1// ratio -*- C++ -*- 2 3// Copyright (C) 2008-2018 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> 40 41#ifdef _GLIBCXX_USE_C99_STDINT_TR1 42 43namespace std _GLIBCXX_VISIBILITY(default) 44{ 45_GLIBCXX_BEGIN_NAMESPACE_VERSION 46 47 /** 48 * @defgroup ratio Rational Arithmetic 49 * @ingroup utilities 50 * 51 * Compile time representation of finite rational numbers. 52 * @{ 53 */ 54 55 template<intmax_t _Pn> 56 struct __static_sign 57 : integral_constant<intmax_t, (_Pn < 0) ? -1 : 1> 58 { }; 59 60 template<intmax_t _Pn> 61 struct __static_abs 62 : integral_constant<intmax_t, _Pn * __static_sign<_Pn>::value> 63 { }; 64 65 template<intmax_t _Pn, intmax_t _Qn> 66 struct __static_gcd 67 : __static_gcd<_Qn, (_Pn % _Qn)> 68 { }; 69 70 template<intmax_t _Pn> 71 struct __static_gcd<_Pn, 0> 72 : integral_constant<intmax_t, __static_abs<_Pn>::value> 73 { }; 74 75 template<intmax_t _Qn> 76 struct __static_gcd<0, _Qn> 77 : integral_constant<intmax_t, __static_abs<_Qn>::value> 78 { }; 79 80 // Let c = 2^(half # of bits in an intmax_t) 81 // then we find a1, a0, b1, b0 s.t. N = a1*c + a0, M = b1*c + b0 82 // The multiplication of N and M becomes, 83 // N * M = (a1 * b1)c^2 + (a0 * b1 + b0 * a1)c + a0 * b0 84 // Multiplication is safe if each term and the sum of the terms 85 // is representable by intmax_t. 86 template<intmax_t _Pn, intmax_t _Qn> 87 struct __safe_multiply 88 { 89 private: 90 static const uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4); 91 92 static const uintmax_t __a0 = __static_abs<_Pn>::value % __c; 93 static const uintmax_t __a1 = __static_abs<_Pn>::value / __c; 94 static const uintmax_t __b0 = __static_abs<_Qn>::value % __c; 95 static const uintmax_t __b1 = __static_abs<_Qn>::value / __c; 96 97 static_assert(__a1 == 0 || __b1 == 0, 98 "overflow in multiplication"); 99 static_assert(__a0 * __b1 + __b0 * __a1 < (__c >> 1), 100 "overflow in multiplication"); 101 static_assert(__b0 * __a0 <= __INTMAX_MAX__, 102 "overflow in multiplication"); 103 static_assert((__a0 * __b1 + __b0 * __a1) * __c 104 <= __INTMAX_MAX__ - __b0 * __a0, 105 "overflow in multiplication"); 106 107 public: 108 static const intmax_t value = _Pn * _Qn; 109 }; 110 111 // Some double-precision utilities, where numbers are represented as 112 // __hi*2^(8*sizeof(uintmax_t)) + __lo. 113 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2> 114 struct __big_less 115 : integral_constant<bool, (__hi1 < __hi2 116 || (__hi1 == __hi2 && __lo1 < __lo2))> 117 { }; 118 119 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2> 120 struct __big_add 121 { 122 static constexpr uintmax_t __lo = __lo1 + __lo2; 123 static constexpr uintmax_t __hi = (__hi1 + __hi2 + 124 (__lo1 + __lo2 < __lo1)); // carry 125 }; 126 127 // Subtract a number from a bigger one. 128 template<uintmax_t __hi1, uintmax_t __lo1, uintmax_t __hi2, uintmax_t __lo2> 129 struct __big_sub 130 { 131 static_assert(!__big_less<__hi1, __lo1, __hi2, __lo2>::value, 132 "Internal library error"); 133 static constexpr uintmax_t __lo = __lo1 - __lo2; 134 static constexpr uintmax_t __hi = (__hi1 - __hi2 - 135 (__lo1 < __lo2)); // carry 136 }; 137 138 // Same principle as __safe_multiply. 139 template<uintmax_t __x, uintmax_t __y> 140 struct __big_mul 141 { 142 private: 143 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4); 144 static constexpr uintmax_t __x0 = __x % __c; 145 static constexpr uintmax_t __x1 = __x / __c; 146 static constexpr uintmax_t __y0 = __y % __c; 147 static constexpr uintmax_t __y1 = __y / __c; 148 static constexpr uintmax_t __x0y0 = __x0 * __y0; 149 static constexpr uintmax_t __x0y1 = __x0 * __y1; 150 static constexpr uintmax_t __x1y0 = __x1 * __y0; 151 static constexpr uintmax_t __x1y1 = __x1 * __y1; 152 static constexpr uintmax_t __mix = __x0y1 + __x1y0; // possible carry... 153 static constexpr uintmax_t __mix_lo = __mix * __c; 154 static constexpr uintmax_t __mix_hi 155 = __mix / __c + ((__mix < __x0y1) ? __c : 0); // ... added here 156 typedef __big_add<__mix_hi, __mix_lo, __x1y1, __x0y0> _Res; 157 public: 158 static constexpr uintmax_t __hi = _Res::__hi; 159 static constexpr uintmax_t __lo = _Res::__lo; 160 }; 161 162 // Adapted from __udiv_qrnnd_c in longlong.h 163 // This version assumes that the high bit of __d is 1. 164 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d> 165 struct __big_div_impl 166 { 167 private: 168 static_assert(__d >= (uintmax_t(1) << (sizeof(intmax_t) * 8 - 1)), 169 "Internal library error"); 170 static_assert(__n1 < __d, "Internal library error"); 171 static constexpr uintmax_t __c = uintmax_t(1) << (sizeof(intmax_t) * 4); 172 static constexpr uintmax_t __d1 = __d / __c; 173 static constexpr uintmax_t __d0 = __d % __c; 174 175 static constexpr uintmax_t __q1x = __n1 / __d1; 176 static constexpr uintmax_t __r1x = __n1 % __d1; 177 static constexpr uintmax_t __m = __q1x * __d0; 178 static constexpr uintmax_t __r1y = __r1x * __c + __n0 / __c; 179 static constexpr uintmax_t __r1z = __r1y + __d; 180 static constexpr uintmax_t __r1 181 = ((__r1y < __m) ? ((__r1z >= __d) && (__r1z < __m)) 182 ? (__r1z + __d) : __r1z : __r1y) - __m; 183 static constexpr uintmax_t __q1 184 = __q1x - ((__r1y < __m) 185 ? ((__r1z >= __d) && (__r1z < __m)) ? 2 : 1 : 0); 186 static constexpr uintmax_t __q0x = __r1 / __d1; 187 static constexpr uintmax_t __r0x = __r1 % __d1; 188 static constexpr uintmax_t __n = __q0x * __d0; 189 static constexpr uintmax_t __r0y = __r0x * __c + __n0 % __c; 190 static constexpr uintmax_t __r0z = __r0y + __d; 191 static constexpr uintmax_t __r0 192 = ((__r0y < __n) ? ((__r0z >= __d) && (__r0z < __n)) 193 ? (__r0z + __d) : __r0z : __r0y) - __n; 194 static constexpr uintmax_t __q0 195 = __q0x - ((__r0y < __n) ? ((__r0z >= __d) 196 && (__r0z < __n)) ? 2 : 1 : 0); 197 198 public: 199 static constexpr uintmax_t __quot = __q1 * __c + __q0; 200 static constexpr uintmax_t __rem = __r0; 201 202 private: 203 typedef __big_mul<__quot, __d> _Prod; 204 typedef __big_add<_Prod::__hi, _Prod::__lo, 0, __rem> _Sum; 205 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0, 206 "Internal library error"); 207 }; 208 209 template<uintmax_t __n1, uintmax_t __n0, uintmax_t __d> 210 struct __big_div 211 { 212 private: 213 static_assert(__d != 0, "Internal library error"); 214 static_assert(sizeof (uintmax_t) == sizeof (unsigned long long), 215 "This library calls __builtin_clzll on uintmax_t, which " 216 "is unsafe on your platform. Please complain to " 217 "http://gcc.gnu.org/bugzilla/"); 218 static constexpr int __shift = __builtin_clzll(__d); 219 static constexpr int __coshift_ = sizeof(uintmax_t) * 8 - __shift; 220 static constexpr int __coshift = (__shift != 0) ? __coshift_ : 0; 221 static constexpr uintmax_t __c1 = uintmax_t(1) << __shift; 222 static constexpr uintmax_t __c2 = uintmax_t(1) << __coshift; 223 static constexpr uintmax_t __new_d = __d * __c1; 224 static constexpr uintmax_t __new_n0 = __n0 * __c1; 225 static constexpr uintmax_t __n1_shifted = (__n1 % __d) * __c1; 226 static constexpr uintmax_t __n0_top = (__shift != 0) ? (__n0 / __c2) : 0; 227 static constexpr uintmax_t __new_n1 = __n1_shifted + __n0_top; 228 typedef __big_div_impl<__new_n1, __new_n0, __new_d> _Res; 229 230 public: 231 static constexpr uintmax_t __quot_hi = __n1 / __d; 232 static constexpr uintmax_t __quot_lo = _Res::__quot; 233 static constexpr uintmax_t __rem = _Res::__rem / __c1; 234 235 private: 236 typedef __big_mul<__quot_lo, __d> _P0; 237 typedef __big_mul<__quot_hi, __d> _P1; 238 typedef __big_add<_P0::__hi, _P0::__lo, _P1::__lo, __rem> _Sum; 239 // No overflow. 240 static_assert(_P1::__hi == 0, "Internal library error"); 241 static_assert(_Sum::__hi >= _P0::__hi, "Internal library error"); 242 // Matches the input data. 243 static_assert(_Sum::__hi == __n1 && _Sum::__lo == __n0, 244 "Internal library error"); 245 static_assert(__rem < __d, "Internal library error"); 246 }; 247 248 /** 249 * @brief Provides compile-time rational arithmetic. 250 * 251 * This class template represents any finite rational number with a 252 * numerator and denominator representable by compile-time constants of 253 * type intmax_t. The ratio is simplified when instantiated. 254 * 255 * For example: 256 * @code 257 * std::ratio<7,-21>::num == -1; 258 * std::ratio<7,-21>::den == 3; 259 * @endcode 260 * 261 */ 262 template<intmax_t _Num, intmax_t _Den = 1> 263 struct ratio 264 { 265 static_assert(_Den != 0, "denominator cannot be zero"); 266 static_assert(_Num >= -__INTMAX_MAX__ && _Den >= -__INTMAX_MAX__, 267 "out of range"); 268 269 // Note: sign(N) * abs(N) == N 270 static constexpr intmax_t num = 271 _Num * __static_sign<_Den>::value / __static_gcd<_Num, _Den>::value; 272 273 static constexpr intmax_t den = 274 __static_abs<_Den>::value / __static_gcd<_Num, _Den>::value; 275 276 typedef ratio<num, den> type; 277 }; 278 279 template<intmax_t _Num, intmax_t _Den> 280 constexpr intmax_t ratio<_Num, _Den>::num; 281 282 template<intmax_t _Num, intmax_t _Den> 283 constexpr intmax_t ratio<_Num, _Den>::den; 284 285 template<typename _R1, typename _R2> 286 struct __ratio_multiply 287 { 288 private: 289 static const intmax_t __gcd1 = 290 __static_gcd<_R1::num, _R2::den>::value; 291 static const intmax_t __gcd2 = 292 __static_gcd<_R2::num, _R1::den>::value; 293 294 public: 295 typedef ratio< 296 __safe_multiply<(_R1::num / __gcd1), 297 (_R2::num / __gcd2)>::value, 298 __safe_multiply<(_R1::den / __gcd2), 299 (_R2::den / __gcd1)>::value> type; 300 301 static constexpr intmax_t num = type::num; 302 static constexpr intmax_t den = type::den; 303 }; 304 305 template<typename _R1, typename _R2> 306 constexpr intmax_t __ratio_multiply<_R1, _R2>::num; 307 308 template<typename _R1, typename _R2> 309 constexpr intmax_t __ratio_multiply<_R1, _R2>::den; 310 311 /// ratio_multiply 312 template<typename _R1, typename _R2> 313 using ratio_multiply = typename __ratio_multiply<_R1, _R2>::type; 314 315 template<typename _R1, typename _R2> 316 struct __ratio_divide 317 { 318 static_assert(_R2::num != 0, "division by 0"); 319 320 typedef typename __ratio_multiply< 321 _R1, 322 ratio<_R2::den, _R2::num>>::type type; 323 324 static constexpr intmax_t num = type::num; 325 static constexpr intmax_t den = type::den; 326 }; 327 328 template<typename _R1, typename _R2> 329 constexpr intmax_t __ratio_divide<_R1, _R2>::num; 330 331 template<typename _R1, typename _R2> 332 constexpr intmax_t __ratio_divide<_R1, _R2>::den; 333 334 /// ratio_divide 335 template<typename _R1, typename _R2> 336 using ratio_divide = typename __ratio_divide<_R1, _R2>::type; 337 338 /// ratio_equal 339 template<typename _R1, typename _R2> 340 struct ratio_equal 341 : integral_constant<bool, _R1::num == _R2::num && _R1::den == _R2::den> 342 { }; 343 344 /// ratio_not_equal 345 template<typename _R1, typename _R2> 346 struct ratio_not_equal 347 : integral_constant<bool, !ratio_equal<_R1, _R2>::value> 348 { }; 349 350 // Both numbers are positive. 351 template<typename _R1, typename _R2, 352 typename _Left = __big_mul<_R1::num,_R2::den>, 353 typename _Right = __big_mul<_R2::num,_R1::den> > 354 struct __ratio_less_impl_1 355 : integral_constant<bool, __big_less<_Left::__hi, _Left::__lo, 356 _Right::__hi, _Right::__lo>::value> 357 { }; 358 359 template<typename _R1, typename _R2, 360 bool = (_R1::num == 0 || _R2::num == 0 361 || (__static_sign<_R1::num>::value 362 != __static_sign<_R2::num>::value)), 363 bool = (__static_sign<_R1::num>::value == -1 364 && __static_sign<_R2::num>::value == -1)> 365 struct __ratio_less_impl 366 : __ratio_less_impl_1<_R1, _R2>::type 367 { }; 368 369 template<typename _R1, typename _R2> 370 struct __ratio_less_impl<_R1, _R2, true, false> 371 : integral_constant<bool, _R1::num < _R2::num> 372 { }; 373 374 template<typename _R1, typename _R2> 375 struct __ratio_less_impl<_R1, _R2, false, true> 376 : __ratio_less_impl_1<ratio<-_R2::num, _R2::den>, 377 ratio<-_R1::num, _R1::den> >::type 378 { }; 379 380 /// ratio_less 381 template<typename _R1, typename _R2> 382 struct ratio_less 383 : __ratio_less_impl<_R1, _R2>::type 384 { }; 385 386 /// ratio_less_equal 387 template<typename _R1, typename _R2> 388 struct ratio_less_equal 389 : integral_constant<bool, !ratio_less<_R2, _R1>::value> 390 { }; 391 392 /// ratio_greater 393 template<typename _R1, typename _R2> 394 struct ratio_greater 395 : integral_constant<bool, ratio_less<_R2, _R1>::value> 396 { }; 397 398 /// ratio_greater_equal 399 template<typename _R1, typename _R2> 400 struct ratio_greater_equal 401 : integral_constant<bool, !ratio_less<_R1, _R2>::value> 402 { }; 403 404#if __cplusplus > 201402L 405 template <typename _R1, typename _R2> 406 inline constexpr bool ratio_equal_v = ratio_equal<_R1, _R2>::value; 407 template <typename _R1, typename _R2> 408 inline constexpr bool ratio_not_equal_v = ratio_not_equal<_R1, _R2>::value; 409 template <typename _R1, typename _R2> 410 inline constexpr bool ratio_less_v = ratio_less<_R1, _R2>::value; 411 template <typename _R1, typename _R2> 412 inline constexpr bool ratio_less_equal_v = 413 ratio_less_equal<_R1, _R2>::value; 414 template <typename _R1, typename _R2> 415 inline constexpr bool ratio_greater_v = ratio_greater<_R1, _R2>::value; 416 template <typename _R1, typename _R2> 417 inline constexpr bool ratio_greater_equal_v 418 = ratio_greater_equal<_R1, _R2>::value; 419#endif // C++17 420 421 template<typename _R1, typename _R2, 422 bool = (_R1::num >= 0), 423 bool = (_R2::num >= 0), 424 bool = ratio_less<ratio<__static_abs<_R1::num>::value, _R1::den>, 425 ratio<__static_abs<_R2::num>::value, _R2::den> >::value> 426 struct __ratio_add_impl 427 { 428 private: 429 typedef typename __ratio_add_impl< 430 ratio<-_R1::num, _R1::den>, 431 ratio<-_R2::num, _R2::den> >::type __t; 432 public: 433 typedef ratio<-__t::num, __t::den> type; 434 }; 435 436 // True addition of nonnegative numbers. 437 template<typename _R1, typename _R2, bool __b> 438 struct __ratio_add_impl<_R1, _R2, true, true, __b> 439 { 440 private: 441 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value; 442 static constexpr uintmax_t __d2 = _R2::den / __g; 443 typedef __big_mul<_R1::den, __d2> __d; 444 typedef __big_mul<_R1::num, _R2::den / __g> __x; 445 typedef __big_mul<_R2::num, _R1::den / __g> __y; 446 typedef __big_add<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n; 447 static_assert(__n::__hi >= __x::__hi, "Internal library error"); 448 typedef __big_div<__n::__hi, __n::__lo, __g> __ng; 449 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value; 450 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final; 451 static_assert(__n_final::__rem == 0, "Internal library error"); 452 static_assert(__n_final::__quot_hi == 0 && 453 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition"); 454 typedef __big_mul<_R1::den / __g2, __d2> __d_final; 455 static_assert(__d_final::__hi == 0 && 456 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition"); 457 public: 458 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type; 459 }; 460 461 template<typename _R1, typename _R2> 462 struct __ratio_add_impl<_R1, _R2, false, true, true> 463 : __ratio_add_impl<_R2, _R1> 464 { }; 465 466 // True subtraction of nonnegative numbers yielding a nonnegative result. 467 template<typename _R1, typename _R2> 468 struct __ratio_add_impl<_R1, _R2, true, false, false> 469 { 470 private: 471 static constexpr uintmax_t __g = __static_gcd<_R1::den, _R2::den>::value; 472 static constexpr uintmax_t __d2 = _R2::den / __g; 473 typedef __big_mul<_R1::den, __d2> __d; 474 typedef __big_mul<_R1::num, _R2::den / __g> __x; 475 typedef __big_mul<-_R2::num, _R1::den / __g> __y; 476 typedef __big_sub<__x::__hi, __x::__lo, __y::__hi, __y::__lo> __n; 477 typedef __big_div<__n::__hi, __n::__lo, __g> __ng; 478 static constexpr uintmax_t __g2 = __static_gcd<__ng::__rem, __g>::value; 479 typedef __big_div<__n::__hi, __n::__lo, __g2> __n_final; 480 static_assert(__n_final::__rem == 0, "Internal library error"); 481 static_assert(__n_final::__quot_hi == 0 && 482 __n_final::__quot_lo <= __INTMAX_MAX__, "overflow in addition"); 483 typedef __big_mul<_R1::den / __g2, __d2> __d_final; 484 static_assert(__d_final::__hi == 0 && 485 __d_final::__lo <= __INTMAX_MAX__, "overflow in addition"); 486 public: 487 typedef ratio<__n_final::__quot_lo, __d_final::__lo> type; 488 }; 489 490 template<typename _R1, typename _R2> 491 struct __ratio_add 492 { 493 typedef typename __ratio_add_impl<_R1, _R2>::type type; 494 static constexpr intmax_t num = type::num; 495 static constexpr intmax_t den = type::den; 496 }; 497 498 template<typename _R1, typename _R2> 499 constexpr intmax_t __ratio_add<_R1, _R2>::num; 500 501 template<typename _R1, typename _R2> 502 constexpr intmax_t __ratio_add<_R1, _R2>::den; 503 504 /// ratio_add 505 template<typename _R1, typename _R2> 506 using ratio_add = typename __ratio_add<_R1, _R2>::type; 507 508 template<typename _R1, typename _R2> 509 struct __ratio_subtract 510 { 511 typedef typename __ratio_add< 512 _R1, 513 ratio<-_R2::num, _R2::den>>::type type; 514 515 static constexpr intmax_t num = type::num; 516 static constexpr intmax_t den = type::den; 517 }; 518 519 template<typename _R1, typename _R2> 520 constexpr intmax_t __ratio_subtract<_R1, _R2>::num; 521 522 template<typename _R1, typename _R2> 523 constexpr intmax_t __ratio_subtract<_R1, _R2>::den; 524 525 /// ratio_subtract 526 template<typename _R1, typename _R2> 527 using ratio_subtract = typename __ratio_subtract<_R1, _R2>::type; 528 529 530 typedef ratio<1, 1000000000000000000> atto; 531 typedef ratio<1, 1000000000000000> femto; 532 typedef ratio<1, 1000000000000> pico; 533 typedef ratio<1, 1000000000> nano; 534 typedef ratio<1, 1000000> micro; 535 typedef ratio<1, 1000> milli; 536 typedef ratio<1, 100> centi; 537 typedef ratio<1, 10> deci; 538 typedef ratio< 10, 1> deca; 539 typedef ratio< 100, 1> hecto; 540 typedef ratio< 1000, 1> kilo; 541 typedef ratio< 1000000, 1> mega; 542 typedef ratio< 1000000000, 1> giga; 543 typedef ratio< 1000000000000, 1> tera; 544 typedef ratio< 1000000000000000, 1> peta; 545 typedef ratio< 1000000000000000000, 1> exa; 546 547 // @} group ratio 548_GLIBCXX_END_NAMESPACE_VERSION 549} // namespace 550 551#endif //_GLIBCXX_USE_C99_STDINT_TR1 552 553#endif // C++11 554 555#endif //_GLIBCXX_RATIO 556