base / third_party / double_conversion / double-conversion / fast-dtoa.cc [blame]

// 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:
//
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//       notice, this list of conditions and the following disclaimer.
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//       copyright notice, this list of conditions and the following
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#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.
  DOUBLE_CONVERSION_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) {
  DOUBLE_CONVERSION_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) {
  DOUBLE_CONVERSION_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.
  if (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) {
  DOUBLE_CONVERSION_ASSERT(low.e() == w.e() && w.e() == high.e());
  DOUBLE_CONVERSION_ASSERT(low.f() + 1 <= high.f() - 1);
  DOUBLE_CONVERSION_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;
    DOUBLE_CONVERSION_ASSERT(digit <= 9);
    buffer[*length] = static_cast<char>('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.
  DOUBLE_CONVERSION_ASSERT(one.e() >= -60);
  DOUBLE_CONVERSION_ASSERT(fractionals < one.f());
  DOUBLE_CONVERSION_ASSERT(DOUBLE_CONVERSION_UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
  for (;;) {
    fractionals *= 10;
    unit *= 10;
    unsafe_interval.set_f(unsafe_interval.f() * 10);
    // Integer division by one.
    int digit = static_cast<int>(fractionals >> -one.e());
    DOUBLE_CONVERSION_ASSERT(digit <= 9);
    buffer[*length] = static_cast<char>('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) {
  DOUBLE_CONVERSION_ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
  DOUBLE_CONVERSION_ASSERT(kMinimalTargetExponent >= -60);
  DOUBLE_CONVERSION_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;
    DOUBLE_CONVERSION_ASSERT(digit <= 9);
    buffer[*length] = static_cast<char>('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.
  DOUBLE_CONVERSION_ASSERT(one.e() >= -60);
  DOUBLE_CONVERSION_ASSERT(fractionals < one.f());
  DOUBLE_CONVERSION_ASSERT(DOUBLE_CONVERSION_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());
    DOUBLE_CONVERSION_ASSERT(digit <= 9);
    buffer[*length] = static_cast<char>('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 {
    DOUBLE_CONVERSION_ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
    float single_v = static_cast<float>(v);
    Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
  }
  DOUBLE_CONVERSION_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);
  DOUBLE_CONVERSION_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);
  DOUBLE_CONVERSION_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);
  DOUBLE_CONVERSION_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) {
  DOUBLE_CONVERSION_ASSERT(v > 0);
  DOUBLE_CONVERSION_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:
      DOUBLE_CONVERSION_UNREACHABLE();
  }
  if (result) {
    *decimal_point = *length + decimal_exponent;
    buffer[*length] = '\0';
  }
  return result;
}

}  // namespace double_conversion