<h2>Editing File: /usr/include/c++/8/ratio</h2><form method='post'><textarea name='content' class='file-edit'>// 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
// &lt;http://www.gnu.org/licenses/&gt;.

/** @file include/ratio
 *  This is a Standard C++ Library header.
 */

#ifndef _GLIBCXX_RATIO
#define _GLIBCXX_RATIO 1

#pragma GCC system_header

#if __cplusplus &lt; 201103L
# include &lt;bits/c++0x_warning.h&gt;
#else

#include &lt;type_traits&gt;
#include &lt;cstdint&gt;

#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&lt;intmax_t _Pn&gt;
    struct __static_sign
    : integral_constant&lt;intmax_t, (_Pn &lt; 0) ? -1 : 1&gt;
    { };

  template&lt;intmax_t _Pn&gt;
    struct __static_abs
    : integral_constant&lt;intmax_t, _Pn * __static_sign&lt;_Pn&gt;::value&gt;
    { };

  template&lt;intmax_t _Pn, intmax_t _Qn&gt;
    struct __static_gcd
    : __static_gcd&lt;_Qn, (_Pn % _Qn)&gt;
    { };

  template&lt;intmax_t _Pn&gt;
    struct __static_gcd&lt;_Pn, 0&gt;
    : integral_constant&lt;intmax_t, __static_abs&lt;_Pn&gt;::value&gt;
    { };

  template&lt;intmax_t _Qn&gt;
    struct __static_gcd&lt;0, _Qn&gt;
    : integral_constant&lt;intmax_t, __static_abs&lt;_Qn&gt;::value&gt;
    { };

  // 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&lt;intmax_t _Pn, intmax_t _Qn&gt;
    struct __safe_multiply
    {
    private:
      static const uintmax_t __c = uintmax_t(1) &lt;&lt; (sizeof(intmax_t) * 4);

      static const uintmax_t __a0 = __static_abs&lt;_Pn&gt;::value % __c;
      static const uintmax_t __a1 = __static_abs&lt;_Pn&gt;::value / __c;
      static const uintmax_t __b0 = __static_abs&lt;_Qn&gt;::value % __c;
      static const uintmax_t __b1 = __static_abs&lt;_Qn&gt;::value / __c;

      static_assert(__a1 == 0 || __b1 == 0,
		    &quot;overflow in multiplication&quot;);
      static_assert(__a0 * __b1 + __b0 * __a1 &lt; (__c &gt;&gt; 1),
		    &quot;overflow in multiplication&quot;);
      static_assert(__b0 * __a0 &lt;= __INTMAX_MAX__,
		    &quot;overflow in multiplication&quot;);
      static_assert((__a0 * __b1 + __b0 * __a1) * __c
		    &lt;= __INTMAX_MAX__ -  __b0 * __a0,
		    &quot;overflow in multiplication&quot;);

    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&lt;uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2&gt;
    struct __big_less
    : integral_constant&lt;bool, (__hi1 &lt; __hi2
			       || (__hi1 == __hi2 &amp;&amp; __lo1 &lt; __lo2))&gt;
    { };

  template&lt;uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2&gt;
    struct __big_add
    {
      static constexpr uintmax_t __lo = __lo1 + __lo2;
      static constexpr uintmax_t __hi = (__hi1 + __hi2 +
					 (__lo1 + __lo2 &lt; __lo1)); // carry
    };

  // Subtract a number from a bigger one.
  template&lt;uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2&gt;
    struct __big_sub
    {
      static_assert(!__big_less&lt;__hi1, __lo1, __hi2, __lo2&gt;::value,
		    &quot;Internal library error&quot;);
      static constexpr uintmax_t __lo = __lo1 - __lo2;
      static constexpr uintmax_t __hi = (__hi1 - __hi2 -
					 (__lo1 &lt; __lo2)); // carry
    };

  // Same principle as __safe_multiply.
  template&lt;uintmax_t __x, uintmax_t __y&gt;
    struct __big_mul
    {
    private:
      static constexpr uintmax_t __c = uintmax_t(1) &lt;&lt; (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 &lt; __x0y1) ? __c : 0); // ... added here
      typedef __big_add&lt;__mix_hi, __mix_lo, __x1y1, __x0y0&gt; _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&lt;uintmax_t __n1, uintmax_t __n0, uintmax_t __d&gt;
    struct __big_div_impl
    {
    private:
      static_assert(__d &gt;= (uintmax_t(1) &lt;&lt; (sizeof(intmax_t) * 8 - 1)),
		    &quot;Internal library error&quot;);
      static_assert(__n1 &lt; __d, &quot;Internal library error&quot;);
      static constexpr uintmax_t __c = uintmax_t(1) &lt;&lt; (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 &lt; __m) ? ((__r1z &gt;= __d) &amp;&amp; (__r1z &lt; __m))
	 ? (__r1z + __d) : __r1z : __r1y) - __m;
      static constexpr uintmax_t __q1
      = __q1x - ((__r1y &lt; __m)
		 ? ((__r1z &gt;= __d) &amp;&amp; (__r1z &lt; __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 &lt; __n) ? ((__r0z &gt;= __d) &amp;&amp; (__r0z &lt; __n))
	 ? (__r0z + __d) : __r0z : __r0y) - __n;
      static constexpr uintmax_t __q0
      = __q0x - ((__r0y &lt; __n) ? ((__r0z &gt;= __d)
				  &amp;&amp; (__r0z &lt; __n)) ? 2 : 1 : 0);

    public:
      static constexpr uintmax_t __quot = __q1 * __c + __q0;
      static constexpr uintmax_t __rem = __r0;

    private:
      typedef __big_mul&lt;__quot, __d&gt; _Prod;
      typedef __big_add&lt;_Prod::__hi, _Prod::__lo, 0, __rem&gt; _Sum;
      static_assert(_Sum::__hi == __n1 &amp;&amp; _Sum::__lo == __n0,
		    &quot;Internal library error&quot;);
  };

  template&lt;uintmax_t __n1, uintmax_t __n0, uintmax_t __d&gt;
    struct __big_div
    {
    private:
      static_assert(__d != 0, &quot;Internal library error&quot;);
      static_assert(sizeof (uintmax_t) == sizeof (unsigned long long),
		    &quot;This library calls __builtin_clzll on uintmax_t, which &quot;
		    &quot;is unsafe on your platform. Please complain to &quot;
		    &quot;http://gcc.gnu.org/bugzilla/&quot;);
      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) &lt;&lt; __shift;
      static constexpr uintmax_t __c2 = uintmax_t(1) &lt;&lt; __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&lt;__new_n1, __new_n0, __new_d&gt; _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&lt;__quot_lo, __d&gt; _P0;
      typedef __big_mul&lt;__quot_hi, __d&gt; _P1;
      typedef __big_add&lt;_P0::__hi, _P0::__lo, _P1::__lo, __rem&gt; _Sum;
      // No overflow.
      static_assert(_P1::__hi == 0, &quot;Internal library error&quot;);
      static_assert(_Sum::__hi &gt;= _P0::__hi, &quot;Internal library error&quot;);
      // Matches the input data.
      static_assert(_Sum::__hi == __n1 &amp;&amp; _Sum::__lo == __n0,
		    &quot;Internal library error&quot;);
      static_assert(__rem &lt; __d, &quot;Internal library error&quot;);
    };

  /**
   *  @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&lt;7,-21&gt;::num == -1;
   *    std::ratio&lt;7,-21&gt;::den == 3;
   *  @endcode
   *
  */
  template&lt;intmax_t _Num, intmax_t _Den = 1&gt;
    struct ratio
    {
      static_assert(_Den != 0, &quot;denominator cannot be zero&quot;);
      static_assert(_Num &gt;= -__INTMAX_MAX__ &amp;&amp; _Den &gt;= -__INTMAX_MAX__,
		    &quot;out of range&quot;);

      // Note: sign(N) * abs(N) == N
      static constexpr intmax_t num =
        _Num * __static_sign&lt;_Den&gt;::value / __static_gcd&lt;_Num, _Den&gt;::value;

      static constexpr intmax_t den =
        __static_abs&lt;_Den&gt;::value / __static_gcd&lt;_Num, _Den&gt;::value;

      typedef ratio&lt;num, den&gt; type;
    };

  template&lt;intmax_t _Num, intmax_t _Den&gt;
    constexpr intmax_t ratio&lt;_Num, _Den&gt;::num;

  template&lt;intmax_t _Num, intmax_t _Den&gt;
    constexpr intmax_t ratio&lt;_Num, _Den&gt;::den;

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_multiply
    {
    private:
      static const intmax_t __gcd1 =
        __static_gcd&lt;_R1::num, _R2::den&gt;::value;
      static const intmax_t __gcd2 =
        __static_gcd&lt;_R2::num, _R1::den&gt;::value;

    public:
      typedef ratio&lt;
        __safe_multiply&lt;(_R1::num / __gcd1),
                        (_R2::num / __gcd2)&gt;::value,
        __safe_multiply&lt;(_R1::den / __gcd2),
                        (_R2::den / __gcd1)&gt;::value&gt; type;

      static constexpr intmax_t num = type::num;
      static constexpr intmax_t den = type::den;
    };

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_multiply&lt;_R1, _R2&gt;::num;

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_multiply&lt;_R1, _R2&gt;::den;

  /// ratio_multiply
  template&lt;typename _R1, typename _R2&gt;
    using ratio_multiply = typename __ratio_multiply&lt;_R1, _R2&gt;::type;

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_divide
    {
      static_assert(_R2::num != 0, &quot;division by 0&quot;);

      typedef typename __ratio_multiply&lt;
        _R1,
        ratio&lt;_R2::den, _R2::num&gt;&gt;::type type;

      static constexpr intmax_t num = type::num;
      static constexpr intmax_t den = type::den;
    };

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_divide&lt;_R1, _R2&gt;::num;

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_divide&lt;_R1, _R2&gt;::den;

  /// ratio_divide
  template&lt;typename _R1, typename _R2&gt;
    using ratio_divide = typename __ratio_divide&lt;_R1, _R2&gt;::type;

  /// ratio_equal
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_equal
    : integral_constant&lt;bool, _R1::num == _R2::num &amp;&amp; _R1::den == _R2::den&gt;
    { };

  /// ratio_not_equal
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_not_equal
    : integral_constant&lt;bool, !ratio_equal&lt;_R1, _R2&gt;::value&gt;
    { };

  // Both numbers are positive.
  template&lt;typename _R1, typename _R2,
           typename _Left = __big_mul&lt;_R1::num,_R2::den&gt;,
           typename _Right = __big_mul&lt;_R2::num,_R1::den&gt; &gt;
    struct __ratio_less_impl_1
    : integral_constant&lt;bool, __big_less&lt;_Left::__hi, _Left::__lo,
           _Right::__hi, _Right::__lo&gt;::value&gt;
    { };

  template&lt;typename _R1, typename _R2,
	   bool = (_R1::num == 0 || _R2::num == 0
		   || (__static_sign&lt;_R1::num&gt;::value
		       != __static_sign&lt;_R2::num&gt;::value)),
	   bool = (__static_sign&lt;_R1::num&gt;::value == -1
		   &amp;&amp; __static_sign&lt;_R2::num&gt;::value == -1)&gt;
    struct __ratio_less_impl
    : __ratio_less_impl_1&lt;_R1, _R2&gt;::type
    { };

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_less_impl&lt;_R1, _R2, true, false&gt;
    : integral_constant&lt;bool, _R1::num &lt; _R2::num&gt;
    { };

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_less_impl&lt;_R1, _R2, false, true&gt;
    : __ratio_less_impl_1&lt;ratio&lt;-_R2::num, _R2::den&gt;,
           ratio&lt;-_R1::num, _R1::den&gt; &gt;::type
    { };

  /// ratio_less
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_less
    : __ratio_less_impl&lt;_R1, _R2&gt;::type
    { };

  /// ratio_less_equal
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_less_equal
    : integral_constant&lt;bool, !ratio_less&lt;_R2, _R1&gt;::value&gt;
    { };

  /// ratio_greater
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_greater
    : integral_constant&lt;bool, ratio_less&lt;_R2, _R1&gt;::value&gt;
    { };

  /// ratio_greater_equal
  template&lt;typename _R1, typename _R2&gt;
    struct ratio_greater_equal
    : integral_constant&lt;bool, !ratio_less&lt;_R1, _R2&gt;::value&gt;
    { };

#if __cplusplus &gt; 201402L
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_equal_v = ratio_equal&lt;_R1, _R2&gt;::value;
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_not_equal_v = ratio_not_equal&lt;_R1, _R2&gt;::value;
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_less_v = ratio_less&lt;_R1, _R2&gt;::value;
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_less_equal_v =
      ratio_less_equal&lt;_R1, _R2&gt;::value;
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_greater_v = ratio_greater&lt;_R1, _R2&gt;::value;
  template &lt;typename _R1, typename _R2&gt;
    inline constexpr bool ratio_greater_equal_v
    = ratio_greater_equal&lt;_R1, _R2&gt;::value;
#endif // C++17

  template&lt;typename _R1, typename _R2,
      bool = (_R1::num &gt;= 0),
      bool = (_R2::num &gt;= 0),
      bool = ratio_less&lt;ratio&lt;__static_abs&lt;_R1::num&gt;::value, _R1::den&gt;,
        ratio&lt;__static_abs&lt;_R2::num&gt;::value, _R2::den&gt; &gt;::value&gt;
    struct __ratio_add_impl
    {
    private:
      typedef typename __ratio_add_impl&lt;
        ratio&lt;-_R1::num, _R1::den&gt;,
        ratio&lt;-_R2::num, _R2::den&gt; &gt;::type __t;
    public:
      typedef ratio&lt;-__t::num, __t::den&gt; type;
    };

  // True addition of nonnegative numbers.
  template&lt;typename _R1, typename _R2, bool __b&gt;
    struct __ratio_add_impl&lt;_R1, _R2, true, true, __b&gt;
    {
    private:
      static constexpr uintmax_t __g = __static_gcd&lt;_R1::den, _R2::den&gt;::value;
      static constexpr uintmax_t __d2 = _R2::den / __g;
      typedef __big_mul&lt;_R1::den, __d2&gt; __d;
      typedef __big_mul&lt;_R1::num, _R2::den / __g&gt; __x;
      typedef __big_mul&lt;_R2::num, _R1::den / __g&gt; __y;
      typedef __big_add&lt;__x::__hi, __x::__lo, __y::__hi, __y::__lo&gt; __n;
      static_assert(__n::__hi &gt;= __x::__hi, &quot;Internal library error&quot;);
      typedef __big_div&lt;__n::__hi, __n::__lo, __g&gt; __ng;
      static constexpr uintmax_t __g2 = __static_gcd&lt;__ng::__rem, __g&gt;::value;
      typedef __big_div&lt;__n::__hi, __n::__lo, __g2&gt; __n_final;
      static_assert(__n_final::__rem == 0, &quot;Internal library error&quot;);
      static_assert(__n_final::__quot_hi == 0 &amp;&amp;
        __n_final::__quot_lo &lt;= __INTMAX_MAX__, &quot;overflow in addition&quot;);
      typedef __big_mul&lt;_R1::den / __g2, __d2&gt; __d_final;
      static_assert(__d_final::__hi == 0 &amp;&amp;
        __d_final::__lo &lt;= __INTMAX_MAX__, &quot;overflow in addition&quot;);
    public:
      typedef ratio&lt;__n_final::__quot_lo, __d_final::__lo&gt; type;
    };

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_add_impl&lt;_R1, _R2, false, true, true&gt;
    : __ratio_add_impl&lt;_R2, _R1&gt;
    { };

  // True subtraction of nonnegative numbers yielding a nonnegative result.
  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_add_impl&lt;_R1, _R2, true, false, false&gt;
    {
    private:
      static constexpr uintmax_t __g = __static_gcd&lt;_R1::den, _R2::den&gt;::value;
      static constexpr uintmax_t __d2 = _R2::den / __g;
      typedef __big_mul&lt;_R1::den, __d2&gt; __d;
      typedef __big_mul&lt;_R1::num, _R2::den / __g&gt; __x;
      typedef __big_mul&lt;-_R2::num, _R1::den / __g&gt; __y;
      typedef __big_sub&lt;__x::__hi, __x::__lo, __y::__hi, __y::__lo&gt; __n;
      typedef __big_div&lt;__n::__hi, __n::__lo, __g&gt; __ng;
      static constexpr uintmax_t __g2 = __static_gcd&lt;__ng::__rem, __g&gt;::value;
      typedef __big_div&lt;__n::__hi, __n::__lo, __g2&gt; __n_final;
      static_assert(__n_final::__rem == 0, &quot;Internal library error&quot;);
      static_assert(__n_final::__quot_hi == 0 &amp;&amp;
        __n_final::__quot_lo &lt;= __INTMAX_MAX__, &quot;overflow in addition&quot;);
      typedef __big_mul&lt;_R1::den / __g2, __d2&gt; __d_final;
      static_assert(__d_final::__hi == 0 &amp;&amp;
        __d_final::__lo &lt;= __INTMAX_MAX__, &quot;overflow in addition&quot;);
    public:
      typedef ratio&lt;__n_final::__quot_lo, __d_final::__lo&gt; type;
    };

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_add
    {
      typedef typename __ratio_add_impl&lt;_R1, _R2&gt;::type type;
      static constexpr intmax_t num = type::num;
      static constexpr intmax_t den = type::den;
    };

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_add&lt;_R1, _R2&gt;::num;

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_add&lt;_R1, _R2&gt;::den;

  /// ratio_add
  template&lt;typename _R1, typename _R2&gt;
    using ratio_add = typename __ratio_add&lt;_R1, _R2&gt;::type;

  template&lt;typename _R1, typename _R2&gt;
    struct __ratio_subtract
    {
      typedef typename __ratio_add&lt;
        _R1,
        ratio&lt;-_R2::num, _R2::den&gt;&gt;::type type;

      static constexpr intmax_t num = type::num;
      static constexpr intmax_t den = type::den;
    };

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_subtract&lt;_R1, _R2&gt;::num;

  template&lt;typename _R1, typename _R2&gt;
    constexpr intmax_t __ratio_subtract&lt;_R1, _R2&gt;::den;

  /// ratio_subtract
  template&lt;typename _R1, typename _R2&gt;
    using ratio_subtract = typename __ratio_subtract&lt;_R1, _R2&gt;::type;


  typedef ratio&lt;1,       1000000000000000000&gt; atto;
  typedef ratio&lt;1,          1000000000000000&gt; femto;
  typedef ratio&lt;1,             1000000000000&gt; pico;
  typedef ratio&lt;1,                1000000000&gt; nano;
  typedef ratio&lt;1,                   1000000&gt; micro;
  typedef ratio&lt;1,                      1000&gt; milli;
  typedef ratio&lt;1,                       100&gt; centi;
  typedef ratio&lt;1,                        10&gt; deci;
  typedef ratio&lt;                       10, 1&gt; deca;
  typedef ratio&lt;                      100, 1&gt; hecto;
  typedef ratio&lt;                     1000, 1&gt; kilo;
  typedef ratio&lt;                  1000000, 1&gt; mega;
  typedef ratio&lt;               1000000000, 1&gt; giga;
  typedef ratio&lt;            1000000000000, 1&gt; tera;
  typedef ratio&lt;         1000000000000000, 1&gt; peta;
  typedef ratio&lt;      1000000000000000000, 1&gt; exa;

  // @} group ratio
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace

#endif //_GLIBCXX_USE_C99_STDINT_TR1

#endif // C++11

#endif //_GLIBCXX_RATIO
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