Boost C++ Libraries

...one of the most highly regarded and expertly designed C++ library projects in the world. Herb Sutter and Andrei Alexandrescu, C++ Coding Standards

boost/multiprecision/number.hpp

///////////////////////////////////////////////////////////////////////////////
//  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_EXTENDED_REAL_HPP
#define BOOST_MATH_EXTENDED_REAL_HPP

#include <boost/cstdint.hpp>
#include <boost/mpl/max.hpp>
#include <boost/mpl/plus.hpp>
#include <boost/mpl/or.hpp>
#include <boost/mpl/find_if.hpp>
#include <boost/assert.hpp>
#include <boost/type_traits/remove_pointer.hpp>
#include <boost/type_traits/is_signed.hpp>
#include <boost/type_traits/is_unsigned.hpp>
#include <boost/type_traits/is_floating_point.hpp>
#include <boost/type_traits/is_integral.hpp>
#include <boost/type_traits/make_unsigned.hpp>
#include <boost/throw_exception.hpp>
#include <boost/multiprecision/detail/generic_interconvert.hpp>
#include <boost/multiprecision/detail/number_compare.hpp>
#include <boost/multiprecision/traits/is_restricted_conversion.hpp>
#include <istream>  // stream operators
#include <cstdio>   // EOF

namespace boost{ namespace multiprecision{

#ifdef BOOST_MSVC
// warning C4127: conditional expression is constant
// warning C4714: function marked as __forceinline not inlined
#pragma warning(push)
#pragma warning(disable:4127 4714)
#endif

template <class Backend, expression_template_option ExpressionTemplates>
class number
{
   typedef number<Backend, ExpressionTemplates> self_type;
public:
   typedef Backend backend_type;
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number() BOOST_NOEXCEPT_IF(noexcept(Backend())) {}
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(const number& e) BOOST_NOEXCEPT_IF(noexcept(Backend(static_cast<const Backend&>(std::declval<Backend>())))) : m_backend(e.m_backend){}
   template <class V>
   BOOST_MP_FORCEINLINE number(const V& v, typename boost::enable_if_c<
            (boost::is_arithmetic<V>::value || is_same<std::string, V>::value || is_convertible<V, const char*>::value)
            && !is_convertible<typename detail::canonical<V, Backend>::type, Backend>::value
            && !detail::is_restricted_conversion<typename detail::canonical<V, Backend>::type, Backend>::value
         >::type* = 0)
   {
      m_backend = canonical_value(v);
   }
   template <class V>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(const V& v, typename boost::enable_if_c<
            is_convertible<typename detail::canonical<V, Backend>::type, Backend>::value
            && !detail::is_restricted_conversion<typename detail::canonical<V, Backend>::type, Backend>::value
         >::type* = 0)
      : m_backend(canonical_value(v)) {}
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(const number& e, unsigned digits10)
      : m_backend(e.m_backend, digits10){}
   template <class V>
   explicit BOOST_MP_FORCEINLINE number(const V& v, typename boost::enable_if_c<
            (boost::is_arithmetic<V>::value || is_same<std::string, V>::value || is_convertible<V, const char*>::value)
            && !detail::is_explicitly_convertible<typename detail::canonical<V, Backend>::type, Backend>::value
            && detail::is_restricted_conversion<typename detail::canonical<V, Backend>::type, Backend>::value
         >::type* = 0)
   {
      m_backend = canonical_value(v);
   }
   template <class V>
   explicit BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(const V& v, typename boost::enable_if_c<
            detail::is_explicitly_convertible<typename detail::canonical<V, Backend>::type, Backend>::value
            && (detail::is_restricted_conversion<typename detail::canonical<V, Backend>::type, Backend>::value
                || !is_convertible<typename detail::canonical<V, Backend>::type, Backend>::value)
         >::type* = 0)
      : m_backend(canonical_value(v)) {}
   /*
   //
   // This conflicts with component based initialization (for rational and complex types)
   // which is arguably more useful.  Disabled for now.
   //
   template <class V>
   number(V v, unsigned digits10, typename boost::enable_if<mpl::or_<boost::is_arithmetic<V>, is_same<std::string, V>, is_convertible<V, const char*> > >::type* dummy1 = 0)
   {
      m_backend.precision(digits10);
      m_backend = canonical_value(v);
   }
   */
   template<expression_template_option ET>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(const number<Backend, ET>& val)
      BOOST_NOEXCEPT_IF(noexcept(Backend(static_cast<const Backend&>(std::declval<Backend>())))) : m_backend(val.backend()) {}

   template <class Other, expression_template_option ET>
   BOOST_MP_FORCEINLINE number(const number<Other, ET>& val,
         typename boost::enable_if_c<(boost::is_convertible<Other, Backend>::value && !detail::is_restricted_conversion<Other, Backend>::value)>::type* = 0)
      BOOST_NOEXCEPT_IF(noexcept(Backend(static_cast<const Other&>(std::declval<Other>()))))
      : m_backend(val.backend()) {}

   template <class Other, expression_template_option ET>
   explicit number(const number<Other, ET>& val, typename boost::enable_if_c<
         (!detail::is_explicitly_convertible<Other, Backend>::value)
         >::type* = 0)
   {
      //
      // Attempt a generic interconvertion:
      //
      detail::generic_interconvert(backend(), val.backend(), number_category<Backend>(), number_category<Other>());
   }
   template <class Other, expression_template_option ET>
   explicit BOOST_MP_FORCEINLINE number(const number<Other, ET>& val, typename boost::enable_if_c<
         (detail::is_explicitly_convertible<Other, Backend>::value
            && (detail::is_restricted_conversion<Other, Backend>::value || !boost::is_convertible<Other, Backend>::value))
         >::type* = 0) BOOST_NOEXCEPT_IF(noexcept(Backend(static_cast<const Other&>(std::declval<Other>()))))
      : m_backend(val.backend()) {}

   template <class V>
   BOOST_MP_FORCEINLINE number(V v1, V v2, typename boost::enable_if<mpl::or_<boost::is_arithmetic<V>, is_same<std::string, V>, is_convertible<V, const char*> > >::type* = 0)
   {
      using default_ops::assign_components;
      assign_components(m_backend, canonical_value(v1), canonical_value(v2));
   }
   template <class Other, expression_template_option ET>
   BOOST_MP_FORCEINLINE number(const number<Other, ET>& v1, const number<Other, ET>& v2, typename boost::enable_if<boost::is_convertible<Other, Backend> >::type* = 0)
   {
      using default_ops::assign_components;
      assign_components(m_backend, v1.backend(), v2.backend());
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   typename boost::enable_if<is_convertible<typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type, self_type>, number&>::type operator=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      typedef typename is_same<number, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type>::type tag_type;
      do_assign(e, tag_type());
      return *this;
   }
   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& assign(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      typedef typename is_same<number, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type>::type tag_type;
      do_assign(e, tag_type());
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator=(const number& e)
      BOOST_NOEXCEPT_IF(noexcept(std::declval<Backend>() = static_cast<const Backend&>(std::declval<Backend>())))
   {
      m_backend = e.m_backend;
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator=(const V& v)
      BOOST_NOEXCEPT_IF(noexcept(std::declval<Backend>() = static_cast<typename boost::multiprecision::detail::canonical<V, Backend>::type const&>(std::declval<typename boost::multiprecision::detail::canonical<V, Backend>::type>())))
   {
      m_backend = canonical_value(v);
      return *this;
   }
   template <class V>
   BOOST_MP_FORCEINLINE number<Backend, ExpressionTemplates>& assign(const V& v)
      BOOST_NOEXCEPT_IF(noexcept(std::declval<Backend>() = static_cast<typename boost::multiprecision::detail::canonical<V, Backend>::type const&>(std::declval<typename boost::multiprecision::detail::canonical<V, Backend>::type>())))
   {
      m_backend = canonical_value(v);
      return *this;
   }
   template <class Other, expression_template_option ET>
   typename boost::disable_if<boost::multiprecision::detail::is_explicitly_convertible<Other, Backend>, number<Backend, ExpressionTemplates>& >::type
      assign(const number<Other, ET>& v)
   {
      //
      // Attempt a generic interconvertion:
      //
      detail::generic_interconvert(backend(), v.backend(), number_category<Backend>(), number_category<Other>());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e, typename boost::enable_if_c<is_convertible<typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type, self_type>::value>::type* = 0)
   {
      *this = e;
   }
   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   explicit number(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e,
      typename boost::enable_if_c<!is_convertible<typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type, self_type>::value
      && boost::multiprecision::detail::is_explicitly_convertible<typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type, self_type>::value>::type* = 0)
   {
      assign(e);
   }

#ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR number(number&& r)
      BOOST_NOEXCEPT_IF(noexcept(Backend(std::declval<Backend>())))
      : m_backend(static_cast<Backend&&>(r.m_backend)){}
   BOOST_MP_FORCEINLINE number& operator=(number&& r) BOOST_NOEXCEPT
   {
      m_backend = static_cast<Backend&&>(r.m_backend);
      return *this;
   }
#endif

   number& operator+=(const self_type& val)
   {
      do_add(detail::expression<detail::terminal, self_type>(val), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator+=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      // Create a copy if e contains this, but not if we're just doing a
      //    x += x
      if(contains_self(e) && !is_self(e))
      {
         self_type temp(e);
         do_add(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_add(e, tag());
      }
      return *this;
   }

   template <class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator+=(const detail::expression<detail::multiply_immediates, Arg1, Arg2, Arg3, Arg4>& e)
   {
      //
      // Fused multiply-add:
      //
      using default_ops::eval_multiply_add;
      eval_multiply_add(m_backend, canonical_value(e.left_ref()), canonical_value(e.right_ref()));
      return *this;
   }

   template <class V>
   typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator+=(const V& v)
   {
      using default_ops::eval_add;
      eval_add(m_backend, canonical_value(v));
      return *this;
   }

   number& operator-=(const self_type& val)
   {
      do_subtract(detail::expression<detail::terminal, self_type>(val), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator-=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      // Create a copy if e contains this:
      if(contains_self(e))
      {
         self_type temp(e);
         do_subtract(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_subtract(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator-=(const V& v)
   {
      using default_ops::eval_subtract;
      eval_subtract(m_backend, canonical_value(v));
      return *this;
   }

   template <class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator-=(const detail::expression<detail::multiply_immediates, Arg1, Arg2, Arg3, Arg4>& e)
   {
      //
      // Fused multiply-subtract:
      //
      using default_ops::eval_multiply_subtract;
      eval_multiply_subtract(m_backend, canonical_value(e.left_ref()), canonical_value(e.right_ref()));
      return *this;
   }


   number& operator *= (const self_type& e)
   {
      do_multiplies(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator*=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      // Create a temporary if the RHS references *this, but not
      // if we're just doing an   x *= x;
      if(contains_self(e) && !is_self(e))
      {
         self_type temp(e);
         do_multiplies(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_multiplies(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator*=(const V& v)
   {
      using default_ops::eval_multiply;
      eval_multiply(m_backend, canonical_value(v));
      return *this;
   }

   number& operator%=(const self_type& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      do_modulus(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }
   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator%=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      // Create a temporary if the RHS references *this:
      if(contains_self(e))
      {
         self_type temp(e);
         do_modulus(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_modulus(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }
   template <class V>
   typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator%=(const V& v)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      using default_ops::eval_modulus;
      eval_modulus(m_backend, canonical_value(v));
      return *this;
   }

   //
   // These operators are *not* proto-ized.
   // The issue is that the increment/decrement must happen
   // even if the result of the operator *is never used*.
   // Possibly we could modify our expression wrapper to
   // execute the increment/decrement on destruction, but
   // correct implementation will be tricky, so defered for now...
   //
   BOOST_MP_FORCEINLINE number& operator++()
   {
      using default_ops::eval_increment;
      eval_increment(m_backend);
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator--()
   {
      using default_ops::eval_decrement;
      eval_decrement(m_backend);
      return *this;
   }

   inline number operator++(int)
   {
      using default_ops::eval_increment;
      self_type temp(*this);
      eval_increment(m_backend);
      return BOOST_MP_MOVE(temp);
   }

   inline number operator--(int)
   {
      using default_ops::eval_decrement;
      self_type temp(*this);
      eval_decrement(m_backend);
      return BOOST_MP_MOVE(temp);
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<is_integral<V>, number&>::type operator <<= (V val)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The left-shift operation is only valid for integer types");
      detail::check_shift_range(val, mpl::bool_<(sizeof(V) > sizeof(std::size_t))>(), is_signed<V>());
      eval_left_shift(m_backend, static_cast<std::size_t>(canonical_value(val)));
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<is_integral<V>, number&>::type operator >>= (V val)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The right-shift operation is only valid for integer types");
      detail::check_shift_range(val, mpl::bool_<(sizeof(V) > sizeof(std::size_t))>(), is_signed<V>());
      eval_right_shift(m_backend, static_cast<std::size_t>(canonical_value(val)));
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator /= (const self_type& e)
   {
      do_divide(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator/=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      // Create a temporary if the RHS references *this:
      if(contains_self(e))
      {
         self_type temp(e);
         do_divide(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_divide(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator/=(const V& v)
   {
      using default_ops::eval_divide;
      eval_divide(m_backend, canonical_value(v));
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator&=(const self_type& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      do_bitwise_and(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator&=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      // Create a temporary if the RHS references *this, but not
      // if we're just doing an   x &= x;
      if(contains_self(e) && !is_self(e))
      {
         self_type temp(e);
         do_bitwise_and(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_bitwise_and(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator&=(const V& v)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      using default_ops::eval_bitwise_and;
      eval_bitwise_and(m_backend, canonical_value(v));
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator|=(const self_type& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      do_bitwise_or(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator|=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      // Create a temporary if the RHS references *this, but not
      // if we're just doing an   x |= x;
      if(contains_self(e) && !is_self(e))
      {
         self_type temp(e);
         do_bitwise_or(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_bitwise_or(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator|=(const V& v)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      using default_ops::eval_bitwise_or;
      eval_bitwise_or(m_backend, canonical_value(v));
      return *this;
   }

   BOOST_MP_FORCEINLINE number& operator^=(const self_type& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      do_bitwise_xor(detail::expression<detail::terminal, self_type>(e), detail::terminal());
      return *this;
   }

   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   number& operator^=(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      if(contains_self(e))
      {
         self_type temp(e);
         do_bitwise_xor(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
      }
      else
      {
         do_bitwise_xor(e, typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::tag_type());
      }
      return *this;
   }

   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<boost::is_convertible<V, self_type>, number<Backend, ExpressionTemplates>& >::type
      operator^=(const V& v)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      using default_ops::eval_bitwise_xor;
      eval_bitwise_xor(m_backend, canonical_value(v));
      return *this;
   }
   //
   // swap:
   //
   BOOST_MP_FORCEINLINE void swap(self_type& other) BOOST_NOEXCEPT
   {
      m_backend.swap(other.backend());
   }
   //
   // Zero and sign:
   //
   BOOST_MP_FORCEINLINE bool is_zero()const
   {
      using default_ops::eval_is_zero;
      return eval_is_zero(m_backend);
   }
   BOOST_MP_FORCEINLINE int sign()const
   {
      using default_ops::eval_get_sign;
      return eval_get_sign(m_backend);
   }
   //
   // String conversion functions:
   //
   std::string str(std::streamsize digits = 0, std::ios_base::fmtflags f = std::ios_base::fmtflags(0))const
   {
      return m_backend.str(digits, f);
   }
   template<class Archive>
   void serialize(Archive & ar, const unsigned int /*version*/)
   {
      ar & m_backend;
   }
private:
   template <class T>
   void convert_to_imp(T* result)const
   {
      using default_ops::eval_convert_to;
      eval_convert_to(result, m_backend);
   }
   template <class B2, expression_template_option ET>
   void convert_to_imp(number<B2, ET>* result)const
   {
      result->assign(*this);
   }
   void convert_to_imp(std::string* result)const
   {
      *result = this->str();
   }
public:
   template <class T>
   T convert_to()const
   {
      T result;
      convert_to_imp(&result);
      return result;
   }
   //
   // Use in boolean context, and explicit conversion operators:
   //
#ifndef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
#  if defined(__GNUC__) && (__GNUC__ == 4) && (__GNUC_MINOR__ < 7)
   //
   // Horrible workaround for gcc-4.6.x which always prefers the template
   // operator bool() rather than the non-template operator when converting to
   // an arithmetic type:
   //
   template <class T, typename boost::enable_if<is_same<T, bool>, int>::type = 0>
   explicit operator T ()const
   {
      using default_ops::eval_is_zero;
      return !eval_is_zero(backend());
   }
   template <class T, typename boost::disable_if_c<is_same<T, bool>::value || is_void<T>::value, int>::type = 0>
   explicit operator T ()const
   {
      return this->template convert_to<T>();
   }
#  else
   template <class T>
   explicit operator T()const
   {
      return this->template convert_to<T>();
   }
   BOOST_MP_FORCEINLINE explicit operator bool()const
   {
      return !is_zero();
   }
   explicit operator void()const {}
#  endif
#else
   typedef bool (self_type::*unmentionable_type)()const;

   BOOST_MP_FORCEINLINE operator unmentionable_type()const
   {
      return is_zero() ? 0 : &self_type::is_zero;
   }
#endif
   //
   // Default precision:
   //
   static unsigned default_precision() BOOST_NOEXCEPT
   {
      return Backend::default_precision();
   }
   static void default_precision(unsigned digits10)
   {
      Backend::default_precision(digits10);
   }
   unsigned precision()const BOOST_NOEXCEPT
   {
      return m_backend.precision();
   }
   void precision(unsigned digits10)
   {
      m_backend.precision(digits10);
   }
   //
   // Comparison:
   //
   BOOST_MP_FORCEINLINE int compare(const number<Backend, ExpressionTemplates>& o)const
      BOOST_NOEXCEPT_IF(noexcept(std::declval<Backend>().compare(std::declval<Backend>())))
   {
      return m_backend.compare(o.m_backend);
   }
   template <class V>
   BOOST_MP_FORCEINLINE typename boost::enable_if<is_arithmetic<V>, int>::type compare(const V& o)const
   {
      using default_ops::eval_get_sign;
      if(o == 0)
         return eval_get_sign(m_backend);
      return m_backend.compare(canonical_value(o));
   }
   BOOST_MP_FORCEINLINE Backend& backend() BOOST_NOEXCEPT
   {
      return m_backend;
   }
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const Backend& backend()const BOOST_NOEXCEPT
   {
      return m_backend;
   }
private:
   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   void do_assign(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e, const mpl::true_&)
   {
      do_assign(e, tag());
   }
   template <class tag, class Arg1, class Arg2, class Arg3, class Arg4>
   void do_assign(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& e, const mpl::false_&)
   {
      // The result of the expression isn't the same type as this -
      // create a temporary result and assign it to *this:
      typedef typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type temp_type;
      temp_type t(e);
      this->assign(t);
   }


   template <class Exp>
   void do_assign(const Exp& e, const detail::add_immediates&)
   {
      using default_ops::eval_add;
      eval_add(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::subtract_immediates&)
   {
      using default_ops::eval_subtract;
      eval_subtract(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::multiply_immediates&)
   {
      using default_ops::eval_multiply;
      eval_multiply(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::multiply_add&)
   {
      using default_ops::eval_multiply_add;
      eval_multiply_add(m_backend, canonical_value(e.left().value()), canonical_value(e.middle().value()), canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::multiply_subtract&)
   {
      using default_ops::eval_multiply_subtract;
      eval_multiply_subtract(m_backend, canonical_value(e.left().value()), canonical_value(e.middle().value()), canonical_value(e.right().value()));
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::divide_immediates&)
   {
      using default_ops::eval_divide;
      eval_divide(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::negate&)
   {
      typedef typename Exp::left_type left_type;
      do_assign(e.left(), typename left_type::tag_type());
      m_backend.negate();
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::plus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_add(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         // Ignore the right node, it's *this, just add the left:
         do_add(e.left(), typename left_type::tag_type());
      }
      else if(bl && br)
      {
         self_type temp(e);
         temp.m_backend.swap(this->m_backend);
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      { // br is always false, but if bl is true we must take the this branch:
         do_assign(e.left(), typename left_type::tag_type());
         do_add(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_add(e.left(), typename left_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::minus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just subtract the right:
         do_subtract(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         // Ignore the right node, it's *this, just subtract the left and negate the result:
         do_subtract(e.left(), typename left_type::tag_type());
         m_backend.negate();
      }
      else if(bl && br)
      {
         self_type temp(e);
         temp.m_backend.swap(this->m_backend);
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      { // br is always false, but if bl is true we must take the this branch:
         do_assign(e.left(), typename left_type::tag_type());
         do_subtract(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_subtract(e.left(), typename left_type::tag_type());
         m_backend.negate();
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::multiplies&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_multiplies(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         // Ignore the right node, it's *this, just add the left:
         do_multiplies(e.left(), typename left_type::tag_type());
      }
      else if(bl && br)
      {
         self_type temp(e);
         temp.m_backend.swap(this->m_backend);
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      { // br is always false, but if bl is true we must take the this branch:
         do_assign(e.left(), typename left_type::tag_type());
         do_multiplies(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_multiplies(e.left(), typename left_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::divides&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_divide(e.right(), typename right_type::tag_type());
      }
      else if(br)
      {
         self_type temp(e);
         temp.m_backend.swap(this->m_backend);
      }
      else
      {
         do_assign(e.left(), typename left_type::tag_type());
         do_divide(e.right(), typename right_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::modulus&)
   {
      //
      // This operation is only valid for integer backends:
      //
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");

      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_modulus(e.right(), typename right_type::tag_type());
      }
      else if(br)
      {
         self_type temp(e);
         temp.m_backend.swap(this->m_backend);
      }
      else
      {
         do_assign(e.left(), typename left_type::tag_type());
         do_modulus(e.right(), typename right_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::modulus_immediates&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      using default_ops::eval_modulus;
      eval_modulus(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_and&)
   {
      //
      // This operation is only valid for integer backends:
      //
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");

      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_bitwise_and(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         do_bitwise_and(e.left(), typename left_type::tag_type());
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      {
         do_assign(e.left(), typename left_type::tag_type());
         do_bitwise_and(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_bitwise_and(e.left(), typename left_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_and_immediates&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");
      using default_ops::eval_bitwise_and;
      eval_bitwise_and(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_or&)
   {
      //
      // This operation is only valid for integer backends:
      //
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");

      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_bitwise_or(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         do_bitwise_or(e.left(), typename left_type::tag_type());
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      {
         do_assign(e.left(), typename left_type::tag_type());
         do_bitwise_or(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_bitwise_or(e.left(), typename left_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_or_immediates&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");
      using default_ops::eval_bitwise_or;
      eval_bitwise_or(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_xor&)
   {
      //
      // This operation is only valid for integer backends:
      //
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");

      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;

      static int const left_depth = left_type::depth;
      static int const right_depth = right_type::depth;

      bool bl = contains_self(e.left());
      bool br = contains_self(e.right());

      if(bl && is_self(e.left()))
      {
         // Ignore the left node, it's *this, just add the right:
         do_bitwise_xor(e.right(), typename right_type::tag_type());
      }
      else if(br && is_self(e.right()))
      {
         do_bitwise_xor(e.left(), typename left_type::tag_type());
      }
      else if(!br && (bl || (left_depth >= right_depth)))
      {
         do_assign(e.left(), typename left_type::tag_type());
         do_bitwise_xor(e.right(), typename right_type::tag_type());
      }
      else
      {
         do_assign(e.right(), typename right_type::tag_type());
         do_bitwise_xor(e.left(), typename left_type::tag_type());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_xor_immediates&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "Bitwise operations are only valid for integer types");
      using default_ops::eval_bitwise_xor;
      eval_bitwise_xor(m_backend, canonical_value(e.left().value()), canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::terminal&)
   {
      if(!is_self(e))
      {
         m_backend = canonical_value(e.value());
      }
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::function&)
   {
      typedef typename Exp::arity tag_type;
      do_assign_function(e, tag_type());
   }
   template <class Exp>
   void do_assign(const Exp& e, const detail::shift_left&)
   {
      // We can only shift by an integer value, not an arbitrary expression:
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      typedef typename right_type::arity right_arity;
      BOOST_STATIC_ASSERT_MSG(right_arity::value == 0, "The left shift operator requires an integer value for the shift operand.");
      typedef typename right_type::result_type right_value_type;
      BOOST_STATIC_ASSERT_MSG(is_integral<right_value_type>::value, "The left shift operator requires an integer value for the shift operand.");
      typedef typename left_type::tag_type tag_type;
      do_assign_left_shift(e.left(), canonical_value(e.right().value()), tag_type());
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::shift_right&)
   {
      // We can only shift by an integer value, not an arbitrary expression:
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      typedef typename right_type::arity right_arity;
      BOOST_STATIC_ASSERT_MSG(right_arity::value == 0, "The left shift operator requires an integer value for the shift operand.");
      typedef typename right_type::result_type right_value_type;
      BOOST_STATIC_ASSERT_MSG(is_integral<right_value_type>::value, "The left shift operator requires an integer value for the shift operand.");
      typedef typename left_type::tag_type tag_type;
      do_assign_right_shift(e.left(), canonical_value(e.right().value()), tag_type());
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::bitwise_complement&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ~ operation is only valid for integer types");
      using default_ops::eval_complement;
      self_type temp(e.left());
      eval_complement(m_backend, temp.backend());
   }

   template <class Exp>
   void do_assign(const Exp& e, const detail::complement_immediates&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ~ operation is only valid for integer types");
      using default_ops::eval_complement;
      eval_complement(m_backend, canonical_value(e.left().value()));
   }

   template <class Exp, class Val>
   void do_assign_right_shift(const Exp& e, const Val& val, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The right shift operation is only valid for integer types");
      using default_ops::eval_right_shift;
      detail::check_shift_range(val, mpl::bool_<(sizeof(Val) > sizeof(std::size_t))>(), is_signed<Val>());
      eval_right_shift(m_backend, canonical_value(e.value()), static_cast<std::size_t>(val));
   }

   template <class Exp, class Val>
   void do_assign_left_shift(const Exp& e, const Val& val, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The left shift operation is only valid for integer types");
      using default_ops::eval_left_shift;
      detail::check_shift_range(val, mpl::bool_<(sizeof(Val) > sizeof(std::size_t))>(), is_signed<Val>());
      eval_left_shift(m_backend, canonical_value(e.value()), static_cast<std::size_t>(val));
   }

   template <class Exp, class Val, class Tag>
   void do_assign_right_shift(const Exp& e, const Val& val, const Tag&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The right shift operation is only valid for integer types");
      using default_ops::eval_right_shift;
      self_type temp(e);
      detail::check_shift_range(val, mpl::bool_<(sizeof(Val) > sizeof(std::size_t))>(), is_signed<Val>());
      eval_right_shift(m_backend, temp.backend(), static_cast<std::size_t>(val));
   }

   template <class Exp, class Val, class Tag>
   void do_assign_left_shift(const Exp& e, const Val& val, const Tag&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The left shift operation is only valid for integer types");
      using default_ops::eval_left_shift;
      self_type temp(e);
      detail::check_shift_range(val, mpl::bool_<(sizeof(Val) > sizeof(std::size_t))>(), is_signed<Val>());
      eval_left_shift(m_backend, temp.backend(), static_cast<std::size_t>(val));
   }

   template <class Exp>
   void do_assign_function(const Exp& e, const mpl::int_<1>&)
   {
      e.left().value()(&m_backend);
   }
   template <class Exp>
   void do_assign_function(const Exp& e, const mpl::int_<2>&)
   {
      typedef typename Exp::right_type right_type;
      typedef typename right_type::tag_type tag_type;
      do_assign_function_1(e.left().value(), e.right_ref(), tag_type());
   }
   template <class F, class Exp>
   void do_assign_function_1(const F& f, const Exp& val, const detail::terminal&)
   {
      f(m_backend, function_arg_value(val));
   }
   template <class F, class Exp, class Tag>
   void do_assign_function_1(const F& f, const Exp& val, const Tag&)
   {
      number t(val);
      f(m_backend, t.backend());
   }
   template <class Exp>
   void do_assign_function(const Exp& e, const mpl::int_<3>&)
   {
      typedef typename Exp::middle_type middle_type;
      typedef typename middle_type::tag_type tag_type;
      typedef typename Exp::right_type end_type;
      typedef typename end_type::tag_type end_tag;
      do_assign_function_2(e.left().value(), e.middle_ref(), e.right_ref(), tag_type(), end_tag());
   }
   template <class F, class Exp1, class Exp2>
   void do_assign_function_2(const F& f, const Exp1& val1, const Exp2& val2, const detail::terminal&, const detail::terminal&)
   {
      f(m_backend, function_arg_value(val1), function_arg_value(val2));
   }
   template <class F, class Exp1, class Exp2, class Tag1>
   void do_assign_function_2(const F& f, const Exp1& val1, const Exp2& val2, const Tag1&, const detail::terminal&)
   {
      self_type temp1(val1);
      f(m_backend, BOOST_MP_MOVE(temp1.backend()), function_arg_value(val2));
   }
   template <class F, class Exp1, class Exp2, class Tag2>
   void do_assign_function_2(const F& f, const Exp1& val1, const Exp2& val2, const detail::terminal&, const Tag2&)
   {
      self_type temp2(val2);
      f(m_backend, function_arg_value(val1), BOOST_MP_MOVE(temp2.backend()));
   }
   template <class F, class Exp1, class Exp2, class Tag1, class Tag2>
   void do_assign_function_2(const F& f, const Exp1& val1, const Exp2& val2, const Tag1&, const Tag2&)
   {
      self_type temp1(val1);
      self_type temp2(val2);
      f(m_backend, BOOST_MP_MOVE(temp1.backend()), BOOST_MP_MOVE(temp2.backend()));
   }

   template <class Exp>
   void do_assign_function(const Exp& e, const mpl::int_<4>&)
   {
      typedef typename Exp::left_middle_type left_type;
      typedef typename left_type::tag_type left_tag_type;
      typedef typename Exp::right_middle_type middle_type;
      typedef typename middle_type::tag_type middle_tag_type;
      typedef typename Exp::right_type right_type;
      typedef typename right_type::tag_type right_tag_type;
      do_assign_function_3a(e.left().value(), e.left_middle_ref(), e.right_middle_ref(), e.right_ref(), left_tag_type(), middle_tag_type(), right_tag_type());
   }
   template <class F, class Exp1, class Exp2, class Exp3, class Tag2, class Tag3>
   void do_assign_function_3a(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const detail::terminal&, const Tag2& t2, const Tag3& t3)
   {
      do_assign_function_3b(f, val1, val2, val3, t2, t3);
   }
   template <class F, class Exp1, class Exp2, class Exp3, class Tag1, class Tag2, class Tag3>
   void do_assign_function_3a(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const Tag1&, const Tag2& t2, const Tag3& t3)
   {
      number t(val1);
      do_assign_function_3b(f, BOOST_MP_MOVE(t), val2, val3, t2, t3);
   }
   template <class F, class Exp1, class Exp2, class Exp3, class Tag3>
   void do_assign_function_3b(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const detail::terminal&, const Tag3& t3)
   {
      do_assign_function_3c(f, val1, val2, val3, t3);
   }
   template <class F, class Exp1, class Exp2, class Exp3, class Tag2, class Tag3>
   void do_assign_function_3b(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const Tag2& /*t2*/, const Tag3& t3)
   {
      number t(val2);
      do_assign_function_3c(f, val1, BOOST_MP_MOVE(t), val3, t3);
   }
   template <class F, class Exp1, class Exp2, class Exp3>
   void do_assign_function_3c(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const detail::terminal&)
   {
      f(m_backend, function_arg_value(val1), function_arg_value(val2), function_arg_value(val3));
   }
   template <class F, class Exp1, class Exp2, class Exp3, class Tag3>
   void do_assign_function_3c(const F& f, const Exp1& val1, const Exp2& val2, const Exp3& val3, const Tag3& /*t3*/)
   {
      number t(val3);
      do_assign_function_3c(f, val1, val2, BOOST_MP_MOVE(t), detail::terminal());
   }

   template <class Exp>
   void do_add(const Exp& e, const detail::terminal&)
   {
      using default_ops::eval_add;
      eval_add(m_backend, canonical_value(e.value()));
   }

   template <class Exp>
   void do_add(const Exp& e, const detail::negate&)
   {
      typedef typename Exp::left_type left_type;
      do_subtract(e.left(), typename left_type::tag_type());
   }

   template <class Exp>
   void do_add(const Exp& e, const detail::plus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_add(e.left(), typename left_type::tag_type());
      do_add(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_add(const Exp& e, const detail::minus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_add(e.left(), typename left_type::tag_type());
      do_subtract(e.right(), typename right_type::tag_type());
   }

   template <class Exp, class unknown>
   void do_add(const Exp& e, const unknown&)
   {
      self_type temp(e);
      do_add(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
   }

   template <class Exp>
   void do_add(const Exp& e, const detail::add_immediates&)
   {
      using default_ops::eval_add;
      eval_add(m_backend, canonical_value(e.left().value()));
      eval_add(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_add(const Exp& e, const detail::subtract_immediates&)
   {
      using default_ops::eval_add;
      using default_ops::eval_subtract;
      eval_add(m_backend, canonical_value(e.left().value()));
      eval_subtract(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_subtract(const Exp& e, const detail::terminal&)
   {
      using default_ops::eval_subtract;
      eval_subtract(m_backend, canonical_value(e.value()));
   }

   template <class Exp>
   void do_subtract(const Exp& e, const detail::negate&)
   {
      typedef typename Exp::left_type left_type;
      do_add(e.left(), typename left_type::tag_type());
   }

   template <class Exp>
   void do_subtract(const Exp& e, const detail::plus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_subtract(e.left(), typename left_type::tag_type());
      do_subtract(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_subtract(const Exp& e, const detail::minus&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_subtract(e.left(), typename left_type::tag_type());
      do_add(e.right(), typename right_type::tag_type());
   }
   template <class Exp>
   void do_subtract(const Exp& e, const detail::add_immediates&)
   {
      using default_ops::eval_subtract;
      eval_subtract(m_backend, canonical_value(e.left().value()));
      eval_subtract(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_subtract(const Exp& e, const detail::subtract_immediates&)
   {
      using default_ops::eval_add;
      using default_ops::eval_subtract;
      eval_subtract(m_backend, canonical_value(e.left().value()));
      eval_add(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp, class unknown>
   void do_subtract(const Exp& e, const unknown&)
   {
      self_type temp(e);
      do_subtract(detail::expression<detail::terminal, self_type>(temp), detail::terminal());
   }

   template <class Exp>
   void do_multiplies(const Exp& e, const detail::terminal&)
   {
      using default_ops::eval_multiply;
      eval_multiply(m_backend, canonical_value(e.value()));
   }

   template <class Exp>
   void do_multiplies(const Exp& e, const detail::negate&)
   {
      typedef typename Exp::left_type left_type;
      do_multiplies(e.left(), typename left_type::tag_type());
      m_backend.negate();
   }

   template <class Exp>
   void do_multiplies(const Exp& e, const detail::multiplies&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_multiplies(e.left(), typename left_type::tag_type());
      do_multiplies(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_multiplies(const Exp& e, const detail::divides&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_multiplies(e.left(), typename left_type::tag_type());
      do_divide(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_multiplies(const Exp& e, const detail::multiply_immediates&)
   {
      using default_ops::eval_multiply;
      eval_multiply(m_backend, canonical_value(e.left().value()));
      eval_multiply(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_multiplies(const Exp& e, const detail::divide_immediates&)
   {
      using default_ops::eval_multiply;
      using default_ops::eval_divide;
      eval_multiply(m_backend, canonical_value(e.left().value()));
      eval_divide(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp, class unknown>
   void do_multiplies(const Exp& e, const unknown&)
   {
      using default_ops::eval_multiply;
      self_type temp(e);
      eval_multiply(m_backend, temp.m_backend);
   }

   template <class Exp>
   void do_divide(const Exp& e, const detail::terminal&)
   {
      using default_ops::eval_divide;
      eval_divide(m_backend, canonical_value(e.value()));
   }

   template <class Exp>
   void do_divide(const Exp& e, const detail::negate&)
   {
      typedef typename Exp::left_type left_type;
      do_divide(e.left(), typename left_type::tag_type());
      m_backend.negate();
   }

   template <class Exp>
   void do_divide(const Exp& e, const detail::multiplies&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_divide(e.left(), typename left_type::tag_type());
      do_divide(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_divide(const Exp& e, const detail::divides&)
   {
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_divide(e.left(), typename left_type::tag_type());
      do_multiplies(e.right(), typename right_type::tag_type());
   }

   template <class Exp>
   void do_divides(const Exp& e, const detail::multiply_immediates&)
   {
      using default_ops::eval_divide;
      eval_divide(m_backend, canonical_value(e.left().value()));
      eval_divide(m_backend, canonical_value(e.right().value()));
   }
   template <class Exp>
   void do_divides(const Exp& e, const detail::divide_immediates&)
   {
      using default_ops::eval_multiply;
      using default_ops::eval_divide;
      eval_divide(m_backend, canonical_value(e.left().value()));
      mutiply(m_backend, canonical_value(e.right().value()));
   }

   template <class Exp, class unknown>
   void do_divide(const Exp& e, const unknown&)
   {
      using default_ops::eval_multiply;
      self_type temp(e);
      eval_divide(m_backend, temp.m_backend);
   }

   template <class Exp>
   void do_modulus(const Exp& e, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      using default_ops::eval_modulus;
      eval_modulus(m_backend, canonical_value(e.value()));
   }

   template <class Exp, class Unknown>
   void do_modulus(const Exp& e, const Unknown&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The modulus operation is only valid for integer types");
      using default_ops::eval_modulus;
      self_type temp(e);
      eval_modulus(m_backend, canonical_value(temp));
   }

   template <class Exp>
   void do_bitwise_and(const Exp& e, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      using default_ops::eval_bitwise_and;
      eval_bitwise_and(m_backend, canonical_value(e.value()));
   }
   template <class Exp>
   void do_bitwise_and(const Exp& e, const detail::bitwise_and&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_bitwise_and(e.left(), typename left_type::tag_type());
      do_bitwise_and(e.right(), typename right_type::tag_type());
   }
   template <class Exp, class unknown>
   void do_bitwise_and(const Exp& e, const unknown&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise & operation is only valid for integer types");
      using default_ops::eval_bitwise_and;
      self_type temp(e);
      eval_bitwise_and(m_backend, temp.m_backend);
   }

   template <class Exp>
   void do_bitwise_or(const Exp& e, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      using default_ops::eval_bitwise_or;
      eval_bitwise_or(m_backend, canonical_value(e.value()));
   }
   template <class Exp>
   void do_bitwise_or(const Exp& e, const detail::bitwise_or&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_bitwise_or(e.left(), typename left_type::tag_type());
      do_bitwise_or(e.right(), typename right_type::tag_type());
   }
   template <class Exp, class unknown>
   void do_bitwise_or(const Exp& e, const unknown&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise | operation is only valid for integer types");
      using default_ops::eval_bitwise_or;
      self_type temp(e);
      eval_bitwise_or(m_backend, temp.m_backend);
   }

   template <class Exp>
   void do_bitwise_xor(const Exp& e, const detail::terminal&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      using default_ops::eval_bitwise_xor;
      eval_bitwise_xor(m_backend, canonical_value(e.value()));
   }
   template <class Exp>
   void do_bitwise_xor(const Exp& e, const detail::bitwise_xor&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      typedef typename Exp::left_type left_type;
      typedef typename Exp::right_type right_type;
      do_bitwise_xor(e.left(), typename left_type::tag_type());
      do_bitwise_xor(e.right(), typename right_type::tag_type());
   }
   template <class Exp, class unknown>
   void do_bitwise_xor(const Exp& e, const unknown&)
   {
      BOOST_STATIC_ASSERT_MSG(number_category<Backend>::value == number_kind_integer, "The bitwise ^ operation is only valid for integer types");
      using default_ops::eval_bitwise_xor;
      self_type temp(e);
      eval_bitwise_xor(m_backend, temp.m_backend);
   }

   // Tests if the expression contains a reference to *this:
   template <class Exp>
   BOOST_MP_FORCEINLINE bool contains_self(const Exp& e)const BOOST_NOEXCEPT
   {
      return contains_self(e, typename Exp::arity());
   }
   template <class Exp>
   BOOST_MP_FORCEINLINE bool contains_self(const Exp& e, mpl::int_<0> const&)const BOOST_NOEXCEPT
   {
      return is_realy_self(e.value());
   }
   template <class Exp>
   BOOST_MP_FORCEINLINE bool contains_self(const Exp& e, mpl::int_<1> const&)const BOOST_NOEXCEPT
   {
      typedef typename Exp::left_type child_type;
      return contains_self(e.left(), typename child_type::arity());
   }
   template <class Exp>
   BOOST_MP_FORCEINLINE bool contains_self(const Exp& e, mpl::int_<2> const&)const BOOST_NOEXCEPT
   {
      typedef typename Exp::left_type child0_type;
      typedef typename Exp::right_type child1_type;
      return contains_self(e.left(), typename child0_type::arity())
         || contains_self(e.right(), typename child1_type::arity());
   }
   template <class Exp>
   BOOST_MP_FORCEINLINE bool contains_self(const Exp& e, mpl::int_<3> const&)const BOOST_NOEXCEPT
   {
      typedef typename Exp::left_type child0_type;
      typedef typename Exp::middle_type child1_type;
      typedef typename Exp::right_type child2_type;
      return contains_self(e.left(), typename child0_type::arity())
         || contains_self(e.middle(), typename child1_type::arity())
         || contains_self(e.right(), typename child2_type::arity());
   }

   // Test if the expression is a reference to *this:
   template <class Exp>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR bool is_self(const Exp& e)const BOOST_NOEXCEPT
   {
      return is_self(e, typename Exp::arity());
   }
   template <class Exp>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR bool is_self(const Exp& e, mpl::int_<0> const&)const BOOST_NOEXCEPT
   {
      return is_realy_self(e.value());
   }
   template <class Exp, int v>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR bool is_self(const Exp&, mpl::int_<v> const&)const BOOST_NOEXCEPT
   {
      return false;
   }

   template <class Val>
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR bool is_realy_self(const Val&)const BOOST_NOEXCEPT{ return false; }
   BOOST_MP_FORCEINLINE BOOST_CONSTEXPR bool is_realy_self(const self_type& v)const BOOST_NOEXCEPT{ return &v == this; }

   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const Backend& function_arg_value(const self_type& v) BOOST_NOEXCEPT {  return v.backend();  }
   template <class V>
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const V& function_arg_value(const V& v) BOOST_NOEXCEPT {  return v;  }
   template <class A1, class A2, class A3, class A4>
   static BOOST_MP_FORCEINLINE const A1& function_arg_value(const detail::expression<detail::terminal, A1, A2, A3, A4>& exp) BOOST_NOEXCEPT { return exp.value(); }
   template <class A2, class A3, class A4>
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const Backend& function_arg_value(const detail::expression<detail::terminal, number<Backend>, A2, A3, A4>& exp) BOOST_NOEXCEPT { return exp.value().backend(); }
   Backend m_backend;

public:
   //
   // These shouldn't really need to be public, or even member functions, but it makes implementing
   // the non-member operators way easier if they are:
   //
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const Backend& canonical_value(const self_type& v) BOOST_NOEXCEPT {  return v.m_backend;  }
   template <class B2, expression_template_option ET>
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR const B2& canonical_value(const number<B2, ET>& v) BOOST_NOEXCEPT {  return v.backend();  }
   template <class V>
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR typename boost::disable_if<is_same<typename detail::canonical<V, Backend>::type, V>, typename detail::canonical<V, Backend>::type>::type
      canonical_value(const V& v) BOOST_NOEXCEPT {  return static_cast<typename detail::canonical<V, Backend>::type>(v);  }
   template <class V>
   static BOOST_MP_FORCEINLINE BOOST_CONSTEXPR typename boost::enable_if<is_same<typename detail::canonical<V, Backend>::type, V>, const V&>::type
      canonical_value(const V& v) BOOST_NOEXCEPT {  return v;  }
   static BOOST_MP_FORCEINLINE typename detail::canonical<std::string, Backend>::type canonical_value(const std::string& v) BOOST_NOEXCEPT {  return v.c_str();  }

};

template <class Backend, expression_template_option ExpressionTemplates>
inline std::ostream& operator << (std::ostream& os, const number<Backend, ExpressionTemplates>& r)
{
   std::streamsize d = os.precision();
   std::string s = r.str(d, os.flags());
   std::streamsize ss = os.width();
   if(ss > static_cast<std::streamsize>(s.size()))
   {
      char fill = os.fill();
      if((os.flags() & std::ios_base::left) == std::ios_base::left)
         s.append(static_cast<std::string::size_type>(ss - s.size()), fill);
      else
         s.insert(0, static_cast<std::string::size_type>(ss - s.size()), fill);
   }
   return os << s;
}

namespace detail{

template <class tag, class A1, class A2, class A3, class A4>
inline std::ostream& operator << (std::ostream& os, const expression<tag, A1, A2, A3, A4>& r)
{
   typedef typename expression<tag, A1, A2, A3, A4>::result_type value_type;
   value_type temp(r);
   return os << temp;
}

} // namespace detail

template <class Backend, expression_template_option ExpressionTemplates>
inline std::istream& operator >> (std::istream& is, number<Backend, ExpressionTemplates>& r)
{
   bool hex_format = (is.flags() & std::ios_base::hex) == std::ios_base::hex;
   bool oct_format = (is.flags() & std::ios_base::oct) == std::ios_base::oct;
   std::string s;
   is >> s;
   if(hex_format && (number_category<Backend>::value == number_kind_integer) && ((s[0] != '0') || (s[1] != 'x')))
      s.insert(s.find_first_not_of("+-"), "0x");
   if(oct_format && (number_category<Backend>::value == number_kind_integer) && (s[0] != '0'))
      s.insert(s.find_first_not_of("+-"), "0");
   r.assign(s);
   return is;
}

template <class Backend, expression_template_option ExpressionTemplates>
BOOST_MP_FORCEINLINE void swap(number<Backend, ExpressionTemplates>& a, number<Backend, ExpressionTemplates>& b)
{
   a.swap(b);
}

}  // namespace multiprecision

template <class T>
class rational;

template <class Backend, multiprecision::expression_template_option ExpressionTemplates>
inline std::istream& operator >> (std::istream& is, rational<multiprecision::number<Backend, ExpressionTemplates> >& r)
{
   std::string s1;
   multiprecision::number<Backend, ExpressionTemplates> v1, v2;
   char c;
   bool have_hex = false;
   bool hex_format = (is.flags() & std::ios_base::hex) == std::ios_base::hex;
   bool oct_format = (is.flags() & std::ios_base::oct) == std::ios_base::oct;

   while((EOF != (c = static_cast<char>(is.peek()))) && (c == 'x' || c == 'X' || c == '-' || c == '+' || (c >= '0' && c <= '9') || (have_hex && (c >= 'a' && c <= 'f')) || (have_hex && (c >= 'A' && c <= 'F'))))
   {
      if(c == 'x' || c == 'X')
         have_hex = true;
      s1.append(1, c);
      is.get();
   }
   if(hex_format && ((s1[0] != '0') || (s1[1] != 'x')))
      s1.insert(0, "0x");
   if(oct_format && (s1[0] != '0'))
      s1.insert(0, "0");
   v1.assign(s1);
   s1.erase();
   if(c == '/')
   {
      is.get();
      while((EOF != (c = static_cast<char>(is.peek()))) && (c == 'x' || c == 'X' || c == '-' || c == '+' || (c >= '0' && c <= '9') || (have_hex && (c >= 'a' && c <= 'f')) || (have_hex && (c >= 'A' && c <= 'F'))))
      {
         if(c == 'x' || c == 'X')
            have_hex = true;
         s1.append(1, c);
         is.get();
      }
      if(hex_format && ((s1[0] != '0') || (s1[1] != 'x')))
         s1.insert(0, "0x");
      if(oct_format && (s1[0] != '0'))
         s1.insert(0, "0");
      v2.assign(s1);
   }
   else
      v2 = 1;
   r.assign(v1, v2);
   return is;
}

template <class T, multiprecision::expression_template_option ExpressionTemplates, class Arithmetic>
typename boost::enable_if<boost::is_arithmetic<Arithmetic>, bool>::type operator == (const rational<multiprecision::number<T, ExpressionTemplates> >& a, const Arithmetic& b)
{
   return a == multiprecision::number<T, ExpressionTemplates>(b);
}

template <class T, multiprecision::expression_template_option ExpressionTemplates, class Arithmetic>
typename boost::enable_if<boost::is_arithmetic<Arithmetic>, bool>::type operator == (const Arithmetic& b, const rational<multiprecision::number<T, ExpressionTemplates> >& a)
{
   return a == multiprecision::number<T, ExpressionTemplates>(b);
}

template <class T, multiprecision::expression_template_option ExpressionTemplates, class Arithmetic>
typename boost::enable_if<boost::is_arithmetic<Arithmetic>, bool>::type operator != (const rational<multiprecision::number<T, ExpressionTemplates> >& a, const Arithmetic& b)
{
   return a != multiprecision::number<T, ExpressionTemplates>(b);
}

template <class T, multiprecision::expression_template_option ExpressionTemplates, class Arithmetic>
typename boost::enable_if<boost::is_arithmetic<Arithmetic>, bool>::type operator != (const Arithmetic& b, const rational<multiprecision::number<T, ExpressionTemplates> >& a)
{
   return a != multiprecision::number<T, ExpressionTemplates>(b);
}

template <class T, multiprecision::expression_template_option ExpressionTemplates>
inline multiprecision::number<T, ExpressionTemplates> numerator(const rational<multiprecision::number<T, ExpressionTemplates> >& a)
{
   return a.numerator();
}

template <class T, multiprecision::expression_template_option ExpressionTemplates>
inline multiprecision::number<T, ExpressionTemplates> denominator(const rational<multiprecision::number<T, ExpressionTemplates> >& a)
{
   return a.denominator();
}

#ifdef BOOST_MSVC
#pragma warning(pop)
#endif

} // namespaces

#include <boost/multiprecision/detail/ublas_interop.hpp>

#endif