A Proposal to add wide_int Template Class

Significant changes to P0539R2 are marked with blue.

Show deleted lines from P0275R2.

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 P0539R2:

"Bits" won in the discussion "Words vs Bytes vs Bits" in Albuquerque
Changed "MachineWords" to "Bits"
Interoperability with other types moved to a separate paper
Removed signedness scoped enum

Differences with P0539R1:

Added a discussion on "Words vs Bytes vs Bits"

Differences with P0539R0:

Reworked the proposal for simpler integration with other Numerics proposals:
- Added an interoperability section [numeric.interop]
- Arithmetic and Integral concepts were moved to [numeric.requirements] as they seem widely useful
- Binary non-member operations that accept Arithmetic or Integral parameter were changed to accept two Arithmetic or Integral parameters respectively and moved to [numeric.interop]
- int128_t and other aliases now depend on P0102R0
Renamed wide_int to wide_integer
wide_integer now uses machine words count as a template parameter, not bits count
Removed not allowed type traits specializations
Added to_chars and from_chars

III. Design and paper limitations

wide_integer is designed to be as close as possible to built-in integral types:

it does not allocate memory
it is a standard layout type
it is trivially copyable
it is trivially desctructible
it is constexpr usable

wide_integer is not a metafunction. Such metafunctions are discussed in P0102R0.

wide_integer does not add noexcept to operations that have UB for built-in types. Operations that have UB traditionally are not marked with noexcept by LWG. Attempt to change that behavior may be done in separate paper.

In this proposal we 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.

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:

Number of machine words
Number of bytes
Number of bits

Where number must be:

Power of 2
Or such as sizeof(wide_integer) is multiply of sizeof(machine word)

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:

x86 users are used to Bits due to wide usage of <cmath> aliases like uint32_t.
Bytes hide the actual bits count. When x86 user specifies 16 bytes he probably wishes to get 128 Bits, not CHAR_BITS * 16 bits.
Bytes are preferable for platforms that have CHAR_BITS % 8 != 0, because code with specified bits may not compile for such platform:
```
void foo(wide_uint<128> ); // OK on x86, does not compile if CHAR_BITS == 9
```

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

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<Bits, S>, then numeric_limits<wide_integer<Bits, 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
  };

  // 26.??.2 class template wide_integer
  template<size_t Bits, typename S> class wide_integer;

  // 26.??.?? type traits specializations
  template<size_t Bits, typename S, size_t Bits2, typename S2>
  struct common_type<wide_integer<Bits, S>, wide_integer<Bits2, S2>>;

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

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

  // 26.??.?? unary operations
  template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator~(const wide_integer<Bits, S>& val) noexcept;
  template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator-(const wide_integer<Bits, S>& val) noexcept(is_unsigned_v<S>);
  template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator+(const wide_integer<Bits, S>& val) noexcept(is_unsigned_v<S>) noexcept;

  // 26.??.?? binary operations
  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator*(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator/(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator+(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept(is_unsigned_v<S>);

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator-(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept(is_unsigned_v<S>);

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator%(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator&(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr operator|(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
    constexpr  operator^(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

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

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

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator<(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator>(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator<=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator>=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator==(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

  template<size_t Bits, typename S, size_t Bits2, typename S2>
    constexpr bool operator!=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

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

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

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

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


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

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

  template <size_t Bits>
  using wide_int = wide_integer<Bits, signed>;

  template <size_t Bits>
  using wide_uint = wide_integer<Bits, unsigned>

[Note to SG6: P0102R0 proposal with exact_2int, fast_2int, least_2int, exact_2uint, fast_2uint, least_2uint requires additional 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 wide_int<128> operator "" _int128(const char*);
  constexpr wide_int<256> operator "" _int256(const char*);
  constexpr wide_int<512> operator "" _int512(const char*);
  constexpr wide_uint<128> operator "" _uint128(const char*);
  constexpr wide_uint<256> operator "" _uint256(const char*);
  constexpr wide_uint<512> 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:
    using int128_t = wide_int<128>;
    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 Bits, typename S>
  class wide_integer {
  public:
    // 26.??.2.?? construct:
    constexpr wide_integer() noexcept = default;
    constexpr wide_integer(const wide_integer<Bits, S>& ) noexcept = default;
    template<typename Arithmetic> constexpr wide_integer(const Arithmetic& other) noexcept;
    template<size_t Bits2, typename S2> constexpr wide_integer(const wide_integer<Bits2, S2>& other) noexcept;

    // 26.??.2.?? assignment:
    constexpr wide_integer<Bits, S>& operator=(const wide_integer<Bits, S>& ) noexcept = default;
    template<typename Arithmetic>
    constexpr wide_integer<Bits, S>& operator=(const Arithmetic& other) noexcept;
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator=(const wide_integer<Bits2, S2>& other) noexcept;

    // 26.??.2.?? compound assignment:
    template<typename Arithmetic>
    constexpr wide_integer<Bits, S>& operator*=(const Arithmetic&);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator*=(const wide_integer<Bits2, S2>&);

    template<typename Arithmetic>
    constexpr wide_integer<Bits, S>& operator/=(const Arithmetic&);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator/=(const wide_integer<Bits2, S2>&);

    template<typename Arithmetic>
    constexpr wide_integer<Bits, S>& operator+=(const Arithmetic&) noexcept(is_unsigned_v<S>);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator+=(const wide_integer<Bits2, S2>&) noexcept(is_unsigned_v<S>);

    template<typename Arithmetic>
    constexpr wide_integer<Bits, S>& operator-=(const Arithmetic&) noexcept(is_unsigned_v<S>);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator-=(const wide_integer<Bits2, S2>&) noexcept(is_unsigned_v<S>);

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator%=(const Integral&);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator%=(const wide_integer<Bits2, S2>&);

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator&=(const Integral&) noexcept;
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator&=(const wide_integer<Bits2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator|=(const Integral&) noexcept;
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator|=(const wide_integer<Bits2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator^=(const Integral&) noexcept;
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator^=(const wide_integer<Bits2, S2>&) noexcept;

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator<<=(const Integral&);
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator<<=(const wide_integer<Bits2, S2>&);

    template<typename Integral>
    constexpr wide_integer<Bits, S>& operator>>=(const Integral&) noexcept;
    template<size_t Bits2, typename S2>
    constexpr wide_integer<Bits, S>& operator>>=(const wide_integer<Bits2, S2>&) noexcept;

    constexpr wide_integer<Bits, S>& operator++() noexcept(is_unsigned_v<S>);
    constexpr wide_integer<Bits, S> operator++(int) noexcept(is_unsigned_v<S>);
    constexpr wide_integer<Bits, S>& operator--() noexcept(is_unsigned_v<S>);
    constexpr wide_integer<Bits, S> operator--(int) noexcept(is_unsigned_v<S>);

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

  private:
    byte data[Bits / CHAR_BITS]; // exposition only
  };
}

The class template wide_integer<size_t Bits, typename S> is a trivial standard layout class that behaves as an integer type of a compile time specified bitness.

Template parameter Bits specifies exact bits count to store the integer value. Bits % (sizeof(int) * CHAR_BITS) is eqaul to 0. sizeof(wide_integer<Bits, unsigned>) and sizeof(wide_integer<Bits, signed>) are required to be equal to Bits * CHAR_BITS.

Template parameter S specifies signedness of the stored integer value and is either signed or unsigned.

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 Bits2, typename S2> constexpr wide_integer(const wide_integer<Bits2, 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<Bits, S>& operator=(const Arithmetic& other) noexcept;

Effects: Constructs an object from other using the integral conversion rules [conv.integral].

template<size_t Bits2, typename S2>
constexpr wide_integer<Bits, S>& operator=(const wide_integer<Bits2, 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 Bits2, typename S2> constexpr wide_integer<Bits, S>& operator*=(const wide_integer<Bits2, S2>&);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator/=(const wide_integer<Bits2, S2>&);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator+=(const wide_integer<Bits2, S2>&) noexcept(is_unsigned_v<S>);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator-=(const wide_integer<Bits2, S2>&) noexcept(is_unsigned_v<S>);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator%=(const wide_integer<Bits2, S2>&);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator&=(const wide_integer<Bits2, S2>&) noexcept;
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator|=(const wide_integer<Bits2, S2>&) noexcept;
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator^=(const wide_integer<Bits2, S2>&) noexcept;
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator<<=(const wide_integer<Bits2, S2>&);
template<size_t Bits2, typename S2> constexpr wide_integer<Bits, S>& operator>>=(const wide_integer<Bits2, S2>&) noexcept;
constexpr wide_integer<Bits, S>& operator++() noexcept(is_unsigned_v<S>);
constexpr wide_integer<Bits, S> operator++(int) noexcept(is_unsigned_v<S>);
constexpr wide_integer<Bits, S>& operator--() noexcept(is_unsigned_v<S>);
constexpr wide_integer<Bits, S> operator--(int) noexcept(is_unsigned_v<S>);

Effects: Behavior of the above operators is similar to operators for built-in integral types.

template<typename Arithmetic> constexpr wide_integer<Bits, S>& operator*=(const Arithmetic&);
template<typename Arithmetic> constexpr wide_integer<Bits, S>& operator/=(const Arithmetic&);
template<typename Arithmetic> constexpr wide_integer<Bits, S>& operator+=(const Arithmetic&) noexcept(is_unsigned_v<S>);
template<typename Arithmetic> constexpr wide_integer<Bits, S>& operator-=(const Arithmetic&) noexcept(is_unsigned_v<S>);

Effects: As if an object wi of type wide_integer<Bits, S> was created from input value and the corresponding operator was called for *this and the wi.

template<typename Integral> constexpr wide_integer<Bits, S>& operator%=(const Integral&);
template<typename Integral> constexpr wide_integer<Bits, S>& operator&=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<Bits, S>& operator|=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<Bits, S>& operator^=(const Integral&) noexcept;
template<typename Integral> constexpr wide_integer<Bits, S>& operator<<=(const Integral&);
template<typename Integral> constexpr wide_integer<Bits, S>& operator>>=(const Integral&) noexcept;

Effects: As if an object wi of type wide_integer<Bits, 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 Bits, typename S, size_t Bits2, typename S2>
struct common_type<wide_integer<Bits, S>, wide_integer<Bits2, S2>> {
  using type = wide_integer<max(Bits, Bits2), see below>;
};

The signed template parameter indicated by this specialization is following:

(is_signed_v<S> && is_signed_v<S2> ? signed : unsigned) if Bits == Bits2
S if Bits > Bits2
S2 otherwise

[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 Bits, typename S, typename Arithmetic>
struct common_type<wide_integer<Bits, S>, Arithmetic> {
  using type = see below;
};

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

The member typedef type is following:

Arithmetic if numeric_limits<Arithmetic>::is_integer is false
wide_integer<Bits, S> if sizeof(wide_integer<Bits, S>) > sizeof(Arithmetic)
Arithmetic if sizeof(wide_integer<Bits, S>) < sizeof(Arithmetic)
Arithmetic if sizeof(wide_integer<Bits, S>) == sizeof(Arithmetic) && is_signed_v<S>
Arithmetic if sizeof(wide_integer<Bits, S>) == sizeof(Arithmetic) && numeric_limits<wide_integer<Bits, S>>::is_signed == numeric_limits<Arithmetic>::is_signed
wide_integer<Bits, S> otherwise

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

template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator~(const wide_integer<Bits, S>& val) noexcept;

Returns: value with inverted significant bits of val.

template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator-(const wide_integer<Bits, S>& val) noexcept(is_unsigned_v<S>);

Returns: val *= -1 if S is true, otherwise the result is unspecified.

template<size_t Bits, typename S> constexpr wide_integer<Bits, S> operator+(const wide_integer<Bits, S>& val) noexcept(is_unsigned_v<S>);

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 Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator*(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

Returns: CT(lhs) *= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator/(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

Returns: CT(lhs) /= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator+(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept(is_unsigned_v<S>);

Returns: CT(lhs) += rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator-(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept(is_unsigned_v<S>);

Returns: CT(lhs) -= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator%(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs);

Returns: CT(lhs) %= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator&(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: CT(lhs) &= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr operator|(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: CT(lhs) |= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  common_type_t<wide_integer<Bits, S>, wide_integer<Bits2, S2>>
  constexpr  operator^(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: CT(lhs) ^= rhs.

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

Returns: CT(lhs) <<= rhs.

template<size_t Bits, typename S>
  common_type_t<wide_integer<Bits, S>, size_t>
  constexpr operator>>(const wide_integer<Bits, S>& lhs, size_t rhs) noexcept;

Returns: CT(lhs) >>= rhs.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator<(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: true if value of CT(lhs) is less than the value of CT(rhs).

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator>(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: true if value of CT(lhs) is greater than the value of CT(rhs).

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator<=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: true if value of CT(lhs) is equal or less than the value of CT(rhs).

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator>=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: true if value of CT(lhs) is equal or greater than the value of CT(rhs).

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator==(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: true if significant bits of CT(lhs) and CT(rhs) are the same.

template<size_t Bits, typename S, size_t Bits2, typename S2>
  constexpr bool operator!=(const wide_integer<Bits, S>& lhs, const wide_integer<Bits2, S2>& rhs) noexcept;

Returns: !(CT(lhs) == CT(rhs)).

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

template<size_t Bits, typename S> std::string to_string(const wide_integer<Bits, S>& val);
template<size_t Bits, typename S> std::wstring to_wstring(const wide_integer<Bits, 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 Bits, typename S>
to_chars_result to_chars(char* first,
                       char* last,
                       const wide_integer<Bits, 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 Bits, typename S>
from_chars_result from_chars(const char* first,
                           const char* last,
                           wide_integer<Bits, 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 Bits, typename S>
basic_ostream<Char, Traits>& operator<<(basic_ostream<Char, Traits>& os, const wide_integer<Bits, S>& val);

Effects: As if by: os << to_string(val).

Returns: os.

template<class Char, class Traits, size_t Bits, typename S>
basic_istream<Char, Traits>& operator>>(basic_istream<Char, Traits>& is, wide_integer<Bits, 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 Bits, typename S> struct hash<wide_integer<Bits, S>>;

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

V. Feature-testing macro

For the purposes of SG10 we recommend the feature-testing macro name __cpp_lib_wide_integer.