1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/mfbt/double-conversion/fast-dtoa.cc Wed Dec 31 06:09:35 2014 +0100
1.3 @@ -0,0 +1,664 @@
1.4 +// Copyright 2012 the V8 project authors. All rights reserved.
1.5 +// Redistribution and use in source and binary forms, with or without
1.6 +// modification, are permitted provided that the following conditions are
1.7 +// met:
1.8 +//
1.9 +// * Redistributions of source code must retain the above copyright
1.10 +// notice, this list of conditions and the following disclaimer.
1.11 +// * Redistributions in binary form must reproduce the above
1.12 +// copyright notice, this list of conditions and the following
1.13 +// disclaimer in the documentation and/or other materials provided
1.14 +// with the distribution.
1.15 +// * Neither the name of Google Inc. nor the names of its
1.16 +// contributors may be used to endorse or promote products derived
1.17 +// from this software without specific prior written permission.
1.18 +//
1.19 +// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
1.20 +// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
1.21 +// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
1.22 +// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
1.23 +// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
1.24 +// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
1.25 +// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
1.26 +// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
1.27 +// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
1.28 +// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
1.29 +// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
1.30 +
1.31 +#include "fast-dtoa.h"
1.32 +
1.33 +#include "cached-powers.h"
1.34 +#include "diy-fp.h"
1.35 +#include "ieee.h"
1.36 +
1.37 +namespace double_conversion {
1.38 +
1.39 +// The minimal and maximal target exponent define the range of w's binary
1.40 +// exponent, where 'w' is the result of multiplying the input by a cached power
1.41 +// of ten.
1.42 +//
1.43 +// A different range might be chosen on a different platform, to optimize digit
1.44 +// generation, but a smaller range requires more powers of ten to be cached.
1.45 +static const int kMinimalTargetExponent = -60;
1.46 +static const int kMaximalTargetExponent = -32;
1.47 +
1.48 +
1.49 +// Adjusts the last digit of the generated number, and screens out generated
1.50 +// solutions that may be inaccurate. A solution may be inaccurate if it is
1.51 +// outside the safe interval, or if we cannot prove that it is closer to the
1.52 +// input than a neighboring representation of the same length.
1.53 +//
1.54 +// Input: * buffer containing the digits of too_high / 10^kappa
1.55 +// * the buffer's length
1.56 +// * distance_too_high_w == (too_high - w).f() * unit
1.57 +// * unsafe_interval == (too_high - too_low).f() * unit
1.58 +// * rest = (too_high - buffer * 10^kappa).f() * unit
1.59 +// * ten_kappa = 10^kappa * unit
1.60 +// * unit = the common multiplier
1.61 +// Output: returns true if the buffer is guaranteed to contain the closest
1.62 +// representable number to the input.
1.63 +// Modifies the generated digits in the buffer to approach (round towards) w.
1.64 +static bool RoundWeed(Vector<char> buffer,
1.65 + int length,
1.66 + uint64_t distance_too_high_w,
1.67 + uint64_t unsafe_interval,
1.68 + uint64_t rest,
1.69 + uint64_t ten_kappa,
1.70 + uint64_t unit) {
1.71 + uint64_t small_distance = distance_too_high_w - unit;
1.72 + uint64_t big_distance = distance_too_high_w + unit;
1.73 + // Let w_low = too_high - big_distance, and
1.74 + // w_high = too_high - small_distance.
1.75 + // Note: w_low < w < w_high
1.76 + //
1.77 + // The real w (* unit) must lie somewhere inside the interval
1.78 + // ]w_low; w_high[ (often written as "(w_low; w_high)")
1.79 +
1.80 + // Basically the buffer currently contains a number in the unsafe interval
1.81 + // ]too_low; too_high[ with too_low < w < too_high
1.82 + //
1.83 + // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.84 + // ^v 1 unit ^ ^ ^ ^
1.85 + // boundary_high --------------------- . . . .
1.86 + // ^v 1 unit . . . .
1.87 + // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
1.88 + // . . ^ . .
1.89 + // . big_distance . . .
1.90 + // . . . . rest
1.91 + // small_distance . . . .
1.92 + // v . . . .
1.93 + // w_high - - - - - - - - - - - - - - - - - - . . . .
1.94 + // ^v 1 unit . . . .
1.95 + // w ---------------------------------------- . . . .
1.96 + // ^v 1 unit v . . .
1.97 + // w_low - - - - - - - - - - - - - - - - - - - - - . . .
1.98 + // . . v
1.99 + // buffer --------------------------------------------------+-------+--------
1.100 + // . .
1.101 + // safe_interval .
1.102 + // v .
1.103 + // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
1.104 + // ^v 1 unit .
1.105 + // boundary_low ------------------------- unsafe_interval
1.106 + // ^v 1 unit v
1.107 + // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.108 + //
1.109 + //
1.110 + // Note that the value of buffer could lie anywhere inside the range too_low
1.111 + // to too_high.
1.112 + //
1.113 + // boundary_low, boundary_high and w are approximations of the real boundaries
1.114 + // and v (the input number). They are guaranteed to be precise up to one unit.
1.115 + // In fact the error is guaranteed to be strictly less than one unit.
1.116 + //
1.117 + // Anything that lies outside the unsafe interval is guaranteed not to round
1.118 + // to v when read again.
1.119 + // Anything that lies inside the safe interval is guaranteed to round to v
1.120 + // when read again.
1.121 + // If the number inside the buffer lies inside the unsafe interval but not
1.122 + // inside the safe interval then we simply do not know and bail out (returning
1.123 + // false).
1.124 + //
1.125 + // Similarly we have to take into account the imprecision of 'w' when finding
1.126 + // the closest representation of 'w'. If we have two potential
1.127 + // representations, and one is closer to both w_low and w_high, then we know
1.128 + // it is closer to the actual value v.
1.129 + //
1.130 + // By generating the digits of too_high we got the largest (closest to
1.131 + // too_high) buffer that is still in the unsafe interval. In the case where
1.132 + // w_high < buffer < too_high we try to decrement the buffer.
1.133 + // This way the buffer approaches (rounds towards) w.
1.134 + // There are 3 conditions that stop the decrementation process:
1.135 + // 1) the buffer is already below w_high
1.136 + // 2) decrementing the buffer would make it leave the unsafe interval
1.137 + // 3) decrementing the buffer would yield a number below w_high and farther
1.138 + // away than the current number. In other words:
1.139 + // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
1.140 + // Instead of using the buffer directly we use its distance to too_high.
1.141 + // Conceptually rest ~= too_high - buffer
1.142 + // We need to do the following tests in this order to avoid over- and
1.143 + // underflows.
1.144 + ASSERT(rest <= unsafe_interval);
1.145 + while (rest < small_distance && // Negated condition 1
1.146 + unsafe_interval - rest >= ten_kappa && // Negated condition 2
1.147 + (rest + ten_kappa < small_distance || // buffer{-1} > w_high
1.148 + small_distance - rest >= rest + ten_kappa - small_distance)) {
1.149 + buffer[length - 1]--;
1.150 + rest += ten_kappa;
1.151 + }
1.152 +
1.153 + // We have approached w+ as much as possible. We now test if approaching w-
1.154 + // would require changing the buffer. If yes, then we have two possible
1.155 + // representations close to w, but we cannot decide which one is closer.
1.156 + if (rest < big_distance &&
1.157 + unsafe_interval - rest >= ten_kappa &&
1.158 + (rest + ten_kappa < big_distance ||
1.159 + big_distance - rest > rest + ten_kappa - big_distance)) {
1.160 + return false;
1.161 + }
1.162 +
1.163 + // Weeding test.
1.164 + // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
1.165 + // Since too_low = too_high - unsafe_interval this is equivalent to
1.166 + // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
1.167 + // Conceptually we have: rest ~= too_high - buffer
1.168 + return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
1.169 +}
1.170 +
1.171 +
1.172 +// Rounds the buffer upwards if the result is closer to v by possibly adding
1.173 +// 1 to the buffer. If the precision of the calculation is not sufficient to
1.174 +// round correctly, return false.
1.175 +// The rounding might shift the whole buffer in which case the kappa is
1.176 +// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
1.177 +//
1.178 +// If 2*rest > ten_kappa then the buffer needs to be round up.
1.179 +// rest can have an error of +/- 1 unit. This function accounts for the
1.180 +// imprecision and returns false, if the rounding direction cannot be
1.181 +// unambiguously determined.
1.182 +//
1.183 +// Precondition: rest < ten_kappa.
1.184 +static bool RoundWeedCounted(Vector<char> buffer,
1.185 + int length,
1.186 + uint64_t rest,
1.187 + uint64_t ten_kappa,
1.188 + uint64_t unit,
1.189 + int* kappa) {
1.190 + ASSERT(rest < ten_kappa);
1.191 + // The following tests are done in a specific order to avoid overflows. They
1.192 + // will work correctly with any uint64 values of rest < ten_kappa and unit.
1.193 + //
1.194 + // If the unit is too big, then we don't know which way to round. For example
1.195 + // a unit of 50 means that the real number lies within rest +/- 50. If
1.196 + // 10^kappa == 40 then there is no way to tell which way to round.
1.197 + if (unit >= ten_kappa) return false;
1.198 + // Even if unit is just half the size of 10^kappa we are already completely
1.199 + // lost. (And after the previous test we know that the expression will not
1.200 + // over/underflow.)
1.201 + if (ten_kappa - unit <= unit) return false;
1.202 + // If 2 * (rest + unit) <= 10^kappa we can safely round down.
1.203 + if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
1.204 + return true;
1.205 + }
1.206 + // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
1.207 + if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
1.208 + // Increment the last digit recursively until we find a non '9' digit.
1.209 + buffer[length - 1]++;
1.210 + for (int i = length - 1; i > 0; --i) {
1.211 + if (buffer[i] != '0' + 10) break;
1.212 + buffer[i] = '0';
1.213 + buffer[i - 1]++;
1.214 + }
1.215 + // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
1.216 + // exception of the first digit all digits are now '0'. Simply switch the
1.217 + // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
1.218 + // the power (the kappa) is increased.
1.219 + if (buffer[0] == '0' + 10) {
1.220 + buffer[0] = '1';
1.221 + (*kappa) += 1;
1.222 + }
1.223 + return true;
1.224 + }
1.225 + return false;
1.226 +}
1.227 +
1.228 +// Returns the biggest power of ten that is less than or equal to the given
1.229 +// number. We furthermore receive the maximum number of bits 'number' has.
1.230 +//
1.231 +// Returns power == 10^(exponent_plus_one-1) such that
1.232 +// power <= number < power * 10.
1.233 +// If number_bits == 0 then 0^(0-1) is returned.
1.234 +// The number of bits must be <= 32.
1.235 +// Precondition: number < (1 << (number_bits + 1)).
1.236 +
1.237 +// Inspired by the method for finding an integer log base 10 from here:
1.238 +// http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
1.239 +static unsigned int const kSmallPowersOfTen[] =
1.240 + {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
1.241 + 1000000000};
1.242 +
1.243 +static void BiggestPowerTen(uint32_t number,
1.244 + int number_bits,
1.245 + uint32_t* power,
1.246 + int* exponent_plus_one) {
1.247 + ASSERT(number < (1u << (number_bits + 1)));
1.248 + // 1233/4096 is approximately 1/lg(10).
1.249 + int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);
1.250 + // We increment to skip over the first entry in the kPowersOf10 table.
1.251 + // Note: kPowersOf10[i] == 10^(i-1).
1.252 + exponent_plus_one_guess++;
1.253 + // We don't have any guarantees that 2^number_bits <= number.
1.254 + // TODO(floitsch): can we change the 'while' into an 'if'? We definitely see
1.255 + // number < (2^number_bits - 1), but I haven't encountered
1.256 + // number < (2^number_bits - 2) yet.
1.257 + while (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
1.258 + exponent_plus_one_guess--;
1.259 + }
1.260 + *power = kSmallPowersOfTen[exponent_plus_one_guess];
1.261 + *exponent_plus_one = exponent_plus_one_guess;
1.262 +}
1.263 +
1.264 +// Generates the digits of input number w.
1.265 +// w is a floating-point number (DiyFp), consisting of a significand and an
1.266 +// exponent. Its exponent is bounded by kMinimalTargetExponent and
1.267 +// kMaximalTargetExponent.
1.268 +// Hence -60 <= w.e() <= -32.
1.269 +//
1.270 +// Returns false if it fails, in which case the generated digits in the buffer
1.271 +// should not be used.
1.272 +// Preconditions:
1.273 +// * low, w and high are correct up to 1 ulp (unit in the last place). That
1.274 +// is, their error must be less than a unit of their last digits.
1.275 +// * low.e() == w.e() == high.e()
1.276 +// * low < w < high, and taking into account their error: low~ <= high~
1.277 +// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
1.278 +// Postconditions: returns false if procedure fails.
1.279 +// otherwise:
1.280 +// * buffer is not null-terminated, but len contains the number of digits.
1.281 +// * buffer contains the shortest possible decimal digit-sequence
1.282 +// such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
1.283 +// correct values of low and high (without their error).
1.284 +// * if more than one decimal representation gives the minimal number of
1.285 +// decimal digits then the one closest to W (where W is the correct value
1.286 +// of w) is chosen.
1.287 +// Remark: this procedure takes into account the imprecision of its input
1.288 +// numbers. If the precision is not enough to guarantee all the postconditions
1.289 +// then false is returned. This usually happens rarely (~0.5%).
1.290 +//
1.291 +// Say, for the sake of example, that
1.292 +// w.e() == -48, and w.f() == 0x1234567890abcdef
1.293 +// w's value can be computed by w.f() * 2^w.e()
1.294 +// We can obtain w's integral digits by simply shifting w.f() by -w.e().
1.295 +// -> w's integral part is 0x1234
1.296 +// w's fractional part is therefore 0x567890abcdef.
1.297 +// Printing w's integral part is easy (simply print 0x1234 in decimal).
1.298 +// In order to print its fraction we repeatedly multiply the fraction by 10 and
1.299 +// get each digit. Example the first digit after the point would be computed by
1.300 +// (0x567890abcdef * 10) >> 48. -> 3
1.301 +// The whole thing becomes slightly more complicated because we want to stop
1.302 +// once we have enough digits. That is, once the digits inside the buffer
1.303 +// represent 'w' we can stop. Everything inside the interval low - high
1.304 +// represents w. However we have to pay attention to low, high and w's
1.305 +// imprecision.
1.306 +static bool DigitGen(DiyFp low,
1.307 + DiyFp w,
1.308 + DiyFp high,
1.309 + Vector<char> buffer,
1.310 + int* length,
1.311 + int* kappa) {
1.312 + ASSERT(low.e() == w.e() && w.e() == high.e());
1.313 + ASSERT(low.f() + 1 <= high.f() - 1);
1.314 + ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
1.315 + // low, w and high are imprecise, but by less than one ulp (unit in the last
1.316 + // place).
1.317 + // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
1.318 + // the new numbers are outside of the interval we want the final
1.319 + // representation to lie in.
1.320 + // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
1.321 + // numbers that are certain to lie in the interval. We will use this fact
1.322 + // later on.
1.323 + // We will now start by generating the digits within the uncertain
1.324 + // interval. Later we will weed out representations that lie outside the safe
1.325 + // interval and thus _might_ lie outside the correct interval.
1.326 + uint64_t unit = 1;
1.327 + DiyFp too_low = DiyFp(low.f() - unit, low.e());
1.328 + DiyFp too_high = DiyFp(high.f() + unit, high.e());
1.329 + // too_low and too_high are guaranteed to lie outside the interval we want the
1.330 + // generated number in.
1.331 + DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
1.332 + // We now cut the input number into two parts: the integral digits and the
1.333 + // fractionals. We will not write any decimal separator though, but adapt
1.334 + // kappa instead.
1.335 + // Reminder: we are currently computing the digits (stored inside the buffer)
1.336 + // such that: too_low < buffer * 10^kappa < too_high
1.337 + // We use too_high for the digit_generation and stop as soon as possible.
1.338 + // If we stop early we effectively round down.
1.339 + DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
1.340 + // Division by one is a shift.
1.341 + uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
1.342 + // Modulo by one is an and.
1.343 + uint64_t fractionals = too_high.f() & (one.f() - 1);
1.344 + uint32_t divisor;
1.345 + int divisor_exponent_plus_one;
1.346 + BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
1.347 + &divisor, &divisor_exponent_plus_one);
1.348 + *kappa = divisor_exponent_plus_one;
1.349 + *length = 0;
1.350 + // Loop invariant: buffer = too_high / 10^kappa (integer division)
1.351 + // The invariant holds for the first iteration: kappa has been initialized
1.352 + // with the divisor exponent + 1. And the divisor is the biggest power of ten
1.353 + // that is smaller than integrals.
1.354 + while (*kappa > 0) {
1.355 + int digit = integrals / divisor;
1.356 + buffer[*length] = '0' + digit;
1.357 + (*length)++;
1.358 + integrals %= divisor;
1.359 + (*kappa)--;
1.360 + // Note that kappa now equals the exponent of the divisor and that the
1.361 + // invariant thus holds again.
1.362 + uint64_t rest =
1.363 + (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
1.364 + // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
1.365 + // Reminder: unsafe_interval.e() == one.e()
1.366 + if (rest < unsafe_interval.f()) {
1.367 + // Rounding down (by not emitting the remaining digits) yields a number
1.368 + // that lies within the unsafe interval.
1.369 + return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
1.370 + unsafe_interval.f(), rest,
1.371 + static_cast<uint64_t>(divisor) << -one.e(), unit);
1.372 + }
1.373 + divisor /= 10;
1.374 + }
1.375 +
1.376 + // The integrals have been generated. We are at the point of the decimal
1.377 + // separator. In the following loop we simply multiply the remaining digits by
1.378 + // 10 and divide by one. We just need to pay attention to multiply associated
1.379 + // data (like the interval or 'unit'), too.
1.380 + // Note that the multiplication by 10 does not overflow, because w.e >= -60
1.381 + // and thus one.e >= -60.
1.382 + ASSERT(one.e() >= -60);
1.383 + ASSERT(fractionals < one.f());
1.384 + ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
1.385 + while (true) {
1.386 + fractionals *= 10;
1.387 + unit *= 10;
1.388 + unsafe_interval.set_f(unsafe_interval.f() * 10);
1.389 + // Integer division by one.
1.390 + int digit = static_cast<int>(fractionals >> -one.e());
1.391 + buffer[*length] = '0' + digit;
1.392 + (*length)++;
1.393 + fractionals &= one.f() - 1; // Modulo by one.
1.394 + (*kappa)--;
1.395 + if (fractionals < unsafe_interval.f()) {
1.396 + return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
1.397 + unsafe_interval.f(), fractionals, one.f(), unit);
1.398 + }
1.399 + }
1.400 +}
1.401 +
1.402 +
1.403 +
1.404 +// Generates (at most) requested_digits digits of input number w.
1.405 +// w is a floating-point number (DiyFp), consisting of a significand and an
1.406 +// exponent. Its exponent is bounded by kMinimalTargetExponent and
1.407 +// kMaximalTargetExponent.
1.408 +// Hence -60 <= w.e() <= -32.
1.409 +//
1.410 +// Returns false if it fails, in which case the generated digits in the buffer
1.411 +// should not be used.
1.412 +// Preconditions:
1.413 +// * w is correct up to 1 ulp (unit in the last place). That
1.414 +// is, its error must be strictly less than a unit of its last digit.
1.415 +// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
1.416 +//
1.417 +// Postconditions: returns false if procedure fails.
1.418 +// otherwise:
1.419 +// * buffer is not null-terminated, but length contains the number of
1.420 +// digits.
1.421 +// * the representation in buffer is the most precise representation of
1.422 +// requested_digits digits.
1.423 +// * buffer contains at most requested_digits digits of w. If there are less
1.424 +// than requested_digits digits then some trailing '0's have been removed.
1.425 +// * kappa is such that
1.426 +// w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
1.427 +//
1.428 +// Remark: This procedure takes into account the imprecision of its input
1.429 +// numbers. If the precision is not enough to guarantee all the postconditions
1.430 +// then false is returned. This usually happens rarely, but the failure-rate
1.431 +// increases with higher requested_digits.
1.432 +static bool DigitGenCounted(DiyFp w,
1.433 + int requested_digits,
1.434 + Vector<char> buffer,
1.435 + int* length,
1.436 + int* kappa) {
1.437 + ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
1.438 + ASSERT(kMinimalTargetExponent >= -60);
1.439 + ASSERT(kMaximalTargetExponent <= -32);
1.440 + // w is assumed to have an error less than 1 unit. Whenever w is scaled we
1.441 + // also scale its error.
1.442 + uint64_t w_error = 1;
1.443 + // We cut the input number into two parts: the integral digits and the
1.444 + // fractional digits. We don't emit any decimal separator, but adapt kappa
1.445 + // instead. Example: instead of writing "1.2" we put "12" into the buffer and
1.446 + // increase kappa by 1.
1.447 + DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
1.448 + // Division by one is a shift.
1.449 + uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
1.450 + // Modulo by one is an and.
1.451 + uint64_t fractionals = w.f() & (one.f() - 1);
1.452 + uint32_t divisor;
1.453 + int divisor_exponent_plus_one;
1.454 + BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
1.455 + &divisor, &divisor_exponent_plus_one);
1.456 + *kappa = divisor_exponent_plus_one;
1.457 + *length = 0;
1.458 +
1.459 + // Loop invariant: buffer = w / 10^kappa (integer division)
1.460 + // The invariant holds for the first iteration: kappa has been initialized
1.461 + // with the divisor exponent + 1. And the divisor is the biggest power of ten
1.462 + // that is smaller than 'integrals'.
1.463 + while (*kappa > 0) {
1.464 + int digit = integrals / divisor;
1.465 + buffer[*length] = '0' + digit;
1.466 + (*length)++;
1.467 + requested_digits--;
1.468 + integrals %= divisor;
1.469 + (*kappa)--;
1.470 + // Note that kappa now equals the exponent of the divisor and that the
1.471 + // invariant thus holds again.
1.472 + if (requested_digits == 0) break;
1.473 + divisor /= 10;
1.474 + }
1.475 +
1.476 + if (requested_digits == 0) {
1.477 + uint64_t rest =
1.478 + (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
1.479 + return RoundWeedCounted(buffer, *length, rest,
1.480 + static_cast<uint64_t>(divisor) << -one.e(), w_error,
1.481 + kappa);
1.482 + }
1.483 +
1.484 + // The integrals have been generated. We are at the point of the decimal
1.485 + // separator. In the following loop we simply multiply the remaining digits by
1.486 + // 10 and divide by one. We just need to pay attention to multiply associated
1.487 + // data (the 'unit'), too.
1.488 + // Note that the multiplication by 10 does not overflow, because w.e >= -60
1.489 + // and thus one.e >= -60.
1.490 + ASSERT(one.e() >= -60);
1.491 + ASSERT(fractionals < one.f());
1.492 + ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
1.493 + while (requested_digits > 0 && fractionals > w_error) {
1.494 + fractionals *= 10;
1.495 + w_error *= 10;
1.496 + // Integer division by one.
1.497 + int digit = static_cast<int>(fractionals >> -one.e());
1.498 + buffer[*length] = '0' + digit;
1.499 + (*length)++;
1.500 + requested_digits--;
1.501 + fractionals &= one.f() - 1; // Modulo by one.
1.502 + (*kappa)--;
1.503 + }
1.504 + if (requested_digits != 0) return false;
1.505 + return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
1.506 + kappa);
1.507 +}
1.508 +
1.509 +
1.510 +// Provides a decimal representation of v.
1.511 +// Returns true if it succeeds, otherwise the result cannot be trusted.
1.512 +// There will be *length digits inside the buffer (not null-terminated).
1.513 +// If the function returns true then
1.514 +// v == (double) (buffer * 10^decimal_exponent).
1.515 +// The digits in the buffer are the shortest representation possible: no
1.516 +// 0.09999999999999999 instead of 0.1. The shorter representation will even be
1.517 +// chosen even if the longer one would be closer to v.
1.518 +// The last digit will be closest to the actual v. That is, even if several
1.519 +// digits might correctly yield 'v' when read again, the closest will be
1.520 +// computed.
1.521 +static bool Grisu3(double v,
1.522 + FastDtoaMode mode,
1.523 + Vector<char> buffer,
1.524 + int* length,
1.525 + int* decimal_exponent) {
1.526 + DiyFp w = Double(v).AsNormalizedDiyFp();
1.527 + // boundary_minus and boundary_plus are the boundaries between v and its
1.528 + // closest floating-point neighbors. Any number strictly between
1.529 + // boundary_minus and boundary_plus will round to v when convert to a double.
1.530 + // Grisu3 will never output representations that lie exactly on a boundary.
1.531 + DiyFp boundary_minus, boundary_plus;
1.532 + if (mode == FAST_DTOA_SHORTEST) {
1.533 + Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
1.534 + } else {
1.535 + ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
1.536 + float single_v = static_cast<float>(v);
1.537 + Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
1.538 + }
1.539 + ASSERT(boundary_plus.e() == w.e());
1.540 + DiyFp ten_mk; // Cached power of ten: 10^-k
1.541 + int mk; // -k
1.542 + int ten_mk_minimal_binary_exponent =
1.543 + kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
1.544 + int ten_mk_maximal_binary_exponent =
1.545 + kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
1.546 + PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
1.547 + ten_mk_minimal_binary_exponent,
1.548 + ten_mk_maximal_binary_exponent,
1.549 + &ten_mk, &mk);
1.550 + ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
1.551 + DiyFp::kSignificandSize) &&
1.552 + (kMaximalTargetExponent >= w.e() + ten_mk.e() +
1.553 + DiyFp::kSignificandSize));
1.554 + // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
1.555 + // 64 bit significand and ten_mk is thus only precise up to 64 bits.
1.556 +
1.557 + // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
1.558 + // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
1.559 + // off by a small amount.
1.560 + // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
1.561 + // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
1.562 + // (f-1) * 2^e < w*10^k < (f+1) * 2^e
1.563 + DiyFp scaled_w = DiyFp::Times(w, ten_mk);
1.564 + ASSERT(scaled_w.e() ==
1.565 + boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
1.566 + // In theory it would be possible to avoid some recomputations by computing
1.567 + // the difference between w and boundary_minus/plus (a power of 2) and to
1.568 + // compute scaled_boundary_minus/plus by subtracting/adding from
1.569 + // scaled_w. However the code becomes much less readable and the speed
1.570 + // enhancements are not terriffic.
1.571 + DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
1.572 + DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
1.573 +
1.574 + // DigitGen will generate the digits of scaled_w. Therefore we have
1.575 + // v == (double) (scaled_w * 10^-mk).
1.576 + // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
1.577 + // integer than it will be updated. For instance if scaled_w == 1.23 then
1.578 + // the buffer will be filled with "123" und the decimal_exponent will be
1.579 + // decreased by 2.
1.580 + int kappa;
1.581 + bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
1.582 + buffer, length, &kappa);
1.583 + *decimal_exponent = -mk + kappa;
1.584 + return result;
1.585 +}
1.586 +
1.587 +
1.588 +// The "counted" version of grisu3 (see above) only generates requested_digits
1.589 +// number of digits. This version does not generate the shortest representation,
1.590 +// and with enough requested digits 0.1 will at some point print as 0.9999999...
1.591 +// Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
1.592 +// therefore the rounding strategy for halfway cases is irrelevant.
1.593 +static bool Grisu3Counted(double v,
1.594 + int requested_digits,
1.595 + Vector<char> buffer,
1.596 + int* length,
1.597 + int* decimal_exponent) {
1.598 + DiyFp w = Double(v).AsNormalizedDiyFp();
1.599 + DiyFp ten_mk; // Cached power of ten: 10^-k
1.600 + int mk; // -k
1.601 + int ten_mk_minimal_binary_exponent =
1.602 + kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
1.603 + int ten_mk_maximal_binary_exponent =
1.604 + kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
1.605 + PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
1.606 + ten_mk_minimal_binary_exponent,
1.607 + ten_mk_maximal_binary_exponent,
1.608 + &ten_mk, &mk);
1.609 + ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
1.610 + DiyFp::kSignificandSize) &&
1.611 + (kMaximalTargetExponent >= w.e() + ten_mk.e() +
1.612 + DiyFp::kSignificandSize));
1.613 + // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
1.614 + // 64 bit significand and ten_mk is thus only precise up to 64 bits.
1.615 +
1.616 + // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
1.617 + // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
1.618 + // off by a small amount.
1.619 + // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
1.620 + // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
1.621 + // (f-1) * 2^e < w*10^k < (f+1) * 2^e
1.622 + DiyFp scaled_w = DiyFp::Times(w, ten_mk);
1.623 +
1.624 + // We now have (double) (scaled_w * 10^-mk).
1.625 + // DigitGen will generate the first requested_digits digits of scaled_w and
1.626 + // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
1.627 + // will not always be exactly the same since DigitGenCounted only produces a
1.628 + // limited number of digits.)
1.629 + int kappa;
1.630 + bool result = DigitGenCounted(scaled_w, requested_digits,
1.631 + buffer, length, &kappa);
1.632 + *decimal_exponent = -mk + kappa;
1.633 + return result;
1.634 +}
1.635 +
1.636 +
1.637 +bool FastDtoa(double v,
1.638 + FastDtoaMode mode,
1.639 + int requested_digits,
1.640 + Vector<char> buffer,
1.641 + int* length,
1.642 + int* decimal_point) {
1.643 + ASSERT(v > 0);
1.644 + ASSERT(!Double(v).IsSpecial());
1.645 +
1.646 + bool result = false;
1.647 + int decimal_exponent = 0;
1.648 + switch (mode) {
1.649 + case FAST_DTOA_SHORTEST:
1.650 + case FAST_DTOA_SHORTEST_SINGLE:
1.651 + result = Grisu3(v, mode, buffer, length, &decimal_exponent);
1.652 + break;
1.653 + case FAST_DTOA_PRECISION:
1.654 + result = Grisu3Counted(v, requested_digits,
1.655 + buffer, length, &decimal_exponent);
1.656 + break;
1.657 + default:
1.658 + UNREACHABLE();
1.659 + }
1.660 + if (result) {
1.661 + *decimal_point = *length + decimal_exponent;
1.662 + buffer[*length] = '\0';
1.663 + }
1.664 + return result;
1.665 +}
1.666 +
1.667 +} // namespace double_conversion
