1 // (C) Copyright John Maddock 2006.
2 // Use, modification and distribution are subject to the
3 // Boost Software License, Version 1.0. (See accompanying file
4 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5
6 #ifndef BOOST_MATH_SPECIAL_ERF_HPP
7 #define BOOST_MATH_SPECIAL_ERF_HPP
8
9 #ifdef _MSC_VER
10 #pragma once
11 #endif
12
13 #include <boost/math/special_functions/math_fwd.hpp>
14 #include <boost/math/tools/config.hpp>
15 #include <boost/math/special_functions/gamma.hpp>
16 #include <boost/math/tools/roots.hpp>
17 #include <boost/math/policies/error_handling.hpp>
18 #include <boost/math/tools/big_constant.hpp>
19
20 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
21 //
22 // This is the only way we can avoid
23 // warning: non-standard suffix on floating constant [-Wpedantic]
24 // when building with -Wall -pedantic. Neither __extension__
25 // nor #pragma diagnostic ignored work :(
26 //
27 #pragma GCC system_header
28 #endif
29
30 namespace boost{ namespace math{
31
32 namespace detail
33 {
34
35 //
36 // Asymptotic series for large z:
37 //
38 template <class T>
39 struct erf_asympt_series_t
40 {
erf_asympt_series_tboost::math::detail::erf_asympt_series_t41 erf_asympt_series_t(T z) : xx(2 * -z * z), tk(1)
42 {
43 BOOST_MATH_STD_USING
44 result = -exp(-z * z) / sqrt(boost::math::constants::pi<T>());
45 result /= z;
46 }
47
48 typedef T result_type;
49
operator ()boost::math::detail::erf_asympt_series_t50 T operator()()
51 {
52 BOOST_MATH_STD_USING
53 T r = result;
54 result *= tk / xx;
55 tk += 2;
56 if( fabs(r) < fabs(result))
57 result = 0;
58 return r;
59 }
60 private:
61 T result;
62 T xx;
63 int tk;
64 };
65 //
66 // How large z has to be in order to ensure that the series converges:
67 //
68 template <class T>
erf_asymptotic_limit_N(const T &)69 inline float erf_asymptotic_limit_N(const T&)
70 {
71 return (std::numeric_limits<float>::max)();
72 }
erf_asymptotic_limit_N(const boost::integral_constant<int,24> &)73 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 24>&)
74 {
75 return 2.8F;
76 }
erf_asymptotic_limit_N(const boost::integral_constant<int,53> &)77 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 53>&)
78 {
79 return 4.3F;
80 }
erf_asymptotic_limit_N(const boost::integral_constant<int,64> &)81 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 64>&)
82 {
83 return 4.8F;
84 }
erf_asymptotic_limit_N(const boost::integral_constant<int,106> &)85 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 106>&)
86 {
87 return 6.5F;
88 }
erf_asymptotic_limit_N(const boost::integral_constant<int,113> &)89 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 113>&)
90 {
91 return 6.8F;
92 }
93
94 template <class T, class Policy>
erf_asymptotic_limit()95 inline T erf_asymptotic_limit()
96 {
97 typedef typename policies::precision<T, Policy>::type precision_type;
98 typedef boost::integral_constant<int,
99 precision_type::value <= 0 ? 0 :
100 precision_type::value <= 24 ? 24 :
101 precision_type::value <= 53 ? 53 :
102 precision_type::value <= 64 ? 64 :
103 precision_type::value <= 113 ? 113 : 0
104 > tag_type;
105 return erf_asymptotic_limit_N(tag_type());
106 }
107
108 template <class T, class Policy, class Tag>
erf_imp(T z,bool invert,const Policy & pol,const Tag & t)109 T erf_imp(T z, bool invert, const Policy& pol, const Tag& t)
110 {
111 BOOST_MATH_STD_USING
112
113 BOOST_MATH_INSTRUMENT_CODE("Generic erf_imp called");
114
115 if(z < 0)
116 {
117 if(!invert)
118 return -erf_imp(T(-z), invert, pol, t);
119 else
120 return 1 + erf_imp(T(-z), false, pol, t);
121 }
122
123 T result;
124
125 if(!invert && (z > detail::erf_asymptotic_limit<T, Policy>()))
126 {
127 detail::erf_asympt_series_t<T> s(z);
128 boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
129 result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, 1);
130 policies::check_series_iterations<T>("boost::math::erf<%1%>(%1%, %1%)", max_iter, pol);
131 }
132 else
133 {
134 T x = z * z;
135 if(x < 0.6)
136 {
137 // Compute P:
138 result = z * exp(-x);
139 result /= sqrt(boost::math::constants::pi<T>());
140 if(result != 0)
141 result *= 2 * detail::lower_gamma_series(T(0.5f), x, pol);
142 }
143 else if(x < 1.1f)
144 {
145 // Compute Q:
146 invert = !invert;
147 result = tgamma_small_upper_part(T(0.5f), x, pol);
148 result /= sqrt(boost::math::constants::pi<T>());
149 }
150 else if(x > 1 / tools::epsilon<T>())
151 {
152 // http://functions.wolfram.com/06.27.06.0006.02
153 invert = !invert;
154 result = exp(-x) / (constants::root_pi<T>() * z);
155 }
156 else
157 {
158 // Compute Q:
159 invert = !invert;
160 result = z * exp(-x);
161 result /= boost::math::constants::root_pi<T>();
162 result *= upper_gamma_fraction(T(0.5f), x, policies::get_epsilon<T, Policy>());
163 }
164 }
165 if(invert)
166 result = 1 - result;
167 return result;
168 }
169
170 template <class T, class Policy>
erf_imp(T z,bool invert,const Policy & pol,const boost::integral_constant<int,53> & t)171 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 53>& t)
172 {
173 BOOST_MATH_STD_USING
174
175 BOOST_MATH_INSTRUMENT_CODE("53-bit precision erf_imp called");
176
177 if ((boost::math::isnan)(z))
178 return policies::raise_denorm_error("boost::math::erf<%1%>(%1%)", "Expected a finite argument but got %1%", z, pol);
179
180 if(z < 0)
181 {
182 if(!invert)
183 return -erf_imp(T(-z), invert, pol, t);
184 else if(z < -0.5)
185 return 2 - erf_imp(T(-z), invert, pol, t);
186 else
187 return 1 + erf_imp(T(-z), false, pol, t);
188 }
189
190 T result;
191
192 //
193 // Big bunch of selection statements now to pick
194 // which implementation to use,
195 // try to put most likely options first:
196 //
197 if(z < 0.5)
198 {
199 //
200 // We're going to calculate erf:
201 //
202 if(z < 1e-10)
203 {
204 if(z == 0)
205 {
206 result = T(0);
207 }
208 else
209 {
210 static const T c = BOOST_MATH_BIG_CONSTANT(T, 53, 0.003379167095512573896158903121545171688);
211 result = static_cast<T>(z * 1.125f + z * c);
212 }
213 }
214 else
215 {
216 // Maximum Deviation Found: 1.561e-17
217 // Expected Error Term: 1.561e-17
218 // Maximum Relative Change in Control Points: 1.155e-04
219 // Max Error found at double precision = 2.961182e-17
220
221 static const T Y = 1.044948577880859375f;
222 static const T P[] = {
223 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0834305892146531832907),
224 BOOST_MATH_BIG_CONSTANT(T, 53, -0.338165134459360935041),
225 BOOST_MATH_BIG_CONSTANT(T, 53, -0.0509990735146777432841),
226 BOOST_MATH_BIG_CONSTANT(T, 53, -0.00772758345802133288487),
227 BOOST_MATH_BIG_CONSTANT(T, 53, -0.000322780120964605683831),
228 };
229 static const T Q[] = {
230 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
231 BOOST_MATH_BIG_CONSTANT(T, 53, 0.455004033050794024546),
232 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0875222600142252549554),
233 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00858571925074406212772),
234 BOOST_MATH_BIG_CONSTANT(T, 53, 0.000370900071787748000569),
235 };
236 T zz = z * z;
237 result = z * (Y + tools::evaluate_polynomial(P, zz) / tools::evaluate_polynomial(Q, zz));
238 }
239 }
240 else if(invert ? (z < 28) : (z < 5.8f))
241 {
242 //
243 // We'll be calculating erfc:
244 //
245 invert = !invert;
246 if(z < 1.5f)
247 {
248 // Maximum Deviation Found: 3.702e-17
249 // Expected Error Term: 3.702e-17
250 // Maximum Relative Change in Control Points: 2.845e-04
251 // Max Error found at double precision = 4.841816e-17
252 static const T Y = 0.405935764312744140625f;
253 static const T P[] = {
254 BOOST_MATH_BIG_CONSTANT(T, 53, -0.098090592216281240205),
255 BOOST_MATH_BIG_CONSTANT(T, 53, 0.178114665841120341155),
256 BOOST_MATH_BIG_CONSTANT(T, 53, 0.191003695796775433986),
257 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0888900368967884466578),
258 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0195049001251218801359),
259 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00180424538297014223957),
260 };
261 static const T Q[] = {
262 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
263 BOOST_MATH_BIG_CONSTANT(T, 53, 1.84759070983002217845),
264 BOOST_MATH_BIG_CONSTANT(T, 53, 1.42628004845511324508),
265 BOOST_MATH_BIG_CONSTANT(T, 53, 0.578052804889902404909),
266 BOOST_MATH_BIG_CONSTANT(T, 53, 0.12385097467900864233),
267 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0113385233577001411017),
268 BOOST_MATH_BIG_CONSTANT(T, 53, 0.337511472483094676155e-5),
269 };
270 BOOST_MATH_INSTRUMENT_VARIABLE(Y);
271 BOOST_MATH_INSTRUMENT_VARIABLE(P[0]);
272 BOOST_MATH_INSTRUMENT_VARIABLE(Q[0]);
273 BOOST_MATH_INSTRUMENT_VARIABLE(z);
274 result = Y + tools::evaluate_polynomial(P, T(z - 0.5)) / tools::evaluate_polynomial(Q, T(z - 0.5));
275 BOOST_MATH_INSTRUMENT_VARIABLE(result);
276 result *= exp(-z * z) / z;
277 BOOST_MATH_INSTRUMENT_VARIABLE(result);
278 }
279 else if(z < 2.5f)
280 {
281 // Max Error found at double precision = 6.599585e-18
282 // Maximum Deviation Found: 3.909e-18
283 // Expected Error Term: 3.909e-18
284 // Maximum Relative Change in Control Points: 9.886e-05
285 static const T Y = 0.50672817230224609375f;
286 static const T P[] = {
287 BOOST_MATH_BIG_CONSTANT(T, 53, -0.0243500476207698441272),
288 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0386540375035707201728),
289 BOOST_MATH_BIG_CONSTANT(T, 53, 0.04394818964209516296),
290 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175679436311802092299),
291 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00323962406290842133584),
292 BOOST_MATH_BIG_CONSTANT(T, 53, 0.000235839115596880717416),
293 };
294 static const T Q[] = {
295 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
296 BOOST_MATH_BIG_CONSTANT(T, 53, 1.53991494948552447182),
297 BOOST_MATH_BIG_CONSTANT(T, 53, 0.982403709157920235114),
298 BOOST_MATH_BIG_CONSTANT(T, 53, 0.325732924782444448493),
299 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0563921837420478160373),
300 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00410369723978904575884),
301 };
302 result = Y + tools::evaluate_polynomial(P, T(z - 1.5)) / tools::evaluate_polynomial(Q, T(z - 1.5));
303 T hi, lo;
304 int expon;
305 hi = floor(ldexp(frexp(z, &expon), 26));
306 hi = ldexp(hi, expon - 26);
307 lo = z - hi;
308 T sq = z * z;
309 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
310 result *= exp(-sq) * exp(-err_sqr) / z;
311 }
312 else if(z < 4.5f)
313 {
314 // Maximum Deviation Found: 1.512e-17
315 // Expected Error Term: 1.512e-17
316 // Maximum Relative Change in Control Points: 2.222e-04
317 // Max Error found at double precision = 2.062515e-17
318 static const T Y = 0.5405750274658203125f;
319 static const T P[] = {
320 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00295276716530971662634),
321 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0137384425896355332126),
322 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00840807615555585383007),
323 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00212825620914618649141),
324 BOOST_MATH_BIG_CONSTANT(T, 53, 0.000250269961544794627958),
325 BOOST_MATH_BIG_CONSTANT(T, 53, 0.113212406648847561139e-4),
326 };
327 static const T Q[] = {
328 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
329 BOOST_MATH_BIG_CONSTANT(T, 53, 1.04217814166938418171),
330 BOOST_MATH_BIG_CONSTANT(T, 53, 0.442597659481563127003),
331 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0958492726301061423444),
332 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0105982906484876531489),
333 BOOST_MATH_BIG_CONSTANT(T, 53, 0.000479411269521714493907),
334 };
335 result = Y + tools::evaluate_polynomial(P, T(z - 3.5)) / tools::evaluate_polynomial(Q, T(z - 3.5));
336 T hi, lo;
337 int expon;
338 hi = floor(ldexp(frexp(z, &expon), 26));
339 hi = ldexp(hi, expon - 26);
340 lo = z - hi;
341 T sq = z * z;
342 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
343 result *= exp(-sq) * exp(-err_sqr) / z;
344 }
345 else
346 {
347 // Max Error found at double precision = 2.997958e-17
348 // Maximum Deviation Found: 2.860e-17
349 // Expected Error Term: 2.859e-17
350 // Maximum Relative Change in Control Points: 1.357e-05
351 static const T Y = 0.5579090118408203125f;
352 static const T P[] = {
353 BOOST_MATH_BIG_CONSTANT(T, 53, 0.00628057170626964891937),
354 BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175389834052493308818),
355 BOOST_MATH_BIG_CONSTANT(T, 53, -0.212652252872804219852),
356 BOOST_MATH_BIG_CONSTANT(T, 53, -0.687717681153649930619),
357 BOOST_MATH_BIG_CONSTANT(T, 53, -2.5518551727311523996),
358 BOOST_MATH_BIG_CONSTANT(T, 53, -3.22729451764143718517),
359 BOOST_MATH_BIG_CONSTANT(T, 53, -2.8175401114513378771),
360 };
361 static const T Q[] = {
362 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
363 BOOST_MATH_BIG_CONSTANT(T, 53, 2.79257750980575282228),
364 BOOST_MATH_BIG_CONSTANT(T, 53, 11.0567237927800161565),
365 BOOST_MATH_BIG_CONSTANT(T, 53, 15.930646027911794143),
366 BOOST_MATH_BIG_CONSTANT(T, 53, 22.9367376522880577224),
367 BOOST_MATH_BIG_CONSTANT(T, 53, 13.5064170191802889145),
368 BOOST_MATH_BIG_CONSTANT(T, 53, 5.48409182238641741584),
369 };
370 result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
371 T hi, lo;
372 int expon;
373 hi = floor(ldexp(frexp(z, &expon), 26));
374 hi = ldexp(hi, expon - 26);
375 lo = z - hi;
376 T sq = z * z;
377 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
378 result *= exp(-sq) * exp(-err_sqr) / z;
379 }
380 }
381 else
382 {
383 //
384 // Any value of z larger than 28 will underflow to zero:
385 //
386 result = 0;
387 invert = !invert;
388 }
389
390 if(invert)
391 {
392 result = 1 - result;
393 }
394
395 return result;
396 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 53>& t)
397
398
399 template <class T, class Policy>
erf_imp(T z,bool invert,const Policy & pol,const boost::integral_constant<int,64> & t)400 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 64>& t)
401 {
402 BOOST_MATH_STD_USING
403
404 BOOST_MATH_INSTRUMENT_CODE("64-bit precision erf_imp called");
405
406 if(z < 0)
407 {
408 if(!invert)
409 return -erf_imp(T(-z), invert, pol, t);
410 else if(z < -0.5)
411 return 2 - erf_imp(T(-z), invert, pol, t);
412 else
413 return 1 + erf_imp(T(-z), false, pol, t);
414 }
415
416 T result;
417
418 //
419 // Big bunch of selection statements now to pick which
420 // implementation to use, try to put most likely options
421 // first:
422 //
423 if(z < 0.5)
424 {
425 //
426 // We're going to calculate erf:
427 //
428 if(z == 0)
429 {
430 result = 0;
431 }
432 else if(z < 1e-10)
433 {
434 static const T c = BOOST_MATH_BIG_CONSTANT(T, 64, 0.003379167095512573896158903121545171688);
435 result = z * 1.125 + z * c;
436 }
437 else
438 {
439 // Max Error found at long double precision = 1.623299e-20
440 // Maximum Deviation Found: 4.326e-22
441 // Expected Error Term: -4.326e-22
442 // Maximum Relative Change in Control Points: 1.474e-04
443 static const T Y = 1.044948577880859375f;
444 static const T P[] = {
445 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0834305892146531988966),
446 BOOST_MATH_BIG_CONSTANT(T, 64, -0.338097283075565413695),
447 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509602734406067204596),
448 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00904906346158537794396),
449 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000489468651464798669181),
450 BOOST_MATH_BIG_CONSTANT(T, 64, -0.200305626366151877759e-4),
451 };
452 static const T Q[] = {
453 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
454 BOOST_MATH_BIG_CONSTANT(T, 64, 0.455817300515875172439),
455 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0916537354356241792007),
456 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0102722652675910031202),
457 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000650511752687851548735),
458 BOOST_MATH_BIG_CONSTANT(T, 64, 0.189532519105655496778e-4),
459 };
460 result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
461 }
462 }
463 else if(invert ? (z < 110) : (z < 6.4f))
464 {
465 //
466 // We'll be calculating erfc:
467 //
468 invert = !invert;
469 if(z < 1.5)
470 {
471 // Max Error found at long double precision = 3.239590e-20
472 // Maximum Deviation Found: 2.241e-20
473 // Expected Error Term: -2.241e-20
474 // Maximum Relative Change in Control Points: 5.110e-03
475 static const T Y = 0.405935764312744140625f;
476 static const T P[] = {
477 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0980905922162812031672),
478 BOOST_MATH_BIG_CONSTANT(T, 64, 0.159989089922969141329),
479 BOOST_MATH_BIG_CONSTANT(T, 64, 0.222359821619935712378),
480 BOOST_MATH_BIG_CONSTANT(T, 64, 0.127303921703577362312),
481 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0384057530342762400273),
482 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00628431160851156719325),
483 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000441266654514391746428),
484 BOOST_MATH_BIG_CONSTANT(T, 64, 0.266689068336295642561e-7),
485 };
486 static const T Q[] = {
487 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
488 BOOST_MATH_BIG_CONSTANT(T, 64, 2.03237474985469469291),
489 BOOST_MATH_BIG_CONSTANT(T, 64, 1.78355454954969405222),
490 BOOST_MATH_BIG_CONSTANT(T, 64, 0.867940326293760578231),
491 BOOST_MATH_BIG_CONSTANT(T, 64, 0.248025606990021698392),
492 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0396649631833002269861),
493 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00279220237309449026796),
494 };
495 result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
496 T hi, lo;
497 int expon;
498 hi = floor(ldexp(frexp(z, &expon), 32));
499 hi = ldexp(hi, expon - 32);
500 lo = z - hi;
501 T sq = z * z;
502 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
503 result *= exp(-sq) * exp(-err_sqr) / z;
504 }
505 else if(z < 2.5)
506 {
507 // Max Error found at long double precision = 3.686211e-21
508 // Maximum Deviation Found: 1.495e-21
509 // Expected Error Term: -1.494e-21
510 // Maximum Relative Change in Control Points: 1.793e-04
511 static const T Y = 0.50672817230224609375f;
512 static const T P[] = {
513 BOOST_MATH_BIG_CONSTANT(T, 64, -0.024350047620769840217),
514 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0343522687935671451309),
515 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0505420824305544949541),
516 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0257479325917757388209),
517 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00669349844190354356118),
518 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00090807914416099524444),
519 BOOST_MATH_BIG_CONSTANT(T, 64, 0.515917266698050027934e-4),
520 };
521 static const T Q[] = {
522 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
523 BOOST_MATH_BIG_CONSTANT(T, 64, 1.71657861671930336344),
524 BOOST_MATH_BIG_CONSTANT(T, 64, 1.26409634824280366218),
525 BOOST_MATH_BIG_CONSTANT(T, 64, 0.512371437838969015941),
526 BOOST_MATH_BIG_CONSTANT(T, 64, 0.120902623051120950935),
527 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0158027197831887485261),
528 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000897871370778031611439),
529 };
530 result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
531 T hi, lo;
532 int expon;
533 hi = floor(ldexp(frexp(z, &expon), 32));
534 hi = ldexp(hi, expon - 32);
535 lo = z - hi;
536 T sq = z * z;
537 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
538 result *= exp(-sq) * exp(-err_sqr) / z;
539 }
540 else if(z < 4.5)
541 {
542 // Maximum Deviation Found: 1.107e-20
543 // Expected Error Term: -1.106e-20
544 // Maximum Relative Change in Control Points: 1.709e-04
545 // Max Error found at long double precision = 1.446908e-20
546 static const T Y = 0.5405750274658203125f;
547 static const T P[] = {
548 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0029527671653097284033),
549 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0141853245895495604051),
550 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0104959584626432293901),
551 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00343963795976100077626),
552 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00059065441194877637899),
553 BOOST_MATH_BIG_CONSTANT(T, 64, 0.523435380636174008685e-4),
554 BOOST_MATH_BIG_CONSTANT(T, 64, 0.189896043050331257262e-5),
555 };
556 static const T Q[] = {
557 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
558 BOOST_MATH_BIG_CONSTANT(T, 64, 1.19352160185285642574),
559 BOOST_MATH_BIG_CONSTANT(T, 64, 0.603256964363454392857),
560 BOOST_MATH_BIG_CONSTANT(T, 64, 0.165411142458540585835),
561 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0259729870946203166468),
562 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00221657568292893699158),
563 BOOST_MATH_BIG_CONSTANT(T, 64, 0.804149464190309799804e-4),
564 };
565 result = Y + tools::evaluate_polynomial(P, T(z - 3.5f)) / tools::evaluate_polynomial(Q, T(z - 3.5f));
566 T hi, lo;
567 int expon;
568 hi = floor(ldexp(frexp(z, &expon), 32));
569 hi = ldexp(hi, expon - 32);
570 lo = z - hi;
571 T sq = z * z;
572 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
573 result *= exp(-sq) * exp(-err_sqr) / z;
574 }
575 else
576 {
577 // Max Error found at long double precision = 7.961166e-21
578 // Maximum Deviation Found: 6.677e-21
579 // Expected Error Term: 6.676e-21
580 // Maximum Relative Change in Control Points: 2.319e-05
581 static const T Y = 0.55825519561767578125f;
582 static const T P[] = {
583 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00593438793008050214106),
584 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280666231009089713937),
585 BOOST_MATH_BIG_CONSTANT(T, 64, -0.141597835204583050043),
586 BOOST_MATH_BIG_CONSTANT(T, 64, -0.978088201154300548842),
587 BOOST_MATH_BIG_CONSTANT(T, 64, -5.47351527796012049443),
588 BOOST_MATH_BIG_CONSTANT(T, 64, -13.8677304660245326627),
589 BOOST_MATH_BIG_CONSTANT(T, 64, -27.1274948720539821722),
590 BOOST_MATH_BIG_CONSTANT(T, 64, -29.2545152747009461519),
591 BOOST_MATH_BIG_CONSTANT(T, 64, -16.8865774499799676937),
592 };
593 static const T Q[] = {
594 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
595 BOOST_MATH_BIG_CONSTANT(T, 64, 4.72948911186645394541),
596 BOOST_MATH_BIG_CONSTANT(T, 64, 23.6750543147695749212),
597 BOOST_MATH_BIG_CONSTANT(T, 64, 60.0021517335693186785),
598 BOOST_MATH_BIG_CONSTANT(T, 64, 131.766251645149522868),
599 BOOST_MATH_BIG_CONSTANT(T, 64, 178.167924971283482513),
600 BOOST_MATH_BIG_CONSTANT(T, 64, 182.499390505915222699),
601 BOOST_MATH_BIG_CONSTANT(T, 64, 104.365251479578577989),
602 BOOST_MATH_BIG_CONSTANT(T, 64, 30.8365511891224291717),
603 };
604 result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
605 T hi, lo;
606 int expon;
607 hi = floor(ldexp(frexp(z, &expon), 32));
608 hi = ldexp(hi, expon - 32);
609 lo = z - hi;
610 T sq = z * z;
611 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
612 result *= exp(-sq) * exp(-err_sqr) / z;
613 }
614 }
615 else
616 {
617 //
618 // Any value of z larger than 110 will underflow to zero:
619 //
620 result = 0;
621 invert = !invert;
622 }
623
624 if(invert)
625 {
626 result = 1 - result;
627 }
628
629 return result;
630 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 64>& t)
631
632
633 template <class T, class Policy>
erf_imp(T z,bool invert,const Policy & pol,const boost::integral_constant<int,113> & t)634 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 113>& t)
635 {
636 BOOST_MATH_STD_USING
637
638 BOOST_MATH_INSTRUMENT_CODE("113-bit precision erf_imp called");
639
640 if(z < 0)
641 {
642 if(!invert)
643 return -erf_imp(T(-z), invert, pol, t);
644 else if(z < -0.5)
645 return 2 - erf_imp(T(-z), invert, pol, t);
646 else
647 return 1 + erf_imp(T(-z), false, pol, t);
648 }
649
650 T result;
651
652 //
653 // Big bunch of selection statements now to pick which
654 // implementation to use, try to put most likely options
655 // first:
656 //
657 if(z < 0.5)
658 {
659 //
660 // We're going to calculate erf:
661 //
662 if(z == 0)
663 {
664 result = 0;
665 }
666 else if(z < 1e-20)
667 {
668 static const T c = BOOST_MATH_BIG_CONSTANT(T, 113, 0.003379167095512573896158903121545171688);
669 result = z * 1.125 + z * c;
670 }
671 else
672 {
673 // Max Error found at long double precision = 2.342380e-35
674 // Maximum Deviation Found: 6.124e-36
675 // Expected Error Term: -6.124e-36
676 // Maximum Relative Change in Control Points: 3.492e-10
677 static const T Y = 1.0841522216796875f;
678 static const T P[] = {
679 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0442269454158250738961589031215451778),
680 BOOST_MATH_BIG_CONSTANT(T, 113, -0.35549265736002144875335323556961233),
681 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0582179564566667896225454670863270393),
682 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0112694696904802304229950538453123925),
683 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000805730648981801146251825329609079099),
684 BOOST_MATH_BIG_CONSTANT(T, 113, -0.566304966591936566229702842075966273e-4),
685 BOOST_MATH_BIG_CONSTANT(T, 113, -0.169655010425186987820201021510002265e-5),
686 BOOST_MATH_BIG_CONSTANT(T, 113, -0.344448249920445916714548295433198544e-7),
687 };
688 static const T Q[] = {
689 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
690 BOOST_MATH_BIG_CONSTANT(T, 113, 0.466542092785657604666906909196052522),
691 BOOST_MATH_BIG_CONSTANT(T, 113, 0.100005087012526447295176964142107611),
692 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0128341535890117646540050072234142603),
693 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00107150448466867929159660677016658186),
694 BOOST_MATH_BIG_CONSTANT(T, 113, 0.586168368028999183607733369248338474e-4),
695 BOOST_MATH_BIG_CONSTANT(T, 113, 0.196230608502104324965623171516808796e-5),
696 BOOST_MATH_BIG_CONSTANT(T, 113, 0.313388521582925207734229967907890146e-7),
697 };
698 result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
699 }
700 }
701 else if(invert ? (z < 110) : (z < 8.65f))
702 {
703 //
704 // We'll be calculating erfc:
705 //
706 invert = !invert;
707 if(z < 1)
708 {
709 // Max Error found at long double precision = 3.246278e-35
710 // Maximum Deviation Found: 1.388e-35
711 // Expected Error Term: 1.387e-35
712 // Maximum Relative Change in Control Points: 6.127e-05
713 static const T Y = 0.371877193450927734375f;
714 static const T P[] = {
715 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0640320213544647969396032886581290455),
716 BOOST_MATH_BIG_CONSTANT(T, 113, 0.200769874440155895637857443946706731),
717 BOOST_MATH_BIG_CONSTANT(T, 113, 0.378447199873537170666487408805779826),
718 BOOST_MATH_BIG_CONSTANT(T, 113, 0.30521399466465939450398642044975127),
719 BOOST_MATH_BIG_CONSTANT(T, 113, 0.146890026406815277906781824723458196),
720 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0464837937749539978247589252732769567),
721 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00987895759019540115099100165904822903),
722 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00137507575429025512038051025154301132),
723 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0001144764551085935580772512359680516),
724 BOOST_MATH_BIG_CONSTANT(T, 113, 0.436544865032836914773944382339900079e-5),
725 };
726 static const T Q[] = {
727 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
728 BOOST_MATH_BIG_CONSTANT(T, 113, 2.47651182872457465043733800302427977),
729 BOOST_MATH_BIG_CONSTANT(T, 113, 2.78706486002517996428836400245547955),
730 BOOST_MATH_BIG_CONSTANT(T, 113, 1.87295924621659627926365005293130693),
731 BOOST_MATH_BIG_CONSTANT(T, 113, 0.829375825174365625428280908787261065),
732 BOOST_MATH_BIG_CONSTANT(T, 113, 0.251334771307848291593780143950311514),
733 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0522110268876176186719436765734722473),
734 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00718332151250963182233267040106902368),
735 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000595279058621482041084986219276392459),
736 BOOST_MATH_BIG_CONSTANT(T, 113, 0.226988669466501655990637599399326874e-4),
737 BOOST_MATH_BIG_CONSTANT(T, 113, 0.270666232259029102353426738909226413e-10),
738 };
739 result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
740 T hi, lo;
741 int expon;
742 hi = floor(ldexp(frexp(z, &expon), 56));
743 hi = ldexp(hi, expon - 56);
744 lo = z - hi;
745 T sq = z * z;
746 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
747 result *= exp(-sq) * exp(-err_sqr) / z;
748 }
749 else if(z < 1.5)
750 {
751 // Max Error found at long double precision = 2.215785e-35
752 // Maximum Deviation Found: 1.539e-35
753 // Expected Error Term: 1.538e-35
754 // Maximum Relative Change in Control Points: 6.104e-05
755 static const T Y = 0.45658016204833984375f;
756 static const T P[] = {
757 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0289965858925328393392496555094848345),
758 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0868181194868601184627743162571779226),
759 BOOST_MATH_BIG_CONSTANT(T, 113, 0.169373435121178901746317404936356745),
760 BOOST_MATH_BIG_CONSTANT(T, 113, 0.13350446515949251201104889028133486),
761 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0617447837290183627136837688446313313),
762 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0185618495228251406703152962489700468),
763 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00371949406491883508764162050169531013),
764 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000485121708792921297742105775823900772),
765 BOOST_MATH_BIG_CONSTANT(T, 113, 0.376494706741453489892108068231400061e-4),
766 BOOST_MATH_BIG_CONSTANT(T, 113, 0.133166058052466262415271732172490045e-5),
767 };
768 static const T Q[] = {
769 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
770 BOOST_MATH_BIG_CONSTANT(T, 113, 2.32970330146503867261275580968135126),
771 BOOST_MATH_BIG_CONSTANT(T, 113, 2.46325715420422771961250513514928746),
772 BOOST_MATH_BIG_CONSTANT(T, 113, 1.55307882560757679068505047390857842),
773 BOOST_MATH_BIG_CONSTANT(T, 113, 0.644274289865972449441174485441409076),
774 BOOST_MATH_BIG_CONSTANT(T, 113, 0.182609091063258208068606847453955649),
775 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0354171651271241474946129665801606795),
776 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00454060370165285246451879969534083997),
777 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000349871943711566546821198612518656486),
778 BOOST_MATH_BIG_CONSTANT(T, 113, 0.123749319840299552925421880481085392e-4),
779 };
780 result = Y + tools::evaluate_polynomial(P, T(z - 1.0f)) / tools::evaluate_polynomial(Q, T(z - 1.0f));
781 T hi, lo;
782 int expon;
783 hi = floor(ldexp(frexp(z, &expon), 56));
784 hi = ldexp(hi, expon - 56);
785 lo = z - hi;
786 T sq = z * z;
787 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
788 result *= exp(-sq) * exp(-err_sqr) / z;
789 }
790 else if(z < 2.25)
791 {
792 // Maximum Deviation Found: 1.418e-35
793 // Expected Error Term: 1.418e-35
794 // Maximum Relative Change in Control Points: 1.316e-04
795 // Max Error found at long double precision = 1.998462e-35
796 static const T Y = 0.50250148773193359375f;
797 static const T P[] = {
798 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0201233630504573402185161184151016606),
799 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0331864357574860196516686996302305002),
800 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0716562720864787193337475444413405461),
801 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0545835322082103985114927569724880658),
802 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0236692635189696678976549720784989593),
803 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00656970902163248872837262539337601845),
804 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00120282643299089441390490459256235021),
805 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000142123229065182650020762792081622986),
806 BOOST_MATH_BIG_CONSTANT(T, 113, 0.991531438367015135346716277792989347e-5),
807 BOOST_MATH_BIG_CONSTANT(T, 113, 0.312857043762117596999398067153076051e-6),
808 };
809 static const T Q[] = {
810 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
811 BOOST_MATH_BIG_CONSTANT(T, 113, 2.13506082409097783827103424943508554),
812 BOOST_MATH_BIG_CONSTANT(T, 113, 2.06399257267556230937723190496806215),
813 BOOST_MATH_BIG_CONSTANT(T, 113, 1.18678481279932541314830499880691109),
814 BOOST_MATH_BIG_CONSTANT(T, 113, 0.447733186643051752513538142316799562),
815 BOOST_MATH_BIG_CONSTANT(T, 113, 0.11505680005657879437196953047542148),
816 BOOST_MATH_BIG_CONSTANT(T, 113, 0.020163993632192726170219663831914034),
817 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00232708971840141388847728782209730585),
818 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000160733201627963528519726484608224112),
819 BOOST_MATH_BIG_CONSTANT(T, 113, 0.507158721790721802724402992033269266e-5),
820 BOOST_MATH_BIG_CONSTANT(T, 113, 0.18647774409821470950544212696270639e-12),
821 };
822 result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
823 T hi, lo;
824 int expon;
825 hi = floor(ldexp(frexp(z, &expon), 56));
826 hi = ldexp(hi, expon - 56);
827 lo = z - hi;
828 T sq = z * z;
829 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
830 result *= exp(-sq) * exp(-err_sqr) / z;
831 }
832 else if (z < 3)
833 {
834 // Maximum Deviation Found: 3.575e-36
835 // Expected Error Term: 3.575e-36
836 // Maximum Relative Change in Control Points: 7.103e-05
837 // Max Error found at long double precision = 5.794737e-36
838 static const T Y = 0.52896785736083984375f;
839 static const T P[] = {
840 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00902152521745813634562524098263360074),
841 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0145207142776691539346923710537580927),
842 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0301681239582193983824211995978678571),
843 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0215548540823305814379020678660434461),
844 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00864683476267958365678294164340749949),
845 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00219693096885585491739823283511049902),
846 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000364961639163319762492184502159894371),
847 BOOST_MATH_BIG_CONSTANT(T, 113, 0.388174251026723752769264051548703059e-4),
848 BOOST_MATH_BIG_CONSTANT(T, 113, 0.241918026931789436000532513553594321e-5),
849 BOOST_MATH_BIG_CONSTANT(T, 113, 0.676586625472423508158937481943649258e-7),
850 };
851 static const T Q[] = {
852 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
853 BOOST_MATH_BIG_CONSTANT(T, 113, 1.93669171363907292305550231764920001),
854 BOOST_MATH_BIG_CONSTANT(T, 113, 1.69468476144051356810672506101377494),
855 BOOST_MATH_BIG_CONSTANT(T, 113, 0.880023580986436640372794392579985511),
856 BOOST_MATH_BIG_CONSTANT(T, 113, 0.299099106711315090710836273697708402),
857 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0690593962363545715997445583603382337),
858 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0108427016361318921960863149875360222),
859 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00111747247208044534520499324234317695),
860 BOOST_MATH_BIG_CONSTANT(T, 113, 0.686843205749767250666787987163701209e-4),
861 BOOST_MATH_BIG_CONSTANT(T, 113, 0.192093541425429248675532015101904262e-5),
862 };
863 result = Y + tools::evaluate_polynomial(P, T(z - 2.25f)) / tools::evaluate_polynomial(Q, T(z - 2.25f));
864 T hi, lo;
865 int expon;
866 hi = floor(ldexp(frexp(z, &expon), 56));
867 hi = ldexp(hi, expon - 56);
868 lo = z - hi;
869 T sq = z * z;
870 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
871 result *= exp(-sq) * exp(-err_sqr) / z;
872 }
873 else if(z < 3.5)
874 {
875 // Maximum Deviation Found: 8.126e-37
876 // Expected Error Term: -8.126e-37
877 // Maximum Relative Change in Control Points: 1.363e-04
878 // Max Error found at long double precision = 1.747062e-36
879 static const T Y = 0.54037380218505859375f;
880 static const T P[] = {
881 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033703486408887424921155540591370375),
882 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0104948043110005245215286678898115811),
883 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0148530118504000311502310457390417795),
884 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00816693029245443090102738825536188916),
885 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00249716579989140882491939681805594585),
886 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0004655591010047353023978045800916647),
887 BOOST_MATH_BIG_CONSTANT(T, 113, 0.531129557920045295895085236636025323e-4),
888 BOOST_MATH_BIG_CONSTANT(T, 113, 0.343526765122727069515775194111741049e-5),
889 BOOST_MATH_BIG_CONSTANT(T, 113, 0.971120407556888763695313774578711839e-7),
890 };
891 static const T Q[] = {
892 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
893 BOOST_MATH_BIG_CONSTANT(T, 113, 1.59911256167540354915906501335919317),
894 BOOST_MATH_BIG_CONSTANT(T, 113, 1.136006830764025173864831382946934),
895 BOOST_MATH_BIG_CONSTANT(T, 113, 0.468565867990030871678574840738423023),
896 BOOST_MATH_BIG_CONSTANT(T, 113, 0.122821824954470343413956476900662236),
897 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0209670914950115943338996513330141633),
898 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00227845718243186165620199012883547257),
899 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000144243326443913171313947613547085553),
900 BOOST_MATH_BIG_CONSTANT(T, 113, 0.407763415954267700941230249989140046e-5),
901 };
902 result = Y + tools::evaluate_polynomial(P, T(z - 3.0f)) / tools::evaluate_polynomial(Q, T(z - 3.0f));
903 T hi, lo;
904 int expon;
905 hi = floor(ldexp(frexp(z, &expon), 56));
906 hi = ldexp(hi, expon - 56);
907 lo = z - hi;
908 T sq = z * z;
909 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
910 result *= exp(-sq) * exp(-err_sqr) / z;
911 }
912 else if(z < 5.5)
913 {
914 // Maximum Deviation Found: 5.804e-36
915 // Expected Error Term: -5.803e-36
916 // Maximum Relative Change in Control Points: 2.475e-05
917 // Max Error found at long double precision = 1.349545e-35
918 static const T Y = 0.55000019073486328125f;
919 static const T P[] = {
920 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00118142849742309772151454518093813615),
921 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0072201822885703318172366893469382745),
922 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0078782276276860110721875733778481505),
923 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00418229166204362376187593976656261146),
924 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00134198400587769200074194304298642705),
925 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000283210387078004063264777611497435572),
926 BOOST_MATH_BIG_CONSTANT(T, 113, 0.405687064094911866569295610914844928e-4),
927 BOOST_MATH_BIG_CONSTANT(T, 113, 0.39348283801568113807887364414008292e-5),
928 BOOST_MATH_BIG_CONSTANT(T, 113, 0.248798540917787001526976889284624449e-6),
929 BOOST_MATH_BIG_CONSTANT(T, 113, 0.929502490223452372919607105387474751e-8),
930 BOOST_MATH_BIG_CONSTANT(T, 113, 0.156161469668275442569286723236274457e-9),
931 };
932 static const T Q[] = {
933 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
934 BOOST_MATH_BIG_CONSTANT(T, 113, 1.52955245103668419479878456656709381),
935 BOOST_MATH_BIG_CONSTANT(T, 113, 1.06263944820093830054635017117417064),
936 BOOST_MATH_BIG_CONSTANT(T, 113, 0.441684612681607364321013134378316463),
937 BOOST_MATH_BIG_CONSTANT(T, 113, 0.121665258426166960049773715928906382),
938 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0232134512374747691424978642874321434),
939 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00310778180686296328582860464875562636),
940 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000288361770756174705123674838640161693),
941 BOOST_MATH_BIG_CONSTANT(T, 113, 0.177529187194133944622193191942300132e-4),
942 BOOST_MATH_BIG_CONSTANT(T, 113, 0.655068544833064069223029299070876623e-6),
943 BOOST_MATH_BIG_CONSTANT(T, 113, 0.11005507545746069573608988651927452e-7),
944 };
945 result = Y + tools::evaluate_polynomial(P, T(z - 4.5f)) / tools::evaluate_polynomial(Q, T(z - 4.5f));
946 T hi, lo;
947 int expon;
948 hi = floor(ldexp(frexp(z, &expon), 56));
949 hi = ldexp(hi, expon - 56);
950 lo = z - hi;
951 T sq = z * z;
952 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
953 result *= exp(-sq) * exp(-err_sqr) / z;
954 }
955 else if(z < 7.5)
956 {
957 // Maximum Deviation Found: 1.007e-36
958 // Expected Error Term: 1.007e-36
959 // Maximum Relative Change in Control Points: 1.027e-03
960 // Max Error found at long double precision = 2.646420e-36
961 static const T Y = 0.5574436187744140625f;
962 static const T P[] = {
963 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000293236907400849056269309713064107674),
964 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00225110719535060642692275221961480162),
965 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190984458121502831421717207849429799),
966 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000747757733460111743833929141001680706),
967 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000170663175280949889583158597373928096),
968 BOOST_MATH_BIG_CONSTANT(T, 113, 0.246441188958013822253071608197514058e-4),
969 BOOST_MATH_BIG_CONSTANT(T, 113, 0.229818000860544644974205957895688106e-5),
970 BOOST_MATH_BIG_CONSTANT(T, 113, 0.134886977703388748488480980637704864e-6),
971 BOOST_MATH_BIG_CONSTANT(T, 113, 0.454764611880548962757125070106650958e-8),
972 BOOST_MATH_BIG_CONSTANT(T, 113, 0.673002744115866600294723141176820155e-10),
973 };
974 static const T Q[] = {
975 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
976 BOOST_MATH_BIG_CONSTANT(T, 113, 1.12843690320861239631195353379313367),
977 BOOST_MATH_BIG_CONSTANT(T, 113, 0.569900657061622955362493442186537259),
978 BOOST_MATH_BIG_CONSTANT(T, 113, 0.169094404206844928112348730277514273),
979 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0324887449084220415058158657252147063),
980 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00419252877436825753042680842608219552),
981 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00036344133176118603523976748563178578),
982 BOOST_MATH_BIG_CONSTANT(T, 113, 0.204123895931375107397698245752850347e-4),
983 BOOST_MATH_BIG_CONSTANT(T, 113, 0.674128352521481412232785122943508729e-6),
984 BOOST_MATH_BIG_CONSTANT(T, 113, 0.997637501418963696542159244436245077e-8),
985 };
986 result = Y + tools::evaluate_polynomial(P, T(z - 6.5f)) / tools::evaluate_polynomial(Q, T(z - 6.5f));
987 T hi, lo;
988 int expon;
989 hi = floor(ldexp(frexp(z, &expon), 56));
990 hi = ldexp(hi, expon - 56);
991 lo = z - hi;
992 T sq = z * z;
993 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
994 result *= exp(-sq) * exp(-err_sqr) / z;
995 }
996 else if(z < 11.5)
997 {
998 // Maximum Deviation Found: 8.380e-36
999 // Expected Error Term: 8.380e-36
1000 // Maximum Relative Change in Control Points: 2.632e-06
1001 // Max Error found at long double precision = 9.849522e-36
1002 static const T Y = 0.56083202362060546875f;
1003 static const T P[] = {
1004 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000282420728751494363613829834891390121),
1005 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00175387065018002823433704079355125161),
1006 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0021344978564889819420775336322920375),
1007 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00124151356560137532655039683963075661),
1008 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000423600733566948018555157026862139644),
1009 BOOST_MATH_BIG_CONSTANT(T, 113, 0.914030340865175237133613697319509698e-4),
1010 BOOST_MATH_BIG_CONSTANT(T, 113, 0.126999927156823363353809747017945494e-4),
1011 BOOST_MATH_BIG_CONSTANT(T, 113, 0.110610959842869849776179749369376402e-5),
1012 BOOST_MATH_BIG_CONSTANT(T, 113, 0.55075079477173482096725348704634529e-7),
1013 BOOST_MATH_BIG_CONSTANT(T, 113, 0.119735694018906705225870691331543806e-8),
1014 };
1015 static const T Q[] = {
1016 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1017 BOOST_MATH_BIG_CONSTANT(T, 113, 1.69889613396167354566098060039549882),
1018 BOOST_MATH_BIG_CONSTANT(T, 113, 1.28824647372749624464956031163282674),
1019 BOOST_MATH_BIG_CONSTANT(T, 113, 0.572297795434934493541628008224078717),
1020 BOOST_MATH_BIG_CONSTANT(T, 113, 0.164157697425571712377043857240773164),
1021 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0315311145224594430281219516531649562),
1022 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00405588922155632380812945849777127458),
1023 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000336929033691445666232029762868642417),
1024 BOOST_MATH_BIG_CONSTANT(T, 113, 0.164033049810404773469413526427932109e-4),
1025 BOOST_MATH_BIG_CONSTANT(T, 113, 0.356615210500531410114914617294694857e-6),
1026 };
1027 result = Y + tools::evaluate_polynomial(P, T(z / 2 - 4.75f)) / tools::evaluate_polynomial(Q, T(z / 2 - 4.75f));
1028 T hi, lo;
1029 int expon;
1030 hi = floor(ldexp(frexp(z, &expon), 56));
1031 hi = ldexp(hi, expon - 56);
1032 lo = z - hi;
1033 T sq = z * z;
1034 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
1035 result *= exp(-sq) * exp(-err_sqr) / z;
1036 }
1037 else
1038 {
1039 // Maximum Deviation Found: 1.132e-35
1040 // Expected Error Term: -1.132e-35
1041 // Maximum Relative Change in Control Points: 4.674e-04
1042 // Max Error found at long double precision = 1.162590e-35
1043 static const T Y = 0.5632686614990234375f;
1044 static const T P[] = {
1045 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000920922048732849448079451574171836943),
1046 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00321439044532288750501700028748922439),
1047 BOOST_MATH_BIG_CONSTANT(T, 113, -0.250455263029390118657884864261823431),
1048 BOOST_MATH_BIG_CONSTANT(T, 113, -0.906807635364090342031792404764598142),
1049 BOOST_MATH_BIG_CONSTANT(T, 113, -8.92233572835991735876688745989985565),
1050 BOOST_MATH_BIG_CONSTANT(T, 113, -21.7797433494422564811782116907878495),
1051 BOOST_MATH_BIG_CONSTANT(T, 113, -91.1451915251976354349734589601171659),
1052 BOOST_MATH_BIG_CONSTANT(T, 113, -144.1279109655993927069052125017673),
1053 BOOST_MATH_BIG_CONSTANT(T, 113, -313.845076581796338665519022313775589),
1054 BOOST_MATH_BIG_CONSTANT(T, 113, -273.11378811923343424081101235736475),
1055 BOOST_MATH_BIG_CONSTANT(T, 113, -271.651566205951067025696102600443452),
1056 BOOST_MATH_BIG_CONSTANT(T, 113, -60.0530577077238079968843307523245547),
1057 };
1058 static const T Q[] = {
1059 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1060 BOOST_MATH_BIG_CONSTANT(T, 113, 3.49040448075464744191022350947892036),
1061 BOOST_MATH_BIG_CONSTANT(T, 113, 34.3563592467165971295915749548313227),
1062 BOOST_MATH_BIG_CONSTANT(T, 113, 84.4993232033879023178285731843850461),
1063 BOOST_MATH_BIG_CONSTANT(T, 113, 376.005865281206894120659401340373818),
1064 BOOST_MATH_BIG_CONSTANT(T, 113, 629.95369438888946233003926191755125),
1065 BOOST_MATH_BIG_CONSTANT(T, 113, 1568.35771983533158591604513304269098),
1066 BOOST_MATH_BIG_CONSTANT(T, 113, 1646.02452040831961063640827116581021),
1067 BOOST_MATH_BIG_CONSTANT(T, 113, 2299.96860633240298708910425594484895),
1068 BOOST_MATH_BIG_CONSTANT(T, 113, 1222.73204392037452750381340219906374),
1069 BOOST_MATH_BIG_CONSTANT(T, 113, 799.359797306084372350264298361110448),
1070 BOOST_MATH_BIG_CONSTANT(T, 113, 72.7415265778588087243442792401576737),
1071 };
1072 result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
1073 T hi, lo;
1074 int expon;
1075 hi = floor(ldexp(frexp(z, &expon), 56));
1076 hi = ldexp(hi, expon - 56);
1077 lo = z - hi;
1078 T sq = z * z;
1079 T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
1080 result *= exp(-sq) * exp(-err_sqr) / z;
1081 }
1082 }
1083 else
1084 {
1085 //
1086 // Any value of z larger than 110 will underflow to zero:
1087 //
1088 result = 0;
1089 invert = !invert;
1090 }
1091
1092 if(invert)
1093 {
1094 result = 1 - result;
1095 }
1096
1097 return result;
1098 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 113>& t)
1099
1100 template <class T, class Policy, class tag>
1101 struct erf_initializer
1102 {
1103 struct init
1104 {
initboost::math::detail::erf_initializer::init1105 init()
1106 {
1107 do_init(tag());
1108 }
do_initboost::math::detail::erf_initializer::init1109 static void do_init(const boost::integral_constant<int, 0>&){}
do_initboost::math::detail::erf_initializer::init1110 static void do_init(const boost::integral_constant<int, 53>&)
1111 {
1112 boost::math::erf(static_cast<T>(1e-12), Policy());
1113 boost::math::erf(static_cast<T>(0.25), Policy());
1114 boost::math::erf(static_cast<T>(1.25), Policy());
1115 boost::math::erf(static_cast<T>(2.25), Policy());
1116 boost::math::erf(static_cast<T>(4.25), Policy());
1117 boost::math::erf(static_cast<T>(5.25), Policy());
1118 }
do_initboost::math::detail::erf_initializer::init1119 static void do_init(const boost::integral_constant<int, 64>&)
1120 {
1121 boost::math::erf(static_cast<T>(1e-12), Policy());
1122 boost::math::erf(static_cast<T>(0.25), Policy());
1123 boost::math::erf(static_cast<T>(1.25), Policy());
1124 boost::math::erf(static_cast<T>(2.25), Policy());
1125 boost::math::erf(static_cast<T>(4.25), Policy());
1126 boost::math::erf(static_cast<T>(5.25), Policy());
1127 }
do_initboost::math::detail::erf_initializer::init1128 static void do_init(const boost::integral_constant<int, 113>&)
1129 {
1130 boost::math::erf(static_cast<T>(1e-22), Policy());
1131 boost::math::erf(static_cast<T>(0.25), Policy());
1132 boost::math::erf(static_cast<T>(1.25), Policy());
1133 boost::math::erf(static_cast<T>(2.125), Policy());
1134 boost::math::erf(static_cast<T>(2.75), Policy());
1135 boost::math::erf(static_cast<T>(3.25), Policy());
1136 boost::math::erf(static_cast<T>(5.25), Policy());
1137 boost::math::erf(static_cast<T>(7.25), Policy());
1138 boost::math::erf(static_cast<T>(11.25), Policy());
1139 boost::math::erf(static_cast<T>(12.5), Policy());
1140 }
force_instantiateboost::math::detail::erf_initializer::init1141 void force_instantiate()const{}
1142 };
1143 static const init initializer;
force_instantiateboost::math::detail::erf_initializer1144 static void force_instantiate()
1145 {
1146 initializer.force_instantiate();
1147 }
1148 };
1149
1150 template <class T, class Policy, class tag>
1151 const typename erf_initializer<T, Policy, tag>::init erf_initializer<T, Policy, tag>::initializer;
1152
1153 } // namespace detail
1154
1155 template <class T, class Policy>
erf(T z,const Policy &)1156 inline typename tools::promote_args<T>::type erf(T z, const Policy& /* pol */)
1157 {
1158 typedef typename tools::promote_args<T>::type result_type;
1159 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1160 typedef typename policies::precision<result_type, Policy>::type precision_type;
1161 typedef typename policies::normalise<
1162 Policy,
1163 policies::promote_float<false>,
1164 policies::promote_double<false>,
1165 policies::discrete_quantile<>,
1166 policies::assert_undefined<> >::type forwarding_policy;
1167
1168 BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1169 BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1170 BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1171
1172 typedef boost::integral_constant<int,
1173 precision_type::value <= 0 ? 0 :
1174 precision_type::value <= 53 ? 53 :
1175 precision_type::value <= 64 ? 64 :
1176 precision_type::value <= 113 ? 113 : 0
1177 > tag_type;
1178
1179 BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1180
1181 detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1182
1183 return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1184 static_cast<value_type>(z),
1185 false,
1186 forwarding_policy(),
1187 tag_type()), "boost::math::erf<%1%>(%1%, %1%)");
1188 }
1189
1190 template <class T, class Policy>
erfc(T z,const Policy &)1191 inline typename tools::promote_args<T>::type erfc(T z, const Policy& /* pol */)
1192 {
1193 typedef typename tools::promote_args<T>::type result_type;
1194 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1195 typedef typename policies::precision<result_type, Policy>::type precision_type;
1196 typedef typename policies::normalise<
1197 Policy,
1198 policies::promote_float<false>,
1199 policies::promote_double<false>,
1200 policies::discrete_quantile<>,
1201 policies::assert_undefined<> >::type forwarding_policy;
1202
1203 BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1204 BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1205 BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1206
1207 typedef boost::integral_constant<int,
1208 precision_type::value <= 0 ? 0 :
1209 precision_type::value <= 53 ? 53 :
1210 precision_type::value <= 64 ? 64 :
1211 precision_type::value <= 113 ? 113 : 0
1212 > tag_type;
1213
1214 BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1215
1216 detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1217
1218 return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1219 static_cast<value_type>(z),
1220 true,
1221 forwarding_policy(),
1222 tag_type()), "boost::math::erfc<%1%>(%1%, %1%)");
1223 }
1224
1225 template <class T>
erf(T z)1226 inline typename tools::promote_args<T>::type erf(T z)
1227 {
1228 return boost::math::erf(z, policies::policy<>());
1229 }
1230
1231 template <class T>
erfc(T z)1232 inline typename tools::promote_args<T>::type erfc(T z)
1233 {
1234 return boost::math::erfc(z, policies::policy<>());
1235 }
1236
1237 } // namespace math
1238 } // namespace boost
1239
1240 #include <boost/math/special_functions/detail/erf_inv.hpp>
1241
1242 #endif // BOOST_MATH_SPECIAL_ERF_HPP
1243
1244
1245
1246
1247