The Tor Browser: comparison mfbt/double-conversion/fast-dtoa.cc

--1:000000000000
+:a7b7b58b6a8a
+// 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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