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 "config.h" 29 30#include "fast-dtoa.h" 31 32#include "cached-powers.h" 33#include "diy-fp.h" 34#include "double.h" 35 36namespace WTF { 37 38namespace double_conversion { 39 40 // The minimal and maximal target exponent define the range of w's binary 41 // exponent, where 'w' is the result of multiplying the input by a cached power 42 // of ten. 43 // 44 // A different range might be chosen on a different platform, to optimize digit 45 // generation, but a smaller range requires more powers of ten to be cached. 46 static const int kMinimalTargetExponent = -60; 47 static const int kMaximalTargetExponent = -32; 48 49 50 // Adjusts the last digit of the generated number, and screens out generated 51 // solutions that may be inaccurate. A solution may be inaccurate if it is 52 // outside the safe interval, or if we cannot prove that it is closer to the 53 // input than a neighboring representation of the same length. 54 // 55 // Input: * buffer containing the digits of too_high / 10^kappa 56 // * the buffer's length 57 // * distance_too_high_w == (too_high - w).f() * unit 58 // * unsafe_interval == (too_high - too_low).f() * unit 59 // * rest = (too_high - buffer * 10^kappa).f() * unit 60 // * ten_kappa = 10^kappa * unit 61 // * unit = the common multiplier 62 // Output: returns true if the buffer is guaranteed to contain the closest 63 // representable number to the input. 64 // Modifies the generated digits in the buffer to approach (round towards) w. 65 static bool RoundWeed(Vector<char> buffer, 66 int length, 67 uint64_t distance_too_high_w, 68 uint64_t unsafe_interval, 69 uint64_t rest, 70 uint64_t ten_kappa, 71 uint64_t unit) { 72 uint64_t small_distance = distance_too_high_w - unit; 73 uint64_t big_distance = distance_too_high_w + unit; 74 // Let w_low = too_high - big_distance, and 75 // w_high = too_high - small_distance. 76 // Note: w_low < w < w_high 77 // 78 // The real w (* unit) must lie somewhere inside the interval 79 // ]w_low; w_high[ (often written as "(w_low; w_high)") 80 81 // Basically the buffer currently contains a number in the unsafe interval 82 // ]too_low; too_high[ with too_low < w < too_high 83 // 84 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 85 // ^v 1 unit ^ ^ ^ ^ 86 // boundary_high --------------------- . . . . 87 // ^v 1 unit . . . . 88 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . 89 // . . ^ . . 90 // . big_distance . . . 91 // . . . . rest 92 // small_distance . . . . 93 // v . . . . 94 // w_high - - - - - - - - - - - - - - - - - - . . . . 95 // ^v 1 unit . . . . 96 // w ---------------------------------------- . . . . 97 // ^v 1 unit v . . . 98 // w_low - - - - - - - - - - - - - - - - - - - - - . . . 99 // . . v 100 // buffer --------------------------------------------------+-------+-------- 101 // . . 102 // safe_interval . 103 // v . 104 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . 105 // ^v 1 unit . 106 // boundary_low ------------------------- unsafe_interval 107 // ^v 1 unit v 108 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 109 // 110 // 111 // Note that the value of buffer could lie anywhere inside the range too_low 112 // to too_high. 113 // 114 // boundary_low, boundary_high and w are approximations of the real boundaries 115 // and v (the input number). They are guaranteed to be precise up to one unit. 116 // In fact the error is guaranteed to be strictly less than one unit. 117 // 118 // Anything that lies outside the unsafe interval is guaranteed not to round 119 // to v when read again. 120 // Anything that lies inside the safe interval is guaranteed to round to v 121 // when read again. 122 // If the number inside the buffer lies inside the unsafe interval but not 123 // inside the safe interval then we simply do not know and bail out (returning 124 // false). 125 // 126 // Similarly we have to take into account the imprecision of 'w' when finding 127 // the closest representation of 'w'. If we have two potential 128 // representations, and one is closer to both w_low and w_high, then we know 129 // it is closer to the actual value v. 130 // 131 // By generating the digits of too_high we got the largest (closest to 132 // too_high) buffer that is still in the unsafe interval. In the case where 133 // w_high < buffer < too_high we try to decrement the buffer. 134 // This way the buffer approaches (rounds towards) w. 135 // There are 3 conditions that stop the decrementation process: 136 // 1) the buffer is already below w_high 137 // 2) decrementing the buffer would make it leave the unsafe interval 138 // 3) decrementing the buffer would yield a number below w_high and farther 139 // away than the current number. In other words: 140 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high 141 // Instead of using the buffer directly we use its distance to too_high. 142 // Conceptually rest ~= too_high - buffer 143 // We need to do the following tests in this order to avoid over- and 144 // underflows. 145 ASSERT(rest <= unsafe_interval); 146 while (rest < small_distance && // Negated condition 1 147 unsafe_interval - rest >= ten_kappa && // Negated condition 2 148 (rest + ten_kappa < small_distance || // buffer{-1} > w_high 149 small_distance - rest >= rest + ten_kappa - small_distance)) { 150 buffer[length - 1]--; 151 rest += ten_kappa; 152 } 153 154 // We have approached w+ as much as possible. We now test if approaching w- 155 // would require changing the buffer. If yes, then we have two possible 156 // representations close to w, but we cannot decide which one is closer. 157 if (rest < big_distance && 158 unsafe_interval - rest >= ten_kappa && 159 (rest + ten_kappa < big_distance || 160 big_distance - rest > rest + ten_kappa - big_distance)) { 161 return false; 162 } 163 164 // Weeding test. 165 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] 166 // Since too_low = too_high - unsafe_interval this is equivalent to 167 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] 168 // Conceptually we have: rest ~= too_high - buffer 169 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); 170 } 171 172 173 // Rounds the buffer upwards if the result is closer to v by possibly adding 174 // 1 to the buffer. If the precision of the calculation is not sufficient to 175 // round correctly, return false. 176 // The rounding might shift the whole buffer in which case the kappa is 177 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. 178 // 179 // If 2*rest > ten_kappa then the buffer needs to be round up. 180 // rest can have an error of +/- 1 unit. This function accounts for the 181 // imprecision and returns false, if the rounding direction cannot be 182 // unambiguously determined. 183 // 184 // Precondition: rest < ten_kappa. 185 static bool RoundWeedCounted(Vector<char> buffer, 186 int length, 187 uint64_t rest, 188 uint64_t ten_kappa, 189 uint64_t unit, 190 int* kappa) { 191 ASSERT(rest < ten_kappa); 192 // The following tests are done in a specific order to avoid overflows. They 193 // will work correctly with any uint64 values of rest < ten_kappa and unit. 194 // 195 // If the unit is too big, then we don't know which way to round. For example 196 // a unit of 50 means that the real number lies within rest +/- 50. If 197 // 10^kappa == 40 then there is no way to tell which way to round. 198 if (unit >= ten_kappa) return false; 199 // Even if unit is just half the size of 10^kappa we are already completely 200 // lost. (And after the previous test we know that the expression will not 201 // over/underflow.) 202 if (ten_kappa - unit <= unit) return false; 203 // If 2 * (rest + unit) <= 10^kappa we can safely round down. 204 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) { 205 return true; 206 } 207 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. 208 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) { 209 // Increment the last digit recursively until we find a non '9' digit. 210 buffer[length - 1]++; 211 for (int i = length - 1; i > 0; --i) { 212 if (buffer[i] != '0' + 10) break; 213 buffer[i] = '0'; 214 buffer[i - 1]++; 215 } 216 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the 217 // exception of the first digit all digits are now '0'. Simply switch the 218 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and 219 // the power (the kappa) is increased. 220 if (buffer[0] == '0' + 10) { 221 buffer[0] = '1'; 222 (*kappa) += 1; 223 } 224 return true; 225 } 226 return false; 227 } 228 229 230 static const uint32_t kTen4 = 10000; 231 static const uint32_t kTen5 = 100000; 232 static const uint32_t kTen6 = 1000000; 233 static const uint32_t kTen7 = 10000000; 234 static const uint32_t kTen8 = 100000000; 235 static const uint32_t kTen9 = 1000000000; 236 237 // Returns the biggest power of ten that is less than or equal to the given 238 // number. We furthermore receive the maximum number of bits 'number' has. 239 // If number_bits == 0 then 0^-1 is returned 240 // The number of bits must be <= 32. 241 // Precondition: number < (1 << (number_bits + 1)). 242 static void BiggestPowerTen(uint32_t number, 243 int number_bits, 244 uint32_t* power, 245 int* exponent) { 246 ASSERT(number < (uint32_t)(1 << (number_bits + 1))); 247 248 switch (number_bits) { 249 case 32: 250 case 31: 251 case 30: 252 if (kTen9 <= number) { 253 *power = kTen9; 254 *exponent = 9; 255 break; 256 } // else fallthrough 257 case 29: 258 case 28: 259 case 27: 260 if (kTen8 <= number) { 261 *power = kTen8; 262 *exponent = 8; 263 break; 264 } // else fallthrough 265 case 26: 266 case 25: 267 case 24: 268 if (kTen7 <= number) { 269 *power = kTen7; 270 *exponent = 7; 271 break; 272 } // else fallthrough 273 case 23: 274 case 22: 275 case 21: 276 case 20: 277 if (kTen6 <= number) { 278 *power = kTen6; 279 *exponent = 6; 280 break; 281 } // else fallthrough 282 case 19: 283 case 18: 284 case 17: 285 if (kTen5 <= number) { 286 *power = kTen5; 287 *exponent = 5; 288 break; 289 } // else fallthrough 290 case 16: 291 case 15: 292 case 14: 293 if (kTen4 <= number) { 294 *power = kTen4; 295 *exponent = 4; 296 break; 297 } // else fallthrough 298 case 13: 299 case 12: 300 case 11: 301 case 10: 302 if (1000 <= number) { 303 *power = 1000; 304 *exponent = 3; 305 break; 306 } // else fallthrough 307 case 9: 308 case 8: 309 case 7: 310 if (100 <= number) { 311 *power = 100; 312 *exponent = 2; 313 break; 314 } // else fallthrough 315 case 6: 316 case 5: 317 case 4: 318 if (10 <= number) { 319 *power = 10; 320 *exponent = 1; 321 break; 322 } // else fallthrough 323 case 3: 324 case 2: 325 case 1: 326 if (1 <= number) { 327 *power = 1; 328 *exponent = 0; 329 break; 330 } // else fallthrough 331 case 0: 332 *power = 0; 333 *exponent = -1; 334 break; 335 default: 336 // Following assignments are here to silence compiler warnings. 337 *power = 0; 338 *exponent = 0; 339 UNREACHABLE(); 340 } 341 } 342 343 344 // Generates the digits of input number w. 345 // w is a floating-point number (DiyFp), consisting of a significand and an 346 // exponent. Its exponent is bounded by kMinimalTargetExponent and 347 // kMaximalTargetExponent. 348 // Hence -60 <= w.e() <= -32. 349 // 350 // Returns false if it fails, in which case the generated digits in the buffer 351 // should not be used. 352 // Preconditions: 353 // * low, w and high are correct up to 1 ulp (unit in the last place). That 354 // is, their error must be less than a unit of their last digits. 355 // * low.e() == w.e() == high.e() 356 // * low < w < high, and taking into account their error: low~ <= high~ 357 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 358 // Postconditions: returns false if procedure fails. 359 // otherwise: 360 // * buffer is not null-terminated, but len contains the number of digits. 361 // * buffer contains the shortest possible decimal digit-sequence 362 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the 363 // correct values of low and high (without their error). 364 // * if more than one decimal representation gives the minimal number of 365 // decimal digits then the one closest to W (where W is the correct value 366 // of w) is chosen. 367 // Remark: this procedure takes into account the imprecision of its input 368 // numbers. If the precision is not enough to guarantee all the postconditions 369 // then false is returned. This usually happens rarely (~0.5%). 370 // 371 // Say, for the sake of example, that 372 // w.e() == -48, and w.f() == 0x1234567890abcdef 373 // w's value can be computed by w.f() * 2^w.e() 374 // We can obtain w's integral digits by simply shifting w.f() by -w.e(). 375 // -> w's integral part is 0x1234 376 // w's fractional part is therefore 0x567890abcdef. 377 // Printing w's integral part is easy (simply print 0x1234 in decimal). 378 // In order to print its fraction we repeatedly multiply the fraction by 10 and 379 // get each digit. Example the first digit after the point would be computed by 380 // (0x567890abcdef * 10) >> 48. -> 3 381 // The whole thing becomes slightly more complicated because we want to stop 382 // once we have enough digits. That is, once the digits inside the buffer 383 // represent 'w' we can stop. Everything inside the interval low - high 384 // represents w. However we have to pay attention to low, high and w's 385 // imprecision. 386 static bool DigitGen(DiyFp low, 387 DiyFp w, 388 DiyFp high, 389 Vector<char> buffer, 390 int* length, 391 int* kappa) { 392 ASSERT(low.e() == w.e() && w.e() == high.e()); 393 ASSERT(low.f() + 1 <= high.f() - 1); 394 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 395 // low, w and high are imprecise, but by less than one ulp (unit in the last 396 // place). 397 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that 398 // the new numbers are outside of the interval we want the final 399 // representation to lie in. 400 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield 401 // numbers that are certain to lie in the interval. We will use this fact 402 // later on. 403 // We will now start by generating the digits within the uncertain 404 // interval. Later we will weed out representations that lie outside the safe 405 // interval and thus _might_ lie outside the correct interval. 406 uint64_t unit = 1; 407 DiyFp too_low = DiyFp(low.f() - unit, low.e()); 408 DiyFp too_high = DiyFp(high.f() + unit, high.e()); 409 // too_low and too_high are guaranteed to lie outside the interval we want the 410 // generated number in. 411 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low); 412 // We now cut the input number into two parts: the integral digits and the 413 // fractionals. We will not write any decimal separator though, but adapt 414 // kappa instead. 415 // Reminder: we are currently computing the digits (stored inside the buffer) 416 // such that: too_low < buffer * 10^kappa < too_high 417 // We use too_high for the digit_generation and stop as soon as possible. 418 // If we stop early we effectively round down. 419 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 420 // Division by one is a shift. 421 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); 422 // Modulo by one is an and. 423 uint64_t fractionals = too_high.f() & (one.f() - 1); 424 uint32_t divisor; 425 int divisor_exponent; 426 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 427 &divisor, &divisor_exponent); 428 *kappa = divisor_exponent + 1; 429 *length = 0; 430 // Loop invariant: buffer = too_high / 10^kappa (integer division) 431 // The invariant holds for the first iteration: kappa has been initialized 432 // with the divisor exponent + 1. And the divisor is the biggest power of ten 433 // that is smaller than integrals. 434 while (*kappa > 0) { 435 int digit = integrals / divisor; 436 buffer[*length] = '0' + digit; 437 (*length)++; 438 integrals %= divisor; 439 (*kappa)--; 440 // Note that kappa now equals the exponent of the divisor and that the 441 // invariant thus holds again. 442 uint64_t rest = 443 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 444 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) 445 // Reminder: unsafe_interval.e() == one.e() 446 if (rest < unsafe_interval.f()) { 447 // Rounding down (by not emitting the remaining digits) yields a number 448 // that lies within the unsafe interval. 449 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), 450 unsafe_interval.f(), rest, 451 static_cast<uint64_t>(divisor) << -one.e(), unit); 452 } 453 divisor /= 10; 454 } 455 456 // The integrals have been generated. We are at the point of the decimal 457 // separator. In the following loop we simply multiply the remaining digits by 458 // 10 and divide by one. We just need to pay attention to multiply associated 459 // data (like the interval or 'unit'), too. 460 // Note that the multiplication by 10 does not overflow, because w.e >= -60 461 // and thus one.e >= -60. 462 ASSERT(one.e() >= -60); 463 ASSERT(fractionals < one.f()); 464 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); 465 while (true) { 466 fractionals *= 10; 467 unit *= 10; 468 unsafe_interval.set_f(unsafe_interval.f() * 10); 469 // Integer division by one. 470 int digit = static_cast<int>(fractionals >> -one.e()); 471 buffer[*length] = '0' + digit; 472 (*length)++; 473 fractionals &= one.f() - 1; // Modulo by one. 474 (*kappa)--; 475 if (fractionals < unsafe_interval.f()) { 476 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit, 477 unsafe_interval.f(), fractionals, one.f(), unit); 478 } 479 } 480 } 481 482 483 484 // Generates (at most) requested_digits digits of input number w. 485 // w is a floating-point number (DiyFp), consisting of a significand and an 486 // exponent. Its exponent is bounded by kMinimalTargetExponent and 487 // kMaximalTargetExponent. 488 // Hence -60 <= w.e() <= -32. 489 // 490 // Returns false if it fails, in which case the generated digits in the buffer 491 // should not be used. 492 // Preconditions: 493 // * w is correct up to 1 ulp (unit in the last place). That 494 // is, its error must be strictly less than a unit of its last digit. 495 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 496 // 497 // Postconditions: returns false if procedure fails. 498 // otherwise: 499 // * buffer is not null-terminated, but length contains the number of 500 // digits. 501 // * the representation in buffer is the most precise representation of 502 // requested_digits digits. 503 // * buffer contains at most requested_digits digits of w. If there are less 504 // than requested_digits digits then some trailing '0's have been removed. 505 // * kappa is such that 506 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. 507 // 508 // Remark: This procedure takes into account the imprecision of its input 509 // numbers. If the precision is not enough to guarantee all the postconditions 510 // then false is returned. This usually happens rarely, but the failure-rate 511 // increases with higher requested_digits. 512 static bool DigitGenCounted(DiyFp w, 513 int requested_digits, 514 Vector<char> buffer, 515 int* length, 516 int* kappa) { 517 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 518 ASSERT(kMinimalTargetExponent >= -60); 519 ASSERT(kMaximalTargetExponent <= -32); 520 // w is assumed to have an error less than 1 unit. Whenever w is scaled we 521 // also scale its error. 522 uint64_t w_error = 1; 523 // We cut the input number into two parts: the integral digits and the 524 // fractional digits. We don't emit any decimal separator, but adapt kappa 525 // instead. Example: instead of writing "1.2" we put "12" into the buffer and 526 // increase kappa by 1. 527 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 528 // Division by one is a shift. 529 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); 530 // Modulo by one is an and. 531 uint64_t fractionals = w.f() & (one.f() - 1); 532 uint32_t divisor; 533 int divisor_exponent; 534 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 535 &divisor, &divisor_exponent); 536 *kappa = divisor_exponent + 1; 537 *length = 0; 538 539 // Loop invariant: buffer = w / 10^kappa (integer division) 540 // The invariant holds for the first iteration: kappa has been initialized 541 // with the divisor exponent + 1. And the divisor is the biggest power of ten 542 // that is smaller than 'integrals'. 543 while (*kappa > 0) { 544 int digit = integrals / divisor; 545 buffer[*length] = '0' + digit; 546 (*length)++; 547 requested_digits--; 548 integrals %= divisor; 549 (*kappa)--; 550 // Note that kappa now equals the exponent of the divisor and that the 551 // invariant thus holds again. 552 if (requested_digits == 0) break; 553 divisor /= 10; 554 } 555 556 if (requested_digits == 0) { 557 uint64_t rest = 558 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 559 return RoundWeedCounted(buffer, *length, rest, 560 static_cast<uint64_t>(divisor) << -one.e(), w_error, 561 kappa); 562 } 563 564 // The integrals have been generated. We are at the point of the decimal 565 // separator. In the following loop we simply multiply the remaining digits by 566 // 10 and divide by one. We just need to pay attention to multiply associated 567 // data (the 'unit'), too. 568 // Note that the multiplication by 10 does not overflow, because w.e >= -60 569 // and thus one.e >= -60. 570 ASSERT(one.e() >= -60); 571 ASSERT(fractionals < one.f()); 572 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); 573 while (requested_digits > 0 && fractionals > w_error) { 574 fractionals *= 10; 575 w_error *= 10; 576 // Integer division by one. 577 int digit = static_cast<int>(fractionals >> -one.e()); 578 buffer[*length] = '0' + digit; 579 (*length)++; 580 requested_digits--; 581 fractionals &= one.f() - 1; // Modulo by one. 582 (*kappa)--; 583 } 584 if (requested_digits != 0) return false; 585 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, 586 kappa); 587 } 588 589 590 // Provides a decimal representation of v. 591 // Returns true if it succeeds, otherwise the result cannot be trusted. 592 // There will be *length digits inside the buffer (not null-terminated). 593 // If the function returns true then 594 // v == (double) (buffer * 10^decimal_exponent). 595 // The digits in the buffer are the shortest representation possible: no 596 // 0.09999999999999999 instead of 0.1. The shorter representation will even be 597 // chosen even if the longer one would be closer to v. 598 // The last digit will be closest to the actual v. That is, even if several 599 // digits might correctly yield 'v' when read again, the closest will be 600 // computed. 601 static bool Grisu3(double v, 602 Vector<char> buffer, 603 int* length, 604 int* decimal_exponent) { 605 DiyFp w = Double(v).AsNormalizedDiyFp(); 606 // boundary_minus and boundary_plus are the boundaries between v and its 607 // closest floating-point neighbors. Any number strictly between 608 // boundary_minus and boundary_plus will round to v when convert to a double. 609 // Grisu3 will never output representations that lie exactly on a boundary. 610 DiyFp boundary_minus, boundary_plus; 611 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); 612 ASSERT(boundary_plus.e() == w.e()); 613 DiyFp ten_mk; // Cached power of ten: 10^-k 614 int mk; // -k 615 int ten_mk_minimal_binary_exponent = 616 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 617 int ten_mk_maximal_binary_exponent = 618 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 619 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 620 ten_mk_minimal_binary_exponent, 621 ten_mk_maximal_binary_exponent, 622 &ten_mk, &mk); 623 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 624 DiyFp::kSignificandSize) && 625 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 626 DiyFp::kSignificandSize)); 627 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 628 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 629 630 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 631 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 632 // off by a small amount. 633 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 634 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 635 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 636 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 637 ASSERT(scaled_w.e() == 638 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); 639 // In theory it would be possible to avoid some recomputations by computing 640 // the difference between w and boundary_minus/plus (a power of 2) and to 641 // compute scaled_boundary_minus/plus by subtracting/adding from 642 // scaled_w. However the code becomes much less readable and the speed 643 // enhancements are not terriffic. 644 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); 645 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); 646 647 // DigitGen will generate the digits of scaled_w. Therefore we have 648 // v == (double) (scaled_w * 10^-mk). 649 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an 650 // integer than it will be updated. For instance if scaled_w == 1.23 then 651 // the buffer will be filled with "123" und the decimal_exponent will be 652 // decreased by 2. 653 int kappa; 654 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, 655 buffer, length, &kappa); 656 *decimal_exponent = -mk + kappa; 657 return result; 658 } 659 660 661 // The "counted" version of grisu3 (see above) only generates requested_digits 662 // number of digits. This version does not generate the shortest representation, 663 // and with enough requested digits 0.1 will at some point print as 0.9999999... 664 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and 665 // therefore the rounding strategy for halfway cases is irrelevant. 666 static bool Grisu3Counted(double v, 667 int requested_digits, 668 Vector<char> buffer, 669 int* length, 670 int* decimal_exponent) { 671 DiyFp w = Double(v).AsNormalizedDiyFp(); 672 DiyFp ten_mk; // Cached power of ten: 10^-k 673 int mk; // -k 674 int ten_mk_minimal_binary_exponent = 675 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 676 int ten_mk_maximal_binary_exponent = 677 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 678 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 679 ten_mk_minimal_binary_exponent, 680 ten_mk_maximal_binary_exponent, 681 &ten_mk, &mk); 682 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 683 DiyFp::kSignificandSize) && 684 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 685 DiyFp::kSignificandSize)); 686 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 687 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 688 689 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 690 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 691 // off by a small amount. 692 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 693 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 694 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 695 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 696 697 // We now have (double) (scaled_w * 10^-mk). 698 // DigitGen will generate the first requested_digits digits of scaled_w and 699 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It 700 // will not always be exactly the same since DigitGenCounted only produces a 701 // limited number of digits.) 702 int kappa; 703 bool result = DigitGenCounted(scaled_w, requested_digits, 704 buffer, length, &kappa); 705 *decimal_exponent = -mk + kappa; 706 return result; 707 } 708 709 710 bool FastDtoa(double v, 711 FastDtoaMode mode, 712 int requested_digits, 713 Vector<char> buffer, 714 int* length, 715 int* decimal_point) { 716 ASSERT(v > 0); 717 ASSERT(!Double(v).IsSpecial()); 718 719 bool result = false; 720 int decimal_exponent = 0; 721 switch (mode) { 722 case FAST_DTOA_SHORTEST: 723 result = Grisu3(v, buffer, length, &decimal_exponent); 724 break; 725 case FAST_DTOA_PRECISION: 726 result = Grisu3Counted(v, requested_digits, 727 buffer, length, &decimal_exponent); 728 break; 729 default: 730 UNREACHABLE(); 731 } 732 if (result) { 733 *decimal_point = *length + decimal_exponent; 734 buffer[*length] = '\0'; 735 } 736 return result; 737 } 738 739} // namespace double_conversion 740 741} // namespace WTF 742