1// ratio -*- C++ -*-
2
3// Copyright (C) 2008-2022 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 _R1, typename _R2>
293 struct __ratio_multiply
294 {
295 private:
296 static const intmax_t __gcd1 =
297 __static_gcd<_R1::num, _R2::den>::value;
298 static const intmax_t __gcd2 =
299 __static_gcd<_R2::num, _R1::den>::value;
300
301 public:
302 typedef ratio<
303 __safe_multiply<(_R1::num / __gcd1),
304 (_R2::num / __gcd2)>::value,
305 __safe_multiply<(_R1::den / __gcd2),
306 (_R2::den / __gcd1)>::value> type;
307
308 static constexpr intmax_t num = type::num;
309 static constexpr intmax_t den = type::den;
310 };
311
312#if ! __cpp_inline_variables
313 template<typename _R1, typename _R2>
314 constexpr intmax_t __ratio_multiply<_R1, _R2>::num;
315
316 template<typename _R1, typename _R2>
317 constexpr intmax_t __ratio_multiply<_R1, _R2>::den;
318#endif
319
320 /// @endcond
321
322 /// ratio_multiply
323 template<typename _R1, typename _R2>
324 using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;
325
326 /// @cond undocumented
327
328 template<typename _R1, typename _R2>
329 struct __ratio_divide
330 {
331 static_assert(_R2::num != 0, "division by 0");
332
333 typedef typename __ratio_multiply<
334 _R1,
335 ratio<_R2::den, _R2::num>>::type type;
336
337 static constexpr intmax_t num = type::num;
338 static constexpr intmax_t den = type::den;
339 };
340
341#if ! __cpp_inline_variables
342 template<typename _R1, typename _R2>
343 constexpr intmax_t __ratio_divide<_R1, _R2>::num;
344
345 template<typename _R1, typename _R2>
346 constexpr intmax_t __ratio_divide<_R1, _R2>::den;
347#endif
348
349 /// @endcond
350
351 /// ratio_divide
352 template<typename _R1, typename _R2>
353 using ratio_divide = typename __ratio_divide<_R1, _R2>::type;
354
355 /// ratio_equal
356 template<typename _R1, typename _R2>
357 struct ratio_equal
358 : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den>
359 { };
360
361 /// ratio_not_equal
362 template<typename _R1, typename _R2>
363 struct ratio_not_equal
364 : integral_constant<bool, !ratio_equal<_R1, _R2>::value>
365 { };
366
367 /// @cond undocumented
368
369 // Both numbers are positive.
370 template<typename _R1, typename _R2,
371 typename _Left = __big_mul<_R1::num,_R2::den>,
372 typename _Right = __big_mul<_R2::num,_R1::den> >
373 struct __ratio_less_impl_1
374 : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo,
375 _Right::__hi, _Right::__lo>::value>
376 { };
377
378 template<typename _R1, typename _R2,
379 bool = (_R1::num == 0 || _R2::num == 0
380 || (__static_sign<_R1::num>::value
381 != __static_sign<_R2::num>::value)),
382 bool = (__static_sign<_R1::num>::value == -1
383 && __static_sign<_R2::num>::value == -1)>
384 struct __ratio_less_impl
385 : __ratio_less_impl_1<_R1, _R2>::type
386 { };
387
388 template<typename _R1, typename _R2>
389 struct __ratio_less_impl<_R1, _R2, true, false>
390 : integral_constant<bool, _R1::num < _R2::num>
391 { };
392
393 template<typename _R1, typename _R2>
394 struct __ratio_less_impl<_R1, _R2, false, true>
395 : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>,
396 ratio<-_R1::num, _R1::den> >::type
397 { };
398
399 /// @endcond
400
401 /// ratio_less
402 template<typename _R1, typename _R2>
403 struct ratio_less
404 : __ratio_less_impl<_R1, _R2>::type
405 { };
406
407 /// ratio_less_equal
408 template<typename _R1, typename _R2>
409 struct ratio_less_equal
410 : integral_constant<bool, !ratio_less<_R2, _R1>::value>
411 { };
412
413 /// ratio_greater
414 template<typename _R1, typename _R2>
415 struct ratio_greater
416 : integral_constant<bool, ratio_less<_R2, _R1>::value>
417 { };
418
419 /// ratio_greater_equal
420 template<typename _R1, typename _R2>
421 struct ratio_greater_equal
422 : integral_constant<bool, !ratio_less<_R1, _R2>::value>
423 { };
424
425#if __cplusplus > 201402L
426 template <typename _R1, typename _R2>
427 inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value;
428 template <typename _R1, typename _R2>
429 inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value;
430 template <typename _R1, typename _R2>
431 inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value;
432 template <typename _R1, typename _R2>
433 inline constexpr bool ratio_less_equal_v =
434 ratio_less_equal<_R1, _R2>::value;
435 template <typename _R1, typename _R2>
436 inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value;
437 template <typename _R1, typename _R2>
438 inline constexpr bool ratio_greater_equal_v
439 = ratio_greater_equal<_R1, _R2>::value;
440#endif // C++17
441
442 /// @cond undocumented
443
444 template<typename _R1, typename _R2,
445 bool = (_R1::num >= 0),
446 bool = (_R2::num >= 0),
447 bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>,
448 ratio<__static_abs<_R2::num>::value, _R2::den> >::value>
449 struct __ratio_add_impl
450 {
451 private:
452 typedef typename __ratio_add_impl<
453 ratio<-_R1::num, _R1::den>,
454 ratio<-_R2::num, _R2::den> >::type __t;
455 public:
456 typedef ratio<-__t::num, __t::den> type;
457 };
458
459 // True addition of nonnegative numbers.
460 template<typename _R1, typename _R2, bool __b>
461 struct __ratio_add_impl<_R1, _R2, true, true, __b>
462 {
463 private:
464 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
465 static constexpr uintmax_t __d2 = _R2::den / __g;
466 typedef __big_mul<_R1::den, __d2> __d;
467 typedef __big_mul<_R1::num, _R2::den / __g> __x;
468 typedef __big_mul<_R2::num, _R1::den / __g> __y;
469 typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
470 static_assert(__n::__hi >= __x::__hi, "Internal library error");
471 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
472 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
473 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
474 static_assert(__n_final::__rem == 0, "Internal library error");
475 static_assert(__n_final::__quot_hi == 0 &&
476 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
477 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
478 static_assert(__d_final::__hi == 0 &&
479 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
480 public:
481 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
482 };
483
484 template<typename _R1, typename _R2>
485 struct __ratio_add_impl<_R1, _R2, false, true, true>
486 : __ratio_add_impl<_R2, _R1>
487 { };
488
489 // True subtraction of nonnegative numbers yielding a nonnegative result.
490 template<typename _R1, typename _R2>
491 struct __ratio_add_impl<_R1, _R2, true, false, false>
492 {
493 private:
494 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
495 static constexpr uintmax_t __d2 = _R2::den / __g;
496 typedef __big_mul<_R1::den, __d2> __d;
497 typedef __big_mul<_R1::num, _R2::den / __g> __x;
498 typedef __big_mul<-_R2::num, _R1::den / __g> __y;
499 typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
500 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
501 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
502 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
503 static_assert(__n_final::__rem == 0, "Internal library error");
504 static_assert(__n_final::__quot_hi == 0 &&
505 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
506 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
507 static_assert(__d_final::__hi == 0 &&
508 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
509 public:
510 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
511 };
512
513 template<typename _R1, typename _R2>
514 struct __ratio_add
515 {
516 typedef typename __ratio_add_impl<_R1, _R2>::type type;
517 static constexpr intmax_t num = type::num;
518 static constexpr intmax_t den = type::den;
519 };
520
521#if ! __cpp_inline_variables
522 template<typename _R1, typename _R2>
523 constexpr intmax_t __ratio_add<_R1, _R2>::num;
524
525 template<typename _R1, typename _R2>
526 constexpr intmax_t __ratio_add<_R1, _R2>::den;
527#endif
528
529 /// @endcond
530
531 /// ratio_add
532 template<typename _R1, typename _R2>
533 using ratio_add = typename __ratio_add<_R1, _R2>::type;
534
535 /// @cond undocumented
536
537 template<typename _R1, typename _R2>
538 struct __ratio_subtract
539 {
540 typedef typename __ratio_add<
541 _R1,
542 ratio<-_R2::num, _R2::den>>::type type;
543
544 static constexpr intmax_t num = type::num;
545 static constexpr intmax_t den = type::den;
546 };
547
548#if ! __cpp_inline_variables
549 template<typename _R1, typename _R2>
550 constexpr intmax_t __ratio_subtract<_R1, _R2>::num;
551
552 template<typename _R1, typename _R2>
553 constexpr intmax_t __ratio_subtract<_R1, _R2>::den;
554#endif
555
556 /// @endcond
557
558 /// ratio_subtract
559 template<typename _R1, typename _R2>
560 using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type;
561
562
563 typedef ratio<1, 1000000000000000000> atto;
564 typedef ratio<1, 1000000000000000> femto;
565 typedef ratio<1, 1000000000000> pico;
566 typedef ratio<1, 1000000000> nano;
567 typedef ratio<1, 1000000> micro;
568 typedef ratio<1, 1000> milli;
569 typedef ratio<1, 100> centi;
570 typedef ratio<1, 10> deci;
571 typedef ratio< 10, 1> deca;
572 typedef ratio< 100, 1> hecto;
573 typedef ratio< 1000, 1> kilo;
574 typedef ratio< 1000000, 1> mega;
575 typedef ratio< 1000000000, 1> giga;
576 typedef ratio< 1000000000000, 1> tera;
577 typedef ratio< 1000000000000000, 1> peta;
578 typedef ratio< 1000000000000000000, 1> exa;
579
580 /// @} group ratio
581_GLIBCXX_END_NAMESPACE_VERSION
582} // namespace
583
584#endif // C++11
585
586#endif //_GLIBCXX_RATIO
587