1// ratio -*- C++ -*- 
2 
3// Copyright (C) 2008-2021 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 template<intmax_t _Num, intmax_t _Den> 
283 constexpr intmax_t ratio<_Num, _Den>::num; 
284 
285 template<intmax_t _Num, intmax_t _Den> 
286 constexpr intmax_t ratio<_Num, _Den>::den; 
287 
288 /// @cond undocumented 
289 
290 template<typename _R1, typename _R2> 
291 struct __ratio_multiply 
292 { 
293 private: 
294 static const intmax_t __gcd1 = 
295 __static_gcd<_R1::num, _R2::den>::value; 
296 static const intmax_t __gcd2 = 
297 __static_gcd<_R2::num, _R1::den>::value; 
298 
299 public: 
300 typedef ratio< 
301 __safe_multiply<(_R1::num / __gcd1), 
302 (_R2::num / __gcd2)>::value, 
303 __safe_multiply<(_R1::den / __gcd2), 
304 (_R2::den / __gcd1)>::value> type; 
305 
306 static constexpr intmax_t num = type::num; 
307 static constexpr intmax_t den = type::den; 
308 }; 
309 
310 template<typename _R1, typename _R2> 
311 constexpr intmax_t __ratio_multiply<_R1, _R2>::num; 
312 
313 template<typename _R1, typename _R2> 
314 constexpr intmax_t __ratio_multiply<_R1, _R2>::den; 
315 
316 /// @endcond 
317 
318 /// ratio_multiply 
319 template<typename _R1, typename _R2> 
320 using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type; 
321 
322 /// @cond undocumented 
323 
324 template<typename _R1, typename _R2> 
325 struct __ratio_divide 
326 { 
327 static_assert(_R2::num != 0, "division by 0"); 
328 
329 typedef typename __ratio_multiply< 
330 _R1, 
331 ratio<_R2::den, _R2::num>>::type type; 
332 
333 static constexpr intmax_t num = type::num; 
334 static constexpr intmax_t den = type::den; 
335 }; 
336 
337 template<typename _R1, typename _R2> 
338 constexpr intmax_t __ratio_divide<_R1, _R2>::num; 
339 
340 template<typename _R1, typename _R2> 
341 constexpr intmax_t __ratio_divide<_R1, _R2>::den; 
342 
343 /// @endcond 
344 
345 /// ratio_divide 
346 template<typename _R1, typename _R2> 
347 using ratio_divide = typename __ratio_divide<_R1, _R2>::type; 
348 
349 /// ratio_equal 
350 template<typename _R1, typename _R2> 
351 struct ratio_equal 
352 : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den> 
353 { }; 
354 
355 /// ratio_not_equal 
356 template<typename _R1, typename _R2> 
357 struct ratio_not_equal 
358 : integral_constant<bool, !ratio_equal<_R1, _R2>::value> 
359 { }; 
360 
361 /// @cond undocumented 
362 
363 // Both numbers are positive. 
364 template<typename _R1, typename _R2, 
365 typename _Left = __big_mul<_R1::num,_R2::den>, 
366 typename _Right = __big_mul<_R2::num,_R1::den> > 
367 struct __ratio_less_impl_1 
368 : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo, 
369 _Right::__hi, _Right::__lo>::value> 
370 { }; 
371 
372 template<typename _R1, typename _R2, 
373 bool = (_R1::num == 0 || _R2::num == 0 
374 || (__static_sign<_R1::num>::value 
375 != __static_sign<_R2::num>::value)), 
376 bool = (__static_sign<_R1::num>::value == -1 
377 && __static_sign<_R2::num>::value == -1)> 
378 struct __ratio_less_impl 
379 : __ratio_less_impl_1<_R1, _R2>::type 
380 { }; 
381 
382 template<typename _R1, typename _R2> 
383 struct __ratio_less_impl<_R1, _R2, true, false> 
384 : integral_constant<bool, _R1::num < _R2::num> 
385 { }; 
386 
387 template<typename _R1, typename _R2> 
388 struct __ratio_less_impl<_R1, _R2, false, true> 
389 : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>, 
390 ratio<-_R1::num, _R1::den> >::type 
391 { }; 
392 
393 /// @endcond 
394 
395 /// ratio_less 
396 template<typename _R1, typename _R2> 
397 struct ratio_less 
398 : __ratio_less_impl<_R1, _R2>::type 
399 { }; 
400 
401 /// ratio_less_equal 
402 template<typename _R1, typename _R2> 
403 struct ratio_less_equal 
404 : integral_constant<bool, !ratio_less<_R2, _R1>::value> 
405 { }; 
406 
407 /// ratio_greater 
408 template<typename _R1, typename _R2> 
409 struct ratio_greater 
410 : integral_constant<bool, ratio_less<_R2, _R1>::value> 
411 { }; 
412 
413 /// ratio_greater_equal 
414 template<typename _R1, typename _R2> 
415 struct ratio_greater_equal 
416 : integral_constant<bool, !ratio_less<_R1, _R2>::value> 
417 { }; 
418 
419#if __cplusplus > 201402L 
420 template <typename _R1, typename _R2> 
421 inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value; 
422 template <typename _R1, typename _R2> 
423 inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value; 
424 template <typename _R1, typename _R2> 
425 inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value; 
426 template <typename _R1, typename _R2> 
427 inline constexpr bool ratio_less_equal_v = 
428 ratio_less_equal<_R1, _R2>::value; 
429 template <typename _R1, typename _R2> 
430 inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value; 
431 template <typename _R1, typename _R2> 
432 inline constexpr bool ratio_greater_equal_v 
433 = ratio_greater_equal<_R1, _R2>::value; 
434#endif // C++17 
435 
436 /// @cond undocumented 
437 
438 template<typename _R1, typename _R2, 
439 bool = (_R1::num >= 0), 
440 bool = (_R2::num >= 0), 
441 bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>, 
442 ratio<__static_abs<_R2::num>::value, _R2::den> >::value> 
443 struct __ratio_add_impl 
444 { 
445 private: 
446 typedef typename __ratio_add_impl< 
447 ratio<-_R1::num, _R1::den>, 
448 ratio<-_R2::num, _R2::den> >::type __t; 
449 public: 
450 typedef ratio<-__t::num, __t::den> type; 
451 }; 
452 
453 // True addition of nonnegative numbers. 
454 template<typename _R1, typename _R2, bool __b> 
455 struct __ratio_add_impl<_R1, _R2, true, true, __b> 
456 { 
457 private: 
458 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value; 
459 static constexpr uintmax_t __d2 = _R2::den / __g; 
460 typedef __big_mul<_R1::den, __d2> __d; 
461 typedef __big_mul<_R1::num, _R2::den / __g> __x; 
462 typedef __big_mul<_R2::num, _R1::den / __g> __y; 
463 typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n; 
464 static_assert(__n::__hi >= __x::__hi, "Internal library error"); 
465 typedef __big_div<__n::__hi, __n::__lo, __g> __ng; 
466 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value; 
467 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final; 
468 static_assert(__n_final::__rem == 0, "Internal library error"); 
469 static_assert(__n_final::__quot_hi == 0 && 
470 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition"); 
471 typedef __big_mul<_R1::den / __g2, __d2> __d_final; 
472 static_assert(__d_final::__hi == 0 && 
473 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition"); 
474 public: 
475 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type; 
476 }; 
477 
478 template<typename _R1, typename _R2> 
479 struct __ratio_add_impl<_R1, _R2, false, true, true> 
480 : __ratio_add_impl<_R2, _R1> 
481 { }; 
482 
483 // True subtraction of nonnegative numbers yielding a nonnegative result. 
484 template<typename _R1, typename _R2> 
485 struct __ratio_add_impl<_R1, _R2, true, false, false> 
486 { 
487 private: 
488 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value; 
489 static constexpr uintmax_t __d2 = _R2::den / __g; 
490 typedef __big_mul<_R1::den, __d2> __d; 
491 typedef __big_mul<_R1::num, _R2::den / __g> __x; 
492 typedef __big_mul<-_R2::num, _R1::den / __g> __y; 
493 typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n; 
494 typedef __big_div<__n::__hi, __n::__lo, __g> __ng; 
495 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value; 
496 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final; 
497 static_assert(__n_final::__rem == 0, "Internal library error"); 
498 static_assert(__n_final::__quot_hi == 0 && 
499 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition"); 
500 typedef __big_mul<_R1::den / __g2, __d2> __d_final; 
501 static_assert(__d_final::__hi == 0 && 
502 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition"); 
503 public: 
504 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type; 
505 }; 
506 
507 template<typename _R1, typename _R2> 
508 struct __ratio_add 
509 { 
510 typedef typename __ratio_add_impl<_R1, _R2>::type type; 
511 static constexpr intmax_t num = type::num; 
512 static constexpr intmax_t den = type::den; 
513 }; 
514 
515 template<typename _R1, typename _R2> 
516 constexpr intmax_t __ratio_add<_R1, _R2>::num; 
517 
518 template<typename _R1, typename _R2> 
519 constexpr intmax_t __ratio_add<_R1, _R2>::den; 
520 
521 /// @endcond 
522 
523 /// ratio_add 
524 template<typename _R1, typename _R2> 
525 using ratio_add = typename __ratio_add<_R1, _R2>::type; 
526 
527 /// @cond undocumented 
528 
529 template<typename _R1, typename _R2> 
530 struct __ratio_subtract 
531 { 
532 typedef typename __ratio_add< 
533 _R1, 
534 ratio<-_R2::num, _R2::den>>::type type; 
535 
536 static constexpr intmax_t num = type::num; 
537 static constexpr intmax_t den = type::den; 
538 }; 
539 
540 template<typename _R1, typename _R2> 
541 constexpr intmax_t __ratio_subtract<_R1, _R2>::num; 
542 
543 template<typename _R1, typename _R2> 
544 constexpr intmax_t __ratio_subtract<_R1, _R2>::den; 
545 
546 /// @endcond 
547 
548 /// ratio_subtract 
549 template<typename _R1, typename _R2> 
550 using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type; 
551 
552 
553 typedef ratio<1, 1000000000000000000> atto; 
554 typedef ratio<1, 1000000000000000> femto; 
555 typedef ratio<1, 1000000000000> pico; 
556 typedef ratio<1, 1000000000> nano; 
557 typedef ratio<1, 1000000> micro; 
558 typedef ratio<1, 1000> milli; 
559 typedef ratio<1, 100> centi; 
560 typedef ratio<1, 10> deci; 
561 typedef ratio< 10, 1> deca; 
562 typedef ratio< 100, 1> hecto; 
563 typedef ratio< 1000, 1> kilo; 
564 typedef ratio< 1000000, 1> mega; 
565 typedef ratio< 1000000000, 1> giga; 
566 typedef ratio< 1000000000000, 1> tera; 
567 typedef ratio< 1000000000000000, 1> peta; 
568 typedef ratio< 1000000000000000000, 1> exa; 
569 
570 /// @} group ratio 
571_GLIBCXX_END_NAMESPACE_VERSION 
572} // namespace 
573 
574#endif // C++11 
575 
576#endif //_GLIBCXX_RATIO 
577