DXR is a code search and navigation tool aimed at making sense of large projects. It supports full-text and regex searches as well as structural queries.

Header

Mercurial (b6057e17f856)

VCS Links

Line Code
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664
// Copyright 2012 the V8 project authors. All rights reserved.
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
//     * Redistributions of source code must retain the above copyright
//       notice, this list of conditions and the following disclaimer.
//     * Redistributions in binary form must reproduce the above
//       copyright notice, this list of conditions and the following
//       disclaimer in the documentation and/or other materials provided
//       with the distribution.
//     * Neither the name of Google Inc. nor the names of its
//       contributors may be used to endorse or promote products derived
//       from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

#include "fast-dtoa.h"

#include "cached-powers.h"
#include "diy-fp.h"
#include "ieee.h"

namespace double_conversion {

// The minimal and maximal target exponent define the range of w's binary
// exponent, where 'w' is the result of multiplying the input by a cached power
// of ten.
//
// A different range might be chosen on a different platform, to optimize digit
// generation, but a smaller range requires more powers of ten to be cached.
static const int kMinimalTargetExponent = -60;
static const int kMaximalTargetExponent = -32;


// Adjusts the last digit of the generated number, and screens out generated
// solutions that may be inaccurate. A solution may be inaccurate if it is
// outside the safe interval, or if we cannot prove that it is closer to the
// input than a neighboring representation of the same length.
//
// Input: * buffer containing the digits of too_high / 10^kappa
//        * the buffer's length
//        * distance_too_high_w == (too_high - w).f() * unit
//        * unsafe_interval == (too_high - too_low).f() * unit
//        * rest = (too_high - buffer * 10^kappa).f() * unit
//        * ten_kappa = 10^kappa * unit
//        * unit = the common multiplier
// Output: returns true if the buffer is guaranteed to contain the closest
//    representable number to the input.
//  Modifies the generated digits in the buffer to approach (round towards) w.
static bool RoundWeed(Vector<char> buffer,
                      int length,
                      uint64_t distance_too_high_w,
                      uint64_t unsafe_interval,
                      uint64_t rest,
                      uint64_t ten_kappa,
                      uint64_t unit) {
  uint64_t small_distance = distance_too_high_w - unit;
  uint64_t big_distance = distance_too_high_w + unit;
  // Let w_low  = too_high - big_distance, and
  //     w_high = too_high - small_distance.
  // Note: w_low < w < w_high
  //
  // The real w (* unit) must lie somewhere inside the interval
  // ]w_low; w_high[ (often written as "(w_low; w_high)")

  // Basically the buffer currently contains a number in the unsafe interval
  // ]too_low; too_high[ with too_low < w < too_high
  //
  //  too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  //                     ^v 1 unit            ^      ^                 ^      ^
  //  boundary_high ---------------------     .      .                 .      .
  //                     ^v 1 unit            .      .                 .      .
  //   - - - - - - - - - - - - - - - - - - -  +  - - + - - - - - -     .      .
  //                                          .      .         ^       .      .
  //                                          .  big_distance  .       .      .
  //                                          .      .         .       .    rest
  //                              small_distance     .         .       .      .
  //                                          v      .         .       .      .
  //  w_high - - - - - - - - - - - - - - - - - -     .         .       .      .
  //                     ^v 1 unit                   .         .       .      .
  //  w ----------------------------------------     .         .       .      .
  //                     ^v 1 unit                   v         .       .      .
  //  w_low  - - - - - - - - - - - - - - - - - - - - -         .       .      .
  //                                                           .       .      v
  //  buffer --------------------------------------------------+-------+--------
  //                                                           .       .
  //                                                  safe_interval    .
  //                                                           v       .
  //   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -     .
  //                     ^v 1 unit                                     .
  //  boundary_low -------------------------                     unsafe_interval
  //                     ^v 1 unit                                     v
  //  too_low  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  //
  //
  // Note that the value of buffer could lie anywhere inside the range too_low
  // to too_high.
  //
  // boundary_low, boundary_high and w are approximations of the real boundaries
  // and v (the input number). They are guaranteed to be precise up to one unit.
  // In fact the error is guaranteed to be strictly less than one unit.
  //
  // Anything that lies outside the unsafe interval is guaranteed not to round
  // to v when read again.
  // Anything that lies inside the safe interval is guaranteed to round to v
  // when read again.
  // If the number inside the buffer lies inside the unsafe interval but not
  // inside the safe interval then we simply do not know and bail out (returning
  // false).
  //
  // Similarly we have to take into account the imprecision of 'w' when finding
  // the closest representation of 'w'. If we have two potential
  // representations, and one is closer to both w_low and w_high, then we know
  // it is closer to the actual value v.
  //
  // By generating the digits of too_high we got the largest (closest to
  // too_high) buffer that is still in the unsafe interval. In the case where
  // w_high < buffer < too_high we try to decrement the buffer.
  // This way the buffer approaches (rounds towards) w.
  // There are 3 conditions that stop the decrementation process:
  //   1) the buffer is already below w_high
  //   2) decrementing the buffer would make it leave the unsafe interval
  //   3) decrementing the buffer would yield a number below w_high and farther
  //      away than the current number. In other words:
  //              (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
  // Instead of using the buffer directly we use its distance to too_high.
  // Conceptually rest ~= too_high - buffer
  // We need to do the following tests in this order to avoid over- and
  // underflows.
  ASSERT(rest <= unsafe_interval);
  while (rest < small_distance &&  // Negated condition 1
         unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
         (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
          small_distance - rest >= rest + ten_kappa - small_distance)) {
    buffer[length - 1]--;
    rest += ten_kappa;
  }

  // We have approached w+ as much as possible. We now test if approaching w-
  // would require changing the buffer. If yes, then we have two possible
  // representations close to w, but we cannot decide which one is closer.
  if (rest < big_distance &&
      unsafe_interval - rest >= ten_kappa &&
      (rest + ten_kappa < big_distance ||
       big_distance - rest > rest + ten_kappa - big_distance)) {
    return false;
  }

  // Weeding test.
  //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
  //   Since too_low = too_high - unsafe_interval this is equivalent to
  //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
  //   Conceptually we have: rest ~= too_high - buffer
  return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
}


// Rounds the buffer upwards if the result is closer to v by possibly adding
// 1 to the buffer. If the precision of the calculation is not sufficient to
// round correctly, return false.
// The rounding might shift the whole buffer in which case the kappa is
// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
//
// If 2*rest > ten_kappa then the buffer needs to be round up.
// rest can have an error of +/- 1 unit. This function accounts for the
// imprecision and returns false, if the rounding direction cannot be
// unambiguously determined.
//
// Precondition: rest < ten_kappa.
static bool RoundWeedCounted(Vector<char> buffer,
                             int length,
                             uint64_t rest,
                             uint64_t ten_kappa,
                             uint64_t unit,
                             int* kappa) {
  ASSERT(rest < ten_kappa);
  // The following tests are done in a specific order to avoid overflows. They
  // will work correctly with any uint64 values of rest < ten_kappa and unit.
  //
  // If the unit is too big, then we don't know which way to round. For example
  // a unit of 50 means that the real number lies within rest +/- 50. If
  // 10^kappa == 40 then there is no way to tell which way to round.
  if (unit >= ten_kappa) return false;
  // Even if unit is just half the size of 10^kappa we are already completely
  // lost. (And after the previous test we know that the expression will not
  // over/underflow.)
  if (ten_kappa - unit <= unit) return false;
  // If 2 * (rest + unit) <= 10^kappa we can safely round down.
  if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
    return true;
  }
  // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
  if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
    // Increment the last digit recursively until we find a non '9' digit.
    buffer[length - 1]++;
    for (int i = length - 1; i > 0; --i) {
      if (buffer[i] != '0' + 10) break;
      buffer[i] = '0';
      buffer[i - 1]++;
    }
    // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
    // exception of the first digit all digits are now '0'. Simply switch the
    // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
    // the power (the kappa) is increased.
    if (buffer[0] == '0' + 10) {
      buffer[0] = '1';
      (*kappa) += 1;
    }
    return true;
  }
  return false;
}

// Returns the biggest power of ten that is less than or equal to the given
// number. We furthermore receive the maximum number of bits 'number' has.
//
// Returns power == 10^(exponent_plus_one-1) such that
//    power <= number < power * 10.
// If number_bits == 0 then 0^(0-1) is returned.
// The number of bits must be <= 32.
// Precondition: number < (1 << (number_bits + 1)).

// Inspired by the method for finding an integer log base 10 from here:
// http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
static unsigned int const kSmallPowersOfTen[] =
    {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
     1000000000};

static void BiggestPowerTen(uint32_t number,
                            int number_bits,
                            uint32_t* power,
                            int* exponent_plus_one) {
  ASSERT(number < (1u << (number_bits + 1)));
  // 1233/4096 is approximately 1/lg(10).
  int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);
  // We increment to skip over the first entry in the kPowersOf10 table.
  // Note: kPowersOf10[i] == 10^(i-1).
  exponent_plus_one_guess++;
  // We don't have any guarantees that 2^number_bits <= number.
  // TODO(floitsch): can we change the 'while' into an 'if'? We definitely see
  // number < (2^number_bits - 1), but I haven't encountered
  // number < (2^number_bits - 2) yet.
  while (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
    exponent_plus_one_guess--;
  }
  *power = kSmallPowersOfTen[exponent_plus_one_guess];
  *exponent_plus_one = exponent_plus_one_guess;
}

// Generates the digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by kMinimalTargetExponent and
// kMaximalTargetExponent.
//       Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
//  * low, w and high are correct up to 1 ulp (unit in the last place). That
//    is, their error must be less than a unit of their last digits.
//  * low.e() == w.e() == high.e()
//  * low < w < high, and taking into account their error: low~ <= high~
//  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
// Postconditions: returns false if procedure fails.
//   otherwise:
//     * buffer is not null-terminated, but len contains the number of digits.
//     * buffer contains the shortest possible decimal digit-sequence
//       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
//       correct values of low and high (without their error).
//     * if more than one decimal representation gives the minimal number of
//       decimal digits then the one closest to W (where W is the correct value
//       of w) is chosen.
// Remark: this procedure takes into account the imprecision of its input
//   numbers. If the precision is not enough to guarantee all the postconditions
//   then false is returned. This usually happens rarely (~0.5%).
//
// Say, for the sake of example, that
//   w.e() == -48, and w.f() == 0x1234567890abcdef
// w's value can be computed by w.f() * 2^w.e()
// We can obtain w's integral digits by simply shifting w.f() by -w.e().
//  -> w's integral part is 0x1234
//  w's fractional part is therefore 0x567890abcdef.
// Printing w's integral part is easy (simply print 0x1234 in decimal).
// In order to print its fraction we repeatedly multiply the fraction by 10 and
// get each digit. Example the first digit after the point would be computed by
//   (0x567890abcdef * 10) >> 48. -> 3
// The whole thing becomes slightly more complicated because we want to stop
// once we have enough digits. That is, once the digits inside the buffer
// represent 'w' we can stop. Everything inside the interval low - high
// represents w. However we have to pay attention to low, high and w's
// imprecision.
static bool DigitGen(DiyFp low,
                     DiyFp w,
                     DiyFp high,
                     Vector<char> buffer,
                     int* length,
                     int* kappa) {
  ASSERT(low.e() == w.e() && w.e() == high.e());
  ASSERT(low.f() + 1 <= high.f() - 1);
  ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
  // low, w and high are imprecise, but by less than one ulp (unit in the last
  // place).
  // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
  // the new numbers are outside of the interval we want the final
  // representation to lie in.
  // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
  // numbers that are certain to lie in the interval. We will use this fact
  // later on.
  // We will now start by generating the digits within the uncertain
  // interval. Later we will weed out representations that lie outside the safe
  // interval and thus _might_ lie outside the correct interval.
  uint64_t unit = 1;
  DiyFp too_low = DiyFp(low.f() - unit, low.e());
  DiyFp too_high = DiyFp(high.f() + unit, high.e());
  // too_low and too_high are guaranteed to lie outside the interval we want the
  // generated number in.
  DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
  // We now cut the input number into two parts: the integral digits and the
  // fractionals. We will not write any decimal separator though, but adapt
  // kappa instead.
  // Reminder: we are currently computing the digits (stored inside the buffer)
  // such that:   too_low < buffer * 10^kappa < too_high
  // We use too_high for the digit_generation and stop as soon as possible.
  // If we stop early we effectively round down.
  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
  // Division by one is a shift.
  uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
  // Modulo by one is an and.
  uint64_t fractionals = too_high.f() & (one.f() - 1);
  uint32_t divisor;
  int divisor_exponent_plus_one;
  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                  &divisor, &divisor_exponent_plus_one);
  *kappa = divisor_exponent_plus_one;
  *length = 0;
  // Loop invariant: buffer = too_high / 10^kappa  (integer division)
  // The invariant holds for the first iteration: kappa has been initialized
  // with the divisor exponent + 1. And the divisor is the biggest power of ten
  // that is smaller than integrals.
  while (*kappa > 0) {
    int digit = integrals / divisor;
    buffer[*length] = '0' + digit;
    (*length)++;
    integrals %= divisor;
    (*kappa)--;
    // Note that kappa now equals the exponent of the divisor and that the
    // invariant thus holds again.
    uint64_t rest =
        (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
    // Reminder: unsafe_interval.e() == one.e()
    if (rest < unsafe_interval.f()) {
      // Rounding down (by not emitting the remaining digits) yields a number
      // that lies within the unsafe interval.
      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
                       unsafe_interval.f(), rest,
                       static_cast<uint64_t>(divisor) << -one.e(), unit);
    }
    divisor /= 10;
  }

  // The integrals have been generated. We are at the point of the decimal
  // separator. In the following loop we simply multiply the remaining digits by
  // 10 and divide by one. We just need to pay attention to multiply associated
  // data (like the interval or 'unit'), too.
  // Note that the multiplication by 10 does not overflow, because w.e >= -60
  // and thus one.e >= -60.
  ASSERT(one.e() >= -60);
  ASSERT(fractionals < one.f());
  ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
  while (true) {
    fractionals *= 10;
    unit *= 10;
    unsafe_interval.set_f(unsafe_interval.f() * 10);
    // Integer division by one.
    int digit = static_cast<int>(fractionals >> -one.e());
    buffer[*length] = '0' + digit;
    (*length)++;
    fractionals &= one.f() - 1;  // Modulo by one.
    (*kappa)--;
    if (fractionals < unsafe_interval.f()) {
      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
                       unsafe_interval.f(), fractionals, one.f(), unit);
    }
  }
}



// Generates (at most) requested_digits digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by kMinimalTargetExponent and
// kMaximalTargetExponent.
//       Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
//  * w is correct up to 1 ulp (unit in the last place). That
//    is, its error must be strictly less than a unit of its last digit.
//  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
//
// Postconditions: returns false if procedure fails.
//   otherwise:
//     * buffer is not null-terminated, but length contains the number of
//       digits.
//     * the representation in buffer is the most precise representation of
//       requested_digits digits.
//     * buffer contains at most requested_digits digits of w. If there are less
//       than requested_digits digits then some trailing '0's have been removed.
//     * kappa is such that
//            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
//
// Remark: This procedure takes into account the imprecision of its input
//   numbers. If the precision is not enough to guarantee all the postconditions
//   then false is returned. This usually happens rarely, but the failure-rate
//   increases with higher requested_digits.
static bool DigitGenCounted(DiyFp w,
                            int requested_digits,
                            Vector<char> buffer,
                            int* length,
                            int* kappa) {
  ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
  ASSERT(kMinimalTargetExponent >= -60);
  ASSERT(kMaximalTargetExponent <= -32);
  // w is assumed to have an error less than 1 unit. Whenever w is scaled we
  // also scale its error.
  uint64_t w_error = 1;
  // We cut the input number into two parts: the integral digits and the
  // fractional digits. We don't emit any decimal separator, but adapt kappa
  // instead. Example: instead of writing "1.2" we put "12" into the buffer and
  // increase kappa by 1.
  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
  // Division by one is a shift.
  uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
  // Modulo by one is an and.
  uint64_t fractionals = w.f() & (one.f() - 1);
  uint32_t divisor;
  int divisor_exponent_plus_one;
  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                  &divisor, &divisor_exponent_plus_one);
  *kappa = divisor_exponent_plus_one;
  *length = 0;

  // Loop invariant: buffer = w / 10^kappa  (integer division)
  // The invariant holds for the first iteration: kappa has been initialized
  // with the divisor exponent + 1. And the divisor is the biggest power of ten
  // that is smaller than 'integrals'.
  while (*kappa > 0) {
    int digit = integrals / divisor;
    buffer[*length] = '0' + digit;
    (*length)++;
    requested_digits--;
    integrals %= divisor;
    (*kappa)--;
    // Note that kappa now equals the exponent of the divisor and that the
    // invariant thus holds again.
    if (requested_digits == 0) break;
    divisor /= 10;
  }

  if (requested_digits == 0) {
    uint64_t rest =
        (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    return RoundWeedCounted(buffer, *length, rest,
                            static_cast<uint64_t>(divisor) << -one.e(), w_error,
                            kappa);
  }

  // The integrals have been generated. We are at the point of the decimal
  // separator. In the following loop we simply multiply the remaining digits by
  // 10 and divide by one. We just need to pay attention to multiply associated
  // data (the 'unit'), too.
  // Note that the multiplication by 10 does not overflow, because w.e >= -60
  // and thus one.e >= -60.
  ASSERT(one.e() >= -60);
  ASSERT(fractionals < one.f());
  ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
  while (requested_digits > 0 && fractionals > w_error) {
    fractionals *= 10;
    w_error *= 10;
    // Integer division by one.
    int digit = static_cast<int>(fractionals >> -one.e());
    buffer[*length] = '0' + digit;
    (*length)++;
    requested_digits--;
    fractionals &= one.f() - 1;  // Modulo by one.
    (*kappa)--;
  }
  if (requested_digits != 0) return false;
  return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
                          kappa);
}


// Provides a decimal representation of v.
// Returns true if it succeeds, otherwise the result cannot be trusted.
// There will be *length digits inside the buffer (not null-terminated).
// If the function returns true then
//        v == (double) (buffer * 10^decimal_exponent).
// The digits in the buffer are the shortest representation possible: no
// 0.09999999999999999 instead of 0.1. The shorter representation will even be
// chosen even if the longer one would be closer to v.
// The last digit will be closest to the actual v. That is, even if several
// digits might correctly yield 'v' when read again, the closest will be
// computed.
static bool Grisu3(double v,
                   FastDtoaMode mode,
                   Vector<char> buffer,
                   int* length,
                   int* decimal_exponent) {
  DiyFp w = Double(v).AsNormalizedDiyFp();
  // boundary_minus and boundary_plus are the boundaries between v and its
  // closest floating-point neighbors. Any number strictly between
  // boundary_minus and boundary_plus will round to v when convert to a double.
  // Grisu3 will never output representations that lie exactly on a boundary.
  DiyFp boundary_minus, boundary_plus;
  if (mode == FAST_DTOA_SHORTEST) {
    Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
  } else {
    ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
    float single_v = static_cast<float>(v);
    Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
  }
  ASSERT(boundary_plus.e() == w.e());
  DiyFp ten_mk;  // Cached power of ten: 10^-k
  int mk;        // -k
  int ten_mk_minimal_binary_exponent =
     kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
  int ten_mk_maximal_binary_exponent =
     kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
  PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
      ten_mk_minimal_binary_exponent,
      ten_mk_maximal_binary_exponent,
      &ten_mk, &mk);
  ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
          DiyFp::kSignificandSize) &&
         (kMaximalTargetExponent >= w.e() + ten_mk.e() +
          DiyFp::kSignificandSize));
  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
  // 64 bit significand and ten_mk is thus only precise up to 64 bits.

  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
  // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
  // off by a small amount.
  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
  //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
  DiyFp scaled_w = DiyFp::Times(w, ten_mk);
  ASSERT(scaled_w.e() ==
         boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
  // In theory it would be possible to avoid some recomputations by computing
  // the difference between w and boundary_minus/plus (a power of 2) and to
  // compute scaled_boundary_minus/plus by subtracting/adding from
  // scaled_w. However the code becomes much less readable and the speed
  // enhancements are not terriffic.
  DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
  DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);

  // DigitGen will generate the digits of scaled_w. Therefore we have
  // v == (double) (scaled_w * 10^-mk).
  // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
  // integer than it will be updated. For instance if scaled_w == 1.23 then
  // the buffer will be filled with "123" und the decimal_exponent will be
  // decreased by 2.
  int kappa;
  bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
                         buffer, length, &kappa);
  *decimal_exponent = -mk + kappa;
  return result;
}


// The "counted" version of grisu3 (see above) only generates requested_digits
// number of digits. This version does not generate the shortest representation,
// and with enough requested digits 0.1 will at some point print as 0.9999999...
// Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
// therefore the rounding strategy for halfway cases is irrelevant.
static bool Grisu3Counted(double v,
                          int requested_digits,
                          Vector<char> buffer,
                          int* length,
                          int* decimal_exponent) {
  DiyFp w = Double(v).AsNormalizedDiyFp();
  DiyFp ten_mk;  // Cached power of ten: 10^-k
  int mk;        // -k
  int ten_mk_minimal_binary_exponent =
     kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
  int ten_mk_maximal_binary_exponent =
     kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
  PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
      ten_mk_minimal_binary_exponent,
      ten_mk_maximal_binary_exponent,
      &ten_mk, &mk);
  ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
          DiyFp::kSignificandSize) &&
         (kMaximalTargetExponent >= w.e() + ten_mk.e() +
          DiyFp::kSignificandSize));
  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
  // 64 bit significand and ten_mk is thus only precise up to 64 bits.

  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
  // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
  // off by a small amount.
  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
  //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
  DiyFp scaled_w = DiyFp::Times(w, ten_mk);

  // We now have (double) (scaled_w * 10^-mk).
  // DigitGen will generate the first requested_digits digits of scaled_w and
  // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
  // will not always be exactly the same since DigitGenCounted only produces a
  // limited number of digits.)
  int kappa;
  bool result = DigitGenCounted(scaled_w, requested_digits,
                                buffer, length, &kappa);
  *decimal_exponent = -mk + kappa;
  return result;
}


bool FastDtoa(double v,
              FastDtoaMode mode,
              int requested_digits,
              Vector<char> buffer,
              int* length,
              int* decimal_point) {
  ASSERT(v > 0);
  ASSERT(!Double(v).IsSpecial());

  bool result = false;
  int decimal_exponent = 0;
  switch (mode) {
    case FAST_DTOA_SHORTEST:
    case FAST_DTOA_SHORTEST_SINGLE:
      result = Grisu3(v, mode, buffer, length, &decimal_exponent);
      break;
    case FAST_DTOA_PRECISION:
      result = Grisu3Counted(v, requested_digits,
                             buffer, length, &decimal_exponent);
      break;
    default:
      UNREACHABLE();
  }
  if (result) {
    *decimal_point = *length + decimal_exponent;
    buffer[*length] = '\0';
  }
  return result;
}

}  // namespace double_conversion