1// ratio -*- C++ -*-
2
3// Copyright (C) 2008-2019 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 */
28
29#ifndef _GLIBCXX_RATIO
30#define _GLIBCXX_RATIO 1
31
32#pragma GCC system_header
33
34#if __cplusplus < 201103L
35# include <bits/c++0x_warning.h>
36#else
37
38#include <type_traits>
39#include <cstdint> // intmax_t, uintmax_t
40
41namespace std _GLIBCXX_VISIBILITY(default)
42{
43_GLIBCXX_BEGIN_NAMESPACE_VERSION
44
45 /**
46 * @defgroup ratio Rational Arithmetic
47 * @ingroup utilities
48 *
49 * Compile time representation of finite rational numbers.
50 * @{
51 */
52
53 template<intmax_t _Pn>
54 struct __static_sign
55 : integral_constant<intmax_t, (_Pn < 0) ? -1 : 1>
56 { };
57
58 template<intmax_t _Pn>
59 struct __static_abs
60 : integral_constant<intmax_t, _Pn * __static_sign<_Pn>::value>
61 { };
62
63 template<intmax_t _Pn, intmax_t _Qn>
64 struct __static_gcd
65 : __static_gcd<_Qn, (_Pn % _Qn)>
66 { };
67
68 template<intmax_t _Pn>
69 struct __static_gcd<_Pn, 0>
70 : integral_constant<intmax_t, __static_abs<_Pn>::value>
71 { };
72
73 template<intmax_t _Qn>
74 struct __static_gcd<0, _Qn>
75 : integral_constant<intmax_t, __static_abs<_Qn>::value>
76 { };
77
78 // Let c = 2^(half # of bits in an intmax_t)
79 // then we find a1, a0, b1, b0 s.t. N = a1*c + a0, M = b1*c + b0
80 // The multiplication of N and M becomes,
81 // N * M = (a1 * b1)c^2 + (a0 * b1 + b0 * a1)c + a0 * b0
82 // Multiplication is safe if each term and the sum of the terms
83 // is representable by intmax_t.
84 template<intmax_t _Pn, intmax_t _Qn>
85 struct __safe_multiply
86 {
87 private:
88 static const uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
89
90 static const uintmax_t __a0 = __static_abs<_Pn>::value % __c;
91 static const uintmax_t __a1 = __static_abs<_Pn>::value / __c;
92 static const uintmax_t __b0 = __static_abs<_Qn>::value % __c;
93 static const uintmax_t __b1 = __static_abs<_Qn>::value / __c;
94
95 static_assert(__a1 == 0 || __b1 == 0,
96 "overflow in multiplication");
97 static_assert(__a0 * __b1 + __b0 * __a1 < (__c >> 1),
98 "overflow in multiplication");
99 static_assert(__b0 * __a0 <= __INTMAX_MAX__,
100 "overflow in multiplication");
101 static_assert((__a0 * __b1 + __b0 * __a1) * __c
102 <= __INTMAX_MAX__ - __b0 * __a0,
103 "overflow in multiplication");
104
105 public:
106 static const intmax_t value = _Pn * _Qn;
107 };
108
109 // Some double-precision utilities, where numbers are represented as
110 // __hi*2^(8*sizeof(uintmax_t)) + __lo.
111 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
112 struct __big_less
113 : integral_constant<bool, (__hi1 < __hi2
114 || (__hi1 == __hi2 && __lo1 < __lo2))>
115 { };
116
117 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
118 struct __big_add
119 {
120 static constexpr uintmax_t __lo = __lo1 + __lo2;
121 static constexpr uintmax_t __hi = (__hi1 + __hi2 +
122 (__lo1 + __lo2 < __lo1)); // carry
123 };
124
125 // Subtract a number from a bigger one.
126 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2>
127 struct __big_sub
128 {
129 static_assert(!__big_less<__hi1, __lo1, __hi2, __lo2>::value,
130 "Internal library error");
131 static constexpr uintmax_t __lo = __lo1 - __lo2;
132 static constexpr uintmax_t __hi = (__hi1 - __hi2 -
133 (__lo1 < __lo2)); // carry
134 };
135
136 // Same principle as __safe_multiply.
137 template<uintmax_t __x, uintmax_t __y>
138 struct __big_mul
139 {
140 private:
141 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
142 static constexpr uintmax_t __x0 = __x % __c;
143 static constexpr uintmax_t __x1 = __x / __c;
144 static constexpr uintmax_t __y0 = __y % __c;
145 static constexpr uintmax_t __y1 = __y / __c;
146 static constexpr uintmax_t __x0y0 = __x0 * __y0;
147 static constexpr uintmax_t __x0y1 = __x0 * __y1;
148 static constexpr uintmax_t __x1y0 = __x1 * __y0;
149 static constexpr uintmax_t __x1y1 = __x1 * __y1;
150 static constexpr uintmax_t __mix = __x0y1 + __x1y0; // possible carry...
151 static constexpr uintmax_t __mix_lo = __mix * __c;
152 static constexpr uintmax_t __mix_hi
153 = __mix / __c + ((__mix < __x0y1) ? __c : 0); // ... added here
154 typedef __big_add<__mix_hi, __mix_lo, __x1y1, __x0y0> _Res;
155 public:
156 static constexpr uintmax_t __hi = _Res::__hi;
157 static constexpr uintmax_t __lo = _Res::__lo;
158 };
159
160 // Adapted from __udiv_qrnnd_c in longlong.h
161 // This version assumes that the high bit of __d is 1.
162 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
163 struct __big_div_impl
164 {
165 private:
166 static_assert(__d >= (uintmax_t(1) << (sizeof(intmax_t) * 8 - 1)),
167 "Internal library error");
168 static_assert(__n1 < __d, "Internal library error");
169 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4);
170 static constexpr uintmax_t __d1 = __d / __c;
171 static constexpr uintmax_t __d0 = __d % __c;
172
173 static constexpr uintmax_t __q1x = __n1 / __d1;
174 static constexpr uintmax_t __r1x = __n1 % __d1;
175 static constexpr uintmax_t __m = __q1x * __d0;
176 static constexpr uintmax_t __r1y = __r1x * __c + __n0 / __c;
177 static constexpr uintmax_t __r1z = __r1y + __d;
178 static constexpr uintmax_t __r1
179 = ((__r1y < __m) ? ((__r1z >= __d) && (__r1z < __m))
180 ? (__r1z + __d) : __r1z : __r1y) - __m;
181 static constexpr uintmax_t __q1
182 = __q1x - ((__r1y < __m)
183 ? ((__r1z >= __d) && (__r1z < __m)) ? 2 : 1 : 0);
184 static constexpr uintmax_t __q0x = __r1 / __d1;
185 static constexpr uintmax_t __r0x = __r1 % __d1;
186 static constexpr uintmax_t __n = __q0x * __d0;
187 static constexpr uintmax_t __r0y = __r0x * __c + __n0 % __c;
188 static constexpr uintmax_t __r0z = __r0y + __d;
189 static constexpr uintmax_t __r0
190 = ((__r0y < __n) ? ((__r0z >= __d) && (__r0z < __n))
191 ? (__r0z + __d) : __r0z : __r0y) - __n;
192 static constexpr uintmax_t __q0
193 = __q0x - ((__r0y < __n) ? ((__r0z >= __d)
194 && (__r0z < __n)) ? 2 : 1 : 0);
195
196 public:
197 static constexpr uintmax_t __quot = __q1 * __c + __q0;
198 static constexpr uintmax_t __rem = __r0;
199
200 private:
201 typedef __big_mul<__quot, __d> _Prod;
202 typedef __big_add<_Prod::__hi, _Prod::__lo, 0, __rem> _Sum;
203 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
204 "Internal library error");
205 };
206
207 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d>
208 struct __big_div
209 {
210 private:
211 static_assert(__d != 0, "Internal library error");
212 static_assert(sizeof (uintmax_t) == sizeof (unsigned long long),
213 "This library calls __builtin_clzll on uintmax_t, which "
214 "is unsafe on your platform. Please complain to "
215 "http://gcc.gnu.org/bugzilla/");
216 static constexpr int __shift = __builtin_clzll(__d);
217 static constexpr int __coshift_ = sizeof(uintmax_t) * 8 - __shift;
218 static constexpr int __coshift = (__shift != 0) ? __coshift_ : 0;
219 static constexpr uintmax_t __c1 = uintmax_t(1) << __shift;
220 static constexpr uintmax_t __c2 = uintmax_t(1) << __coshift;
221 static constexpr uintmax_t __new_d = __d * __c1;
222 static constexpr uintmax_t __new_n0 = __n0 * __c1;
223 static constexpr uintmax_t __n1_shifted = (__n1 % __d) * __c1;
224 static constexpr uintmax_t __n0_top = (__shift != 0) ? (__n0 / __c2) : 0;
225 static constexpr uintmax_t __new_n1 = __n1_shifted + __n0_top;
226 typedef __big_div_impl<__new_n1, __new_n0, __new_d> _Res;
227
228 public:
229 static constexpr uintmax_t __quot_hi = __n1 / __d;
230 static constexpr uintmax_t __quot_lo = _Res::__quot;
231 static constexpr uintmax_t __rem = _Res::__rem / __c1;
232
233 private:
234 typedef __big_mul<__quot_lo, __d> _P0;
235 typedef __big_mul<__quot_hi, __d> _P1;
236 typedef __big_add<_P0::__hi, _P0::__lo, _P1::__lo, __rem> _Sum;
237 // No overflow.
238 static_assert(_P1::__hi == 0, "Internal library error");
239 static_assert(_Sum::__hi >= _P0::__hi, "Internal library error");
240 // Matches the input data.
241 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0,
242 "Internal library error");
243 static_assert(__rem < __d, "Internal library error");
244 };
245
246 /**
247 * @brief Provides compile-time rational arithmetic.
248 *
249 * This class template represents any finite rational number with a
250 * numerator and denominator representable by compile-time constants of
251 * type intmax_t. The ratio is simplified when instantiated.
252 *
253 * For example:
254 * @code
255 * std::ratio<7,-21>::num == -1;
256 * std::ratio<7,-21>::den == 3;
257 * @endcode
258 *
259 */
260 template<intmax_t _Num, intmax_t _Den = 1>
261 struct ratio
262 {
263 static_assert(_Den != 0, "denominator cannot be zero");
264 static_assert(_Num >= -__INTMAX_MAX__ && _Den >= -__INTMAX_MAX__,
265 "out of range");
266
267 // Note: sign(N) * abs(N) == N
268 static constexpr intmax_t num =
269 _Num * __static_sign<_Den>::value / __static_gcd<_Num, _Den>::value;
270
271 static constexpr intmax_t den =
272 __static_abs<_Den>::value / __static_gcd<_Num, _Den>::value;
273
274 typedef ratio<num, den> type;
275 };
276
277 template<intmax_t _Num, intmax_t _Den>
278 constexpr intmax_t ratio<_Num, _Den>::num;
279
280 template<intmax_t _Num, intmax_t _Den>
281 constexpr intmax_t ratio<_Num, _Den>::den;
282
283 template<typename _R1, typename _R2>
284 struct __ratio_multiply
285 {
286 private:
287 static const intmax_t __gcd1 =
288 __static_gcd<_R1::num, _R2::den>::value;
289 static const intmax_t __gcd2 =
290 __static_gcd<_R2::num, _R1::den>::value;
291
292 public:
293 typedef ratio<
294 __safe_multiply<(_R1::num / __gcd1),
295 (_R2::num / __gcd2)>::value,
296 __safe_multiply<(_R1::den / __gcd2),
297 (_R2::den / __gcd1)>::value> type;
298
299 static constexpr intmax_t num = type::num;
300 static constexpr intmax_t den = type::den;
301 };
302
303 template<typename _R1, typename _R2>
304 constexpr intmax_t __ratio_multiply<_R1, _R2>::num;
305
306 template<typename _R1, typename _R2>
307 constexpr intmax_t __ratio_multiply<_R1, _R2>::den;
308
309 /// ratio_multiply
310 template<typename _R1, typename _R2>
311 using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type;
312
313 template<typename _R1, typename _R2>
314 struct __ratio_divide
315 {
316 static_assert(_R2::num != 0, "division by 0");
317
318 typedef typename __ratio_multiply<
319 _R1,
320 ratio<_R2::den, _R2::num>>::type type;
321
322 static constexpr intmax_t num = type::num;
323 static constexpr intmax_t den = type::den;
324 };
325
326 template<typename _R1, typename _R2>
327 constexpr intmax_t __ratio_divide<_R1, _R2>::num;
328
329 template<typename _R1, typename _R2>
330 constexpr intmax_t __ratio_divide<_R1, _R2>::den;
331
332 /// ratio_divide
333 template<typename _R1, typename _R2>
334 using ratio_divide = typename __ratio_divide<_R1, _R2>::type;
335
336 /// ratio_equal
337 template<typename _R1, typename _R2>
338 struct ratio_equal
339 : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den>
340 { };
341
342 /// ratio_not_equal
343 template<typename _R1, typename _R2>
344 struct ratio_not_equal
345 : integral_constant<bool, !ratio_equal<_R1, _R2>::value>
346 { };
347
348 // Both numbers are positive.
349 template<typename _R1, typename _R2,
350 typename _Left = __big_mul<_R1::num,_R2::den>,
351 typename _Right = __big_mul<_R2::num,_R1::den> >
352 struct __ratio_less_impl_1
353 : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo,
354 _Right::__hi, _Right::__lo>::value>
355 { };
356
357 template<typename _R1, typename _R2,
358 bool = (_R1::num == 0 || _R2::num == 0
359 || (__static_sign<_R1::num>::value
360 != __static_sign<_R2::num>::value)),
361 bool = (__static_sign<_R1::num>::value == -1
362 && __static_sign<_R2::num>::value == -1)>
363 struct __ratio_less_impl
364 : __ratio_less_impl_1<_R1, _R2>::type
365 { };
366
367 template<typename _R1, typename _R2>
368 struct __ratio_less_impl<_R1, _R2, true, false>
369 : integral_constant<bool, _R1::num < _R2::num>
370 { };
371
372 template<typename _R1, typename _R2>
373 struct __ratio_less_impl<_R1, _R2, false, true>
374 : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>,
375 ratio<-_R1::num, _R1::den> >::type
376 { };
377
378 /// ratio_less
379 template<typename _R1, typename _R2>
380 struct ratio_less
381 : __ratio_less_impl<_R1, _R2>::type
382 { };
383
384 /// ratio_less_equal
385 template<typename _R1, typename _R2>
386 struct ratio_less_equal
387 : integral_constant<bool, !ratio_less<_R2, _R1>::value>
388 { };
389
390 /// ratio_greater
391 template<typename _R1, typename _R2>
392 struct ratio_greater
393 : integral_constant<bool, ratio_less<_R2, _R1>::value>
394 { };
395
396 /// ratio_greater_equal
397 template<typename _R1, typename _R2>
398 struct ratio_greater_equal
399 : integral_constant<bool, !ratio_less<_R1, _R2>::value>
400 { };
401
402#if __cplusplus > 201402L
403 template <typename _R1, typename _R2>
404 inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value;
405 template <typename _R1, typename _R2>
406 inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value;
407 template <typename _R1, typename _R2>
408 inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value;
409 template <typename _R1, typename _R2>
410 inline constexpr bool ratio_less_equal_v =
411 ratio_less_equal<_R1, _R2>::value;
412 template <typename _R1, typename _R2>
413 inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value;
414 template <typename _R1, typename _R2>
415 inline constexpr bool ratio_greater_equal_v
416 = ratio_greater_equal<_R1, _R2>::value;
417#endif // C++17
418
419 template<typename _R1, typename _R2,
420 bool = (_R1::num >= 0),
421 bool = (_R2::num >= 0),
422 bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>,
423 ratio<__static_abs<_R2::num>::value, _R2::den> >::value>
424 struct __ratio_add_impl
425 {
426 private:
427 typedef typename __ratio_add_impl<
428 ratio<-_R1::num, _R1::den>,
429 ratio<-_R2::num, _R2::den> >::type __t;
430 public:
431 typedef ratio<-__t::num, __t::den> type;
432 };
433
434 // True addition of nonnegative numbers.
435 template<typename _R1, typename _R2, bool __b>
436 struct __ratio_add_impl<_R1, _R2, true, true, __b>
437 {
438 private:
439 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
440 static constexpr uintmax_t __d2 = _R2::den / __g;
441 typedef __big_mul<_R1::den, __d2> __d;
442 typedef __big_mul<_R1::num, _R2::den / __g> __x;
443 typedef __big_mul<_R2::num, _R1::den / __g> __y;
444 typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
445 static_assert(__n::__hi >= __x::__hi, "Internal library error");
446 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
447 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
448 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
449 static_assert(__n_final::__rem == 0, "Internal library error");
450 static_assert(__n_final::__quot_hi == 0 &&
451 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
452 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
453 static_assert(__d_final::__hi == 0 &&
454 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
455 public:
456 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
457 };
458
459 template<typename _R1, typename _R2>
460 struct __ratio_add_impl<_R1, _R2, false, true, true>
461 : __ratio_add_impl<_R2, _R1>
462 { };
463
464 // True subtraction of nonnegative numbers yielding a nonnegative result.
465 template<typename _R1, typename _R2>
466 struct __ratio_add_impl<_R1, _R2, true, false, false>
467 {
468 private:
469 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value;
470 static constexpr uintmax_t __d2 = _R2::den / __g;
471 typedef __big_mul<_R1::den, __d2> __d;
472 typedef __big_mul<_R1::num, _R2::den / __g> __x;
473 typedef __big_mul<-_R2::num, _R1::den / __g> __y;
474 typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n;
475 typedef __big_div<__n::__hi, __n::__lo, __g> __ng;
476 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value;
477 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final;
478 static_assert(__n_final::__rem == 0, "Internal library error");
479 static_assert(__n_final::__quot_hi == 0 &&
480 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition");
481 typedef __big_mul<_R1::den / __g2, __d2> __d_final;
482 static_assert(__d_final::__hi == 0 &&
483 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition");
484 public:
485 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type;
486 };
487
488 template<typename _R1, typename _R2>
489 struct __ratio_add
490 {
491 typedef typename __ratio_add_impl<_R1, _R2>::type type;
492 static constexpr intmax_t num = type::num;
493 static constexpr intmax_t den = type::den;
494 };
495
496 template<typename _R1, typename _R2>
497 constexpr intmax_t __ratio_add<_R1, _R2>::num;
498
499 template<typename _R1, typename _R2>
500 constexpr intmax_t __ratio_add<_R1, _R2>::den;
501
502 /// ratio_add
503 template<typename _R1, typename _R2>
504 using ratio_add = typename __ratio_add<_R1, _R2>::type;
505
506 template<typename _R1, typename _R2>
507 struct __ratio_subtract
508 {
509 typedef typename __ratio_add<
510 _R1,
511 ratio<-_R2::num, _R2::den>>::type type;
512
513 static constexpr intmax_t num = type::num;
514 static constexpr intmax_t den = type::den;
515 };
516
517 template<typename _R1, typename _R2>
518 constexpr intmax_t __ratio_subtract<_R1, _R2>::num;
519
520 template<typename _R1, typename _R2>
521 constexpr intmax_t __ratio_subtract<_R1, _R2>::den;
522
523 /// ratio_subtract
524 template<typename _R1, typename _R2>
525 using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type;
526
527
528 typedef ratio<1, 1000000000000000000> atto;
529 typedef ratio<1, 1000000000000000> femto;
530 typedef ratio<1, 1000000000000> pico;
531 typedef ratio<1, 1000000000> nano;
532 typedef ratio<1, 1000000> micro;
533 typedef ratio<1, 1000> milli;
534 typedef ratio<1, 100> centi;
535 typedef ratio<1, 10> deci;
536 typedef ratio< 10, 1> deca;
537 typedef ratio< 100, 1> hecto;
538 typedef ratio< 1000, 1> kilo;
539 typedef ratio< 1000000, 1> mega;
540 typedef ratio< 1000000000, 1> giga;
541 typedef ratio< 1000000000000, 1> tera;
542 typedef ratio< 1000000000000000, 1> peta;
543 typedef ratio< 1000000000000000000, 1> exa;
544
545 /// @} group ratio
546_GLIBCXX_END_NAMESPACE_VERSION
547} // namespace
548
549#endif // C++11
550
551#endif //_GLIBCXX_RATIO
552