ratio
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// ratio -*- C++ -*-
// Copyright (C) 2008-2021 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library. This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
// <http://www.gnu.org/licenses/>.
/** @file include/ratio
* This is a Standard C++ Library header.
* @ingroup ratio
*/
#ifndef _GLIBCXX_RATIO
#define _GLIBCXX_RATIO 1
#pragma GCC system_header
#if __cplusplus < 201103L
# include <bits/c++0x_warning.h>
#else
#include <type_traits>
#include <cstdint> // intmax_t, uintmax_t
namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @defgroup ratio Rational Arithmetic
* @ingroup utilities
*
* Compile time representation of finite rational numbers.
* @{
*/
/// @cond undocumented
template<intmax_t _Pn>
struct __static_sign
: integral_constant<intmax_t, (_Pn < 0) ? -1 : 1>
{ };
template<intmax_t _Pn>
struct __static_abs
: integral_constant<intmax_t, _Pn * __static_sign<_Pn>::value>
{ };
template<intmax_t _Pn, intmax_t _Qn>
struct __static_gcd
: __static_gcd<_Qn, (_Pn % _Qn)>
{ };
template<intmax_t _Pn>
struct __static_gcd<_Pn, 0>
: integral_constant<intmax_t, __static_abs<_Pn>::value>
{ };
template<intmax_t _Qn>
struct __static_gcd<0, _Qn>
: integral_constant<intmax_t, __static_abs<_Qn>::value>
{ };
// Let c = 2^(half # of bits in an intmax_t)
// then we find a1, a0, b1, b0 s.t. N = a1*c + a0, M = b1*c + b0
// The multiplication of N and M becomes,
// N * M = (a1 * b1)c^2 + (a0 * b1 + b0 * a1)c + a0 * b0
// Multiplication is safe if each term and the sum of the terms
// is representable by intmax_t.
template<intmax_t _Pn, intmax_t _Qn>
struct __safe_multiply
{
private:
static const uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
static const uintmax_t __a0 = __static_abs<_Pn>::value % __c;
static const uintmax_t __a1 = __static_abs<_Pn>::value / __c;
static const uintmax_t __b0 = __static_abs<_Qn>::value % __c;
static const uintmax_t __b1 = __static_abs<_Qn>::value / __c;
static_assert(__a1 == 0 || __b1 == 0,
"overflow in multiplication");
static_assert(__a0 * __b1 + __b0 * __a1 < (__c >> 1),
"overflow in multiplication");
static_assert(__b0 * __a0 <= __INTMAX_MAX__,
"overflow in multiplication");
static_assert((__a0 * __b1 + __b0 * __a1) * __c
<= __INTMAX_MAX__ - __b0 * __a0,
"overflow in multiplication");
public:
static const intmax_t value = _Pn * _Qn;
};
// Some double-precision utilities, where numbers are represented as
// __hi*2^(8*sizeof(uintmax_t)) + __lo.
template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
struct __big_less
: integral_constant<bool, (__hi1 < __hi2
|| (__hi1 == __hi2 && __lo1 < __lo2))>
{ };
template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
struct __big_add
{
static constexpr uintmax_t __lo = __lo1 + __lo2;
static constexpr uintmax_t __hi = (__hi1 + __hi2 +
(__lo1 + __lo2 < __lo1)); // carry
};
// Subtract a number from a bigger one.
template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
struct __big_sub
{
static_assert(!__big_less<__hi1, __lo1, __hi2, __lo2>::value,
"Internal library error");
static constexpr uintmax_t __lo = __lo1 - __lo2;
static constexpr uintmax_t __hi = (__hi1 - __hi2 -
(__lo1 < __lo2)); // carry
};
// Same principle as __safe_multiply.
template<uintmax_t __x, uintmax_t __y>
struct __big_mul
{
private:
static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
static constexpr uintmax_t __x0 = __x % __c;
static constexpr uintmax_t __x1 = __x / __c;
static constexpr uintmax_t __y0 = __y % __c;
static constexpr uintmax_t __y1 = __y / __c;
static constexpr uintmax_t __x0y0 = __x0 * __y0;
static constexpr uintmax_t __x0y1 = __x0 * __y1;
static constexpr uintmax_t __x1y0 = __x1 * __y0;
static constexpr uintmax_t __x1y1 = __x1 * __y1;
static constexpr uintmax_t __mix = __x0y1 + __x1y0; // possible carry...
static constexpr uintmax_t __mix_lo = __mix * __c;
static constexpr uintmax_t __mix_hi
= __mix / __c + ((__mix < __x0y1) ? __c : 0); // ... added here
typedef __big_add<__mix_hi, __mix_lo, __x1y1, __x0y0> _Res;
public:
static constexpr uintmax_t __hi = _Res::__hi;
static constexpr uintmax_t __lo = _Res::__lo;
};
// Adapted from __udiv_qrnnd_c in longlong.h
// This version assumes that the high bit of __d is 1.
template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
struct __big_div_impl
{
private:
static_assert(__d >= (uintmax_t(1) << (sizeof(intmax_t) * 8 - 1)),
"Internal library error");
static_assert(__n1 < __d, "Internal library error");
static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
static constexpr uintmax_t __d1 = __d / __c;
static constexpr uintmax_t __d0 = __d % __c;
static constexpr uintmax_t __q1x = __n1 / __d1;
static constexpr uintmax_t __r1x = __n1 % __d1;
static constexpr uintmax_t __m = __q1x * __d0;
static constexpr uintmax_t __r1y = __r1x * __c + __n0 / __c;
static constexpr uintmax_t __r1z = __r1y + __d;
static constexpr uintmax_t __r1
= ((__r1y < __m) ? ((__r1z >= __d) && (__r1z < __m))
? (__r1z + __d) : __r1z : __r1y) - __m;
static constexpr uintmax_t __q1
= __q1x - ((__r1y < __m)
? ((__r1z >= __d) && (__r1z < __m)) ? 2 : 1 : 0);
static constexpr uintmax_t __q0x = __r1 / __d1;
static constexpr uintmax_t __r0x = __r1 % __d1;
static constexpr uintmax_t __n = __q0x * __d0;
static constexpr uintmax_t __r0y = __r0x * __c + __n0 % __c;
static constexpr uintmax_t __r0z = __r0y + __d;
static constexpr uintmax_t __r0
= ((__r0y < __n) ? ((__r0z >= __d) && (__r0z < __n))
? (__r0z + __d) : __r0z : __r0y) - __n;
static constexpr uintmax_t __q0
= __q0x - ((__r0y < __n) ? ((__r0z >= __d)
&& (__r0z < __n)) ? 2 : 1 : 0);
public:
static constexpr uintmax_t __quot = __q1 * __c + __q0;
static constexpr uintmax_t __rem = __r0;
private:
typedef __big_mul<__quot, __d> _Prod;
typedef __big_add<_Prod::__hi, _Prod::__lo, 0, __rem> _Sum;
static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
"Internal library error");
};
template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
struct __big_div
{
private:
static_assert(__d != 0, "Internal library error");
static_assert(sizeof (uintmax_t) == sizeof (unsigned long long),
"This library calls __builtin_clzll on uintmax_t, which "
"is unsafe on your platform. Please complain to "
"http://gcc.gnu.org/bugzilla/");
static constexpr int __shift = __builtin_clzll(__d);
static constexpr int __coshift_ = sizeof(uintmax_t) * 8 - __shift;
static constexpr int __coshift = (__shift != 0) ? __coshift_ : 0;
static constexpr uintmax_t __c1 = uintmax_t(1) << __shift;
static constexpr uintmax_t __c2 = uintmax_t(1) << __coshift;
static constexpr uintmax_t __new_d = __d * __c1;
static constexpr uintmax_t __new_n0 = __n0 * __c1;
static constexpr uintmax_t __n1_shifted = (__n1 % __d) * __c1;
static constexpr uintmax_t __n0_top = (__shift != 0) ? (__n0 / __c2) : 0;
static constexpr uintmax_t __new_n1 = __n1_shifted + __n0_top;
typedef __big_div_impl<__new_n1, __new_n0, __new_d> _Res;
public:
static constexpr uintmax_t __quot_hi = __n1 / __d;
static constexpr uintmax_t __quot_lo = _Res::__quot;
static constexpr uintmax_t __rem = _Res::__rem / __c1;
private:
typedef __big_mul<__quot_lo, __d> _P0;
typedef __big_mul<__quot_hi, __d> _P1;
typedef __big_add<_P0::__hi, _P0::__lo, _P1::__lo, __rem> _Sum;
// No overflow.
static_assert(_P1::__hi == 0, "Internal library error");
static_assert(_Sum::__hi >= _P0::__hi, "Internal library error");
// Matches the input data.
static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
"Internal library error");
static_assert(__rem < __d, "Internal library error");
};
/// @endcond
/**
* @brief Provides compile-time rational arithmetic.
*
* This class template represents any finite rational number with a
* numerator and denominator representable by compile-time constants of
* type intmax_t. The ratio is simplified when instantiated.
*
* For example:
* @code
* std::ratio<7,-21>::num == -1;
* std::ratio<7,-21>::den == 3;
* @endcode
*
*/
template<intmax_t _Num, intmax_t _Den = 1>
struct ratio
{
static_assert(_Den != 0, "denominator cannot be zero");
static_assert(_Num >= -__INTMAX_MAX__ && _Den >= -__INTMAX_MAX__,
"out of range");
// Note: sign(N) * abs(N) == N
static constexpr intmax_t num =
_Num * __static_sign<_Den>::value / __static_gcd<_Num, _Den>::value;
static constexpr intmax_t den =
__static_abs<_Den>::value / __static_gcd<_Num, _Den>::value;
typedef ratio<num, den> type;
};
template<intmax_t _Num, intmax_t _Den>
constexpr intmax_t ratio<_Num, _Den>::num;
template<intmax_t _Num, intmax_t _Den>
constexpr intmax_t ratio<_Num, _Den>::den;
/// @cond undocumented
template<typename _R1, typename _R2>
struct __ratio_multiply
{
private:
static const intmax_t __gcd1 =
__static_gcd<_R1::num, _R2::den>::value;
static const intmax_t __gcd2 =
__static_gcd<_R2::num, _R1::den>::value;
public:
typedef ratio<
__safe_multiply<(_R1::num / __gcd1),
(_R2::num / __gcd2)>::value,
__safe_multiply<(_R1::den / __gcd2),
(_R2::den / __gcd1)>::value> type;
static constexpr intmax_t num = type::num;
static constexpr intmax_t den = type::den;
};
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_multiply<_R1, _R2>::num;
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_multiply<_R1, _R2>::den;
/// @endcond
/// ratio_multiply
template<typename _R1, typename _R2>
using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;
/// @cond undocumented
template<typename _R1, typename _R2>
struct __ratio_divide
{
static_assert(_R2::num != 0, "division by 0");
typedef typename __ratio_multiply<
_R1,
ratio<_R2::den, _R2::num>>::type type;
static constexpr intmax_t num = type::num;
static constexpr intmax_t den = type::den;
};
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_divide<_R1, _R2>::num;
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_divide<_R1, _R2>::den;
/// @endcond
/// ratio_divide
template<typename _R1, typename _R2>
using ratio_divide = typename __ratio_divide<_R1, _R2>::type;
/// ratio_equal
template<typename _R1, typename _R2>
struct ratio_equal
: integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den>
{ };
/// ratio_not_equal
template<typename _R1, typename _R2>
struct ratio_not_equal
: integral_constant<bool, !ratio_equal<_R1, _R2>::value>
{ };
/// @cond undocumented
// Both numbers are positive.
template<typename _R1, typename _R2,
typename _Left = __big_mul<_R1::num,_R2::den>,
typename _Right = __big_mul<_R2::num,_R1::den> >
struct __ratio_less_impl_1
: integral_constant<bool, __big_less<_Left::__hi, _Left::__lo,
_Right::__hi, _Right::__lo>::value>
{ };
template<typename _R1, typename _R2,
bool = (_R1::num == 0 || _R2::num == 0
|| (__static_sign<_R1::num>::value
!= __static_sign<_R2::num>::value)),
bool = (__static_sign<_R1::num>::value == -1
&& __static_sign<_R2::num>::value == -1)>
struct __ratio_less_impl
: __ratio_less_impl_1<_R1, _R2>::type
{ };
template<typename _R1, typename _R2>
struct __ratio_less_impl<_R1, _R2, true, false>
: integral_constant<bool, _R1::num < _R2::num>
{ };
template<typename _R1, typename _R2>
struct __ratio_less_impl<_R1, _R2, false, true>
: __ratio_less_impl_1<ratio<-_R2::num, _R2::den>,
ratio<-_R1::num, _R1::den> >::type
{ };
/// @endcond
/// ratio_less
template<typename _R1, typename _R2>
struct ratio_less
: __ratio_less_impl<_R1, _R2>::type
{ };
/// ratio_less_equal
template<typename _R1, typename _R2>
struct ratio_less_equal
: integral_constant<bool, !ratio_less<_R2, _R1>::value>
{ };
/// ratio_greater
template<typename _R1, typename _R2>
struct ratio_greater
: integral_constant<bool, ratio_less<_R2, _R1>::value>
{ };
/// ratio_greater_equal
template<typename _R1, typename _R2>
struct ratio_greater_equal
: integral_constant<bool, !ratio_less<_R1, _R2>::value>
{ };
#if __cplusplus > 201402L
template <typename _R1, typename _R2>
inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value;
template <typename _R1, typename _R2>
inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value;
template <typename _R1, typename _R2>
inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value;
template <typename _R1, typename _R2>
inline constexpr bool ratio_less_equal_v =
ratio_less_equal<_R1, _R2>::value;
template <typename _R1, typename _R2>
inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value;
template <typename _R1, typename _R2>
inline constexpr bool ratio_greater_equal_v
= ratio_greater_equal<_R1, _R2>::value;
#endif // C++17
/// @cond undocumented
template<typename _R1, typename _R2,
bool = (_R1::num >= 0),
bool = (_R2::num >= 0),
bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>,
ratio<__static_abs<_R2::num>::value, _R2::den> >::value>
struct __ratio_add_impl
{
private:
typedef typename __ratio_add_impl<
ratio<-_R1::num, _R1::den>,
ratio<-_R2::num, _R2::den> >::type __t;
public:
typedef ratio<-__t::num, __t::den> type;
};
// True addition of nonnegative numbers.
template<typename _R1, typename _R2, bool __b>
struct __ratio_add_impl<_R1, _R2, true, true, __b>
{
private:
static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
static constexpr uintmax_t __d2 = _R2::den / __g;
typedef __big_mul<_R1::den, __d2> __d;
typedef __big_mul<_R1::num, _R2::den / __g> __x;
typedef __big_mul<_R2::num, _R1::den / __g> __y;
typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
static_assert(__n::__hi >= __x::__hi, "Internal library error");
typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
static_assert(__n_final::__rem == 0, "Internal library error");
static_assert(__n_final::__quot_hi == 0 &&
__n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
typedef __big_mul<_R1::den / __g2, __d2> __d_final;
static_assert(__d_final::__hi == 0 &&
__d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
public:
typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
};
template<typename _R1, typename _R2>
struct __ratio_add_impl<_R1, _R2, false, true, true>
: __ratio_add_impl<_R2, _R1>
{ };
// True subtraction of nonnegative numbers yielding a nonnegative result.
template<typename _R1, typename _R2>
struct __ratio_add_impl<_R1, _R2, true, false, false>
{
private:
static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
static constexpr uintmax_t __d2 = _R2::den / __g;
typedef __big_mul<_R1::den, __d2> __d;
typedef __big_mul<_R1::num, _R2::den / __g> __x;
typedef __big_mul<-_R2::num, _R1::den / __g> __y;
typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
static_assert(__n_final::__rem == 0, "Internal library error");
static_assert(__n_final::__quot_hi == 0 &&
__n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
typedef __big_mul<_R1::den / __g2, __d2> __d_final;
static_assert(__d_final::__hi == 0 &&
__d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
public:
typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
};
template<typename _R1, typename _R2>
struct __ratio_add
{
typedef typename __ratio_add_impl<_R1, _R2>::type type;
static constexpr intmax_t num = type::num;
static constexpr intmax_t den = type::den;
};
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_add<_R1, _R2>::num;
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_add<_R1, _R2>::den;
/// @endcond
/// ratio_add
template<typename _R1, typename _R2>
using ratio_add = typename __ratio_add<_R1, _R2>::type;
/// @cond undocumented
template<typename _R1, typename _R2>
struct __ratio_subtract
{
typedef typename __ratio_add<
_R1,
ratio<-_R2::num, _R2::den>>::type type;
static constexpr intmax_t num = type::num;
static constexpr intmax_t den = type::den;
};
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_subtract<_R1, _R2>::num;
template<typename _R1, typename _R2>
constexpr intmax_t __ratio_subtract<_R1, _R2>::den;
/// @endcond
/// ratio_subtract
template<typename _R1, typename _R2>
using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type;
typedef ratio<1, 1000000000000000000> atto;
typedef ratio<1, 1000000000000000> femto;
typedef ratio<1, 1000000000000> pico;
typedef ratio<1, 1000000000> nano;
typedef ratio<1, 1000000> micro;
typedef ratio<1, 1000> milli;
typedef ratio<1, 100> centi;
typedef ratio<1, 10> deci;
typedef ratio< 10, 1> deca;
typedef ratio< 100, 1> hecto;
typedef ratio< 1000, 1> kilo;
typedef ratio< 1000000, 1> mega;
typedef ratio< 1000000000, 1> giga;
typedef ratio< 1000000000000, 1> tera;
typedef ratio< 1000000000000000, 1> peta;
typedef ratio< 1000000000000000000, 1> exa;
/// @} group ratio
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace
#endif // C++11
#endif //_GLIBCXX_RATIO