libstdc++
ratio
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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 
42 namespace 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