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Edit File : ratio \| Size : 19.42 KB
// ratio -- C++ -- // Copyright (C) 2008-2018 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. / #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> #ifdef _GLIBCXX_USE_C99_STDINT_TR1 namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION /* * @defgroup ratio Rational Arithmetic * @ingroup utilities * * Compile time representation of finite rational numbers. * @{ / 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 = a1c + a0, M = b1c + 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 // __hi2^(8sizeof(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"); }; /** * @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; 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; /// ratio_multiply template<typename _R1, typename _R2> using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type; 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; /// 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> { }; // 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 { }; /// 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 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; /// ratio_add template<typename _R1, typename _R2> using ratio_add = typename __ratio_add<_R1, _R2>::type; 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; /// 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 //_GLIBCXX_USE_C99_STDINT_TR1 #endif // C++11 #endif //_GLIBCXX_RATIO Back

Edit File : ratio | Size : 19.42 KB

// ratio -*- C++ -*-

// Copyright (C) 2008-2018 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.
 */

#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>

#ifdef _GLIBCXX_USE_C99_STDINT_TR1

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

/**
   * @defgroup ratio Rational Arithmetic
   * @ingroup utilities
   *
   * Compile time representation of finite rational numbers.
   * @{
   */

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");
    };

/**
   *  @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;

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;

/// ratio_multiply
  template<typename _R1, typename _R2>
    using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;

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;

/// 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>
    { };

// 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
    { };

/// 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

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;

/// ratio_add
  template<typename _R1, typename _R2>
    using ratio_add = typename __ratio_add<_R1, _R2>::type;

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;

/// 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 //_GLIBCXX_USE_C99_STDINT_TR1

#endif // C++11

#endif //_GLIBCXX_RATIO

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