boost/math/special_functions/detail/erf_inv.hpp
// (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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.000508781949658280665617), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00836874819741736770379), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0334806625409744615033), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0126926147662974029034), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0365637971411762664006), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0219878681111168899165), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00822687874676915743155), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00538772965071242932965) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, -0.970005043303290640362), BOOST_MATH_BIG_CONSTANT(T, 64, -1.56574558234175846809), BOOST_MATH_BIG_CONSTANT(T, 64, 1.56221558398423026363), BOOST_MATH_BIG_CONSTANT(T, 64, 0.662328840472002992063), BOOST_MATH_BIG_CONSTANT(T, 64, -0.71228902341542847553), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0527396382340099713954), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0795283687341571680018), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00233393759374190016776), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000886216390456424707504) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.202433508355938759655), BOOST_MATH_BIG_CONSTANT(T, 64, 0.105264680699391713268), BOOST_MATH_BIG_CONSTANT(T, 64, 8.37050328343119927838), BOOST_MATH_BIG_CONSTANT(T, 64, 17.6447298408374015486), BOOST_MATH_BIG_CONSTANT(T, 64, -18.8510648058714251895), BOOST_MATH_BIG_CONSTANT(T, 64, -44.6382324441786960818), BOOST_MATH_BIG_CONSTANT(T, 64, 17.445385985570866523), BOOST_MATH_BIG_CONSTANT(T, 64, 21.1294655448340526258), BOOST_MATH_BIG_CONSTANT(T, 64, -3.67192254707729348546) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 6.24264124854247537712), BOOST_MATH_BIG_CONSTANT(T, 64, 3.9713437953343869095), BOOST_MATH_BIG_CONSTANT(T, 64, -28.6608180499800029974), BOOST_MATH_BIG_CONSTANT(T, 64, -20.1432634680485188801), BOOST_MATH_BIG_CONSTANT(T, 64, 48.5609213108739935468), BOOST_MATH_BIG_CONSTANT(T, 64, 10.8268667355460159008), BOOST_MATH_BIG_CONSTANT(T, 64, -22.6436933413139721736), BOOST_MATH_BIG_CONSTANT(T, 64, 1.72114765761200282724) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.131102781679951906451), BOOST_MATH_BIG_CONSTANT(T, 64, -0.163794047193317060787), BOOST_MATH_BIG_CONSTANT(T, 64, 0.117030156341995252019), BOOST_MATH_BIG_CONSTANT(T, 64, 0.387079738972604337464), BOOST_MATH_BIG_CONSTANT(T, 64, 0.337785538912035898924), BOOST_MATH_BIG_CONSTANT(T, 64, 0.142869534408157156766), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0290157910005329060432), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00214558995388805277169), BOOST_MATH_BIG_CONSTANT(T, 64, -0.679465575181126350155e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.285225331782217055858e-7), BOOST_MATH_BIG_CONSTANT(T, 64, -0.681149956853776992068e-9) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 3.46625407242567245975), BOOST_MATH_BIG_CONSTANT(T, 64, 5.38168345707006855425), BOOST_MATH_BIG_CONSTANT(T, 64, 4.77846592945843778382), BOOST_MATH_BIG_CONSTANT(T, 64, 2.59301921623620271374), BOOST_MATH_BIG_CONSTANT(T, 64, 0.848854343457902036425), BOOST_MATH_BIG_CONSTANT(T, 64, 0.152264338295331783612), BOOST_MATH_BIG_CONSTANT(T, 64, 0.01105924229346489121) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.0350353787183177984712), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00222426529213447927281), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0185573306514231072324), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00950804701325919603619), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00187123492819559223345), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000157544617424960554631), BOOST_MATH_BIG_CONSTANT(T, 64, 0.460469890584317994083e-5), BOOST_MATH_BIG_CONSTANT(T, 64, -0.230404776911882601748e-9), BOOST_MATH_BIG_CONSTANT(T, 64, 0.266339227425782031962e-11) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 1.3653349817554063097), BOOST_MATH_BIG_CONSTANT(T, 64, 0.762059164553623404043), BOOST_MATH_BIG_CONSTANT(T, 64, 0.220091105764131249824), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0341589143670947727934), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00263861676657015992959), BOOST_MATH_BIG_CONSTANT(T, 64, 0.764675292302794483503e-4) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.0167431005076633737133), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00112951438745580278863), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00105628862152492910091), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000209386317487588078668), BOOST_MATH_BIG_CONSTANT(T, 64, 0.149624783758342370182e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.449696789927706453732e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.462596163522878599135e-8), BOOST_MATH_BIG_CONSTANT(T, 64, -0.281128735628831791805e-13), BOOST_MATH_BIG_CONSTANT(T, 64, 0.99055709973310326855e-16) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.591429344886417493481), BOOST_MATH_BIG_CONSTANT(T, 64, 0.138151865749083321638), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0160746087093676504695), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000964011807005165528527), BOOST_MATH_BIG_CONSTANT(T, 64, 0.275335474764726041141e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.282243172016108031869e-6) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.0024978212791898131227), BOOST_MATH_BIG_CONSTANT(T, 64, -0.779190719229053954292e-5), BOOST_MATH_BIG_CONSTANT(T, 64, 0.254723037413027451751e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.162397777342510920873e-5), BOOST_MATH_BIG_CONSTANT(T, 64, 0.396341011304801168516e-7), BOOST_MATH_BIG_CONSTANT(T, 64, 0.411632831190944208473e-9), BOOST_MATH_BIG_CONSTANT(T, 64, 0.145596286718675035587e-11), BOOST_MATH_BIG_CONSTANT(T, 64, -0.116765012397184275695e-17) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.207123112214422517181), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0169410838120975906478), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000690538265622684595676), BOOST_MATH_BIG_CONSTANT(T, 64, 0.145007359818232637924e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.144437756628144157666e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.509761276599778486139e-9) }; 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[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.000539042911019078575891), BOOST_MATH_BIG_CONSTANT(T, 64, -0.28398759004727721098e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.899465114892291446442e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.229345859265920864296e-7), BOOST_MATH_BIG_CONSTANT(T, 64, 0.225561444863500149219e-9), BOOST_MATH_BIG_CONSTANT(T, 64, 0.947846627503022684216e-12), BOOST_MATH_BIG_CONSTANT(T, 64, 0.135880130108924861008e-14), BOOST_MATH_BIG_CONSTANT(T, 64, -0.348890393399948882918e-21) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0845746234001899436914), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00282092984726264681981), BOOST_MATH_BIG_CONSTANT(T, 64, 0.468292921940894236786e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.399968812193862100054e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.161809290887904476097e-8), BOOST_MATH_BIG_CONSTANT(T, 64, 0.231558608310259605225e-11) }; 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 { boost::math::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 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); } 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) { boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); 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, max_iter); } 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, max_iter); } policies::check_root_iterations<T>("boost::math::erf_inv<%1%>", max_iter, pol); } else { result = guess; } return result; } template <class T, class Policy> struct erf_inv_initializer { struct init { init() { do_init(); } static bool is_value_non_zero(T); static void do_init() { boost::math::erf_inv(static_cast<T>(0.25), Policy()); boost::math::erf_inv(static_cast<T>(0.55), Policy()); boost::math::erf_inv(static_cast<T>(0.95), Policy()); boost::math::erfc_inv(static_cast<T>(1e-15), Policy()); // These following initializations must not be called if // type T can not hold the relevant values without // underflow to zero. We check this at runtime because // some tools such as valgrind silently change the precision // of T at runtime, and numeric_limits basically lies! if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)))) boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)), Policy()); // Some compilers choke on constants that would underflow, even in code that isn't instantiated // so try and filter these cases out in the preprocessor: #if LDBL_MAX_10_EXP >= 800 if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)))) boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)), Policy()); if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)))) boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)), Policy()); #else if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)))) boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)), Policy()); if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)))) boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)), Policy()); #endif } void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class Policy> const typename erf_inv_initializer<T, Policy>::init erf_inv_initializer<T, Policy>::initializer; template <class T, class Policy> bool erf_inv_initializer<T, Policy>::init::is_value_non_zero(T v) { // This needs to be non-inline to detect whether v is non zero at runtime // rather than at compile time, only relevant when running under valgrind // which changes long double's to double's on the fly. return v != 0; } } // 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)) return 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; detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate(); // // 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)) return 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; detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate(); // // 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