fast-dtoa.cc revision 25f6136652d8341ed047e7fc1a450af5bd218ea9
1// Copyright 2010 the V8 project authors. All rights reserved. 2// Redistribution and use in source and binary forms, with or without 3// modification, are permitted provided that the following conditions are 4// met: 5// 6// * Redistributions of source code must retain the above copyright 7// notice, this list of conditions and the following disclaimer. 8// * Redistributions in binary form must reproduce the above 9// copyright notice, this list of conditions and the following 10// disclaimer in the documentation and/or other materials provided 11// with the distribution. 12// * Neither the name of Google Inc. nor the names of its 13// contributors may be used to endorse or promote products derived 14// from this software without specific prior written permission. 15// 16// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 17// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 18// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 19// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT 20// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 21// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT 22// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 23// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 24// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 25// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 26// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 27 28#include "v8.h" 29 30#include "fast-dtoa.h" 31 32#include "cached-powers.h" 33#include "diy-fp.h" 34#include "double.h" 35 36namespace v8 { 37namespace internal { 38 39// The minimal and maximal target exponent define the range of w's binary 40// exponent, where 'w' is the result of multiplying the input by a cached power 41// of ten. 42// 43// A different range might be chosen on a different platform, to optimize digit 44// generation, but a smaller range requires more powers of ten to be cached. 45static const int minimal_target_exponent = -60; 46static const int maximal_target_exponent = -32; 47 48 49// Adjusts the last digit of the generated number, and screens out generated 50// solutions that may be inaccurate. A solution may be inaccurate if it is 51// outside the safe interval, or if we ctannot prove that it is closer to the 52// input than a neighboring representation of the same length. 53// 54// Input: * buffer containing the digits of too_high / 10^kappa 55// * the buffer's length 56// * distance_too_high_w == (too_high - w).f() * unit 57// * unsafe_interval == (too_high - too_low).f() * unit 58// * rest = (too_high - buffer * 10^kappa).f() * unit 59// * ten_kappa = 10^kappa * unit 60// * unit = the common multiplier 61// Output: returns true if the buffer is guaranteed to contain the closest 62// representable number to the input. 63// Modifies the generated digits in the buffer to approach (round towards) w. 64bool RoundWeed(Vector<char> buffer, 65 int length, 66 uint64_t distance_too_high_w, 67 uint64_t unsafe_interval, 68 uint64_t rest, 69 uint64_t ten_kappa, 70 uint64_t unit) { 71 uint64_t small_distance = distance_too_high_w - unit; 72 uint64_t big_distance = distance_too_high_w + unit; 73 // Let w_low = too_high - big_distance, and 74 // w_high = too_high - small_distance. 75 // Note: w_low < w < w_high 76 // 77 // The real w (* unit) must lie somewhere inside the interval 78 // ]w_low; w_low[ (often written as "(w_low; w_low)") 79 80 // Basically the buffer currently contains a number in the unsafe interval 81 // ]too_low; too_high[ with too_low < w < too_high 82 // 83 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 84 // ^v 1 unit ^ ^ ^ ^ 85 // boundary_high --------------------- . . . . 86 // ^v 1 unit . . . . 87 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . 88 // . . ^ . . 89 // . big_distance . . . 90 // . . . . rest 91 // small_distance . . . . 92 // v . . . . 93 // w_high - - - - - - - - - - - - - - - - - - . . . . 94 // ^v 1 unit . . . . 95 // w ---------------------------------------- . . . . 96 // ^v 1 unit v . . . 97 // w_low - - - - - - - - - - - - - - - - - - - - - . . . 98 // . . v 99 // buffer --------------------------------------------------+-------+-------- 100 // . . 101 // safe_interval . 102 // v . 103 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . 104 // ^v 1 unit . 105 // boundary_low ------------------------- unsafe_interval 106 // ^v 1 unit v 107 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 108 // 109 // 110 // Note that the value of buffer could lie anywhere inside the range too_low 111 // to too_high. 112 // 113 // boundary_low, boundary_high and w are approximations of the real boundaries 114 // and v (the input number). They are guaranteed to be precise up to one unit. 115 // In fact the error is guaranteed to be strictly less than one unit. 116 // 117 // Anything that lies outside the unsafe interval is guaranteed not to round 118 // to v when read again. 119 // Anything that lies inside the safe interval is guaranteed to round to v 120 // when read again. 121 // If the number inside the buffer lies inside the unsafe interval but not 122 // inside the safe interval then we simply do not know and bail out (returning 123 // false). 124 // 125 // Similarly we have to take into account the imprecision of 'w' when rounding 126 // the buffer. If we have two potential representations we need to make sure 127 // that the chosen one is closer to w_low and w_high since v can be anywhere 128 // between them. 129 // 130 // By generating the digits of too_high we got the largest (closest to 131 // too_high) buffer that is still in the unsafe interval. In the case where 132 // w_high < buffer < too_high we try to decrement the buffer. 133 // This way the buffer approaches (rounds towards) w. 134 // There are 3 conditions that stop the decrementation process: 135 // 1) the buffer is already below w_high 136 // 2) decrementing the buffer would make it leave the unsafe interval 137 // 3) decrementing the buffer would yield a number below w_high and farther 138 // away than the current number. In other words: 139 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high 140 // Instead of using the buffer directly we use its distance to too_high. 141 // Conceptually rest ~= too_high - buffer 142 while (rest < small_distance && // Negated condition 1 143 unsafe_interval - rest >= ten_kappa && // Negated condition 2 144 (rest + ten_kappa < small_distance || // buffer{-1} > w_high 145 small_distance - rest >= rest + ten_kappa - small_distance)) { 146 buffer[length - 1]--; 147 rest += ten_kappa; 148 } 149 150 // We have approached w+ as much as possible. We now test if approaching w- 151 // would require changing the buffer. If yes, then we have two possible 152 // representations close to w, but we cannot decide which one is closer. 153 if (rest < big_distance && 154 unsafe_interval - rest >= ten_kappa && 155 (rest + ten_kappa < big_distance || 156 big_distance - rest > rest + ten_kappa - big_distance)) { 157 return false; 158 } 159 160 // Weeding test. 161 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] 162 // Since too_low = too_high - unsafe_interval this is equivalent to 163 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] 164 // Conceptually we have: rest ~= too_high - buffer 165 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); 166} 167 168 169 170static const uint32_t kTen4 = 10000; 171static const uint32_t kTen5 = 100000; 172static const uint32_t kTen6 = 1000000; 173static const uint32_t kTen7 = 10000000; 174static const uint32_t kTen8 = 100000000; 175static const uint32_t kTen9 = 1000000000; 176 177// Returns the biggest power of ten that is less than or equal than the given 178// number. We furthermore receive the maximum number of bits 'number' has. 179// If number_bits == 0 then 0^-1 is returned 180// The number of bits must be <= 32. 181// Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)). 182static void BiggestPowerTen(uint32_t number, 183 int number_bits, 184 uint32_t* power, 185 int* exponent) { 186 switch (number_bits) { 187 case 32: 188 case 31: 189 case 30: 190 if (kTen9 <= number) { 191 *power = kTen9; 192 *exponent = 9; 193 break; 194 } // else fallthrough 195 case 29: 196 case 28: 197 case 27: 198 if (kTen8 <= number) { 199 *power = kTen8; 200 *exponent = 8; 201 break; 202 } // else fallthrough 203 case 26: 204 case 25: 205 case 24: 206 if (kTen7 <= number) { 207 *power = kTen7; 208 *exponent = 7; 209 break; 210 } // else fallthrough 211 case 23: 212 case 22: 213 case 21: 214 case 20: 215 if (kTen6 <= number) { 216 *power = kTen6; 217 *exponent = 6; 218 break; 219 } // else fallthrough 220 case 19: 221 case 18: 222 case 17: 223 if (kTen5 <= number) { 224 *power = kTen5; 225 *exponent = 5; 226 break; 227 } // else fallthrough 228 case 16: 229 case 15: 230 case 14: 231 if (kTen4 <= number) { 232 *power = kTen4; 233 *exponent = 4; 234 break; 235 } // else fallthrough 236 case 13: 237 case 12: 238 case 11: 239 case 10: 240 if (1000 <= number) { 241 *power = 1000; 242 *exponent = 3; 243 break; 244 } // else fallthrough 245 case 9: 246 case 8: 247 case 7: 248 if (100 <= number) { 249 *power = 100; 250 *exponent = 2; 251 break; 252 } // else fallthrough 253 case 6: 254 case 5: 255 case 4: 256 if (10 <= number) { 257 *power = 10; 258 *exponent = 1; 259 break; 260 } // else fallthrough 261 case 3: 262 case 2: 263 case 1: 264 if (1 <= number) { 265 *power = 1; 266 *exponent = 0; 267 break; 268 } // else fallthrough 269 case 0: 270 *power = 0; 271 *exponent = -1; 272 break; 273 default: 274 // Following assignments are here to silence compiler warnings. 275 *power = 0; 276 *exponent = 0; 277 UNREACHABLE(); 278 } 279} 280 281 282// Generates the digits of input number w. 283// w is a floating-point number (DiyFp), consisting of a significand and an 284// exponent. Its exponent is bounded by minimal_target_exponent and 285// maximal_target_exponent. 286// Hence -60 <= w.e() <= -32. 287// 288// Returns false if it fails, in which case the generated digits in the buffer 289// should not be used. 290// Preconditions: 291// * low, w and high are correct up to 1 ulp (unit in the last place). That 292// is, their error must be less that a unit of their last digits. 293// * low.e() == w.e() == high.e() 294// * low < w < high, and taking into account their error: low~ <= high~ 295// * minimal_target_exponent <= w.e() <= maximal_target_exponent 296// Postconditions: returns false if procedure fails. 297// otherwise: 298// * buffer is not null-terminated, but len contains the number of digits. 299// * buffer contains the shortest possible decimal digit-sequence 300// such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the 301// correct values of low and high (without their error). 302// * if more than one decimal representation gives the minimal number of 303// decimal digits then the one closest to W (where W is the correct value 304// of w) is chosen. 305// Remark: this procedure takes into account the imprecision of its input 306// numbers. If the precision is not enough to guarantee all the postconditions 307// then false is returned. This usually happens rarely (~0.5%). 308// 309// Say, for the sake of example, that 310// w.e() == -48, and w.f() == 0x1234567890abcdef 311// w's value can be computed by w.f() * 2^w.e() 312// We can obtain w's integral digits by simply shifting w.f() by -w.e(). 313// -> w's integral part is 0x1234 314// w's fractional part is therefore 0x567890abcdef. 315// Printing w's integral part is easy (simply print 0x1234 in decimal). 316// In order to print its fraction we repeatedly multiply the fraction by 10 and 317// get each digit. Example the first digit after the point would be computed by 318// (0x567890abcdef * 10) >> 48. -> 3 319// The whole thing becomes slightly more complicated because we want to stop 320// once we have enough digits. That is, once the digits inside the buffer 321// represent 'w' we can stop. Everything inside the interval low - high 322// represents w. However we have to pay attention to low, high and w's 323// imprecision. 324bool DigitGen(DiyFp low, 325 DiyFp w, 326 DiyFp high, 327 Vector<char> buffer, 328 int* length, 329 int* kappa) { 330 ASSERT(low.e() == w.e() && w.e() == high.e()); 331 ASSERT(low.f() + 1 <= high.f() - 1); 332 ASSERT(minimal_target_exponent <= w.e() && w.e() <= maximal_target_exponent); 333 // low, w and high are imprecise, but by less than one ulp (unit in the last 334 // place). 335 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that 336 // the new numbers are outside of the interval we want the final 337 // representation to lie in. 338 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield 339 // numbers that are certain to lie in the interval. We will use this fact 340 // later on. 341 // We will now start by generating the digits within the uncertain 342 // interval. Later we will weed out representations that lie outside the safe 343 // interval and thus _might_ lie outside the correct interval. 344 uint64_t unit = 1; 345 DiyFp too_low = DiyFp(low.f() - unit, low.e()); 346 DiyFp too_high = DiyFp(high.f() + unit, high.e()); 347 // too_low and too_high are guaranteed to lie outside the interval we want the 348 // generated number in. 349 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low); 350 // We now cut the input number into two parts: the integral digits and the 351 // fractionals. We will not write any decimal separator though, but adapt 352 // kappa instead. 353 // Reminder: we are currently computing the digits (stored inside the buffer) 354 // such that: too_low < buffer * 10^kappa < too_high 355 // We use too_high for the digit_generation and stop as soon as possible. 356 // If we stop early we effectively round down. 357 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 358 // Division by one is a shift. 359 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); 360 // Modulo by one is an and. 361 uint64_t fractionals = too_high.f() & (one.f() - 1); 362 uint32_t divider; 363 int divider_exponent; 364 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 365 ÷r, ÷r_exponent); 366 *kappa = divider_exponent + 1; 367 *length = 0; 368 // Loop invariant: buffer = too_high / 10^kappa (integer division) 369 // The invariant holds for the first iteration: kappa has been initialized 370 // with the divider exponent + 1. And the divider is the biggest power of ten 371 // that is smaller than integrals. 372 while (*kappa > 0) { 373 int digit = integrals / divider; 374 buffer[*length] = '0' + digit; 375 (*length)++; 376 integrals %= divider; 377 (*kappa)--; 378 // Note that kappa now equals the exponent of the divider and that the 379 // invariant thus holds again. 380 uint64_t rest = 381 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 382 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) 383 // Reminder: unsafe_interval.e() == one.e() 384 if (rest < unsafe_interval.f()) { 385 // Rounding down (by not emitting the remaining digits) yields a number 386 // that lies within the unsafe interval. 387 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), 388 unsafe_interval.f(), rest, 389 static_cast<uint64_t>(divider) << -one.e(), unit); 390 } 391 divider /= 10; 392 } 393 394 // The integrals have been generated. We are at the point of the decimal 395 // separator. In the following loop we simply multiply the remaining digits by 396 // 10 and divide by one. We just need to pay attention to multiply associated 397 // data (like the interval or 'unit'), too. 398 // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and 399 // increase its (imaginary) exponent. At the same time we decrease the 400 // divider's (one's) exponent and shift its significand. 401 // Basically, if fractionals was a DiyFp (with fractionals.e == one.e): 402 // fractionals.f *= 10; 403 // fractionals.f >>= 1; fractionals.e++; // value remains unchanged. 404 // one.f >>= 1; one.e++; // value remains unchanged. 405 // and we have again fractionals.e == one.e which allows us to divide 406 // fractionals.f() by one.f() 407 // We simply combine the *= 10 and the >>= 1. 408 while (true) { 409 fractionals *= 5; 410 unit *= 5; 411 unsafe_interval.set_f(unsafe_interval.f() * 5); 412 unsafe_interval.set_e(unsafe_interval.e() + 1); // Will be optimized out. 413 one.set_f(one.f() >> 1); 414 one.set_e(one.e() + 1); 415 // Integer division by one. 416 int digit = static_cast<int>(fractionals >> -one.e()); 417 buffer[*length] = '0' + digit; 418 (*length)++; 419 fractionals &= one.f() - 1; // Modulo by one. 420 (*kappa)--; 421 if (fractionals < unsafe_interval.f()) { 422 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit, 423 unsafe_interval.f(), fractionals, one.f(), unit); 424 } 425 } 426} 427 428 429// Provides a decimal representation of v. 430// Returns true if it succeeds, otherwise the result cannot be trusted. 431// There will be *length digits inside the buffer (not null-terminated). 432// If the function returns true then 433// v == (double) (buffer * 10^decimal_exponent). 434// The digits in the buffer are the shortest representation possible: no 435// 0.09999999999999999 instead of 0.1. The shorter representation will even be 436// chosen even if the longer one would be closer to v. 437// The last digit will be closest to the actual v. That is, even if several 438// digits might correctly yield 'v' when read again, the closest will be 439// computed. 440bool grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) { 441 DiyFp w = Double(v).AsNormalizedDiyFp(); 442 // boundary_minus and boundary_plus are the boundaries between v and its 443 // closest floating-point neighbors. Any number strictly between 444 // boundary_minus and boundary_plus will round to v when convert to a double. 445 // Grisu3 will never output representations that lie exactly on a boundary. 446 DiyFp boundary_minus, boundary_plus; 447 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); 448 ASSERT(boundary_plus.e() == w.e()); 449 DiyFp ten_mk; // Cached power of ten: 10^-k 450 int mk; // -k 451 GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent, 452 maximal_target_exponent, &mk, &ten_mk); 453 ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() + 454 DiyFp::kSignificandSize && 455 maximal_target_exponent >= w.e() + ten_mk.e() + 456 DiyFp::kSignificandSize); 457 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 458 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 459 460 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 461 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 462 // off by a small amount. 463 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 464 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 465 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 466 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 467 ASSERT(scaled_w.e() == 468 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); 469 // In theory it would be possible to avoid some recomputations by computing 470 // the difference between w and boundary_minus/plus (a power of 2) and to 471 // compute scaled_boundary_minus/plus by subtracting/adding from 472 // scaled_w. However the code becomes much less readable and the speed 473 // enhancements are not terriffic. 474 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); 475 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); 476 477 // DigitGen will generate the digits of scaled_w. Therefore we have 478 // v == (double) (scaled_w * 10^-mk). 479 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an 480 // integer than it will be updated. For instance if scaled_w == 1.23 then 481 // the buffer will be filled with "123" und the decimal_exponent will be 482 // decreased by 2. 483 int kappa; 484 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, 485 buffer, length, &kappa); 486 *decimal_exponent = -mk + kappa; 487 return result; 488} 489 490 491bool FastDtoa(double v, 492 Vector<char> buffer, 493 int* length, 494 int* point) { 495 ASSERT(v > 0); 496 ASSERT(!Double(v).IsSpecial()); 497 498 int decimal_exponent; 499 bool result = grisu3(v, buffer, length, &decimal_exponent); 500 *point = *length + decimal_exponent; 501 buffer[*length] = '\0'; 502 return result; 503} 504 505} } // namespace v8::internal 506