libstdc++
ratio
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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 
41 namespace 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