text: ///////////////////////////////////////////////////////////////////////////////
  //  Copyright 2011 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_0.txt)
  
  #ifndef BOOST_MATH_BIG_NUM_BASE_HPP
  #define BOOST_MATH_BIG_NUM_BASE_HPP
  
  #include <limits>
  #include <boost/utility/enable_if.hpp>
  #include <boost/type_traits/is_convertible.hpp>
  #include <boost/type_traits/decay.hpp>
  #ifdef BOOST_MSVC
  #  pragma warning(push)
  #  pragma warning(disable:4307)
  #endif
  #include <boost/lexical_cast.hpp>
  #ifdef BOOST_MSVC
  #  pragma warning(pop)
  #endif
  
  #if defined(NDEBUG) && !defined(_DEBUG)
  #  define BOOST_MP_FORCEINLINE BOOST_FORCEINLINE
  #else
  #  define BOOST_MP_FORCEINLINE inline
  #endif
  
  namespace boost{ namespace multiprecision{
  
  enum expression_template_option
  {
     et_off  = 0,
     et_on   = 1
  };
  
  template <class Backend>
  struct expression_template_default
  {
     static const expression_template_option value = et_on;
  };
  
  template <class Backend, expression_template_option ExpressionTemplates = expression_template_default<Backend>::value>
  class number;
  
  template <class T>
  struct is_number : public mpl::false_ {};
  
  template <class Backend, expression_template_option ExpressionTemplates>
  struct is_number<number<Backend, ExpressionTemplates> > : public mpl::true_ {};
  
  namespace detail{
  
  // Forward-declare an expression wrapper
  template<class tag, class Arg1 = void, class Arg2 = void, class Arg3 = void, class Arg4 = void>
  struct expression;
  
  } // namespace detail
  
  template <class T>
  struct is_number_expression : public mpl::false_ {};
  
  template<class tag, class Arg1, class Arg2, class Arg3, class Arg4>
  struct is_number_expression<detail::expression<tag, Arg1, Arg2, Arg3, Arg4> > : public mpl::true_ {};
  
  template <class T, class Num>
  struct is_compatible_arithmetic_type
     : public mpl::bool_<
           is_convertible<T, Num>::value
           && !is_same<T, Num>::value
           && !is_number_expression<T>::value>
  {};
  
  namespace detail{
  //
  // Workaround for missing abs(long long) and abs(__int128) on some compilers:
  //
  template <class T>
  BOOST_CONSTEXPR typename enable_if_c<(is_signed<T>::value || is_floating_point<T>::value), T>::type abs(T t) BOOST_NOEXCEPT
  {
     // This strange expression avoids a hardware trap in the corner case
     // that val is the most negative value permitted in long long.
     // See https://svn.boost.org/trac/boost/ticket/9740.
     return t < 0 ? T(1u) + T(-(t + 1)) : t;
  }
  template <class T>
  BOOST_CONSTEXPR typename enable_if_c<(is_unsigned<T>::value), T>::type abs(T t) BOOST_NOEXCEPT
  {
     return t;
  }
  
  #define BOOST_MP_USING_ABS using boost::multiprecision::detail::abs;
  
  template <class T>
  BOOST_CONSTEXPR typename enable_if_c<(is_signed<T>::value || is_floating_point<T>::value), typename make_unsigned<T>::type>::type unsigned_abs(T t) BOOST_NOEXCEPT
  {
     // This strange expression avoids a hardware trap in the corner case
     // that val is the most negative value permitted in long long.
     // See https://svn.boost.org/trac/boost/ticket/9740.
     return t < 0 ? static_cast<typename make_unsigned<T>::type>(1u) + static_cast<typename make_unsigned<T>::type>(-(t + 1)) : static_cast<typename make_unsigned<T>::type>(t);
  }
  template <class T>
  BOOST_CONSTEXPR typename enable_if_c<(is_unsigned<T>::value), T>::type unsigned_abs(T t) BOOST_NOEXCEPT
  {
     return t;
  }
  
  //
  // Move support:
  //
  #ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
  #  define BOOST_MP_MOVE(x) std::move(x)
  #else
  #  define BOOST_MP_MOVE(x) x
  #endif
  
  template <class T>
  struct bits_of
  {
     BOOST_STATIC_ASSERT(is_integral<T>::value || is_enum<T>::value || std::numeric_limits<T>::is_specialized);
     static const unsigned value =
        std::numeric_limits<T>::is_specialized ?
           std::numeric_limits<T>::digits
           : sizeof(T) * CHAR_BIT - (is_signed<T>::value ? 1 : 0);
  };
  
  template <int b>
  struct has_enough_bits
  {
     template <class T>
     struct type : public mpl::bool_<bits_of<T>::value>= b>{};
  };
  
  template <class Val, class Backend, class Tag>
  struct canonical_imp
  {
     typedef typename remove_cv<typename decay<const Val>::type>::type type;
  };
  template <class B, class Backend, class Tag>
  struct canonical_imp<number<B, et_on>, Backend, Tag>
  {
     typedef B type;
  };
  template <class B, class Backend, class Tag>
  struct canonical_imp<number<B, et_off>, Backend, Tag>
  {
     typedef B type;
  };
  template <class Val, class Backend>
  struct canonical_imp<Val, Backend, mpl::int_<0> >
  {
     typedef typename has_enough_bits<bits_of<Val>::value>::template type<mpl::_> pred_type;
     typedef typename mpl::find_if<
        typename Backend::signed_types,
        pred_type
     >::type iter_type;
     typedef typename mpl::deref<iter_type>::type type;
  };
  template <class Val, class Backend>
  struct canonical_imp<Val, Backend, mpl::int_<1> >
  {
     typedef typename has_enough_bits<bits_of<Val>::value>::template type<mpl::_> pred_type;
     typedef typename mpl::find_if<
        typename Backend::unsigned_types,
        pred_type
     >::type iter_type;
     typedef typename mpl::deref<iter_type>::type type;
  };
  template <class Val, class Backend>
  struct canonical_imp<Val, Backend, mpl::int_<2> >
  {
     typedef typename has_enough_bits<bits_of<Val>::value>::template type<mpl::_> pred_type;
     typedef typename mpl::find_if<
        typename Backend::float_types,
        pred_type
     >::type iter_type;
     typedef typename mpl::deref<iter_type>::type type;
  };
  template <class Val, class Backend>
  struct canonical_imp<Val, Backend, mpl::int_<3> >
  {
     typedef const char* type;
  };
  
  template <class Val, class Backend>
  struct canonical
  {
     typedef typename mpl::if_<
        is_signed<Val>,
        mpl::int_<0>,
        typename mpl::if_<
           is_unsigned<Val>,
           mpl::int_<1>,
           typename mpl::if_<
              is_floating_point<Val>,
              mpl::int_<2>,
              typename mpl::if_<
                 mpl::or_<
                    is_convertible<Val, const char*>,
                    is_same<Val, std::string>
                 >,
                 mpl::int_<3>,
                 mpl::int_<4>
              >::type
           >::type
        >::type
     >::type tag_type;
  
     typedef typename canonical_imp<Val, Backend, tag_type>::type type;
  };
  
  struct terminal{};
  struct negate{};
  struct plus{};
  struct minus{};
  struct multiplies{};
  struct divides{};
  struct modulus{};
  struct shift_left{};
  struct shift_right{};
  struct bitwise_and{};
  struct bitwise_or{};
  struct bitwise_xor{};
  struct bitwise_complement{};
  struct add_immediates{};
  struct subtract_immediates{};
  struct multiply_immediates{};
  struct divide_immediates{};
  struct modulus_immediates{};
  struct bitwise_and_immediates{};
  struct bitwise_or_immediates{};
  struct bitwise_xor_immediates{};
  struct complement_immediates{};
  struct function{};
  struct multiply_add{};
  struct multiply_subtract{};
  
  template <class T>
  struct backend_type;
  
  template <class T, expression_template_option ExpressionTemplates>
  struct backend_type<number<T, ExpressionTemplates> >
  {
     typedef T type;
  };
  
  template <class tag, class A1, class A2, class A3, class A4>
  struct backend_type<expression<tag, A1, A2, A3, A4> >
  {
     typedef typename backend_type<typename expression<tag, A1, A2, A3, A4>::result_type>::type type;
  };
  
  
  template <class T1, class T2>
  struct combine_expression
  {
  #ifdef BOOST_NO_CXX11_DECLTYPE
     typedef typename mpl::if_c<(sizeof(T1() + T2()) == sizeof(T1)), T1, T2>::type type;
  #else
     typedef decltype(T1() + T2()) type;
  #endif
  };
  
  template <class T1, expression_template_option ExpressionTemplates, class T2>
  struct combine_expression<number<T1, ExpressionTemplates>, T2>
  {
     typedef number<T1, ExpressionTemplates> type;
  };
  
  template <class T1, class T2, expression_template_option ExpressionTemplates>
  struct combine_expression<T1, number<T2, ExpressionTemplates> >
  {
     typedef number<T2, ExpressionTemplates> type;
  };
  
  template <class T, expression_template_option ExpressionTemplates>
  struct combine_expression<number<T, ExpressionTemplates>, number<T, ExpressionTemplates> >
  {
     typedef number<T, ExpressionTemplates> type;
  };
  
  template <class T1, expression_template_option ExpressionTemplates1, class T2, expression_template_option ExpressionTemplates2>
  struct combine_expression<number<T1, ExpressionTemplates1>, number<T2, ExpressionTemplates2> >
  {
     typedef typename mpl::if_c<
        is_convertible<number<T2, ExpressionTemplates2>, number<T1, ExpressionTemplates2> >::value,
        number<T1, ExpressionTemplates1>,
        number<T2, ExpressionTemplates2>
        >::type type;
  };
  
  template <class T>
  struct arg_type
  {
     typedef expression<terminal, T> type;
  };
  
  template <class Tag, class Arg1, class Arg2, class Arg3, class Arg4>
  struct arg_type<expression<Tag, Arg1, Arg2, Arg3, Arg4> >
  {
     typedef expression<Tag, Arg1, Arg2, Arg3, Arg4> type;
  };
  
  struct unmentionable
  {
     unmentionable* proc(){ return 0; }
  };
  
  typedef unmentionable* (unmentionable::*unmentionable_type)();
  
  template <class T>
  struct expression_storage
  {
     typedef const T& type;
  };
  
  template <class T>
  struct expression_storage<T*>
  {
     typedef T* type;
  };
  
  template <class T>
  struct expression_storage<const T*>
  {
     typedef const T* type;
  };
  
  template <class tag, class A1, class A2, class A3, class A4>
  struct expression_storage<expression<tag, A1, A2, A3, A4> >
  {
     typedef expression<tag, A1, A2, A3, A4> type;
  };
  
  template<class tag, class Arg1>
  struct expression<tag, Arg1, void, void, void>
  {
     typedef mpl::int_<1> arity;
     typedef typename arg_type<Arg1>::type left_type;
     typedef typename left_type::result_type left_result_type;
     typedef typename left_type::result_type result_type;
     typedef tag tag_type;
  
     explicit expression(const Arg1& a) : arg(a) {}
  
     left_type left()const { return left_type(arg); }
  
     const Arg1& left_ref()const BOOST_NOEXCEPT { return arg; }
  
     static const unsigned depth = left_type::depth + 1;
  #ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
     explicit operator bool()const
     {
        result_type r(*this);
        return static_cast<bool>(r);
     }
  #else
     operator unmentionable_type()const
     {
        result_type r(*this);
        return r ? &unmentionable::proc : 0;
     }
  #endif
  
  private:
     typename expression_storage<Arg1>::type arg;
     expression& operator=(const expression&);
  };
  
  template<class Arg1>
  struct expression<terminal, Arg1, void, void, void>
  {
     typedef mpl::int_<0> arity;
     typedef Arg1 result_type;
     typedef terminal tag_type;
  
     explicit expression(const Arg1& a) : arg(a) {}
  
     const Arg1& value()const BOOST_NOEXCEPT { return arg; }
  
     static const unsigned depth = 0;
  
  #ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
     explicit operator bool()const
     {
        return static_cast<bool>(arg);
     }
  #else
     operator unmentionable_type()const
     {
        return arg ? &unmentionable::proc : 0;
     }
  #endif
  
  private:
     typename expression_storage<Arg1>::type arg;
     expression& operator=(const expression&);
  };
  
  template <class tag, class Arg1, class Arg2>
  struct expression<tag, Arg1, Arg2, void, void>
  {
     typedef mpl::int_<2> arity;
     typedef typename arg_type<Arg1>::type left_type;
     typedef typename arg_type<Arg2>::type right_type;
     typedef typename left_type::result_type left_result_type;
     typedef typename right_type::result_type right_result_type;
     typedef typename combine_expression<left_result_type, right_result_type>::type result_type;
     typedef tag tag_type;
  
     expression(const Arg1& a1, const Arg2& a2) : arg1(a1), arg2(a2) {}
  
     left_type left()const { return left_type(arg1); }
     right_type right()const { return right_type(arg2); }
     const Arg1& left_ref()const BOOST_NOEXCEPT { return arg1; }
     const Arg2& right_ref()const BOOST_NOEXCEPT { return arg2; }
  
  #ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
     explicit operator bool()const
     {
        result_type r(*this);
        return static_cast<bool>(r);
     }
  #else
     operator unmentionable_type()const
     {
        result_type r(*this);
        return r ? &unmentionable::proc : 0;
     }
  #endif
     static const unsigned left_depth = left_type::depth + 1;
     static const unsigned right_depth = right_type::depth + 1;
     static const unsigned depth = left_depth > right_depth ? left_depth : right_depth;
  private:
     typename expression_storage<Arg1>::type arg1;
     typename expression_storage<Arg2>::type arg2;
     expression& operator=(const expression&);
  };
  
  template <class tag, class Arg1, class Arg2, class Arg3>
  struct expression<tag, Arg1, Arg2, Arg3, void>
  {
     typedef mpl::int_<3> arity;
     typedef typename arg_type<Arg1>::type left_type;
     typedef typename arg_type<Arg2>::type middle_type;
     typedef typename arg_type<Arg3>::type right_type;
     typedef typename left_type::result_type left_result_type;
     typedef typename middle_type::result_type middle_result_type;
     typedef typename right_type::result_type right_result_type;
     typedef typename combine_expression<
        left_result_type,
        typename combine_expression<right_result_type, middle_result_type>::type
     >::type result_type;
     typedef tag tag_type;
  
     expression(const Arg1& a1, const Arg2& a2, const Arg3& a3) : arg1(a1), arg2(a2), arg3(a3) {}
  
     left_type left()const { return left_type(arg1); }
     middle_type middle()const { return middle_type(arg2); }
     right_type right()const { return right_type(arg3); }
     const Arg1& left_ref()const BOOST_NOEXCEPT { return arg1; }
     const Arg2& middle_ref()const BOOST_NOEXCEPT { return arg2; }
     const Arg3& right_ref()const BOOST_NOEXCEPT { return arg3; }
  
  #ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
     explicit operator bool()const
     {
        result_type r(*this);
        return static_cast<bool>(r);
     }
  #else
     operator unmentionable_type()const
     {
        result_type r(*this);
        return r ? &unmentionable::proc : 0;
     }
  #endif
     static const unsigned left_depth = left_type::depth + 1;
     static const unsigned middle_depth = middle_type::depth + 1;
     static const unsigned right_depth = right_type::depth + 1;
     static const unsigned depth = left_depth > right_depth ? (left_depth > middle_depth ? left_depth : middle_depth) : (right_depth > middle_depth ? right_depth : middle_depth);
  private:
     typename expression_storage<Arg1>::type arg1;
     typename expression_storage<Arg2>::type arg2;
     typename expression_storage<Arg3>::type arg3;
     expression& operator=(const expression&);
  };
  
  template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
  struct expression
  {
     typedef mpl::int_<4> arity;
     typedef typename arg_type<Arg1>::type left_type;
     typedef typename arg_type<Arg2>::type left_middle_type;
     typedef typename arg_type<Arg3>::type right_middle_type;
     typedef typename arg_type<Arg4>::type right_type;
     typedef typename left_type::result_type left_result_type;
     typedef typename left_middle_type::result_type left_middle_result_type;
     typedef typename right_middle_type::result_type right_middle_result_type;
     typedef typename right_type::result_type right_result_type;
     typedef typename combine_expression<
        typename combine_expression<
           typename combine_expression<left_result_type, left_middle_result_type>::type,
           right_middle_result_type
        >::type,
        right_result_type
     >::type result_type;
     typedef tag tag_type;
  
     expression(const Arg1& a1, const Arg2& a2, const Arg3& a3, const Arg4& a4) : arg1(a1), arg2(a2), arg3(a3), arg4(a4) {}
  
     left_type left()const { return left_type(arg1); }
     left_middle_type left_middle()const { return left_middle_type(arg2); }
     right_middle_type right_middle()const { return right_middle_type(arg3); }
     right_type right()const { return right_type(arg4); }
     const Arg1& left_ref()const BOOST_NOEXCEPT { return arg1; }
     const Arg2& left_middle_ref()const BOOST_NOEXCEPT { return arg2; }
     const Arg3& right_middle_ref()const BOOST_NOEXCEPT { return arg3; }
     const Arg4& right_ref()const BOOST_NOEXCEPT { return arg4; }
  
  #ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
     explicit operator bool()const
     {
        result_type r(*this);
        return static_cast<bool>(r);
     }
  #else
     operator unmentionable_type()const
     {
        result_type r(*this);
        return r ? &unmentionable::proc : 0;
     }
  #endif
     static const unsigned left_depth = left_type::depth + 1;
     static const unsigned left_middle_depth = left_middle_type::depth + 1;
     static const unsigned right_middle_depth = right_middle_type::depth + 1;
     static const unsigned right_depth = right_type::depth + 1;
  
     static const unsigned left_max_depth = left_depth > left_middle_depth ? left_depth : left_middle_depth;
     static const unsigned right_max_depth = right_depth > right_middle_depth ? right_depth : right_middle_depth;
  
     static const unsigned depth = left_max_depth > right_max_depth ? left_max_depth : right_max_depth;
  private:
     typename expression_storage<Arg1>::type arg1;
     typename expression_storage<Arg2>::type arg2;
     typename expression_storage<Arg3>::type arg3;
     typename expression_storage<Arg4>::type arg4;
     expression& operator=(const expression&);
  };
  
  template <class T>
  struct digits2
  {
     BOOST_STATIC_ASSERT(std::numeric_limits<T>::is_specialized);
     BOOST_STATIC_ASSERT((std::numeric_limits<T>::radix == 2) || (std::numeric_limits<T>::radix == 10));
     // If we really have so many digits that this fails, then we're probably going to hit other problems anyway:
     BOOST_STATIC_ASSERT(LONG_MAX / 1000 > (std::numeric_limits<T>::digits + 1));
     static const long value = std::numeric_limits<T>::radix == 10 ?  (((std::numeric_limits<T>::digits + 1) * 1000L) / 301L) : std::numeric_limits<T>::digits;
  };
  
  #ifndef BOOST_MP_MIN_EXPONENT_DIGITS
  #ifdef _MSC_VER
  #  define BOOST_MP_MIN_EXPONENT_DIGITS 2
  #else
  #  define BOOST_MP_MIN_EXPONENT_DIGITS 2
  #endif
  #endif
  
  template <class S>
  void format_float_string(S& str, boost::intmax_t my_exp, boost::intmax_t digits, std::ios_base::fmtflags f, bool iszero)
  {
     typedef typename S::size_type size_type;
     bool scientific = (f & std::ios_base::scientific) == std::ios_base::scientific;
     bool fixed      = (f & std::ios_base::fixed) == std::ios_base::fixed;
     bool showpoint  = (f & std::ios_base::showpoint) == std::ios_base::showpoint;
     bool showpos     = (f & std::ios_base::showpos) == std::ios_base::showpos;
  
     bool neg = str.size() && (str[0] == '-');
  
     if(neg)
        str.erase(0, 1);
  
     if(digits == 0)
     {
        digits = (std::max)(str.size(), size_type(16));
     }
  
     if(iszero || str.empty() || (str.find_first_not_of('0') == S::npos))
     {
        // We will be printing zero, even though the value might not
        // actually be zero (it just may have been rounded to zero).
        str = "0";
        if(scientific || fixed)
        {
           str.append(1, '.');
           str.append(size_type(digits), '0');
           if(scientific)
              str.append("e+00");
        }
        else
        {
           if(showpoint)
           {
              str.append(1, '.');
              if(digits > 1)
                 str.append(size_type(digits - 1), '0');
           }
        }
        if(neg)
           str.insert(0, 1, '-');
        else if(showpos)
           str.insert(0, 1, '+');
        return;
     }
  
     if(!fixed && !scientific && !showpoint)
     {
        //
        // Suppress trailing zeros:
        //
        std::string::iterator pos = str.end();
        while(pos != str.begin() && *--pos == '0'){}
        if(pos != str.end())
           ++pos;
        str.erase(pos, str.end());
        if(str.empty())
           str = '0';
     }
     else if(!fixed || (my_exp >= 0))
     {
        //
        // Pad out the end with zero's if we need to:
        //
        boost::intmax_t chars = str.size();
        chars = digits - chars;
        if(scientific)
           ++chars;
        if(chars > 0)
        {
           str.append(static_cast<std::string::size_type>(chars), '0');
        }
     }
  
     if(fixed || (!scientific && (my_exp >= -4) && (my_exp < digits)))
     {
        if(1 + my_exp > static_cast<boost::intmax_t>(str.size()))
        {
           // Just pad out the end with zeros:
           str.append(static_cast<std::string::size_type>(1 + my_exp - str.size()), '0');
           if(showpoint || fixed)
              str.append(".");
        }
        else if(my_exp + 1 < static_cast<boost::intmax_t>(str.size()))
        {
           if(my_exp < 0)
           {
              str.insert(0, static_cast<std::string::size_type>(-1 - my_exp), '0');
              str.insert(0, "0.");
           }
           else
           {
              // Insert the decimal point:
              str.insert(static_cast<std::string::size_type>(my_exp + 1), 1, '.');
           }
        }
        else if(showpoint || fixed) // we have exactly the digits we require to left of the point
           str += ".";
  
        if(fixed)
        {
           // We may need to add trailing zeros:
           boost::intmax_t l = str.find('.') + 1;
           l = digits - (str.size() - l);
           if(l > 0)
              str.append(size_type(l), '0');
        }
     }
     else
     {
        BOOST_MP_USING_ABS
        // Scientific format:
        if(showpoint || (str.size() > 1))
           str.insert(1, 1, '.');
        str.append(1, 'e');
        S e = boost::lexical_cast<S>(abs(my_exp));
        if(e.size() < BOOST_MP_MIN_EXPONENT_DIGITS)
           e.insert(0, BOOST_MP_MIN_EXPONENT_DIGITS-e.size(), '0');
        if(my_exp < 0)
           e.insert(0, 1, '-');
        else
           e.insert(0, 1, '+');
        str.append(e);
     }
     if(neg)
        str.insert(0, 1, '-');
     else if(showpos)
        str.insert(0, 1, '+');
  }
  
  template <class V>
  void check_shift_range(V val, const mpl::true_&, const mpl::true_&)
  {
     if(val > (std::numeric_limits<std::size_t>::max)())
        BOOST_THROW_EXCEPTION(std::out_of_range("Can not shift by a value greater than std::numeric_limits<std::size_t>::max()."));
     if(val < 0)
        BOOST_THROW_EXCEPTION(std::out_of_range("Can not shift by a negative value."));
  }
  template <class V>
  void check_shift_range(V val, const mpl::false_&, const mpl::true_&)
  {
     if(val < 0)
        BOOST_THROW_EXCEPTION(std::out_of_range("Can not shift by a negative value."));
  }
  template <class V>
  void check_shift_range(V val, const mpl::true_&, const mpl::false_&)
  {
     if(val > (std::numeric_limits<std::size_t>::max)())
        BOOST_THROW_EXCEPTION(std::out_of_range("Can not shift by a value greater than std::numeric_limits<std::size_t>::max()."));
  }
  template <class V>
  void check_shift_range(V, const mpl::false_&, const mpl::false_&) BOOST_NOEXCEPT{}
  
  } // namespace detail
  
  //
  // Traits class, lets us know what kind of number we have, defaults to a floating point type:
  //
  enum number_category_type
  {
     number_kind_unknown = -1,
     number_kind_integer = 0,
     number_kind_floating_point = 1,
     number_kind_rational = 2,
     number_kind_fixed_point = 3
  };
  
  template <class Num>
  struct number_category : public mpl::int_<std::numeric_limits<Num>::is_integer ? number_kind_integer : (std::numeric_limits<Num>::max_exponent ? number_kind_floating_point : number_kind_unknown)> {};
  template <class Backend, expression_template_option ExpressionTemplates>
  struct number_category<number<Backend, ExpressionTemplates> > : public number_category<Backend>{};
  template <class tag, class A1, class A2, class A3, class A4>
  struct number_category<detail::expression<tag, A1, A2, A3, A4> > : public number_category<typename detail::expression<tag, A1, A2, A3, A4>::result_type>{};
  
  template <class T>
  struct component_type;
  template <class T, expression_template_option ExpressionTemplates>
  struct component_type<number<T, ExpressionTemplates> > : public component_type<T>{};
  template <class tag, class A1, class A2, class A3, class A4>
  struct component_type<detail::expression<tag, A1, A2, A3, A4> > : public component_type<typename detail::expression<tag, A1, A2, A3, A4>::result_type>{};
  
  template <class T>
  struct is_unsigned_number : public mpl::false_{};
  template <class Backend, expression_template_option ExpressionTemplates>
  struct is_unsigned_number<number<Backend, ExpressionTemplates> > : public is_unsigned_number<Backend> {};
  template <class T>
  struct is_signed_number : public mpl::bool_<!is_unsigned_number<T>::value> {};
  template <class T>
  struct is_interval_number : public mpl::false_ {};
  template <class Backend, expression_template_option ExpressionTemplates>
  struct is_interval_number<number<Backend, ExpressionTemplates> > : public is_interval_number<Backend>{};
  
  }} // namespaces
  
  namespace boost{ namespace math{ namespace tools{
  
  template <class T>
  struct promote_arg;
  
  template <class tag, class A1, class A2, class A3, class A4>
  struct promote_arg<boost::multiprecision::detail::expression<tag, A1, A2, A3, A4> >
  {
     typedef typename boost::multiprecision::detail::expression<tag, A1, A2, A3, A4>::result_type type;
  };
  
  template <class R, class B, boost::multiprecision::expression_template_option ET>
  inline R real_cast(const boost::multiprecision::number<B, ET>& val)
  {
     return val.template convert_to<R>();
  }
  
  template <class R, class tag, class A1, class A2, class A3, class A4>
  inline R real_cast(const boost::multiprecision::detail::expression<tag, A1, A2, A3, A4>& val)
  {
     typedef typename boost::multiprecision::detail::expression<tag, A1, A2, A3, A4>::result_type val_type;
     return val_type(val).template convert_to<R>();
  }
  
  
  }
  
  namespace constants{
  
     template <class T>
     struct is_explicitly_convertible_from_string;
  
     template <class B, boost::multiprecision::expression_template_option ET>
     struct is_explicitly_convertible_from_string<boost::multiprecision::number<B, ET> >
     {
        static const bool value = true;
     };
  
  }
  
  }}
  
  #endif // BOOST_MATH_BIG_NUM_BASE_HPP
  
  
  