Chris@16: //  (C) Copyright John Maddock 2006.
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_SF_ERF_INV_HPP
Chris@16: #define BOOST_MATH_SF_ERF_INV_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: namespace boost{ namespace math{ 
Chris@16: 
Chris@16: namespace detail{
Chris@16: //
Chris@16: // The inverse erf and erfc functions share a common implementation,
Chris@16: // this version is for 80-bit long double's and smaller:
Chris@16: //
Chris@16: template <class T, class Policy>
Chris@16: T erf_inv_imp(const T& p, const T& q, const Policy&, const boost::mpl::int_<64>*)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING // for ADL of std names.
Chris@16: 
Chris@16:    T result = 0;
Chris@16:    
Chris@16:    if(p <= 0.5)
Chris@16:    {
Chris@16:       //
Chris@16:       // Evaluate inverse erf using the rational approximation:
Chris@16:       //
Chris@16:       // x = p(p+10)(Y+R(p))
Chris@16:       //
Chris@16:       // Where Y is a constant, and R(p) is optimised for a low
Chris@16:       // absolute error compared to |Y|.
Chris@16:       //
Chris@16:       // double: Max error found: 2.001849e-18
Chris@16:       // long double: Max error found: 1.017064e-20
Chris@16:       // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
Chris@16:       //
Chris@16:       static const float Y = 0.0891314744949340820313f;
Chris@16:       static const T P[] = {    
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000508781949658280665617),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00836874819741736770379),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0334806625409744615033),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0126926147662974029034),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0365637971411762664006),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0219878681111168899165),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00822687874676915743155),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00538772965071242932965)
Chris@16:       };
Chris@16:       static const T Q[] = {    
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.970005043303290640362),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -1.56574558234175846809),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 1.56221558398423026363),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.662328840472002992063),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.71228902341542847553),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0527396382340099713954),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0795283687341571680018),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00233393759374190016776),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000886216390456424707504)
Chris@16:       };
Chris@16:       T g = p * (p + 10);
Chris@16:       T r = tools::evaluate_polynomial(P, p) / tools::evaluate_polynomial(Q, p);
Chris@16:       result = g * Y + g * r;
Chris@16:    }
Chris@16:    else if(q >= 0.25)
Chris@16:    {
Chris@16:       //
Chris@16:       // Rational approximation for 0.5 > q >= 0.25
Chris@16:       //
Chris@16:       // x = sqrt(-2*log(q)) / (Y + R(q))
Chris@16:       //
Chris@16:       // Where Y is a constant, and R(q) is optimised for a low
Chris@16:       // absolute error compared to Y.
Chris@16:       //
Chris@16:       // double : Max error found: 7.403372e-17
Chris@16:       // long double : Max error found: 6.084616e-20
Chris@16:       // Maximum Deviation Found (error term) 4.811e-20
Chris@16:       //
Chris@16:       static const float Y = 2.249481201171875f;
Chris@16:       static const T P[] = {    
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -0.202433508355938759655),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 0.105264680699391713268),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 8.37050328343119927838),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 17.6447298408374015486),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -18.8510648058714251895),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -44.6382324441786960818),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 17.445385985570866523),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 21.1294655448340526258),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -3.67192254707729348546)
Chris@16:       };
Chris@16:       static const T Q[] = {    
Chris@101:          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 6.24264124854247537712),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 3.9713437953343869095),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -28.6608180499800029974),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -20.1432634680485188801),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 48.5609213108739935468),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 10.8268667355460159008),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, -22.6436933413139721736),
Chris@16:          BOOST_MATH_BIG_CONSTANT(T, 64, 1.72114765761200282724)
Chris@16:       };
Chris@16:       T g = sqrt(-2 * log(q));
Chris@101:       T xs = q - 0.25f;
Chris@16:       T r = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:       result = g / (Y + r);
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       //
Chris@16:       // For q < 0.25 we have a series of rational approximations all
Chris@16:       // of the general form:
Chris@16:       //
Chris@16:       // let: x = sqrt(-log(q))
Chris@16:       //
Chris@16:       // Then the result is given by:
Chris@16:       //
Chris@16:       // x(Y+R(x-B))
Chris@16:       //
Chris@16:       // where Y is a constant, B is the lowest value of x for which 
Chris@16:       // the approximation is valid, and R(x-B) is optimised for a low
Chris@16:       // absolute error compared to Y.
Chris@16:       //
Chris@16:       // Note that almost all code will really go through the first
Chris@16:       // or maybe second approximation.  After than we're dealing with very
Chris@16:       // small input values indeed: 80 and 128 bit long double's go all the
Chris@16:       // way down to ~ 1e-5000 so the "tail" is rather long...
Chris@16:       //
Chris@16:       T x = sqrt(-log(q));
Chris@16:       if(x < 3)
Chris@16:       {
Chris@16:          // Max error found: 1.089051e-20
Chris@16:          static const float Y = 0.807220458984375f;
Chris@16:          static const T P[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.131102781679951906451),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.163794047193317060787),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.117030156341995252019),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.387079738972604337464),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.337785538912035898924),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.142869534408157156766),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0290157910005329060432),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00214558995388805277169),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.679465575181126350155e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.285225331782217055858e-7),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.681149956853776992068e-9)
Chris@16:          };
Chris@16:          static const T Q[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 3.46625407242567245975),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 5.38168345707006855425),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 4.77846592945843778382),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 2.59301921623620271374),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.848854343457902036425),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.152264338295331783612),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.01105924229346489121)
Chris@16:          };
Chris@101:          T xs = x - 1.125f;
Chris@16:          T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:          result = Y * x + R * x;
Chris@16:       }
Chris@16:       else if(x < 6)
Chris@16:       {
Chris@16:          // Max error found: 8.389174e-21
Chris@16:          static const float Y = 0.93995571136474609375f;
Chris@16:          static const T P[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0350353787183177984712),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.00222426529213447927281),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0185573306514231072324),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00950804701325919603619),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00187123492819559223345),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000157544617424960554631),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.460469890584317994083e-5),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.230404776911882601748e-9),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.266339227425782031962e-11)
Chris@16:          };
Chris@16:          static const T Q[] = {    
Chris@101:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.3653349817554063097),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.762059164553623404043),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.220091105764131249824),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0341589143670947727934),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00263861676657015992959),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.764675292302794483503e-4)
Chris@16:          };
Chris@16:          T xs = x - 3;
Chris@16:          T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:          result = Y * x + R * x;
Chris@16:       }
Chris@16:       else if(x < 18)
Chris@16:       {
Chris@16:          // Max error found: 1.481312e-19
Chris@16:          static const float Y = 0.98362827301025390625f;
Chris@16:          static const T P[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0167431005076633737133),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.00112951438745580278863),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00105628862152492910091),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000209386317487588078668),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.149624783758342370182e-4),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.449696789927706453732e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.462596163522878599135e-8),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.281128735628831791805e-13),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.99055709973310326855e-16)
Chris@16:          };
Chris@16:          static const T Q[] = {    
Chris@101:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.591429344886417493481),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.138151865749083321638),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0160746087093676504695),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000964011807005165528527),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.275335474764726041141e-4),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.282243172016108031869e-6)
Chris@16:          };
Chris@16:          T xs = x - 6;
Chris@16:          T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:          result = Y * x + R * x;
Chris@16:       }
Chris@16:       else if(x < 44)
Chris@16:       {
Chris@16:          // Max error found: 5.697761e-20
Chris@16:          static const float Y = 0.99714565277099609375f;
Chris@16:          static const T P[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0024978212791898131227),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.779190719229053954292e-5),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.254723037413027451751e-4),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.162397777342510920873e-5),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.396341011304801168516e-7),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.411632831190944208473e-9),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.145596286718675035587e-11),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.116765012397184275695e-17)
Chris@16:          };
Chris@16:          static const T Q[] = {    
Chris@101:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.207123112214422517181),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0169410838120975906478),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000690538265622684595676),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.145007359818232637924e-4),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.144437756628144157666e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.509761276599778486139e-9)
Chris@16:          };
Chris@16:          T xs = x - 18;
Chris@16:          T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:          result = Y * x + R * x;
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          // Max error found: 1.279746e-20
Chris@16:          static const float Y = 0.99941349029541015625f;
Chris@16:          static const T P[] = {    
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.000539042911019078575891),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.28398759004727721098e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.899465114892291446442e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.229345859265920864296e-7),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.225561444863500149219e-9),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.947846627503022684216e-12),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.135880130108924861008e-14),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, -0.348890393399948882918e-21)
Chris@16:          };
Chris@16:          static const T Q[] = {    
Chris@101:             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0845746234001899436914),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00282092984726264681981),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.468292921940894236786e-4),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.399968812193862100054e-6),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.161809290887904476097e-8),
Chris@16:             BOOST_MATH_BIG_CONSTANT(T, 64, 0.231558608310259605225e-11)
Chris@16:          };
Chris@16:          T xs = x - 44;
Chris@16:          T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
Chris@16:          result = Y * x + R * x;
Chris@16:       }
Chris@16:    }
Chris@16:    return result;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: struct erf_roots
Chris@16: {
Chris@16:    boost::math::tuple<T,T,T> operator()(const T& guess)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       T derivative = sign * (2 / sqrt(constants::pi<T>())) * exp(-(guess * guess));
Chris@16:       T derivative2 = -2 * guess * derivative;
Chris@16:       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);
Chris@16:    }
Chris@16:    erf_roots(T z, int s) : target(z), sign(s) {}
Chris@16: private:
Chris@16:    T target;
Chris@16:    int sign;
Chris@16: };
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: T erf_inv_imp(const T& p, const T& q, const Policy& pol, const boost::mpl::int_<0>*)
Chris@16: {
Chris@16:    //
Chris@16:    // Generic version, get a guess that's accurate to 64-bits (10^-19)
Chris@16:    //
Chris@16:    T guess = erf_inv_imp(p, q, pol, static_cast<mpl::int_<64> const*>(0));
Chris@16:    T result;
Chris@16:    //
Chris@16:    // If T has more bit's than 64 in it's mantissa then we need to iterate,
Chris@16:    // otherwise we can just return the result:
Chris@16:    //
Chris@16:    if(policies::digits<T, Policy>() > 64)
Chris@16:    {
Chris@16:       boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
Chris@16:       if(p <= 0.5)
Chris@16:       {
Chris@16:          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);
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          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);
Chris@16:       }
Chris@16:       policies::check_root_iterations<T>("boost::math::erf_inv<%1%>", max_iter, pol);
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       result = guess;
Chris@16:    }
Chris@16:    return result;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: struct erf_inv_initializer
Chris@16: {
Chris@16:    struct init
Chris@16:    {
Chris@16:       init()
Chris@16:       {
Chris@16:          do_init();
Chris@16:       }
Chris@101:       static bool is_value_non_zero(T);
Chris@16:       static void do_init()
Chris@16:       {
Chris@16:          boost::math::erf_inv(static_cast<T>(0.25), Policy());
Chris@16:          boost::math::erf_inv(static_cast<T>(0.55), Policy());
Chris@16:          boost::math::erf_inv(static_cast<T>(0.95), Policy());
Chris@16:          boost::math::erfc_inv(static_cast<T>(1e-15), Policy());
Chris@101:          // These following initializations must not be called if
Chris@101:          // type T can not hold the relevant values without
Chris@101:          // underflow to zero.  We check this at runtime because
Chris@101:          // some tools such as valgrind silently change the precision
Chris@101:          // of T at runtime, and numeric_limits basically lies!
Chris@101:          if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130))))
Chris@16:             boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-130)), Policy());
Chris@16: 
Chris@16:          // Some compilers choke on constants that would underflow, even in code that isn't instantiated
Chris@16:          // so try and filter these cases out in the preprocessor:
Chris@16: #if LDBL_MAX_10_EXP >= 800
Chris@101:          if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800))))
Chris@16:             boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-800)), Policy());
Chris@101:          if(is_value_non_zero(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900))))
Chris@16:             boost::math::erfc_inv(static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1e-900)), Policy());
Chris@16: #else
Chris@101:          if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800))))
Chris@16:             boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-800)), Policy());
Chris@101:          if(is_value_non_zero(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900))))
Chris@16:             boost::math::erfc_inv(static_cast<T>(BOOST_MATH_HUGE_CONSTANT(T, 64, 1e-900)), Policy());
Chris@16: #endif
Chris@16:       }
Chris@16:       void force_instantiate()const{}
Chris@16:    };
Chris@16:    static const init initializer;
Chris@16:    static void force_instantiate()
Chris@16:    {
Chris@16:       initializer.force_instantiate();
Chris@16:    }
Chris@16: };
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: const typename erf_inv_initializer<T, Policy>::init erf_inv_initializer<T, Policy>::initializer;
Chris@16: 
Chris@101: template <class T, class Policy>
Chris@101: bool erf_inv_initializer<T, Policy>::init::is_value_non_zero(T v)
Chris@101: {
Chris@101:    // This needs to be non-inline to detect whether v is non zero at runtime
Chris@101:    // rather than at compile time, only relevant when running under valgrind
Chris@101:    // which changes long double's to double's on the fly.
Chris@101:    return v != 0;
Chris@101: }
Chris@101: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: typename tools::promote_args<T>::type erfc_inv(T z, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T>::type result_type;
Chris@16: 
Chris@16:    //
Chris@16:    // Begin by testing for domain errors, and other special cases:
Chris@16:    //
Chris@16:    static const char* function = "boost::math::erfc_inv<%1%>(%1%, %1%)";
Chris@16:    if((z < 0) || (z > 2))
Chris@101:       return policies::raise_domain_error<result_type>(function, "Argument outside range [0,2] in inverse erfc function (got p=%1%).", z, pol);
Chris@16:    if(z == 0)
Chris@16:       return policies::raise_overflow_error<result_type>(function, 0, pol);
Chris@16:    if(z == 2)
Chris@16:       return -policies::raise_overflow_error<result_type>(function, 0, pol);
Chris@16:    //
Chris@16:    // Normalise the input, so it's in the range [0,1], we will
Chris@16:    // negate the result if z is outside that range.  This is a simple
Chris@16:    // application of the erfc reflection formula: erfc(-z) = 2 - erfc(z)
Chris@16:    //
Chris@16:    result_type p, q, s;
Chris@16:    if(z > 1)
Chris@16:    {
Chris@16:       q = 2 - z;
Chris@16:       p = 1 - q;
Chris@16:       s = -1;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       p = 1 - z;
Chris@16:       q = z;
Chris@16:       s = 1;
Chris@16:    }
Chris@16:    //
Chris@16:    // A bit of meta-programming to figure out which implementation
Chris@16:    // to use, based on the number of bits in the mantissa of T:
Chris@16:    //
Chris@16:    typedef typename policies::precision<result_type, Policy>::type precision_type;
Chris@16:    typedef typename mpl::if_<
Chris@16:       mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
Chris@16:       mpl::int_<0>,
Chris@16:       mpl::int_<64>
Chris@16:    >::type tag_type;
Chris@16:    //
Chris@16:    // Likewise use internal promotion, so we evaluate at a higher
Chris@16:    // precision internally if it's appropriate:
Chris@16:    //
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type eval_type;
Chris@16:    typedef typename policies::normalise<
Chris@16:       Policy, 
Chris@16:       policies::promote_float<false>, 
Chris@16:       policies::promote_double<false>, 
Chris@16:       policies::discrete_quantile<>,
Chris@16:       policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:    detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
Chris@16: 
Chris@16:    //
Chris@16:    // And get the result, negating where required:
Chris@16:    //
Chris@16:    return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
Chris@16:       detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: typename tools::promote_args<T>::type erf_inv(T z, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T>::type result_type;
Chris@16: 
Chris@16:    //
Chris@16:    // Begin by testing for domain errors, and other special cases:
Chris@16:    //
Chris@16:    static const char* function = "boost::math::erf_inv<%1%>(%1%, %1%)";
Chris@16:    if((z < -1) || (z > 1))
Chris@101:       return policies::raise_domain_error<result_type>(function, "Argument outside range [-1, 1] in inverse erf function (got p=%1%).", z, pol);
Chris@16:    if(z == 1)
Chris@16:       return policies::raise_overflow_error<result_type>(function, 0, pol);
Chris@16:    if(z == -1)
Chris@16:       return -policies::raise_overflow_error<result_type>(function, 0, pol);
Chris@16:    if(z == 0)
Chris@16:       return 0;
Chris@16:    //
Chris@16:    // Normalise the input, so it's in the range [0,1], we will
Chris@16:    // negate the result if z is outside that range.  This is a simple
Chris@16:    // application of the erf reflection formula: erf(-z) = -erf(z)
Chris@16:    //
Chris@16:    result_type p, q, s;
Chris@16:    if(z < 0)
Chris@16:    {
Chris@16:       p = -z;
Chris@16:       q = 1 - p;
Chris@16:       s = -1;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       p = z;
Chris@16:       q = 1 - z;
Chris@16:       s = 1;
Chris@16:    }
Chris@16:    //
Chris@16:    // A bit of meta-programming to figure out which implementation
Chris@16:    // to use, based on the number of bits in the mantissa of T:
Chris@16:    //
Chris@16:    typedef typename policies::precision<result_type, Policy>::type precision_type;
Chris@16:    typedef typename mpl::if_<
Chris@16:       mpl::or_<mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<64> > >,
Chris@16:       mpl::int_<0>,
Chris@16:       mpl::int_<64>
Chris@16:    >::type tag_type;
Chris@16:    //
Chris@16:    // Likewise use internal promotion, so we evaluate at a higher
Chris@16:    // precision internally if it's appropriate:
Chris@16:    //
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type eval_type;
Chris@16:    typedef typename policies::normalise<
Chris@16:       Policy, 
Chris@16:       policies::promote_float<false>, 
Chris@16:       policies::promote_double<false>, 
Chris@16:       policies::discrete_quantile<>,
Chris@16:       policies::assert_undefined<> >::type forwarding_policy;
Chris@16:    //
Chris@16:    // Likewise use internal promotion, so we evaluate at a higher
Chris@16:    // precision internally if it's appropriate:
Chris@16:    //
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type eval_type;
Chris@16: 
Chris@16:    detail::erf_inv_initializer<eval_type, forwarding_policy>::force_instantiate();
Chris@16:    //
Chris@16:    // And get the result, negating where required:
Chris@16:    //
Chris@16:    return s * policies::checked_narrowing_cast<result_type, forwarding_policy>(
Chris@16:       detail::erf_inv_imp(static_cast<eval_type>(p), static_cast<eval_type>(q), forwarding_policy(), static_cast<tag_type const*>(0)), function);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type erfc_inv(T z)
Chris@16: {
Chris@16:    return erfc_inv(z, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type erf_inv(T z)
Chris@16: {
Chris@16:    return erf_inv(z, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif // BOOST_MATH_SF_ERF_INV_HPP
Chris@16: