23 General utilities library [utilities]

23.16 Compile-time rational arithmetic [ratio]

23.16.1 In general [ratio.general]

This subclause describes the ratio library.
It provides a class template ratio which exactly represents any finite rational number with a numerator and denominator representable by compile-time constants of type intmax_­t.
Throughout this subclause, the names of template parameters are used to express type requirements.
If a template parameter is named R1 or R2, and the template argument is not a specialization of the ratio template, the program is ill-formed.

23.16.2 Header <ratio> synopsis [ratio.syn]

namespace std {
  // [ratio.ratio], class template ratio
  template <intmax_t N, intmax_t D = 1> class ratio;

  // [ratio.arithmetic], ratio arithmetic
  template <class R1, class R2> using ratio_add = see below;
  template <class R1, class R2> using ratio_subtract = see below;
  template <class R1, class R2> using ratio_multiply = see below;
  template <class R1, class R2> using ratio_divide = see below;

  // [ratio.comparison], ratio comparison
  template <class R1, class R2> struct ratio_equal;
  template <class R1, class R2> struct ratio_not_equal;
  template <class R1, class R2> struct ratio_less;
  template <class R1, class R2> struct ratio_less_equal;
  template <class R1, class R2> struct ratio_greater;
  template <class R1, class R2> struct ratio_greater_equal;

  template <class R1, class R2>
    inline constexpr bool ratio_equal_v = ratio_equal<R1, R2>::value;
  template <class R1, class R2>
    inline constexpr bool ratio_not_equal_v = ratio_not_equal<R1, R2>::value;
  template <class R1, class R2>
    inline constexpr bool ratio_less_v = ratio_less<R1, R2>::value;
  template <class R1, class R2>
    inline constexpr bool ratio_less_equal_v = ratio_less_equal<R1, R2>::value;
  template <class R1, class R2>
    inline constexpr bool ratio_greater_v = ratio_greater<R1, R2>::value;
  template <class R1, class R2>
    inline constexpr bool ratio_greater_equal_v = ratio_greater_equal<R1, R2>::value;

  // [ratio.si], convenience SI typedefs
  using yocto = ratio<1, 1'000'000'000'000'000'000'000'000>;  // see below
  using zepto = ratio<1,     1'000'000'000'000'000'000'000>;  // see below
  using atto  = ratio<1,         1'000'000'000'000'000'000>;
  using femto = ratio<1,             1'000'000'000'000'000>;
  using pico  = ratio<1,                 1'000'000'000'000>;
  using nano  = ratio<1,                     1'000'000'000>;
  using micro = ratio<1,                         1'000'000>;
  using milli = ratio<1,                             1'000>;
  using centi = ratio<1,                               100>;
  using deci  = ratio<1,                                10>;
  using deca  = ratio<                               10, 1>;
  using hecto = ratio<                              100, 1>;
  using kilo  = ratio<                            1'000, 1>;
  using mega  = ratio<                        1'000'000, 1>;
  using giga  = ratio<                    1'000'000'000, 1>;
  using tera  = ratio<                1'000'000'000'000, 1>;
  using peta  = ratio<            1'000'000'000'000'000, 1>;
  using exa   = ratio<        1'000'000'000'000'000'000, 1>;
  using zetta = ratio<    1'000'000'000'000'000'000'000, 1>;  // see below
  using yotta = ratio<1'000'000'000'000'000'000'000'000, 1>;  // see below
}

23.16.3 Class template ratio [ratio.ratio]

namespace std {
  template <intmax_t N, intmax_t D = 1> class ratio {
  public:
    static constexpr intmax_t num;
    static constexpr intmax_t den;
    using type = ratio<num, den>;
  };
}
If the template argument D is zero or the absolute values of either of the template arguments N and D is not representable by type intmax_­t, the program is ill-formed.
[Note
:
These rules ensure that infinite ratios are avoided and that for any negative input, there exists a representable value of its absolute value which is positive.
In a two's complement representation, this excludes the most negative value.
end note
]
The static data members num and den shall have the following values, where gcd represents the greatest common divisor of the absolute values of N and D:
  • num shall have the value sign(N) * sign(D) * abs(N) / gcd.
  • den shall have the value abs(D) / gcd.

23.16.4 Arithmetic on ratios [ratio.arithmetic]

Each of the alias templates ratio_­add, ratio_­subtract, ratio_­multiply, and ratio_­divide denotes the result of an arithmetic computation on two ratios R1 and R2.
With X and Y computed (in the absence of arithmetic overflow) as specified by Table 51, each alias denotes a ratio<U, V> such that U is the same as ratio<X, Y>​::​num and V is the same as ratio<X, Y>​::​den.
If it is not possible to represent U or V with intmax_­t, the program is ill-formed.
Otherwise, an implementation should yield correct values of U and V.
If it is not possible to represent X or Y with intmax_­t, the program is ill-formed unless the implementation yields correct values of U and V.
Table 51 — Expressions used to perform ratio arithmetic
Type
Value of X
Value of Y
ratio_­add<R1, R2>
R1​::​num * R2​::​den +
R1​::​den * R2​::​den
R2​::​num * R1​::​den
ratio_­subtract<R1, R2>
R1​::​num * R2​::​den -
R1​::​den * R2​::​den
R2​::​num * R1​::​den
ratio_­multiply<R1, R2>
R1​::​num * R2​::​num
R1​::​den * R2​::​den
ratio_­divide<R1, R2>
R1​::​num * R2​::​den
R1​::​den * R2​::​num
[Example
:
static_assert(ratio_add<ratio<1, 3>, ratio<1, 6>>::num == 1, "1/3+1/6 == 1/2");
static_assert(ratio_add<ratio<1, 3>, ratio<1, 6>>::den == 2, "1/3+1/6 == 1/2");
static_assert(ratio_multiply<ratio<1, 3>, ratio<3, 2>>::num == 1, "1/3*3/2 == 1/2");
static_assert(ratio_multiply<ratio<1, 3>, ratio<3, 2>>::den == 2, "1/3*3/2 == 1/2");

// The following cases may cause the program to be ill-formed under some implementations
static_assert(ratio_add<ratio<1, INT_MAX>, ratio<1, INT_MAX>>::num == 2,
  "1/MAX+1/MAX == 2/MAX");
static_assert(ratio_add<ratio<1, INT_MAX>, ratio<1, INT_MAX>>::den == INT_MAX,
  "1/MAX+1/MAX == 2/MAX");
static_assert(ratio_multiply<ratio<1, INT_MAX>, ratio<INT_MAX, 2>>::num == 1,
  "1/MAX * MAX/2 == 1/2");
static_assert(ratio_multiply<ratio<1, INT_MAX>, ratio<INT_MAX, 2>>::den == 2,
  "1/MAX * MAX/2 == 1/2");
end example
]

23.16.5 Comparison of ratios [ratio.comparison]

template <class R1, class R2> struct ratio_equal : bool_constant<R1::num == R2::num && R1::den == R2::den> { }; template <class R1, class R2> struct ratio_not_equal : bool_constant<!ratio_equal_v<R1, R2>> { }; template <class R1, class R2> struct ratio_less : bool_constant<see below> { };
If R1​::​num × R2​::​den is less than R2​::​num × R1​::​den, ratio_­less<R1, R2> shall be derived from bool_­constant<true>; otherwise it shall be derived from bool_­constant<false>.
Implementations may use other algorithms to compute this relationship to avoid overflow.
If overflow occurs, the program is ill-formed.
template <class R1, class R2> struct ratio_less_equal : bool_constant<!ratio_less_v<R2, R1>> { }; template <class R1, class R2> struct ratio_greater : bool_constant<ratio_less_v<R2, R1>> { }; template <class R1, class R2> struct ratio_greater_equal : bool_constant<!ratio_less_v<R1, R2>> { };

23.16.6 SI types for ratio [ratio.si]

For each of the typedef-names yocto, zepto, zetta, and yotta, if both of the constants used in its specification are representable by intmax_­t, the typedef shall be defined; if either of the constants is not representable by intmax_­t, the typedef shall not be defined.