boost/multiprecision/cpp_int.hpp

//////////////////3/////////////////////////////////////////////
//  Copyright 2012 John Maddock. Distributed under the Boost
//  Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at https://www.boost.org/LICENSE_1_0.txt

#ifndef BOOST_MP_CPP_INT_HPP
#define BOOST_MP_CPP_INT_HPP

#include <iostream>
#include <iomanip>
#include <type_traits>
#include <cstdint>
#include <boost/multiprecision/detail/endian.hpp>
#include <boost/multiprecision/number.hpp>
#include <boost/multiprecision/detail/integer_ops.hpp>
#include <boost/multiprecision/detail/rebind.hpp>
#include <boost/core/empty_value.hpp>
#include <boost/multiprecision/cpp_int/cpp_int_config.hpp>
#include <boost/multiprecision/rational_adaptor.hpp>
#include <boost/multiprecision/traits/is_byte_container.hpp>
#include <boost/integer/static_min_max.hpp>
#include <boost/multiprecision/cpp_int/checked.hpp>
#include <boost/multiprecision/detail/constexpr.hpp>
#include <boost/multiprecision/cpp_int/value_pack.hpp>

namespace boost {
namespace multiprecision {
namespace backends {

#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable : 4307) // integral constant overflow (oveflow is in a branch not taken when it would overflow)
#pragma warning(disable : 4127) // conditional expression is constant
#pragma warning(disable : 4702) // Unreachable code (reachability depends on template params)
#endif

template <unsigned MinBits = 0, unsigned MaxBits = 0, boost::multiprecision::cpp_integer_type SignType = signed_magnitude, cpp_int_check_type Checked = unchecked, class Allocator = typename std::conditional<MinBits && (MinBits == MaxBits), void, std::allocator<limb_type> >::type>
struct cpp_int_backend;

} // namespace backends

namespace detail {

template <unsigned MinBits, unsigned MaxBits, boost::multiprecision::cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct is_byte_container<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> > : public std::false_type
{};

} // namespace detail

namespace backends {

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator, bool trivial = false>
struct cpp_int_base;
//
// Traits class determines the maximum and minimum precision values:
//
template <class T>
struct max_precision;

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct max_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
{
   static constexpr const unsigned value = std::is_void<Allocator>::value ? static_unsigned_max<MinBits, MaxBits>::value
                                                           : (((MaxBits >= MinBits) && MaxBits) ? MaxBits : UINT_MAX);
};

template <class T>
struct min_precision;

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct min_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
{
   static constexpr const unsigned value = (std::is_void<Allocator>::value ? static_unsigned_max<MinBits, MaxBits>::value : MinBits);
};
//
// Traits class determines whether the number of bits precision requested could fit in a native type,
// we call this a "trivial" cpp_int:
//
template <class T>
struct is_trivial_cpp_int
{
   static constexpr const bool value = false;
};

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct is_trivial_cpp_int<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
{
   using self = cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>;
   static constexpr const bool                                             value = std::is_void<Allocator>::value && (max_precision<self>::value <= (sizeof(double_limb_type) * CHAR_BIT) - (SignType == signed_packed ? 1 : 0));
};

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct is_trivial_cpp_int<cpp_int_base<MinBits, MaxBits, SignType, Checked, Allocator, true> >
{
   static constexpr const bool value = true;
};

} // namespace backends
//
// Traits class to determine whether a cpp_int_backend is signed or not:
//
template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct is_unsigned_number<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
    : public std::integral_constant<bool, (SignType == unsigned_magnitude) || (SignType == unsigned_packed)>
{};

namespace backends {
//
// Traits class determines whether T should be implicitly convertible to U, or
// whether the constructor should be made explicit.  The latter happens if we
// are losing the sign, or have fewer digits precision in the target type:
//
template <class T, class U>
struct is_implicit_cpp_int_conversion;

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator, unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
struct is_implicit_cpp_int_conversion<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>, cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >
{
   using t1 = cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>     ;
   using t2 = cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>;
   static constexpr const bool                                                  value =
       (is_signed_number<t2>::value || !is_signed_number<t1>::value) && (max_precision<t1>::value <= max_precision<t2>::value);
};

//
// Traits class to determine whether operations on a cpp_int may throw:
//
template <class T>
struct is_non_throwing_cpp_int : public std::integral_constant<bool, false>
{};
template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType>
struct is_non_throwing_cpp_int<cpp_int_backend<MinBits, MaxBits, SignType, unchecked, void> > : public std::integral_constant<bool, true>
{};

//
// Traits class, determines whether the cpp_int is fixed precision or not:
//
template <class T>
struct is_fixed_precision;
template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct is_fixed_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
    : public std::integral_constant<bool, max_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value != UINT_MAX>
{};

namespace detail {

inline BOOST_MP_CXX14_CONSTEXPR void verify_new_size(unsigned new_size, unsigned min_size, const std::integral_constant<int, checked>&)
{
   if (new_size < min_size)
      BOOST_THROW_EXCEPTION(std::overflow_error("Unable to allocate sufficient storage for the value of the result: value overflows the maximum allowable magnitude."));
}
inline BOOST_MP_CXX14_CONSTEXPR void verify_new_size(unsigned /*new_size*/, unsigned /*min_size*/, const std::integral_constant<int, unchecked>&) {}

template <class U>
inline BOOST_MP_CXX14_CONSTEXPR void verify_limb_mask(bool b, U limb, U mask, const std::integral_constant<int, checked>&)
{
   // When we mask out "limb" with "mask", do we loose bits?  If so it's an overflow error:
   if (b && (limb & ~mask))
      BOOST_THROW_EXCEPTION(std::overflow_error("Overflow in cpp_int arithmetic: there is insufficient precision in the target type to hold all of the bits of the result."));
}
template <class U>
inline BOOST_MP_CXX14_CONSTEXPR void verify_limb_mask(bool /*b*/, U /*limb*/, U /*mask*/, const std::integral_constant<int, unchecked>&) {}

} // namespace detail

//
// Now define the various data layouts that are possible as partial specializations of the base class,
// starting with the default arbitrary precision signed integer type:
//
template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
struct cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>
    : private boost::empty_value<typename detail::rebind<limb_type, Allocator>::type>
{
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2, bool trivial2>
   friend struct cpp_int_base;

   using allocator_type = typename detail::rebind<limb_type, Allocator>::type;
   using limb_pointer = typename std::allocator_traits<allocator_type>::pointer      ;
   using const_limb_pointer = typename std::allocator_traits<allocator_type>::const_pointer;
   using checked_type = std::integral_constant<int, Checked>;

   //
   // Interface invariants:
   //
   static_assert(!std::is_void<Allocator>::value, "Allocator must not be void here");

   using base_type = boost::empty_value<allocator_type>;

private:
   struct limb_data
   {
      unsigned        capacity;
      limb_pointer    data;
   };

 public:
   static constexpr unsigned limb_bits = sizeof(limb_type) * CHAR_BIT;
   static constexpr limb_type max_limb_value = ~static_cast<limb_type>(0u);
   static constexpr limb_type sign_bit_mask = static_cast<limb_type>(1u) << (limb_bits - 1);
   static constexpr unsigned internal_limb_count =
                                       MinBits
                                           ? (MinBits / limb_bits + ((MinBits % limb_bits) ? 1 : 0))
                                           : (sizeof(limb_data) / sizeof(limb_type)) > 1 ? (sizeof(limb_data) / sizeof(limb_type)) : 2;
 private:
   union data_type
   {
      limb_data        ld;
      limb_type        la[internal_limb_count];
      limb_type        first;
      double_limb_type double_first;

      constexpr data_type() noexcept : first(0) {}
      constexpr data_type(limb_type i) noexcept : first(i) {}
      constexpr data_type(signed_limb_type i) noexcept : first(i < 0 ? static_cast<limb_type>(boost::multiprecision::detail::unsigned_abs(i)) : i) {}
#if BOOST_MP_ENDIAN_LITTLE_BYTE
      constexpr data_type(double_limb_type i) noexcept : double_first(i)
      {}
      constexpr data_type(signed_double_limb_type i) noexcept : double_first(i < 0 ? static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) : i) {}
#endif
#if !defined(BOOST_NO_CXX11_UNIFIED_INITIALIZATION_SYNTAX) && !(defined(BOOST_MSVC) && (BOOST_MSVC < 1900))
      constexpr data_type(limb_type* limbs, unsigned len) noexcept : ld{ len, limbs }
      {}
#else
      constexpr data_type(limb_type* limbs, unsigned len) noexcept
      {
         ld.capacity = len;
         ld.data = limbs;
      }
#endif
   };

   data_type m_data;
   unsigned  m_limbs;
   bool      m_sign, m_internal, m_alias;

 public:
   //
   // Direct construction:
   //
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(limb_type i) noexcept
       : m_data(i),
         m_limbs(1),
         m_sign(false),
         m_internal(true),
         m_alias(false) {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(signed_limb_type i) noexcept
       : m_data(i),
         m_limbs(1),
         m_sign(i < 0),
         m_internal(true),
         m_alias(false) {}
#if BOOST_MP_ENDIAN_LITTLE_BYTE && !defined(BOOST_MP_TEST_NO_LE)
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(double_limb_type i) noexcept
       : m_data(i),
         m_limbs(i > max_limb_value ? 2 : 1),
         m_sign(false),
         m_internal(true),
         m_alias(false)
   {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(signed_double_limb_type i) noexcept
       : m_data(i),
         m_limbs(i < 0 ? (static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) > static_cast<double_limb_type>(max_limb_value) ? 2 : 1) : (i > max_limb_value ? 2 : 1)),
         m_sign(i < 0),
         m_internal(true),
         m_alias(false) {}
#endif
   //
   // Aliasing constructor aliases data:
   //
   struct scoped_shared_storage : private boost::empty_value<allocator_type>
   {
    private:
      limb_type*      data;
      unsigned        capacity;
      unsigned        allocated;
      bool            is_alias;
      allocator_type& allocator() noexcept { return boost::empty_value<allocator_type>::get(); }

    public:
      scoped_shared_storage(const allocator_type& a, unsigned len)
          : boost::empty_value<allocator_type>(boost::empty_init_t(), a), capacity(len), allocated(0), is_alias(false)
      {
         data = allocator().allocate(len);
      }
      scoped_shared_storage(const cpp_int_base& i, unsigned len)
          : boost::empty_value<allocator_type>(boost::empty_init_t(), i.allocator()), capacity(len), allocated(0), is_alias(false)
      {
         data = allocator().allocate(len);
      }
      scoped_shared_storage(limb_type* limbs, unsigned n) : data(limbs), capacity(n), allocated(0), is_alias(true) {}
      ~scoped_shared_storage()
      {
         if(!is_alias)
            allocator().deallocate(data, capacity);
      }
      limb_type* allocate(unsigned n) noexcept
      {
         limb_type* result = data + allocated;
         allocated += n;
         BOOST_ASSERT(allocated <= capacity);
         return result;
      }
      void deallocate(unsigned n)
      {
         BOOST_ASSERT(n <= allocated);
         allocated -= n;
      }
   };
   explicit constexpr cpp_int_base(limb_type* data, unsigned offset, unsigned len) noexcept
       : m_data(data + offset, len),
         m_limbs(len),
         m_sign(false),
         m_internal(false),
         m_alias(true) {}
   // This next constructor is for constructing const objects from const limb_type*'s only.
   // Unfortunately we appear to have no way to assert that within the language, and the const_cast
   // is a side effect of that :(
   explicit constexpr cpp_int_base(const limb_type* data, unsigned offset, unsigned len) noexcept
       : m_data(const_cast<limb_type*>(data) + offset, len),
         m_limbs(len),
         m_sign(false),
         m_internal(false),
         m_alias(true) {}
   explicit cpp_int_base(scoped_shared_storage& data, unsigned len) noexcept
       : m_data(data.allocate(len), len),
         m_limbs(len),
         m_sign(false),
         m_internal(false),
         m_alias(true) {}
   //
   // Helper functions for getting at our internal data, and manipulating storage:
   //
   BOOST_MP_FORCEINLINE allocator_type&       allocator() noexcept { return base_type::get(); }
   BOOST_MP_FORCEINLINE const allocator_type& allocator() const noexcept { return base_type::get(); }
   BOOST_MP_FORCEINLINE unsigned              size() const noexcept { return m_limbs; }
   BOOST_MP_FORCEINLINE limb_pointer          limbs() noexcept { return m_internal ? m_data.la : m_data.ld.data; }
   BOOST_MP_FORCEINLINE const_limb_pointer    limbs() const noexcept { return m_internal ? m_data.la : m_data.ld.data; }
   BOOST_MP_FORCEINLINE unsigned              capacity() const noexcept { return m_internal ? internal_limb_count : m_data.ld.capacity; }
   BOOST_MP_FORCEINLINE bool                  sign() const noexcept { return m_sign; }
   void                                       sign(bool b) noexcept
   {
      m_sign = b;
      // Check for zero value:
      if (m_sign && (m_limbs == 1))
      {
         if (limbs()[0] == 0)
            m_sign = false;
      }
   }
   void resize(unsigned new_size, unsigned min_size)
   {
      constexpr const unsigned max_limbs = MaxBits / (CHAR_BIT * sizeof(limb_type)) + ((MaxBits % (CHAR_BIT * sizeof(limb_type))) ? 1 : 0);
      // We never resize beyond MaxSize:
      if (new_size > max_limbs)
         new_size = max_limbs;
      detail::verify_new_size(new_size, min_size, checked_type());
      // See if we have enough capacity already:
      unsigned cap = capacity();
      if (new_size > cap)
      {
         // We must not be an alias, memory allocation here defeats the whole point of aliasing:
         BOOST_ASSERT(!m_alias);
         // Allocate a new buffer and copy everything over:
         cap             = (std::min)((std::max)(cap * 4, new_size), max_limbs);
         limb_pointer pl = allocator().allocate(cap);
         std::memcpy(pl, limbs(), size() * sizeof(limbs()[0]));
         if (!m_internal && !m_alias)
            allocator().deallocate(limbs(), capacity());
         else
            m_internal = false;
         m_limbs            = new_size;
         m_data.ld.capacity = cap;
         m_data.ld.data     = pl;
      }
      else
      {
         m_limbs = new_size;
      }
   }
   BOOST_MP_FORCEINLINE void normalize() noexcept
   {
      limb_pointer p = limbs();
      while ((m_limbs - 1) && !p[m_limbs - 1])
         --m_limbs;
   }
   BOOST_MP_FORCEINLINE constexpr cpp_int_base() noexcept : m_data(), m_limbs(1), m_sign(false), m_internal(true), m_alias(false){}
   BOOST_MP_FORCEINLINE                 cpp_int_base(const cpp_int_base& o) : base_type(o), m_limbs(o.m_alias ? o.m_limbs : 0), m_sign(o.m_sign), m_internal(o.m_alias ? false : true), m_alias(o.m_alias)
   {
      if (m_alias)
      {
         m_data.ld = o.m_data.ld;
      }
      else
      {
         resize(o.size(), o.size());
         std::memcpy(limbs(), o.limbs(), o.size() * sizeof(limbs()[0]));
      }
   }
   // rvalue copy:
   cpp_int_base(cpp_int_base&& o)
      : base_type(static_cast<base_type&&>(o)), m_limbs(o.m_limbs), m_sign(o.m_sign), m_internal(o.m_internal), m_alias(o.m_alias)
   {
      if (m_internal)
      {
         std::memcpy(limbs(), o.limbs(), o.size() * sizeof(limbs()[0]));
      }
      else
      {
         m_data.ld = o.m_data.ld;
         o.m_limbs = 0;
         o.m_internal = true;
      }
   }
   cpp_int_base& operator=(cpp_int_base&& o) noexcept
   {
      if (!m_internal && !m_alias)
         allocator().deallocate(m_data.ld.data, m_data.ld.capacity);
      *static_cast<base_type*>(this) = static_cast<base_type&&>(o);
      m_limbs = o.m_limbs;
      m_sign = o.m_sign;
      m_internal = o.m_internal;
      m_alias = o.m_alias;
      if (m_internal)
      {
         std::memcpy(limbs(), o.limbs(), o.size() * sizeof(limbs()[0]));
      }
      else
      {
         m_data.ld = o.m_data.ld;
         o.m_limbs = 0;
         o.m_internal = true;
      }
      return *this;
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_int_check_type Checked2>
   cpp_int_base& operator=(cpp_int_base<MinBits2, MaxBits2, signed_magnitude, Checked2, Allocator>&& o) noexcept
   {
      if(o.m_internal)
      {
         m_sign = o.m_sign;
         this->resize(o.size(), o.size());
         std::memcpy(this->limbs(), o.limbs(), o.size() * sizeof(*(o.limbs())));
         return *this;
      }
      if (!m_internal && !m_alias)
         allocator().deallocate(m_data.ld.data, m_data.ld.capacity);
      *static_cast<base_type*>(this) = static_cast<typename cpp_int_base<MinBits2, MaxBits2, signed_magnitude, Checked2, Allocator>::base_type&&>(o);
      m_limbs                        = o.m_limbs;
      m_sign                         = o.m_sign;
      m_internal                     = o.m_internal;
      m_alias                        = o.m_alias;
      m_data.ld.capacity             = o.m_data.ld.capacity;
      m_data.ld.data                 = o.limbs();
      o.m_limbs                      = 0;
      o.m_internal                   = true;
      return *this;
   }
   BOOST_MP_FORCEINLINE ~cpp_int_base() noexcept
   {
      if (!m_internal && !m_alias)
         allocator().deallocate(limbs(), capacity());
   }
   void assign(const cpp_int_base& o)
   {
      if (this != &o)
      {
         static_cast<base_type&>(*this) = static_cast<const base_type&>(o);
         m_limbs                        = 0;
         resize(o.size(), o.size());
         std::memcpy(limbs(), o.limbs(), o.size() * sizeof(limbs()[0]));
         m_sign = o.m_sign;
      }
   }
   BOOST_MP_FORCEINLINE void negate() noexcept
   {
      m_sign = !m_sign;
      // Check for zero value:
      if (m_sign && (m_limbs == 1))
      {
         if (limbs()[0] == 0)
            m_sign = false;
      }
   }
   BOOST_MP_FORCEINLINE bool isneg() const noexcept
   {
      return m_sign;
   }
   BOOST_MP_FORCEINLINE void do_swap(cpp_int_base& o) noexcept
   {
      std::swap(m_data, o.m_data);
      std::swap(m_sign, o.m_sign);
      std::swap(m_internal, o.m_internal);
      std::swap(m_limbs, o.m_limbs);
      std::swap(m_alias, o.m_alias);
   }

 protected:
   template <class A>
   void check_in_range(const A&) noexcept {}
};

template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
const unsigned cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::limb_bits;
template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
const limb_type cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::max_limb_value;
template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
const limb_type cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::sign_bit_mask;
template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
const unsigned cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::internal_limb_count;

template <unsigned MinBits, unsigned MaxBits, cpp_int_check_type Checked, class Allocator>
struct cpp_int_base<MinBits, MaxBits, unsigned_magnitude, Checked, Allocator, false>
    : private boost::empty_value<typename detail::rebind<limb_type, Allocator>::type>
{
   //
   // There is currently no support for unsigned arbitrary precision arithmetic, largely
   // because it's not clear what subtraction should do:
   //
   static_assert(((sizeof(Allocator) == 0) && !std::is_void<Allocator>::value), "There is curently no support for unsigned arbitrary precision integers.");
};
//
// Fixed precision (i.e. no allocator), signed-magnitude type with limb-usage count:
//
template <unsigned MinBits, cpp_int_check_type Checked>
struct cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>
{
   using limb_pointer = limb_type*        ;
   using const_limb_pointer = const limb_type*  ;
   using checked_type = std::integral_constant<int, Checked>;

   struct scoped_shared_storage
   {
      BOOST_MP_CXX14_CONSTEXPR  scoped_shared_storage(const cpp_int_base&, unsigned) {}
      BOOST_MP_CXX14_CONSTEXPR void deallocate(unsigned) {}
   };

   //
   // Interface invariants:
   //
   static_assert(MinBits > sizeof(double_limb_type) * CHAR_BIT, "Template parameter MinBits is inconsistent with the parameter trivial - did you mistakingly try to override the trivial parameter?");

 public:
   static constexpr unsigned limb_bits = sizeof(limb_type) * CHAR_BIT;
   static constexpr limb_type max_limb_value = ~static_cast<limb_type>(0u);
   static constexpr limb_type sign_bit_mask = static_cast<limb_type>(1u) << (limb_bits - 1);
   static constexpr unsigned  internal_limb_count = MinBits / limb_bits + ((MinBits % limb_bits) ? 1 : 0);
   static constexpr limb_type upper_limb_mask = (MinBits % limb_bits) ? (limb_type(1) << (MinBits % limb_bits)) - 1 : (~limb_type(0));
   static_assert(internal_limb_count >= 2, "A fixed precision integer type must have at least 2 limbs");

 private:
   union data_type
   {
      limb_type        m_data[internal_limb_count];
      limb_type        m_first_limb;
      double_limb_type m_double_first_limb;

      constexpr data_type()
          : m_data{0}
      {}
      constexpr data_type(limb_type i)
          : m_data{i}
      {}
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
      constexpr data_type(limb_type i, limb_type j) : m_data{i, j}
      {}
#endif
      constexpr data_type(double_limb_type i) : m_double_first_limb(i)
      {
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
         if (BOOST_MP_IS_CONST_EVALUATED(m_double_first_limb))
         {
            data_type t(static_cast<limb_type>(i & max_limb_value), static_cast<limb_type>(i >> limb_bits));
            *this = t;
         }
#endif
      }
      template <limb_type... VALUES>
      constexpr data_type(literals::detail::value_pack<VALUES...>) : m_data{VALUES...}
      {}
   } m_wrapper;
   std::uint16_t m_limbs;
   bool            m_sign;

 public:
   //
   // Direct construction:
   //
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(limb_type i) noexcept
       : m_wrapper(i),
         m_limbs(1),
         m_sign(false) {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(signed_limb_type i) noexcept
       : m_wrapper(limb_type(i < 0 ? static_cast<limb_type>(-static_cast<signed_double_limb_type>(i)) : i)),
         m_limbs(1),
         m_sign(i < 0) {}
#if BOOST_MP_ENDIAN_LITTLE_BYTE && !defined(BOOST_MP_TEST_NO_LE)
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(double_limb_type i) noexcept
       : m_wrapper(i),
         m_limbs(i > max_limb_value ? 2 : 1),
         m_sign(false)
   {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(signed_double_limb_type i) noexcept
       : m_wrapper(double_limb_type(i < 0 ? static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) : i)),
         m_limbs(i < 0 ? (static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) > max_limb_value ? 2 : 1) : (i > max_limb_value ? 2 : 1)),
         m_sign(i < 0) {}
#endif
   template <limb_type... VALUES>
   constexpr cpp_int_base(literals::detail::value_pack<VALUES...> i)
       : m_wrapper(i), m_limbs(sizeof...(VALUES)), m_sign(false)
   {}
   constexpr cpp_int_base(literals::detail::value_pack<> i)
       : m_wrapper(i), m_limbs(1), m_sign(false) {}
   constexpr cpp_int_base(const cpp_int_base& a, const literals::detail::negate_tag&)
       : m_wrapper(a.m_wrapper), m_limbs(a.m_limbs), m_sign((a.m_limbs == 1) && (*a.limbs() == 0) ? false : !a.m_sign) {}
   explicit constexpr cpp_int_base(scoped_shared_storage&, unsigned) noexcept : m_wrapper(), m_limbs(0), m_sign(false)
   {}
   //
   // These are deprecated in C++20 unless we make them explicit:
   //
   BOOST_MP_CXX14_CONSTEXPR cpp_int_base& operator=(const cpp_int_base&) = default;
   //
   // Helper functions for getting at our internal data, and manipulating storage:
   //
   BOOST_MP_FORCEINLINE constexpr unsigned              size() const noexcept { return m_limbs; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR limb_pointer limbs() noexcept { return m_wrapper.m_data; }
   BOOST_MP_FORCEINLINE constexpr const_limb_pointer    limbs() const noexcept { return m_wrapper.m_data; }
   BOOST_MP_FORCEINLINE constexpr bool                  sign() const noexcept { return m_sign; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void         sign(bool b) noexcept
   {
      m_sign = b;
      // Check for zero value:
      if (m_sign && (m_limbs == 1))
      {
         if (limbs()[0] == 0)
            m_sign = false;
      }
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void resize(unsigned new_size, unsigned min_size) noexcept((Checked == unchecked))
   {
      m_limbs = static_cast<std::uint16_t>((std::min)(new_size, internal_limb_count));
      detail::verify_new_size(m_limbs, min_size, checked_type());
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void normalize() noexcept((Checked == unchecked))
   {
      limb_pointer p = limbs();
      detail::verify_limb_mask(m_limbs == internal_limb_count, p[m_limbs - 1], upper_limb_mask, checked_type());
      p[internal_limb_count - 1] &= upper_limb_mask;
      while ((m_limbs - 1) && !p[m_limbs - 1])
         --m_limbs;
      if ((m_limbs == 1) && (!*p))
         m_sign = false; // zero is always unsigned
   }

   BOOST_MP_FORCEINLINE constexpr cpp_int_base() noexcept : m_wrapper(limb_type(0u)), m_limbs(1), m_sign(false) {}
   // Not defaulted, it breaks constexpr support in the Intel compiler for some reason:
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(const cpp_int_base& o) noexcept
       : m_wrapper(o.m_wrapper),
         m_limbs(o.m_limbs),
         m_sign(o.m_sign) {}
   // Defaulted functions:
   //~cpp_int_base() noexcept {}

   void BOOST_MP_CXX14_CONSTEXPR assign(const cpp_int_base& o) noexcept
   {
      if (this != &o)
      {
         m_limbs = o.m_limbs;
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
         if (BOOST_MP_IS_CONST_EVALUATED(m_limbs))
         {
            for (unsigned i = 0; i < m_limbs; ++i)
               limbs()[i] = o.limbs()[i];
         }
         else
#endif
            std::memcpy(limbs(), o.limbs(), o.size() * sizeof(o.limbs()[0]));
         m_sign = o.m_sign;
      }
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void negate() noexcept
   {
      m_sign = !m_sign;
      // Check for zero value:
      if (m_sign && (m_limbs == 1))
      {
         if (limbs()[0] == 0)
            m_sign = false;
      }
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR bool isneg() const noexcept
   {
      return m_sign;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_swap(cpp_int_base& o) noexcept
   {
      for (unsigned i = 0; i < (std::max)(size(), o.size()); ++i)
         std_constexpr::swap(m_wrapper.m_data[i], o.m_wrapper.m_data[i]);
      std_constexpr::swap(m_sign, o.m_sign);
      std_constexpr::swap(m_limbs, o.m_limbs);
   }

 protected:
   template <class A>
   BOOST_MP_CXX14_CONSTEXPR void check_in_range(const A&) noexcept {}
};

template <unsigned MinBits, cpp_int_check_type Checked>
const unsigned cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::limb_bits;
template <unsigned MinBits, cpp_int_check_type Checked>
const limb_type cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::max_limb_value;
template <unsigned MinBits, cpp_int_check_type Checked>
const limb_type cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::sign_bit_mask;
template <unsigned MinBits, cpp_int_check_type Checked>
const unsigned cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::internal_limb_count;
//
// Fixed precision (i.e. no allocator), unsigned type with limb-usage count:
//
template <unsigned MinBits, cpp_int_check_type Checked>
struct cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>
{
   using limb_pointer = limb_type*        ;
   using const_limb_pointer = const limb_type*  ;
   using checked_type = std::integral_constant<int, Checked>;

   struct scoped_shared_storage
   {
      BOOST_MP_CXX14_CONSTEXPR scoped_shared_storage(const cpp_int_base&, unsigned) {}
      BOOST_MP_CXX14_CONSTEXPR void deallocate(unsigned) {}
   };
   //
   // Interface invariants:
   //
   static_assert(MinBits > sizeof(double_limb_type) * CHAR_BIT, "Template parameter MinBits is inconsistent with the parameter trivial - did you mistakingly try to override the trivial parameter?");

 public:
   static constexpr unsigned limb_bits = sizeof(limb_type) * CHAR_BIT;
   static constexpr limb_type max_limb_value = ~static_cast<limb_type>(0u);
   static constexpr limb_type sign_bit_mask = static_cast<limb_type>(1u) << (limb_bits - 1);
   static constexpr unsigned internal_limb_count = MinBits / limb_bits + ((MinBits % limb_bits) ? 1 : 0);
   static constexpr limb_type upper_limb_mask = (MinBits % limb_bits) ? (limb_type(1) << (MinBits % limb_bits)) - 1 : (~limb_type(0));
   static_assert(internal_limb_count >= 2, "A fixed precision integer type must have at least 2 limbs");

 private:
   union data_type
   {
      limb_type        m_data[internal_limb_count];
      limb_type        m_first_limb;
      double_limb_type m_double_first_limb;

      constexpr data_type()
          : m_data{0}
      {}
      constexpr data_type(limb_type i)
          : m_data{i}
      {}
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
      constexpr data_type(limb_type i, limb_type j) : m_data{i, j}
      {}
#endif
      constexpr data_type(double_limb_type i) : m_double_first_limb(i)
      {
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
         if (BOOST_MP_IS_CONST_EVALUATED(m_double_first_limb))
         {
            data_type t(static_cast<limb_type>(i & max_limb_value), static_cast<limb_type>(i >> limb_bits));
            *this = t;
         }
#endif
      }
      template <limb_type... VALUES>
      constexpr data_type(literals::detail::value_pack<VALUES...>) : m_data{VALUES...}
      {}
   } m_wrapper;
   limb_type m_limbs;

 public:
   //
   // Direct construction:
   //
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(limb_type i) noexcept
       : m_wrapper(i),
         m_limbs(1) {}
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(signed_limb_type i) noexcept((Checked == unchecked))
       : m_wrapper(limb_type(i < 0 ? static_cast<limb_type>(-static_cast<signed_double_limb_type>(i)) : i)), m_limbs(1)
   {
      if (i < 0)
         negate();
   }
#if BOOST_MP_ENDIAN_LITTLE_BYTE && !defined(BOOST_MP_TEST_NO_LE)
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(double_limb_type i) noexcept
       : m_wrapper(i),
         m_limbs(i > max_limb_value ? 2 : 1)
   {}
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(signed_double_limb_type i) noexcept((Checked == unchecked))
       : m_wrapper(double_limb_type(i < 0 ? static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) : i)),
         m_limbs(i < 0 ? (static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i)) > max_limb_value ? 2 : 1) : (i > max_limb_value ? 2 : 1))
   {
      if (i < 0)
         negate();
   }
#endif
   template <limb_type... VALUES>
   constexpr cpp_int_base(literals::detail::value_pack<VALUES...> i)
       : m_wrapper(i), m_limbs(sizeof...(VALUES))
   {}
   constexpr cpp_int_base(literals::detail::value_pack<>)
       : m_wrapper(static_cast<limb_type>(0u)), m_limbs(1) {}
   explicit constexpr cpp_int_base(scoped_shared_storage&, unsigned) noexcept : m_wrapper(), m_limbs(1)
   {}
       //
   // Helper functions for getting at our internal data, and manipulating storage:
   //
   BOOST_MP_FORCEINLINE constexpr unsigned              size() const noexcept { return m_limbs; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR limb_pointer limbs() noexcept { return m_wrapper.m_data; }
   BOOST_MP_FORCEINLINE constexpr const_limb_pointer    limbs() const noexcept { return m_wrapper.m_data; }
   BOOST_MP_FORCEINLINE constexpr bool                  sign() const noexcept { return false; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void         sign(bool b) noexcept((Checked == unchecked))
   {
      if (b)
         negate();
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void resize(unsigned new_size, unsigned min_size) noexcept((Checked == unchecked))
   {
      m_limbs = (std::min)(new_size, internal_limb_count);
      detail::verify_new_size(m_limbs, min_size, checked_type());
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void normalize() noexcept((Checked == unchecked))
   {
      limb_pointer p = limbs();
      detail::verify_limb_mask(m_limbs == internal_limb_count, p[internal_limb_count - 1], upper_limb_mask, checked_type());
      p[internal_limb_count - 1] &= upper_limb_mask;
      while ((m_limbs - 1) && !p[m_limbs - 1])
         --m_limbs;
   }

   BOOST_MP_FORCEINLINE constexpr cpp_int_base() noexcept
       : m_wrapper(limb_type(0u)),
         m_limbs(1) {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(const cpp_int_base& o) noexcept
       : m_wrapper(o.m_wrapper),
         m_limbs(o.m_limbs) {}
   // Defaulted functions:
   //~cpp_int_base() noexcept {}
   //
   // These are deprecated in C++20 unless we make them explicit:
   //
   BOOST_MP_CXX14_CONSTEXPR cpp_int_base& operator=(const cpp_int_base&) = default;

   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void assign(const cpp_int_base& o) noexcept
   {
      if (this != &o)
      {
         m_limbs = o.m_limbs;
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
         if (BOOST_MP_IS_CONST_EVALUATED(m_limbs))
         {
            for (unsigned i = 0; i < m_limbs; ++i)
               limbs()[i] = o.limbs()[i];
         }
         else
#endif
            std::memcpy(limbs(), o.limbs(), o.size() * sizeof(limbs()[0]));
      }
   }

 private:
   void check_negate(const std::integral_constant<int, checked>&)
   {
      BOOST_THROW_EXCEPTION(std::range_error("Attempt to negate an unsigned number."));
   }
   BOOST_MP_CXX14_CONSTEXPR void check_negate(const std::integral_constant<int, unchecked>&) {}

 public:
   BOOST_MP_CXX14_CONSTEXPR void negate() noexcept((Checked == unchecked))
   {
      // Not so much a negate as a complement - this gets called when subtraction
      // would result in a "negative" number:
      if ((m_limbs == 1) && (m_wrapper.m_data[0] == 0))
         return; // negating zero is always zero, and always OK.
      check_negate(checked_type());
      unsigned i = m_limbs;
      for (; i < internal_limb_count; ++i)
         m_wrapper.m_data[i] = 0;
      m_limbs = internal_limb_count;
      for (i = 0; i < internal_limb_count; ++i)
         m_wrapper.m_data[i] = ~m_wrapper.m_data[i];
      normalize();
      eval_increment(static_cast<cpp_int_backend<MinBits, MinBits, unsigned_magnitude, Checked, void>&>(*this));
   }
   BOOST_MP_FORCEINLINE constexpr bool isneg() const noexcept
   {
      return false;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_swap(cpp_int_base& o) noexcept
   {
      for (unsigned i = 0; i < (std::max)(size(), o.size()); ++i)
         std_constexpr::swap(m_wrapper.m_data[i], o.m_wrapper.m_data[i]);
      std_constexpr::swap(m_limbs, o.m_limbs);
   }

 protected:
   template <class A>
   BOOST_MP_CXX14_CONSTEXPR void check_in_range(const A&) noexcept {}
};

template <unsigned MinBits, cpp_int_check_type Checked>
const unsigned cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::limb_bits;
template <unsigned MinBits, cpp_int_check_type Checked>
const limb_type cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::max_limb_value;
template <unsigned MinBits, cpp_int_check_type Checked>
const limb_type cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::sign_bit_mask;
template <unsigned MinBits, cpp_int_check_type Checked>
const unsigned cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::internal_limb_count;
//
// Traits classes to figure out a native type with N bits, these vary from boost::uint_t<N> only
// because some platforms have native integer types longer than boost::long_long_type, "really boost::long_long_type" anyone??
//
template <unsigned N, bool s>
struct trivial_limb_type_imp
{
   using type = double_limb_type;
};

template <unsigned N>
struct trivial_limb_type_imp<N, true>
{
   using type = typename boost::uint_t<N>::least;
};

template <unsigned N>
struct trivial_limb_type : public trivial_limb_type_imp<N, N <= sizeof(boost::long_long_type) * CHAR_BIT>
{};
//
// Backend for fixed precision signed-magnitude type which will fit entirely inside a "double_limb_type":
//
template <unsigned MinBits, cpp_int_check_type Checked>
struct cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, true>
{
   using local_limb_type = typename trivial_limb_type<MinBits>::type;
   using limb_pointer = local_limb_type*                         ;
   using const_limb_pointer = const local_limb_type*                   ;
   using checked_type = std::integral_constant<int, Checked>                       ;

   struct scoped_shared_storage
   {
      BOOST_MP_CXX14_CONSTEXPR scoped_shared_storage(const cpp_int_base&, unsigned) {}
      BOOST_MP_CXX14_CONSTEXPR void deallocate(unsigned) {}
   };

 protected:
   static constexpr unsigned limb_bits = sizeof(local_limb_type) * CHAR_BIT;
   static constexpr local_limb_type limb_mask = (MinBits < limb_bits) ? local_limb_type((local_limb_type(~local_limb_type(0))) >> (limb_bits - MinBits)) : local_limb_type(~local_limb_type(0));

 private:
   local_limb_type m_data;
   bool            m_sign;

   //
   // Interface invariants:
   //
   static_assert(MinBits <= sizeof(double_limb_type) * CHAR_BIT, "Template parameter MinBits is inconsistent with the parameter trivial - did you mistakingly try to override the trivial parameter?");

 protected:
   template <class T>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<!(!boost::multiprecision::detail::is_integral<T>::value || (std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::digits <= (int)MinBits)))>::type
   check_in_range(T val, const std::integral_constant<int, checked>&)
   {
      using common_type = typename std::common_type<typename boost::multiprecision::detail::make_unsigned<T>::type, local_limb_type>::type;

      if (static_cast<common_type>(boost::multiprecision::detail::unsigned_abs(val)) > static_cast<common_type>(limb_mask))
         BOOST_THROW_EXCEPTION(std::range_error("The argument to a cpp_int constructor exceeded the largest value it can represent."));
   }
   template <class T>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<!(boost::multiprecision::detail::is_integral<T>::value || (std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::digits <= (int)MinBits)))>::type
   check_in_range(T val, const std::integral_constant<int, checked>&)
   {
      using std::abs;
      using common_type = typename std::common_type<T, local_limb_type>::type;

      if (static_cast<common_type>(abs(val)) > static_cast<common_type>(limb_mask))
         BOOST_THROW_EXCEPTION(std::range_error("The argument to a cpp_int constructor exceeded the largest value it can represent."));
   }
   template <class T, int C>
   BOOST_MP_CXX14_CONSTEXPR void check_in_range(T, const std::integral_constant<int, C>&) noexcept {}

   template <class T>
   BOOST_MP_CXX14_CONSTEXPR void check_in_range(T val) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<T>(), checked_type())))
   {
      check_in_range(val, checked_type());
   }

 public:
   //
   // Direct construction:
   //
   template <class SI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == unchecked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<SI>())))
       : m_data(i < 0 ? static_cast<local_limb_type>(static_cast<typename boost::multiprecision::detail::make_unsigned<SI>::type>(boost::multiprecision::detail::unsigned_abs(i)) & limb_mask) : static_cast<local_limb_type>(i & limb_mask)), m_sign(i < 0) {}
   template <class SI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<SI>())))
       : m_data(i < 0 ? (static_cast<local_limb_type>(static_cast<typename boost::multiprecision::detail::make_unsigned<SI>::type>(boost::multiprecision::detail::unsigned_abs(i)) & limb_mask)) : static_cast<local_limb_type>(i & limb_mask)), m_sign(i < 0)
   {
      check_in_range(i);
   }
   template <class UI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(static_cast<local_limb_type>(i) & limb_mask),
         m_sign(false) {}
   template <class UI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<UI>())))
       : m_data(static_cast<local_limb_type>(i) & limb_mask), m_sign(false) { check_in_range(i); }
#if !(defined(__clang__) && defined(__MINGW32__))
   template <class F>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(F i, typename std::enable_if<std::is_floating_point<F>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(static_cast<local_limb_type>(std::fabs(i)) & limb_mask),
         m_sign(i < 0) {}
   template <class F>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(F i, typename std::enable_if<std::is_floating_point<F>::value && (Checked == checked)>::type const* = 0)
       : m_data(static_cast<local_limb_type>(std::fabs(i)) & limb_mask), m_sign(i < 0) { check_in_range(i); }
#else
   //
   // conversion from float to __int128 is broken on clang/mingw,
   // see: https://bugs.llvm.org/show_bug.cgi?id=48940
   // Since no floating point type has more than 64 bits of
   // precision, we can simply cast to an intermediate type to
   // solve the issue:
   //
   template <class F>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(F i, typename std::enable_if<std::is_floating_point<F>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(static_cast<local_limb_type>(static_cast<std::uint64_t>(std::fabs(i))) & limb_mask),
         m_sign(i < 0) {}
   template <class F>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(F i, typename std::enable_if<std::is_floating_point<F>::value && (Checked == checked)>::type const* = 0)
       : m_data(static_cast<local_limb_type>(static_cast<std::uint64_t>(std::fabs(i))) & limb_mask), m_sign(i < 0) { check_in_range(i); }
#endif

   constexpr cpp_int_base(literals::detail::value_pack<>) noexcept
       : m_data(static_cast<local_limb_type>(0u)),
         m_sign(false)
   {}
   template <limb_type a>
   constexpr cpp_int_base(literals::detail::value_pack<a>) noexcept
       : m_data(static_cast<local_limb_type>(a)),
         m_sign(false) {}
   template <limb_type a, limb_type b>
   constexpr cpp_int_base(literals::detail::value_pack<a, b>) noexcept
       : m_data(static_cast<local_limb_type>(a) | (static_cast<local_limb_type>(b) << bits_per_limb)),
         m_sign(false) {}
   constexpr cpp_int_base(const cpp_int_base& a, const literals::detail::negate_tag&) noexcept
       : m_data(a.m_data),
         m_sign(a.m_data ? !a.m_sign : false) {}
   //
   // These are deprecated in C++20 unless we make them explicit:
   //
   BOOST_MP_CXX14_CONSTEXPR cpp_int_base& operator=(const cpp_int_base&) = default;

   explicit constexpr cpp_int_base(scoped_shared_storage&, unsigned) noexcept : m_data(0), m_sign(false)
   {}
       //
   // Helper functions for getting at our internal data, and manipulating storage:
   //
   BOOST_MP_FORCEINLINE constexpr unsigned              size() const noexcept { return 1; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR limb_pointer limbs() noexcept { return &m_data; }
   BOOST_MP_FORCEINLINE constexpr const_limb_pointer    limbs() const noexcept { return &m_data; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR bool         sign() const noexcept { return m_sign; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void         sign(bool b) noexcept
   {
      m_sign = b;
      // Check for zero value:
      if (m_sign && !m_data)
      {
         m_sign = false;
      }
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void resize(unsigned /* new_size */, unsigned min_size)
   {
      detail::verify_new_size(2, min_size, checked_type());
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void normalize() noexcept((Checked == unchecked))
   {
      if (!m_data)
         m_sign = false; // zero is always unsigned
      detail::verify_limb_mask(true, m_data, limb_mask, checked_type());
      m_data &= limb_mask;
   }

   BOOST_MP_FORCEINLINE constexpr cpp_int_base() noexcept : m_data(0), m_sign(false) {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(const cpp_int_base& o) noexcept
       : m_data(o.m_data),
         m_sign(o.m_sign) {}
   //~cpp_int_base() noexcept {}
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void assign(const cpp_int_base& o) noexcept
   {
      m_data = o.m_data;
      m_sign = o.m_sign;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void negate() noexcept
   {
      m_sign = !m_sign;
      // Check for zero value:
      if (m_data == 0)
      {
         m_sign = false;
      }
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR bool isneg() const noexcept
   {
      return m_sign;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_swap(cpp_int_base& o) noexcept
   {
      std_constexpr::swap(m_sign, o.m_sign);
      std_constexpr::swap(m_data, o.m_data);
   }
};
//
// Backend for unsigned fixed precision (i.e. no allocator) type which will fit entirely inside a "double_limb_type":
//
template <unsigned MinBits, cpp_int_check_type Checked>
struct cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, true>
{
   using local_limb_type = typename trivial_limb_type<MinBits>::type;
   using limb_pointer = local_limb_type*                         ;
   using const_limb_pointer = const local_limb_type*                   ;

   struct scoped_shared_storage
   {
      BOOST_MP_CXX14_CONSTEXPR scoped_shared_storage(const cpp_int_base&, unsigned) {}
      BOOST_MP_CXX14_CONSTEXPR void deallocate(unsigned) {}
   };

 private:
   static constexpr unsigned limb_bits = sizeof(local_limb_type) * CHAR_BIT;
   static constexpr local_limb_type limb_mask = limb_bits != MinBits ? static_cast<local_limb_type>(static_cast<local_limb_type>(~local_limb_type(0)) >> (limb_bits - MinBits))
                                                                           : static_cast<local_limb_type>(~local_limb_type(0));

   local_limb_type m_data;

   using checked_type = std::integral_constant<int, Checked>;

   //
   // Interface invariants:
   //
   static_assert(MinBits <= sizeof(double_limb_type) * CHAR_BIT, "Template parameter MinBits is inconsistent with the parameter trivial - did you mistakingly try to override the trivial parameter?");

 protected:
   template <class T>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if< !(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::digits <= (int)MinBits))>::type
   check_in_range(T val, const std::integral_constant<int, checked>&, const std::integral_constant<bool, false>&)
   {
      using common_type = typename std::common_type<T, local_limb_type>::type;

      if (static_cast<common_type>(val) > limb_mask)
         BOOST_THROW_EXCEPTION(std::range_error("The argument to a cpp_int constructor exceeded the largest value it can represent."));
   }
   template <class T>
   BOOST_MP_CXX14_CONSTEXPR void check_in_range(T val, const std::integral_constant<int, checked>&, const std::integral_constant<bool, true>&)
   {
      using common_type = typename std::common_type<T, local_limb_type>::type;

      if (static_cast<common_type>(val) > static_cast<common_type>(limb_mask))
         BOOST_THROW_EXCEPTION(std::range_error("The argument to a cpp_int constructor exceeded the largest value it can represent."));
      if (val < 0)
         BOOST_THROW_EXCEPTION(std::range_error("The argument to an unsigned cpp_int constructor was negative."));
   }
   template <class T, int C, bool B>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void check_in_range(T, const std::integral_constant<int, C>&, const std::integral_constant<bool, B>&) noexcept {}

   template <class T>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void check_in_range(T val) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<T>(), checked_type(), boost::multiprecision::detail::is_signed<T>())))
   {
      check_in_range(val, checked_type(), boost::multiprecision::detail::is_signed<T>());
   }

 public:
   //
   // Direct construction:
   //
#ifdef __MSVC_RUNTIME_CHECKS
   template <class SI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(i < 0 ? (1 + ~static_cast<local_limb_type>(-i & limb_mask)) & limb_mask : static_cast<local_limb_type>(i & limb_mask))
   {}
   template <class SI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<SI>())))
       : m_data(i < 0 ? 1 + ~static_cast<local_limb_type>(-i & limb_mask) : static_cast<local_limb_type>(i & limb_mask)) { check_in_range(i); }
   template <class UI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(static_cast<local_limb_type>(i& limb_mask)) {}
   template <class UI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<UI>())))
       : m_data(static_cast<local_limb_type>(i & limb_mask)) { check_in_range(i); }
#else
   template <class SI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(i < 0 ? (1 + ~static_cast<local_limb_type>(-i)) & limb_mask : static_cast<local_limb_type>(i) & limb_mask)
   {}
   template <class SI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(SI i, typename std::enable_if<boost::multiprecision::detail::is_signed<SI>::value && boost::multiprecision::detail::is_integral<SI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<SI>())))
       : m_data(i < 0 ? 1 + ~static_cast<local_limb_type>(-i) : static_cast<local_limb_type>(i)) { check_in_range(i); }
   template <class UI>
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == unchecked)>::type const* = 0) noexcept
       : m_data(static_cast<local_limb_type>(i) & limb_mask) {}
   template <class UI>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(UI i, typename std::enable_if<boost::multiprecision::detail::is_unsigned<UI>::value && (Checked == checked)>::type const* = 0) noexcept(noexcept(std::declval<cpp_int_base>().check_in_range(std::declval<UI>())))
       : m_data(static_cast<local_limb_type>(i)) { check_in_range(i); }
#endif
   template <class F>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_base(F i, typename std::enable_if<std::is_floating_point<F>::value >::type const* = 0) noexcept((Checked == unchecked))
       : m_data(static_cast<local_limb_type>(std::fabs(i)) & limb_mask)
   {
      check_in_range(i);
      if (i < 0)
         negate();
   }
   constexpr cpp_int_base(literals::detail::value_pack<>) noexcept
       : m_data(static_cast<local_limb_type>(0u))
   {}
   template <limb_type a>
   constexpr cpp_int_base(literals::detail::value_pack<a>) noexcept
       : m_data(static_cast<local_limb_type>(a)) {}
   template <limb_type a, limb_type b>
   constexpr cpp_int_base(literals::detail::value_pack<a, b>) noexcept
       : m_data(static_cast<local_limb_type>(a) | (static_cast<local_limb_type>(b) << bits_per_limb)) {}
   //
   // These are deprecated in C++20 unless we make them explicit:
   //
   BOOST_MP_CXX14_CONSTEXPR cpp_int_base& operator=(const cpp_int_base&) = default;

   explicit constexpr cpp_int_base(scoped_shared_storage&, unsigned) noexcept : m_data(0)
   {}
       //
   // Helper functions for getting at our internal data, and manipulating storage:
   //
   BOOST_MP_FORCEINLINE constexpr unsigned              size() const noexcept { return 1; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR limb_pointer limbs() noexcept { return &m_data; }
   BOOST_MP_FORCEINLINE constexpr const_limb_pointer    limbs() const noexcept { return &m_data; }
   BOOST_MP_FORCEINLINE constexpr bool                  sign() const noexcept { return false; }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void         sign(bool b) noexcept((Checked == unchecked))
   {
      if (b)
         negate();
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void resize(unsigned, unsigned min_size)
   {
      detail::verify_new_size(2, min_size, checked_type());
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void normalize() noexcept((Checked == unchecked))
   {
      detail::verify_limb_mask(true, m_data, limb_mask, checked_type());
      m_data &= limb_mask;
   }

   BOOST_MP_FORCEINLINE constexpr cpp_int_base() noexcept : m_data(0) {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_base(const cpp_int_base& o) noexcept
       : m_data(o.m_data) {}
   //~cpp_int_base() noexcept {}
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void assign(const cpp_int_base& o) noexcept
   {
      m_data = o.m_data;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void negate() noexcept((Checked == unchecked))
   {
      BOOST_IF_CONSTEXPR(Checked == checked)
      {
         BOOST_THROW_EXCEPTION(std::range_error("Attempt to negate an unsigned type."));
      }
      m_data = ~m_data;
      ++m_data;
   }
   BOOST_MP_FORCEINLINE constexpr bool isneg() const noexcept
   {
      return false;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_swap(cpp_int_base& o) noexcept
   {
      std_constexpr::swap(m_data, o.m_data);
   }
};
//
// Traits class, lets us know whether type T can be directly converted to the base type,
// used to enable/disable constructors etc:
//
template <class Arg, class Base>
struct is_allowed_cpp_int_base_conversion : public std::conditional<
                                                std::is_same<Arg, limb_type>::value || std::is_same<Arg, signed_limb_type>::value
#if BOOST_MP_ENDIAN_LITTLE_BYTE && !defined(BOOST_MP_TEST_NO_LE)
                                                    || std::is_same<Arg, double_limb_type>::value || std::is_same<Arg, signed_double_limb_type>::value
#endif
                                                    || literals::detail::is_value_pack<Arg>::value || (is_trivial_cpp_int<Base>::value && boost::multiprecision::detail::is_arithmetic<Arg>::value),
                                                std::integral_constant<bool, true>,
                                                std::integral_constant<bool, false>>::type
{};
//
// Now the actual backend, normalising parameters passed to the base class:
//
template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct cpp_int_backend
    : public cpp_int_base<
          min_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value,
          max_precision<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value,
          SignType,
          Checked,
          Allocator,
          is_trivial_cpp_int<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value>
{
   using self_type = cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>;
   using base_type = cpp_int_base<
       min_precision<self_type>::value,
       max_precision<self_type>::value,
       SignType,
       Checked,
       Allocator,
       is_trivial_cpp_int<self_type>::value>;
   using trivial_tag = std::integral_constant<bool, is_trivial_cpp_int<self_type>::value>;

 public:
   using signed_types = typename std::conditional<
       is_trivial_cpp_int<self_type>::value,
       std::tuple<
           signed char, short, int, long,
           boost::long_long_type, signed_double_limb_type>,
       std::tuple<signed_limb_type, signed_double_limb_type> >::type;
   using unsigned_types = typename std::conditional<
       is_trivial_cpp_int<self_type>::value,
       std::tuple<unsigned char, unsigned short, unsigned,
                 unsigned long, boost::ulong_long_type, double_limb_type>,
       std::tuple<limb_type, double_limb_type> >::type;
   using float_types = typename std::conditional<
       is_trivial_cpp_int<self_type>::value,
       std::tuple<float, double, long double>,
       std::tuple<long double> >::type;
   using checked_type = std::integral_constant<int, Checked>        ;

   BOOST_MP_FORCEINLINE constexpr cpp_int_backend() noexcept {}
   BOOST_MP_FORCEINLINE constexpr cpp_int_backend(const cpp_int_backend& o) noexcept(std::is_void<Allocator>::value) : base_type(o) {}
   // rvalue copy:
   BOOST_MP_FORCEINLINE constexpr cpp_int_backend(cpp_int_backend&& o) noexcept
       : base_type(static_cast<base_type&&>(o))
   {}
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2>
   BOOST_MP_FORCEINLINE BOOST_CXX14_CONSTEXPR cpp_int_backend(cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2>&& o, typename std::enable_if<is_implicit_cpp_int_conversion<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2>, self_type>::value>::type* = 0) noexcept
   {
      *this = static_cast<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2>&&>(o);
   }
   //
   // Direct construction from arithmetic type:
   //
   template <class Arg>
   BOOST_MP_FORCEINLINE constexpr cpp_int_backend(Arg i, typename std::enable_if<is_allowed_cpp_int_base_conversion<Arg, base_type>::value>::type const* = 0) noexcept(noexcept(base_type(std::declval<Arg>())))
       : base_type(i) {}
   //
   // Aliasing constructor: the result will alias the memory referenced, unless
   // we have fixed precision and storage, in which case we copy the memory:
   //
   explicit constexpr cpp_int_backend(limb_type* data, unsigned offset, unsigned len) noexcept
       : base_type(data, offset, len) {}
   explicit cpp_int_backend(const limb_type* data, unsigned offset, unsigned len) noexcept
       : base_type(data, offset, len) { this->normalize(); }
   explicit constexpr cpp_int_backend(typename base_type::scoped_shared_storage& data, unsigned len) noexcept
       : base_type(data, len) {}

 private:
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR void do_assign(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other, std::integral_constant<bool, true> const&, std::integral_constant<bool, true> const&)
   {
      // Assigning trivial type to trivial type:
      this->check_in_range(*other.limbs());
      *this->limbs() = static_cast<typename self_type::local_limb_type>(*other.limbs());
      this->sign(other.sign());
      this->normalize();
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR void do_assign(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other, std::integral_constant<bool, true> const&, std::integral_constant<bool, false> const&)
   {
      // non-trivial to trivial narrowing conversion:
      double_limb_type v = *other.limbs();
      if (other.size() > 1)
      {
         v |= static_cast<double_limb_type>(other.limbs()[1]) << bits_per_limb;
         BOOST_IF_CONSTEXPR(Checked == checked)
         {
            if (other.size() > 2)
            {
               BOOST_THROW_EXCEPTION(std::range_error("Assignment of a cpp_int that is out of range for the target type."));
            }
         }
      }
      *this = v;
      this->sign(other.sign());
      this->normalize();
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR void do_assign(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other, std::integral_constant<bool, false> const&, std::integral_constant<bool, true> const&)
   {
      // trivial to non-trivial, treat the trivial argument as if it were an unsigned arithmetic type, then set the sign afterwards:
      *this = static_cast<
          typename boost::multiprecision::detail::canonical<
              typename cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>::local_limb_type,
              cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::type>(*other.limbs());
      this->sign(other.sign());
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR void do_assign(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other, std::integral_constant<bool, false> const&, std::integral_constant<bool, false> const&)
   {
      // regular non-trivial to non-trivial assign:
      this->resize(other.size(), other.size());

#if !defined(BOOST_MP_HAS_IS_CONSTANT_EVALUATED) && !defined(BOOST_MP_HAS_BUILTIN_IS_CONSTANT_EVALUATED) && !defined(BOOST_NO_CXX14_CONSTEXPR)
      unsigned count = (std::min)(other.size(), this->size());
      for (unsigned i = 0; i < count; ++i)
         this->limbs()[i] = other.limbs()[i];
#else
#ifndef BOOST_MP_NO_CONSTEXPR_DETECTION
      if (BOOST_MP_IS_CONST_EVALUATED(other.size()))
      {
         unsigned count = (std::min)(other.size(), this->size());
         for (unsigned i = 0; i < count; ++i)
            this->limbs()[i] = other.limbs()[i];
      }
      else
#endif
         std::memcpy(this->limbs(), other.limbs(), (std::min)(other.size(), this->size()) * sizeof(this->limbs()[0]));
#endif
      this->sign(other.sign());
      this->normalize();
   }

 public:
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR cpp_int_backend(
       const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other,
       typename std::enable_if<is_implicit_cpp_int_conversion<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>, self_type>::value>::type* = 0)
       : base_type()
   {
      do_assign(
          other,
          std::integral_constant<bool, is_trivial_cpp_int<self_type>::value>(),
          std::integral_constant<bool, is_trivial_cpp_int<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >::value>());
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   explicit BOOST_MP_CXX14_CONSTEXPR cpp_int_backend(
       const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other,
       typename std::enable_if< !(is_implicit_cpp_int_conversion<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>, self_type>::value)>::type* = 0)
       : base_type()
   {
      do_assign(
          other,
          std::integral_constant<bool, is_trivial_cpp_int<self_type>::value>(),
          std::integral_constant<bool, is_trivial_cpp_int<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >::value>());
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR cpp_int_backend& operator=(
       const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& other)
   {
      do_assign(
          other,
          std::integral_constant<bool, is_trivial_cpp_int<self_type>::value>(),
          std::integral_constant<bool, is_trivial_cpp_int<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >::value>());
      return *this;
   }
   constexpr cpp_int_backend(const cpp_int_backend& a, const literals::detail::negate_tag& tag)
       : base_type(static_cast<const base_type&>(a), tag)
   {}

   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_backend& operator=(const cpp_int_backend& o) noexcept(noexcept(std::declval<cpp_int_backend>().assign(std::declval<const cpp_int_backend&>())))
   {
      this->assign(o);
      return *this;
   }
   // rvalue copy:
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR cpp_int_backend& operator=(cpp_int_backend&& o) noexcept(noexcept(std::declval<base_type&>() = std::declval<base_type>()))
   {
      *static_cast<base_type*>(this) = static_cast<base_type&&>(o);
      return *this;
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<((MaxBits2 <= MaxBits) || (MaxBits == 0)) && !std::is_void<Allocator>::value, cpp_int_backend&>::type operator=(cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator>&& o) noexcept
   {
      *static_cast<base_type*>(this) = static_cast<typename cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2>::base_type&&>(o);
      return *this;
   }
 private:
   template <class A>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<boost::multiprecision::detail::is_unsigned<A>::value>::type do_assign_arithmetic(A val, const std::integral_constant<bool, true>&)
       noexcept(noexcept(std::declval<cpp_int_backend>().check_in_range(std::declval<A>())))
   {
      this->check_in_range(val);
      *this->limbs() = static_cast<typename self_type::local_limb_type>(val);
      this->sign(false);
      this->normalize();
   }
   template <class A>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<!(boost::multiprecision::detail::is_unsigned<A>::value || !boost::multiprecision::detail::is_integral<A>::value)>::type do_assign_arithmetic(A val, const std::integral_constant<bool, true>&)
       noexcept(noexcept(std::declval<cpp_int_backend>().check_in_range(std::declval<A>())) && noexcept(std::declval<cpp_int_backend>().sign(true)))
   {
      this->check_in_range(val);
      *this->limbs() = (val < 0) ? static_cast<typename self_type::local_limb_type>(boost::multiprecision::detail::unsigned_abs(val)) : static_cast<typename self_type::local_limb_type>(val);
      this->sign(val < 0);
      this->normalize();
   }
   template <class A>
   BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<!boost::multiprecision::detail::is_integral<A>::value>::type do_assign_arithmetic(A val, const std::integral_constant<bool, true>&)
   {
      this->check_in_range(val);
      *this->limbs() = (val < 0) ? static_cast<typename self_type::local_limb_type>(boost::multiprecision::detail::abs(val)) : static_cast<typename self_type::local_limb_type>(val);
      this->sign(val < 0);
      this->normalize();
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_assign_arithmetic(limb_type i, const std::integral_constant<bool, false>&) noexcept
   {
      this->resize(1, 1);
      *this->limbs() = i;
      this->sign(false);
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void do_assign_arithmetic(signed_limb_type i, const std::integral_constant<bool, false>&) noexcept(noexcept(std::declval<cpp_int_backend>().sign(true)))
   {
      this->resize(1, 1);
      *this->limbs() = static_cast<limb_type>(boost::multiprecision::detail::unsigned_abs(i));
      this->sign(i < 0);
   }
   BOOST_MP_CXX14_CONSTEXPR void do_assign_arithmetic(double_limb_type i, const std::integral_constant<bool, false>&) noexcept
   {
      static_assert(sizeof(i) == 2 * sizeof(limb_type), "Failed integer size check");
      static_assert(base_type::internal_limb_count >= 2, "Failed internal limb count");
      typename base_type::limb_pointer p = this->limbs();
#ifdef __MSVC_RUNTIME_CHECKS
      *p = static_cast<limb_type>(i & ~static_cast<limb_type>(0));
#else
      *p                        = static_cast<limb_type>(i);
#endif
      p[1] = static_cast<limb_type>(i >> base_type::limb_bits);
      this->resize(p[1] ? 2 : 1, p[1] ? 2 : 1);
      this->sign(false);
   }
   BOOST_MP_CXX14_CONSTEXPR void do_assign_arithmetic(signed_double_limb_type i, const std::integral_constant<bool, false>&) noexcept(noexcept(std::declval<cpp_int_backend>().sign(true)))
   {
      static_assert(sizeof(i) == 2 * sizeof(limb_type), "double limb type size check failed");
      static_assert(base_type::internal_limb_count >= 2, "Failed internal limb count check");
      bool s = false;
      if (i < 0)
         s = true;
      double_limb_type                 ui = static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i));
      typename base_type::limb_pointer p  = this->limbs();
#ifdef __MSVC_RUNTIME_CHECKS
      *p = static_cast<limb_type>(ui & ~static_cast<limb_type>(0));
#else
      *p                        = static_cast<limb_type>(ui);
#endif
      p[1] = static_cast<limb_type>(ui >> base_type::limb_bits);
      this->resize(p[1] ? 2 : 1, p[1] ? 2 : 1);
      this->sign(s);
   }

   BOOST_MP_CXX14_CONSTEXPR void do_assign_arithmetic(long double a, const std::integral_constant<bool, false>&)
   {
      using default_ops::eval_add;
      using default_ops::eval_subtract;
      using std::floor;
      using std::frexp;
      using std::ldexp;

      if (a < 0)
      {
         do_assign_arithmetic(-a, std::integral_constant<bool, false>());
         this->sign(true);
         return;
      }

      if (a == 0)
      {
         *this = static_cast<limb_type>(0u);
      }

      if (a == 1)
      {
         *this = static_cast<limb_type>(1u);
      }

      if ((boost::math::isinf)(a) || (boost::math::isnan)(a))
      {
         BOOST_THROW_EXCEPTION(std::runtime_error("Cannot convert a non-finite number to an integer."));
      }

      int         e = 0;
      long double f(0), term(0);
      *this = static_cast<limb_type>(0u);

      f = frexp(a, &e);

#if !(defined(__clang__) && (__clang_major__ <= 7))
      constexpr limb_type shift = std::numeric_limits<limb_type>::digits;
#else
      // clang 7 has an issue converting long double to unsigned long long in
      // release mode (bits get dropped, conversion appears to go via float)
      // Never extract more than double bits at a time:
      constexpr limb_type shift = std::numeric_limits<limb_type>::digits > std::numeric_limits<double>::digits
            ? std::numeric_limits<double>::digits : std::numeric_limits<limb_type>::digits;
#endif

      while (f)
      {
         // extract int sized bits from f:
         f    = ldexp(f, shift);
         term = floor(f);
         e -= shift;
         eval_left_shift(*this, shift);
#if !(defined(__clang__) && (__clang_major__ <= 7))
         if (term > 0)
            eval_add(*this, static_cast<limb_type>(term));
         else
            eval_subtract(*this, static_cast<limb_type>(-term));
#else
         // clang 7 requires extra cast to double to avoid buggy code generation:
         if (term > 0)
            eval_add(*this, static_cast<limb_type>(static_cast<double>(term)));
         else
            eval_subtract(*this, static_cast<limb_type>(static_cast<double>(-term)));
#endif
         f -= term;
      }
      if (e > 0)
         eval_left_shift(*this, e);
      else if (e < 0)
         eval_right_shift(*this, -e);
   }

 public:
   template <class Arithmetic>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<!boost::multiprecision::detail::is_byte_container<Arithmetic>::value, cpp_int_backend&>::type operator=(Arithmetic val) noexcept(noexcept(std::declval<cpp_int_backend>().do_assign_arithmetic(std::declval<Arithmetic>(), trivial_tag())))
   {
      do_assign_arithmetic(val, trivial_tag());
      return *this;
   }

 private:
   void do_assign_string(const char* s, const std::integral_constant<bool, true>&)
   {
      std::size_t n  = s ? std::strlen(s) : 0;
      *this          = 0;
      unsigned radix = 10;
      bool     isneg = false;
      if (n && (*s == '-'))
      {
         --n;
         ++s;
         isneg = true;
      }
      if (n && (*s == '0'))
      {
         if ((n > 1) && ((s[1] == 'x') || (s[1] == 'X')))
         {
            radix = 16;
            s += 2;
            n -= 2;
         }
         else
         {
            radix = 8;
            n -= 1;
         }
      }
      if (n)
      {
         unsigned val;
         while (*s)
         {
            if (*s >= '0' && *s <= '9')
               val = *s - '0';
            else if (*s >= 'a' && *s <= 'f')
               val = 10 + *s - 'a';
            else if (*s >= 'A' && *s <= 'F')
               val = 10 + *s - 'A';
            else
               val = radix + 1;
            if (val >= radix)
            {
               BOOST_THROW_EXCEPTION(std::runtime_error("Unexpected content found while parsing character string."));
            }
            *this->limbs() = detail::checked_multiply(*this->limbs(), static_cast<typename base_type::local_limb_type>(radix), checked_type());
            *this->limbs() = detail::checked_add(*this->limbs(), static_cast<typename base_type::local_limb_type>(val), checked_type());
            ++s;
         }
      }
      if (isneg)
         this->negate();
   }
   void do_assign_string(const char* s, const std::integral_constant<bool, false>&)
   {
      using default_ops::eval_add;
      using default_ops::eval_multiply;
      std::size_t n  = s ? std::strlen(s) : 0;
      *this          = static_cast<limb_type>(0u);
      unsigned radix = 10;
      bool     isneg = false;
      if (n && (*s == '-'))
      {
         --n;
         ++s;
         isneg = true;
      }
      if (n && (*s == '0'))
      {
         if ((n > 1) && ((s[1] == 'x') || (s[1] == 'X')))
         {
            radix = 16;
            s += 2;
            n -= 2;
         }
         else
         {
            radix = 8;
            n -= 1;
         }
      }
      //
      // Exception guarantee: create the result in stack variable "result"
      // then do a swap at the end.  In the event of a throw, *this will
      // be left unchanged.
      //
      cpp_int_backend result;
      if (n)
      {
         if (radix == 16)
         {
            while (*s == '0')
               ++s;
            std::size_t bitcount = 4 * std::strlen(s);
            limb_type   val;
            std::size_t limb, shift;
            if (bitcount > 4)
               bitcount -= 4;
            else
               bitcount = 0;
            std::size_t newsize = bitcount / (sizeof(limb_type) * CHAR_BIT) + 1;
            result.resize(static_cast<unsigned>(newsize), static_cast<unsigned>(newsize)); // will throw if this is a checked integer that cannot be resized
            std::memset(result.limbs(), 0, result.size() * sizeof(limb_type));
            while (*s)
            {
               if (*s >= '0' && *s <= '9')
                  val = *s - '0';
               else if (*s >= 'a' && *s <= 'f')
                  val = 10 + *s - 'a';
               else if (*s >= 'A' && *s <= 'F')
                  val = 10 + *s - 'A';
               else
               {
                  BOOST_THROW_EXCEPTION(std::runtime_error("Unexpected content found while parsing character string."));
               }
               limb  = bitcount / (sizeof(limb_type) * CHAR_BIT);
               shift = bitcount % (sizeof(limb_type) * CHAR_BIT);
               val <<= shift;
               if (result.size() > limb)
               {
                  result.limbs()[limb] |= val;
               }
               ++s;
               bitcount -= 4;
            }
            result.normalize();
         }
         else if (radix == 8)
         {
            while (*s == '0')
               ++s;
            std::size_t bitcount = 3 * std::strlen(s);
            limb_type   val;
            std::size_t limb, shift;
            if (bitcount > 3)
               bitcount -= 3;
            else
               bitcount = 0;
            std::size_t newsize = bitcount / (sizeof(limb_type) * CHAR_BIT) + 1;
            result.resize(static_cast<unsigned>(newsize), static_cast<unsigned>(newsize)); // will throw if this is a checked integer that cannot be resized
            std::memset(result.limbs(), 0, result.size() * sizeof(limb_type));
            while (*s)
            {
               if (*s >= '0' && *s <= '7')
                  val = *s - '0';
               else
               {
                  BOOST_THROW_EXCEPTION(std::runtime_error("Unexpected content found while parsing character string."));
               }
               limb  = bitcount / (sizeof(limb_type) * CHAR_BIT);
               shift = bitcount % (sizeof(limb_type) * CHAR_BIT);
               if (result.size() > limb)
               {
                  result.limbs()[limb] |= (val << shift);
                  if (shift > sizeof(limb_type) * CHAR_BIT - 3)
                  {
                     // Deal with the bits in val that overflow into the next limb:
                     val >>= (sizeof(limb_type) * CHAR_BIT - shift);
                     if (val)
                     {
                        // If this is the most-significant-limb, we may need to allocate an extra one for the overflow:
                        if (limb + 1 == newsize)
                           result.resize(static_cast<unsigned>(newsize + 1), static_cast<unsigned>(newsize + 1));
                        if (result.size() > limb + 1)
                        {
                           result.limbs()[limb + 1] |= val;
                        }
                     }
                  }
               }
               ++s;
               bitcount -= 3;
            }
            result.normalize();
         }
         else
         {
            // Base 10, we extract blocks of size 10^9 at a time, that way
            // the number of multiplications is kept to a minimum:
            limb_type block_mult = max_block_10;
            while (*s)
            {
               limb_type block = 0;
               for (unsigned i = 0; i < digits_per_block_10; ++i)
               {
                  limb_type val;
                  if (*s >= '0' && *s <= '9')
                     val = *s - '0';
                  else
                     BOOST_THROW_EXCEPTION(std::runtime_error("Unexpected character encountered in input."));
                  block *= 10;
                  block += val;
                  if (!*++s)
                  {
                     block_mult = block_multiplier(i);
                     break;
                  }
               }
               eval_multiply(result, block_mult);
               eval_add(result, block);
            }
         }
      }
      if (isneg)
         result.negate();
      result.swap(*this);
   }

 public:
   cpp_int_backend& operator=(const char* s)
   {
      do_assign_string(s, trivial_tag());
      return *this;
   }
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR void swap(cpp_int_backend& o) noexcept
   {
      this->do_swap(o);
   }

 private:
   std::string do_get_trivial_string(std::ios_base::fmtflags f, const std::integral_constant<bool, false>&) const
   {
      using io_type = typename std::conditional<sizeof(typename base_type::local_limb_type) == 1, unsigned, typename base_type::local_limb_type>::type;
      if (this->sign() && (((f & std::ios_base::hex) == std::ios_base::hex) || ((f & std::ios_base::oct) == std::ios_base::oct)))
         BOOST_THROW_EXCEPTION(std::runtime_error("Base 8 or 16 printing of negative numbers is not supported."));
      std::stringstream ss;
      ss.flags(f & ~std::ios_base::showpos);
      ss << static_cast<io_type>(*this->limbs());
      std::string result;
      if (this->sign())
         result += '-';
      else if (f & std::ios_base::showpos)
         result += '+';
      result += ss.str();
      return result;
   }
   std::string do_get_trivial_string(std::ios_base::fmtflags f, const std::integral_constant<bool, true>&) const
   {
      // Even though we have only one limb, we can't do IO on it :-(
      int base = 10;
      if ((f & std::ios_base::oct) == std::ios_base::oct)
         base = 8;
      else if ((f & std::ios_base::hex) == std::ios_base::hex)
         base = 16;
      std::string result;

      unsigned Bits = sizeof(typename base_type::local_limb_type) * CHAR_BIT;

      if (base == 8 || base == 16)
      {
         if (this->sign())
            BOOST_THROW_EXCEPTION(std::runtime_error("Base 8 or 16 printing of negative numbers is not supported."));
         limb_type                           shift = base == 8 ? 3 : 4;
         limb_type                           mask  = static_cast<limb_type>((1u << shift) - 1);
         typename base_type::local_limb_type v     = *this->limbs();
         result.assign(Bits / shift + (Bits % shift ? 1 : 0), '0');
         std::string::difference_type pos      = result.size() - 1;
         char                         letter_a = f & std::ios_base::uppercase ? 'A' : 'a';
         for (unsigned i = 0; i < Bits / shift; ++i)
         {
            char c = '0' + static_cast<char>(v & mask);
            if (c > '9')
               c += letter_a - '9' - 1;
            result[pos--] = c;
            v >>= shift;
         }
         if (Bits % shift)
         {
            mask   = static_cast<limb_type>((1u << (Bits % shift)) - 1);
            char c = '0' + static_cast<char>(v & mask);
            if (c > '9')
               c += letter_a - '9';
            result[pos] = c;
         }
         //
         // Get rid of leading zeros:
         //
         std::string::size_type n = result.find_first_not_of('0');
         if (!result.empty() && (n == std::string::npos))
            n = result.size() - 1;
         result.erase(0, n);
         if (f & std::ios_base::showbase)
         {
            const char* pp = base == 8 ? "0" : (f & std::ios_base::uppercase) ? "0X" : "0x";
            result.insert(static_cast<std::string::size_type>(0), pp);
         }
      }
      else
      {
         result.assign(Bits / 3 + 1, '0');
         std::string::difference_type        pos = result.size() - 1;
         typename base_type::local_limb_type v(*this->limbs());
         bool                                neg = false;
         if (this->sign())
         {
            neg = true;
         }
         while (v)
         {
            result[pos] = (v % 10) + '0';
            --pos;
            v /= 10;
         }
         std::string::size_type n = result.find_first_not_of('0');
         result.erase(0, n);
         if (result.empty())
            result = "0";
         if (neg)
            result.insert(static_cast<std::string::size_type>(0), 1, '-');
         else if (f & std::ios_base::showpos)
            result.insert(static_cast<std::string::size_type>(0), 1, '+');
      }
      return result;
   }
   std::string do_get_string(std::ios_base::fmtflags f, const std::integral_constant<bool, true>&) const
   {
#ifdef BOOST_MP_NO_DOUBLE_LIMB_TYPE_IO
      return do_get_trivial_string(f, std::integral_constant<bool, std::is_same<typename base_type::local_limb_type, double_limb_type>::value>());
#else
      return do_get_trivial_string(f, std::integral_constant<bool, false>());
#endif
   }
   std::string do_get_string(std::ios_base::fmtflags f, const std::integral_constant<bool, false>&) const
   {
      using default_ops::eval_get_sign;
      int base = 10;
      if ((f & std::ios_base::oct) == std::ios_base::oct)
         base = 8;
      else if ((f & std::ios_base::hex) == std::ios_base::hex)
         base = 16;
      std::string result;

      unsigned Bits = this->size() * base_type::limb_bits;

      if (base == 8 || base == 16)
      {
         if (this->sign())
            BOOST_THROW_EXCEPTION(std::runtime_error("Base 8 or 16 printing of negative numbers is not supported."));
         limb_type       shift = base == 8 ? 3 : 4;
         limb_type       mask  = static_cast<limb_type>((1u << shift) - 1);
         cpp_int_backend t(*this);
         result.assign(Bits / shift + ((Bits % shift) ? 1 : 0), '0');
         std::string::difference_type pos      = result.size() - 1;
         char                         letter_a = f & std::ios_base::uppercase ? 'A' : 'a';
         for (unsigned i = 0; i < Bits / shift; ++i)
         {
            char c = '0' + static_cast<char>(t.limbs()[0] & mask);
            if (c > '9')
               c += letter_a - '9' - 1;
            result[pos--] = c;
            eval_right_shift(t, shift);
         }
         if (Bits % shift)
         {
            mask   = static_cast<limb_type>((1u << (Bits % shift)) - 1);
            char c = '0' + static_cast<char>(t.limbs()[0] & mask);
            if (c > '9')
               c += letter_a - '9';
            result[pos] = c;
         }
         //
         // Get rid of leading zeros:
         //
         std::string::size_type n = result.find_first_not_of('0');
         if (!result.empty() && (n == std::string::npos))
            n = result.size() - 1;
         result.erase(0, n);
         if (f & std::ios_base::showbase)
         {
            const char* pp = base == 8 ? "0" : (f & std::ios_base::uppercase) ? "0X" : "0x";
            result.insert(static_cast<std::string::size_type>(0), pp);
         }
      }
      else
      {
         result.assign(Bits / 3 + 1, '0');
         std::string::difference_type pos = result.size() - 1;
         cpp_int_backend              t(*this);
         cpp_int_backend              r;
         bool                         neg = false;
         if (t.sign())
         {
            t.negate();
            neg = true;
         }
         if (this->size() == 1)
         {
            result = boost::lexical_cast<std::string>(t.limbs()[0]);
         }
         else
         {
            cpp_int_backend block10;
            block10 = max_block_10;
            while (eval_get_sign(t) != 0)
            {
               cpp_int_backend t2;
               divide_unsigned_helper(&t2, t, block10, r);
               t           = t2;
               limb_type v = r.limbs()[0];
               for (unsigned i = 0; i < digits_per_block_10; ++i)
               {
                  char c = '0' + v % 10;
                  v /= 10;
                  result[pos] = c;
                  if (pos-- == 0)
                     break;
               }
            }
         }
         std::string::size_type n = result.find_first_not_of('0');
         result.erase(0, n);
         if (result.empty())
            result = "0";
         if (neg)
            result.insert(static_cast<std::string::size_type>(0), 1, '-');
         else if (f & std::ios_base::showpos)
            result.insert(static_cast<std::string::size_type>(0), 1, '+');
      }
      return result;
   }

 public:
   std::string str(std::streamsize /*digits*/, std::ios_base::fmtflags f) const
   {
      return do_get_string(f, trivial_tag());
   }

 private:
   template <class Container>
   void construct_from_container(const Container& c, const std::integral_constant<bool, false>&)
   {
      //
      // We assume that c is a sequence of (unsigned) bytes with the most significant byte first:
      //
      unsigned newsize = static_cast<unsigned>(c.size() / sizeof(limb_type));
      if (c.size() % sizeof(limb_type))
      {
         ++newsize;
      }
      if (newsize)
      {
         this->resize(newsize, newsize); // May throw
         std::memset(this->limbs(), 0, this->size());
         typename Container::const_iterator i(c.begin()), j(c.end());
         unsigned                           byte_location = static_cast<unsigned>(c.size() - 1);
         while (i != j)
         {
            unsigned limb  = byte_location / sizeof(limb_type);
            unsigned shift = (byte_location % sizeof(limb_type)) * CHAR_BIT;
            if (this->size() > limb)
               this->limbs()[limb] |= static_cast<limb_type>(static_cast<unsigned char>(*i)) << shift;
            ++i;
            --byte_location;
         }
      }
   }
   template <class Container>
   BOOST_MP_CXX14_CONSTEXPR void construct_from_container(const Container& c, const std::integral_constant<bool, true>&)
   {
      //
      // We assume that c is a sequence of (unsigned) bytes with the most significant byte first:
      //
      using local_limb_type = typename base_type::local_limb_type;
      *this->limbs() = 0;
      if (c.size())
      {
         typename Container::const_iterator i(c.begin()), j(c.end());
         unsigned                           byte_location = static_cast<unsigned>(c.size() - 1);
         while (i != j)
         {
            unsigned limb  = byte_location / sizeof(local_limb_type);
            unsigned shift = (byte_location % sizeof(local_limb_type)) * CHAR_BIT;
            if (limb == 0)
               this->limbs()[0] |= static_cast<limb_type>(static_cast<unsigned char>(*i)) << shift;
            ++i;
            --byte_location;
         }
      }
   }

 public:
   template <class Container>
   BOOST_MP_CXX14_CONSTEXPR cpp_int_backend(const Container& c, typename std::enable_if<boost::multiprecision::detail::is_byte_container<Container>::value>::type const* = 0)
   {
      //
      // We assume that c is a sequence of (unsigned) bytes with the most significant byte first:
      //
      construct_from_container(c, trivial_tag());
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare_imp(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o, const std::integral_constant<bool, false>&, const std::integral_constant<bool, false>&) const noexcept
   {
      if (this->sign() != o.sign())
         return this->sign() ? -1 : 1;

      // Only do the compare if the same sign:
      int result = compare_unsigned(o);

      if (this->sign())
         result = -result;
      return result;
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare_imp(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o, const std::integral_constant<bool, true>&, const std::integral_constant<bool, false>&) const
   {
      cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> t(*this);
      return t.compare(o);
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare_imp(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o, const std::integral_constant<bool, false>&, const std::integral_constant<bool, true>&) const
   {
      cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> t(o);
      return compare(t);
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare_imp(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o, const std::integral_constant<bool, true>&, const std::integral_constant<bool, true>&) const noexcept
   {
      if (this->sign())
      {
         if (o.sign())
         {
            return *this->limbs() < *o.limbs() ? 1 : (*this->limbs() > *o.limbs() ? -1 : 0);
         }
         else
            return -1;
      }
      else
      {
         if (o.sign())
            return 1;
         return *this->limbs() < *o.limbs() ? -1 : (*this->limbs() > *o.limbs() ? 1 : 0);
      }
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o) const noexcept
   {
      using t1 = std::integral_constant<bool, is_trivial_cpp_int<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value>     ;
      using t2 = std::integral_constant<bool, is_trivial_cpp_int<cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >::value>;
      return compare_imp(o, t1(), t2());
   }
   template <unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
   BOOST_MP_CXX14_CONSTEXPR int compare_unsigned(const cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>& o) const noexcept
   {
      if (this->size() != o.size())
      {
         return this->size() > o.size() ? 1 : -1;
      }
      typename base_type::const_limb_pointer pa = this->limbs();
      typename base_type::const_limb_pointer pb = o.limbs();
      for (int i = this->size() - 1; i >= 0; --i)
      {
         if (pa[i] != pb[i])
            return pa[i] > pb[i] ? 1 : -1;
      }
      return 0;
   }
   template <class Arithmetic>
   BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<boost::multiprecision::detail::is_arithmetic<Arithmetic>::value, int>::type compare(Arithmetic i) const
   {
      // braindead version:
      cpp_int_backend t;
      t = i;
      return compare(t);
   }
};

} // namespace backends

namespace default_ops {

template <class Backend>
struct double_precision_type;

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct double_precision_type<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
{
   using type = typename std::conditional<
       backends::is_fixed_precision<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value,
       backends::cpp_int_backend<
           (std::is_void<Allocator>::value ? 2 * backends::max_precision<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value
                                      : MinBits),
           2 * backends::max_precision<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::value,
           SignType,
           Checked,
           Allocator>,
       backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >::type;
};

} // namespace default_ops

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator, unsigned MinBits2, unsigned MaxBits2, cpp_integer_type SignType2, cpp_int_check_type Checked2, class Allocator2>
struct is_equivalent_number_type<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>, backends::cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2> >
   : public std::integral_constant<bool, std::numeric_limits<number<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator>, et_on> >::digits == std::numeric_limits<number<backends::cpp_int_backend<MinBits2, MaxBits2, SignType2, Checked2, Allocator2>, et_on> >::digits>{};

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked>
struct expression_template_default<backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, void> >
{
   static constexpr const expression_template_option value = et_off;
};

using boost::multiprecision::backends::cpp_int_backend;

template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
struct number_category<cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> > : public std::integral_constant<int, number_kind_integer>
{};

using cpp_int = number<cpp_int_backend<> >          ;
using cpp_rational_backend = rational_adaptor<cpp_int_backend<> >;
using cpp_rational = number<cpp_rational_backend>        ;

// Fixed precision unsigned types:
using uint128_t = number<cpp_int_backend<128, 128, unsigned_magnitude, unchecked, void> >  ;
using uint256_t = number<cpp_int_backend<256, 256, unsigned_magnitude, unchecked, void> >  ;
using uint512_t = number<cpp_int_backend<512, 512, unsigned_magnitude, unchecked, void> >  ;
using uint1024_t = number<cpp_int_backend<1024, 1024, unsigned_magnitude, unchecked, void> >;

// Fixed precision signed types:
using int128_t = number<cpp_int_backend<128, 128, signed_magnitude, unchecked, void> >  ;
using int256_t = number<cpp_int_backend<256, 256, signed_magnitude, unchecked, void> >  ;
using int512_t = number<cpp_int_backend<512, 512, signed_magnitude, unchecked, void> >  ;
using int1024_t = number<cpp_int_backend<1024, 1024, signed_magnitude, unchecked, void> >;

// Over again, but with checking enabled this time:
using checked_cpp_int = number<cpp_int_backend<0, 0, signed_magnitude, checked> >          ;
using checked_cpp_rational_backend = rational_adaptor<cpp_int_backend<0, 0, signed_magnitude, checked> >;
using checked_cpp_rational = number<checked_cpp_rational_backend>                               ;
// Fixed precision unsigned types:
using checked_uint128_t = number<cpp_int_backend<128, 128, unsigned_magnitude, checked, void> >  ;
using checked_uint256_t = number<cpp_int_backend<256, 256, unsigned_magnitude, checked, void> >  ;
using checked_uint512_t = number<cpp_int_backend<512, 512, unsigned_magnitude, checked, void> >  ;
using checked_uint1024_t = number<cpp_int_backend<1024, 1024, unsigned_magnitude, checked, void> >;

// Fixed precision signed types:
using checked_int128_t = number<cpp_int_backend<128, 128, signed_magnitude, checked, void> >  ;
using checked_int256_t = number<cpp_int_backend<256, 256, signed_magnitude, checked, void> >  ;
using checked_int512_t = number<cpp_int_backend<512, 512, signed_magnitude, checked, void> >  ;
using checked_int1024_t = number<cpp_int_backend<1024, 1024, signed_magnitude, checked, void> >;

#ifdef _MSC_VER
#pragma warning(pop)
#endif

}} // namespace boost::multiprecision

//
// Last of all we include the implementations of all the eval_* non member functions:
//
#include <boost/multiprecision/cpp_int/limits.hpp>
#include <boost/multiprecision/cpp_int/comparison.hpp>
#include <boost/multiprecision/cpp_int/add.hpp>
#include <boost/multiprecision/cpp_int/multiply.hpp>
#include <boost/multiprecision/cpp_int/divide.hpp>
#include <boost/multiprecision/cpp_int/bitwise.hpp>
#include <boost/multiprecision/cpp_int/misc.hpp>
#include <boost/multiprecision/cpp_int/literals.hpp>
#include <boost/multiprecision/cpp_int/serialize.hpp>
#include <boost/multiprecision/cpp_int/import_export.hpp>

#endif