text: // Copyright 2010 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 "v8.h"
  
  #include "fast-dtoa.h"
  
  #include "cached-powers.h"
  #include "diy-fp.h"
  #include "double.h"
  
  namespace v8 {
  namespace internal {
  
  // 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 ctannot 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;
  }
  
  
  static const uint32_t kTen4 = 10000;
  static const uint32_t kTen5 = 100000;
  static const uint32_t kTen6 = 1000000;
  static const uint32_t kTen7 = 10000000;
  static const uint32_t kTen8 = 100000000;
  static const uint32_t kTen9 = 1000000000;
  
  // Returns the biggest power of ten that is less than or equal than the given
  // number. We furthermore receive the maximum number of bits 'number' has.
  // If number_bits == 0 then 0^-1 is returned
  // The number of bits must be <= 32.
  // Precondition: number < (1 << (number_bits + 1)).
  static void BiggestPowerTen(uint32_t number,
                              int number_bits,
                              uint32_t* power,
                              int* exponent) {
    switch (number_bits) {
      case 32:
      case 31:
      case 30:
        if (kTen9 <= number) {
          *power = kTen9;
          *exponent = 9;
          break;
        }  // else fallthrough
      case 29:
      case 28:
      case 27:
        if (kTen8 <= number) {
          *power = kTen8;
          *exponent = 8;
          break;
        }  // else fallthrough
      case 26:
      case 25:
      case 24:
        if (kTen7 <= number) {
          *power = kTen7;
          *exponent = 7;
          break;
        }  // else fallthrough
      case 23:
      case 22:
      case 21:
      case 20:
        if (kTen6 <= number) {
          *power = kTen6;
          *exponent = 6;
          break;
        }  // else fallthrough
      case 19:
      case 18:
      case 17:
        if (kTen5 <= number) {
          *power = kTen5;
          *exponent = 5;
          break;
        }  // else fallthrough
      case 16:
      case 15:
      case 14:
        if (kTen4 <= number) {
          *power = kTen4;
          *exponent = 4;
          break;
        }  // else fallthrough
      case 13:
      case 12:
      case 11:
      case 10:
        if (1000 <= number) {
          *power = 1000;
          *exponent = 3;
          break;
        }  // else fallthrough
      case 9:
      case 8:
      case 7:
        if (100 <= number) {
          *power = 100;
          *exponent = 2;
          break;
        }  // else fallthrough
      case 6:
      case 5:
      case 4:
        if (10 <= number) {
          *power = 10;
          *exponent = 1;
          break;
        }  // else fallthrough
      case 3:
      case 2:
      case 1:
        if (1 <= number) {
          *power = 1;
          *exponent = 0;
          break;
        }  // else fallthrough
      case 0:
        *power = 0;
        *exponent = -1;
        break;
      default:
        // Following assignments are here to silence compiler warnings.
        *power = 0;
        *exponent = 0;
        UNREACHABLE();
    }
  }
  
  
  // 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;
    BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                    &divisor, &divisor_exponent);
    *kappa = divisor_exponent + 1;
    *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(V8_2PART_UINT64_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 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;
    BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                    &divisor, &divisor_exponent);
    *kappa = divisor_exponent + 1;
    *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(V8_2PART_UINT64_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,
                     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;
    Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
    ASSERT(boundary_plus.e() == w.e());
    DiyFp ten_mk;  // Cached power of ten: 10^-k
    int mk;        // -k
    GetCachedPower(w.e() + DiyFp::kSignificandSize, kMinimalTargetExponent,
                   kMaximalTargetExponent, &mk, &ten_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
    GetCachedPower(w.e() + DiyFp::kSignificandSize, kMinimalTargetExponent,
                   kMaximalTargetExponent, &mk, &ten_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:
        result = Grisu3(v, 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 v8::internal
  