The Tor Browser: mfbt/double-conversion/fast-dtoa.cc@97036ab72558 (annotated)

mfbt/double-conversion/fast-dtoa.cc@97036ab72558 (annotated)

mfbt/double-conversion/fast-dtoa.cc

Tue, 06 Jan 2015 21:39:09 +0100

author: Michael Schloh von Bennewitz <michael@schloh.com>
date: Tue, 06 Jan 2015 21:39:09 +0100
branch: TOR_BUG_9701
changeset 8: 97036ab72558
permissions: -rw-r--r--

Conditionally force memory storage according to privacy.thirdparty.isolate;
This solves Tor bug #9701, complying with disk avoidance documented in
https://www.torproject.org/projects/torbrowser/design/#disk-avoidance.

 // Copyright 2012 the V8 project authors. All rights reserved.
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are
 // met:
 //
 //     * Redistributions of source code must retain the above copyright
 //       notice, this list of conditions and the following disclaimer.
 //     * Redistributions in binary form must reproduce the above
 //       copyright notice, this list of conditions and the following
 //       disclaimer in the documentation and/or other materials provided
 //       with the distribution.
 //     * Neither the name of Google Inc. nor the names of its
 //       contributors may be used to endorse or promote products derived
 //       from this software without specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
 // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 #include "fast-dtoa.h"
 #include "cached-powers.h"
 #include "diy-fp.h"
 #include "ieee.h"
 namespace double_conversion {
 // The minimal and maximal target exponent define the range of w's binary
 // exponent, where 'w' is the result of multiplying the input by a cached power
 // of ten.
 //
 // A different range might be chosen on a different platform, to optimize digit
 // generation, but a smaller range requires more powers of ten to be cached.
 static const int kMinimalTargetExponent = -60;
 static const int kMaximalTargetExponent = -32;
 // Adjusts the last digit of the generated number, and screens out generated
 // solutions that may be inaccurate. A solution may be inaccurate if it is
 // outside the safe interval, or if we cannot prove that it is closer to the
 // input than a neighboring representation of the same length.
 //
 // Input: * buffer containing the digits of too_high / 10^kappa
 //        * the buffer's length
 //        * distance_too_high_w == (too_high - w).f() * unit
 //        * unsafe_interval == (too_high - too_low).f() * unit
 //        * rest = (too_high - buffer * 10^kappa).f() * unit
 //        * ten_kappa = 10^kappa * unit
 //        * unit = the common multiplier
 // Output: returns true if the buffer is guaranteed to contain the closest
 //    representable number to the input.
 //  Modifies the generated digits in the buffer to approach (round towards) w.
 static bool RoundWeed(Vector<char> buffer,
                       int length,
                       uint64_t distance_too_high_w,
                       uint64_t unsafe_interval,
                       uint64_t rest,
                       uint64_t ten_kappa,
                       uint64_t unit) {
   uint64_t small_distance = distance_too_high_w - unit;
   uint64_t big_distance = distance_too_high_w + unit;
   // Let w_low  = too_high - big_distance, and
   //     w_high = too_high - small_distance.
   // Note: w_low < w < w_high
   //
   // The real w (* unit) must lie somewhere inside the interval
   // ]w_low; w_high[ (often written as "(w_low; w_high)")
   // Basically the buffer currently contains a number in the unsafe interval
   // ]too_low; too_high[ with too_low < w < too_high
   //
   //  too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   //                     ^v 1 unit            ^      ^                 ^      ^
   //  boundary_high ---------------------     .      .                 .      .
   //                     ^v 1 unit            .      .                 .      .
   //   - - - - - - - - - - - - - - - - - - -  +  - - + - - - - - -     .      .
   //                                          .      .         ^       .      .
   //                                          .  big_distance  .       .      .
   //                                          .      .         .       .    rest
   //                              small_distance     .         .       .      .
   //                                          v      .         .       .      .
   //  w_high - - - - - - - - - - - - - - - - - -     .         .       .      .
   //                     ^v 1 unit                   .         .       .      .
   //  w ----------------------------------------     .         .       .      .
   //                     ^v 1 unit                   v         .       .      .
   //  w_low  - - - - - - - - - - - - - - - - - - - - -         .       .      .
   //                                                           .       .      v
   //  buffer --------------------------------------------------+-------+--------
   //                                                           .       .
   //                                                  safe_interval    .
   //                                                           v       .
   //   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -     .
   //                     ^v 1 unit                                     .
   //  boundary_low -------------------------                     unsafe_interval
   //                     ^v 1 unit                                     v
   //  too_low  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   //
   //
   // Note that the value of buffer could lie anywhere inside the range too_low
   // to too_high.
   //
   // boundary_low, boundary_high and w are approximations of the real boundaries
   // and v (the input number). They are guaranteed to be precise up to one unit.
   // In fact the error is guaranteed to be strictly less than one unit.
   //
   // Anything that lies outside the unsafe interval is guaranteed not to round
   // to v when read again.
   // Anything that lies inside the safe interval is guaranteed to round to v
   // when read again.
   // If the number inside the buffer lies inside the unsafe interval but not
   // inside the safe interval then we simply do not know and bail out (returning
   // false).
   //
   // Similarly we have to take into account the imprecision of 'w' when finding
   // the closest representation of 'w'. If we have two potential
   // representations, and one is closer to both w_low and w_high, then we know
   // it is closer to the actual value v.
   //
   // By generating the digits of too_high we got the largest (closest to
   // too_high) buffer that is still in the unsafe interval. In the case where
   // w_high < buffer < too_high we try to decrement the buffer.
   // This way the buffer approaches (rounds towards) w.
   // There are 3 conditions that stop the decrementation process:
   //   1) the buffer is already below w_high
   //   2) decrementing the buffer would make it leave the unsafe interval
   //   3) decrementing the buffer would yield a number below w_high and farther
   //      away than the current number. In other words:
   //              (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
   // Instead of using the buffer directly we use its distance to too_high.
   // Conceptually rest ~= too_high - buffer
   // We need to do the following tests in this order to avoid over- and
   // underflows.
   ASSERT(rest <= unsafe_interval);
   while (rest < small_distance &&  // Negated condition 1
          unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
          (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
           small_distance - rest >= rest + ten_kappa - small_distance)) {
     buffer[length - 1]--;
     rest += ten_kappa;
   }
   // We have approached w+ as much as possible. We now test if approaching w-
   // would require changing the buffer. If yes, then we have two possible
   // representations close to w, but we cannot decide which one is closer.
   if (rest < big_distance &&
       unsafe_interval - rest >= ten_kappa &&
       (rest + ten_kappa < big_distance ||
        big_distance - rest > rest + ten_kappa - big_distance)) {
     return false;
   }
   // Weeding test.
   //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
   //   Since too_low = too_high - unsafe_interval this is equivalent to
   //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
   //   Conceptually we have: rest ~= too_high - buffer
   return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
 }
 // Rounds the buffer upwards if the result is closer to v by possibly adding
 // 1 to the buffer. If the precision of the calculation is not sufficient to
 // round correctly, return false.
 // The rounding might shift the whole buffer in which case the kappa is
 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
 //
 // If 2*rest > ten_kappa then the buffer needs to be round up.
 // rest can have an error of +/- 1 unit. This function accounts for the
 // imprecision and returns false, if the rounding direction cannot be
 // unambiguously determined.
 //
 // Precondition: rest < ten_kappa.
 static bool RoundWeedCounted(Vector<char> buffer,
                              int length,
                              uint64_t rest,
                              uint64_t ten_kappa,
                              uint64_t unit,
                              int* kappa) {
   ASSERT(rest < ten_kappa);
   // The following tests are done in a specific order to avoid overflows. They
   // will work correctly with any uint64 values of rest < ten_kappa and unit.
   //
   // If the unit is too big, then we don't know which way to round. For example
   // a unit of 50 means that the real number lies within rest +/- 50. If
   // 10^kappa == 40 then there is no way to tell which way to round.
   if (unit >= ten_kappa) return false;
   // Even if unit is just half the size of 10^kappa we are already completely
   // lost. (And after the previous test we know that the expression will not
   // over/underflow.)
   if (ten_kappa - unit <= unit) return false;
   // If 2 * (rest + unit) <= 10^kappa we can safely round down.
   if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
     return true;
   }
   // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
   if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
     // Increment the last digit recursively until we find a non '9' digit.
     buffer[length - 1]++;
     for (int i = length - 1; i > 0; --i) {
       if (buffer[i] != '0' + 10) break;
       buffer[i] = '0';
       buffer[i - 1]++;
     }
     // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
     // exception of the first digit all digits are now '0'. Simply switch the
     // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
     // the power (the kappa) is increased.
     if (buffer[0] == '0' + 10) {
       buffer[0] = '1';
       (*kappa) += 1;
     }
     return true;
   }
   return false;
 }
 // Returns the biggest power of ten that is less than or equal to the given
 // number. We furthermore receive the maximum number of bits 'number' has.
 //
 // Returns power == 10^(exponent_plus_one-1) such that
 //    power <= number < power * 10.
 // If number_bits == 0 then 0^(0-1) is returned.
 // The number of bits must be <= 32.
 // Precondition: number < (1 << (number_bits + 1)).
 // Inspired by the method for finding an integer log base 10 from here:
 // http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
 static unsigned int const kSmallPowersOfTen[] =
     {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
      1000000000};
 static void BiggestPowerTen(uint32_t number,
                             int number_bits,
                             uint32_t* power,
                             int* exponent_plus_one) {
   ASSERT(number < (1u << (number_bits + 1)));
   // 1233/4096 is approximately 1/lg(10).
   int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);
   // We increment to skip over the first entry in the kPowersOf10 table.
   // Note: kPowersOf10[i] == 10^(i-1).
   exponent_plus_one_guess++;
   // We don't have any guarantees that 2^number_bits <= number.
   // TODO(floitsch): can we change the 'while' into an 'if'? We definitely see
   // number < (2^number_bits - 1), but I haven't encountered
   // number < (2^number_bits - 2) yet.
   while (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
     exponent_plus_one_guess--;
   }
   *power = kSmallPowersOfTen[exponent_plus_one_guess];
   *exponent_plus_one = exponent_plus_one_guess;
 }
 // Generates the digits of input number w.
 // w is a floating-point number (DiyFp), consisting of a significand and an
 // exponent. Its exponent is bounded by kMinimalTargetExponent and
 // kMaximalTargetExponent.
 //       Hence -60 <= w.e() <= -32.
 //
 // Returns false if it fails, in which case the generated digits in the buffer
 // should not be used.
 // Preconditions:
 //  * low, w and high are correct up to 1 ulp (unit in the last place). That
 //    is, their error must be less than a unit of their last digits.
 //  * low.e() == w.e() == high.e()
 //  * low < w < high, and taking into account their error: low~ <= high~
 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
 // Postconditions: returns false if procedure fails.
 //   otherwise:
 //     * buffer is not null-terminated, but len contains the number of digits.
 //     * buffer contains the shortest possible decimal digit-sequence
 //       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
 //       correct values of low and high (without their error).
 //     * if more than one decimal representation gives the minimal number of
 //       decimal digits then the one closest to W (where W is the correct value
 //       of w) is chosen.
 // Remark: this procedure takes into account the imprecision of its input
 //   numbers. If the precision is not enough to guarantee all the postconditions
 //   then false is returned. This usually happens rarely (~0.5%).
 //
 // Say, for the sake of example, that
 //   w.e() == -48, and w.f() == 0x1234567890abcdef
 // w's value can be computed by w.f() * 2^w.e()
 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
 //  -> w's integral part is 0x1234
 //  w's fractional part is therefore 0x567890abcdef.
 // Printing w's integral part is easy (simply print 0x1234 in decimal).
 // In order to print its fraction we repeatedly multiply the fraction by 10 and
 // get each digit. Example the first digit after the point would be computed by
 //   (0x567890abcdef * 10) >> 48. -> 3
 // The whole thing becomes slightly more complicated because we want to stop
 // once we have enough digits. That is, once the digits inside the buffer
 // represent 'w' we can stop. Everything inside the interval low - high
 // represents w. However we have to pay attention to low, high and w's
 // imprecision.
 static bool DigitGen(DiyFp low,
                      DiyFp w,
                      DiyFp high,
                      Vector<char> buffer,
                      int* length,
                      int* kappa) {
   ASSERT(low.e() == w.e() && w.e() == high.e());
   ASSERT(low.f() + 1 <= high.f() - 1);
   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
   // low, w and high are imprecise, but by less than one ulp (unit in the last
   // place).
   // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
   // the new numbers are outside of the interval we want the final
   // representation to lie in.
   // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
   // numbers that are certain to lie in the interval. We will use this fact
   // later on.
   // We will now start by generating the digits within the uncertain
   // interval. Later we will weed out representations that lie outside the safe
   // interval and thus _might_ lie outside the correct interval.
   uint64_t unit = 1;
   DiyFp too_low = DiyFp(low.f() - unit, low.e());
   DiyFp too_high = DiyFp(high.f() + unit, high.e());
   // too_low and too_high are guaranteed to lie outside the interval we want the
   // generated number in.
   DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
   // We now cut the input number into two parts: the integral digits and the
   // fractionals. We will not write any decimal separator though, but adapt
   // kappa instead.
   // Reminder: we are currently computing the digits (stored inside the buffer)
   // such that:   too_low < buffer * 10^kappa < too_high
   // We use too_high for the digit_generation and stop as soon as possible.
   // If we stop early we effectively round down.
   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
   // Division by one is a shift.
   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
   // Modulo by one is an and.
   uint64_t fractionals = too_high.f() & (one.f() - 1);
   uint32_t divisor;
   int divisor_exponent_plus_one;
   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                   &divisor, &divisor_exponent_plus_one);
   *kappa = divisor_exponent_plus_one;
   *length = 0;
   // Loop invariant: buffer = too_high / 10^kappa  (integer division)
   // The invariant holds for the first iteration: kappa has been initialized
   // with the divisor exponent + 1. And the divisor is the biggest power of ten
   // that is smaller than integrals.
   while (*kappa > 0) {
     int digit = integrals / divisor;
     buffer[*length] = '0' + digit;
     (*length)++;
     integrals %= divisor;
     (*kappa)--;
     // Note that kappa now equals the exponent of the divisor and that the
     // invariant thus holds again.
     uint64_t rest =
         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
     // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
     // Reminder: unsafe_interval.e() == one.e()
     if (rest < unsafe_interval.f()) {
       // Rounding down (by not emitting the remaining digits) yields a number
       // that lies within the unsafe interval.
       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
                        unsafe_interval.f(), rest,
                        static_cast<uint64_t>(divisor) << -one.e(), unit);
     }
     divisor /= 10;
   }
   // The integrals have been generated. We are at the point of the decimal
   // separator. In the following loop we simply multiply the remaining digits by
   // 10 and divide by one. We just need to pay attention to multiply associated
   // data (like the interval or 'unit'), too.
   // Note that the multiplication by 10 does not overflow, because w.e >= -60
   // and thus one.e >= -60.
   ASSERT(one.e() >= -60);
   ASSERT(fractionals < one.f());
   ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
   while (true) {
     fractionals *= 10;
     unit *= 10;
     unsafe_interval.set_f(unsafe_interval.f() * 10);
     // Integer division by one.
     int digit = static_cast<int>(fractionals >> -one.e());
     buffer[*length] = '0' + digit;
     (*length)++;
     fractionals &= one.f() - 1;  // Modulo by one.
     (*kappa)--;
     if (fractionals < unsafe_interval.f()) {
       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
                        unsafe_interval.f(), fractionals, one.f(), unit);
     }
   }
 }
 // Generates (at most) requested_digits digits of input number w.
 // w is a floating-point number (DiyFp), consisting of a significand and an
 // exponent. Its exponent is bounded by kMinimalTargetExponent and
 // kMaximalTargetExponent.
 //       Hence -60 <= w.e() <= -32.
 //
 // Returns false if it fails, in which case the generated digits in the buffer
 // should not be used.
 // Preconditions:
 //  * w is correct up to 1 ulp (unit in the last place). That
 //    is, its error must be strictly less than a unit of its last digit.
 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
 //
 // Postconditions: returns false if procedure fails.
 //   otherwise:
 //     * buffer is not null-terminated, but length contains the number of
 //       digits.
 //     * the representation in buffer is the most precise representation of
 //       requested_digits digits.
 //     * buffer contains at most requested_digits digits of w. If there are less
 //       than requested_digits digits then some trailing '0's have been removed.
 //     * kappa is such that
 //            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
 //
 // Remark: This procedure takes into account the imprecision of its input
 //   numbers. If the precision is not enough to guarantee all the postconditions
 //   then false is returned. This usually happens rarely, but the failure-rate
 //   increases with higher requested_digits.
 static bool DigitGenCounted(DiyFp w,
                             int requested_digits,
                             Vector<char> buffer,
                             int* length,
                             int* kappa) {
   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
   ASSERT(kMinimalTargetExponent >= -60);
   ASSERT(kMaximalTargetExponent <= -32);
   // w is assumed to have an error less than 1 unit. Whenever w is scaled we
   // also scale its error.
   uint64_t w_error = 1;
   // We cut the input number into two parts: the integral digits and the
   // fractional digits. We don't emit any decimal separator, but adapt kappa
   // instead. Example: instead of writing "1.2" we put "12" into the buffer and
   // increase kappa by 1.
   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
   // Division by one is a shift.
   uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
   // Modulo by one is an and.
   uint64_t fractionals = w.f() & (one.f() - 1);
   uint32_t divisor;
   int divisor_exponent_plus_one;
   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                   &divisor, &divisor_exponent_plus_one);
   *kappa = divisor_exponent_plus_one;
   *length = 0;
   // Loop invariant: buffer = w / 10^kappa  (integer division)
   // The invariant holds for the first iteration: kappa has been initialized
   // with the divisor exponent + 1. And the divisor is the biggest power of ten
   // that is smaller than 'integrals'.
   while (*kappa > 0) {
     int digit = integrals / divisor;
     buffer[*length] = '0' + digit;
     (*length)++;
     requested_digits--;
     integrals %= divisor;
     (*kappa)--;
     // Note that kappa now equals the exponent of the divisor and that the
     // invariant thus holds again.
     if (requested_digits == 0) break;
     divisor /= 10;
   }
   if (requested_digits == 0) {
     uint64_t rest =
         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
     return RoundWeedCounted(buffer, *length, rest,
                             static_cast<uint64_t>(divisor) << -one.e(), w_error,
                             kappa);
   }
   // The integrals have been generated. We are at the point of the decimal
   // separator. In the following loop we simply multiply the remaining digits by
   // 10 and divide by one. We just need to pay attention to multiply associated
   // data (the 'unit'), too.
   // Note that the multiplication by 10 does not overflow, because w.e >= -60
   // and thus one.e >= -60.
   ASSERT(one.e() >= -60);
   ASSERT(fractionals < one.f());
   ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
   while (requested_digits > 0 && fractionals > w_error) {
     fractionals *= 10;
     w_error *= 10;
     // Integer division by one.
     int digit = static_cast<int>(fractionals >> -one.e());
     buffer[*length] = '0' + digit;
     (*length)++;
     requested_digits--;
     fractionals &= one.f() - 1;  // Modulo by one.
     (*kappa)--;
   }
   if (requested_digits != 0) return false;
   return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
                           kappa);
 }
 // Provides a decimal representation of v.
 // Returns true if it succeeds, otherwise the result cannot be trusted.
 // There will be *length digits inside the buffer (not null-terminated).
 // If the function returns true then
 //        v == (double) (buffer * 10^decimal_exponent).
 // The digits in the buffer are the shortest representation possible: no
 // 0.09999999999999999 instead of 0.1. The shorter representation will even be
 // chosen even if the longer one would be closer to v.
 // The last digit will be closest to the actual v. That is, even if several
 // digits might correctly yield 'v' when read again, the closest will be
 // computed.
 static bool Grisu3(double v,
                    FastDtoaMode mode,
                    Vector<char> buffer,
                    int* length,
                    int* decimal_exponent) {
   DiyFp w = Double(v).AsNormalizedDiyFp();
   // boundary_minus and boundary_plus are the boundaries between v and its
   // closest floating-point neighbors. Any number strictly between
   // boundary_minus and boundary_plus will round to v when convert to a double.
   // Grisu3 will never output representations that lie exactly on a boundary.
   DiyFp boundary_minus, boundary_plus;
   if (mode == FAST_DTOA_SHORTEST) {
     Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
   } else {
     ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
     float single_v = static_cast<float>(v);
     Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
   }
   ASSERT(boundary_plus.e() == w.e());
   DiyFp ten_mk;  // Cached power of ten: 10^-k
   int mk;        // -k
   int ten_mk_minimal_binary_exponent =
      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
   int ten_mk_maximal_binary_exponent =
      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
       ten_mk_minimal_binary_exponent,
       ten_mk_maximal_binary_exponent,
       &ten_mk, &mk);
   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
           DiyFp::kSignificandSize) &&
          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
           DiyFp::kSignificandSize));
   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
   // off by a small amount.
   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
   ASSERT(scaled_w.e() ==
          boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
   // In theory it would be possible to avoid some recomputations by computing
   // the difference between w and boundary_minus/plus (a power of 2) and to
   // compute scaled_boundary_minus/plus by subtracting/adding from
   // scaled_w. However the code becomes much less readable and the speed
   // enhancements are not terriffic.
   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
   DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);
   // DigitGen will generate the digits of scaled_w. Therefore we have
   // v == (double) (scaled_w * 10^-mk).
   // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
   // integer than it will be updated. For instance if scaled_w == 1.23 then
   // the buffer will be filled with "123" und the decimal_exponent will be
   // decreased by 2.
   int kappa;
   bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
                          buffer, length, &kappa);
   *decimal_exponent = -mk + kappa;
   return result;
 }
 // The "counted" version of grisu3 (see above) only generates requested_digits
 // number of digits. This version does not generate the shortest representation,
 // and with enough requested digits 0.1 will at some point print as 0.9999999...
 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
 // therefore the rounding strategy for halfway cases is irrelevant.
 static bool Grisu3Counted(double v,
                           int requested_digits,
                           Vector<char> buffer,
                           int* length,
                           int* decimal_exponent) {
   DiyFp w = Double(v).AsNormalizedDiyFp();
   DiyFp ten_mk;  // Cached power of ten: 10^-k
   int mk;        // -k
   int ten_mk_minimal_binary_exponent =
      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
   int ten_mk_maximal_binary_exponent =
      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
       ten_mk_minimal_binary_exponent,
       ten_mk_maximal_binary_exponent,
       &ten_mk, &mk);
   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
           DiyFp::kSignificandSize) &&
          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
           DiyFp::kSignificandSize));
   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
   // off by a small amount.
   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
   // We now have (double) (scaled_w * 10^-mk).
   // DigitGen will generate the first requested_digits digits of scaled_w and
   // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
   // will not always be exactly the same since DigitGenCounted only produces a
   // limited number of digits.)
   int kappa;
   bool result = DigitGenCounted(scaled_w, requested_digits,
                                 buffer, length, &kappa);
   *decimal_exponent = -mk + kappa;
   return result;
 }
 bool FastDtoa(double v,
               FastDtoaMode mode,
               int requested_digits,
               Vector<char> buffer,
               int* length,
               int* decimal_point) {
   ASSERT(v > 0);
   ASSERT(!Double(v).IsSpecial());
   bool result = false;
   int decimal_exponent = 0;
   switch (mode) {
     case FAST_DTOA_SHORTEST:
     case FAST_DTOA_SHORTEST_SINGLE:
       result = Grisu3(v, mode, buffer, length, &decimal_exponent);
       break;
     case FAST_DTOA_PRECISION:
       result = Grisu3Counted(v, requested_digits,
                              buffer, length, &decimal_exponent);
       break;
     default:
       UNREACHABLE();
   }
   if (result) {
     *decimal_point = *length + decimal_exponent;
     buffer[*length] = '\0';
   }
   return result;
 }
 }  // namespace double_conversion

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mfbt/double-conversion/fast-dtoa.cc