Imported Upstream version 1.49.0upstream/1.49.0

1 files changed, 1447 insertions, 0 deletions

diff --git a/boost/math/special_functions/beta.hpp b/boost/math/special_functions/beta.hpp
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+//  (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_SPECIAL_BETA_HPP
+#define BOOST_MATH_SPECIAL_BETA_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/tools/config.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+#include <boost/math/special_functions/factorials.hpp>
+#include <boost/math/special_functions/erf.hpp>
+#include <boost/math/special_functions/log1p.hpp>
+#include <boost/math/special_functions/expm1.hpp>
+#include <boost/math/special_functions/trunc.hpp>
+#include <boost/math/tools/roots.hpp>
+#include <boost/static_assert.hpp>
+#include <boost/config/no_tr1/cmath.hpp>
+
+namespace boost{ namespace math{
+
+namespace detail{
+
+//
+// Implementation of Beta(a,b) using the Lanczos approximation:
+//
+template <class T, class Lanczos, class Policy>
+T beta_imp(T a, T b, const Lanczos&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING  // for ADL of std names
+
+   if(a <= 0)
+      policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got a=%1%).", a, pol);
+   if(b <= 0)
+      policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got b=%1%).", b, pol);
+
+   T result;
+
+   T prefix = 1;
+   T c = a + b;
+
+   // Special cases:
+   if((c == a) && (b < tools::epsilon<T>()))
+      return boost::math::tgamma(b, pol);
+   else if((c == b) && (a < tools::epsilon<T>()))
+      return boost::math::tgamma(a, pol);
+   if(b == 1)
+      return 1/a;
+   else if(a == 1)
+      return 1/b;
+
+   /*
+   //
+   // This code appears to be no longer necessary: it was
+   // used to offset errors introduced from the Lanczos
+   // approximation, but the current Lanczos approximations
+   // are sufficiently accurate for all z that we can ditch
+   // this.  It remains in the file for future reference...
+   //
+   // If a or b are less than 1, shift to greater than 1:
+   if(a < 1)
+   {
+      prefix *= c / a;
+      c += 1;
+      a += 1;
+   }
+   if(b < 1)
+   {
+      prefix *= c / b;
+      c += 1;
+      b += 1;
+   }
+   */
+
+   if(a < b)
+      std::swap(a, b);
+
+   // Lanczos calculation:
+   T agh = a + Lanczos::g() - T(0.5);
+   T bgh = b + Lanczos::g() - T(0.5);
+   T cgh = c + Lanczos::g() - T(0.5);
+   result = Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b) / Lanczos::lanczos_sum_expG_scaled(c);
+   T ambh = a - T(0.5) - b;
+   if((fabs(b * ambh) < (cgh * 100)) && (a > 100))
+   {
+      // Special case where the base of the power term is close to 1
+      // compute (1+x)^y instead:
+      result *= exp(ambh * boost::math::log1p(-b / cgh, pol));
+   }
+   else
+   {
+      result *= pow(agh / cgh, a - T(0.5) - b);
+   }
+   if(cgh > 1e10f)
+      // this avoids possible overflow, but appears to be marginally less accurate:
+      result *= pow((agh / cgh) * (bgh / cgh), b);
+   else
+      result *= pow((agh * bgh) / (cgh * cgh), b);
+   result *= sqrt(boost::math::constants::e<T>() / bgh);
+
+   // If a and b were originally less than 1 we need to scale the result:
+   result *= prefix;
+
+   return result;
+} // template <class T, class Lanczos> beta_imp(T a, T b, const Lanczos&)
+
+//
+// Generic implementation of Beta(a,b) without Lanczos approximation support
+// (Caution this is slow!!!):
+//
+template <class T, class Policy>
+T beta_imp(T a, T b, const lanczos::undefined_lanczos& /* l */, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   if(a <= 0)
+      policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got a=%1%).", a, pol);
+   if(b <= 0)
+      policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got b=%1%).", b, pol);
+
+   T result;
+
+   T prefix = 1;
+   T c = a + b;
+
+   // special cases:
+   if((c == a) && (b < tools::epsilon<T>()))
+      return boost::math::tgamma(b, pol);
+   else if((c == b) && (a < tools::epsilon<T>()))
+      return boost::math::tgamma(a, pol);
+   if(b == 1)
+      return 1/a;
+   else if(a == 1)
+      return 1/b;
+
+   // shift to a and b > 1 if required:
+   if(a < 1)
+   {
+      prefix *= c / a;
+      c += 1;
+      a += 1;
+   }
+   if(b < 1)
+   {
+      prefix *= c / b;
+      c += 1;
+      b += 1;
+   }
+   if(a < b)
+      std::swap(a, b);
+
+   // set integration limits:
+   T la = (std::max)(T(10), a);
+   T lb = (std::max)(T(10), b);
+   T lc = (std::max)(T(10), T(a+b));
+
+   // calculate the fraction parts:
+   T sa = detail::lower_gamma_series(a, la, pol) / a;
+   sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+   T sb = detail::lower_gamma_series(b, lb, pol) / b;
+   sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+   T sc = detail::lower_gamma_series(c, lc, pol) / c;
+   sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+
+   // and the exponent part:
+   result = exp(lc - la - lb) * pow(la/lc, a) * pow(lb/lc, b);
+
+   // and combine:
+   result *= sa * sb / sc;
+
+   // if a and b were originally less than 1 we need to scale the result:
+   result *= prefix;
+
+   return result;
+} // template <class T>T beta_imp(T a, T b, const lanczos::undefined_lanczos& l)
+
+
+//
+// Compute the leading power terms in the incomplete Beta:
+//
+// (x^a)(y^b)/Beta(a,b) when normalised, and
+// (x^a)(y^b) otherwise.
+//
+// Almost all of the error in the incomplete beta comes from this
+// function: particularly when a and b are large. Computing large
+// powers are *hard* though, and using logarithms just leads to
+// horrendous cancellation errors.
+//
+template <class T, class Lanczos, class Policy>
+T ibeta_power_terms(T a,
+                        T b,
+                        T x,
+                        T y,
+                        const Lanczos&,
+                        bool normalised,
+                        const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   if(!normalised)
+   {
+      // can we do better here?
+      return pow(x, a) * pow(y, b);
+   }
+
+   T result;
+
+   T prefix = 1;
+   T c = a + b;
+
+   // combine power terms with Lanczos approximation:
+   T agh = a + Lanczos::g() - T(0.5);
+   T bgh = b + Lanczos::g() - T(0.5);
+   T cgh = c + Lanczos::g() - T(0.5);
+   result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
+
+   // l1 and l2 are the base of the exponents minus one:
+   T l1 = (x * b - y * agh) / agh;
+   T l2 = (y * a - x * bgh) / bgh;
+   if(((std::min)(fabs(l1), fabs(l2)) < 0.2))
+   {
+      // when the base of the exponent is very near 1 we get really
+      // gross errors unless extra care is taken:
+      if((l1 * l2 > 0) || ((std::min)(a, b) < 1))
+      {
+         //
+         // This first branch handles the simple cases where either: 
+         //
+         // * The two power terms both go in the same direction 
+         // (towards zero or towards infinity).  In this case if either 
+         // term overflows or underflows, then the product of the two must 
+         // do so also.  
+         // *Alternatively if one exponent is less than one, then we 
+         // can't productively use it to eliminate overflow or underflow 
+         // from the other term.  Problems with spurious overflow/underflow 
+         // can't be ruled out in this case, but it is *very* unlikely 
+         // since one of the power terms will evaluate to a number close to 1.
+         //
+         if(fabs(l1) < 0.1)
+         {
+            result *= exp(a * boost::math::log1p(l1, pol));
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+         else
+         {
+            result *= pow((x * cgh) / agh, a);
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+         if(fabs(l2) < 0.1)
+         {
+            result *= exp(b * boost::math::log1p(l2, pol));
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+         else
+         {
+            result *= pow((y * cgh) / bgh, b);
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+      }
+      else if((std::max)(fabs(l1), fabs(l2)) < 0.5)
+      {
+         //
+         // Both exponents are near one and both the exponents are 
+         // greater than one and further these two 
+         // power terms tend in opposite directions (one towards zero, 
+         // the other towards infinity), so we have to combine the terms 
+         // to avoid any risk of overflow or underflow.
+         //
+         // We do this by moving one power term inside the other, we have:
+         //
+         //    (1 + l1)^a * (1 + l2)^b
+         //  = ((1 + l1)*(1 + l2)^(b/a))^a
+         //  = (1 + l1 + l3 + l1*l3)^a   ;  l3 = (1 + l2)^(b/a) - 1
+         //                                    = exp((b/a) * log(1 + l2)) - 1
+         //
+         // The tricky bit is deciding which term to move inside :-)
+         // By preference we move the larger term inside, so that the
+         // size of the largest exponent is reduced.  However, that can
+         // only be done as long as l3 (see above) is also small.
+         //
+         bool small_a = a < b;
+         T ratio = b / a;
+         if((small_a && (ratio * l2 < 0.1)) || (!small_a && (l1 / ratio > 0.1)))
+         {
+            T l3 = boost::math::expm1(ratio * boost::math::log1p(l2, pol), pol);
+            l3 = l1 + l3 + l3 * l1;
+            l3 = a * boost::math::log1p(l3, pol);
+            result *= exp(l3);
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+         else
+         {
+            T l3 = boost::math::expm1(boost::math::log1p(l1, pol) / ratio, pol);
+            l3 = l2 + l3 + l3 * l2;
+            l3 = b * boost::math::log1p(l3, pol);
+            result *= exp(l3);
+            BOOST_MATH_INSTRUMENT_VARIABLE(result);
+         }
+      }
+      else if(fabs(l1) < fabs(l2))
+      {
+         // First base near 1 only:
+         T l = a * boost::math::log1p(l1, pol)
+            + b * log((y * cgh) / bgh);
+         result *= exp(l);
+         BOOST_MATH_INSTRUMENT_VARIABLE(result);
+      }
+      else
+      {
+         // Second base near 1 only:
+         T l = b * boost::math::log1p(l2, pol)
+            + a * log((x * cgh) / agh);
+         result *= exp(l);
+         BOOST_MATH_INSTRUMENT_VARIABLE(result);
+      }
+   }
+   else
+   {
+      // general case:
+      T b1 = (x * cgh) / agh;
+      T b2 = (y * cgh) / bgh;
+      l1 = a * log(b1);
+      l2 = b * log(b2);
+      BOOST_MATH_INSTRUMENT_VARIABLE(b1);
+      BOOST_MATH_INSTRUMENT_VARIABLE(b2);
+      BOOST_MATH_INSTRUMENT_VARIABLE(l1);
+      BOOST_MATH_INSTRUMENT_VARIABLE(l2);
+      if((l1 >= tools::log_max_value<T>())
+         || (l1 <= tools::log_min_value<T>())
+         || (l2 >= tools::log_max_value<T>())
+         || (l2 <= tools::log_min_value<T>())
+         )
+      {
+         // Oops, overflow, sidestep:
+         if(a < b)
+            result *= pow(pow(b2, b/a) * b1, a);
+         else
+            result *= pow(pow(b1, a/b) * b2, b);
+         BOOST_MATH_INSTRUMENT_VARIABLE(result);
+      }
+      else
+      {
+         // finally the normal case:
+         result *= pow(b1, a) * pow(b2, b);
+         BOOST_MATH_INSTRUMENT_VARIABLE(result);
+      }
+   }
+   // combine with the leftover terms from the Lanczos approximation:
+   result *= sqrt(bgh / boost::math::constants::e<T>());
+   result *= sqrt(agh / cgh);
+   result *= prefix;
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(result);
+
+   return result;
+}
+//
+// Compute the leading power terms in the incomplete Beta:
+//
+// (x^a)(y^b)/Beta(a,b) when normalised, and
+// (x^a)(y^b) otherwise.
+//
+// Almost all of the error in the incomplete beta comes from this
+// function: particularly when a and b are large. Computing large
+// powers are *hard* though, and using logarithms just leads to
+// horrendous cancellation errors.
+//
+// This version is generic, slow, and does not use the Lanczos approximation.
+//
+template <class T, class Policy>
+T ibeta_power_terms(T a,
+                        T b,
+                        T x,
+                        T y,
+                        const boost::math::lanczos::undefined_lanczos&,
+                        bool normalised,
+                        const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   if(!normalised)
+   {
+      return pow(x, a) * pow(y, b);
+   }
+
+   T result= 0; // assignment here silences warnings later
+
+   T c = a + b;
+
+   // integration limits for the gamma functions:
+   //T la = (std::max)(T(10), a);
+   //T lb = (std::max)(T(10), b);
+   //T lc = (std::max)(T(10), a+b);
+   T la = a + 5;
+   T lb = b + 5;
+   T lc = a + b + 5;
+   // gamma function partials:
+   T sa = detail::lower_gamma_series(a, la, pol) / a;
+   sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+   T sb = detail::lower_gamma_series(b, lb, pol) / b;
+   sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+   T sc = detail::lower_gamma_series(c, lc, pol) / c;
+   sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+   // gamma function powers combined with incomplete beta powers:
+
+   T b1 = (x * lc) / la;
+   T b2 = (y * lc) / lb;
+   T e1 = lc - la - lb;
+   T lb1 = a * log(b1);
+   T lb2 = b * log(b2);
+
+   if((lb1 >= tools::log_max_value<T>())
+      || (lb1 <= tools::log_min_value<T>())
+      || (lb2 >= tools::log_max_value<T>())
+      || (lb2 <= tools::log_min_value<T>())
+      || (e1 >= tools::log_max_value<T>())
+      || (e1 <= tools::log_min_value<T>())
+      )
+   {
+      result = exp(lb1 + lb2 - e1);
+   }
+   else
+   {
+      T p1, p2;
+      if((fabs(b1 - 1) * a < 10) && (a > 1))
+         p1 = exp(a * boost::math::log1p((x * b - y * la) / la, pol));
+      else
+         p1 = pow(b1, a);
+      if((fabs(b2 - 1) * b < 10) && (b > 1))
+         p2 = exp(b * boost::math::log1p((y * a - x * lb) / lb, pol));
+      else
+         p2 = pow(b2, b);
+      T p3 = exp(e1);
+      result = p1 * p2 / p3;
+   }
+   // and combine with the remaining gamma function components:
+   result /= sa * sb / sc;
+
+   return result;
+}
+//
+// Series approximation to the incomplete beta:
+//
+template <class T>
+struct ibeta_series_t
+{
+   typedef T result_type;
+   ibeta_series_t(T a_, T b_, T x_, T mult) : result(mult), x(x_), apn(a_), poch(1-b_), n(1) {}
+   T operator()()
+   {
+      T r = result / apn;
+      apn += 1;
+      result *= poch * x / n;
+      ++n;
+      poch += 1;
+      return r;
+   }
+private:
+   T result, x, apn, poch;
+   int n;
+};
+
+template <class T, class Lanczos, class Policy>
+T ibeta_series(T a, T b, T x, T s0, const Lanczos&, bool normalised, T* p_derivative, T y, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   T result;
+
+   BOOST_ASSERT((p_derivative == 0) || normalised);
+
+   if(normalised)
+   {
+      T c = a + b;
+
+      // incomplete beta power term, combined with the Lanczos approximation:
+      T agh = a + Lanczos::g() - T(0.5);
+      T bgh = b + Lanczos::g() - T(0.5);
+      T cgh = c + Lanczos::g() - T(0.5);
+      result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
+      if(a * b < bgh * 10)
+         result *= exp((b - 0.5f) * boost::math::log1p(a / bgh, pol));
+      else
+         result *= pow(cgh / bgh, b - 0.5f);
+      result *= pow(x * cgh / agh, a);
+      result *= sqrt(agh / boost::math::constants::e<T>());
+
+      if(p_derivative)
+      {
+         *p_derivative = result * pow(y, b);
+         BOOST_ASSERT(*p_derivative >= 0);
+      }
+   }
+   else
+   {
+      // Non-normalised, just compute the power:
+      result = pow(x, a);
+   }
+   if(result < tools::min_value<T>())
+      return s0; // Safeguard: series can't cope with denorms.
+   ibeta_series_t<T> s(a, b, x, result);
+   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+   result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, s0);
+   policies::check_series_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (with lanczos)", max_iter, pol);
+   return result;
+}
+//
+// Incomplete Beta series again, this time without Lanczos support:
+//
+template <class T, class Policy>
+T ibeta_series(T a, T b, T x, T s0, const boost::math::lanczos::undefined_lanczos&, bool normalised, T* p_derivative, T y, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   T result;
+   BOOST_ASSERT((p_derivative == 0) || normalised);
+
+   if(normalised)
+   {
+      T c = a + b;
+
+      // figure out integration limits for the gamma function:
+      //T la = (std::max)(T(10), a);
+      //T lb = (std::max)(T(10), b);
+      //T lc = (std::max)(T(10), a+b);
+      T la = a + 5;
+      T lb = b + 5;
+      T lc = a + b + 5;
+
+      // calculate the gamma parts:
+      T sa = detail::lower_gamma_series(a, la, pol) / a;
+      sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+      T sb = detail::lower_gamma_series(b, lb, pol) / b;
+      sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+      T sc = detail::lower_gamma_series(c, lc, pol) / c;
+      sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+
+      // and their combined power-terms:
+      T b1 = (x * lc) / la;
+      T b2 = lc/lb;
+      T e1 = lc - la - lb;
+      T lb1 = a * log(b1);
+      T lb2 = b * log(b2);
+
+      if((lb1 >= tools::log_max_value<T>())
+         || (lb1 <= tools::log_min_value<T>())
+         || (lb2 >= tools::log_max_value<T>())
+         || (lb2 <= tools::log_min_value<T>())
+         || (e1 >= tools::log_max_value<T>())
+         || (e1 <= tools::log_min_value<T>()) )
+      {
+         T p = lb1 + lb2 - e1;
+         result = exp(p);
+      }
+      else
+      {
+         result = pow(b1, a);
+         if(a * b < lb * 10)
+            result *= exp(b * boost::math::log1p(a / lb, pol));
+         else
+            result *= pow(b2, b);
+         result /= exp(e1);
+      }
+      // and combine the results:
+      result /= sa * sb / sc;
+
+      if(p_derivative)
+      {
+         *p_derivative = result * pow(y, b);
+         BOOST_ASSERT(*p_derivative >= 0);
+      }
+   }
+   else
+   {
+      // Non-normalised, just compute the power:
+      result = pow(x, a);
+   }
+   if(result < tools::min_value<T>())
+      return s0; // Safeguard: series can't cope with denorms.
+   ibeta_series_t<T> s(a, b, x, result);
+   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+   result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, s0);
+   policies::check_series_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (without lanczos)", max_iter, pol);
+   return result;
+}
+
+//
+// Continued fraction for the incomplete beta:
+//
+template <class T>
+struct ibeta_fraction2_t
+{
+   typedef std::pair<T, T> result_type;
+
+   ibeta_fraction2_t(T a_, T b_, T x_) : a(a_), b(b_), x(x_), m(0) {}
+
+   result_type operator()()
+   {
+      T aN = (a + m - 1) * (a + b + m - 1) * m * (b - m) * x * x;
+      T denom = (a + 2 * m - 1);
+      aN /= denom * denom;
+
+      T bN = m;
+      bN += (m * (b - m) * x) / (a + 2*m - 1);
+      bN += ((a + m) * (a - (a + b) * x + 1 + m *(2 - x))) / (a + 2*m + 1);
+
+      ++m;
+
+      return std::make_pair(aN, bN);
+   }
+
+private:
+   T a, b, x;
+   int m;
+};
+//
+// Evaluate the incomplete beta via the continued fraction representation:
+//
+template <class T, class Policy>
+inline T ibeta_fraction2(T a, T b, T x, T y, const Policy& pol, bool normalised, T* p_derivative)
+{
+   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+   BOOST_MATH_STD_USING
+   T result = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
+   if(p_derivative)
+   {
+      *p_derivative = result;
+      BOOST_ASSERT(*p_derivative >= 0);
+   }
+   if(result == 0)
+      return result;
+
+   ibeta_fraction2_t<T> f(a, b, x);
+   T fract = boost::math::tools::continued_fraction_b(f, boost::math::policies::get_epsilon<T, Policy>());
+   return result / fract;
+}
+//
+// Computes the difference between ibeta(a,b,x) and ibeta(a+k,b,x):
+//
+template <class T, class Policy>
+T ibeta_a_step(T a, T b, T x, T y, int k, const Policy& pol, bool normalised, T* p_derivative)
+{
+   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(k);
+
+   T prefix = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
+   if(p_derivative)
+   {
+      *p_derivative = prefix;
+      BOOST_ASSERT(*p_derivative >= 0);
+   }
+   prefix /= a;
+   if(prefix == 0)
+      return prefix;
+   T sum = 1;
+   T term = 1;
+   // series summation from 0 to k-1:
+   for(int i = 0; i < k-1; ++i)
+   {
+      term *= (a+b+i) * x / (a+i+1);
+      sum += term;
+   }
+   prefix *= sum;
+
+   return prefix;
+}
+//
+// This function is only needed for the non-regular incomplete beta,
+// it computes the delta in:
+// beta(a,b,x) = prefix + delta * beta(a+k,b,x)
+// it is currently only called for small k.
+//
+template <class T>
+inline T rising_factorial_ratio(T a, T b, int k)
+{
+   // calculate:
+   // (a)(a+1)(a+2)...(a+k-1)
+   // _______________________
+   // (b)(b+1)(b+2)...(b+k-1)
+
+   // This is only called with small k, for large k
+   // it is grossly inefficient, do not use outside it's
+   // intended purpose!!!
+   BOOST_MATH_INSTRUMENT_VARIABLE(k);
+   if(k == 0)
+      return 1;
+   T result = 1;
+   for(int i = 0; i < k; ++i)
+      result *= (a+i) / (b+i);
+   return result;
+}
+//
+// Routine for a > 15, b < 1
+//
+// Begin by figuring out how large our table of Pn's should be,
+// quoted accuracies are "guestimates" based on empiracal observation.
+// Note that the table size should never exceed the size of our
+// tables of factorials.
+//
+template <class T>
+struct Pn_size
+{
+   // This is likely to be enough for ~35-50 digit accuracy
+   // but it's hard to quantify exactly:
+   BOOST_STATIC_CONSTANT(unsigned, value = 50);
+   BOOST_STATIC_ASSERT(::boost::math::max_factorial<T>::value >= 100);
+};
+template <>
+struct Pn_size<float>
+{
+   BOOST_STATIC_CONSTANT(unsigned, value = 15); // ~8-15 digit accuracy
+   BOOST_STATIC_ASSERT(::boost::math::max_factorial<float>::value >= 30);
+};
+template <>
+struct Pn_size<double>
+{
+   BOOST_STATIC_CONSTANT(unsigned, value = 30); // 16-20 digit accuracy
+   BOOST_STATIC_ASSERT(::boost::math::max_factorial<double>::value >= 60);
+};
+template <>
+struct Pn_size<long double>
+{
+   BOOST_STATIC_CONSTANT(unsigned, value = 50); // ~35-50 digit accuracy
+   BOOST_STATIC_ASSERT(::boost::math::max_factorial<long double>::value >= 100);
+};
+
+template <class T, class Policy>
+T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const Policy& pol, bool normalised)
+{
+   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+   BOOST_MATH_STD_USING
+   //
+   // This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
+   //
+   // Some values we'll need later, these are Eq 9.1:
+   //
+   T bm1 = b - 1;
+   T t = a + bm1 / 2;
+   T lx, u;
+   if(y < 0.35)
+      lx = boost::math::log1p(-y, pol);
+   else
+      lx = log(x);
+   u = -t * lx;
+   // and from from 9.2:
+   T prefix;
+   T h = regularised_gamma_prefix(b, u, pol, lanczos_type());
+   if(h <= tools::min_value<T>())
+      return s0;
+   if(normalised)
+   {
+      prefix = h / boost::math::tgamma_delta_ratio(a, b, pol);
+      prefix /= pow(t, b);
+   }
+   else
+   {
+      prefix = full_igamma_prefix(b, u, pol) / pow(t, b);
+   }
+   prefix *= mult;
+   //
+   // now we need the quantity Pn, unfortunatately this is computed
+   // recursively, and requires a full history of all the previous values
+   // so no choice but to declare a big table and hope it's big enough...
+   //
+   T p[ ::boost::math::detail::Pn_size<T>::value ] = { 1 };  // see 9.3.
+   //
+   // Now an initial value for J, see 9.6:
+   //
+   T j = boost::math::gamma_q(b, u, pol) / h;
+   //
+   // Now we can start to pull things together and evaluate the sum in Eq 9:
+   //
+   T sum = s0 + prefix * j;  // Value at N = 0
+   // some variables we'll need:
+   unsigned tnp1 = 1; // 2*N+1
+   T lx2 = lx / 2;
+   lx2 *= lx2;
+   T lxp = 1;
+   T t4 = 4 * t * t;
+   T b2n = b;
+
+   for(unsigned n = 1; n < sizeof(p)/sizeof(p[0]); ++n)
+   {
+      /*
+      // debugging code, enable this if you want to determine whether
+      // the table of Pn's is large enough...
+      //
+      static int max_count = 2;
+      if(n > max_count)
+      {
+         max_count = n;
+         std::cerr << "Max iterations in BGRAT was " << n << std::endl;
+      }
+      */
+      //
+      // begin by evaluating the next Pn from Eq 9.4:
+      //
+      tnp1 += 2;
+      p[n] = 0;
+      T mbn = b - n;
+      unsigned tmp1 = 3;
+      for(unsigned m = 1; m < n; ++m)
+      {
+         mbn = m * b - n;
+         p[n] += mbn * p[n-m] / boost::math::unchecked_factorial<T>(tmp1);
+         tmp1 += 2;
+      }
+      p[n] /= n;
+      p[n] += bm1 / boost::math::unchecked_factorial<T>(tnp1);
+      //
+      // Now we want Jn from Jn-1 using Eq 9.6:
+      //
+      j = (b2n * (b2n + 1) * j + (u + b2n + 1) * lxp) / t4;
+      lxp *= lx2;
+      b2n += 2;
+      //
+      // pull it together with Eq 9:
+      //
+      T r = prefix * p[n] * j;
+      sum += r;
+      if(r > 1)
+      {
+         if(fabs(r) < fabs(tools::epsilon<T>() * sum))
+            break;
+      }
+      else
+      {
+         if(fabs(r / tools::epsilon<T>()) < fabs(sum))
+            break;
+      }
+   }
+   return sum;
+} // template <class T, class Lanczos>T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const Lanczos& l, bool normalised)
+
+//
+// For integer arguments we can relate the incomplete beta to the
+// complement of the binomial distribution cdf and use this finite sum.
+//
+template <class T>
+inline T binomial_ccdf(T n, T k, T x, T y)
+{
+   BOOST_MATH_STD_USING // ADL of std names
+   T result = pow(x, n);
+   T term = result;
+   for(unsigned i = itrunc(T(n - 1)); i > k; --i)
+   {
+      term *= ((i + 1) * y) / ((n - i) * x) ;
+      result += term;
+   }
+
+   return result;
+}
+
+
+//
+// The incomplete beta function implementation:
+// This is just a big bunch of spagetti code to divide up the
+// input range and select the right implementation method for
+// each domain:
+//
+template <class T, class Policy>
+T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, bool normalised, T* p_derivative)
+{
+   static const char* function = "boost::math::ibeta<%1%>(%1%, %1%, %1%)";
+   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+   BOOST_MATH_STD_USING // for ADL of std math functions.
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(a);
+   BOOST_MATH_INSTRUMENT_VARIABLE(b);
+   BOOST_MATH_INSTRUMENT_VARIABLE(x);
+   BOOST_MATH_INSTRUMENT_VARIABLE(inv);
+   BOOST_MATH_INSTRUMENT_VARIABLE(normalised);
+
+   bool invert = inv;
+   T fract;
+   T y = 1 - x;
+
+   BOOST_ASSERT((p_derivative == 0) || normalised);
+
+   if(p_derivative)
+      *p_derivative = -1; // value not set.
+
+   if((x < 0) || (x > 1))
+      policies::raise_domain_error<T>(function, "Parameter x outside the range [0,1] in the incomplete beta function (got x=%1%).", x, pol);
+
+   if(normalised)
+   {
+      if(a < 0)
+         policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be >= zero (got a=%1%).", a, pol);
+      if(b < 0)
+         policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be >= zero (got b=%1%).", b, pol);
+      // extend to a few very special cases:
+      if(a == 0)
+      {
+         if(b == 0)
+            policies::raise_domain_error<T>(function, "The arguments a and b to the incomplete beta function cannot both be zero, with x=%1%.", x, pol);
+         if(b > 0)
+            return inv ? 0 : 1;
+      }
+      else if(b == 0)
+      {
+         if(a > 0)
+            return inv ? 1 : 0;
+      }
+   }
+   else
+   {
+      if(a <= 0)
+         policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be greater than zero (got a=%1%).", a, pol);
+      if(b <= 0)
+         policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be greater than zero (got b=%1%).", b, pol);
+   }
+
+   if(x == 0)
+   {
+      if(p_derivative)
+      {
+         *p_derivative = (a == 1) ? (T)1 : (a < 1) ? T(tools::max_value<T>() / 2) : T(tools::min_value<T>() * 2);
+      }
+      return (invert ? (normalised ? T(1) : boost::math::beta(a, b, pol)) : T(0));
+   }
+   if(x == 1)
+   {
+      if(p_derivative)
+      {
+         *p_derivative = (b == 1) ? T(1) : (b < 1) ? T(tools::max_value<T>() / 2) : T(tools::min_value<T>() * 2);
+      }
+      return (invert == 0 ? (normalised ? 1 : boost::math::beta(a, b, pol)) : 0);
+   }
+
+   if((std::min)(a, b) <= 1)
+   {
+      if(x > 0.5)
+      {
+         std::swap(a, b);
+         std::swap(x, y);
+         invert = !invert;
+         BOOST_MATH_INSTRUMENT_VARIABLE(invert);
+      }
+      if((std::max)(a, b) <= 1)
+      {
+         // Both a,b < 1:
+         if((a >= (std::min)(T(0.2), b)) || (pow(x, a) <= 0.9))
+         {
+            if(!invert)
+            {
+               fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+            else
+            {
+               fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+               invert = false;
+               fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+         }
+         else
+         {
+            std::swap(a, b);
+            std::swap(x, y);
+            invert = !invert;
+            if(y >= 0.3)
+            {
+               if(!invert)
+               {
+                  fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+               else
+               {
+                  fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+                  invert = false;
+                  fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+            }
+            else
+            {
+               // Sidestep on a, and then use the series representation:
+               T prefix;
+               if(!normalised)
+               {
+                  prefix = rising_factorial_ratio(T(a+b), a, 20);
+               }
+               else
+               {
+                  prefix = 1;
+               }
+               fract = ibeta_a_step(a, b, x, y, 20, pol, normalised, p_derivative);
+               if(!invert)
+               {
+                  fract = beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+               else
+               {
+                  fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+                  invert = false;
+                  fract = -beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+            }
+         }
+      }
+      else
+      {
+         // One of a, b < 1 only:
+         if((b <= 1) || ((x < 0.1) && (pow(b * x, a) <= 0.7)))
+         {
+            if(!invert)
+            {
+               fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+            else
+            {
+               fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+               invert = false;
+               fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+         }
+         else
+         {
+            std::swap(a, b);
+            std::swap(x, y);
+            invert = !invert;
+
+            if(y >= 0.3)
+            {
+               if(!invert)
+               {
+                  fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+               else
+               {
+                  fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+                  invert = false;
+                  fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+            }
+            else if(a >= 15)
+            {
+               if(!invert)
+               {
+                  fract = beta_small_b_large_a_series(a, b, x, y, T(0), T(1), pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+               else
+               {
+                  fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+                  invert = false;
+                  fract = -beta_small_b_large_a_series(a, b, x, y, fract, T(1), pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+            }
+            else
+            {
+               // Sidestep to improve errors:
+               T prefix;
+               if(!normalised)
+               {
+                  prefix = rising_factorial_ratio(T(a+b), a, 20);
+               }
+               else
+               {
+                  prefix = 1;
+               }
+               fract = ibeta_a_step(a, b, x, y, 20, pol, normalised, p_derivative);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               if(!invert)
+               {
+                  fract = beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+               else
+               {
+                  fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+                  invert = false;
+                  fract = -beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+                  BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+               }
+            }
+         }
+      }
+   }
+   else
+   {
+      // Both a,b >= 1:
+      T lambda;
+      if(a < b)
+      {
+         lambda = a - (a + b) * x;
+      }
+      else
+      {
+         lambda = (a + b) * y - b;
+      }
+      if(lambda < 0)
+      {
+         std::swap(a, b);
+         std::swap(x, y);
+         invert = !invert;
+         BOOST_MATH_INSTRUMENT_VARIABLE(invert);
+      }
+      
+      if(b < 40)
+      {
+         if((floor(a) == a) && (floor(b) == b))
+         {
+            // relate to the binomial distribution and use a finite sum:
+            T k = a - 1;
+            T n = b + k;
+            fract = binomial_ccdf(n, k, x, y);
+            if(!normalised)
+               fract *= boost::math::beta(a, b, pol);
+            BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+         }
+         else if(b * x <= 0.7)
+         {
+            if(!invert)
+            {
+               fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+            else
+            {
+               fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+               invert = false;
+               fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+            }
+         }
+         else if(a > 15)
+         {
+            // sidestep so we can use the series representation:
+            int n = itrunc(T(floor(b)), pol);
+            if(n == b)
+               --n;
+            T bbar = b - n;
+            T prefix;
+            if(!normalised)
+            {
+               prefix = rising_factorial_ratio(T(a+bbar), bbar, n);
+            }
+            else
+            {
+               prefix = 1;
+            }
+            fract = ibeta_a_step(bbar, a, y, x, n, pol, normalised, static_cast<T*>(0));
+            fract = beta_small_b_large_a_series(a,  bbar, x, y, fract, T(1), pol, normalised);
+            fract /= prefix;
+            BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+         }
+         else if(normalised)
+         {
+            // the formula here for the non-normalised case is tricky to figure
+            // out (for me!!), and requires two pochhammer calculations rather
+            // than one, so leave it for now....
+            int n = itrunc(T(floor(b)), pol);
+            T bbar = b - n;
+            if(bbar <= 0)
+            {
+               --n;
+               bbar += 1;
+            }
+            fract = ibeta_a_step(bbar, a, y, x, n, pol, normalised, static_cast<T*>(0));
+            fract += ibeta_a_step(a, bbar, x, y, 20, pol, normalised, static_cast<T*>(0));
+            if(invert)
+               fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+            //fract = ibeta_series(a+20, bbar, x, fract, l, normalised, p_derivative, y);
+            fract = beta_small_b_large_a_series(T(a+20),  bbar, x, y, fract, T(1), pol, normalised);
+            if(invert)
+            {
+               fract = -fract;
+               invert = false;
+            }
+            BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+         }
+         else
+         {
+            fract = ibeta_fraction2(a, b, x, y, pol, normalised, p_derivative);
+            BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+         }
+      }
+      else
+      {
+         fract = ibeta_fraction2(a, b, x, y, pol, normalised, p_derivative);
+         BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+      }
+   }
+   if(p_derivative)
+   {
+      if(*p_derivative < 0)
+      {
+         *p_derivative = ibeta_power_terms(a, b, x, y, lanczos_type(), true, pol);
+      }
+      T div = y * x;
+
+      if(*p_derivative != 0)
+      {
+         if((tools::max_value<T>() * div < *p_derivative))
+         {
+            // overflow, return an arbitarily large value:
+            *p_derivative = tools::max_value<T>() / 2;
+         }
+         else
+         {
+            *p_derivative /= div;
+         }
+      }
+   }
+   return invert ? (normalised ? 1 : boost::math::beta(a, b, pol)) - fract : fract;
+} // template <class T, class Lanczos>T ibeta_imp(T a, T b, T x, const Lanczos& l, bool inv, bool normalised)
+
+template <class T, class Policy>
+inline T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, bool normalised)
+{
+   return ibeta_imp(a, b, x, pol, inv, normalised, static_cast<T*>(0));
+}
+
+template <class T, class Policy>
+T ibeta_derivative_imp(T a, T b, T x, const Policy& pol)
+{
+   static const char* function = "ibeta_derivative<%1%>(%1%,%1%,%1%)";
+   //
+   // start with the usual error checks:
+   //
+   if(a <= 0)
+      policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be greater than zero (got a=%1%).", a, pol);
+   if(b <= 0)
+      policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be greater than zero (got b=%1%).", b, pol);
+   if((x < 0) || (x > 1))
+      policies::raise_domain_error<T>(function, "Parameter x outside the range [0,1] in the incomplete beta function (got x=%1%).", x, pol);
+   //
+   // Now the corner cases:
+   //
+   if(x == 0)
+   {
+      return (a > 1) ? 0 : 
+         (a == 1) ? 1 / boost::math::beta(a, b, pol) : policies::raise_overflow_error<T>(function, 0, pol);
+   }
+   else if(x == 1)
+   {
+      return (b > 1) ? 0 :
+         (b == 1) ? 1 / boost::math::beta(a, b, pol) : policies::raise_overflow_error<T>(function, 0, pol);
+   }
+   //
+   // Now the regular cases:
+   //
+   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+   T f1 = ibeta_power_terms<T>(a, b, x, 1 - x, lanczos_type(), true, pol);
+   T y = (1 - x) * x;
+
+   if(f1 == 0)
+      return 0;
+   
+   if((tools::max_value<T>() * y < f1))
+   {
+      // overflow:
+      return policies::raise_overflow_error<T>(function, 0, pol);
+   }
+
+   f1 /= y;
+
+   return f1;
+}
+//
+// Some forwarding functions that dis-ambiguate the third argument type:
+//
+template <class RT1, class RT2, class Policy>
+inline typename tools::promote_args<RT1, RT2>::type 
+   beta(RT1 a, RT2 b, const Policy&, const mpl::true_*)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::beta_imp(static_cast<value_type>(a), static_cast<value_type>(b), evaluation_type(), forwarding_policy()), "boost::math::beta<%1%>(%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   beta(RT1 a, RT2 b, RT3 x, const mpl::false_*)
+{
+   return boost::math::beta(a, b, x, policies::policy<>());
+}
+} // namespace detail
+
+//
+// The actual function entry-points now follow, these just figure out
+// which Lanczos approximation to use
+// and forward to the implementation functions:
+//
+template <class RT1, class RT2, class A>
+inline typename tools::promote_args<RT1, RT2, A>::type 
+   beta(RT1 a, RT2 b, A arg)
+{
+   typedef typename policies::is_policy<A>::type tag;
+   return boost::math::detail::beta(a, b, arg, static_cast<tag*>(0));
+}
+
+template <class RT1, class RT2>
+inline typename tools::promote_args<RT1, RT2>::type 
+   beta(RT1 a, RT2 b)
+{
+   return boost::math::beta(a, b, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   beta(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), false, false), "boost::math::beta<%1%>(%1%,%1%,%1%)");
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   betac(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), true, false), "boost::math::betac<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   betac(RT1 a, RT2 b, RT3 x)
+{
+   return boost::math::betac(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibeta(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), false, true), "boost::math::ibeta<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibeta(RT1 a, RT2 b, RT3 x)
+{
+   return boost::math::ibeta(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibetac(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), true, true), "boost::math::ibetac<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibetac(RT1 a, RT2 b, RT3 x)
+{
+   return boost::math::ibetac(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibeta_derivative(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+
+   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_derivative_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy()), "boost::math::ibeta_derivative<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type 
+   ibeta_derivative(RT1 a, RT2 b, RT3 x)
+{
+   return boost::math::ibeta_derivative(a, b, x, policies::policy<>());
+}
+
+} // namespace math
+} // namespace boost
+
+#include <boost/math/special_functions/detail/ibeta_inverse.hpp>
+#include <boost/math/special_functions/detail/ibeta_inv_ab.hpp>
+
+#endif // BOOST_MATH_SPECIAL_BETA_HPP
+
+
+
+
+