21 Metaprogramming library [meta]

21.4 Compile-time rational arithmetic [ratio]

21.4.1 In general [ratio.general]

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Subclause [ratio] 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.
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Throughout subclause [ratio], 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.

21.4.2 Header <ratio> synopsis [ratio.syn]

// all freestanding 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> constexpr bool ratio_equal_v = ratio_equal<R1, R2>::value; template<class R1, class R2> constexpr bool ratio_not_equal_v = ratio_not_equal<R1, R2>::value; template<class R1, class R2> constexpr bool ratio_less_v = ratio_less<R1, R2>::value; template<class R1, class R2> constexpr bool ratio_less_equal_v = ratio_less_equal<R1, R2>::value; template<class R1, class R2> constexpr bool ratio_greater_v = ratio_greater<R1, R2>::value; template<class R1, class R2> constexpr bool ratio_greater_equal_v = ratio_greater_equal<R1, R2>::value; // [ratio.si], convenience SI typedefs using quecto = ratio<1, 1'000'000'000'000'000'000'000'000'000'000>; // see below using ronto = ratio<1, 1'000'000'000'000'000'000'000'000'000>; // see below 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 using ronna = ratio< 1'000'000'000'000'000'000'000'000'000, 1>; // see below using quetta = ratio<1'000'000'000'000'000'000'000'000'000'000, 1>; // see below }

21.4.3 Class template ratio [ratio.ratio]

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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>; }; }
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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 1: 
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.
This excludes the most negative value.
— end note]
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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:
  • (2.1)
    num shall have the value sgn(N) * sgn(D) * abs(N) / gcd.
  • (2.2)
    den shall have the value abs(D) / gcd.

21.4.4 Arithmetic on ratios [ratio.arithmetic]

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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 56, 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.
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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 56: Expressions used to perform ratio arithmetic [tab: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
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[Example 1: 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]

21.4.5 Comparison of ratios [ratio.comparison]

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template<class R1, class R2> struct ratio_equal : bool_constant<R1::num == R2::num && R1::den == R2::den> { };
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template<class R1, class R2> struct ratio_not_equal : bool_constant<!ratio_equal_v<R1, R2>> { };
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template<class R1, class R2> struct ratio_less : bool_constant<see below> { };
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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.
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template<class R1, class R2> struct ratio_less_equal : bool_constant<!ratio_less_v<R2, R1>> { };
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template<class R1, class R2> struct ratio_greater : bool_constant<ratio_less_v<R2, R1>> { };
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template<class R1, class R2> struct ratio_greater_equal : bool_constant<!ratio_less_v<R1, R2>> { };

21.4.6 SI types for ratio [ratio.si]

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