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_SF_ERF_INV_HPP
7 #define BOOST_MATH_SF_ERF_INV_HPP
8
9 #ifdef _MSC_VER
10 #pragma once
11 #pragma warning(push)
12 #pragma warning(disable:4127) // Conditional expression is constant
13 #pragma warning(disable:4702) // Unreachable code: optimization warning
14 #endif
15
16 namespace boost{ namespace math{
17
18 namespace detail{
19 //
20 // The inverse erf and erfc functions share a common implementation,
21 // this version is for 80-bit long double's and smaller:
22 //
23 template <class T, class Policy>
24 T erf_inv_imp(const T& p, const T& q, const Policy&, const boost::mpl::int_<64>*)
25 {
26 BOOST_MATH_STD_USING // for ADL of std names.
27
28 T result = 0;
29
30 if(p <= 0.5)
31 {
32 //
33 // Evaluate inverse erf using the rational approximation:
34 //
35 // x = p(p+10)(Y+R(p))
36 //
37 // Where Y is a constant, and R(p) is optimised for a low
38 // absolute error compared to |Y|.
39 //
40 // double: Max error found: 2.001849e-18
41 // long double: Max error found: 1.017064e-20
42 // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
43 //
44 static const float Y = 0.0891314744949340820313f;
45 static const T P[] = {
46 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000508781949658280665617),
47 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00836874819741736770379),
48 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0334806625409744615033),
49 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0126926147662974029034),
50 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0365637971411762664006),
51 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0219878681111168899165),
52 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00822687874676915743155),
53 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00538772965071242932965)
54 };
55 static const T Q[] = {
56 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
57 BOOST_MATH_BIG_CONSTANT(T, 64, -0.970005043303290640362),
58 BOOST_MATH_BIG_CONSTANT(T, 64, -1.56574558234175846809),
59 BOOST_MATH_BIG_CONSTANT(T, 64, 1.56221558398423026363),
60 BOOST_MATH_BIG_CONSTANT(T, 64, 0.662328840472002992063),
61 BOOST_MATH_BIG_CONSTANT(T, 64, -0.71228902341542847553),
62 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0527396382340099713954),
63 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0795283687341571680018),
64 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00233393759374190016776),
65 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000886216390456424707504)
66 };
67 T g = p * (p + 10);
68 T r = tools::evaluate_polynomial(P, p) / tools::evaluate_polynomial(Q, p);
69 result = g * Y + g * r;
70 }
71 else if(q >= 0.25)
72 {
73 //
74 // Rational approximation for 0.5 > q >= 0.25
75 //
76 // x = sqrt(-2*log(q)) / (Y + R(q))
77 //
78 // Where Y is a constant, and R(q) is optimised for a low
79 // absolute error compared to Y.
80 //
81 // double : Max error found: 7.403372e-17
82 // long double : Max error found: 6.084616e-20
83 // Maximum Deviation Found (error term) 4.811e-20
84 //
85 static const float Y = 2.249481201171875f;
86 static const T P[] = {
87 BOOST_MATH_BIG_CONSTANT(T, 64, -0.202433508355938759655),
88 BOOST_MATH_BIG_CONSTANT(T, 64, 0.105264680699391713268),
89 BOOST_MATH_BIG_CONSTANT(T, 64, 8.37050328343119927838),
90 BOOST_MATH_BIG_CONSTANT(T, 64, 17.6447298408374015486),
91 BOOST_MATH_BIG_CONSTANT(T, 64, -18.8510648058714251895),
92 BOOST_MATH_BIG_CONSTANT(T, 64, -44.6382324441786960818),
93 BOOST_MATH_BIG_CONSTANT(T, 64, 17.445385985570866523),
94 BOOST_MATH_BIG_CONSTANT(T, 64, 21.1294655448340526258),
95 BOOST_MATH_BIG_CONSTANT(T, 64, -3.67192254707729348546)
96 };
97 static const T Q[] = {
98 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
99 BOOST_MATH_BIG_CONSTANT(T, 64, 6.24264124854247537712),
100 BOOST_MATH_BIG_CONSTANT(T, 64, 3.9713437953343869095),
101 BOOST_MATH_BIG_CONSTANT(T, 64, -28.6608180499800029974),
102 BOOST_MATH_BIG_CONSTANT(T, 64, -20.1432634680485188801),
103 BOOST_MATH_BIG_CONSTANT(T, 64, 48.5609213108739935468),
104 BOOST_MATH_BIG_CONSTANT(T, 64, 10.8268667355460159008),
105 BOOST_MATH_BIG_CONSTANT(T, 64, -22.6436933413139721736),
106 BOOST_MATH_BIG_CONSTANT(T, 64, 1.72114765761200282724)
107 };
108 T g = sqrt(-2 * log(q));
109 T xs = q - 0.25f;
110 T r = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
111 result = g / (Y + r);
112 }
113 else
114 {
115 //
116 // For q < 0.25 we have a series of rational approximations all
117 // of the general form:
118 //
119 // let: x = sqrt(-log(q))
120 //
121 // Then the result is given by:
122 //
123 // x(Y+R(x-B))
124 //
125 // where Y is a constant, B is the lowest value of x for which
126 // the approximation is valid, and R(x-B) is optimised for a low
127 // absolute error compared to Y.
128 //
129 // Note that almost all code will really go through the first
130 // or maybe second approximation. After than we're dealing with very
131 // small input values indeed: 80 and 128 bit long double's go all the
132 // way down to ~ 1e-5000 so the "tail" is rather long...
133 //
134 T x = sqrt(-log(q));
135 if(x < 3)
136 {
137 // Max error found: 1.089051e-20
138 static const float Y = 0.807220458984375f;
139 static const T P[] = {
140 BOOST_MATH_BIG_CONSTANT(T, 64, -0.131102781679951906451),
141 BOOST_MATH_BIG_CONSTANT(T, 64, -0.163794047193317060787),
142 BOOST_MATH_BIG_CONSTANT(T, 64, 0.117030156341995252019),
143 BOOST_MATH_BIG_CONSTANT(T, 64, 0.387079738972604337464),
144 BOOST_MATH_BIG_CONSTANT(T, 64, 0.337785538912035898924),
145 BOOST_MATH_BIG_CONSTANT(T, 64, 0.142869534408157156766),
146 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0290157910005329060432),
147 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00214558995388805277169),
148 BOOST_MATH_BIG_CONSTANT(T, 64, -0.679465575181126350155e-6),
149 BOOST_MATH_BIG_CONSTANT(T, 64, 0.285225331782217055858e-7),
150 BOOST_MATH_BIG_CONSTANT(T, 64, -0.681149956853776992068e-9)
151 };
152 static const T Q[] = {
153 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
154 BOOST_MATH_BIG_CONSTANT(T, 64, 3.46625407242567245975),
155 BOOST_MATH_BIG_CONSTANT(T, 64, 5.38168345707006855425),
156 BOOST_MATH_BIG_CONSTANT(T, 64, 4.77846592945843778382),
157 BOOST_MATH_BIG_CONSTANT(T, 64, 2.59301921623620271374),
158 BOOST_MATH_BIG_CONSTANT(T, 64, 0.848854343457902036425),
159 BOOST_MATH_BIG_CONSTANT(T, 64, 0.152264338295331783612),
160 BOOST_MATH_BIG_CONSTANT(T, 64, 0.01105924229346489121)
161 };
162 T xs = x - 1.125f;
163 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
164 result = Y * x + R * x;
165 }
166 else if(x < 6)
167 {
168 // Max error found: 8.389174e-21
169 static const float Y = 0.93995571136474609375f;
170 static const T P[] = {
171 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0350353787183177984712),
172 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00222426529213447927281),
173 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0185573306514231072324),
174 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00950804701325919603619),
175 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00187123492819559223345),
176 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000157544617424960554631),
177 BOOST_MATH_BIG_CONSTANT(T, 64, 0.460469890584317994083e-5),
178 BOOST_MATH_BIG_CONSTANT(T, 64, -0.230404776911882601748e-9),
179 BOOST_MATH_BIG_CONSTANT(T, 64, 0.266339227425782031962e-11)
180 };
181 static const T Q[] = {
182 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
183 BOOST_MATH_BIG_CONSTANT(T, 64, 1.3653349817554063097),
184 BOOST_MATH_BIG_CONSTANT(T, 64, 0.762059164553623404043),
185 BOOST_MATH_BIG_CONSTANT(T, 64, 0.220091105764131249824),
186 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0341589143670947727934),
187 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00263861676657015992959),
188 BOOST_MATH_BIG_CONSTANT(T, 64, 0.764675292302794483503e-4)
189 };
190 T xs = x - 3;
191 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
192 result = Y * x + R * x;
193 }
194 else if(x < 18)
195 {
196 // Max error found: 1.481312e-19
197 static const float Y = 0.98362827301025390625f;
198 static const T P[] = {
199 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0167431005076633737133),
200 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00112951438745580278863),
201 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00105628862152492910091),
202 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000209386317487588078668),
203 BOOST_MATH_BIG_CONSTANT(T, 64, 0.149624783758342370182e-4),
204 BOOST_MATH_BIG_CONSTANT(T, 64, 0.449696789927706453732e-6),
205 BOOST_MATH_BIG_CONSTANT(T, 64, 0.462596163522878599135e-8),
206 BOOST_MATH_BIG_CONSTANT(T, 64, -0.281128735628831791805e-13),
207 BOOST_MATH_BIG_CONSTANT(T, 64, 0.99055709973310326855e-16)
208 };
209 static const T Q[] = {
210 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
211 BOOST_MATH_BIG_CONSTANT(T, 64, 0.591429344886417493481),
212 BOOST_MATH_BIG_CONSTANT(T, 64, 0.138151865749083321638),
213 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0160746087093676504695),
214 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000964011807005165528527),
215 BOOST_MATH_BIG_CONSTANT(T, 64, 0.275335474764726041141e-4),
216 BOOST_MATH_BIG_CONSTANT(T, 64, 0.282243172016108031869e-6)
217 };
218 T xs = x - 6;
219 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
220 result = Y * x + R * x;
221 }
222 else if(x < 44)
223 {
224 // Max error found: 5.697761e-20
225 static const float Y = 0.99714565277099609375f;
226 static const T P[] = {
227 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0024978212791898131227),
228 BOOST_MATH_BIG_CONSTANT(T, 64, -0.779190719229053954292e-5),
229 BOOST_MATH_BIG_CONSTANT(T, 64, 0.254723037413027451751e-4),
230 BOOST_MATH_BIG_CONSTANT(T, 64, 0.162397777342510920873e-5),
231 BOOST_MATH_BIG_CONSTANT(T, 64, 0.396341011304801168516e-7),
232 BOOST_MATH_BIG_CONSTANT(T, 64, 0.411632831190944208473e-9),
233 BOOST_MATH_BIG_CONSTANT(T, 64, 0.145596286718675035587e-11),
234 BOOST_MATH_BIG_CONSTANT(T, 64, -0.116765012397184275695e-17)
235 };
236 static const T Q[] = {
237 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
238 BOOST_MATH_BIG_CONSTANT(T, 64, 0.207123112214422517181),
239 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0169410838120975906478),
240 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000690538265622684595676),
241 BOOST_MATH_BIG_CONSTANT(T, 64, 0.145007359818232637924e-4),
242 BOOST_MATH_BIG_CONSTANT(T, 64, 0.144437756628144157666e-6),
243 BOOST_MATH_BIG_CONSTANT(T, 64, 0.509761276599778486139e-9)
244 };
245 T xs = x - 18;
246 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
247 result = Y * x + R * x;
248 }
249 else
250 {
251 // Max error found: 1.279746e-20
252 static const float Y = 0.99941349029541015625f;
253 static const T P[] = {
254 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000539042911019078575891),
255 BOOST_MATH_BIG_CONSTANT(T, 64, -0.28398759004727721098e-6),
256 BOOST_MATH_BIG_CONSTANT(T, 64, 0.899465114892291446442e-6),
257 BOOST_MATH_BIG_CONSTANT(T, 64, 0.229345859265920864296e-7),
258 BOOST_MATH_BIG_CONSTANT(T, 64, 0.225561444863500149219e-9),
259 BOOST_MATH_BIG_CONSTANT(T, 64, 0.947846627503022684216e-12),
260 BOOST_MATH_BIG_CONSTANT(T, 64, 0.135880130108924861008e-14),
261 BOOST_MATH_BIG_CONSTANT(T, 64, -0.348890393399948882918e-21)
262 };
263 static const T Q[] = {
264 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
265 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0845746234001899436914),
266 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00282092984726264681981),
267 BOOST_MATH_BIG_CONSTANT(T, 64, 0.468292921940894236786e-4),
268 BOOST_MATH_BIG_CONSTANT(T, 64, 0.399968812193862100054e-6),
269 BOOST_MATH_BIG_CONSTANT(T, 64, 0.161809290887904476097e-8),
270 BOOST_MATH_BIG_CONSTANT(T, 64, 0.231558608310259605225e-11)
271 };
272 T xs = x - 44;
273 T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
274 result = Y * x + R * x;
275 }
276 }
277 return result;
278 }
279
280 template <class T, class Policy>
281 struct erf_roots
282 {
operator ()boost::math::detail::erf_roots283 boost::math::tuple<T,T,T> operator()(const T& guess)
284 {
285 BOOST_MATH_STD_USING
286 T derivative = sign * (2 / sqrt(constants::pi<T>())) * exp(-(guess * guess));
287 T derivative2 = -2 * guess * derivative;
288 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);
289 }
erf_rootsboost::math::detail::erf_roots290 erf_roots(T z, int s) : target(z), sign(s) {}
291 private:
292 T target;
293 int sign;
294 };
295
296 template <class T, class Policy>
297 T erf_inv_imp(const T& p, const T& q, const Policy& pol, const boost::mpl::int_<0>*)
298 {
299 //
300 // Generic version, get a guess that's accurate to 64-bits (10^-19)
301 //
302 T guess = erf_inv_imp(p, q, pol, static_cast<mpl::int_<64> const*>(0));
303 T result;
304 //
305 // If T has more bit's than 64 in it's mantissa then we need to iterate,
306 // otherwise we can just return the result:
307 //
308 if(policies::digits<T, Policy>() > 64)
309 {
310 boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
311 if(p <= 0.5)
312 {
313 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 }
315 else
316 {
317 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);
318 }
319 policies::check_root_iterations<T>("boost::math::erf_inv<%1%>", max_iter, pol);
320 }
321 else
322 {
323 result = guess;
324 }
325 return result;
326 }
327
328 template <class T, class Policy>
329 struct erf_inv_initializer
330 {
331 struct init
332 {
initboost::math::detail::erf_inv_initializer::init333 init()
334 {
335 do_init();
336 }
337 static bool is_value_non_zero(T);
do_initboost::math::detail::erf_inv_initializer::init338 static void do_init()
339 {
340 // If std::numeric_limits<T>::digits is zero, we must not call
341 // our inituialization code here as the precision presumably
342 // varies at runtime, and will not have been set yet.
343 if(std::numeric_limits<T>::digits)
344 {
345 boost::math::erf_inv(static_cast<T>(0.25), Policy());
346 boost::math::erf_inv(static_cast<T>(0.55), Policy());
347 boost::math::erf_inv(static_cast<T>(0.95), Policy());
348 boost::math::erfc_inv(static_cast<T>(1e-15), Policy());
349 // These following initializations must not be called if
350 // type T can not hold the relevant values without
351 // underflow to zero. We check this at runtime because
352 // some tools such as valgrind silently change the precision
353 // of T at runtime, and numeric_limits basically lies!
354 if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130))))
355 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)), Policy());
356
357 // Some compilers choke on constants that would underflow, even in code that isn't instantiated
358 // so try and filter these cases out in the preprocessor:
359 #if LDBL_MAX_10_EXP >= 800
360 if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800))))
361 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)), Policy());
362 if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900))))
363 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)), Policy());
364 #else
365 if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800))))
366 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)), Policy());
367 if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900))))
368 boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)), Policy());
369 #endif
370 }
371 }
force_instantiateboost::math::detail::erf_inv_initializer::init372 void force_instantiate()const{}
373 };
374 static const init initializer;
force_instantiateboost::math::detail::erf_inv_initializer375 static void force_instantiate()
376 {
377 initializer.force_instantiate();
378 }
379 };
380
381 template <class T, class Policy>
382 const typename erf_inv_initializer<T, Policy>::init erf_inv_initializer<T, Policy>::initializer;
383
384 template <class T, class Policy>
is_value_non_zero(T v)385 bool erf_inv_initializer<T, Policy>::init::is_value_non_zero(T v)
386 {
387 // This needs to be non-inline to detect whether v is non zero at runtime
388 // rather than at compile time, only relevant when running under valgrind
389 // which changes long double's to double's on the fly.
390 return v != 0;
391 }
392
393 } // namespace detail
394
395 template <class T, class Policy>
erfc_inv(T z,const Policy & pol)396 typename tools::promote_args<T>::type erfc_inv(T z, const Policy& pol)
397 {
398 typedef typename tools::promote_args<T>::type result_type;
399
400 //
401 // Begin by testing for domain errors, and other special cases:
402 //
403 static const char* function = "boost::math::erfc_inv<%1%>(%1%, %1%)";
404 if((z < 0) || (z > 2))
405 return policies::raise_domain_error<result_type>(function, "Argument outside range [0,2] in inverse erfc function (got p=%1%).", z, pol);
406 if(z == 0)
407 return policies::raise_overflow_error<result_type>(function, 0, pol);
408 if(z == 2)
409 return -policies::raise_overflow_error<result_type>(function, 0, pol);
410 //
411 // Normalise the input, so it's in the range [0,1], we will
412 // negate the result if z is outside that range. This is a simple
413 // application of the erfc reflection formula: erfc(-z) = 2 - erfc(z)
414 //
415 result_type p, q, s;
416 if(z > 1)
417 {
418 q = 2 - z;
419 p = 1 - q;
420 s = -1;
421 }
422 else
423 {
424 p = 1 - z;
425 q = z;
426 s = 1;
427 }
428 //
429 // A bit of meta-programming to figure out which implementation
430 // to use, based on the number of bits in the mantissa of T:
431 //
432 typedef typename policies::precision<result_type, Policy>::type precision_type;
433 typedef typename mpl::if_<
434 mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
435 mpl::int_<0>,
436 mpl::int_<64>
437 >::type tag_type;
438 //
439 // Likewise use internal promotion, so we evaluate at a higher
440 // precision internally if it's appropriate:
441 //
442 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
443 typedef typename policies::normalise<
444 Policy,
445 policies::promote_float<false>,
446 policies::promote_double<false>,
447 policies::discrete_quantile<>,
448 policies::assert_undefined<> >::type forwarding_policy;
449
450 detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
451
452 //
453 // And get the result, negating where required:
454 //
455 return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
456 detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
457 }
458
459 template <class T, class Policy>
erf_inv(T z,const Policy & pol)460 typename tools::promote_args<T>::type erf_inv(T z, const Policy& pol)
461 {
462 typedef typename tools::promote_args<T>::type result_type;
463
464 //
465 // Begin by testing for domain errors, and other special cases:
466 //
467 static const char* function = "boost::math::erf_inv<%1%>(%1%, %1%)";
468 if((z < -1) || (z > 1))
469 return policies::raise_domain_error<result_type>(function, "Argument outside range [-1, 1] in inverse erf function (got p=%1%).", z, pol);
470 if(z == 1)
471 return policies::raise_overflow_error<result_type>(function, 0, pol);
472 if(z == -1)
473 return -policies::raise_overflow_error<result_type>(function, 0, pol);
474 if(z == 0)
475 return 0;
476 //
477 // Normalise the input, so it's in the range [0,1], we will
478 // negate the result if z is outside that range. This is a simple
479 // application of the erf reflection formula: erf(-z) = -erf(z)
480 //
481 result_type p, q, s;
482 if(z < 0)
483 {
484 p = -z;
485 q = 1 - p;
486 s = -1;
487 }
488 else
489 {
490 p = z;
491 q = 1 - z;
492 s = 1;
493 }
494 //
495 // A bit of meta-programming to figure out which implementation
496 // to use, based on the number of bits in the mantissa of T:
497 //
498 typedef typename policies::precision<result_type, Policy>::type precision_type;
499 typedef typename mpl::if_<
500 mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
501 mpl::int_<0>,
502 mpl::int_<64>
503 >::type tag_type;
504 //
505 // Likewise use internal promotion, so we evaluate at a higher
506 // precision internally if it's appropriate:
507 //
508 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
509 typedef typename policies::normalise<
510 Policy,
511 policies::promote_float<false>,
512 policies::promote_double<false>,
513 policies::discrete_quantile<>,
514 policies::assert_undefined<> >::type forwarding_policy;
515 //
516 // Likewise use internal promotion, so we evaluate at a higher
517 // precision internally if it's appropriate:
518 //
519 typedef typename policies::evaluation<result_type, Policy>::type eval_type;
520
521 detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
522 //
523 // And get the result, negating where required:
524 //
525 return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
526 detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
527 }
528
529 template <class T>
erfc_inv(T z)530 inline typename tools::promote_args<T>::type erfc_inv(T z)
531 {
532 return erfc_inv(z, policies::policy<>());
533 }
534
535 template <class T>
erf_inv(T z)536 inline typename tools::promote_args<T>::type erf_inv(T z)
537 {
538 return erf_inv(z, policies::policy<>());
539 }
540
541 } // namespace math
542 } // namespace boost
543
544 #ifdef _MSC_VER
545 #pragma warning(pop)
546 #endif
547
548 #endif // BOOST_MATH_SF_ERF_INV_HPP
549
550