| // (C) Copyright John Maddock 2006. |
| // Use, modification and distribution are subject to the |
| // Boost Software License, Version 1.0. (See accompanying file |
| // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) |
| |
| #ifndef BOOST_MATH_SF_ERF_INV_HPP |
| #define BOOST_MATH_SF_ERF_INV_HPP |
| |
| #ifdef _MSC_VER |
| #pragma once |
| #endif |
| |
| namespace boost{ namespace math{ |
| |
| namespace detail{ |
| // |
| // The inverse erf and erfc functions share a common implementation, |
| // this version is for 80-bit long double's and smaller: |
| // |
| template <class T, class Policy> |
| T erf_inv_imp(const T& p, const T& q, const Policy&, const boost::mpl::int_<64>*) |
| { |
| BOOST_MATH_STD_USING // for ADL of std names. |
| |
| T result = 0; |
| |
| if(p <= 0.5) |
| { |
| // |
| // Evaluate inverse erf using the rational approximation: |
| // |
| // x = p(p+10)(Y+R(p)) |
| // |
| // Where Y is a constant, and R(p) is optimised for a low |
| // absolute error compared to |Y|. |
| // |
| // double: Max error found: 2.001849e-18 |
| // long double: Max error found: 1.017064e-20 |
| // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21 |
| // |
| static const float Y = 0.0891314744949340820313f; |
| static const T P[] = { |
| -0.000508781949658280665617L, |
| -0.00836874819741736770379L, |
| 0.0334806625409744615033L, |
| -0.0126926147662974029034L, |
| -0.0365637971411762664006L, |
| 0.0219878681111168899165L, |
| 0.00822687874676915743155L, |
| -0.00538772965071242932965L |
| }; |
| static const T Q[] = { |
| 1, |
| -0.970005043303290640362L, |
| -1.56574558234175846809L, |
| 1.56221558398423026363L, |
| 0.662328840472002992063L, |
| -0.71228902341542847553L, |
| -0.0527396382340099713954L, |
| 0.0795283687341571680018L, |
| -0.00233393759374190016776L, |
| 0.000886216390456424707504L |
| }; |
| T g = p * (p + 10); |
| T r = tools::evaluate_polynomial(P, p) / tools::evaluate_polynomial(Q, p); |
| result = g * Y + g * r; |
| } |
| else if(q >= 0.25) |
| { |
| // |
| // Rational approximation for 0.5 > q >= 0.25 |
| // |
| // x = sqrt(-2*log(q)) / (Y + R(q)) |
| // |
| // Where Y is a constant, and R(q) is optimised for a low |
| // absolute error compared to Y. |
| // |
| // double : Max error found: 7.403372e-17 |
| // long double : Max error found: 6.084616e-20 |
| // Maximum Deviation Found (error term) 4.811e-20 |
| // |
| static const float Y = 2.249481201171875f; |
| static const T P[] = { |
| -0.202433508355938759655L, |
| 0.105264680699391713268L, |
| 8.37050328343119927838L, |
| 17.6447298408374015486L, |
| -18.8510648058714251895L, |
| -44.6382324441786960818L, |
| 17.445385985570866523L, |
| 21.1294655448340526258L, |
| -3.67192254707729348546L |
| }; |
| static const T Q[] = { |
| 1L, |
| 6.24264124854247537712L, |
| 3.9713437953343869095L, |
| -28.6608180499800029974L, |
| -20.1432634680485188801L, |
| 48.5609213108739935468L, |
| 10.8268667355460159008L, |
| -22.6436933413139721736L, |
| 1.72114765761200282724L |
| }; |
| T g = sqrt(-2 * log(q)); |
| T xs = q - 0.25; |
| T r = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = g / (Y + r); |
| } |
| else |
| { |
| // |
| // For q < 0.25 we have a series of rational approximations all |
| // of the general form: |
| // |
| // let: x = sqrt(-log(q)) |
| // |
| // Then the result is given by: |
| // |
| // x(Y+R(x-B)) |
| // |
| // where Y is a constant, B is the lowest value of x for which |
| // the approximation is valid, and R(x-B) is optimised for a low |
| // absolute error compared to Y. |
| // |
| // Note that almost all code will really go through the first |
| // or maybe second approximation. After than we're dealing with very |
| // small input values indeed: 80 and 128 bit long double's go all the |
| // way down to ~ 1e-5000 so the "tail" is rather long... |
| // |
| T x = sqrt(-log(q)); |
| if(x < 3) |
| { |
| // Max error found: 1.089051e-20 |
| static const float Y = 0.807220458984375f; |
| static const T P[] = { |
| -0.131102781679951906451L, |
| -0.163794047193317060787L, |
| 0.117030156341995252019L, |
| 0.387079738972604337464L, |
| 0.337785538912035898924L, |
| 0.142869534408157156766L, |
| 0.0290157910005329060432L, |
| 0.00214558995388805277169L, |
| -0.679465575181126350155e-6L, |
| 0.285225331782217055858e-7L, |
| -0.681149956853776992068e-9L |
| }; |
| static const T Q[] = { |
| 1, |
| 3.46625407242567245975L, |
| 5.38168345707006855425L, |
| 4.77846592945843778382L, |
| 2.59301921623620271374L, |
| 0.848854343457902036425L, |
| 0.152264338295331783612L, |
| 0.01105924229346489121L |
| }; |
| T xs = x - 1.125; |
| T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 6) |
| { |
| // Max error found: 8.389174e-21 |
| static const float Y = 0.93995571136474609375f; |
| static const T P[] = { |
| -0.0350353787183177984712L, |
| -0.00222426529213447927281L, |
| 0.0185573306514231072324L, |
| 0.00950804701325919603619L, |
| 0.00187123492819559223345L, |
| 0.000157544617424960554631L, |
| 0.460469890584317994083e-5L, |
| -0.230404776911882601748e-9L, |
| 0.266339227425782031962e-11L |
| }; |
| static const T Q[] = { |
| 1L, |
| 1.3653349817554063097L, |
| 0.762059164553623404043L, |
| 0.220091105764131249824L, |
| 0.0341589143670947727934L, |
| 0.00263861676657015992959L, |
| 0.764675292302794483503e-4L |
| }; |
| T xs = x - 3; |
| T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 18) |
| { |
| // Max error found: 1.481312e-19 |
| static const float Y = 0.98362827301025390625f; |
| static const T P[] = { |
| -0.0167431005076633737133L, |
| -0.00112951438745580278863L, |
| 0.00105628862152492910091L, |
| 0.000209386317487588078668L, |
| 0.149624783758342370182e-4L, |
| 0.449696789927706453732e-6L, |
| 0.462596163522878599135e-8L, |
| -0.281128735628831791805e-13L, |
| 0.99055709973310326855e-16L |
| }; |
| static const T Q[] = { |
| 1L, |
| 0.591429344886417493481L, |
| 0.138151865749083321638L, |
| 0.0160746087093676504695L, |
| 0.000964011807005165528527L, |
| 0.275335474764726041141e-4L, |
| 0.282243172016108031869e-6L |
| }; |
| T xs = x - 6; |
| T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 44) |
| { |
| // Max error found: 5.697761e-20 |
| static const float Y = 0.99714565277099609375f; |
| static const T P[] = { |
| -0.0024978212791898131227L, |
| -0.779190719229053954292e-5L, |
| 0.254723037413027451751e-4L, |
| 0.162397777342510920873e-5L, |
| 0.396341011304801168516e-7L, |
| 0.411632831190944208473e-9L, |
| 0.145596286718675035587e-11L, |
| -0.116765012397184275695e-17L |
| }; |
| static const T Q[] = { |
| 1L, |
| 0.207123112214422517181L, |
| 0.0169410838120975906478L, |
| 0.000690538265622684595676L, |
| 0.145007359818232637924e-4L, |
| 0.144437756628144157666e-6L, |
| 0.509761276599778486139e-9L |
| }; |
| T xs = x - 18; |
| T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else |
| { |
| // Max error found: 1.279746e-20 |
| static const float Y = 0.99941349029541015625f; |
| static const T P[] = { |
| -0.000539042911019078575891L, |
| -0.28398759004727721098e-6L, |
| 0.899465114892291446442e-6L, |
| 0.229345859265920864296e-7L, |
| 0.225561444863500149219e-9L, |
| 0.947846627503022684216e-12L, |
| 0.135880130108924861008e-14L, |
| -0.348890393399948882918e-21L |
| }; |
| static const T Q[] = { |
| 1L, |
| 0.0845746234001899436914L, |
| 0.00282092984726264681981L, |
| 0.468292921940894236786e-4L, |
| 0.399968812193862100054e-6L, |
| 0.161809290887904476097e-8L, |
| 0.231558608310259605225e-11L |
| }; |
| T xs = x - 44; |
| T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| } |
| return result; |
| } |
| |
| template <class T, class Policy> |
| struct erf_roots |
| { |
| std::tr1::tuple<T,T,T> operator()(const T& guess) |
| { |
| BOOST_MATH_STD_USING |
| T derivative = sign * (2 / sqrt(constants::pi<T>())) * exp(-(guess * guess)); |
| T derivative2 = -2 * guess * derivative; |
| return std::tr1::make_tuple(((sign > 0) ? boost::math::erf(guess, Policy()) : boost::math::erfc(guess, Policy())) - target, derivative, derivative2); |
| } |
| erf_roots(T z, int s) : target(z), sign(s) {} |
| private: |
| T target; |
| int sign; |
| }; |
| |
| template <class T, class Policy> |
| T erf_inv_imp(const T& p, const T& q, const Policy& pol, const boost::mpl::int_<0>*) |
| { |
| // |
| // Generic version, get a guess that's accurate to 64-bits (10^-19) |
| // |
| T guess = erf_inv_imp(p, q, pol, static_cast<mpl::int_<64> const*>(0)); |
| T result; |
| // |
| // If T has more bit's than 64 in it's mantissa then we need to iterate, |
| // otherwise we can just return the result: |
| // |
| if(policies::digits<T, Policy>() > 64) |
| { |
| if(p <= 0.5) |
| { |
| 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); |
| } |
| else |
| { |
| 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); |
| } |
| } |
| else |
| { |
| result = guess; |
| } |
| return result; |
| } |
| |
| } // namespace detail |
| |
| template <class T, class Policy> |
| typename tools::promote_args<T>::type erfc_inv(T z, const Policy& pol) |
| { |
| typedef typename tools::promote_args<T>::type result_type; |
| // |
| // Begin by testing for domain errors, and other special cases: |
| // |
| static const char* function = "boost::math::erfc_inv<%1%>(%1%, %1%)"; |
| if((z < 0) || (z > 2)) |
| policies::raise_domain_error<result_type>(function, "Argument outside range [0,2] in inverse erfc function (got p=%1%).", z, pol); |
| if(z == 0) |
| return policies::raise_overflow_error<result_type>(function, 0, pol); |
| if(z == 2) |
| return -policies::raise_overflow_error<result_type>(function, 0, pol); |
| // |
| // Normalise the input, so it's in the range [0,1], we will |
| // negate the result if z is outside that range. This is a simple |
| // application of the erfc reflection formula: erfc(-z) = 2 - erfc(z) |
| // |
| result_type p, q, s; |
| if(z > 1) |
| { |
| q = 2 - z; |
| p = 1 - q; |
| s = -1; |
| } |
| else |
| { |
| p = 1 - z; |
| q = z; |
| s = 1; |
| } |
| // |
| // A bit of meta-programming to figure out which implementation |
| // to use, based on the number of bits in the mantissa of T: |
| // |
| typedef typename policies::precision<result_type, Policy>::type precision_type; |
| typedef typename mpl::if_< |
| mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >, |
| mpl::int_<0>, |
| mpl::int_<64> |
| >::type tag_type; |
| // |
| // Likewise use internal promotion, so we evaluate at a higher |
| // precision internally if it's appropriate: |
| // |
| typedef typename policies::evaluation<result_type, Policy>::type eval_type; |
| typedef typename policies::normalise< |
| Policy, |
| policies::promote_float<false>, |
| policies::promote_double<false>, |
| policies::discrete_quantile<>, |
| policies::assert_undefined<> >::type forwarding_policy; |
| |
| // |
| // And get the result, negating where required: |
| // |
| return s * policies::checked_narrowing_cast<result_type, forwarding_policy>( |
| detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function); |
| } |
| |
| template <class T, class Policy> |
| typename tools::promote_args<T>::type erf_inv(T z, const Policy& pol) |
| { |
| typedef typename tools::promote_args<T>::type result_type; |
| // |
| // Begin by testing for domain errors, and other special cases: |
| // |
| static const char* function = "boost::math::erf_inv<%1%>(%1%, %1%)"; |
| if((z < -1) || (z > 1)) |
| policies::raise_domain_error<result_type>(function, "Argument outside range [-1, 1] in inverse erf function (got p=%1%).", z, pol); |
| if(z == 1) |
| return policies::raise_overflow_error<result_type>(function, 0, pol); |
| if(z == -1) |
| return -policies::raise_overflow_error<result_type>(function, 0, pol); |
| if(z == 0) |
| return 0; |
| // |
| // Normalise the input, so it's in the range [0,1], we will |
| // negate the result if z is outside that range. This is a simple |
| // application of the erf reflection formula: erf(-z) = -erf(z) |
| // |
| result_type p, q, s; |
| if(z < 0) |
| { |
| p = -z; |
| q = 1 - p; |
| s = -1; |
| } |
| else |
| { |
| p = z; |
| q = 1 - z; |
| s = 1; |
| } |
| // |
| // A bit of meta-programming to figure out which implementation |
| // to use, based on the number of bits in the mantissa of T: |
| // |
| typedef typename policies::precision<result_type, Policy>::type precision_type; |
| typedef typename mpl::if_< |
| mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >, |
| mpl::int_<0>, |
| mpl::int_<64> |
| >::type tag_type; |
| // |
| // Likewise use internal promotion, so we evaluate at a higher |
| // precision internally if it's appropriate: |
| // |
| typedef typename policies::evaluation<result_type, Policy>::type eval_type; |
| typedef typename policies::normalise< |
| Policy, |
| policies::promote_float<false>, |
| policies::promote_double<false>, |
| policies::discrete_quantile<>, |
| policies::assert_undefined<> >::type forwarding_policy; |
| // |
| // Likewise use internal promotion, so we evaluate at a higher |
| // precision internally if it's appropriate: |
| // |
| typedef typename policies::evaluation<result_type, Policy>::type eval_type; |
| // |
| // And get the result, negating where required: |
| // |
| return s * policies::checked_narrowing_cast<result_type, forwarding_policy>( |
| detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function); |
| } |
| |
| template <class T> |
| inline typename tools::promote_args<T>::type erfc_inv(T z) |
| { |
| return erfc_inv(z, policies::policy<>()); |
| } |
| |
| template <class T> |
| inline typename tools::promote_args<T>::type erf_inv(T z) |
| { |
| return erf_inv(z, policies::policy<>()); |
| } |
| |
| } // namespace math |
| } // namespace boost |
| |
| #endif // BOOST_MATH_SF_ERF_INV_HPP |
| |
