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 // Copyright 2012 the V8 project authors. All rights reserved.

 // Redistribution and use in source and binary forms, with or without

 // modification, are permitted provided that the following conditions are

 // met:

 //

 //     * Redistributions of source code must retain the above copyright

 //       notice, this list of conditions and the following disclaimer.

 //     * Redistributions in binary form must reproduce the above

 //       copyright notice, this list of conditions and the following

 //       disclaimer in the documentation and/or other materials provided

 //       with the distribution.

 //     * Neither the name of Google Inc. nor the names of its

 //       contributors may be used to endorse or promote products derived

 //       from this software without specific prior written permission.

 //

 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS

 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR

 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT

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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.

   ASSERT(rest <= unsafe_interval);

   while (rest < small_distance &&  // Negated condition 1

          unsafe_interval - rest >= ten_kappa &&  // Negated condition 2

          (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high

           small_distance - rest >= rest + ten_kappa - small_distance)) {

     buffer[length - 1]--;

     rest += ten_kappa;

   }

   // We have approached w+ as much as possible. We now test if approaching w-

   // would require changing the buffer. If yes, then we have two possible

   // representations close to w, but we cannot decide which one is closer.

   if (rest < big_distance &&

       unsafe_interval - rest >= ten_kappa &&

       (rest + ten_kappa < big_distance ||

        big_distance - rest > rest + ten_kappa - big_distance)) {

     return false;

   }

   // Weeding test.

   //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]

   //   Since too_low = too_high - unsafe_interval this is equivalent to

   //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]

   //   Conceptually we have: rest ~= too_high - buffer

   return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);

 }

 // Rounds the buffer upwards if the result is closer to v by possibly adding

 // 1 to the buffer. If the precision of the calculation is not sufficient to

 // round correctly, return false.

 // The rounding might shift the whole buffer in which case the kappa is

 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.

 //

 // If 2*rest > ten_kappa then the buffer needs to be round up.

 // rest can have an error of +/- 1 unit. This function accounts for the

 // imprecision and returns false, if the rounding direction cannot be

 // unambiguously determined.

 //

 // Precondition: rest < ten_kappa.

 static bool RoundWeedCounted(Vector<char> buffer,

                              int length,

                              uint64_t rest,

                              uint64_t ten_kappa,

                              uint64_t unit,

                              int* kappa) {

   ASSERT(rest < ten_kappa);

   // The following tests are done in a specific order to avoid overflows. They

   // will work correctly with any uint64 values of rest < ten_kappa and unit.

   //

   // If the unit is too big, then we don't know which way to round. For example

   // a unit of 50 means that the real number lies within rest +/- 50. If

   // 10^kappa == 40 then there is no way to tell which way to round.

   if (unit >= ten_kappa) return false;

   // Even if unit is just half the size of 10^kappa we are already completely

   // lost. (And after the previous test we know that the expression will not

   // over/underflow.)

   if (ten_kappa - unit <= unit) return false;

   // If 2 * (rest + unit) <= 10^kappa we can safely round down.

   if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {

     return true;

   }

   // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.

   if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {

     // Increment the last digit recursively until we find a non '9' digit.

     buffer[length - 1]++;

     for (int i = length - 1; i > 0; --i) {

       if (buffer[i] != '0' + 10) break;

       buffer[i] = '0';

       buffer[i - 1]++;

     }

     // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the

     // exception of the first digit all digits are now '0'. Simply switch the

     // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and

     // the power (the kappa) is increased.

     if (buffer[0] == '0' + 10) {

       buffer[0] = '1';

       (*kappa) += 1;

     }

     return true;

   }

   return false;

 }

 // Returns the biggest power of ten that is less than or equal to the given

 // number. We furthermore receive the maximum number of bits 'number' has.

 //

 // Returns power == 10^(exponent_plus_one-1) such that

 //    power <= number < power * 10.

 // If number_bits == 0 then 0^(0-1) is returned.

 // The number of bits must be <= 32.

 // Precondition: number < (1 << (number_bits + 1)).

 // Inspired by the method for finding an integer log base 10 from here:

 // http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10

 static unsigned int const kSmallPowersOfTen[] =

     {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,

      1000000000};

 static void BiggestPowerTen(uint32_t number,

                             int number_bits,

                             uint32_t* power,

                             int* exponent_plus_one) {

   ASSERT(number < (1u << (number_bits + 1)));

   // 1233/4096 is approximately 1/lg(10).

   int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);

   // We increment to skip over the first entry in the kPowersOf10 table.

   // Note: kPowersOf10[i] == 10^(i-1).

   exponent_plus_one_guess++;

   // We don't have any guarantees that 2^number_bits <= number.

   // TODO(floitsch): can we change the 'while' into an 'if'? We definitely see

   // number < (2^number_bits - 1), but I haven't encountered

   // number < (2^number_bits - 2) yet.

   while (number < kSmallPowersOfTen[exponent_plus_one_guess]) {

     exponent_plus_one_guess--;

   }

   *power = kSmallPowersOfTen[exponent_plus_one_guess];

   *exponent_plus_one = exponent_plus_one_guess;

 }

 // Generates the digits of input number w.

 // w is a floating-point number (DiyFp), consisting of a significand and an

 // exponent. Its exponent is bounded by kMinimalTargetExponent and

 // kMaximalTargetExponent.

 //       Hence -60 <= w.e() <= -32.

 //

 // Returns false if it fails, in which case the generated digits in the buffer

 // should not be used.

 // Preconditions:

 //  * low, w and high are correct up to 1 ulp (unit in the last place). That

 //    is, their error must be less than a unit of their last digits.

 //  * low.e() == w.e() == high.e()

 //  * low < w < high, and taking into account their error: low~ <= high~

 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent

 // Postconditions: returns false if procedure fails.

 //   otherwise:

 //     * buffer is not null-terminated, but len contains the number of digits.

 //     * buffer contains the shortest possible decimal digit-sequence

 //       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the

 //       correct values of low and high (without their error).

 //     * if more than one decimal representation gives the minimal number of

 //       decimal digits then the one closest to W (where W is the correct value

 //       of w) is chosen.

 // Remark: this procedure takes into account the imprecision of its input

 //   numbers. If the precision is not enough to guarantee all the postconditions

 //   then false is returned. This usually happens rarely (~0.5%).

 //

 // Say, for the sake of example, that

 //   w.e() == -48, and w.f() == 0x1234567890abcdef

 // w's value can be computed by w.f() * 2^w.e()

 // We can obtain w's integral digits by simply shifting w.f() by -w.e().

 //  -> w's integral part is 0x1234

 //  w's fractional part is therefore 0x567890abcdef.

 // Printing w's integral part is easy (simply print 0x1234 in decimal).

 // In order to print its fraction we repeatedly multiply the fraction by 10 and

 // get each digit. Example the first digit after the point would be computed by

 //   (0x567890abcdef * 10) >> 48. -> 3

 // The whole thing becomes slightly more complicated because we want to stop

 // once we have enough digits. That is, once the digits inside the buffer

 // represent 'w' we can stop. Everything inside the interval low - high

 // represents w. However we have to pay attention to low, high and w's

 // imprecision.

 static bool DigitGen(DiyFp low,

                      DiyFp w,

                      DiyFp high,

                      Vector<char> buffer,

                      int* length,

                      int* kappa) {

   ASSERT(low.e() == w.e() && w.e() == high.e());

   ASSERT(low.f() + 1 <= high.f() - 1);

   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);

   // low, w and high are imprecise, but by less than one ulp (unit in the last

   // place).

   // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that

   // the new numbers are outside of the interval we want the final

   // representation to lie in.

   // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield

   // numbers that are certain to lie in the interval. We will use this fact

   // later on.

   // We will now start by generating the digits within the uncertain

   // interval. Later we will weed out representations that lie outside the safe

   // interval and thus _might_ lie outside the correct interval.

   uint64_t unit = 1;

   DiyFp too_low = DiyFp(low.f() - unit, low.e());

   DiyFp too_high = DiyFp(high.f() + unit, high.e());

   // too_low and too_high are guaranteed to lie outside the interval we want the

   // generated number in.

   DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);

   // We now cut the input number into two parts: the integral digits and the

   // fractionals. We will not write any decimal separator though, but adapt

   // kappa instead.

   // Reminder: we are currently computing the digits (stored inside the buffer)

   // such that:   too_low < buffer * 10^kappa < too_high

   // We use too_high for the digit_generation and stop as soon as possible.

   // If we stop early we effectively round down.

   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());

   // Division by one is a shift.

   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());

   // Modulo by one is an and.

   uint64_t fractionals = too_high.f() & (one.f() - 1);

   uint32_t divisor;

   int divisor_exponent_plus_one;

   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),

                   &divisor, &divisor_exponent_plus_one);

   *kappa = divisor_exponent_plus_one;

   *length = 0;

   // Loop invariant: buffer = too_high / 10^kappa  (integer division)

   // The invariant holds for the first iteration: kappa has been initialized

   // with the divisor exponent + 1. And the divisor is the biggest power of ten

   // that is smaller than integrals.

   while (*kappa > 0) {

     int digit = integrals / divisor;

     buffer[*length] = '0' + digit;

     (*length)++;

     integrals %= divisor;

     (*kappa)--;

     // Note that kappa now equals the exponent of the divisor and that the

     // invariant thus holds again.

     uint64_t rest =

         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;

     // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())

     // Reminder: unsafe_interval.e() == one.e()

     if (rest < unsafe_interval.f()) {

       // Rounding down (by not emitting the remaining digits) yields a number

       // that lies within the unsafe interval.

       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),

                        unsafe_interval.f(), rest,

                        static_cast<uint64_t>(divisor) << -one.e(), unit);

     }

     divisor /= 10;

   }

   // The integrals have been generated. We are at the point of the decimal

   // separator. In the following loop we simply multiply the remaining digits by

   // 10 and divide by one. We just need to pay attention to multiply associated

   // data (like the interval or 'unit'), too.

   // Note that the multiplication by 10 does not overflow, because w.e >= -60

   // and thus one.e >= -60.

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

   ASSERT(fractionals < one.f());

   ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());

   while (true) {

     fractionals *= 10;

     unit *= 10;

     unsafe_interval.set_f(unsafe_interval.f() * 10);

     // Integer division by one.

     int digit = static_cast<int>(fractionals >> -one.e());

     buffer[*length] = '0' + digit;

     (*length)++;

     fractionals &= one.f() - 1;  // Modulo by one.

     (*kappa)--;

     if (fractionals < unsafe_interval.f()) {

       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,

                        unsafe_interval.f(), fractionals, one.f(), unit);

     }

   }

 }

 // Generates (at most) requested_digits digits of input number w.

 // w is a floating-point number (DiyFp), consisting of a significand and an

 // exponent. Its exponent is bounded by kMinimalTargetExponent and

 // kMaximalTargetExponent.

 //       Hence -60 <= w.e() <= -32.

 //

 // Returns false if it fails, in which case the generated digits in the buffer

 // should not be used.

 // Preconditions:

 //  * w is correct up to 1 ulp (unit in the last place). That

 //    is, its error must be strictly less than a unit of its last digit.

 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent

 //

 // Postconditions: returns false if procedure fails.

 //   otherwise:

 //     * buffer is not null-terminated, but length contains the number of

 //       digits.

 //     * the representation in buffer is the most precise representation of

 //       requested_digits digits.

 //     * buffer contains at most requested_digits digits of w. If there are less

 //       than requested_digits digits then some trailing '0's have been removed.

 //     * kappa is such that

 //            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.

 //

 // Remark: This procedure takes into account the imprecision of its input

 //   numbers. If the precision is not enough to guarantee all the postconditions

 //   then false is returned. This usually happens rarely, but the failure-rate

 //   increases with higher requested_digits.

 static bool DigitGenCounted(DiyFp w,

                             int requested_digits,

                             Vector<char> buffer,

                             int* length,

                             int* kappa) {

   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);

   ASSERT(kMinimalTargetExponent >= -60);

   ASSERT(kMaximalTargetExponent <= -32);

   // w is assumed to have an error less than 1 unit. Whenever w is scaled we

   // also scale its error.

   uint64_t w_error = 1;

   // We cut the input number into two parts: the integral digits and the

   // fractional digits. We don't emit any decimal separator, but adapt kappa

   // instead. Example: instead of writing "1.2" we put "12" into the buffer and

   // increase kappa by 1.

   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());

   // Division by one is a shift.

   uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());

   // Modulo by one is an and.

   uint64_t fractionals = w.f() & (one.f() - 1);

   uint32_t divisor;

   int divisor_exponent_plus_one;

   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),

                   &divisor, &divisor_exponent_plus_one);

   *kappa = divisor_exponent_plus_one;

   *length = 0;

   // Loop invariant: buffer = w / 10^kappa  (integer division)

   // The invariant holds for the first iteration: kappa has been initialized

   // with the divisor exponent + 1. And the divisor is the biggest power of ten

   // that is smaller than 'integrals'.

   while (*kappa > 0) {

     int digit = integrals / divisor;

     buffer[*length] = '0' + digit;

     (*length)++;

     requested_digits--;

     integrals %= divisor;

     (*kappa)--;

     // Note that kappa now equals the exponent of the divisor and that the

     // invariant thus holds again.

     if (requested_digits == 0) break;

     divisor /= 10;

   }

   if (requested_digits == 0) {

     uint64_t rest =

         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;

     return RoundWeedCounted(buffer, *length, rest,

                             static_cast<uint64_t>(divisor) << -one.e(), w_error,

                             kappa);

   }

   // The integrals have been generated. We are at the point of the decimal

   // separator. In the following loop we simply multiply the remaining digits by

   // 10 and divide by one. We just need to pay attention to multiply associated

   // data (the 'unit'), too.

   // Note that the multiplication by 10 does not overflow, because w.e >= -60

   // and thus one.e >= -60.

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

   ASSERT(fractionals < one.f());

   ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());

   while (requested_digits > 0 && fractionals > w_error) {

     fractionals *= 10;

     w_error *= 10;

     // Integer division by one.

     int digit = static_cast<int>(fractionals >> -one.e());

     buffer[*length] = '0' + digit;

     (*length)++;

     requested_digits--;

     fractionals &= one.f() - 1;  // Modulo by one.

     (*kappa)--;

   }

   if (requested_digits != 0) return false;

   return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,

                           kappa);

 }

 // Provides a decimal representation of v.

 // Returns true if it succeeds, otherwise the result cannot be trusted.

 // There will be *length digits inside the buffer (not null-terminated).

 // If the function returns true then

 //        v == (double) (buffer * 10^decimal_exponent).

 // The digits in the buffer are the shortest representation possible: no

 // 0.09999999999999999 instead of 0.1. The shorter representation will even be

 // chosen even if the longer one would be closer to v.

 // The last digit will be closest to the actual v. That is, even if several

 // digits might correctly yield 'v' when read again, the closest will be

 // computed.

 static bool Grisu3(double v,

                    FastDtoaMode mode,

                    Vector<char> buffer,

                    int* length,

                    int* decimal_exponent) {

   DiyFp w = Double(v).AsNormalizedDiyFp();

   // boundary_minus and boundary_plus are the boundaries between v and its

   // closest floating-point neighbors. Any number strictly between

   // boundary_minus and boundary_plus will round to v when convert to a double.

   // Grisu3 will never output representations that lie exactly on a boundary.

   DiyFp boundary_minus, boundary_plus;

   if (mode == FAST_DTOA_SHORTEST) {

     Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);

   } else {

     ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);

     float single_v = static_cast<float>(v);

     Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);

   }

   ASSERT(boundary_plus.e() == w.e());

   DiyFp ten_mk;  // Cached power of ten: 10^-k

   int mk;        // -k

   int ten_mk_minimal_binary_exponent =

      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);

   int ten_mk_maximal_binary_exponent =

      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);

   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

       ten_mk_minimal_binary_exponent,

       ten_mk_maximal_binary_exponent,

       &ten_mk, &mk);

   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +

           DiyFp::kSignificandSize) &&

          (kMaximalTargetExponent >= w.e() + ten_mk.e() +

           DiyFp::kSignificandSize));

   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a

   // 64 bit significand and ten_mk is thus only precise up to 64 bits.

   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated

   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now

   // off by a small amount.

   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.

   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then

   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e

   DiyFp scaled_w = DiyFp::Times(w, ten_mk);

   ASSERT(scaled_w.e() ==

          boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);

   // In theory it would be possible to avoid some recomputations by computing

   // the difference between w and boundary_minus/plus (a power of 2) and to

   // compute scaled_boundary_minus/plus by subtracting/adding from

   // scaled_w. However the code becomes much less readable and the speed

   // enhancements are not terriffic.

   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);

   DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);

   // DigitGen will generate the digits of scaled_w. Therefore we have

   // v == (double) (scaled_w * 10^-mk).

   // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an

   // integer than it will be updated. For instance if scaled_w == 1.23 then

   // the buffer will be filled with "123" und the decimal_exponent will be

   // decreased by 2.

   int kappa;

   bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,

                          buffer, length, &kappa);

   *decimal_exponent = -mk + kappa;

   return result;

 }

 // The "counted" version of grisu3 (see above) only generates requested_digits

 // number of digits. This version does not generate the shortest representation,

 // and with enough requested digits 0.1 will at some point print as 0.9999999...

 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and

 // therefore the rounding strategy for halfway cases is irrelevant.

 static bool Grisu3Counted(double v,

                           int requested_digits,

                           Vector<char> buffer,

                           int* length,

                           int* decimal_exponent) {

   DiyFp w = Double(v).AsNormalizedDiyFp();

   DiyFp ten_mk;  // Cached power of ten: 10^-k

   int mk;        // -k

   int ten_mk_minimal_binary_exponent =

      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);

   int ten_mk_maximal_binary_exponent =

      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);

   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

       ten_mk_minimal_binary_exponent,

       ten_mk_maximal_binary_exponent,

       &ten_mk, &mk);

   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +

           DiyFp::kSignificandSize) &&

          (kMaximalTargetExponent >= w.e() + ten_mk.e() +

           DiyFp::kSignificandSize));

   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a

   // 64 bit significand and ten_mk is thus only precise up to 64 bits.

   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated

   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now

   // off by a small amount.

   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.

   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then

   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e

   DiyFp scaled_w = DiyFp::Times(w, ten_mk);

   // We now have (double) (scaled_w * 10^-mk).

   // DigitGen will generate the first requested_digits digits of scaled_w and

   // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It

   // will not always be exactly the same since DigitGenCounted only produces a

   // limited number of digits.)

   int kappa;

   bool result = DigitGenCounted(scaled_w, requested_digits,

                                 buffer, length, &kappa);

   *decimal_exponent = -mk + kappa;

   return result;

 }

 bool FastDtoa(double v,

               FastDtoaMode mode,

               int requested_digits,

               Vector<char> buffer,

               int* length,

               int* decimal_point) {

   ASSERT(v > 0);

   ASSERT(!Double(v).IsSpecial());

   bool result = false;

   int decimal_exponent = 0;

   switch (mode) {

     case FAST_DTOA_SHORTEST:

     case FAST_DTOA_SHORTEST_SINGLE:

       result = Grisu3(v, mode, buffer, length, &decimal_exponent);

       break;

     case FAST_DTOA_PRECISION:

       result = Grisu3Counted(v, requested_digits,

                              buffer, length, &decimal_exponent);

       break;

     default:

       UNREACHABLE();

   }

   if (result) {

     *decimal_point = *length + decimal_exponent;

     buffer[*length] = '\0';

   }

   return result;

 }

 }  // namespace double_conversion