comparison DEPENDENCIES/generic/include/boost/math/bindings/rr.hpp @ 16:2665513ce2d3

Add boost headers

+//  Copyright John Maddock 2007.
+//  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_NTL_RR_HPP
+#define BOOST_MATH_NTL_RR_HPP
+
+#include <boost/config.hpp>
+#include <boost/limits.hpp>
+#include <boost/math/tools/real_cast.hpp>
+#include <boost/math/tools/precision.hpp>
+#include <boost/math/constants/constants.hpp>
+#include <boost/math/tools/roots.hpp>
+#include <boost/math/special_functions/fpclassify.hpp>
+#include <boost/math/bindings/detail/big_digamma.hpp>
+#include <boost/math/bindings/detail/big_lanczos.hpp>
+
+#include <ostream>
+#include <istream>
+#include <boost/config/no_tr1/cmath.hpp>
+#include <NTL/RR.h>
+
+namespace boost{ namespace math{
+
+namespace ntl
+{
+
+class RR;
+
+RR ldexp(RR r, int exp);
+RR frexp(RR r, int* exp);
+
+class RR
+{
+public:
+   // Constructors:
+   RR() {}
+   RR(const ::NTL::RR& c) : m_value(c){}
+   RR(char c)
+   {
+      m_value = c;
+   }
+#ifndef BOOST_NO_INTRINSIC_WCHAR_T
+   RR(wchar_t c)
+   {
+      m_value = c;
+   }
+#endif
+   RR(unsigned char c)
+   {
+      m_value = c;
+   }
+   RR(signed char c)
+   {
+      m_value = c;
+   }
+   RR(unsigned short c)
+   {
+      m_value = c;
+   }
+   RR(short c)
+   {
+      m_value = c;
+   }
+   RR(unsigned int c)
+   {
+      assign_large_int(c);
+   }
+   RR(int c)
+   {
+      assign_large_int(c);
+   }
+   RR(unsigned long c)
+   {
+      assign_large_int(c);
+   }
+   RR(long c)
+   {
+      assign_large_int(c);
+   }
+#ifdef BOOST_HAS_LONG_LONG
+   RR(boost::ulong_long_type c)
+   {
+      assign_large_int(c);
+   }
+   RR(boost::long_long_type c)
+   {
+      assign_large_int(c);
+   }
+#endif
+   RR(float c)
+   {
+      m_value = c;
+   }
+   RR(double c)
+   {
+      m_value = c;
+   }
+   RR(long double c)
+   {
+      assign_large_real(c);
+   }
+
+   // Assignment:
+   RR& operator=(char c) { m_value = c; return *this; }
+   RR& operator=(unsigned char c) { m_value = c; return *this; }
+   RR& operator=(signed char c) { m_value = c; return *this; }
+#ifndef BOOST_NO_INTRINSIC_WCHAR_T
+   RR& operator=(wchar_t c) { m_value = c; return *this; }
+#endif
+   RR& operator=(short c) { m_value = c; return *this; }
+   RR& operator=(unsigned short c) { m_value = c; return *this; }
+   RR& operator=(int c) { assign_large_int(c); return *this; }
+   RR& operator=(unsigned int c) { assign_large_int(c); return *this; }
+   RR& operator=(long c) { assign_large_int(c); return *this; }
+   RR& operator=(unsigned long c) { assign_large_int(c); return *this; }
+#ifdef BOOST_HAS_LONG_LONG
+   RR& operator=(boost::long_long_type c) { assign_large_int(c); return *this; }
+   RR& operator=(boost::ulong_long_type c) { assign_large_int(c); return *this; }
+#endif
+   RR& operator=(float c) { m_value = c; return *this; }
+   RR& operator=(double c) { m_value = c; return *this; }
+   RR& operator=(long double c) { assign_large_real(c); return *this; }
+
+   // Access:
+   NTL::RR& value(){ return m_value; }
+   NTL::RR const& value()const{ return m_value; }
+
+   // Member arithmetic:
+   RR& operator+=(const RR& other)
+   { m_value += other.value(); return *this; }
+   RR& operator-=(const RR& other)
+   { m_value -= other.value(); return *this; }
+   RR& operator*=(const RR& other)
+   { m_value *= other.value(); return *this; }
+   RR& operator/=(const RR& other)
+   { m_value /= other.value(); return *this; }
+   RR operator-()const
+   { return -m_value; }
+   RR const& operator+()const
+   { return *this; }
+
+   // RR compatibity:
+   const ::NTL::ZZ& mantissa() const
+   { return m_value.mantissa(); }
+   long exponent() const
+   { return m_value.exponent(); }
+
+   static void SetPrecision(long p)
+   { ::NTL::RR::SetPrecision(p); }
+
+   static long precision()
+   { return ::NTL::RR::precision(); }
+
+   static void SetOutputPrecision(long p)
+   { ::NTL::RR::SetOutputPrecision(p); }
+   static long OutputPrecision()
+   { return ::NTL::RR::OutputPrecision(); }
+
+
+private:
+   ::NTL::RR m_value;
+
+   template <class V>
+   void assign_large_real(const V& a)
+   {
+      using std::frexp;
+      using std::ldexp;
+      using std::floor;
+      if (a == 0) {
+         clear(m_value);
+         return;
+      }
+
+      if (a == 1) {
+         NTL::set(m_value);
+         return;
+      }
+
+      if (!(boost::math::isfinite)(a))
+      {
+         throw std::overflow_error("Cannot construct an instance of NTL::RR with an infinite value.");
+      }
+
+      int e;
+      long double f, term;
+      ::NTL::RR t;
+      clear(m_value);
+
+      f = frexp(a, &e);
+
+      while(f)
+      {
+         // extract 30 bits from f:
+         f = ldexp(f, 30);
+         term = floor(f);
+         e -= 30;
+         conv(t.x, (int)term);
+         t.e = e;
+         m_value += t;
+         f -= term;
+      }
+   }
+
+   template <class V>
+   void assign_large_int(V a)
+   {
+#ifdef BOOST_MSVC
+#pragma warning(push)
+#pragma warning(disable:4146)
+#endif
+      clear(m_value);
+      int exp = 0;
+      NTL::RR t;
+      bool neg = a < V(0) ? true : false;
+      if(neg) 
+         a = -a;
+      while(a)
+      {
+         t = static_cast<double>(a & 0xffff);
+         m_value += ldexp(RR(t), exp).value();
+         a >>= 16;
+         exp += 16;
+      }
+      if(neg)
+         m_value = -m_value;
+#ifdef BOOST_MSVC
+#pragma warning(pop)
+#endif
+   }
+};
+
+// Non-member arithmetic:
+inline RR operator+(const RR& a, const RR& b)
+{
+   RR result(a);
+   result += b;
+   return result;
+}
+inline RR operator-(const RR& a, const RR& b)
+{
+   RR result(a);
+   result -= b;
+   return result;
+}
+inline RR operator*(const RR& a, const RR& b)
+{
+   RR result(a);
+   result *= b;
+   return result;
+}
+inline RR operator/(const RR& a, const RR& b)
+{
+   RR result(a);
+   result /= b;
+   return result;
+}
+
+// Comparison:
+inline bool operator == (const RR& a, const RR& b)
+{ return a.value() == b.value() ? true : false; }
+inline bool operator != (const RR& a, const RR& b)
+{ return a.value() != b.value() ? true : false;}
+inline bool operator < (const RR& a, const RR& b)
+{ return a.value() < b.value() ? true : false; }
+inline bool operator <= (const RR& a, const RR& b)
+{ return a.value() <= b.value() ? true : false; }
+inline bool operator > (const RR& a, const RR& b)
+{ return a.value() > b.value() ? true : false; }
+inline bool operator >= (const RR& a, const RR& b)
+{ return a.value() >= b.value() ? true : false; }
+
+#if 0
+// Non-member mixed compare:
+template <class T>
+inline bool operator == (const T& a, const RR& b)
+{
+   return a == b.value();
+}
+template <class T>
+inline bool operator != (const T& a, const RR& b)
+{
+   return a != b.value();
+}
+template <class T>
+inline bool operator < (const T& a, const RR& b)
+{
+   return a < b.value();
+}
+template <class T>
+inline bool operator > (const T& a, const RR& b)
+{
+   return a > b.value();
+}
+template <class T>
+inline bool operator <= (const T& a, const RR& b)
+{
+   return a <= b.value();
+}
+template <class T>
+inline bool operator >= (const T& a, const RR& b)
+{
+   return a >= b.value();
+}
+#endif  // Non-member mixed compare:
+
+// Non-member functions:
+/*
+inline RR acos(RR a)
+{ return ::NTL::acos(a.value()); }
+*/
+inline RR cos(RR a)
+{ return ::NTL::cos(a.value()); }
+/*
+inline RR asin(RR a)
+{ return ::NTL::asin(a.value()); }
+inline RR atan(RR a)
+{ return ::NTL::atan(a.value()); }
+inline RR atan2(RR a, RR b)
+{ return ::NTL::atan2(a.value(), b.value()); }
+*/
+inline RR ceil(RR a)
+{ return ::NTL::ceil(a.value()); }
+/*
+inline RR fmod(RR a, RR b)
+{ return ::NTL::fmod(a.value(), b.value()); }
+inline RR cosh(RR a)
+{ return ::NTL::cosh(a.value()); }
+*/
+inline RR exp(RR a)
+{ return ::NTL::exp(a.value()); }
+inline RR fabs(RR a)
+{ return ::NTL::fabs(a.value()); }
+inline RR abs(RR a)
+{ return ::NTL::abs(a.value()); }
+inline RR floor(RR a)
+{ return ::NTL::floor(a.value()); }
+/*
+inline RR modf(RR a, RR* ipart)
+{
+   ::NTL::RR ip;
+   RR result = modf(a.value(), &ip);
+   *ipart = ip;
+   return result;
+}
+inline RR frexp(RR a, int* expon)
+{ return ::NTL::frexp(a.value(), expon); }
+inline RR ldexp(RR a, int expon)
+{ return ::NTL::ldexp(a.value(), expon); }
+*/
+inline RR log(RR a)
+{ return ::NTL::log(a.value()); }
+inline RR log10(RR a)
+{ return ::NTL::log10(a.value()); }
+/*
+inline RR tan(RR a)
+{ return ::NTL::tan(a.value()); }
+*/
+inline RR pow(RR a, RR b)
+{ return ::NTL::pow(a.value(), b.value()); }
+inline RR pow(RR a, int b)
+{ return ::NTL::power(a.value(), b); }
+inline RR sin(RR a)
+{ return ::NTL::sin(a.value()); }
+/*
+inline RR sinh(RR a)
+{ return ::NTL::sinh(a.value()); }
+*/
+inline RR sqrt(RR a)
+{ return ::NTL::sqrt(a.value()); }
+/*
+inline RR tanh(RR a)
+{ return ::NTL::tanh(a.value()); }
+*/
+   inline RR pow(const RR& r, long l)
+   {
+      return ::NTL::power(r.value(), l);
+   }
+   inline RR tan(const RR& a)
+   {
+      return sin(a)/cos(a);
+   }
+   inline RR frexp(RR r, int* exp)
+   {
+      *exp = r.value().e;
+      r.value().e = 0;
+      while(r >= 1)
+      {
+         *exp += 1;
+         r.value().e -= 1;
+      }
+      while(r < 0.5)
+      {
+         *exp -= 1;
+         r.value().e += 1;
+      }
+      BOOST_ASSERT(r < 1);
+      BOOST_ASSERT(r >= 0.5);
+      return r;
+   }
+   inline RR ldexp(RR r, int exp)
+   {
+      r.value().e += exp;
+      return r;
+   }
+
+// Streaming:
+template <class charT, class traits>
+inline std::basic_ostream<charT, traits>& operator<<(std::basic_ostream<charT, traits>& os, const RR& a)
+{
+   return os << a.value();
+}
+template <class charT, class traits>
+inline std::basic_istream<charT, traits>& operator>>(std::basic_istream<charT, traits>& is, RR& a)
+{
+   ::NTL::RR v;
+   is >> v;
+   a = v;
+   return is;
+}
+
+} // namespace ntl
+
+namespace lanczos{
+
+struct ntl_lanczos
+{
+   static ntl::RR lanczos_sum(const ntl::RR& z)
+   {
+      unsigned long p = ntl::RR::precision();
+      if(p <= 72)
+         return lanczos13UDT::lanczos_sum(z);
+      else if(p <= 120)
+         return lanczos22UDT::lanczos_sum(z);
+      else if(p <= 170)
+         return lanczos31UDT::lanczos_sum(z);
+      else //if(p <= 370) approx 100 digit precision:
+         return lanczos61UDT::lanczos_sum(z);
+   }
+   static ntl::RR lanczos_sum_expG_scaled(const ntl::RR& z)
+   {
+      unsigned long p = ntl::RR::precision();
+      if(p <= 72)
+         return lanczos13UDT::lanczos_sum_expG_scaled(z);
+      else if(p <= 120)
+         return lanczos22UDT::lanczos_sum_expG_scaled(z);
+      else if(p <= 170)
+         return lanczos31UDT::lanczos_sum_expG_scaled(z);
+      else //if(p <= 370) approx 100 digit precision:
+         return lanczos61UDT::lanczos_sum_expG_scaled(z);
+   }
+   static ntl::RR lanczos_sum_near_1(const ntl::RR& z)
+   {
+      unsigned long p = ntl::RR::precision();
+      if(p <= 72)
+         return lanczos13UDT::lanczos_sum_near_1(z);
+      else if(p <= 120)
+         return lanczos22UDT::lanczos_sum_near_1(z);
+      else if(p <= 170)
+         return lanczos31UDT::lanczos_sum_near_1(z);
+      else //if(p <= 370) approx 100 digit precision:
+         return lanczos61UDT::lanczos_sum_near_1(z);
+   }
+   static ntl::RR lanczos_sum_near_2(const ntl::RR& z)
+   {
+      unsigned long p = ntl::RR::precision();
+      if(p <= 72)
+         return lanczos13UDT::lanczos_sum_near_2(z);
+      else if(p <= 120)
+         return lanczos22UDT::lanczos_sum_near_2(z);
+      else if(p <= 170)
+         return lanczos31UDT::lanczos_sum_near_2(z);
+      else //if(p <= 370) approx 100 digit precision:
+         return lanczos61UDT::lanczos_sum_near_2(z);
+   }
+   static ntl::RR g()
+   { 
+      unsigned long p = ntl::RR::precision();
+      if(p <= 72)
+         return lanczos13UDT::g();
+      else if(p <= 120)
+         return lanczos22UDT::g();
+      else if(p <= 170)
+         return lanczos31UDT::g();
+      else //if(p <= 370) approx 100 digit precision:
+         return lanczos61UDT::g();
+   }
+};
+
+template<class Policy>
+struct lanczos<ntl::RR, Policy>
+{
+   typedef ntl_lanczos type;
+};
+
+} // namespace lanczos
+
+namespace tools
+{
+
+template<>
+inline int digits<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   return ::NTL::RR::precision();
+}
+
+template <>
+inline float real_cast<float, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   double r;
+   conv(r, t.value());
+   return static_cast<float>(r);
+}
+template <>
+inline double real_cast<double, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   double r;
+   conv(r, t.value());
+   return r;
+}
+
+namespace detail{
+
+template<class I>
+void convert_to_long_result(NTL::RR const& r, I& result)
+{
+   result = 0;
+   I last_result(0);
+   NTL::RR t(r);
+   double term;
+   do
+   {
+      conv(term, t);
+      last_result = result;
+      result += static_cast<I>(term);
+      t -= term;
+   }while(result != last_result);
+}
+
+}
+
+template <>
+inline long double real_cast<long double, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   long double result(0);
+   detail::convert_to_long_result(t.value(), result);
+   return result;
+}
+template <>
+inline boost::math::ntl::RR real_cast<boost::math::ntl::RR, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   return t;
+}
+template <>
+inline unsigned real_cast<unsigned, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   unsigned result;
+   detail::convert_to_long_result(t.value(), result);
+   return result;
+}
+template <>
+inline int real_cast<int, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   int result;
+   detail::convert_to_long_result(t.value(), result);
+   return result;
+}
+template <>
+inline long real_cast<long, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   long result;
+   detail::convert_to_long_result(t.value(), result);
+   return result;
+}
+template <>
+inline long long real_cast<long long, boost::math::ntl::RR>(boost::math::ntl::RR t)
+{
+   long long result;
+   detail::convert_to_long_result(t.value(), result);
+   return result;
+}
+
+template <>
+inline boost::math::ntl::RR max_value<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   static bool has_init = false;
+   static NTL::RR val;
+   if(!has_init)
+   {
+      val = 1;
+      val.e = NTL_OVFBND-20;
+      has_init = true;
+   }
+   return val;
+}
+
+template <>
+inline boost::math::ntl::RR min_value<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   static bool has_init = false;
+   static NTL::RR val;
+   if(!has_init)
+   {
+      val = 1;
+      val.e = -NTL_OVFBND+20;
+      has_init = true;
+   }
+   return val;
+}
+
+template <>
+inline boost::math::ntl::RR log_max_value<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   static bool has_init = false;
+   static NTL::RR val;
+   if(!has_init)
+   {
+      val = 1;
+      val.e = NTL_OVFBND-20;
+      val = log(val);
+      has_init = true;
+   }
+   return val;
+}
+
+template <>
+inline boost::math::ntl::RR log_min_value<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   static bool has_init = false;
+   static NTL::RR val;
+   if(!has_init)
+   {
+      val = 1;
+      val.e = -NTL_OVFBND+20;
+      val = log(val);
+      has_init = true;
+   }
+   return val;
+}
+
+template <>
+inline boost::math::ntl::RR epsilon<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+   return ldexp(boost::math::ntl::RR(1), 1-boost::math::policies::digits<boost::math::ntl::RR, boost::math::policies::policy<> >());
+}
+
+} // namespace tools
+
+//
+// The number of digits precision in RR can vary with each call
+// so we need to recalculate these with each call:
+//
+namespace constants{
+
+template<> inline boost::math::ntl::RR pi<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+    NTL::RR result;
+    ComputePi(result);
+    return result;
+}
+template<> inline boost::math::ntl::RR e<boost::math::ntl::RR>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(boost::math::ntl::RR))
+{
+    NTL::RR result;
+    result = 1;
+    return exp(result);
+}
+
+} // namespace constants
+
+namespace ntl{
+   //
+   // These are some fairly brain-dead versions of the math
+   // functions that NTL fails to provide.
+   //
+
+
+   //
+   // Inverse trig functions:
+   //
+   struct asin_root
+   {
+      asin_root(RR const& target) : t(target){}
+
+      boost::math::tuple<RR, RR, RR> operator()(RR const& p)
+      {
+         RR f0 = sin(p);
+         RR f1 = cos(p);
+         RR f2 = -f0;
+         f0 -= t;
+         return boost::math::make_tuple(f0, f1, f2);
+      }
+   private:
+      RR t;
+   };
+
+   inline RR asin(RR z)
+   {
+      double r;
+      conv(r, z.value());
+      return boost::math::tools::halley_iterate(
+         asin_root(z), 
+         RR(std::asin(r)), 
+         RR(-boost::math::constants::pi<RR>()/2),
+         RR(boost::math::constants::pi<RR>()/2),
+         NTL::RR::precision());
+   }
+
+   struct acos_root
+   {
+      acos_root(RR const& target) : t(target){}
+
+      boost::math::tuple<RR, RR, RR> operator()(RR const& p)
+      {
+         RR f0 = cos(p);
+         RR f1 = -sin(p);
+         RR f2 = -f0;
+         f0 -= t;
+         return boost::math::make_tuple(f0, f1, f2);
+      }
+   private:
+      RR t;
+   };
+
+   inline RR acos(RR z)
+   {
+      double r;
+      conv(r, z.value());
+      return boost::math::tools::halley_iterate(
+         acos_root(z), 
+         RR(std::acos(r)), 
+         RR(-boost::math::constants::pi<RR>()/2),
+         RR(boost::math::constants::pi<RR>()/2),
+         NTL::RR::precision());
+   }
+
+   struct atan_root
+   {
+      atan_root(RR const& target) : t(target){}
+
+      boost::math::tuple<RR, RR, RR> operator()(RR const& p)
+      {
+         RR c = cos(p);
+         RR ta = tan(p);
+         RR f0 = ta - t;
+         RR f1 = 1 / (c * c);
+         RR f2 = 2 * ta / (c * c);
+         return boost::math::make_tuple(f0, f1, f2);
+      }
+   private:
+      RR t;
+   };
+
+   inline RR atan(RR z)
+   {
+      double r;
+      conv(r, z.value());
+      return boost::math::tools::halley_iterate(
+         atan_root(z), 
+         RR(std::atan(r)), 
+         -boost::math::constants::pi<RR>()/2,
+         boost::math::constants::pi<RR>()/2,
+         NTL::RR::precision());
+   }
+
+   inline RR atan2(RR y, RR x)
+   {
+      if(x > 0)
+         return atan(y / x);
+      if(x < 0)
+      {
+         return y < 0 ? atan(y / x) - boost::math::constants::pi<RR>() : atan(y / x) + boost::math::constants::pi<RR>();
+      }
+      return y < 0 ? -boost::math::constants::half_pi<RR>() : boost::math::constants::half_pi<RR>() ;
+   }
+
+   inline RR sinh(RR z)
+   {
+      return (expm1(z.value()) - expm1(-z.value())) / 2;
+   }
+
+   inline RR cosh(RR z)
+   {
+      return (exp(z) + exp(-z)) / 2;
+   }
+
+   inline RR tanh(RR z)
+   {
+      return sinh(z) / cosh(z);
+   }
+
+   inline RR fmod(RR x, RR y)
+   {
+      // This is a really crummy version of fmod, we rely on lots
+      // of digits to get us out of trouble...
+      RR factor = floor(x/y);
+      return x - factor * y;
+   }
+
+   template <class Policy>
+   inline int iround(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<int>(round(x, pol));
+   }
+
+   template <class Policy>
+   inline long lround(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<long>(round(x, pol));
+   }
+
+   template <class Policy>
+   inline long long llround(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<long long>(round(x, pol));
+   }
+
+   template <class Policy>
+   inline int itrunc(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<int>(trunc(x, pol));
+   }
+
+   template <class Policy>
+   inline long ltrunc(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<long>(trunc(x, pol));
+   }
+
+   template <class Policy>
+   inline long long lltrunc(RR const& x, const Policy& pol)
+   {
+      return tools::real_cast<long long>(trunc(x, pol));
+   }
+
+} // namespace ntl
+
+namespace detail{
+
+template <class Policy>
+ntl::RR digamma_imp(ntl::RR x, const mpl::int_<0>* , const Policy& pol)
+{
+   //
+   // This handles reflection of negative arguments, and all our
+   // error handling, then forwards to the T-specific approximation.
+   //
+   BOOST_MATH_STD_USING // ADL of std functions.
+
+   ntl::RR result = 0;
+   //
+   // Check for negative arguments and use reflection:
+   //
+   if(x < 0)
+   {
+      // Reflect:
+      x = 1 - x;
+      // Argument reduction for tan:
+      ntl::RR remainder = x - floor(x);
+      // Shift to negative if > 0.5:
+      if(remainder > 0.5)
+      {
+         remainder -= 1;
+      }
+      //
+      // check for evaluation at a negative pole:
+      //
+      if(remainder == 0)
+      {
+         return policies::raise_pole_error<ntl::RR>("boost::math::digamma<%1%>(%1%)", 0, (1-x), pol);
+      }
+      result = constants::pi<ntl::RR>() / tan(constants::pi<ntl::RR>() * remainder);
+   }
+   result += big_digamma(x);
+   return result;
+}
+
+} // namespace detail
+
+} // namespace math
+} // namespace boost
+
+#endif // BOOST_MATH_REAL_CONCEPT_HPP
+
+