Document number: D0539R2
Project: Programming Language C++
Audience: SG6 Numerics
 
Igor Klevanets <cerevra@yandex.ru>, <cerevra@yandex-team.ru>
Antony Polukhin <antoshkka@gmail.com>, <antoshkka@yandex-team.ru>
 
Date: 2017-10-10

A Proposal to add wide_int Template Class

Significant changes to P0539R1 are marked with blue.

Green lines are notes for the editor or for the SG6 that must not be treated as part of the wording.

I. Introduction and Motivation

Current standard provides signed and unsigned int8_t, int16_t, int32_t, int64_t. It is usually enough for every day tasks, but sometimes appears a need in big numbers: for cryptography, IPv6, very big counters etc. Non-standard type __int128 which is provided by gcc and clang illuminates this need. But there is no cross-platform solution and no way to satisfy future needs in even more big numbers.

This is an attempt to solve the problem in a generic way on a library level and provide wording for P0104R0: Multi-Word Integer Operations and Types.

A proof of concept implementation available at: https://github.com/cerevra/int/tree/master/v3.

II. Changelog

Differences with P0539R1:

Differences with P0539R0:

III. Discussion about the way to set the wide_integer size

In this proposal we would like to concentrate on the wide_integer class that uses machine words under the cover. Such implementations allow you to get best performance and leave behind some design questions, like "Is (sizeof(wide_integer<X>) == X / CHAR_BITS) or not?".

However, we do not wish to shut the door close for extending the abilities of the wide_integer class. Some users may wish to see unit40_t, or unit48_t.

We insist on interface that allows specifying integers not representable by machine words count. Such extensions of functionality may be discussed in separate papers.

[Note to SG6: The "Words vs Bytes vs Bits" interface problem seems to rise very often. It may be worth discussing it in a bigger group or at least discuss it after the remaining parts of the paper are tentatively accepted. - end note]

There are several possible meanings of the first template argument of wide_integer:

Where number must be:

As was noted above, we would like to use machine words under the hood, but we would like to avoid any mention of machine words in the interface. Machine word is a strong restriction for interface, which does not allow to extend functionality in the future.

"Power of 2" restriction seems over restrictive. If machine word is 32 bits users may wish to have uint96_t. Supporting such type is quite easy.

So the actual choice can be Bytes or Bits where sizeof(wide_integer) is multiply of sizeof(machine word).

Pros and cons for Bits vs Bytes:

There was also an idea to provide a way for a user to replace the underlying type that represents machine word (allow to specify unsigned int or long or long long). Such idea seems to unnecessary complicate the interface and expose the internals of the wide_integer type. We assume that the decision must be done by the library implementor rather than by user.

IV. Proposed wording

[Note to SG6: Following 4 paragraphs affect all the new numeric class proposals. - end note]

18.3.4 Class template numeric_limits[numeric.limits]

[Note to editor: Add the following sentence after the sentence "Specializations shall be provided for each arithmetic type, both floating-point and integer, including bool." (first sentence in fourth paragraph in [numeric.limits]) - end note]

Specializations shall be also provided for wide_integer type. [Note: If there is a built-in integral type Integral that has the same signedness and width as wide_integer<MachineWords, S>, then numeric_limits<wide_integer<MachineWords, S>> specialized in the same way as numeric_limits<Integral>- end note]

26.2 Definitions[numerics.defns]

[Note to editor: Add the following paragraph to the end of [numerics.defns] - end note]

Define CT as common_type_t<A, B>, where A and B are the types of the two function arguments.

26.3 Numeric type requirements[numeric.requirements]

[Note to editor: Add the following paragraphs to the end of [numeric.requirements] - end note]

Functions that accept template parameters starting with Arithmetic shall not participate in overload resolution unless std::numeric_limits<Arithmetic>::is_specialized is true.

Functions that accept template parameters starting with Integral shall not participate in overload resolution unless std::numeric_limits<Integral>::is_integer is true.

26.4 Numeric types interoperability[numeric.interop]

[Note to SG6: This is an early attempt to make it possible for different numeric types to inter-operate. It works for wide_integer and built-in types, but this approach requires additional study and common_type fixes for new Arithmetic types. - end note]

Following operators are defined for types T and U if T and U have a defined common_type_t<T, U> and satisfy the Arithmetic or Integral requirements from [numeric.requirements] [Note: Implementations are encouraged to provide optimized specializations of the following operators - end note]:

[Note to SG6: We may add to Arithmetic and Integral concepts additional requirement that std::numeric_limits<Integral-or-Arithmetic>::is_interoprable shall be true. In that case no user code will be broken even if user already added following operators. However this seems to be an overkill. - end note]

template<typename Arithmetic1, typename Arithmetic2>
  common_type_t<Arithmetic1, Arithmetic2>
  constexpr operator*(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) * CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  common_type_t<Arithmetic1, Arithmetic2>
  constexpr operator/(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) / CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  common_type_t<Arithmetic1, Arithmetic2>
  constexpr operator+(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) + CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  common_type_t<Arithmetic1, Arithmetic2>
  constexpr operator-(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) - CT(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, Integral2>
  constexpr operator%(const Integral1& lhs, const Integral2& rhs);
Returns: CT(lhs) % CT(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, Integral2>
  constexpr operator&(const Integral1& lhs, const Integral2& rhs);
Returns: CT(lhs) & CT(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, Integral2>
  constexpr operator|(const Integral1& lhs, const Integral2& rhs);
Returns: CT(lhs) | CT(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, Integral2>
  constexpr operator^(const Integral1& lhs, const Integral2& rhs);
Returns: CT(lhs) ^ CT(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, size_t>
  constexpr operator<<(const Integral1& lhs, const Integral2& rhs);
Returns: common_type_t<Integral1, size_t>(lhs) << static_cast<size_t>(rhs).
template<typename Integral1, typename Integral2>
  common_type_t<Integral1, size_t>
  constexpr operator>>(const Integral1& lhs, const Integral2& rhs);
Returns: common_type_t<Integral1, size_t>(lhs) >> static_cast<size_t>(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator<(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) < CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator>(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) > CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator<=(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) <= CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator>=(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) >= CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator==(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: CT(lhs) == CT(rhs).
template<typename Arithmetic1, typename Arithmetic2>
  constexpr bool operator!=(const Arithmetic1& lhs, const Arithmetic2& rhs);
Returns: !(lhs == rhs).

26.??.1 Header <wide_integer> synopsis[numeric.wide_integer.syn]

namespace std {
  enum class signedness {
      Unsigned,
      Signed
  };

  [Note to SG6: We leave MachineWords in the interface until the "Words vs Bytes vs Bits" interface problem is not solved. - end note]
  // 26.??.2 class template wide_integer
  template<size_t MachineWords, signedness S> class wide_integer;

  // 26.??.?? type traits specializations
  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  struct common_type<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>;

  template<size_t MachineWords, signedness S, typename Arithmetic>
  struct common_type<wide_integer<MachineWords, S>, Arithmetic>;

  template<typename Arithmetic, size_t MachineWords, signedness S>
  struct common_type<Arithmetic, wide_integer<MachineWords, S>>
    : common_type<wide_integer<MachineWords, S>, Arithmetic>
  ;

  // 26.??.?? unary operations
  template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator~(const wide_integer<MachineWords, S>& val) noexcept;
  template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator-(const wide_integer<MachineWords, S>& val) noexcept(S == signedness::Unsigned);
  template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator+(const wide_integer<MachineWords, S>& val) noexcept(S == signedness::Unsigned) noexcept;

  // 26.??.?? binary operations
  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator*(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator/(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator+(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept(S == signedness::Unsigned);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator-(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept(S == signedness::Unsigned);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator%(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator&(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr operator|(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
    constexpr  operator^(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S>
    common_type_t<wide_integer<MachineWords, S>, size_t>
    constexpr  operator<<(const wide_integer<MachineWords, S>& lhs, size_t rhs);

  template<size_t MachineWords, signedness S>
    common_type_t<wide_integer<MachineWords, S>, size_t>
    constexpr  operator>>(const wide_integer<MachineWords, S>& lhs, size_t rhs);

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator<(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator>(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator<=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator>=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator==(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
    constexpr bool operator!=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;

  // 26.??.?? numeric conversions
  template<size_t MachineWords, signedness S> std::string to_string(const wide_integer<MachineWords, S>& val);
  template<size_t MachineWords, signedness S> std::wstring to_wstring(const wide_integer<MachineWords, S>& val);

  // 26.??.?? iostream specializations
  template<class Char, class Traits, size_t MachineWords, signedness S>
  basic_ostream<Char, Traits>& operator<<(basic_ostream<Char, Traits>& os, const wide_integer<MachineWords, S>& val);

  template<class Char, class Traits, size_t MachineWords, signedness S>
  basic_istream<Char, Traits>& operator>>(basic_istream<Char, Traits>& is, wide_integer<MachineWords, S>& val) noexcept;

  // 26.??.?? hash support
  template<class T> struct hash;
  template<size_t MachineWords, signedness S> struct hash<wide_integer<MachineWords, S>>;


  template <size_t MachineWords, signedness S>
  to_chars_result to_chars(char* first,
                           char* last,
                           const wide_integer<MachineWords, S>& value,
                           int base = 10);

  template <size_t MachineWords, signedness S>
  from_chars_result from_chars(const char* first,
                               const char* last,
                               wide_integer<MachineWords, S>& value,
                               int base = 10);

  template <size_t MachineWords>
  using wide_int = wide_integer<MachineWords, signedness::Signed>;

  template <size_t MachineWords>
  using wide_uint = wide_integer<MachineWords, signedness::Unsigned>

[Note to SG6: Following part requires updated P0102R0 proposal with exact_2int, fast_2int, least_2int, exact_2uint, fast_2uint, least_2uint specializations for wide_integer. - end note]

  // optional aliases
  using int128_t  = exact_2int<128>;      // wide_int<platform-specific> or platform-specific build in type
  using uint128_t = exact_2uint<128>;     // wide_uint<platform-specific> or platform-specific build in type

  using int256_t  = exact_2int<256>;      // wide_int<platform-specific> or platform-specific build in type
  using uint256_t = exact_2uint<256>;     // wide_uint<platform-specific> or platform-specific build in type

  using int512_t  = exact_2int<512>;      // wide_int<platform-specific> or platform-specific build in type
  using uint512_t = exact_2uint<512>;     // wide_uint<platform-specific> or platform-specific build in type


  // non optional aliases
  using int_fast128_t  = fast_2int<128>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_fast128_t = fast_2uint<128>;  // wide_uint<platform-specific> or platform-specific build in type

  using int_fast256_t  = fast_2int<256>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_fast256_t = fast_2uint<256>;  // wide_uint<platform-specific> or platform-specific build in type

  using int_fast512_t  = fast_2int<512>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_fast512_t = fast_2uint<512>;  // wide_uint<platform-specific> or platform-specific build in type


  using int_least128_t  = least_2int<128>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_least128_t = least_2uint<128>;  // wide_uint<platform-specific> or platform-specific build in type

  using int_least256_t  = least_2int<256>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_least256_t = least_2uint<256>;  // wide_uint<platform-specific> or platform-specific build in type

  using int_least512_t  = least_2int<512>;   // wide_int<platform-specific> or platform-specific build in type
  using uint_least512_t = least_2uint<512>;  // wide_uint<platform-specific> or platform-specific build in type


  // optional literals
  inline namespace literals {
  inline namespace wide_int_literals {

  constexpr int128_t operator "" _int128(const char*);
  constexpr int256_t operator "" _int256(const char*);
  constexpr int512_t operator "" _int512(const char*);
  constexpr uint128_t operator "" _uint128(const char*);
  constexpr uint256_t operator "" _uint256(const char*);
  constexpr uint512_t operator "" _uint512(const char*);

  } // namespace wide_int_literals
  } // namespace literals
}

The header <wide_integer> defines class template wide_integer and a set of operators for representing and manipulating integers of specified width.

[Example:
    constexpr int128_t c = std::numeric_limits<int128_t>::min();
    static_assert(c == 0x80000000000000000000000000000000_uint128);

    int256_t a = 13;
    a += 0xFF;
    a *= 2.0;
    a -= 12_int128;
    assert(a > 0);
]

26.??.2 Template class wide_integer overview[numeric.wide_integer.overview]

namespace std {
  template<size_t MachineWords, signedness S>
  class wide_integer {
  public:
    // 26.??.2.?? construct:
    constexpr wide_integer() noexcept = default;
    template<typename Arithmetic> constexpr wide_integer(const Arithmetic& other) noexcept;
    template<size_t MachineWords2, signedness S2> constexpr wide_integer(const wide_integer<MachineWords2, S2>& other) noexcept;

    // 26.??.2.?? assignment:
    template<typename Arithmetic>
    constexpr wide_integer<MachineWords, S>& operator=(const Arithmetic& other) noexcept;
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator=(const wide_integer<MachineWords2, S2>& other) noexcept;

    // 26.??.2.?? compound assignment:
    template<typename Arithmetic>
    constexpr wide_integer<MachineWords, S>& operator*=(const Arithmetic&);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator*=(const wide_integer<MachineWords2, S2>&);

    template<typename Arithmetic>
    constexpr wide_integer<MachineWords, S>& operator/=(const Arithmetic&);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator/=(const wide_integer<MachineWords2, S2>&);

    template<typename Arithmetic>
    constexpr wide_integer<MachineWords, S>& operator+=(const Arithmetic&) noexcept(S == signedness::Unsigned);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator+=(const wide_integer<MachineWords2, S2>&) noexcept(S == signedness::Unsigned);

    template<typename Arithmetic>
    constexpr wide_integer<MachineWords, S>& operator-=(const Arithmetic&) noexcept(S == signedness::Unsigned);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator-=(const wide_integer<MachineWords2, S2>&) noexcept(S == signedness::Unsigned);

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator%=(const Integral&);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator%=(const wide_integer<MachineWords2, S2>&);

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator&=(const Integral&) noexcept;
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator&=(const wide_integer<MachineWords2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator|=(const Integral&) noexcept;
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator|=(const wide_integer<MachineWords2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator^=(const Integral&) noexcept;
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator^=(const wide_integer<MachineWords2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator<<=(const Integral&);
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator<<=(const wide_integer<MachineWords2, S2>&);

    template<typename Integral>
    constexpr wide_integer<MachineWords, S>& operator>>=(const Integral&) noexcept;
    template<size_t MachineWords2, signedness S2>
    constexpr wide_integer<MachineWords, S>& operator>>=(const wide_integer<MachineWords2, S2>&) noexcept;

    constexpr wide_integer<MachineWords, S>& operator++() noexcept(S == signedness::Unsigned);
    constexpr wide_integer<MachineWords, S> operator++(int) noexcept(S == signedness::Unsigned);
    constexpr wide_integer<MachineWords, S>& operator--() noexcept(S == signedness::Unsigned);
    constexpr wide_integer<MachineWords, S> operator--(int) noexcept(S == signedness::Unsigned);

    // 26.??.2.?? observers:
    template <typename Arithmetic> constexpr operator Arithmetic() const noexcept;
    constexpr explicit operator bool() const noexcept;

  private:
    uint32_t data[MachineWords / sizeof(uint32_t) + 1]; // exposition only
  };
}

The class template wide_integer<size_t MachineWords, signedness S> is a POD class that behaves as an integer type of a compile time specified width. Template parameter MachineWords specifies exact machine words count to store the integer value. sizeof(wide_integer<MachineWords, signedness::Unsigned>) and sizeof(wide_integer<MachineWords, signedness::Signed>) are required to be equal to MachineWords * sizeof(machine-word). Template parameter S specifies signedness of the stored integer value.

[Note to SG6: Definition for machine words must be provided somewhere and referenced from the above paragraph. - end note]

26.??.2.?? wide_integer constructors [numeric.wide_integer.cons]

constexpr wide_integer() noexcept = default;
Effects: A Constructs an object with undefined value.
template<typename Arithmetic> constexpr wide_integer(const Arithmetic& other) noexcept;
Effects: Constructs an object from other using the integral conversion rules [conv.integral].
template<size_t MachineWords2, signedness S2> constexpr wide_integer(const wide_integer<MachineWords2, S2>& other) noexcept;
Effects: Constructs an object from other using the integral conversion rules [conv.integral].

26.??.2.?? wide_integer assignments [numeric.wide_integer.assign]

template<typename Arithmetic>
constexpr wide_integer<MachineWords, S>& operator=(const Arithmetic& other) noexcept;
Effects: Constructs an object from other using the integral conversion rules [conv.integral].
template<size_t MachineWords2, signedness S2>
constexpr wide_integer<MachineWords, S>& operator=(const wide_integer<MachineWords2, S2>& other) noexcept;
Effects: Constructs an object from other using the integral conversion rules [conv.integral].

26.??.2.?? wide_integer compound assignments [numeric.wide_integer.cassign]

template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator*=(const wide_integer<MachineWords2, S2>&);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator/=(const wide_integer<MachineWords2, S2>&);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator+=(const wide_integer<MachineWords2, S2>&) noexcept(S == signedness::Unsigned);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator-=(const wide_integer<MachineWords2, S2>&) noexcept(S == signedness::Unsigned);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator%=(const wide_integer<MachineWords2, S2>&);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator&=(const wide_integer<MachineWords2, S2>&) noexcept;
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator|=(const wide_integer<MachineWords2, S2>&) noexcept;
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator^=(const wide_integer<MachineWords2, S2>&) noexcept;
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator<<=(const wide_integer<MachineWords2, S2>&);
template<size_t MachineWords2, signedness S2> constexpr wide_integer<MachineWords, S>& operator>>=(const wide_integer<MachineWords2, S2>&) noexcept;
constexpr wide_integer<MachineWords, S>& operator++() noexcept(S == signedness::Unsigned);
constexpr wide_integer<MachineWords, S> operator++(int) noexcept(S == signedness::Unsigned);
constexpr wide_integer<MachineWords, S>& operator--() noexcept(S == signedness::Unsigned);
constexpr wide_integer<MachineWords, S> operator--(int) noexcept(S == signedness::Unsigned);
Effects: Behavior of the above operators is similar to operators for built-in integral types.
template<typename Arithmetic> constexpr wide_integer<MachineWords, S>& operator*=(const Arithmetic&);
template<typename Arithmetic> constexpr wide_integer<MachineWords, S>& operator/=(const Arithmetic&);
template<typename Arithmetic> constexpr wide_integer<MachineWords, S>& operator+=(const Arithmetic&) noexcept(S == signedness::Unsigned);
template<typename Arithmetic> constexpr wide_integer<MachineWords, S>& operator-=(const Arithmetic&) noexcept(S == signedness::Unsigned);
Effects: As if an object wi of type wide_integer<MachineWords, S> was created from input value and the corresponding operator was called for *this and the wi.
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator%=(const Integral&);
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator&=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator|=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator^=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator<<=(const Integral&);
template<typename Integral> constexpr wide_integer<MachineWords, S>& operator>>=(const Integral&) noexcept;
Effects: As if an object wi of type wide_integer<MachineWords, S> was created from input value and the corresponding operator was called for *this and the wi.

26.??.2.?? wide_integer observers [numeric.wide_integer.observers]

template <typename Arithmetic> constexpr operator Arithmetic() const noexcept;
Returns: If Arithmetic type is an integral type then it is constructed from *this using the integral conversion rules [conv.integral]. Otherwise Arithmetic is constructed from *this using the floating-integral conversion rules [conv.fpint].
constexpr explicit operator bool() const noexcept;
Returns: true if *this is not equal to 0.

26.??.?? Specializations of common_type [numeric.wide_integer.traits.specializations]

template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
struct common_type<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>> {
  using type = wide_integer<max(MachineWords, MachineWords2), see below>;
};

The signed template parameter indicated by this specialization is following:

[Note: common_type follows the usual arithmetic conversions design. - end note]

[Note to SG6: common_type attempts to follow the usual arithmetic conversions design here for interoperability between different numeric types. Following two specializations must be moved to a more generic place and enriched with usual arithmetic conversion rules for all the other numeric classes that specialize std::numeric_limits- end note]

template<size_t MachineWords, signedness S, typename Arithmetic>
struct common_type<wide_integer<MachineWords, S>, Arithmetic> {
  using type = see below;
};
template<typename Arithmetic, size_t MachineWords, signedness S>
struct common_type<Arithmetic, wide_integer<MachineWords, S>>
  : common_type<wide_integer<MachineWords, S>, Arithmetic>;

The member typedef type is following:

26.??.?? Unary operators [numeric.wide_integer.unary_ops]

template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator~(const wide_integer<MachineWords, S>& val) noexcept;
Returns: value with inverted significant bits of val.
template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator-(const wide_integer<MachineWords, S>& val) noexcept(S == signedness::Unsigned);
Returns: val *= -1 if S is true, otherwise the result is unspecified.
template<size_t MachineWords, signedness S> constexpr wide_integer<MachineWords, S> operator+(const wide_integer<MachineWords, S>& val) noexcept(S == signedness::Unsigned);
Returns: val.

26.??.?? Binary operators [numeric.wide_integer.binary_ops]

In the function descriptions that follow, CT represents common_type_t<A, B>, where A and B are the types of the two arguments to the function.

template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator*(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);
Returns: CT(lhs) *= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator/(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);
Returns: CT(lhs) /= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator+(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept(S == signedness::Unsigned);
Returns: CT(lhs) += rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator-(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept(S == signedness::Unsigned);
Returns: CT(lhs) -= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator%(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs);
Returns: CT(lhs) %= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator&(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: CT(lhs) &= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr operator|(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: CT(lhs) |= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  common_type_t<wide_integer<MachineWords, S>, wide_integer<MachineWords2, S2>>
  constexpr  operator^(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: CT(lhs) ^= rhs.
template<size_t MachineWords, signedness S>
  common_type_t<wide_integer<MachineWords, S>, size_t>
  constexpr operator<<(const wide_integer<MachineWords, S>& lhs, size_t rhs);
Returns: CT(lhs) <<= rhs.
template<size_t MachineWords, signedness S>
  common_type_t<wide_integer<MachineWords, S>, size_t>
  constexpr operator>>(const wide_integer<MachineWords, S>& lhs, size_t rhs) noexcept;
Returns: CT(lhs) >>= rhs.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator<(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: true if value of CT(lhs) is less than the value of CT(rhs).
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator>(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: true if value of CT(lhs) is greater than the value of CT(rhs).
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator<=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: true if value of CT(lhs) is equal or less than the value of CT(rhs).
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator>=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: true if value of CT(lhs) is equal or greater than the value of CT(rhs).
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator==(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: true if significant bits of CT(lhs) and CT(rhs) are the same.
template<size_t MachineWords, signedness S, size_t MachineWords2, signedness S2>
  constexpr bool operator!=(const wide_integer<MachineWords, S>& lhs, const wide_integer<MachineWords2, S2>& rhs) noexcept;
Returns: !(CT(lhs) == CT(rhs)).

26.??.?? Numeric conversions [numeric.wide_integer.conversions]

template<size_t MachineWords, signedness S> std::string to_string(const wide_integer<MachineWords, S>& val);
template<size_t MachineWords, signedness S> std::wstring to_wstring(const wide_integer<MachineWords, S>& val);
Returns: Each function returns an object holding the character representation of the value of its argument. All the significant bits of the argument are outputed as a signed decimal in the style [-]dddd.
template <size_t MachineWords, signedness S>
to_chars_result to_chars(char* first,
                       char* last,
                       const wide_integer<MachineWords, S>& value,
                       int base = 10);

Behavior of wide_integer overload is subject to the usual rules of primitive numeric output conversion functions [utility.to.chars].

template <size_t MachineWords, signedness S>
from_chars_result from_chars(const char* first,
                           const char* last,
                           wide_integer<MachineWords, S>& value,
                           int base = 10);

Behavior of wide_integer overload is subject to the usual rules of primitive numeric input conversion functions [utility.from.chars].

26.??.?? iostream specializations [numeric.wide_integer.io]

template<class Char, class Traits, size_t MachineWords, signedness S>
basic_ostream<Char, Traits>& operator<<(basic_ostream<Char, Traits>& os, const wide_integer<MachineWords, S>& val);
Effects: As if by: os << to_string(val).
Returns: os.
template<class Char, class Traits, size_t MachineWords, signedness S>
basic_istream<Char, Traits>& operator>>(basic_istream<Char, Traits>& is, wide_integer<MachineWords, S>& val);
Effects: Extracts a wide_integer that is represented as a decimal number in the is. If bad input is encountered, calls is.setstate(ios_base::failbit) (which may throw ios::failure ([iostate.flags])).
Returns: is.

26.??.?? Hash support [numeric.wide_integer.hash]

template<size_t MachineWords, signedness S> struct hash<wide_integer<MachineWords, S>>;

The specialization is enabled (20.14.14). If there is a built-in integral type Integral that has the same signedness and width as wide_integer<MachineWords, S>, and wi is an object of type wide_integer<MachineWords, S>, then hash<wide_integer<MachineWords, S>>()(wi) == hash<Integral>()(Integral(wi)).

V. Feature-testing macro

For the purposes of SG10we recommend the feature-testing macro name __cpp_lib_wide_integer.