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 ////////////////////////////////////////////////////////////////
 //  Copyright 2012 - 2022 John Maddock.
 //  Copyright 2022 Christopher Kormanyos.
 //  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 <cstdint>
 #include <cstring>
 #include <iostream>
 #include <iomanip>
 #include <type_traits>
 #include <string>
 #include <boost/multiprecision/detail/standalone_config.hpp>
 #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/multiprecision/cpp_int/cpp_int_config.hpp>
 #include <boost/multiprecision/rational_adaptor.hpp>
 #include <boost/multiprecision/traits/is_byte_container.hpp>
 #include <boost/multiprecision/cpp_int/checked.hpp>
 #include <boost/multiprecision/detail/constexpr.hpp>
 #include <boost/multiprecision/detail/float128_functions.hpp>
 #include <boost/multiprecision/cpp_int/value_pack.hpp>
 #include <boost/multiprecision/detail/empty_value.hpp>
 #include <boost/multiprecision/detail/no_exceptions_support.hpp>
 #include <boost/multiprecision/detail/assert.hpp>
 #include <boost/multiprecision/detail/fpclassify.hpp>
 
 namespace boost {
 namespace multiprecision {
 
 #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
 #if defined(__GNUC__) && !defined(__clang__)
 // see https://github.com/boostorg/multiprecision/issues/413
 // and https://github.com/boostorg/multiprecision/issues/431
 #pragma GCC diagnostic push
 #pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
 #endif
 
 namespace detail {
 
 template <std::size_t MinBits, std::size_t 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 {
 
 namespace detail {
    template <std::size_t Value1, std::size_t Value2>
    struct static_unsigned_max
    {
       static constexpr std::size_t value = (Value1 > Value2) ? Value1 : Value2;
    };
 } // Namespace detail
 
 template <std::size_t MinBits, std::size_t 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 <std::size_t MinBits, std::size_t 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 std::size_t value = std::is_void<Allocator>::value ? detail::static_unsigned_max<MinBits, MaxBits>::value
                                                                        : (((MaxBits >= MinBits) && MaxBits) ? MaxBits : SIZE_MAX);
 };
 
 template <class T>
 struct min_precision;
 
 template <std::size_t MinBits, std::size_t 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 std::size_t value = (std::is_void<Allocator>::value ? detail::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 bool value = false;
 };
 
 template <std::size_t MinBits, std::size_t 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 bool value = std::is_void<Allocator>::value && (max_precision<self>::value <= (sizeof(double_limb_type) * CHAR_BIT) - (SignType == signed_packed ? 1 : 0));
 };
 
 template <std::size_t MinBits, std::size_t 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 bool value = true;
 };
 
 } // namespace backends
 //
 // Traits class to determine whether a cpp_int_backend is signed or not:
 //
 template <std::size_t MinBits, std::size_t 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 <std::size_t MinBits, std::size_t MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator, std::size_t MinBits2, std::size_t 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 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 <std::size_t MinBits, std::size_t 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 <std::size_t MinBits, std::size_t 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 != SIZE_MAX>
 {};
 
 namespace detail {
 
 inline BOOST_MP_CXX14_CONSTEXPR void verify_new_size(std::size_t new_size, std::size_t min_size, const std::integral_constant<int, checked>&)
 {
    if (new_size < min_size)
       BOOST_MP_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(std::size_t /*new_size*/, std::size_t /*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_MP_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 <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 struct cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>
     : private boost::multiprecision::detail::empty_value<typename detail::rebind<limb_type, Allocator>::type>
 {
    template <std::size_t MinBits2, std::size_t 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::multiprecision::detail::empty_value<allocator_type>;
 
 private:
    struct limb_data
    {
       std::size_t        capacity;
       limb_pointer    data;
    };
 
  public:
    static constexpr std::size_t 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 std::size_t 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(static_cast<limb_type>(boost::multiprecision::detail::unsigned_abs(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(static_cast<double_limb_type>(boost::multiprecision::detail::unsigned_abs(i))) {}
 #endif
 #if !defined(BOOST_NO_CXX11_UNIFIED_INITIALIZATION_SYNTAX) && !(defined(BOOST_MSVC) && (BOOST_MSVC < 1900))
       constexpr data_type(limb_type* limbs, std::size_t len) noexcept : ld{ len, limbs }
       {}
 #else
       constexpr data_type(limb_type* limbs, std::size_t len) noexcept
       {
          ld.capacity = len;
          ld.data = limbs;
       }
 #endif
    };
 
    data_type m_data;
    std::size_t  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::multiprecision::detail::empty_value<allocator_type>
    {
     private:
       limb_type*      data;
       std::size_t        capacity;
       std::size_t        allocated;
       bool            is_alias;
       allocator_type& allocator() noexcept { return boost::multiprecision::detail::empty_value<allocator_type>::get(); }
 
     public:
       scoped_shared_storage(const allocator_type& a, std::size_t len)
           : boost::multiprecision::detail::empty_value<allocator_type>(boost::multiprecision::detail::empty_init_t(), a), capacity(len), allocated(0), is_alias(false)
       {
          data = allocator().allocate(len);
       }
       scoped_shared_storage(const cpp_int_base& i, std::size_t len)
           : boost::multiprecision::detail::empty_value<allocator_type>(boost::multiprecision::detail::empty_init_t(), i.allocator()), capacity(len), allocated(0), is_alias(false)
       {
          data = allocator().allocate(len);
       }
       scoped_shared_storage(limb_type* limbs, std::size_t n) : data(limbs), capacity(n), allocated(0), is_alias(true) {}
       ~scoped_shared_storage()
       {
          if(!is_alias)
             allocator().deallocate(data, capacity);
       }
       limb_type* allocate(std::size_t n) noexcept 
       {
          limb_type* result = data + allocated;
          allocated += n;
          BOOST_MP_ASSERT(allocated <= capacity);
          return result; 
       }
       void deallocate(std::size_t n)
       {
          BOOST_MP_ASSERT(n <= allocated);
          allocated -= n;
       }
    };
    explicit constexpr cpp_int_base(limb_type* data, std::size_t offset, std::size_t 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, std::size_t offset, std::size_t 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, std::size_t 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 std::size_t              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 std::size_t              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(std::size_t new_size, std::size_t min_size)
    {
       constexpr std::size_t 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:
       std::size_t cap = capacity();
       if (new_size > cap)
       {
          // We must not be an alias, memory allocation here defeats the whole point of aliasing:
          BOOST_MP_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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 constexpr std::size_t cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::limb_bits;
 template <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 constexpr limb_type cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::max_limb_value;
 template <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 constexpr limb_type cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::sign_bit_mask;
 template <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 constexpr std::size_t cpp_int_base<MinBits, MaxBits, signed_magnitude, Checked, Allocator, false>::internal_limb_count;
 
 template <std::size_t MinBits, std::size_t MaxBits, cpp_int_check_type Checked, class Allocator>
 struct cpp_int_base<MinBits, MaxBits, unsigned_magnitude, Checked, Allocator, false>
     : private boost::multiprecision::detail::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 <std::size_t 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&, std::size_t) {}
       BOOST_MP_CXX14_CONSTEXPR void deallocate(std::size_t) {}
    };
 
    //
    // 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 std::size_t 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 std::size_t  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&, std::size_t) 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 std::size_t              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(std::size_t new_size, std::size_t 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 (std::size_t 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 (std::size_t 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 <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr std::size_t cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::limb_bits;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr limb_type cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::max_limb_value;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr limb_type cpp_int_base<MinBits, MinBits, signed_magnitude, Checked, void, false>::sign_bit_mask;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr std::size_t 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 <std::size_t 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&, std::size_t) {}
       BOOST_MP_CXX14_CONSTEXPR void deallocate(std::size_t) {}
    };
    //
    // 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 std::size_t 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 std::size_t 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::size_t 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(static_cast<limb_type>(boost::multiprecision::detail::unsigned_abs(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&, std::size_t) noexcept : m_wrapper(), m_limbs(1)
    {}
        //
    // Helper functions for getting at our internal data, and manipulating storage:
    //
    BOOST_MP_FORCEINLINE constexpr std::size_t              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(std::size_t new_size, std::size_t 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 (std::size_t 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_MP_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());
       std::size_t 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 (std::size_t 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 <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr std::size_t cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::limb_bits;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr limb_type cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::max_limb_value;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr limb_type cpp_int_base<MinBits, MinBits, unsigned_magnitude, Checked, void, false>::sign_bit_mask;
 template <std::size_t MinBits, cpp_int_check_type Checked>
 constexpr std::size_t 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 long long, "really long long" 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::multiprecision::detail::uint_t<N>::least;
 };
 
 template <unsigned N>
 struct trivial_limb_type : public trivial_limb_type_imp<N, N <= sizeof(long long) * CHAR_BIT>
 {};
 //
 // Backend for fixed precision signed-magnitude type which will fit entirely inside a "double_limb_type":
 //
 template <std::size_t 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<static_cast<unsigned>(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&, std::size_t) {}
       BOOST_MP_CXX14_CONSTEXPR void deallocate(std::size_t) {}
    };
 
  protected:
    static constexpr std::size_t 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 <= static_cast<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_MP_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 <= static_cast<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_MP_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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) noexcept
        : m_data(static_cast<local_limb_type>(i < 0 ? -i : 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* = nullptr)
        : m_data(static_cast<local_limb_type>(i < 0 ? -i : 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* = nullptr) noexcept
        : m_data(static_cast<local_limb_type>(static_cast<std::uint64_t>(i < 0 ? -i : 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* = nullptr)
        : m_data(static_cast<local_limb_type>(static_cast<std::uint64_t>(i < 0 ? -i : 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&, std::size_t) noexcept : m_data(0), m_sign(false)
    {}
        //
    // Helper functions for getting at our internal data, and manipulating storage:
    //
    BOOST_MP_FORCEINLINE constexpr std::size_t              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(std::size_t /* new_size */, std::size_t 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 <std::size_t 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<static_cast<unsigned>(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&, std::size_t) {}
       BOOST_MP_CXX14_CONSTEXPR void deallocate(std::size_t) {}
    };
 
  private:
    static constexpr std::size_t 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 <= static_cast<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_MP_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_MP_THROW_EXCEPTION(std::range_error("The argument to a cpp_int constructor exceeded the largest value it can represent."));
       if (val < 0)
          BOOST_MP_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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) 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* = nullptr) noexcept((Checked == unchecked))
        : m_data(static_cast<local_limb_type>(i < 0 ? -i : 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&, std::size_t) noexcept : m_data(0)
    {}
        //
    // Helper functions for getting at our internal data, and manipulating storage:
    //
    BOOST_MP_FORCEINLINE constexpr std::size_t              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, std::size_t 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()
    #if !defined(BOOST_NO_CXX17_IF_CONSTEXPR)
    noexcept((Checked == unchecked))
    #endif
    {
       BOOST_IF_CONSTEXPR(Checked == checked)
       {
          BOOST_MP_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 <std::size_t MinBits, std::size_t 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,
            long long, 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, unsigned long long, 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 <std::size_t MinBits2, std::size_t 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* = nullptr) 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* = nullptr) 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, std::size_t offset, std::size_t len) noexcept
        : base_type(data, offset, len) {}
    explicit cpp_int_backend(const limb_type* data, std::size_t offset, std::size_t len) noexcept
        : base_type(data, offset, len) { this->normalize(); }
    explicit constexpr cpp_int_backend(typename base_type::scoped_shared_storage& data, std::size_t len) noexcept
        : base_type(data, len) {}
 
  private:
    template <std::size_t MinBits2, std::size_t 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::true_type const&, std::true_type 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 <std::size_t MinBits2, std::size_t 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::true_type const&, std::false_type 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_MP_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 <std::size_t MinBits2, std::size_t 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::false_type const&, std::true_type 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 <std::size_t MinBits2, std::size_t 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::false_type const&, std::false_type 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)
       std::size_t count = (std::min)(other.size(), this->size());
       for (std::size_t 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()))
       {
          std::size_t count = (std::min)(other.size(), this->size());
          for (std::size_t i = 0; i < count; ++i)
             this->limbs()[i] = other.limbs()[i];
       }
       else
 #endif
       {
          static_assert(sizeof(other.limbs()[0]) == sizeof(this->limbs()[0]), "This method requires equal limb sizes");
          std::memcpy(this->limbs(), other.limbs(), (std::min)(other.size() * sizeof(other.limbs()[0]), this->size() * sizeof(this->limbs()[0])));
       }
 #endif
       this->sign(other.sign());
       this->normalize();
    }
 
  public:
    template <std::size_t MinBits2, std::size_t 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* = nullptr)
        : 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 <std::size_t MinBits2, std::size_t 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* = nullptr)
        : 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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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;
    }
 
    template <class A>
    BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<
          boost::multiprecision::detail::is_unsigned<A>::value
          && trivial_tag::value, cpp_int_backend&>::type 
       operator=(const A& val)
          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();
       return *this;
    }
    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)
          && trivial_tag::value, cpp_int_backend&>::type
       operator=(const A& val)
          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();
       return *this;
    }
    template <class A>
    BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<
          std::is_convertible<A, limb_type>::value 
          && !boost::multiprecision::detail::is_integral<A>::value
          && trivial_tag::value, cpp_int_backend&>::type 
       operator=(const A& val)
    {
       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();
       return *this;
    }
    template <class A>
    BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if<
          std::is_same<A, limb_type>::value && !trivial_tag::value, cpp_int_backend&>::type
       operator=(A i) noexcept
    {
       this->resize(1, 1);
       *this->limbs() = i;
       this->sign(false);
       return *this;
    }
    template <class A>
    BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if <
          std::is_same<A, signed_limb_type>::value && !trivial_tag::value, cpp_int_backend&>::type
       operator=(A i) 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);
       return *this;
    }
    template <class A>
    BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if <
          std::is_same<A, double_limb_type>::value && !trivial_tag::value, cpp_int_backend&>::type
       operator=(A i) 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);
       return *this;
    }
    template <class A>
    BOOST_MP_CXX14_CONSTEXPR typename std::enable_if <
          std::is_same<A, signed_double_limb_type>::value && !trivial_tag::value, cpp_int_backend&>::type
       operator=(A i) 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);
       return *this;
    }
 private:
    template <class F>
    BOOST_MP_CXX14_CONSTEXPR void do_assign_float(F a)
    {
       using default_ops::eval_add;
       using default_ops::eval_subtract;
       BOOST_MP_FLOAT128_USING using std::floor; using std::frexp; using std::ldexp;
 
       if (a < 0)
       {
          do_assign_float(-a);
          this->sign(true);
          return;
       }
 
       if (a == 0)
       {
          *this = static_cast<limb_type>(0u);
       }
 
       if (a == 1)
       {
          *this = static_cast<limb_type>(1u);
       }
 
       if (!BOOST_MP_ISFINITE(a))
       {
          BOOST_MP_THROW_EXCEPTION(std::runtime_error("Cannot convert a non-finite number to an integer."));
       }
 
       int         e = 0;
       F 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 != static_cast<F>(0.0f))
       {
          // extract int sized bits from f:
          f    = ldexp(f, shift);
          term = floor(f);
          e = e - static_cast<int>(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, static_cast<unsigned int>(e));
       else if (e < 0)
          eval_right_shift(*this, static_cast<unsigned int>(-e));
    }
 public:
    template <class A>
    BOOST_MP_FORCEINLINE BOOST_MP_CXX14_CONSTEXPR typename std::enable_if <
       std::is_floating_point<A>::value && !trivial_tag::value, cpp_int_backend&>::type
       operator=(A a)
    {
       do_assign_float(a);
       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 = static_cast<unsigned>(*s - '0');
             else if (*s >= 'a' && *s <= 'f')
                val = 10u + static_cast<unsigned>(*s - 'a');
             else if (*s >= 'A' && *s <= 'F')
                val = 10u + static_cast<unsigned>(*s - 'A');
             else
                val = radix + 1u;
             if (val >= radix)
             {
                BOOST_MP_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 = static_cast<unsigned>(*s - '0');
                else if (*s >= 'a' && *s <= 'f')
                   val = 10u + static_cast<unsigned>(*s - 'a');
                else if (*s >= 'A' && *s <= 'F')
                   val = 10u + static_cast<unsigned>(*s - 'A');
                else
                {
                   #if defined(BOOST_NO_EXCEPTIONS)
                   val = static_cast<unsigned>('0');
                   #endif
 
                   BOOST_MP_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 = static_cast<unsigned>(*s - '0');
                else
                {
                   #if defined(BOOST_NO_EXCEPTIONS)
                   val = static_cast<unsigned>('0');
                   #endif
 
                   BOOST_MP_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));
                            result.limbs()[limb + 1] = 0;
                         }
                         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 = static_cast<limb_type>(*s - '0');
                   else
                   {
                      #if defined(BOOST_NO_EXCEPTIONS)
                      val = static_cast<unsigned>('0');
                      #endif
 
                      BOOST_MP_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_MP_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;
 
       std::size_t Bits = sizeof(typename base_type::local_limb_type) * CHAR_BIT;
 
       if (base == 8 || base == 16)
       {
          if (this->sign())
             BOOST_MP_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      = static_cast<std::string::difference_type>(result.size() - 1u);
          char                         letter_a = f & std::ios_base::uppercase ? 'A' : 'a';
          for (std::size_t i = 0; i < Bits / shift; ++i)
          {
             char c = static_cast<char>('0' + static_cast<char>(v & mask));
             if (c > '9')
                c = static_cast<char>(c + letter_a - '9' - 1);
             result[static_cast<std::size_t>(pos)] = c;
             --pos;
             v >>= shift;
          }
          if (Bits % shift)
          {
             mask   = static_cast<limb_type>((1u << (Bits % shift)) - 1);
             char c = static_cast<char>('0' + static_cast<char>(v & mask));
             if (c > '9')
                c = static_cast<char>(c + letter_a - '9');
             result[static_cast<std::size_t>(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 = static_cast<std::string::difference_type>(result.size() - 1u);
          typename base_type::local_limb_type v(*this->limbs());
          bool                                neg = false;
          if (this->sign())
          {
             neg = true;
          }
          while (v)
          {
             result[static_cast<std::string::size_type>(pos)] = static_cast<char>(static_cast<char>(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;
 
       std::size_t Bits = this->size() * base_type::limb_bits;
 
       if (base == 8 || base == 16)
       {
          if (this->sign())
             BOOST_MP_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      = static_cast<std::string::difference_type>(result.size() - 1u);
          char                         letter_a = f & std::ios_base::uppercase ? 'A' : 'a';
          for (std::size_t i = 0; i < Bits / shift; ++i)
          {
             char c = static_cast<char>('0' + static_cast<char>(t.limbs()[0] & mask));
             if (c > '9')
                c = static_cast<char>(c + letter_a - '9' - 1);
             result[static_cast<std::size_t>(pos)] = c;
             --pos;
             eval_right_shift(t, shift);
          }
          if (Bits % shift)
          {
             mask   = static_cast<limb_type>((1u << (Bits % shift)) - 1);
             char c = static_cast<char>('0' + static_cast<char>(t.limbs()[0] & mask));
             if (c > '9')
                c = static_cast<char>(c + letter_a - '9');
             result[static_cast<std::size_t>(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 = static_cast<std::string::difference_type>(result.size() - 1u);
          cpp_int_backend              t(*this);
          cpp_int_backend              r;
          bool                         neg = false;
          if (t.sign())
          {
             t.negate();
             neg = true;
          }
          if (this->size() == 1)
          {
             result = std::to_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 (std::size_t i = 0; i < digits_per_block_10; ++i)
                {
                   char c = static_cast<char>('0' + static_cast<char>(v % 10));
                   v /= 10;
                   result[static_cast<std::size_t>(pos)] = c;
                   if (pos-- == 0u)
                      break;
                }
             }
          }
          std::string::size_type n = result.find_first_not_of('0');
          result.erase(0, n);
          if (result.empty())
             result = std::string(static_cast<std::size_t>(1u), '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:
       //
       std::size_t 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());
          std::size_t                           byte_location = static_cast<unsigned>(c.size() - 1);
          while (i != j)
          {
             std::size_t limb  = byte_location / sizeof(limb_type);
             std::size_t 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());
          std::size_t                           byte_location = static_cast<unsigned>(c.size() - 1);
          while (i != j)
          {
             std::size_t limb  = byte_location / sizeof(local_limb_type);
             std::size_t 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* = nullptr)
    {
       //
       // We assume that c is a sequence of (unsigned) bytes with the most significant byte first:
       //
       construct_from_container(c, trivial_tag());
    }
    template <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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 <std::size_t MinBits2, std::size_t 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 (std::ptrdiff_t i = static_cast<std::ptrdiff_t>(static_cast<std::ptrdiff_t>(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 <std::size_t MinBits, std::size_t 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 <std::size_t MinBits, std::size_t MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator, std::size_t MinBits2, std::size_t 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 <std::size_t MinBits, std::size_t 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>
 {};
 
 #ifdef BOOST_HAS_INT128
 
 namespace detail {
 
 template <std::size_t MinBits, std::size_t MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
 struct is_convertible_arithmetic<int128_type, backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
 {
    static constexpr bool value = true;
 };
 template <std::size_t MinBits, std::size_t MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked, class Allocator>
 struct is_convertible_arithmetic<uint128_type, backends::cpp_int_backend<MinBits, MaxBits, SignType, Checked, Allocator> >
 {
    static constexpr bool value = true;
 };
 
 }
 
 #endif
 
 #if defined(__GNUC__) && !defined(__clang__)
 // see https://github.com/boostorg/multiprecision/issues/413
 // and https://github.com/boostorg/multiprecision/issues/431
 #pragma GCC diagnostic pop
 #endif
 #ifdef BOOST_MSVC
 #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