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
45namespace 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
652