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// Copyright 2012 the V8 project authors. All rights reserved.
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above
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// copyright notice, this list of conditions and the following
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// disclaimer in the documentation and/or other materials provided
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// with the distribution.
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// * Neither the name of Google Inc. nor the names of its
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// contributors may be used to endorse or promote products derived
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// from this software without specific prior written permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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#include "fast-dtoa.h"
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#include "cached-powers.h"
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namespace double_conversion {
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// The minimal and maximal target exponent define the range of w's binary
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// exponent, where 'w' is the result of multiplying the input by a cached power
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// A different range might be chosen on a different platform, to optimize digit
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// generation, but a smaller range requires more powers of ten to be cached.
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static const int kMinimalTargetExponent = -60;
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static const int kMaximalTargetExponent = -32;
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// Adjusts the last digit of the generated number, and screens out generated
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// solutions that may be inaccurate. A solution may be inaccurate if it is
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// outside the safe interval, or if we cannot prove that it is closer to the
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// input than a neighboring representation of the same length.
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// Input: * buffer containing the digits of too_high / 10^kappa
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// * the buffer's length
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// * distance_too_high_w == (too_high - w).f() * unit
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// * unsafe_interval == (too_high - too_low).f() * unit
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// * rest = (too_high - buffer * 10^kappa).f() * unit
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// * ten_kappa = 10^kappa * unit
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// * unit = the common multiplier
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// Output: returns true if the buffer is guaranteed to contain the closest
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// representable number to the input.
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// Modifies the generated digits in the buffer to approach (round towards) w.
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static bool RoundWeed(Vector<char> buffer,
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uint64_t distance_too_high_w,
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uint64_t unsafe_interval,
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uint64_t small_distance = distance_too_high_w - unit;
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uint64_t big_distance = distance_too_high_w + unit;
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// Let w_low = too_high - big_distance, and
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// w_high = too_high - small_distance.
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// Note: w_low < w < w_high
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// The real w (* unit) must lie somewhere inside the interval
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// ]w_low; w_high[ (often written as "(w_low; w_high)")
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// Basically the buffer currently contains a number in the unsafe interval
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// ]too_low; too_high[ with too_low < w < too_high
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// too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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// boundary_high --------------------- . . . .
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// - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
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// . big_distance . . .
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// small_distance . . . .
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// w_high - - - - - - - - - - - - - - - - - - . . . .
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// w ---------------------------------------- . . . .
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// w_low - - - - - - - - - - - - - - - - - - - - - . . .
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// buffer --------------------------------------------------+-------+--------
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// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
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// boundary_low ------------------------- unsafe_interval
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// too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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// Note that the value of buffer could lie anywhere inside the range too_low
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// boundary_low, boundary_high and w are approximations of the real boundaries
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// and v (the input number). They are guaranteed to be precise up to one unit.
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// In fact the error is guaranteed to be strictly less than one unit.
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// Anything that lies outside the unsafe interval is guaranteed not to round
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// to v when read again.
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// Anything that lies inside the safe interval is guaranteed to round to v
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// If the number inside the buffer lies inside the unsafe interval but not
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// inside the safe interval then we simply do not know and bail out (returning
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// Similarly we have to take into account the imprecision of 'w' when finding
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// the closest representation of 'w'. If we have two potential
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// representations, and one is closer to both w_low and w_high, then we know
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// it is closer to the actual value v.
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// By generating the digits of too_high we got the largest (closest to
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// too_high) buffer that is still in the unsafe interval. In the case where
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// w_high < buffer < too_high we try to decrement the buffer.
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// This way the buffer approaches (rounds towards) w.
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// There are 3 conditions that stop the decrementation process:
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// 1) the buffer is already below w_high
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// 2) decrementing the buffer would make it leave the unsafe interval
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// 3) decrementing the buffer would yield a number below w_high and farther
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// away than the current number. In other words:
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// (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
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// Instead of using the buffer directly we use its distance to too_high.
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// Conceptually rest ~= too_high - buffer
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// We need to do the following tests in this order to avoid over- and
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ASSERT(rest <= unsafe_interval);
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while (rest < small_distance && // Negated condition 1
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unsafe_interval - rest >= ten_kappa && // Negated condition 2
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(rest + ten_kappa < small_distance || // buffer{-1} > w_high
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small_distance - rest >= rest + ten_kappa - small_distance)) {
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buffer[length - 1]--;
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// We have approached w+ as much as possible. We now test if approaching w-
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// would require changing the buffer. If yes, then we have two possible
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// representations close to w, but we cannot decide which one is closer.
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if (rest < big_distance &&
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unsafe_interval - rest >= ten_kappa &&
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(rest + ten_kappa < big_distance ||
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big_distance - rest > rest + ten_kappa - big_distance)) {
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// The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
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// Since too_low = too_high - unsafe_interval this is equivalent to
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// [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
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// Conceptually we have: rest ~= too_high - buffer
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return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
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// Rounds the buffer upwards if the result is closer to v by possibly adding
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// 1 to the buffer. If the precision of the calculation is not sufficient to
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// round correctly, return false.
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// The rounding might shift the whole buffer in which case the kappa is
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// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
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// If 2*rest > ten_kappa then the buffer needs to be round up.
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// rest can have an error of +/- 1 unit. This function accounts for the
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// imprecision and returns false, if the rounding direction cannot be
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// unambiguously determined.
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// Precondition: rest < ten_kappa.
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static bool RoundWeedCounted(Vector<char> buffer,
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ASSERT(rest < ten_kappa);
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// The following tests are done in a specific order to avoid overflows. They
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// will work correctly with any uint64 values of rest < ten_kappa and unit.
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// If the unit is too big, then we don't know which way to round. For example
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// a unit of 50 means that the real number lies within rest +/- 50. If
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// 10^kappa == 40 then there is no way to tell which way to round.
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if (unit >= ten_kappa) return false;
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// Even if unit is just half the size of 10^kappa we are already completely
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// lost. (And after the previous test we know that the expression will not
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if (ten_kappa - unit <= unit) return false;
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// If 2 * (rest + unit) <= 10^kappa we can safely round down.
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if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
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// If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
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if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
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// Increment the last digit recursively until we find a non '9' digit.
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buffer[length - 1]++;
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for (int i = length - 1; i > 0; --i) {
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if (buffer[i] != '0' + 10) break;
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// If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
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// exception of the first digit all digits are now '0'. Simply switch the
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// first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
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// the power (the kappa) is increased.
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if (buffer[0] == '0' + 10) {
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// Returns the biggest power of ten that is less than or equal to the given
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// number. We furthermore receive the maximum number of bits 'number' has.
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// Returns power == 10^(exponent_plus_one-1) such that
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// power <= number < power * 10.
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// If number_bits == 0 then 0^(0-1) is returned.
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// The number of bits must be <= 32.
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// Precondition: number < (1 << (number_bits + 1)).
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// Inspired by the method for finding an integer log base 10 from here:
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// http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
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static unsigned int const kSmallPowersOfTen[] =
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{0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
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static void BiggestPowerTen(uint32_t number,
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int* exponent_plus_one) {
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ASSERT(number < (1u << (number_bits + 1)));
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// 1233/4096 is approximately 1/lg(10).
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int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);
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// We increment to skip over the first entry in the kPowersOf10 table.
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// Note: kPowersOf10[i] == 10^(i-1).
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exponent_plus_one_guess++;
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// We don't have any guarantees that 2^number_bits <= number.
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// TODO(floitsch): can we change the 'while' into an 'if'? We definitely see
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// number < (2^number_bits - 1), but I haven't encountered
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// number < (2^number_bits - 2) yet.
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while (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
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exponent_plus_one_guess--;
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*power = kSmallPowersOfTen[exponent_plus_one_guess];
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*exponent_plus_one = exponent_plus_one_guess;
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// Generates the digits of input number w.
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// w is a floating-point number (DiyFp), consisting of a significand and an
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// exponent. Its exponent is bounded by kMinimalTargetExponent and
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// kMaximalTargetExponent.
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// Hence -60 <= w.e() <= -32.
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// Returns false if it fails, in which case the generated digits in the buffer
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// should not be used.
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// * low, w and high are correct up to 1 ulp (unit in the last place). That
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// is, their error must be less than a unit of their last digits.
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// * low.e() == w.e() == high.e()
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// * low < w < high, and taking into account their error: low~ <= high~
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// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
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// Postconditions: returns false if procedure fails.
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// * buffer is not null-terminated, but len contains the number of digits.
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// * buffer contains the shortest possible decimal digit-sequence
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// such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
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// correct values of low and high (without their error).
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// * if more than one decimal representation gives the minimal number of
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// decimal digits then the one closest to W (where W is the correct value
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// Remark: this procedure takes into account the imprecision of its input
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// numbers. If the precision is not enough to guarantee all the postconditions
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// then false is returned. This usually happens rarely (~0.5%).
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// Say, for the sake of example, that
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// w.e() == -48, and w.f() == 0x1234567890abcdef
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// w's value can be computed by w.f() * 2^w.e()
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// We can obtain w's integral digits by simply shifting w.f() by -w.e().
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// -> w's integral part is 0x1234
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// w's fractional part is therefore 0x567890abcdef.
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// Printing w's integral part is easy (simply print 0x1234 in decimal).
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// In order to print its fraction we repeatedly multiply the fraction by 10 and
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// get each digit. Example the first digit after the point would be computed by
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// (0x567890abcdef * 10) >> 48. -> 3
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// The whole thing becomes slightly more complicated because we want to stop
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// once we have enough digits. That is, once the digits inside the buffer
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// represent 'w' we can stop. Everything inside the interval low - high
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// represents w. However we have to pay attention to low, high and w's
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static bool DigitGen(DiyFp low,
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ASSERT(low.e() == w.e() && w.e() == high.e());
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ASSERT(low.f() + 1 <= high.f() - 1);
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ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
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// low, w and high are imprecise, but by less than one ulp (unit in the last
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// If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
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// the new numbers are outside of the interval we want the final
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// representation to lie in.
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// Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
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// numbers that are certain to lie in the interval. We will use this fact
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// We will now start by generating the digits within the uncertain
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// interval. Later we will weed out representations that lie outside the safe
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// interval and thus _might_ lie outside the correct interval.
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DiyFp too_low = DiyFp(low.f() - unit, low.e());
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DiyFp too_high = DiyFp(high.f() + unit, high.e());
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// too_low and too_high are guaranteed to lie outside the interval we want the
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// generated number in.
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DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
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// We now cut the input number into two parts: the integral digits and the
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// fractionals. We will not write any decimal separator though, but adapt
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// Reminder: we are currently computing the digits (stored inside the buffer)
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// such that: too_low < buffer * 10^kappa < too_high
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// We use too_high for the digit_generation and stop as soon as possible.
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// If we stop early we effectively round down.
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DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
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// Division by one is a shift.
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uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
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// Modulo by one is an and.
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uint64_t fractionals = too_high.f() & (one.f() - 1);
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int divisor_exponent_plus_one;
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BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
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&divisor, &divisor_exponent_plus_one);
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*kappa = divisor_exponent_plus_one;
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// Loop invariant: buffer = too_high / 10^kappa (integer division)
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// The invariant holds for the first iteration: kappa has been initialized
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// with the divisor exponent + 1. And the divisor is the biggest power of ten
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// that is smaller than integrals.
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int digit = integrals / divisor;
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buffer[*length] = '0' + digit;
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integrals %= divisor;
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// Note that kappa now equals the exponent of the divisor and that the
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// invariant thus holds again.
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(static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
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// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
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// Reminder: unsafe_interval.e() == one.e()
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if (rest < unsafe_interval.f()) {
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// Rounding down (by not emitting the remaining digits) yields a number
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// that lies within the unsafe interval.
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return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
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unsafe_interval.f(), rest,
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static_cast<uint64_t>(divisor) << -one.e(), unit);
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// The integrals have been generated. We are at the point of the decimal
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// separator. In the following loop we simply multiply the remaining digits by
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// 10 and divide by one. We just need to pay attention to multiply associated
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// data (like the interval or 'unit'), too.
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// Note that the multiplication by 10 does not overflow, because w.e >= -60
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// and thus one.e >= -60.
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ASSERT(one.e() >= -60);
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ASSERT(fractionals < one.f());
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ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
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unsafe_interval.set_f(unsafe_interval.f() * 10);
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// Integer division by one.
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int digit = static_cast<int>(fractionals >> -one.e());
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buffer[*length] = '0' + digit;
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fractionals &= one.f() - 1; // Modulo by one.
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if (fractionals < unsafe_interval.f()) {
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return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
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unsafe_interval.f(), fractionals, one.f(), unit);
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// Generates (at most) requested_digits digits of input number w.
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// w is a floating-point number (DiyFp), consisting of a significand and an
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// exponent. Its exponent is bounded by kMinimalTargetExponent and
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// kMaximalTargetExponent.
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// Hence -60 <= w.e() <= -32.
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// Returns false if it fails, in which case the generated digits in the buffer
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// should not be used.
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// * w is correct up to 1 ulp (unit in the last place). That
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// is, its error must be strictly less than a unit of its last digit.
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// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
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// Postconditions: returns false if procedure fails.
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// * buffer is not null-terminated, but length contains the number of
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// * the representation in buffer is the most precise representation of
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// requested_digits digits.
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// * buffer contains at most requested_digits digits of w. If there are less
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// than requested_digits digits then some trailing '0's have been removed.
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// * kappa is such that
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// w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
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// Remark: This procedure takes into account the imprecision of its input
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// numbers. If the precision is not enough to guarantee all the postconditions
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// then false is returned. This usually happens rarely, but the failure-rate
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// increases with higher requested_digits.
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static bool DigitGenCounted(DiyFp w,
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int requested_digits,
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ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
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ASSERT(kMinimalTargetExponent >= -60);
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ASSERT(kMaximalTargetExponent <= -32);
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// w is assumed to have an error less than 1 unit. Whenever w is scaled we
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// also scale its error.
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uint64_t w_error = 1;
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// We cut the input number into two parts: the integral digits and the
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// fractional digits. We don't emit any decimal separator, but adapt kappa
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// instead. Example: instead of writing "1.2" we put "12" into the buffer and
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// increase kappa by 1.
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DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
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// Division by one is a shift.
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uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
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// Modulo by one is an and.
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uint64_t fractionals = w.f() & (one.f() - 1);
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int divisor_exponent_plus_one;
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BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
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&divisor, &divisor_exponent_plus_one);
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*kappa = divisor_exponent_plus_one;
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// Loop invariant: buffer = w / 10^kappa (integer division)
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// The invariant holds for the first iteration: kappa has been initialized
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// with the divisor exponent + 1. And the divisor is the biggest power of ten
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// that is smaller than 'integrals'.
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int digit = integrals / divisor;
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buffer[*length] = '0' + digit;
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integrals %= divisor;
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// Note that kappa now equals the exponent of the divisor and that the
468
// invariant thus holds again.
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if (requested_digits == 0) break;
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if (requested_digits == 0) {
475
(static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
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return RoundWeedCounted(buffer, *length, rest,
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static_cast<uint64_t>(divisor) << -one.e(), w_error,
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// The integrals have been generated. We are at the point of the decimal
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// separator. In the following loop we simply multiply the remaining digits by
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// 10 and divide by one. We just need to pay attention to multiply associated
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// data (the 'unit'), too.
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// Note that the multiplication by 10 does not overflow, because w.e >= -60
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// and thus one.e >= -60.
487
ASSERT(one.e() >= -60);
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ASSERT(fractionals < one.f());
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ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
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while (requested_digits > 0 && fractionals > w_error) {
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// Integer division by one.
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int digit = static_cast<int>(fractionals >> -one.e());
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buffer[*length] = '0' + digit;
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fractionals &= one.f() - 1; // Modulo by one.
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if (requested_digits != 0) return false;
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return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
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// Provides a decimal representation of v.
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// Returns true if it succeeds, otherwise the result cannot be trusted.
509
// There will be *length digits inside the buffer (not null-terminated).
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// If the function returns true then
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// v == (double) (buffer * 10^decimal_exponent).
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// The digits in the buffer are the shortest representation possible: no
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// 0.09999999999999999 instead of 0.1. The shorter representation will even be
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// chosen even if the longer one would be closer to v.
515
// The last digit will be closest to the actual v. That is, even if several
516
// digits might correctly yield 'v' when read again, the closest will be
518
static bool Grisu3(double v,
522
int* decimal_exponent) {
523
DiyFp w = Double(v).AsNormalizedDiyFp();
524
// boundary_minus and boundary_plus are the boundaries between v and its
525
// closest floating-point neighbors. Any number strictly between
526
// boundary_minus and boundary_plus will round to v when convert to a double.
527
// Grisu3 will never output representations that lie exactly on a boundary.
528
DiyFp boundary_minus, boundary_plus;
529
if (mode == FAST_DTOA_SHORTEST) {
530
Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
532
assert(mode == FAST_DTOA_SHORTEST_SINGLE);
533
float single_v = static_cast<float>(v);
534
Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
536
ASSERT(boundary_plus.e() == w.e());
537
DiyFp ten_mk; // Cached power of ten: 10^-k
539
int ten_mk_minimal_binary_exponent =
540
kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
541
int ten_mk_maximal_binary_exponent =
542
kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
543
PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
544
ten_mk_minimal_binary_exponent,
545
ten_mk_maximal_binary_exponent,
547
ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
548
DiyFp::kSignificandSize) &&
549
(kMaximalTargetExponent >= w.e() + ten_mk.e() +
550
DiyFp::kSignificandSize));
551
// Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
552
// 64 bit significand and ten_mk is thus only precise up to 64 bits.
554
// The DiyFp::Times procedure rounds its result, and ten_mk is approximated
555
// too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
556
// off by a small amount.
557
// In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
558
// In other words: let f = scaled_w.f() and e = scaled_w.e(), then
559
// (f-1) * 2^e < w*10^k < (f+1) * 2^e
560
DiyFp scaled_w = DiyFp::Times(w, ten_mk);
561
ASSERT(scaled_w.e() ==
562
boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
563
// In theory it would be possible to avoid some recomputations by computing
564
// the difference between w and boundary_minus/plus (a power of 2) and to
565
// compute scaled_boundary_minus/plus by subtracting/adding from
566
// scaled_w. However the code becomes much less readable and the speed
567
// enhancements are not terriffic.
568
DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
569
DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
571
// DigitGen will generate the digits of scaled_w. Therefore we have
572
// v == (double) (scaled_w * 10^-mk).
573
// Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
574
// integer than it will be updated. For instance if scaled_w == 1.23 then
575
// the buffer will be filled with "123" und the decimal_exponent will be
578
bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
579
buffer, length, &kappa);
580
*decimal_exponent = -mk + kappa;
585
// The "counted" version of grisu3 (see above) only generates requested_digits
586
// number of digits. This version does not generate the shortest representation,
587
// and with enough requested digits 0.1 will at some point print as 0.9999999...
588
// Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
589
// therefore the rounding strategy for halfway cases is irrelevant.
590
static bool Grisu3Counted(double v,
591
int requested_digits,
594
int* decimal_exponent) {
595
DiyFp w = Double(v).AsNormalizedDiyFp();
596
DiyFp ten_mk; // Cached power of ten: 10^-k
598
int ten_mk_minimal_binary_exponent =
599
kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
600
int ten_mk_maximal_binary_exponent =
601
kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
602
PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
603
ten_mk_minimal_binary_exponent,
604
ten_mk_maximal_binary_exponent,
606
ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
607
DiyFp::kSignificandSize) &&
608
(kMaximalTargetExponent >= w.e() + ten_mk.e() +
609
DiyFp::kSignificandSize));
610
// Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
611
// 64 bit significand and ten_mk is thus only precise up to 64 bits.
613
// The DiyFp::Times procedure rounds its result, and ten_mk is approximated
614
// too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
615
// off by a small amount.
616
// In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
617
// In other words: let f = scaled_w.f() and e = scaled_w.e(), then
618
// (f-1) * 2^e < w*10^k < (f+1) * 2^e
619
DiyFp scaled_w = DiyFp::Times(w, ten_mk);
621
// We now have (double) (scaled_w * 10^-mk).
622
// DigitGen will generate the first requested_digits digits of scaled_w and
623
// return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
624
// will not always be exactly the same since DigitGenCounted only produces a
625
// limited number of digits.)
627
bool result = DigitGenCounted(scaled_w, requested_digits,
628
buffer, length, &kappa);
629
*decimal_exponent = -mk + kappa;
634
bool FastDtoa(double v,
636
int requested_digits,
639
int* decimal_point) {
641
ASSERT(!Double(v).IsSpecial());
644
int decimal_exponent = 0;
646
case FAST_DTOA_SHORTEST:
647
case FAST_DTOA_SHORTEST_SINGLE:
648
result = Grisu3(v, mode, buffer, length, &decimal_exponent);
650
case FAST_DTOA_PRECISION:
651
result = Grisu3Counted(v, requested_digits,
652
buffer, length, &decimal_exponent);
658
*decimal_point = *length + decimal_exponent;
659
buffer[*length] = '\0';
664
} // namespace double_conversion
added
removed
mozilla/mfbt/double-conversion/fast-dtoa.cc
