
786 // The round-off b is limited to ~ 0.5 * ulp(746) ~ 5.68e-14
787 // and we can use an approximation for expm1 (x/1! + x^2/2! + ...)
788 // The second term is required for the expm1 result but the
789 // bits are not significant to change the product with exp(a)
790
791 final double ea = Math.exp(a);
792 // b ~ expm1(b)
793 return ea * b + ea;
794 }
795
796 // Extended precision multiplication specialised for the square adapted from:
797 // org.apache.commons.numbers.core.ExtendedPrecision
798
799 /**
800 * Compute the low part of the double length number {@code (z,zz)} for the exact
801 * square of {@code x} using Dekker's mult12 algorithm. The standard precision product
802 * {@code x*x} must be provided. The number {@code x} is split into high and low parts
803 * using Dekker's algorithm.
804 *
805 * <p>Warning: This method does not perform scaling in Dekker's split and large
806 * finite numbers can create NaN results.
807 *
808 * @param x Number to square
809 * @param xx Standard precision product {@code x*x}
810 * @return the low part of the square double length number
811 */
812 private static double squareLowUnscaled(double x, double xx) {
813 // Split the numbers using Dekker's algorithm without scaling
814 final double hx = highPartUnscaled(x);
815 final double lx = x - hx;
816
817 return squareLow(hx, lx, xx);
818 }
819
820 /**
821 * Implement Dekker's method to split a value into two parts. Multiplying by (2^s + 1) creates
822 * a big value from which to derive the two split parts.
823 * <pre>
824 * c = (2^s + 1) * a
825 * a_big = c - a
826 * a_hi = c - a_big
827 * a_lo = a - a_hi
828 * a = a_hi + a_lo
829 * </pre>
830 *
831 * <p>The multiplicand allows a p-bit value to be split into
832 * (p-s)-bit value {@code a_hi} and a non-overlapping (s-1)-bit value {@code a_lo}.
833 * Combined they have (p-1) bits of significand but the sign bit of {@code a_lo}
834 * contains a bit of information. The constant is chosen so that s is ceil(p/2) where
835 * the precision p for a double is 53-bits (1-bit of the mantissa is assumed to be
836 * 1 for a non sub-normal number) and s is 27.
837 *
838 * <p>This conversion does not use scaling and the result of overflow is NaN. Overflow
839 * may occur when the exponent of the input value is above 996.
840 *
841 * <p>Splitting a NaN or infinite value will return NaN.
842 *
843 * @param value Value.
844 * @return the high part of the value.
845 * @see Math#getExponent(double)
846 */
847 private static double highPartUnscaled(double value) {
848 final double c = MULTIPLIER * value;
849 return c - (c - value);
850 }
851
852 /**
853 * Compute the low part of the double length number {@code (z,zz)} for the exact
854 * square of {@code x} using Dekker's mult12 algorithm. The standard
855 * precision product {@code x*x} must be provided. The number {@code x}
856 * should already be split into low and high parts.
857 *
858 * <p>Note: This uses the high part of the result {@code (z,zz)} as {@code x * x} and not
859 * {@code hx * hx + hx * lx + lx * hx} as specified in Dekker's original paper.
860 * See Shewchuk (1997) for working examples.
861 *
862 * @param hx High part of factor.
863 * @param lx Low part of factor.
864 * @param xx Square of the factor.
865 * @return <code>lx * ly - (((xy - hx * hy) - lx * hy) - hx * ly)</code>
866 * @see <a href="http://www-2.cs.cmu.edu/afs/cs/project/quake/public/papers/robust-arithmetic.ps">
867 * Shewchuk (1997) Theorum 18</a>
868 */
869 private static double squareLow(double hx, double lx, double xx) {
870 // Compute the multiply low part:
871 // err1 = xy - hx * hy
872 // err2 = err1 - lx * hy
873 // err3 = err2 - hx * ly
874 // low = lx * ly - err3
875 return lx * lx - ((xx - hx * hx) - 2 * lx * hx);
876 }
877 }
878