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

// Copyright (C) 2008, 2009, 2010, 2011, 2012 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

// .

/** @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 

#else

#include 

#include 

#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

    struct __static_sign

    : integral_constant

    { };

  template

    struct __static_abs

    : integral_constant::value>

    { };

  template

    struct __static_gcd

    : __static_gcd<_Qn, (_Pn % _Qn)>

    { };

  template

    struct __static_gcd<_Pn, 0>

    : integral_constant::value>

    { };

  template

    struct __static_gcd<0, _Qn>

    : integral_constant::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

    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

    struct __big_less

    : integral_constant

                               || (__hi1 == __hi2 && __lo1 < __lo2))>

    { };

  template

    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

    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

    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

    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

    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

    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 type;

    };

  template

    constexpr intmax_t ratio<_Num, _Den>::num;

  template

    constexpr intmax_t ratio<_Num, _Den>::den;

  template

    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

    constexpr intmax_t __ratio_multiply<_R1, _R2>::num;

  template

    constexpr intmax_t __ratio_multiply<_R1, _R2>::den;

  /// ratio_multiply

  template

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

  template

    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

    constexpr intmax_t __ratio_divide<_R1, _R2>::num;

  template

    constexpr intmax_t __ratio_divide<_R1, _R2>::den;

  /// ratio_divide

  template

    using ratio_divide = typename __ratio_divide<_R1, _R2>::type;

  /// ratio_equal

  template

    struct ratio_equal

    : integral_constant

    { };

  /// ratio_not_equal

  template

    struct ratio_not_equal

    : integral_constant::value>

    { };

  // Both numbers are positive.

  template

           typename _Left = __big_mul<_R1::num,_R2::den>,

           typename _Right = __big_mul<_R2::num,_R1::den> >

    struct __ratio_less_impl_1

    : integral_constant

           _Right::__hi, _Right::__lo>::value>

    { };

  template

           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

    struct __ratio_less_impl<_R1, _R2, true, false>

    : integral_constant

    { };

  template

    struct __ratio_less_impl<_R1, _R2, false, true>

    : __ratio_less_impl_1,

           ratio<-_R1::num, _R1::den> >::type

    { };

  /// ratio_less

  template

    struct ratio_less

    : __ratio_less_impl<_R1, _R2>::type

    { };

  /// ratio_less_equal

  template

    struct ratio_less_equal

    : integral_constant::value>

    { };

  /// ratio_greater

  template

    struct ratio_greater

    : integral_constant::value>

    { };

  /// ratio_greater_equal

  template

    struct ratio_greater_equal

    : integral_constant::value>

    { };

  template

      bool = (_R1::num >= 0),

      bool = (_R2::num >= 0),

      bool = ratio_less::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

    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

    struct __ratio_add_impl<_R1, _R2, false, true, true>

    : __ratio_add_impl<_R2, _R1>

    { };

  // True subtraction of nonnegative numbers yielding a nonnegative result.

  template

    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

    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

    constexpr intmax_t __ratio_add<_R1, _R2>::num;

  template

    constexpr intmax_t __ratio_add<_R1, _R2>::den;

  /// ratio_add

  template

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

  template

    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

    constexpr intmax_t __ratio_subtract<_R1, _R2>::num;

  template

    constexpr intmax_t __ratio_subtract<_R1, _R2>::den;

  /// ratio_subtract

  template

    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