Boost C++ Libraries

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
#pragma warning(push)
#pragma warning(disable:4127) // Conditional expression is constant
#pragma warning(disable:4702) // Unreachable code: optimization warning
#endif

#include <type_traits>

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 std::integral_constant<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
      //
      // LCOV_EXCL_START
      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)
      };
      // LCOV_EXCL_STOP
      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
      //
      // LCOV_EXCL_START
      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)
      };
      // LCOV_EXCL_STOP
      T g = sqrt(-2 * log(q));
      T xs = q - 0.25f;
      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)
      {
         // LCOV_EXCL_START
         // 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)
         };
         // LCOV_EXCL_STOP
         T xs = x - 1.125f;
         T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
         result = Y * x + R * x;
      }
      else if(x < 6)
      {
         // LCOV_EXCL_START
         // 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)
         };
         // LCOV_EXCL_STOP
         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)
      {
         // LCOV_EXCL_START
         // 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)
         };
         // LCOV_EXCL_STOP
         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)
      {
         // LCOV_EXCL_START
         // 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)
         };
         // LCOV_EXCL_STOP
         T xs = x - 18;
         T R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs);
         result = Y * x + R * x;
      }
      else
      {
         // LCOV_EXCL_START
         // 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)
         };
         // LCOV_EXCL_STOP
         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 std::integral_constant<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<std::integral_constant<int, 64> const*>(nullptr));
   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)
   {
      std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
      if(p <= 0.5)
      {
         result = tools::halley_iterate(detail::erf_roots<typename std::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 std::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;
}

} // 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, nullptr, pol);
   if(z == 2)
      return -policies::raise_overflow_error<result_type>(function, nullptr, 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 std::integral_constant<int,
      precision_type::value <= 0 ? 0 :
      precision_type::value <= 64 ? 64 : 0
   > 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*>(nullptr)), 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, nullptr, pol);
   if(z == -1)
      return -policies::raise_overflow_error<result_type>(function, nullptr, 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 std::integral_constant<int,
      precision_type::value <= 0 ? 0 :
      precision_type::value <= 64 ? 64 : 0
   > 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*>(nullptr)), 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

#ifdef _MSC_VER
#pragma warning(pop)
#endif

#endif // BOOST_MATH_SF_ERF_INV_HPP