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Diffstat (limited to 'js/src/v8-dtoa/fast-dtoa.cc')
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diff --git a/js/src/v8-dtoa/fast-dtoa.cc b/js/src/v8-dtoa/fast-dtoa.cc new file mode 100644 index 0000000..b4b7be0 --- /dev/null +++ b/js/src/v8-dtoa/fast-dtoa.cc @@ -0,0 +1,505 @@ +// Copyright 2010 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 "v8.h" + +#include "fast-dtoa.h" + +#include "cached-powers.h" +#include "diy-fp.h" +#include "double.h" + +namespace v8 { +namespace internal { + +// 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 minimal_target_exponent = -60; +static const int maximal_target_exponent = -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 ctannot 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. +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_low[ (often written as "(w_low; w_low)") + + // 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 rounding + // the buffer. If we have two potential representations we need to make sure + // that the chosen one is closer to w_low and w_high since v can be anywhere + // between them. + // + // 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 + 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); +} + + + +static const uint32_t kTen4 = 10000; +static const uint32_t kTen5 = 100000; +static const uint32_t kTen6 = 1000000; +static const uint32_t kTen7 = 10000000; +static const uint32_t kTen8 = 100000000; +static const uint32_t kTen9 = 1000000000; + +// Returns the biggest power of ten that is less than or equal than the given +// number. We furthermore receive the maximum number of bits 'number' has. +// If number_bits == 0 then 0^-1 is returned +// The number of bits must be <= 32. +// Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)). +static void BiggestPowerTen(uint32_t number, + int number_bits, + uint32_t* power, + int* exponent) { + switch (number_bits) { + case 32: + case 31: + case 30: + if (kTen9 <= number) { + *power = kTen9; + *exponent = 9; + break; + } // else fallthrough + case 29: + case 28: + case 27: + if (kTen8 <= number) { + *power = kTen8; + *exponent = 8; + break; + } // else fallthrough + case 26: + case 25: + case 24: + if (kTen7 <= number) { + *power = kTen7; + *exponent = 7; + break; + } // else fallthrough + case 23: + case 22: + case 21: + case 20: + if (kTen6 <= number) { + *power = kTen6; + *exponent = 6; + break; + } // else fallthrough + case 19: + case 18: + case 17: + if (kTen5 <= number) { + *power = kTen5; + *exponent = 5; + break; + } // else fallthrough + case 16: + case 15: + case 14: + if (kTen4 <= number) { + *power = kTen4; + *exponent = 4; + break; + } // else fallthrough + case 13: + case 12: + case 11: + case 10: + if (1000 <= number) { + *power = 1000; + *exponent = 3; + break; + } // else fallthrough + case 9: + case 8: + case 7: + if (100 <= number) { + *power = 100; + *exponent = 2; + break; + } // else fallthrough + case 6: + case 5: + case 4: + if (10 <= number) { + *power = 10; + *exponent = 1; + break; + } // else fallthrough + case 3: + case 2: + case 1: + if (1 <= number) { + *power = 1; + *exponent = 0; + break; + } // else fallthrough + case 0: + *power = 0; + *exponent = -1; + break; + default: + // Following assignments are here to silence compiler warnings. + *power = 0; + *exponent = 0; + UNREACHABLE(); + } +} + + +// 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 minimal_target_exponent and +// maximal_target_exponent. +// 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 that a unit of their last digits. +// * low.e() == w.e() == high.e() +// * low < w < high, and taking into account their error: low~ <= high~ +// * minimal_target_exponent <= w.e() <= maximal_target_exponent +// 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. +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(minimal_target_exponent <= w.e() && w.e() <= maximal_target_exponent); + // 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 divider; + int divider_exponent; + BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), + ÷r, ÷r_exponent); + *kappa = divider_exponent + 1; + *length = 0; + // Loop invariant: buffer = too_high / 10^kappa (integer division) + // The invariant holds for the first iteration: kappa has been initialized + // with the divider exponent + 1. And the divider is the biggest power of ten + // that is smaller than integrals. + while (*kappa > 0) { + int digit = integrals / divider; + buffer[*length] = '0' + digit; + (*length)++; + integrals %= divider; + (*kappa)--; + // Note that kappa now equals the exponent of the divider 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>(divider) << -one.e(), unit); + } + divider /= 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. + // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and + // increase its (imaginary) exponent. At the same time we decrease the + // divider's (one's) exponent and shift its significand. + // Basically, if fractionals was a DiyFp (with fractionals.e == one.e): + // fractionals.f *= 10; + // fractionals.f >>= 1; fractionals.e++; // value remains unchanged. + // one.f >>= 1; one.e++; // value remains unchanged. + // and we have again fractionals.e == one.e which allows us to divide + // fractionals.f() by one.f() + // We simply combine the *= 10 and the >>= 1. + while (true) { + fractionals *= 5; + unit *= 5; + unsafe_interval.set_f(unsafe_interval.f() * 5); + unsafe_interval.set_e(unsafe_interval.e() + 1); // Will be optimized out. + one.set_f(one.f() >> 1); + one.set_e(one.e() + 1); + // 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); + } + } +} + + +// 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. +bool grisu3(double v, 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; + Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); + ASSERT(boundary_plus.e() == w.e()); + DiyFp ten_mk; // Cached power of ten: 10^-k + int mk; // -k + GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent, + maximal_target_exponent, &mk, &ten_mk); + ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() + + DiyFp::kSignificandSize && + maximal_target_exponent >= 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; +} + + +bool FastDtoa(double v, + Vector<char> buffer, + int* length, + int* point) { + ASSERT(v > 0); + ASSERT(!Double(v).IsSpecial()); + + int decimal_exponent; + bool result = grisu3(v, buffer, length, &decimal_exponent); + *point = *length + decimal_exponent; + buffer[*length] = '\0'; + return result; +} + +} } // namespace v8::internal |