libstdc++
ratio
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1 // ratio -*- C++ -*-
2 
3 // Copyright (C) 2008-2024 Free Software Foundation, Inc.
4 //
5 // This file is part of the GNU ISO C++ Library. This library is free
6 // software; you can redistribute it and/or modify it under the
7 // terms of the GNU General Public License as published by the
8 // Free Software Foundation; either version 3, or (at your option)
9 // any later version.
10 
11 // This library is distributed in the hope that it will be useful,
12 // but WITHOUT ANY WARRANTY; without even the implied warranty of
13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 // GNU General Public License for more details.
15 
16 // Under Section 7 of GPL version 3, you are granted additional
17 // permissions described in the GCC Runtime Library Exception, version
18 // 3.1, as published by the Free Software Foundation.
19 
20 // You should have received a copy of the GNU General Public License and
21 // a copy of the GCC Runtime Library Exception along with this program;
22 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
23 // <http://www.gnu.org/licenses/>.
24 
25 /** @file include/ratio
26  * This is a Standard C++ Library header.
27  * @ingroup ratio
28  */
29 
30 #ifndef _GLIBCXX_RATIO
31 #define _GLIBCXX_RATIO 1
32 
33 #pragma GCC system_header
34 
35 #if __cplusplus < 201103L
36 # include <bits/c++0x_warning.h>
37 #else
38 
39 #include <type_traits>
40 #include <cstdint> // intmax_t, uintmax_t
41 
42 #define __glibcxx_want_ratio
43 #include <bits/version.h>
44 
45 namespace std _GLIBCXX_VISIBILITY(default)
46 {
47 _GLIBCXX_BEGIN_NAMESPACE_VERSION
48 
49  /**
50  * @defgroup ratio Rational Arithmetic
51  * @ingroup utilities
52  *
53  * Compile time representation of finite rational numbers.
54  * @{
55  */
56 
57  /// @cond undocumented
58 
59  template<intmax_t _Pn>
60  struct __static_sign
61  : integral_constant<intmax_t, (_Pn < 0) ? -1 : 1>
62  { };
63 
64  template<intmax_t _Pn>
65  struct __static_abs
66  : integral_constant<intmax_t, _Pn * __static_sign<_Pn>::value>
67  { };
68 
69  template<intmax_t _Pn, intmax_t _Qn>
70  struct __static_gcd
71  : __static_gcd<_Qn, (_Pn % _Qn)>
72  { };
73 
74  template<intmax_t _Pn>
75  struct __static_gcd<_Pn, 0>
76  : integral_constant<intmax_t, __static_abs<_Pn>::value>
77  { };
78 
79  template<intmax_t _Qn>
80  struct __static_gcd<0, _Qn>
81  : integral_constant<intmax_t, __static_abs<_Qn>::value>
82  { };
83 
84  // Let c = 2^(half # of bits in an intmax_t)
85  // then we find a1, a0, b1, b0 s.t. N = a1*c + a0, M = b1*c + b0
86  // The multiplication of N and M becomes,
87  // N * M = (a1 * b1)c^2 + (a0 * b1 + b0 * a1)c + a0 * b0
88  // Multiplication is safe if each term and the sum of the terms
89  // is representable by intmax_t.
90  template<intmax_t _Pn, intmax_t _Qn>
91  struct __safe_multiply
92  {
93  private:
94  static const uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
95 
96  static const uintmax_t __a0 = __static_abs<_Pn>::value % __c;
97  static const uintmax_t __a1 = __static_abs<_Pn>::value / __c;
98  static const uintmax_t __b0 = __static_abs<_Qn>::value % __c;
99  static const uintmax_t __b1 = __static_abs<_Qn>::value / __c;
100 
101  static_assert(__a1 == 0 || __b1 == 0,
102  "overflow in multiplication");
103  static_assert(__a0 * __b1 + __b0 * __a1 < (__c >> 1),
104  "overflow in multiplication");
105  static_assert(__b0 * __a0 <= __INTMAX_MAX__,
106  "overflow in multiplication");
107  static_assert((__a0 * __b1 + __b0 * __a1) * __c
108  <= __INTMAX_MAX__ - __b0 * __a0,
109  "overflow in multiplication");
110 
111  public:
112  static const intmax_t value = _Pn * _Qn;
113  };
114 
115  // Some double-precision utilities, where numbers are represented as
116  // __hi*2^(8*sizeof(uintmax_t)) + __lo.
117  template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
118  struct __big_less
119  : integral_constant<bool, (__hi1 < __hi2
120  || (__hi1 == __hi2 && __lo1 < __lo2))>
121  { };
122 
123  template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
124  struct __big_add
125  {
126  static constexpr uintmax_t __lo = __lo1 + __lo2;
127  static constexpr uintmax_t __hi = (__hi1 + __hi2 +
128  (__lo1 + __lo2 < __lo1)); // carry
129  };
130 
131  // Subtract a number from a bigger one.
132  template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
133  struct __big_sub
134  {
135  static_assert(!__big_less<__hi1, __lo1, __hi2, __lo2>::value,
136  "Internal library error");
137  static constexpr uintmax_t __lo = __lo1 - __lo2;
138  static constexpr uintmax_t __hi = (__hi1 - __hi2 -
139  (__lo1 < __lo2)); // carry
140  };
141 
142  // Same principle as __safe_multiply.
143  template<uintmax_t __x, uintmax_t __y>
144  struct __big_mul
145  {
146  private:
147  static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
148  static constexpr uintmax_t __x0 = __x % __c;
149  static constexpr uintmax_t __x1 = __x / __c;
150  static constexpr uintmax_t __y0 = __y % __c;
151  static constexpr uintmax_t __y1 = __y / __c;
152  static constexpr uintmax_t __x0y0 = __x0 * __y0;
153  static constexpr uintmax_t __x0y1 = __x0 * __y1;
154  static constexpr uintmax_t __x1y0 = __x1 * __y0;
155  static constexpr uintmax_t __x1y1 = __x1 * __y1;
156  static constexpr uintmax_t __mix = __x0y1 + __x1y0; // possible carry...
157  static constexpr uintmax_t __mix_lo = __mix * __c;
158  static constexpr uintmax_t __mix_hi
159  = __mix / __c + ((__mix < __x0y1) ? __c : 0); // ... added here
160  typedef __big_add<__mix_hi, __mix_lo, __x1y1, __x0y0> _Res;
161  public:
162  static constexpr uintmax_t __hi = _Res::__hi;
163  static constexpr uintmax_t __lo = _Res::__lo;
164  };
165 
166  // Adapted from __udiv_qrnnd_c in longlong.h
167  // This version assumes that the high bit of __d is 1.
168  template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
169  struct __big_div_impl
170  {
171  private:
172  static_assert(__d >= (uintmax_t(1) << (sizeof(intmax_t) * 8 - 1)),
173  "Internal library error");
174  static_assert(__n1 < __d, "Internal library error");
175  static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
176  static constexpr uintmax_t __d1 = __d / __c;
177  static constexpr uintmax_t __d0 = __d % __c;
178 
179  static constexpr uintmax_t __q1x = __n1 / __d1;
180  static constexpr uintmax_t __r1x = __n1 % __d1;
181  static constexpr uintmax_t __m = __q1x * __d0;
182  static constexpr uintmax_t __r1y = __r1x * __c + __n0 / __c;
183  static constexpr uintmax_t __r1z = __r1y + __d;
184  static constexpr uintmax_t __r1
185  = ((__r1y < __m) ? ((__r1z >= __d) && (__r1z < __m))
186  ? (__r1z + __d) : __r1z : __r1y) - __m;
187  static constexpr uintmax_t __q1
188  = __q1x - ((__r1y < __m)
189  ? ((__r1z >= __d) && (__r1z < __m)) ? 2 : 1 : 0);
190  static constexpr uintmax_t __q0x = __r1 / __d1;
191  static constexpr uintmax_t __r0x = __r1 % __d1;
192  static constexpr uintmax_t __n = __q0x * __d0;
193  static constexpr uintmax_t __r0y = __r0x * __c + __n0 % __c;
194  static constexpr uintmax_t __r0z = __r0y + __d;
195  static constexpr uintmax_t __r0
196  = ((__r0y < __n) ? ((__r0z >= __d) && (__r0z < __n))
197  ? (__r0z + __d) : __r0z : __r0y) - __n;
198  static constexpr uintmax_t __q0
199  = __q0x - ((__r0y < __n) ? ((__r0z >= __d)
200  && (__r0z < __n)) ? 2 : 1 : 0);
201 
202  public:
203  static constexpr uintmax_t __quot = __q1 * __c + __q0;
204  static constexpr uintmax_t __rem = __r0;
205 
206  private:
207  typedef __big_mul<__quot, __d> _Prod;
208  typedef __big_add<_Prod::__hi, _Prod::__lo, 0, __rem> _Sum;
209  static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
210  "Internal library error");
211  };
212 
213  template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
214  struct __big_div
215  {
216  private:
217  static_assert(__d != 0, "Internal library error");
218  static_assert(sizeof (uintmax_t) == sizeof (unsigned long long),
219  "This library calls __builtin_clzll on uintmax_t, which "
220  "is unsafe on your platform. Please complain to "
221  "http://gcc.gnu.org/bugzilla/");
222  static constexpr int __shift = __builtin_clzll(__d);
223  static constexpr int __coshift_ = sizeof(uintmax_t) * 8 - __shift;
224  static constexpr int __coshift = (__shift != 0) ? __coshift_ : 0;
225  static constexpr uintmax_t __c1 = uintmax_t(1) << __shift;
226  static constexpr uintmax_t __c2 = uintmax_t(1) << __coshift;
227  static constexpr uintmax_t __new_d = __d * __c1;
228  static constexpr uintmax_t __new_n0 = __n0 * __c1;
229  static constexpr uintmax_t __n1_shifted = (__n1 % __d) * __c1;
230  static constexpr uintmax_t __n0_top = (__shift != 0) ? (__n0 / __c2) : 0;
231  static constexpr uintmax_t __new_n1 = __n1_shifted + __n0_top;
232  typedef __big_div_impl<__new_n1, __new_n0, __new_d> _Res;
233 
234  public:
235  static constexpr uintmax_t __quot_hi = __n1 / __d;
236  static constexpr uintmax_t __quot_lo = _Res::__quot;
237  static constexpr uintmax_t __rem = _Res::__rem / __c1;
238 
239  private:
240  typedef __big_mul<__quot_lo, __d> _P0;
241  typedef __big_mul<__quot_hi, __d> _P1;
242  typedef __big_add<_P0::__hi, _P0::__lo, _P1::__lo, __rem> _Sum;
243  // No overflow.
244  static_assert(_P1::__hi == 0, "Internal library error");
245  static_assert(_Sum::__hi >= _P0::__hi, "Internal library error");
246  // Matches the input data.
247  static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
248  "Internal library error");
249  static_assert(__rem < __d, "Internal library error");
250  };
251 
252  /// @endcond
253 
254  /**
255  * @brief Provides compile-time rational arithmetic.
256  *
257  * This class template represents any finite rational number with a
258  * numerator and denominator representable by compile-time constants of
259  * type intmax_t. The ratio is simplified when instantiated.
260  *
261  * For example:
262  * @code
263  * std::ratio<7,-21>::num == -1;
264  * std::ratio<7,-21>::den == 3;
265  * @endcode
266  *
267  */
268  template<intmax_t _Num, intmax_t _Den = 1>
269  struct ratio
270  {
271  static_assert(_Den != 0, "denominator cannot be zero");
272  static_assert(_Num >= -__INTMAX_MAX__ && _Den >= -__INTMAX_MAX__,
273  "out of range");
274 
275  // Note: sign(N) * abs(N) == N
276  static constexpr intmax_t num =
277  _Num * __static_sign<_Den>::value / __static_gcd<_Num, _Den>::value;
278 
279  static constexpr intmax_t den =
280  __static_abs<_Den>::value / __static_gcd<_Num, _Den>::value;
281 
282  typedef ratio<num, den> type;
283  };
284 
285 #if ! __cpp_inline_variables
286  template<intmax_t _Num, intmax_t _Den>
287  constexpr intmax_t ratio<_Num, _Den>::num;
288 
289  template<intmax_t _Num, intmax_t _Den>
290  constexpr intmax_t ratio<_Num, _Den>::den;
291 #endif
292 
293  /// @cond undocumented
294 
295  template<typename _Tp>
296  struct __is_ratio
297  : std::false_type
298  { };
299 
300  template<intmax_t _Num, intmax_t _Den>
301  struct __is_ratio<ratio<_Num, _Den>>
302  : std::true_type
303  { };
304 
305 #if __cpp_variable_templates
306  template<typename _Tp>
307  constexpr bool __is_ratio_v = false;
308  template<intmax_t _Num, intmax_t _Den>
309  constexpr bool __is_ratio_v<ratio<_Num, _Den>> = true;
310 #endif
311 
312  template<typename _R1, typename _R2>
313  constexpr bool
314  __are_both_ratios() noexcept
315  {
316 #if __cpp_variable_templates && __cpp_if_constexpr
317  if constexpr (__is_ratio_v<_R1>)
318  if constexpr (__is_ratio_v<_R2>)
319  return true;
320  return false;
321 #else
322  return __and_<__is_ratio<_R1>, __is_ratio<_R2>>::value;
323 #endif
324  }
325 
326  template<typename _R1, typename _R2>
327  struct __ratio_multiply
328  {
329  static_assert(std::__are_both_ratios<_R1, _R2>(),
330  "both template arguments must be a std::ratio");
331 
332  private:
333  static const intmax_t __gcd1 =
334  __static_gcd<_R1::num, _R2::den>::value;
335  static const intmax_t __gcd2 =
336  __static_gcd<_R2::num, _R1::den>::value;
337 
338  public:
339  typedef ratio<
340  __safe_multiply<(_R1::num / __gcd1),
341  (_R2::num / __gcd2)>::value,
342  __safe_multiply<(_R1::den / __gcd2),
343  (_R2::den / __gcd1)>::value> type;
344 
345  static constexpr intmax_t num = type::num;
346  static constexpr intmax_t den = type::den;
347  };
348 
349 #if ! __cpp_inline_variables
350  template<typename _R1, typename _R2>
351  constexpr intmax_t __ratio_multiply<_R1, _R2>::num;
352 
353  template<typename _R1, typename _R2>
354  constexpr intmax_t __ratio_multiply<_R1, _R2>::den;
355 #endif
356 
357  /// @endcond
358 
359  /// ratio_multiply
360  template<typename _R1, typename _R2>
361  using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;
362 
363  /// @cond undocumented
364 
365  template<typename _R1, typename _R2>
366  struct __ratio_divide
367  {
368  static_assert(_R2::num != 0, "division by 0");
369 
370  typedef typename __ratio_multiply<
371  _R1,
372  ratio<_R2::den, _R2::num>>::type type;
373 
374  static constexpr intmax_t num = type::num;
375  static constexpr intmax_t den = type::den;
376  };
377 
378 #if ! __cpp_inline_variables
379  template<typename _R1, typename _R2>
380  constexpr intmax_t __ratio_divide<_R1, _R2>::num;
381 
382  template<typename _R1, typename _R2>
383  constexpr intmax_t __ratio_divide<_R1, _R2>::den;
384 #endif
385 
386  /// @endcond
387 
388  /// ratio_divide
389  template<typename _R1, typename _R2>
390  using ratio_divide = typename __ratio_divide<_R1, _R2>::type;
391 
392  /// ratio_equal
393  template<typename _R1, typename _R2>
394  struct ratio_equal
395  : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den>
396  {
397  static_assert(std::__are_both_ratios<_R1, _R2>(),
398  "both template arguments must be a std::ratio");
399  };
400 
401  /// ratio_not_equal
402  template<typename _R1, typename _R2>
403  struct ratio_not_equal
404  : integral_constant<bool, !ratio_equal<_R1, _R2>::value>
405  { };
406 
407  /// @cond undocumented
408 
409  // Both numbers are positive.
410  template<typename _R1, typename _R2,
411  typename _Left = __big_mul<_R1::num,_R2::den>,
412  typename _Right = __big_mul<_R2::num,_R1::den> >
413  struct __ratio_less_impl_1
414  : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo,
415  _Right::__hi, _Right::__lo>::value>
416  { };
417 
418  template<typename _R1, typename _R2,
419  bool = (_R1::num == 0 || _R2::num == 0
420  || (__static_sign<_R1::num>::value
421  != __static_sign<_R2::num>::value)),
422  bool = (__static_sign<_R1::num>::value == -1
423  && __static_sign<_R2::num>::value == -1)>
424  struct __ratio_less_impl
425  : __ratio_less_impl_1<_R1, _R2>::type
426  { };
427 
428  template<typename _R1, typename _R2>
429  struct __ratio_less_impl<_R1, _R2, true, false>
430  : integral_constant<bool, _R1::num < _R2::num>
431  { };
432 
433  template<typename _R1, typename _R2>
434  struct __ratio_less_impl<_R1, _R2, false, true>
435  : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>,
436  ratio<-_R1::num, _R1::den> >::type
437  { };
438 
439  /// @endcond
440 
441  /// ratio_less
442  template<typename _R1, typename _R2>
443  struct ratio_less
444  : __ratio_less_impl<_R1, _R2>::type
445  {
446  static_assert(std::__are_both_ratios<_R1, _R2>(),
447  "both template arguments must be a std::ratio");
448  };
449 
450  /// ratio_less_equal
451  template<typename _R1, typename _R2>
452  struct ratio_less_equal
453  : integral_constant<bool, !ratio_less<_R2, _R1>::value>
454  { };
455 
456  /// ratio_greater
457  template<typename _R1, typename _R2>
458  struct ratio_greater
459  : integral_constant<bool, ratio_less<_R2, _R1>::value>
460  { };
461 
462  /// ratio_greater_equal
463  template<typename _R1, typename _R2>
464  struct ratio_greater_equal
465  : integral_constant<bool, !ratio_less<_R1, _R2>::value>
466  { };
467 
468 #if __cplusplus > 201402L
469  template <typename _R1, typename _R2>
470  inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value;
471  template <typename _R1, typename _R2>
472  inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value;
473  template <typename _R1, typename _R2>
474  inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value;
475  template <typename _R1, typename _R2>
476  inline constexpr bool ratio_less_equal_v
477  = ratio_less_equal<_R1, _R2>::value;
478  template <typename _R1, typename _R2>
479  inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value;
480  template <typename _R1, typename _R2>
481  inline constexpr bool ratio_greater_equal_v
482  = ratio_greater_equal<_R1, _R2>::value;
483 #endif // C++17
484 
485  /// @cond undocumented
486 
487  template<typename _R1, typename _R2,
488  bool = (_R1::num >= 0),
489  bool = (_R2::num >= 0),
490  bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>,
491  ratio<__static_abs<_R2::num>::value, _R2::den> >::value>
492  struct __ratio_add_impl
493  {
494  private:
495  typedef typename __ratio_add_impl<
496  ratio<-_R1::num, _R1::den>,
497  ratio<-_R2::num, _R2::den> >::type __t;
498  public:
499  typedef ratio<-__t::num, __t::den> type;
500  };
501 
502  // True addition of nonnegative numbers.
503  template<typename _R1, typename _R2, bool __b>
504  struct __ratio_add_impl<_R1, _R2, true, true, __b>
505  {
506  private:
507  static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
508  static constexpr uintmax_t __d2 = _R2::den / __g;
509  typedef __big_mul<_R1::den, __d2> __d;
510  typedef __big_mul<_R1::num, _R2::den / __g> __x;
511  typedef __big_mul<_R2::num, _R1::den / __g> __y;
512  typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
513  static_assert(__n::__hi >= __x::__hi, "Internal library error");
514  typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
515  static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
516  typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
517  static_assert(__n_final::__rem == 0, "Internal library error");
518  static_assert(__n_final::__quot_hi == 0 &&
519  __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
520  typedef __big_mul<_R1::den / __g2, __d2> __d_final;
521  static_assert(__d_final::__hi == 0 &&
522  __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
523  public:
524  typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
525  };
526 
527  template<typename _R1, typename _R2>
528  struct __ratio_add_impl<_R1, _R2, false, true, true>
529  : __ratio_add_impl<_R2, _R1>
530  { };
531 
532  // True subtraction of nonnegative numbers yielding a nonnegative result.
533  template<typename _R1, typename _R2>
534  struct __ratio_add_impl<_R1, _R2, true, false, false>
535  {
536  private:
537  static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
538  static constexpr uintmax_t __d2 = _R2::den / __g;
539  typedef __big_mul<_R1::den, __d2> __d;
540  typedef __big_mul<_R1::num, _R2::den / __g> __x;
541  typedef __big_mul<-_R2::num, _R1::den / __g> __y;
542  typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
543  typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
544  static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
545  typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
546  static_assert(__n_final::__rem == 0, "Internal library error");
547  static_assert(__n_final::__quot_hi == 0 &&
548  __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
549  typedef __big_mul<_R1::den / __g2, __d2> __d_final;
550  static_assert(__d_final::__hi == 0 &&
551  __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
552  public:
553  typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
554  };
555 
556  template<typename _R1, typename _R2>
557  struct __ratio_add
558  {
559  static_assert(std::__are_both_ratios<_R1, _R2>(),
560  "both template arguments must be a std::ratio");
561 
562  typedef typename __ratio_add_impl<_R1, _R2>::type type;
563  static constexpr intmax_t num = type::num;
564  static constexpr intmax_t den = type::den;
565  };
566 
567 #if ! __cpp_inline_variables
568  template<typename _R1, typename _R2>
569  constexpr intmax_t __ratio_add<_R1, _R2>::num;
570 
571  template<typename _R1, typename _R2>
572  constexpr intmax_t __ratio_add<_R1, _R2>::den;
573 #endif
574 
575  /// @endcond
576 
577  /// ratio_add
578  template<typename _R1, typename _R2>
579  using ratio_add = typename __ratio_add<_R1, _R2>::type;
580 
581  /// @cond undocumented
582 
583  template<typename _R1, typename _R2>
584  struct __ratio_subtract
585  {
586  typedef typename __ratio_add<
587  _R1,
588  ratio<-_R2::num, _R2::den>>::type type;
589 
590  static constexpr intmax_t num = type::num;
591  static constexpr intmax_t den = type::den;
592  };
593 
594 #if ! __cpp_inline_variables
595  template<typename _R1, typename _R2>
596  constexpr intmax_t __ratio_subtract<_R1, _R2>::num;
597 
598  template<typename _R1, typename _R2>
599  constexpr intmax_t __ratio_subtract<_R1, _R2>::den;
600 #endif
601 
602  /// @endcond
603 
604  /// ratio_subtract
605  template<typename _R1, typename _R2>
606  using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type;
607 
608 #if __INTMAX_WIDTH__ >= 96
609 # if __cpp_lib_ratio >= 202306L
610 # if __INTMAX_WIDTH__ >= 128
611  using quecto = ratio< 1, 1000000000000000000000000000000>;
612 # endif
613  using ronto = ratio< 1, 1000000000000000000000000000>;
614 # endif
615  using yocto = ratio< 1, 1000000000000000000000000>;
616  using zepto = ratio< 1, 1000000000000000000000>;
617 #endif
618  using atto = ratio< 1, 1000000000000000000>;
619  using femto = ratio< 1, 1000000000000000>;
620  using pico = ratio< 1, 1000000000000>;
621  using nano = ratio< 1, 1000000000>;
622  using micro = ratio< 1, 1000000>;
623  using milli = ratio< 1, 1000>;
624  using centi = ratio< 1, 100>;
625  using deci = ratio< 1, 10>;
626  using deca = ratio< 10, 1>;
627  using hecto = ratio< 100, 1>;
628  using kilo = ratio< 1000, 1>;
629  using mega = ratio< 1000000, 1>;
630  using giga = ratio< 1000000000, 1>;
631  using tera = ratio< 1000000000000, 1>;
632  using peta = ratio< 1000000000000000, 1>;
633  using exa = ratio< 1000000000000000000, 1>;
634 #if __INTMAX_WIDTH__ >= 96
635  using zetta = ratio< 1000000000000000000000, 1>;
636  using yotta = ratio<1000000000000000000000000, 1>;
637 # if __cpp_lib_ratio >= 202306L
638  using ronna = ratio<1000000000000000000000000000, 1>;
639 # if __INTMAX_WIDTH__ >= 128
640  using quetta = ratio<1000000000000000000000000000000, 1>;
641 # endif
642 # endif
643 #endif
644 
645  /// @} group ratio
646 _GLIBCXX_END_NAMESPACE_VERSION
647 } // namespace
648 
649 #endif // C++11
650 
651 #endif //_GLIBCXX_RATIO