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