1// ratio -*- C++ -*-
2
3// Copyright (C) 2008-2023 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
42namespace std _GLIBCXX_VISIBILITY(default)
43{
44_GLIBCXX_BEGIN_NAMESPACE_VERSION
45
46 /**
47 * @defgroup ratio Rational Arithmetic
48 * @ingroup utilities
49 *
50 * Compile time representation of finite rational numbers.
51 * @{
52 */
53
54 /// @cond undocumented
55
56 template<intmax_t _Pn>
57 struct __static_sign
58 : integral_constant<intmax_t, (_Pn < 0) ? -1 : 1>
59 { };
60
61 template<intmax_t _Pn>
62 struct __static_abs
63 : integral_constant<intmax_t, _Pn * __static_sign<_Pn>::value>
64 { };
65
66 template<intmax_t _Pn, intmax_t _Qn>
67 struct __static_gcd
68 : __static_gcd<_Qn, (_Pn % _Qn)>
69 { };
70
71 template<intmax_t _Pn>
72 struct __static_gcd<_Pn, 0>
73 : integral_constant<intmax_t, __static_abs<_Pn>::value>
74 { };
75
76 template<intmax_t _Qn>
77 struct __static_gcd<0, _Qn>
78 : integral_constant<intmax_t, __static_abs<_Qn>::value>
79 { };
80
81 // Let c = 2^(half # of bits in an intmax_t)
82 // then we find a1, a0, b1, b0 s.t. N = a1*c + a0, M = b1*c + b0
83 // The multiplication of N and M becomes,
84 // N * M = (a1 * b1)c^2 + (a0 * b1 + b0 * a1)c + a0 * b0
85 // Multiplication is safe if each term and the sum of the terms
86 // is representable by intmax_t.
87 template<intmax_t _Pn, intmax_t _Qn>
88 struct __safe_multiply
89 {
90 private:
91 static const uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
92
93 static const uintmax_t __a0 = __static_abs<_Pn>::value % __c;
94 static const uintmax_t __a1 = __static_abs<_Pn>::value / __c;
95 static const uintmax_t __b0 = __static_abs<_Qn>::value % __c;
96 static const uintmax_t __b1 = __static_abs<_Qn>::value / __c;
97
98 static_assert(__a1 == 0 || __b1 == 0,
99 "overflow in multiplication");
100 static_assert(__a0 * __b1 + __b0 * __a1 < (__c >> 1),
101 "overflow in multiplication");
102 static_assert(__b0 * __a0 <= __INTMAX_MAX__,
103 "overflow in multiplication");
104 static_assert((__a0 * __b1 + __b0 * __a1) * __c
105 <= __INTMAX_MAX__ - __b0 * __a0,
106 "overflow in multiplication");
107
108 public:
109 static const intmax_t value = _Pn * _Qn;
110 };
111
112 // Some double-precision utilities, where numbers are represented as
113 // __hi*2^(8*sizeof(uintmax_t)) + __lo.
114 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
115 struct __big_less
116 : integral_constant<bool, (__hi1 < __hi2
117 || (__hi1 == __hi2 && __lo1 < __lo2))>
118 { };
119
120 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
121 struct __big_add
122 {
123 static constexpr uintmax_t __lo = __lo1 + __lo2;
124 static constexpr uintmax_t __hi = (__hi1 + __hi2 +
125 (__lo1 + __lo2 < __lo1)); // carry
126 };
127
128 // Subtract a number from a bigger one.
129 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
130 struct __big_sub
131 {
132 static_assert(!__big_less<__hi1, __lo1, __hi2, __lo2>::value,
133 "Internal library error");
134 static constexpr uintmax_t __lo = __lo1 - __lo2;
135 static constexpr uintmax_t __hi = (__hi1 - __hi2 -
136 (__lo1 < __lo2)); // carry
137 };
138
139 // Same principle as __safe_multiply.
140 template<uintmax_t __x, uintmax_t __y>
141 struct __big_mul
142 {
143 private:
144 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
145 static constexpr uintmax_t __x0 = __x % __c;
146 static constexpr uintmax_t __x1 = __x / __c;
147 static constexpr uintmax_t __y0 = __y % __c;
148 static constexpr uintmax_t __y1 = __y / __c;
149 static constexpr uintmax_t __x0y0 = __x0 * __y0;
150 static constexpr uintmax_t __x0y1 = __x0 * __y1;
151 static constexpr uintmax_t __x1y0 = __x1 * __y0;
152 static constexpr uintmax_t __x1y1 = __x1 * __y1;
153 static constexpr uintmax_t __mix = __x0y1 + __x1y0; // possible carry...
154 static constexpr uintmax_t __mix_lo = __mix * __c;
155 static constexpr uintmax_t __mix_hi
156 = __mix / __c + ((__mix < __x0y1) ? __c : 0); // ... added here
157 typedef __big_add<__mix_hi, __mix_lo, __x1y1, __x0y0> _Res;
158 public:
159 static constexpr uintmax_t __hi = _Res::__hi;
160 static constexpr uintmax_t __lo = _Res::__lo;
161 };
162
163 // Adapted from __udiv_qrnnd_c in longlong.h
164 // This version assumes that the high bit of __d is 1.
165 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
166 struct __big_div_impl
167 {
168 private:
169 static_assert(__d >= (uintmax_t(1) << (sizeof(intmax_t) * 8 - 1)),
170 "Internal library error");
171 static_assert(__n1 < __d, "Internal library error");
172 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
173 static constexpr uintmax_t __d1 = __d / __c;
174 static constexpr uintmax_t __d0 = __d % __c;
175
176 static constexpr uintmax_t __q1x = __n1 / __d1;
177 static constexpr uintmax_t __r1x = __n1 % __d1;
178 static constexpr uintmax_t __m = __q1x * __d0;
179 static constexpr uintmax_t __r1y = __r1x * __c + __n0 / __c;
180 static constexpr uintmax_t __r1z = __r1y + __d;
181 static constexpr uintmax_t __r1
182 = ((__r1y < __m) ? ((__r1z >= __d) && (__r1z < __m))
183 ? (__r1z + __d) : __r1z : __r1y) - __m;
184 static constexpr uintmax_t __q1
185 = __q1x - ((__r1y < __m)
186 ? ((__r1z >= __d) && (__r1z < __m)) ? 2 : 1 : 0);
187 static constexpr uintmax_t __q0x = __r1 / __d1;
188 static constexpr uintmax_t __r0x = __r1 % __d1;
189 static constexpr uintmax_t __n = __q0x * __d0;
190 static constexpr uintmax_t __r0y = __r0x * __c + __n0 % __c;
191 static constexpr uintmax_t __r0z = __r0y + __d;
192 static constexpr uintmax_t __r0
193 = ((__r0y < __n) ? ((__r0z >= __d) && (__r0z < __n))
194 ? (__r0z + __d) : __r0z : __r0y) - __n;
195 static constexpr uintmax_t __q0
196 = __q0x - ((__r0y < __n) ? ((__r0z >= __d)
197 && (__r0z < __n)) ? 2 : 1 : 0);
198
199 public:
200 static constexpr uintmax_t __quot = __q1 * __c + __q0;
201 static constexpr uintmax_t __rem = __r0;
202
203 private:
204 typedef __big_mul<__quot, __d> _Prod;
205 typedef __big_add<_Prod::__hi, _Prod::__lo, 0, __rem> _Sum;
206 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
207 "Internal library error");
208 };
209
210 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
211 struct __big_div
212 {
213 private:
214 static_assert(__d != 0, "Internal library error");
215 static_assert(sizeof (uintmax_t) == sizeof (unsigned long long),
216 "This library calls __builtin_clzll on uintmax_t, which "
217 "is unsafe on your platform. Please complain to "
218 "http://gcc.gnu.org/bugzilla/");
219 static constexpr int __shift = __builtin_clzll(__d);
220 static constexpr int __coshift_ = sizeof(uintmax_t) * 8 - __shift;
221 static constexpr int __coshift = (__shift != 0) ? __coshift_ : 0;
222 static constexpr uintmax_t __c1 = uintmax_t(1) << __shift;
223 static constexpr uintmax_t __c2 = uintmax_t(1) << __coshift;
224 static constexpr uintmax_t __new_d = __d * __c1;
225 static constexpr uintmax_t __new_n0 = __n0 * __c1;
226 static constexpr uintmax_t __n1_shifted = (__n1 % __d) * __c1;
227 static constexpr uintmax_t __n0_top = (__shift != 0) ? (__n0 / __c2) : 0;
228 static constexpr uintmax_t __new_n1 = __n1_shifted + __n0_top;
229 typedef __big_div_impl<__new_n1, __new_n0, __new_d> _Res;
230
231 public:
232 static constexpr uintmax_t __quot_hi = __n1 / __d;
233 static constexpr uintmax_t __quot_lo = _Res::__quot;
234 static constexpr uintmax_t __rem = _Res::__rem / __c1;
235
236 private:
237 typedef __big_mul<__quot_lo, __d> _P0;
238 typedef __big_mul<__quot_hi, __d> _P1;
239 typedef __big_add<_P0::__hi, _P0::__lo, _P1::__lo, __rem> _Sum;
240 // No overflow.
241 static_assert(_P1::__hi == 0, "Internal library error");
242 static_assert(_Sum::__hi >= _P0::__hi, "Internal library error");
243 // Matches the input data.
244 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
245 "Internal library error");
246 static_assert(__rem < __d, "Internal library error");
247 };
248
249 /// @endcond
250
251 /**
252 * @brief Provides compile-time rational arithmetic.
253 *
254 * This class template represents any finite rational number with a
255 * numerator and denominator representable by compile-time constants of
256 * type intmax_t. The ratio is simplified when instantiated.
257 *
258 * For example:
259 * @code
260 * std::ratio<7,-21>::num == -1;
261 * std::ratio<7,-21>::den == 3;
262 * @endcode
263 *
264 */
265 template<intmax_t _Num, intmax_t _Den = 1>
266 struct ratio
267 {
268 static_assert(_Den != 0, "denominator cannot be zero");
269 static_assert(_Num >= -__INTMAX_MAX__ && _Den >= -__INTMAX_MAX__,
270 "out of range");
271
272 // Note: sign(N) * abs(N) == N
273 static constexpr intmax_t num =
274 _Num * __static_sign<_Den>::value / __static_gcd<_Num, _Den>::value;
275
276 static constexpr intmax_t den =
277 __static_abs<_Den>::value / __static_gcd<_Num, _Den>::value;
278
279 typedef ratio<num, den> type;
280 };
281
282#if ! __cpp_inline_variables
283 template<intmax_t _Num, intmax_t _Den>
284 constexpr intmax_t ratio<_Num, _Den>::num;
285
286 template<intmax_t _Num, intmax_t _Den>
287 constexpr intmax_t ratio<_Num, _Den>::den;
288#endif
289
290 /// @cond undocumented
291
292 template<typename _Tp>
293 struct __is_ratio
294 : std::false_type
295 { };
296
297 template<intmax_t _Num, intmax_t _Den>
298 struct __is_ratio<ratio<_Num, _Den>>
299 : std::true_type
300 { };
301
302#if __cpp_variable_templates
303 template<typename _Tp>
304 constexpr bool __is_ratio_v = false;
305 template<intmax_t _Num, intmax_t _Den>
306 constexpr bool __is_ratio_v<ratio<_Num, _Den>> = true;
307#endif
308
309 template<typename _R1, typename _R2>
310 constexpr bool
311 __are_both_ratios() noexcept
312 {
313#if __cpp_variable_templates && __cpp_if_constexpr
314 if constexpr (__is_ratio_v<_R1>)
315 if constexpr (__is_ratio_v<_R2>)
316 return true;
317 return false;
318#else
319 return __and_<__is_ratio<_R1>, __is_ratio<_R2>>::value;
320#endif
321 }
322
323 template<typename _R1, typename _R2>
324 struct __ratio_multiply
325 {
326 static_assert(std::__are_both_ratios<_R1, _R2>(),
327 "both template arguments must be a std::ratio");
328
329 private:
330 static const intmax_t __gcd1 =
331 __static_gcd<_R1::num, _R2::den>::value;
332 static const intmax_t __gcd2 =
333 __static_gcd<_R2::num, _R1::den>::value;
334
335 public:
336 typedef ratio<
337 __safe_multiply<(_R1::num / __gcd1),
338 (_R2::num / __gcd2)>::value,
339 __safe_multiply<(_R1::den / __gcd2),
340 (_R2::den / __gcd1)>::value> type;
341
342 static constexpr intmax_t num = type::num;
343 static constexpr intmax_t den = type::den;
344 };
345
346#if ! __cpp_inline_variables
347 template<typename _R1, typename _R2>
348 constexpr intmax_t __ratio_multiply<_R1, _R2>::num;
349
350 template<typename _R1, typename _R2>
351 constexpr intmax_t __ratio_multiply<_R1, _R2>::den;
352#endif
353
354 /// @endcond
355
356 /// ratio_multiply
357 template<typename _R1, typename _R2>
358 using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;
359
360 /// @cond undocumented
361
362 template<typename _R1, typename _R2>
363 struct __ratio_divide
364 {
365 static_assert(_R2::num != 0, "division by 0");
366
367 typedef typename __ratio_multiply<
368 _R1,
369 ratio<_R2::den, _R2::num>>::type type;
370
371 static constexpr intmax_t num = type::num;
372 static constexpr intmax_t den = type::den;
373 };
374
375#if ! __cpp_inline_variables
376 template<typename _R1, typename _R2>
377 constexpr intmax_t __ratio_divide<_R1, _R2>::num;
378
379 template<typename _R1, typename _R2>
380 constexpr intmax_t __ratio_divide<_R1, _R2>::den;
381#endif
382
383 /// @endcond
384
385 /// ratio_divide
386 template<typename _R1, typename _R2>
387 using ratio_divide = typename __ratio_divide<_R1, _R2>::type;
388
389 /// ratio_equal
390 template<typename _R1, typename _R2>
391 struct ratio_equal
392 : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den>
393 {
394 static_assert(std::__are_both_ratios<_R1, _R2>(),
395 "both template arguments must be a std::ratio");
396 };
397
398 /// ratio_not_equal
399 template<typename _R1, typename _R2>
400 struct ratio_not_equal
401 : integral_constant<bool, !ratio_equal<_R1, _R2>::value>
402 { };
403
404 /// @cond undocumented
405
406 // Both numbers are positive.
407 template<typename _R1, typename _R2,
408 typename _Left = __big_mul<_R1::num,_R2::den>,
409 typename _Right = __big_mul<_R2::num,_R1::den> >
410 struct __ratio_less_impl_1
411 : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo,
412 _Right::__hi, _Right::__lo>::value>
413 { };
414
415 template<typename _R1, typename _R2,
416 bool = (_R1::num == 0 || _R2::num == 0
417 || (__static_sign<_R1::num>::value
418 != __static_sign<_R2::num>::value)),
419 bool = (__static_sign<_R1::num>::value == -1
420 && __static_sign<_R2::num>::value == -1)>
421 struct __ratio_less_impl
422 : __ratio_less_impl_1<_R1, _R2>::type
423 { };
424
425 template<typename _R1, typename _R2>
426 struct __ratio_less_impl<_R1, _R2, true, false>
427 : integral_constant<bool, _R1::num < _R2::num>
428 { };
429
430 template<typename _R1, typename _R2>
431 struct __ratio_less_impl<_R1, _R2, false, true>
432 : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>,
433 ratio<-_R1::num, _R1::den> >::type
434 { };
435
436 /// @endcond
437
438 /// ratio_less
439 template<typename _R1, typename _R2>
440 struct ratio_less
441 : __ratio_less_impl<_R1, _R2>::type
442 {
443 static_assert(std::__are_both_ratios<_R1, _R2>(),
444 "both template arguments must be a std::ratio");
445 };
446
447 /// ratio_less_equal
448 template<typename _R1, typename _R2>
449 struct ratio_less_equal
450 : integral_constant<bool, !ratio_less<_R2, _R1>::value>
451 { };
452
453 /// ratio_greater
454 template<typename _R1, typename _R2>
455 struct ratio_greater
456 : integral_constant<bool, ratio_less<_R2, _R1>::value>
457 { };
458
459 /// ratio_greater_equal
460 template<typename _R1, typename _R2>
461 struct ratio_greater_equal
462 : integral_constant<bool, !ratio_less<_R1, _R2>::value>
463 { };
464
465#if __cplusplus > 201402L
466 template <typename _R1, typename _R2>
467 inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value;
468 template <typename _R1, typename _R2>
469 inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value;
470 template <typename _R1, typename _R2>
471 inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value;
472 template <typename _R1, typename _R2>
473 inline constexpr bool ratio_less_equal_v
474 = ratio_less_equal<_R1, _R2>::value;
475 template <typename _R1, typename _R2>
476 inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value;
477 template <typename _R1, typename _R2>
478 inline constexpr bool ratio_greater_equal_v
479 = ratio_greater_equal<_R1, _R2>::value;
480#endif // C++17
481
482 /// @cond undocumented
483
484 template<typename _R1, typename _R2,
485 bool = (_R1::num >= 0),
486 bool = (_R2::num >= 0),
487 bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>,
488 ratio<__static_abs<_R2::num>::value, _R2::den> >::value>
489 struct __ratio_add_impl
490 {
491 private:
492 typedef typename __ratio_add_impl<
493 ratio<-_R1::num, _R1::den>,
494 ratio<-_R2::num, _R2::den> >::type __t;
495 public:
496 typedef ratio<-__t::num, __t::den> type;
497 };
498
499 // True addition of nonnegative numbers.
500 template<typename _R1, typename _R2, bool __b>
501 struct __ratio_add_impl<_R1, _R2, true, true, __b>
502 {
503 private:
504 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
505 static constexpr uintmax_t __d2 = _R2::den / __g;
506 typedef __big_mul<_R1::den, __d2> __d;
507 typedef __big_mul<_R1::num, _R2::den / __g> __x;
508 typedef __big_mul<_R2::num, _R1::den / __g> __y;
509 typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
510 static_assert(__n::__hi >= __x::__hi, "Internal library error");
511 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
512 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
513 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
514 static_assert(__n_final::__rem == 0, "Internal library error");
515 static_assert(__n_final::__quot_hi == 0 &&
516 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
517 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
518 static_assert(__d_final::__hi == 0 &&
519 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
520 public:
521 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
522 };
523
524 template<typename _R1, typename _R2>
525 struct __ratio_add_impl<_R1, _R2, false, true, true>
526 : __ratio_add_impl<_R2, _R1>
527 { };
528
529 // True subtraction of nonnegative numbers yielding a nonnegative result.
530 template<typename _R1, typename _R2>
531 struct __ratio_add_impl<_R1, _R2, true, false, false>
532 {
533 private:
534 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
535 static constexpr uintmax_t __d2 = _R2::den / __g;
536 typedef __big_mul<_R1::den, __d2> __d;
537 typedef __big_mul<_R1::num, _R2::den / __g> __x;
538 typedef __big_mul<-_R2::num, _R1::den / __g> __y;
539 typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
540 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
541 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
542 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
543 static_assert(__n_final::__rem == 0, "Internal library error");
544 static_assert(__n_final::__quot_hi == 0 &&
545 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
546 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
547 static_assert(__d_final::__hi == 0 &&
548 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
549 public:
550 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
551 };
552
553 template<typename _R1, typename _R2>
554 struct __ratio_add
555 {
556 static_assert(std::__are_both_ratios<_R1, _R2>(),
557 "both template arguments must be a std::ratio");
558
559 typedef typename __ratio_add_impl<_R1, _R2>::type type;
560 static constexpr intmax_t num = type::num;
561 static constexpr intmax_t den = type::den;
562 };
563
564#if ! __cpp_inline_variables
565 template<typename _R1, typename _R2>
566 constexpr intmax_t __ratio_add<_R1, _R2>::num;
567
568 template<typename _R1, typename _R2>
569 constexpr intmax_t __ratio_add<_R1, _R2>::den;
570#endif
571
572 /// @endcond
573
574 /// ratio_add
575 template<typename _R1, typename _R2>
576 using ratio_add = typename __ratio_add<_R1, _R2>::type;
577
578 /// @cond undocumented
579
580 template<typename _R1, typename _R2>
581 struct __ratio_subtract
582 {
583 typedef typename __ratio_add<
584 _R1,
585 ratio<-_R2::num, _R2::den>>::type type;
586
587 static constexpr intmax_t num = type::num;
588 static constexpr intmax_t den = type::den;
589 };
590
591#if ! __cpp_inline_variables
592 template<typename _R1, typename _R2>
593 constexpr intmax_t __ratio_subtract<_R1, _R2>::num;
594
595 template<typename _R1, typename _R2>
596 constexpr intmax_t __ratio_subtract<_R1, _R2>::den;
597#endif
598
599 /// @endcond
600
601 /// ratio_subtract
602 template<typename _R1, typename _R2>
603 using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type;
604
605
606 typedef ratio<1, 1000000000000000000> atto;
607 typedef ratio<1, 1000000000000000> femto;
608 typedef ratio<1, 1000000000000> pico;
609 typedef ratio<1, 1000000000> nano;
610 typedef ratio<1, 1000000> micro;
611 typedef ratio<1, 1000> milli;
612 typedef ratio<1, 100> centi;
613 typedef ratio<1, 10> deci;
614 typedef ratio< 10, 1> deca;
615 typedef ratio< 100, 1> hecto;
616 typedef ratio< 1000, 1> kilo;
617 typedef ratio< 1000000, 1> mega;
618 typedef ratio< 1000000000, 1> giga;
619 typedef ratio< 1000000000000, 1> tera;
620 typedef ratio< 1000000000000000, 1> peta;
621 typedef ratio< 1000000000000000000, 1> exa;
622
623 /// @} group ratio
624_GLIBCXX_END_NAMESPACE_VERSION
625} // namespace
626
627#endif // C++11
628
629#endif //_GLIBCXX_RATIO
630