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)
6 #ifndef BOOST_MATH_SF_ERF_INV_HPP
7 #define BOOST_MATH_SF_ERF_INV_HPP
13 namespace boost{ namespace math{
17 // The inverse erf and erfc functions share a common implementation,
18 // this version is for 80-bit long double's and smaller:
20 template <class T, class Policy>
21 T erf_inv_imp(const T& p, const T& q, const Policy&, const boost::mpl::int_<64>*)
23 BOOST_MATH_STD_USING // for ADL of std names.
30 // Evaluate inverse erf using the rational approximation:
32 // x = p(p+10)(Y+R(p))
34 // Where Y is a constant, and R(p) is optimised for a low
35 // absolute error compared to |Y|.
37 // double: Max error found: 2.001849e-18
38 // long double: Max error found: 1.017064e-20
39 // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
41 static const float Y = 0.0891314744949340820313f;
42 static const T P[] = {
43 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000508781949658280665617),
44 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00836874819741736770379),
45 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0334806625409744615033),
46 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0126926147662974029034),
47 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0365637971411762664006),
48 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0219878681111168899165),
49 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00822687874676915743155),
50 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00538772965071242932965)
52 static const T Q[] = {
53 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
54 BOOST_MATH_BIG_CONSTANT(T, 64, -0.970005043303290640362),
55 BOOST_MATH_BIG_CONSTANT(T, 64, -1.56574558234175846809),
56 BOOST_MATH_BIG_CONSTANT(T, 64, 1.56221558398423026363),
57 BOOST_MATH_BIG_CONSTANT(T, 64, 0.662328840472002992063),
58 BOOST_MATH_BIG_CONSTANT(T, 64, -0.71228902341542847553),
59 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0527396382340099713954),
60 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0795283687341571680018),
61 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00233393759374190016776),
62 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000886216390456424707504)
65 T r = tools::evaluate_polynomial(P, p) / tools::evaluate_polynomial(Q, p);
66 result = g * Y + g * r;
71 // Rational approximation for 0.5 > q >= 0.25
73 // x = sqrt(-2*log(q)) / (Y + R(q))
75 // Where Y is a constant, and R(q) is optimised for a low
76 // absolute error compared to Y.
78 // double : Max error found: 7.403372e-17
79 // long double : Max error found: 6.084616e-20
80 // Maximum Deviation Found (error term) 4.811e-20
82 static const float Y = 2.249481201171875f;
83 static const T P[] = {
84 BOOST_MATH_BIG_CONSTANT(T, 64, -0.202433508355938759655),
85 BOOST_MATH_BIG_CONSTANT(T, 64, 0.105264680699391713268),
86 BOOST_MATH_BIG_CONSTANT(T, 64, 8.37050328343119927838),
87 BOOST_MATH_BIG_CONSTANT(T, 64, 17.6447298408374015486),
88 BOOST_MATH_BIG_CONSTANT(T, 64, -18.8510648058714251895),
89 BOOST_MATH_BIG_CONSTANT(T, 64, -44.6382324441786960818),
90 BOOST_MATH_BIG_CONSTANT(T, 64, 17.445385985570866523),
91 BOOST_MATH_BIG_CONSTANT(T, 64, 21.1294655448340526258),
92 BOOST_MATH_BIG_CONSTANT(T, 64, -3.67192254707729348546)
94 static const T Q[] = {
95 BOOST_MATH_BIG_CONSTANT(T, 64, 1),
96 BOOST_MATH_BIG_CONSTANT(T, 64, 6.24264124854247537712),
97 BOOST_MATH_BIG_CONSTANT(T, 64, 3.9713437953343869095),
98 BOOST_MATH_BIG_CONSTANT(T, 64, -28.6608180499800029974),
99 BOOST_MATH_BIG_CONSTANT(T, 64, -20.1432634680485188801),
100 BOOST_MATH_BIG_CONSTANT(T, 64, 48.5609213108739935468),
101 BOOST_MATH_BIG_CONSTANT(T, 64, 10.8268667355460159008),
102 BOOST_MATH_BIG_CONSTANT(T, 64, -22.6436933413139721736),
103 BOOST_MATH_BIG_CONSTANT(T, 64, 1.72114765761200282724)
105 T g = sqrt(-2 * log(q));
107 T r = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
108 result = g / (Y + r);
113 // For q < 0.25 we have a series of rational approximations all
114 // of the general form:
116 // let: x = sqrt(-log(q))
118 // Then the result is given by:
122 // where Y is a constant, B is the lowest value of x for which
123 // the approximation is valid, and R(x-B) is optimised for a low
124 // absolute error compared to Y.
126 // Note that almost all code will really go through the first
127 // or maybe second approximation. After than we're dealing with very
128 // small input values indeed: 80 and 128 bit long double's go all the
129 // way down to ~ 1e-5000 so the "tail" is rather long...
134 // Max error found: 1.089051e-20
135 static const float Y = 0.807220458984375f;
136 static const T P[] = {
137 BOOST_MATH_BIG_CONSTANT(T, 64, -0.131102781679951906451),
138 BOOST_MATH_BIG_CONSTANT(T, 64, -0.163794047193317060787),
139 BOOST_MATH_BIG_CONSTANT(T, 64, 0.117030156341995252019),
140 BOOST_MATH_BIG_CONSTANT(T, 64, 0.387079738972604337464),
141 BOOST_MATH_BIG_CONSTANT(T, 64, 0.337785538912035898924),
142 BOOST_MATH_BIG_CONSTANT(T, 64, 0.142869534408157156766),
143 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0290157910005329060432),
144 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00214558995388805277169),
145 BOOST_MATH_BIG_CONSTANT(T, 64, -0.679465575181126350155e-6),
146 BOOST_MATH_BIG_CONSTANT(T, 64, 0.285225331782217055858e-7),
147 BOOST_MATH_BIG_CONSTANT(T, 64, -0.681149956853776992068e-9)
149 static const T Q[] = {
150 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
151 BOOST_MATH_BIG_CONSTANT(T, 64, 3.46625407242567245975),
152 BOOST_MATH_BIG_CONSTANT(T, 64, 5.38168345707006855425),
153 BOOST_MATH_BIG_CONSTANT(T, 64, 4.77846592945843778382),
154 BOOST_MATH_BIG_CONSTANT(T, 64, 2.59301921623620271374),
155 BOOST_MATH_BIG_CONSTANT(T, 64, 0.848854343457902036425),
156 BOOST_MATH_BIG_CONSTANT(T, 64, 0.152264338295331783612),
157 BOOST_MATH_BIG_CONSTANT(T, 64, 0.01105924229346489121)
160 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
161 result = Y * x + R * x;
165 // Max error found: 8.389174e-21
166 static const float Y = 0.93995571136474609375f;
167 static const T P[] = {
168 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0350353787183177984712),
169 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00222426529213447927281),
170 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0185573306514231072324),
171 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00950804701325919603619),
172 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00187123492819559223345),
173 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000157544617424960554631),
174 BOOST_MATH_BIG_CONSTANT(T, 64, 0.460469890584317994083e-5),
175 BOOST_MATH_BIG_CONSTANT(T, 64, -0.230404776911882601748e-9),
176 BOOST_MATH_BIG_CONSTANT(T, 64, 0.266339227425782031962e-11)
178 static const T Q[] = {
179 BOOST_MATH_BIG_CONSTANT(T, 64, 1),
180 BOOST_MATH_BIG_CONSTANT(T, 64, 1.3653349817554063097),
181 BOOST_MATH_BIG_CONSTANT(T, 64, 0.762059164553623404043),
182 BOOST_MATH_BIG_CONSTANT(T, 64, 0.220091105764131249824),
183 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0341589143670947727934),
184 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00263861676657015992959),
185 BOOST_MATH_BIG_CONSTANT(T, 64, 0.764675292302794483503e-4)
188 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
189 result = Y * x + R * x;
193 // Max error found: 1.481312e-19
194 static const float Y = 0.98362827301025390625f;
195 static const T P[] = {
196 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0167431005076633737133),
197 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00112951438745580278863),
198 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00105628862152492910091),
199 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000209386317487588078668),
200 BOOST_MATH_BIG_CONSTANT(T, 64, 0.149624783758342370182e-4),
201 BOOST_MATH_BIG_CONSTANT(T, 64, 0.449696789927706453732e-6),
202 BOOST_MATH_BIG_CONSTANT(T, 64, 0.462596163522878599135e-8),
203 BOOST_MATH_BIG_CONSTANT(T, 64, -0.281128735628831791805e-13),
204 BOOST_MATH_BIG_CONSTANT(T, 64, 0.99055709973310326855e-16)
206 static const T Q[] = {
207 BOOST_MATH_BIG_CONSTANT(T, 64, 1),
208 BOOST_MATH_BIG_CONSTANT(T, 64, 0.591429344886417493481),
209 BOOST_MATH_BIG_CONSTANT(T, 64, 0.138151865749083321638),
210 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0160746087093676504695),
211 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000964011807005165528527),
212 BOOST_MATH_BIG_CONSTANT(T, 64, 0.275335474764726041141e-4),
213 BOOST_MATH_BIG_CONSTANT(T, 64, 0.282243172016108031869e-6)
216 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
217 result = Y * x + R * x;
221 // Max error found: 5.697761e-20
222 static const float Y = 0.99714565277099609375f;
223 static const T P[] = {
224 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0024978212791898131227),
225 BOOST_MATH_BIG_CONSTANT(T, 64, -0.779190719229053954292e-5),
226 BOOST_MATH_BIG_CONSTANT(T, 64, 0.254723037413027451751e-4),
227 BOOST_MATH_BIG_CONSTANT(T, 64, 0.162397777342510920873e-5),
228 BOOST_MATH_BIG_CONSTANT(T, 64, 0.396341011304801168516e-7),
229 BOOST_MATH_BIG_CONSTANT(T, 64, 0.411632831190944208473e-9),
230 BOOST_MATH_BIG_CONSTANT(T, 64, 0.145596286718675035587e-11),
231 BOOST_MATH_BIG_CONSTANT(T, 64, -0.116765012397184275695e-17)
233 static const T Q[] = {
234 BOOST_MATH_BIG_CONSTANT(T, 64, 1),
235 BOOST_MATH_BIG_CONSTANT(T, 64, 0.207123112214422517181),
236 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0169410838120975906478),
237 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000690538265622684595676),
238 BOOST_MATH_BIG_CONSTANT(T, 64, 0.145007359818232637924e-4),
239 BOOST_MATH_BIG_CONSTANT(T, 64, 0.144437756628144157666e-6),
240 BOOST_MATH_BIG_CONSTANT(T, 64, 0.509761276599778486139e-9)
243 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
244 result = Y * x + R * x;
248 // Max error found: 1.279746e-20
249 static const float Y = 0.99941349029541015625f;
250 static const T P[] = {
251 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000539042911019078575891),
252 BOOST_MATH_BIG_CONSTANT(T, 64, -0.28398759004727721098e-6),
253 BOOST_MATH_BIG_CONSTANT(T, 64, 0.899465114892291446442e-6),
254 BOOST_MATH_BIG_CONSTANT(T, 64, 0.229345859265920864296e-7),
255 BOOST_MATH_BIG_CONSTANT(T, 64, 0.225561444863500149219e-9),
256 BOOST_MATH_BIG_CONSTANT(T, 64, 0.947846627503022684216e-12),
257 BOOST_MATH_BIG_CONSTANT(T, 64, 0.135880130108924861008e-14),
258 BOOST_MATH_BIG_CONSTANT(T, 64, -0.348890393399948882918e-21)
260 static const T Q[] = {
261 BOOST_MATH_BIG_CONSTANT(T, 64, 1),
262 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0845746234001899436914),
263 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00282092984726264681981),
264 BOOST_MATH_BIG_CONSTANT(T, 64, 0.468292921940894236786e-4),
265 BOOST_MATH_BIG_CONSTANT(T, 64, 0.399968812193862100054e-6),
266 BOOST_MATH_BIG_CONSTANT(T, 64, 0.161809290887904476097e-8),
267 BOOST_MATH_BIG_CONSTANT(T, 64, 0.231558608310259605225e-11)
270 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
271 result = Y * x + R * x;
277 template <class T, class Policy>
280 boost::math::tuple<T,T,T> operator()(const T& guess)
283 T derivative = sign * (2 / sqrt(constants::pi<T>())) * exp(-(guess * guess));
284 T derivative2 = -2 * guess * derivative;
285 return boost::math::make_tuple(((sign > 0) ? static_cast<T>(boost::math::erf(guess, Policy()) - target) : static_cast<T>(boost::math::erfc(guess, Policy())) - target), derivative, derivative2);
287 erf_roots(T z, int s) : target(z), sign(s) {}
293 template <class T, class Policy>
294 T erf_inv_imp(const T& p, const T& q, const Policy& pol, const boost::mpl::int_<0>*)
297 // Generic version, get a guess that's accurate to 64-bits (10^-19)
299 T guess = erf_inv_imp(p, q, pol, static_cast<mpl::int_<64> const*>(0));
302 // If T has more bit's than 64 in it's mantissa then we need to iterate,
303 // otherwise we can just return the result:
305 if(policies::digits<T, Policy>() > 64)
307 boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
310 result = tools::halley_iterate(detail::erf_roots<typename remove_cv<T>::type, Policy>(p, 1), guess, static_cast<T>(0), tools::max_value<T>(), (policies::digits<T, Policy>() * 2) / 3, max_iter);
314 result = tools::halley_iterate(detail::erf_roots<typename remove_cv<T>::type, Policy>(q, -1), guess, static_cast<T>(0), tools::max_value<T>(), (policies::digits<T, Policy>() * 2) / 3, max_iter);
316 policies::check_root_iterations<T>("boost::math::erf_inv<%1%>", max_iter, pol);
325 template <class T, class Policy>
326 struct erf_inv_initializer
334 static void do_init()
336 boost::math::erf_inv(static_cast<T>(0.25), Policy());
337 boost::math::erf_inv(static_cast<T>(0.55), Policy());
338 boost::math::erf_inv(static_cast<T>(0.95), Policy());
339 boost::math::erfc_inv(static_cast<T>(1e-15), Policy());
340 if(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)) != 0)
341 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)), Policy());
343 // Some compilers choke on constants that would underflow, even in code that isn't instantiated
344 // so try and filter these cases out in the preprocessor:
345 #if LDBL_MAX_10_EXP >= 800
346 if(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)) != 0)
347 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)), Policy());
348 if(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)) != 0)
349 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)), Policy());
351 if(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)) != 0)
352 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)), Policy());
353 if(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)) != 0)
354 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)), Policy());
357 void force_instantiate()const{}
359 static const init initializer;
360 static void force_instantiate()
362 initializer.force_instantiate();
366 template <class T, class Policy>
367 const typename erf_inv_initializer<T, Policy>::init erf_inv_initializer<T, Policy>::initializer;
369 } // namespace detail
371 template <class T, class Policy>
372 typename tools::promote_args<T>::type erfc_inv(T z, const Policy& pol)
374 typedef typename tools::promote_args<T>::type result_type;
377 // Begin by testing for domain errors, and other special cases:
379 static const char* function = "boost::math::erfc_inv<%1%>(%1%, %1%)";
380 if((z < 0) || (z > 2))
381 policies::raise_domain_error<result_type>(function, "Argument outside range [0,2] in inverse erfc function (got p=%1%).", z, pol);
383 return policies::raise_overflow_error<result_type>(function, 0, pol);
385 return -policies::raise_overflow_error<result_type>(function, 0, pol);
387 // Normalise the input, so it's in the range [0,1], we will
388 // negate the result if z is outside that range. This is a simple
389 // application of the erfc reflection formula: erfc(-z) = 2 - erfc(z)
405 // A bit of meta-programming to figure out which implementation
406 // to use, based on the number of bits in the mantissa of T:
408 typedef typename policies::precision<result_type, Policy>::type precision_type;
409 typedef typename mpl::if_<
410 mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
415 // Likewise use internal promotion, so we evaluate at a higher
416 // precision internally if it's appropriate:
418 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
419 typedef typename policies::normalise<
421 policies::promote_float<false>,
422 policies::promote_double<false>,
423 policies::discrete_quantile<>,
424 policies::assert_undefined<> >::type forwarding_policy;
426 detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
429 // And get the result, negating where required:
431 return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
432 detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
435 template <class T, class Policy>
436 typename tools::promote_args<T>::type erf_inv(T z, const Policy& pol)
438 typedef typename tools::promote_args<T>::type result_type;
441 // Begin by testing for domain errors, and other special cases:
443 static const char* function = "boost::math::erf_inv<%1%>(%1%, %1%)";
444 if((z < -1) || (z > 1))
445 policies::raise_domain_error<result_type>(function, "Argument outside range [-1, 1] in inverse erf function (got p=%1%).", z, pol);
447 return policies::raise_overflow_error<result_type>(function, 0, pol);
449 return -policies::raise_overflow_error<result_type>(function, 0, pol);
453 // Normalise the input, so it's in the range [0,1], we will
454 // negate the result if z is outside that range. This is a simple
455 // application of the erf reflection formula: erf(-z) = -erf(z)
471 // A bit of meta-programming to figure out which implementation
472 // to use, based on the number of bits in the mantissa of T:
474 typedef typename policies::precision<result_type, Policy>::type precision_type;
475 typedef typename mpl::if_<
476 mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
481 // Likewise use internal promotion, so we evaluate at a higher
482 // precision internally if it's appropriate:
484 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
485 typedef typename policies::normalise<
487 policies::promote_float<false>,
488 policies::promote_double<false>,
489 policies::discrete_quantile<>,
490 policies::assert_undefined<> >::type forwarding_policy;
492 // Likewise use internal promotion, so we evaluate at a higher
493 // precision internally if it's appropriate:
495 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
497 detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
499 // And get the result, negating where required:
501 return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
502 detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
506 inline typename tools::promote_args<T>::type erfc_inv(T z)
508 return erfc_inv(z, policies::policy<>());
512 inline typename tools::promote_args<T>::type erf_inv(T z)
514 return erf_inv(z, policies::policy<>());
520 #endif // BOOST_MATH_SF_ERF_INV_HPP