| // ratio -*- C++ -*- |
| |
| // Copyright (C) 2008-2020 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 |