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#include <double-conversion/fast-dtoa.h>

#include <double-conversion/cached-powers.h>

#include <double-conversion/diy-fp.h>

#include <double-conversion/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.

  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) {

  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;

    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.

  ASSERT(one.e() >= -60);

  ASSERT(fractionals < one.f());

  ASSERT(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());

    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) {

  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;

    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.

  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());

    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 {

    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