boost/multiprecision/cpp_bin_float.hpp

///////////////////////////////////////////////////////////////
//  Copyright 2013 John Maddock. Distributed under the Boost
//  Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_

#ifndef BOOST_MATH_CPP_BIN_FLOAT_HPP
#define BOOST_MATH_CPP_BIN_FLOAT_HPP

#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/integer.hpp>
#include <boost/math/special_functions/trunc.hpp>
#include <boost/multiprecision/detail/float_string_cvt.hpp>

namespace boost{ namespace multiprecision{ namespace backends{

enum digit_base_type
{
   digit_base_2 = 2,
   digit_base_10 = 10
};

#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:4522 6326)  // multiple assignment operators specified, comparison of two constants
#endif

namespace detail{

template <class U>
inline typename enable_if_c<is_unsigned<U>::value, bool>::type is_negative(U) { return false; }
template <class S>
inline typename disable_if_c<is_unsigned<S>::value, bool>::type is_negative(S s) { return s < 0; }

}

template <unsigned Digits, digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
class cpp_bin_float
{
public:
   static const unsigned bit_count = DigitBase == digit_base_2 ? Digits : (Digits * 1000uL) / 301uL + (((Digits * 1000uL) % 301) ? 2u : 1u);
   typedef cpp_int_backend<is_void<Allocator>::value ? bit_count : 0, bit_count, is_void<Allocator>::value ? unsigned_magnitude : signed_magnitude, unchecked, Allocator> rep_type;
   typedef cpp_int_backend<is_void<Allocator>::value ? 2 * bit_count : 0, 2 * bit_count, is_void<Allocator>::value ? unsigned_magnitude : signed_magnitude, unchecked, Allocator> double_rep_type;

   typedef typename rep_type::signed_types                        signed_types;
   typedef typename rep_type::unsigned_types                      unsigned_types;
   typedef boost::mpl::list<double, long double>                  float_types;
   typedef Exponent                                               exponent_type;

   static const exponent_type max_exponent_limit = boost::integer_traits<exponent_type>::const_max - 2 * static_cast<exponent_type>(bit_count);
   static const exponent_type min_exponent_limit = boost::integer_traits<exponent_type>::const_min + 2 * static_cast<exponent_type>(bit_count);

   BOOST_STATIC_ASSERT_MSG(MinExponent >= min_exponent_limit, "Template parameter MinExponent is too negative for our internal logic to function correctly, sorry!");
   BOOST_STATIC_ASSERT_MSG(MaxExponent <= max_exponent_limit, "Template parameter MaxExponent is too large for our internal logic to function correctly, sorry!");
   BOOST_STATIC_ASSERT_MSG(MinExponent <= 0, "Template parameter MinExponent can not be positive!");
   BOOST_STATIC_ASSERT_MSG(MaxExponent >= 0, "Template parameter MaxExponent can not be negative!");

   static const exponent_type max_exponent = MaxExponent == 0 ? max_exponent_limit : MaxExponent;
   static const exponent_type min_exponent = MinExponent == 0 ? min_exponent_limit : MinExponent;

   static const exponent_type exponent_zero = max_exponent + 1;
   static const exponent_type exponent_infinity = max_exponent + 2;
   static const exponent_type exponent_nan = max_exponent + 3;

private:

   rep_type m_data;
   exponent_type m_exponent;
   bool m_sign;
public:
   cpp_bin_float() BOOST_MP_NOEXCEPT_IF(noexcept(rep_type())) : m_data(), m_exponent(exponent_nan), m_sign(false) {}

   cpp_bin_float(const cpp_bin_float &o) BOOST_MP_NOEXCEPT_IF(noexcept(rep_type(std::declval<const rep_type&>())))
      : m_data(o.m_data), m_exponent(o.m_exponent), m_sign(o.m_sign) {}

   template <unsigned D, digit_base_type B, class A, class E, E MinE, E MaxE>
   cpp_bin_float(const cpp_bin_float<D, B, A, E, MinE, MaxE> &o, typename boost::enable_if_c<(bit_count >= cpp_bin_float<D, B, A, E, MinE, MaxE>::bit_count)>::type const* = 0)
      : m_exponent(o.exponent()), m_sign(o.sign())
   {
      typename cpp_bin_float<D, B, A, E, MinE, MaxE>::rep_type b(o.bits());
      this->sign() = o.sign();
      this->exponent() = o.exponent() + (int)bit_count - (int)cpp_bin_float<D, B, A, E, MinE, MaxE>::bit_count;
      copy_and_round(*this, b);
   }

   template <unsigned D, digit_base_type B, class A, class E, E MinE, E MaxE>
   explicit cpp_bin_float(const cpp_bin_float<D, B, A, E, MinE, MaxE> &o, typename boost::disable_if_c<(bit_count >= cpp_bin_float<D, B, A, E, MinE, MaxE>::bit_count)>::type const* = 0)
      : m_exponent(o.exponent()), m_sign(o.sign())
   {
      typename cpp_bin_float<D, B, A, E, MinE, MaxE>::rep_type b(o.bits());
      this->sign() = o.sign();
      this->exponent() = o.exponent() - (int)(cpp_bin_float<D, B, A, E, MinE, MaxE>::bit_count - bit_count);
      copy_and_round(*this, b);
   }

   template <class Float>
   cpp_bin_float(const Float& f,
      typename boost::enable_if_c<
         (number_category<Float>::value == number_kind_floating_point)
         && (std::numeric_limits<Float>::digits <= (int)bit_count)
         && (std::numeric_limits<Float>::radix == 2)
      >::type const* = 0)
      : m_data(), m_exponent(0), m_sign(false)
   {
      this->assign_float(f);
   }

   cpp_bin_float& operator=(const cpp_bin_float &o) BOOST_MP_NOEXCEPT_IF(noexcept(std::declval<rep_type&>() = std::declval<const rep_type&>()))
   {
      m_data = o.m_data;
      m_exponent = o.m_exponent;
      m_sign = o.m_sign;
      return *this;
   }

   template <unsigned D, digit_base_type B, class A, class E, E MinE, E MaxE>
   cpp_bin_float& operator=(const cpp_bin_float<D, B, A, E, MinE, MaxE> &o)
   {
      typename cpp_bin_float<D, B, A, E, MinE, MaxE>::rep_type b(o.bits());
      this->exponent() = o.exponent() + (int)bit_count - (int)cpp_bin_float<D, B, A, E, MinE, MaxE>::bit_count;
      this->sign() = o.sign();
      copy_and_round(*this, b);
      return *this;
   }

   template <class Float>
   typename boost::enable_if_c<
      (number_category<Float>::value == number_kind_floating_point)
      && (std::numeric_limits<Float>::digits <= (int)bit_count)
      && (std::numeric_limits<Float>::radix == 2), cpp_bin_float&>::type operator=(const Float& f)
   {
      return assign_float(f);
   }

   template <class Float>
   typename boost::enable_if_c<is_floating_point<Float>::value, cpp_bin_float&>::type assign_float(Float f)
   {
      BOOST_MATH_STD_USING
      using default_ops::eval_add;
      typedef typename boost::multiprecision::detail::canonical<int, cpp_bin_float>::type bf_int_type;

      switch((boost::math::fpclassify)(f))
      {
      case FP_ZERO:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_zero;
         return *this;
      case FP_NAN:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_nan;
         return *this;
      case FP_INFINITE:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_infinity;
         return *this;
      }
      if(f < 0)
      {
         *this = -f;
         this->negate();
         return *this;
      }

      typedef typename mpl::front<unsigned_types>::type ui_type;
      m_data = static_cast<ui_type>(0u);
      m_sign = false;
      m_exponent = 0;

      static const int bits = sizeof(int) * CHAR_BIT - 1;
      int e;
      f = frexp(f, &e);
      while(f)
      {
         f = ldexp(f, bits);
         e -= bits;
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
         int ipart = itrunc(f);
#else
         int ipart = static_cast<int>(f);
#endif
         f -= ipart;
         m_exponent += bits;
         cpp_bin_float t;
         t = static_cast<bf_int_type>(ipart);
         eval_add(*this, t);
      }
      m_exponent += static_cast<Exponent>(e);
      return *this;
   }

   template <class Float>
   typename boost::enable_if_c<
      (number_category<Float>::value == number_kind_floating_point)
         && !is_floating_point<Float>::value
         /*&& (std::numeric_limits<number<Float> >::radix == 2)*/,
      cpp_bin_float&>::type assign_float(Float f)
   {
      BOOST_MATH_STD_USING
      using default_ops::eval_add;
      using default_ops::eval_get_sign;
      using default_ops::eval_convert_to;
      using default_ops::eval_subtract;

      typedef typename boost::multiprecision::detail::canonical<int, Float>::type f_int_type;
      typedef typename boost::multiprecision::detail::canonical<int, cpp_bin_float>::type bf_int_type;

      switch(eval_fpclassify(f))
      {
      case FP_ZERO:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_zero;
         return *this;
      case FP_NAN:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_nan;
         return *this;
      case FP_INFINITE:
         m_data = limb_type(0);
         m_sign = false;
         m_exponent = exponent_infinity;
         return *this;
      }
      if(eval_get_sign(f) < 0)
      {
         f.negate();
         *this = f;
         this->negate();
         return *this;
      }

      typedef typename mpl::front<unsigned_types>::type ui_type;
      m_data = static_cast<ui_type>(0u);
      m_sign = false;
      m_exponent = 0;

      static const int bits = sizeof(int) * CHAR_BIT - 1;
      int e;
      eval_frexp(f, f, &e);
      while(eval_get_sign(f) != 0)
      {
         eval_ldexp(f, f, bits);
         e -= bits;
         int ipart;
         eval_convert_to(&ipart, f);
         eval_subtract(f, static_cast<f_int_type>(ipart));
         m_exponent += bits;
         eval_add(*this, static_cast<bf_int_type>(ipart));
      }
      m_exponent += e;
      if(m_exponent > max_exponent)
         m_exponent = exponent_infinity;
      if(m_exponent < min_exponent)
      {
         m_data = limb_type(0u);
         m_exponent = exponent_zero;
         m_sign = false;
      }
      else if(eval_get_sign(m_data) == 0)
      {
         m_exponent = exponent_zero;
         m_sign = false;
      }
      return *this;
   }

   template <class I>
   typename boost::enable_if<is_integral<I>, cpp_bin_float&>::type operator=(const I& i)
   {
      using default_ops::eval_bit_test;
      if(!i)
      {
         m_data = static_cast<limb_type>(0);
         m_exponent = exponent_zero;
         m_sign = false;
      }
      else
      {
         typedef typename make_unsigned<I>::type ui_type;
         ui_type fi = static_cast<ui_type>(boost::multiprecision::detail::unsigned_abs(i));
         typedef typename boost::multiprecision::detail::canonical<ui_type, rep_type>::type ar_type;
         m_data = static_cast<ar_type>(fi);
         unsigned shift = msb(fi);
         if(shift >= bit_count)
         {
            m_exponent = static_cast<Exponent>(shift);
            m_data = static_cast<ar_type>(fi >> (shift + 1 - bit_count));
         }
         else
         {
            m_exponent = static_cast<Exponent>(shift);
            eval_left_shift(m_data, bit_count - shift - 1);
         }
         BOOST_ASSERT(eval_bit_test(m_data, bit_count-1));
         m_sign = detail::is_negative(i);
      }
      return *this;
   }

   cpp_bin_float& operator=(const char *s);

   void swap(cpp_bin_float &o) BOOST_NOEXCEPT
   {
      m_data.swap(o.m_data);
      std::swap(m_exponent, o.m_exponent);
      std::swap(m_sign, o.m_sign);
   }

   std::string str(std::streamsize dig, std::ios_base::fmtflags f) const;

   void negate()
   {
      if((m_exponent != exponent_zero) && (m_exponent != exponent_nan))
         m_sign = !m_sign;
   }

   int compare(const cpp_bin_float &o) const BOOST_NOEXCEPT
   {
      if(m_sign != o.m_sign)
         return m_sign ? -1 : 1;
      int result;
      if(m_exponent == exponent_nan)
         return -1;
      else if(m_exponent != o.m_exponent)
      {
         if(m_exponent == exponent_zero)
            result = -1;
         else if(o.m_exponent == exponent_zero)
            result = 1;
         else
            result = m_exponent > o.m_exponent ? 1 : -1;
      }
      else
         result = m_data.compare(o.m_data);
      if(m_sign)
         result = -result;
      return result;
   }
   template <class A>
   int compare(const A& o) const BOOST_NOEXCEPT
   {
      cpp_bin_float b;
      b = o;
      return compare(b);
   }

   rep_type& bits() { return m_data; }
   const rep_type& bits()const { return m_data; }
   exponent_type& exponent() { return m_exponent; }
   const exponent_type& exponent()const { return m_exponent; }
   bool& sign() { return m_sign; }
   const bool& sign()const { return m_sign; }
   void check_invariants()
   {
      using default_ops::eval_bit_test;
      using default_ops::eval_is_zero;
      if((m_exponent <= max_exponent) && (m_exponent >= min_exponent))
      {
         BOOST_ASSERT(eval_bit_test(m_data, bit_count - 1));
      }
      else
      {
         BOOST_ASSERT(m_exponent > max_exponent);
         BOOST_ASSERT(m_exponent <= exponent_nan);
         BOOST_ASSERT(eval_is_zero(m_data));
      }
   }
   template<class Archive>
   void serialize(Archive & ar, const unsigned int /*version*/)
   {
      ar & m_data;
      ar & m_exponent;
      ar & m_sign;
   }
};

#ifdef BOOST_MSVC
#pragma warning(pop)
#endif

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class Int>
inline void copy_and_round(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, Int &arg)
{
   // Precondition: exponent of res must have been set before this function is called
   // as we may need to adjust it based on how many cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count in arg are set.
   using default_ops::eval_msb;
   using default_ops::eval_lsb;
   using default_ops::eval_left_shift;
   using default_ops::eval_bit_test;
   using default_ops::eval_right_shift;
   using default_ops::eval_increment;
   using default_ops::eval_get_sign;

   // cancellation may have resulted in arg being all zeros:
   if(eval_get_sign(arg) == 0)
   {
      res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero;
      res.sign() = false;
      res.bits() = static_cast<limb_type>(0u);
      return;
   }
   int msb = eval_msb(arg);
   if(static_cast<int>(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count) > msb + 1)
   {
      // Must have had cancellation in subtraction, shift left and copy:
      res.bits() = arg;
      eval_left_shift(res.bits(), cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - msb - 1);
      res.exponent() -= static_cast<Exponent>(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - msb - 1);
   }
   else if(static_cast<int>(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count) < msb + 1)
   {
      // We have more cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count than we need, so round as required,
      // first get the rounding bit:
      bool roundup = eval_bit_test(arg, msb - cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count);
      // Then check for a tie:
      if(roundup && (msb - cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count == eval_lsb(arg)))
      {
         // Ties round towards even:
         if(!eval_bit_test(arg, msb - cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count + 1))
            roundup = false;
      }
      // Shift off the cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count we don't need:
      eval_right_shift(arg, msb - cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count + 1);
      res.exponent() += static_cast<Exponent>(msb - (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count + 1);
      if(roundup)
      {
         eval_increment(arg);
         if(eval_bit_test(arg, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count))
         {
            // This happens very very rairly:
            eval_right_shift(arg, 1u);
            ++res.exponent();
         }
      }
      res.bits() = arg;
   }
   else
   {
      res.bits() = arg;
   }
   BOOST_ASSERT((eval_msb(res.bits()) == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1));

   if(res.exponent() > cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent)
   {
      // Overflow:
      res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity;
      res.bits() = static_cast<limb_type>(0u);
   }
   else if(res.exponent() < cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::min_exponent)
   {
      // Underflow:
      res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero;
      res.bits() = static_cast<limb_type>(0u);
      res.sign() = false;
   }
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void do_eval_add(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   using default_ops::eval_add;
   using default_ops::eval_bit_test;

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type dt;

   // Special cases first:
   switch(a.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      res = b;
      if(res.sign())
         res.negate();
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      if(b.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan)
         res = b;
      else
         res = a;
      return; // result is still infinite.
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = a;
      return; // result is still a NaN.
   }
   switch(b.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = b;
      if(res.sign())
         res.negate();
      return; // result is infinite.
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = b;
      return; // result is a NaN.
   }

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type e_diff = a.exponent() - b.exponent();
   bool s = a.sign();
   if(e_diff >= 0)
   {
      dt = a.bits();
      if(e_diff < (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
      {
         eval_left_shift(dt, e_diff);
         res.exponent() = a.exponent() - e_diff;
         eval_add(dt, b.bits());
      }
      else
         res.exponent() = a.exponent();
   }
   else
   {
      dt= b.bits();
      if(-e_diff < (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
      {
         eval_left_shift(dt, -e_diff);
         res.exponent() = b.exponent() + e_diff;
         eval_add(dt, a.bits());
      }
      else
         res.exponent() = b.exponent();
   }

   copy_and_round(res, dt);
   res.check_invariants();
   if(res.sign() != s)
      res.negate();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void do_eval_subtract(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   using default_ops::eval_subtract;
   using default_ops::eval_bit_test;

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type dt;

   // Special cases first:
   switch(a.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      if(b.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan)
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      else
      {
         res = b;
         if(!res.sign())
            res.negate();
      }
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      if((b.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan) || (b.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity))
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      else
         res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = a;
      return; // result is still a NaN.
   }
   switch(b.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan;
      res.sign() = false;
      res.bits() = static_cast<limb_type>(0u);
      return; // result is a NaN.
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = b;
      return; // result is still a NaN.
   }

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type e_diff = a.exponent() - b.exponent();
   bool s = a.sign();
   if((e_diff > 0) || ((e_diff == 0) && a.bits().compare(b.bits()) >= 0))
   {
      dt = a.bits();
      if(e_diff < (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
      {
         eval_left_shift(dt, e_diff);
         res.exponent() = a.exponent() - e_diff;
         eval_subtract(dt, b.bits());
      }
      else
         res.exponent() = a.exponent();
   }
   else
   {
      dt = b.bits();
      if(-e_diff < (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
      {
         eval_left_shift(dt, -e_diff);
         res.exponent() = b.exponent() + e_diff;
         eval_subtract(dt, a.bits());
      }
      else
         res.exponent() = b.exponent();
      s = !s;
   }

   copy_and_round(res, dt);
   if(res.sign() != s)
      res.negate();
   res.check_invariants();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_add(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   if(a.sign() == b.sign())
      do_eval_add(res, a, b);
   else
      do_eval_subtract(res, a, b);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_add(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a)
{
   return eval_add(res, res, a);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_subtract(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   if(a.sign() != b.sign())
      do_eval_add(res, a, b);
   else
      do_eval_subtract(res, a, b);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_subtract(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a)
{
   return eval_subtract(res, res, a);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   using default_ops::eval_bit_test;
   using default_ops::eval_multiply;

   // Special cases first:
   switch(a.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      if(b.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan)
         res = b;
      else
         res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      switch(b.exponent())
      {
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
         break;
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
         res = b;
         break;
      default:
         res = a;
         break;
      }
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = a;
      return;
   }
   if(b.exponent() > cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent)
   {
      res = b;
      return;
   }
   if((a.exponent() > 0) && (b.exponent() > 0))
   {
      if(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent + 2 - a.exponent() < b.exponent())
      {
         // We will certainly overflow:
         res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity;
         res.sign() = a.sign() != b.sign();
         res.bits() = static_cast<limb_type>(0u);
         return;
      }
   }
   if((a.exponent() < 0) && (b.exponent() < 0))
   {
      if(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::min_exponent - 2 - a.exponent() > b.exponent())
      {
         // We will certainly underflow:
         res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero;
         res.sign() = false;
         res.bits() = static_cast<limb_type>(0u);
         return;
      }
   }

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type dt;
   eval_multiply(dt, a.bits(), b.bits());
   res.exponent() = a.exponent() + b.exponent() - cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count + 1;
   copy_and_round(res, dt);
   res.check_invariants();
   res.sign() = a.sign() != b.sign();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a)
{
   eval_multiply(res, res, a);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class U>
inline typename enable_if_c<is_unsigned<U>::value>::type eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const U &b)
{
   using default_ops::eval_bit_test;
   using default_ops::eval_multiply;

   // Special cases first:
   switch(a.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      if(b == 0)
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      else
         res = a;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = a;
      return;
   }

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type dt;
   typedef typename boost::multiprecision::detail::canonical<U, typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type>::type canon_ui_type;
   eval_multiply(dt, a.bits(), static_cast<canon_ui_type>(b));
   res.exponent() = a.exponent();
   copy_and_round(res, dt);
   res.check_invariants();
   res.sign() = a.sign();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class U>
inline typename enable_if_c<is_unsigned<U>::value>::type eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const U &b)
{
   eval_multiply(res, res, b);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class S>
inline typename enable_if_c<is_signed<S>::value>::type eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, const S &b)
{
   typedef typename make_unsigned<S>::type ui_type;
   eval_multiply(res, a, static_cast<ui_type>(boost::multiprecision::detail::unsigned_abs(b)));
   if(b < 0)
      res.negate();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class S>
inline typename enable_if_c<is_signed<S>::value>::type eval_multiply(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const S &b)
{
   eval_multiply(res, res, b);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &u, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &v)
{
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:6326)  // comparison of two constants
#endif
   using default_ops::eval_subtract;
   using default_ops::eval_qr;
   using default_ops::eval_bit_test;
   using default_ops::eval_get_sign;
   using default_ops::eval_increment;

   //
   // Special cases first:
   //
   switch(u.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      switch(v.exponent())
      {
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
         return;
      }
      res = u;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      switch(v.exponent())
      {
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
         return;
      }
      res = u;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      return;
   }
   switch(v.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      {
      bool s = u.sign() != v.sign();
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity().backend();
      res.sign() = s;
      return;
      }
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res.exponent() = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero;
      res.bits() = limb_type(0);
      res.sign() = false;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      return;
   }

   // We can scale u and v so that both are integers, then perform integer
   // division to obtain quotient q and remainder r, such that:
   //
   // q * v + r = u
   //
   // and hense:
   //
   // q + r/v = u/v
   //
   // From this, assuming q has cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count
   // bits we only need to determine whether
   // r/v is less than, equal to, or greater than 0.5 to determine rounding -
   // this we can do with a shift and comparison.
   //
   // We can set the exponent and sign of the result up front:
   //
   res.exponent() = u.exponent() - v.exponent() - 1;
   res.sign() = u.sign() != v.sign();
   //
   // Now get the quotient and remainder:
   //
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type t(u.bits()), t2(v.bits()), q, r;
   eval_left_shift(t, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count);
   eval_qr(t, t2, q, r);
   //
   // We now have either "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count"
   // or "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count+1" significant
   // bits in q.
   //
   static const unsigned limb_bits = sizeof(limb_type) * CHAR_BIT;
   if(eval_bit_test(q, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count))
   {
      //
      // OK we have cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count+1 bits,
      // so we already have rounding info,
      // we just need to changes things if the last bit is 1 and either the
      // remainder is non-zero (ie we do not have a tie) or the quotient would
      // be odd if it were shifted to the correct number of bits (ie a tiebreak).
      //
      BOOST_ASSERT((eval_msb(q) == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count));
      if((q.limbs()[0] & 1u) && (eval_get_sign(r) || (q.limbs()[0] & 2u)))
      {
         eval_increment(q);
      }
   }
   else
   {
      //
      // We have exactly "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count" bits in q.
      // Get rounding info, which we can get by comparing 2r with v.
      // We want to call copy_and_round to handle rounding and general cleanup,
      // so we'll left shift q and add some fake digits on the end to represent
      // how we'll be rounding.
      //
      BOOST_ASSERT((eval_msb(q) == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1));
      static const unsigned lshift = (cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count < limb_bits) ? 2 : limb_bits;
      eval_left_shift(q, lshift);
      res.exponent() -= lshift;
      eval_left_shift(r, 1u);
      int c = r.compare(v.bits());
      if(c == 0)
         q.limbs()[0] |= static_cast<limb_type>(1u) << (lshift - 1);
      else if(c > 0)
         q.limbs()[0] |= (static_cast<limb_type>(1u) << (lshift - 1)) + static_cast<limb_type>(1u);
   }
   copy_and_round(res, q);
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   eval_divide(res, res, arg);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class U>
inline typename enable_if_c<is_unsigned<U>::value>::type eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &u, const U &v)
{
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:6326)  // comparison of two constants
#endif
   using default_ops::eval_subtract;
   using default_ops::eval_qr;
   using default_ops::eval_bit_test;
   using default_ops::eval_get_sign;
   using default_ops::eval_increment;

   //
   // Special cases first:
   //
   switch(u.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      if(v == 0)
      {
         res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
         return;
      }
      res = u;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = u;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      return;
   }
   if(v == 0)
   {
      bool s = u.sign();
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity().backend();
      res.sign() = s;
      return;
   }

   // We can scale u and v so that both are integers, then perform integer
   // division to obtain quotient q and remainder r, such that:
   //
   // q * v + r = u
   //
   // and hense:
   //
   // q + r/v = u/v
   //
   // From this, assuming q has "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count" cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count, we only need to determine whether
   // r/v is less than, equal to, or greater than 0.5 to determine rounding -
   // this we can do with a shift and comparison.
   //
   // We can set the exponent and sign of the result up front:
   //
   int gb = msb(v);
   res.exponent() = u.exponent() - static_cast<Exponent>(gb) - static_cast<Exponent>(1);
   res.sign() = u.sign();
   //
   // Now get the quotient and remainder:
   //
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type t(u.bits()), q, r;
   eval_left_shift(t, gb + 1);
   eval_qr(t, number<typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type>::canonical_value(v), q, r);
   //
   // We now have either "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count" or "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count+1" significant cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count in q.
   //
   static const unsigned limb_bits = sizeof(limb_type) * CHAR_BIT;
   if(eval_bit_test(q, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count))
   {
      //
      // OK we have cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count+1 cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count, so we already have rounding info,
      // we just need to changes things if the last bit is 1 and the
      // remainder is non-zero (ie we do not have a tie).
      //
      BOOST_ASSERT((eval_msb(q) == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count));
      if((q.limbs()[0] & 1u) && eval_get_sign(r))
      {
         eval_increment(q);
      }
   }
   else
   {
      //
      // We have exactly "cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count" cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count in q.
      // Get rounding info, which we can get by comparing 2r with v.
      // We want to call copy_and_round to handle rounding and general cleanup,
      // so we'll left shift q and add some fake cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count on the end to represent
      // how we'll be rounding.
      //
      BOOST_ASSERT((eval_msb(q) == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1));
      static const unsigned lshift = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count < limb_bits ? 2 : limb_bits;
      eval_left_shift(q, lshift);
      res.exponent() -= lshift;
      eval_left_shift(r, 1u);
      int c = r.compare(number<typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type>::canonical_value(v));
      if(c == 0)
         q.limbs()[0] |= static_cast<limb_type>(1u) << (lshift - 1);
      else if(c > 0)
         q.limbs()[0] |= (static_cast<limb_type>(1u) << (lshift - 1)) + static_cast<limb_type>(1u);
   }
   copy_and_round(res, q);
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class U>
inline typename enable_if_c<is_unsigned<U>::value>::type eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const U &v)
{
   eval_divide(res, res, v);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class S>
inline typename enable_if_c<is_signed<S>::value>::type eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &u, const S &v)
{
   typedef typename make_unsigned<S>::type ui_type;
   eval_divide(res, u, static_cast<ui_type>(boost::multiprecision::detail::unsigned_abs(v)));
   if(v < 0)
      res.negate();
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class S>
inline typename enable_if_c<is_signed<S>::value>::type eval_divide(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const S &v)
{
   eval_divide(res, res, v);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline int eval_get_sign(const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   return arg.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero ? 0 : arg.sign() ? -1 : 1;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline bool eval_is_zero(const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   return arg.exponent() == cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline bool eval_eq(const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &a, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &b)
{
   return (a.exponent() == b.exponent())
      && (a.sign() == b.sign())
      && (a.bits().compare(b.bits()) == 0)
      && (a.exponent() != cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_convert_to(boost::long_long_type *res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      *res = 0;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      BOOST_THROW_EXCEPTION(std::runtime_error("Could not convert NaN to integer."));
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      *res = (std::numeric_limits<boost::long_long_type>::max)();
      if(arg.sign())
         *res = -*res;
      return;
   }
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::rep_type man(arg.bits());
   typename mpl::if_c<sizeof(typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type) < sizeof(int), int, typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type>::type shift
      = (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1 - arg.exponent();
   if(shift > (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1)
   {
      *res = 0;
      return;
   }
   if(arg.sign() && (arg.compare((std::numeric_limits<boost::long_long_type>::min)()) <= 0))
   {
      *res = (std::numeric_limits<boost::long_long_type>::min)();
      return;
   }
   else if(!arg.sign() && (arg.compare((std::numeric_limits<boost::long_long_type>::max)()) >= 0))
   {
      *res = (std::numeric_limits<boost::long_long_type>::max)();
      return;
   }
   eval_right_shift(man, shift);
   eval_convert_to(res, man);
   if(arg.sign())
   {
      *res = -*res;
   }
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_convert_to(boost::ulong_long_type *res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      *res = 0;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      BOOST_THROW_EXCEPTION(std::runtime_error("Could not convert NaN to integer."));
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      *res = (std::numeric_limits<boost::ulong_long_type>::max)();
      return;
   }
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::rep_type man(arg.bits());
   typename mpl::if_c<sizeof(typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type) < sizeof(int), int, typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type>::type shift
      = (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1 - arg.exponent();
   if(shift > (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1)
   {
      *res = 0;
      return;
   }
   else if(shift < 0)
   {
      // TODO: what if we have fewer cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count than a boost::long_long_type?
      *res = (std::numeric_limits<boost::long_long_type>::max)();
      return;
   }
   eval_right_shift(man, shift);
   eval_convert_to(res, man);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_convert_to(long double *res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      *res = 0;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      *res = std::numeric_limits<long double>::quiet_NaN();
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      *res = (std::numeric_limits<long double>::infinity)();
      if(arg.sign())
         *res = -*res;
      return;
   }
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type e = arg.exponent();
   e -= cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1;
   *res = std::ldexp(static_cast<long double>(*arg.bits().limbs()), e);
   for(unsigned i = 1; i < arg.bits().size(); ++i)
   {
      e += sizeof(*arg.bits().limbs()) * CHAR_BIT;
      *res += std::ldexp(static_cast<long double>(arg.bits().limbs()[i]), e);
   }
   if(arg.sign())
      *res = -*res;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_frexp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg, Exponent *e)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      *e = 0;
      res = arg;
      return;
   }
   res = arg;
   *e = arg.exponent() + 1;
   res.exponent() = -1;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class I>
inline void eval_frexp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg, I *pe)
{
   Exponent e;
   eval_frexp(res, arg, &e);
   if((e > (std::numeric_limits<I>::max)()) || (e < (std::numeric_limits<I>::min)()))
   {
      BOOST_THROW_EXCEPTION(std::runtime_error("Exponent was outside of the range of the argument type to frexp."));
   }
   *pe = static_cast<I>(e);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_ldexp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg, Exponent e)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = arg;
      return;
   }
   if((e > 0) && (cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent - e < arg.exponent()))
   {
      // Overflow:
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity().backend();
      res.sign() = arg.sign();
   }
   else if((e < 0) && (cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::min_exponent - e > arg.exponent()))
   {
      // Underflow:
      res = limb_type(0);
   }
   else
   {
      res = arg;
      res.exponent() += e;
   }
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class I>
inline typename enable_if_c<is_unsigned<I>::value>::type eval_ldexp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg, I e)
{
   typedef typename make_signed<I>::type si_type;
   if(e > static_cast<I>((std::numeric_limits<si_type>::max)()))
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity().backend();
   else
      eval_ldexp(res, arg, static_cast<si_type>(e));
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, class I>
inline typename enable_if_c<is_signed<I>::value>::type eval_ldexp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg, I e)
{
   if((e > (std::numeric_limits<Exponent>::max)()) || (e < (std::numeric_limits<Exponent>::min)()))
   {
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity().backend();
      if(e < 0)
         res.negate();
   }
   else
      eval_ldexp(res, arg, static_cast<Exponent>(e));
}

/*
* Sign manipulation
*/

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_abs(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   res = arg;
   res.sign() = false;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_fabs(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   res = arg;
   res.sign() = false;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline int eval_fpclassify(const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
      return FP_ZERO;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      return FP_INFINITE;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      return FP_NAN;
   }
   return FP_NORMAL;
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_sqrt(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   using default_ops::eval_integer_sqrt;
   using default_ops::eval_bit_test;
   using default_ops::eval_increment;
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
      res = arg;
      return;
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      return;
   }
   if(arg.sign())
   {
      res = std::numeric_limits<number<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN().backend();
      return;
   }

   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::double_rep_type t(arg.bits()), r, s;
   eval_left_shift(t, arg.exponent() & 1 ? cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count : cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1);
   eval_integer_sqrt(s, r, t);

   if(!eval_bit_test(s, cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count))
   {
      // We have exactly the right number of cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count in the result, round as required:
      if(s.compare(r) < 0)
      {
         eval_increment(s);
      }
   }
   typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type ae = arg.exponent();
   res.exponent() = ae / 2;
   if((ae & 1) && (ae < 0))
      --res.exponent();
   copy_and_round(res, s);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_floor(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   using default_ops::eval_increment;
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = arg;
      return;
   }
   typename mpl::if_c<sizeof(typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type) < sizeof(int), int, typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type>::type shift =
      (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - arg.exponent() - 1;
   if((arg.exponent() > (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent) || (shift <= 0))
   {
      // Either arg is already an integer, or a special value:
      res = arg;
      return;
   }
   if(shift >= (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
   {
      res = static_cast<signed_limb_type>(arg.sign() ? -1 : 0);
      return;
   }
   bool fractional = (int)eval_lsb(arg.bits()) < shift;
   res = arg;
   eval_right_shift(res.bits(), shift);
   if(fractional && res.sign())
   {
      eval_increment(res.bits());
      if(eval_msb(res.bits()) != cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1 - shift)
      {
         // Must have extended result by one bit in the increment:
         --shift;
         ++res.exponent();
      }
   }
   eval_left_shift(res.bits(), shift);
}

template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
inline void eval_ceil(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
{
   using default_ops::eval_increment;
   switch(arg.exponent())
   {
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_zero:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_nan:
   case cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity:
      res = arg;
      return;
   }
   typename mpl::if_c<sizeof(typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type) < sizeof(int), int, typename cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type>::type shift = (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - arg.exponent() - 1;
   if((arg.exponent() > (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent) || (shift <= 0))
   {
      // Either arg is already an integer, or a special value:
      res = arg;
      return;
   }
   if(shift >= (int)cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count)
   {
      res = static_cast<signed_limb_type>(arg.sign() ? 0 : 1);
      return;
   }
   bool fractional = (int)eval_lsb(arg.bits()) < shift;
   res = arg;
   eval_right_shift(res.bits(), shift);
   if(fractional && !res.sign())
   {
      eval_increment(res.bits());
      if(eval_msb(res.bits()) != cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count - 1 - shift)
      {
         // Must have extended result by one bit in the increment:
         --shift;
         ++res.exponent();
      }
   }
   eval_left_shift(res.bits(), shift);
}

} // namespace backends

#ifdef BOOST_NO_SFINAE_EXPR

namespace detail{

template<unsigned D1, backends::digit_base_type B1, class A1, class E1, E1 M1, E1 M2, unsigned D2, backends::digit_base_type B2, class A2, class E2, E2 M3, E2 M4>
struct is_explicitly_convertible<backends::cpp_bin_float<D1, B1, A1, E1, M1, M2>, backends::cpp_bin_float<D2, B2, A2, E2, M3, M4> > : public mpl::true_ {};

}
#endif


using backends::cpp_bin_float;
using backends::digit_base_2;
using backends::digit_base_10;

template<unsigned Digits, backends::digit_base_type DigitBase, class Exponent, Exponent MinE, Exponent MaxE, class Allocator>
struct number_category<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > : public boost::mpl::int_<boost::multiprecision::number_kind_floating_point>{};

template<unsigned Digits, backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
struct expression_template_default<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> >
{
   static const expression_template_option value = is_void<Allocator>::value ? et_off : et_on;
};

typedef number<backends::cpp_bin_float<50> > cpp_bin_float_50;
typedef number<backends::cpp_bin_float<100> > cpp_bin_float_100;

typedef number<backends::cpp_bin_float<24, backends::digit_base_2, void, boost::int16_t, -126, 127>, et_off> cpp_bin_float_single;
typedef number<backends::cpp_bin_float<53, backends::digit_base_2, void, boost::int16_t, -1022, 1023>, et_off> cpp_bin_float_double;
typedef number<backends::cpp_bin_float<64, backends::digit_base_2, void, boost::int16_t, -16382, 16383>, et_off> cpp_bin_float_double_extended;
typedef number<backends::cpp_bin_float<113, backends::digit_base_2, void, boost::int16_t, -16382, 16383>, et_off> cpp_bin_float_quad;

}} // namespaces

#include <boost/multiprecision/cpp_bin_float/io.hpp>
#include <boost/multiprecision/cpp_bin_float/transcendental.hpp>

namespace std{

//
// numeric_limits [partial] specializations for the types declared in this header:
//
template<unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
class numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >
{
   typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> number_type;
public:
   BOOST_STATIC_CONSTEXPR bool is_specialized = true;
   static number_type (min)()
   {
      initializer.do_nothing();
      static std::pair<bool, number_type> value;
      if(!value.first)
      {
         value.first = true;
         value.second = 1u;
         value.second.backend().exponent() = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::min_exponent;
      }
      return value.second;
   }
   static number_type (max)()
   {
      initializer.do_nothing();
      static std::pair<bool, number_type> value;
      if(!value.first)
      {
         value.first = true;
         eval_complement(value.second.backend().bits(), value.second.backend().bits());
         value.second.backend().exponent() = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent;
      }
      return value.second;
   }
   BOOST_STATIC_CONSTEXPR number_type lowest()
   {
      return -(max)();
   }
   BOOST_STATIC_CONSTEXPR int digits = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count;
   BOOST_STATIC_CONSTEXPR int digits10 = (digits - 1) * 301 / 1000;
   // Is this really correct???
   BOOST_STATIC_CONSTEXPR int max_digits10 = (digits * 301 / 1000) + 2;
   BOOST_STATIC_CONSTEXPR bool is_signed = true;
   BOOST_STATIC_CONSTEXPR bool is_integer = false;
   BOOST_STATIC_CONSTEXPR bool is_exact = false;
   BOOST_STATIC_CONSTEXPR int radix = 2;
   static number_type epsilon()
   {
      initializer.do_nothing();
      static std::pair<bool, number_type> value;
      if(!value.first)
      {
         value.first = true;
         value.second = 1;
         value.second = ldexp(value.second, 1 - (int)digits);
      }
      return value.second;
   }
   // What value should this be????
   static number_type round_error()
   {
      // returns 0.5
      initializer.do_nothing();
      static std::pair<bool, number_type> value;
      if(!value.first)
      {
         value.first = true;
         value.second = 1;
         value.second = ldexp(value.second, -1);
      }
      return value.second;
   }
   BOOST_STATIC_CONSTEXPR typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type min_exponent = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::min_exponent;
   BOOST_STATIC_CONSTEXPR typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type min_exponent10 = (min_exponent / 1000) * 301L;
   BOOST_STATIC_CONSTEXPR typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type max_exponent = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::max_exponent;
   BOOST_STATIC_CONSTEXPR typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type max_exponent10 = (max_exponent / 1000) * 301L;
   BOOST_STATIC_CONSTEXPR bool has_infinity = true;
   BOOST_STATIC_CONSTEXPR bool has_quiet_NaN = true;
   BOOST_STATIC_CONSTEXPR bool has_signaling_NaN = false;
   BOOST_STATIC_CONSTEXPR float_denorm_style has_denorm = denorm_absent;
   BOOST_STATIC_CONSTEXPR bool has_denorm_loss = false;
   static number_type infinity()
   {
      // returns epsilon/2
      initializer.do_nothing();
      static std::pair<bool, number_type> value;
      if(!value.first)
      {
         value.first = true;
         value.second.backend().exponent() = boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_infinity;
      }
      return value.second;
   }
   static number_type quiet_NaN()
   {
      return number_type();
   }
   BOOST_STATIC_CONSTEXPR number_type signaling_NaN()
   {
      return number_type(0);
   }
   BOOST_STATIC_CONSTEXPR number_type denorm_min() { return number_type(0); }
   BOOST_STATIC_CONSTEXPR bool is_iec559 = false;
   BOOST_STATIC_CONSTEXPR bool is_bounded = true;
   BOOST_STATIC_CONSTEXPR bool is_modulo = false;
   BOOST_STATIC_CONSTEXPR bool traps = true;
   BOOST_STATIC_CONSTEXPR bool tinyness_before = false;
   BOOST_STATIC_CONSTEXPR float_round_style round_style = round_to_nearest;
private:
   struct data_initializer
   {
      data_initializer()
      {
         std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::epsilon();
         std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::round_error();
         (std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::min)();
         (std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::max)();
         std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::infinity();
         std::numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> > >::quiet_NaN();
      }
      void do_nothing()const{}
   };
   static const data_initializer initializer;
};

template<unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
const typename numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::data_initializer numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::initializer;

#ifndef BOOST_NO_INCLASS_MEMBER_INITIALIZATION

template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST int numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::digits;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST int numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::digits10;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST int numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::max_digits10;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_signed;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_integer;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_exact;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST int numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::radix;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::min_exponent;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::min_exponent10;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::max_exponent;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST typename boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::exponent_type numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::max_exponent10;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::has_infinity;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::has_quiet_NaN;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::has_signaling_NaN;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST float_denorm_style numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::has_denorm;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::has_denorm_loss;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_iec559;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_bounded;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::is_modulo;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::traps;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST bool numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::tinyness_before;
template <unsigned Digits, boost::multiprecision::backends::digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE, boost::multiprecision::expression_template_option ExpressionTemplates>
BOOST_CONSTEXPR_OR_CONST float_round_style numeric_limits<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>, ExpressionTemplates> >::round_style;

#endif

} // namespace std

#endif