fast-dtoa.cc | Explore in Territory

// Copyright 2011 the V8 project authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

#include "src/base/numbers/fast-dtoa.h"

#include <stdint.h>

#include "src/base/logging.h"
#include "src/base/numbers/cached-powers.h"
#include "src/base/numbers/diy-fp.h"
#include "src/base/numbers/double.h"

namespace v8 {
namespace base {

// The minimal and maximal target exponent define the range of w's binary
// exponent, where 'w' is the result of multiplying the input by a cached power
// of ten.
//
// A different range might be chosen on a different platform, to optimize digit
// generation, but a smaller range requires more powers of ten to be cached.
static const int kMinimalTargetExponent = …;
static const int kMaximalTargetExponent = …;

// 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.
static bool RoundWeed(char* last_digit, uint64_t distance_too_high_w,
                      uint64_t unsafe_interval, uint64_t rest,
                      uint64_t ten_kappa, uint64_t unit) { … }

// Rounds the buffer upwards if the result is closer to v by possibly adding
// 1 to the buffer. If the precision of the calculation is not sufficient to
// round correctly, return false.
// The rounding might shift the whole buffer in which case the kappa is
// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
//
// If 2*rest > ten_kappa then the buffer needs to be round up.
// rest can have an error of +/- 1 unit. This function accounts for the
// imprecision and returns false, if the rounding direction cannot be
// unambiguously determined.
//
// Precondition: rest < ten_kappa.
static bool RoundWeedCounted(Vector<char> buffer, int length, uint64_t rest,
                             uint64_t ten_kappa, uint64_t unit, int* kappa) { … }

static const uint32_t kTen4 = …;
static const uint32_t kTen5 = …;
static const uint32_t kTen6 = …;
static const uint32_t kTen7 = …;
static const uint32_t kTen8 = …;
static const uint32_t kTen9 = …;

struct DivMagic { … };

// This table was computed by libdivide. Essentially, the shift is
// floor(log2(x)), and the mul is 2^(33 + shift) / x, rounded up and truncated
// to 32 bits.
static const DivMagic div[] = …;

// Returns *val / divisor, and does *val %= divisor. d must be the DivMagic
// corresponding to the divisor.
//
// This algorithm is exactly the same as libdivide's branch-free u32 algorithm,
// except that we add back a branch anyway to support 1.
//
// GCC/Clang uses a slightly different algorithm that doesn't need
// the extra rounding step (and that would allow us to do 1 without
// a branch), but it requires a pre-shift for the case of 10000,
// so it ends up slower, at least on x86-64.
//
// Note that this is actually a small loss for certain CPUs with
// a very fast divider (e.g. Zen 3), but a significant win for most
// others (including the entire Skylake family).
static inline uint32_t fast_divmod(uint32_t* val, uint32_t divisor,
                                   const DivMagic& d) { … }

// 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: number < (1 << (number_bits + 1)).
static inline void BiggestPowerTen(uint32_t number, int number_bits,
                                   uint32_t* power, unsigned* exponent) { … }

// Generates the digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by kMinimalTargetExponent and
// kMaximalTargetExponent.
//       Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
//  * low, w and high are correct up to 1 ulp (unit in the last place). That
//    is, their error must be less than a unit of their last digits.
//  * low.e() == w.e() == high.e()
//  * low < w < high, and taking into account their error: low~ <= high~
//  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
// Postconditions: returns false if procedure fails.
//   otherwise:
//     * buffer is not null-terminated, but len contains the number of digits.
//     * buffer contains the shortest possible decimal digit-sequence
//       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
//       correct values of low and high (without their error).
//     * if more than one decimal representation gives the minimal number of
//       decimal digits then the one closest to W (where W is the correct value
//       of w) is chosen.
// Remark: this procedure takes into account the imprecision of its input
//   numbers. If the precision is not enough to guarantee all the postconditions
//   then false is returned. This usually happens rarely (~0.5%).
//
// Say, for the sake of example, that
//   w.e() == -48, and w.f() == 0x1234567890ABCDEF
// w's value can be computed by w.f() * 2^w.e()
// We can obtain w's integral digits by simply shifting w.f() by -w.e().
//  -> w's integral part is 0x1234
//  w's fractional part is therefore 0x567890ABCDEF.
// Printing w's integral part is easy (simply print 0x1234 in decimal).
// In order to print its fraction we repeatedly multiply the fraction by 10 and
// get each digit. Example the first digit after the point would be computed by
//   (0x567890ABCDEF * 10) >> 48. -> 3
// The whole thing becomes slightly more complicated because we want to stop
// once we have enough digits. That is, once the digits inside the buffer
// represent 'w' we can stop. Everything inside the interval low - high
// represents w. However we have to pay attention to low, high and w's
// imprecision.
static bool DigitGen(DiyFp low, DiyFp w, DiyFp high, char** outptr,
                     int* kappa) { … }

// Generates (at most) requested_digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by kMinimalTargetExponent and
// kMaximalTargetExponent.
//       Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
//  * w is correct up to 1 ulp (unit in the last place). That
//    is, its error must be strictly less than a unit of its last digit.
//  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
//
// Postconditions: returns false if procedure fails.
//   otherwise:
//     * buffer is not null-terminated, but length contains the number of
//       digits.
//     * the representation in buffer is the most precise representation of
//       requested_digits digits.
//     * buffer contains at most requested_digits digits of w. If there are less
//       than requested_digits digits then some trailing '0's have been removed.
//     * kappa is such that
//            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
//
// Remark: This procedure takes into account the imprecision of its input
//   numbers. If the precision is not enough to guarantee all the postconditions
//   then false is returned. This usually happens rarely, but the failure-rate
//   increases with higher requested_digits.
static bool DigitGenCounted(DiyFp w, int requested_digits, Vector<char> buffer,
                            int* length, int* kappa) { … }

// Provides a decimal representation of v.
// Returns true if it succeeds, otherwise the result cannot be trusted.
// There will be *length digits inside the buffer (not null-terminated).
// If the function returns true then
//        v == (double) (buffer * 10^decimal_exponent).
// The digits in the buffer are the shortest representation possible: no
// 0.09999999999999999 instead of 0.1. The shorter representation will even be
// chosen even if the longer one would be closer to v.
// The last digit will be closest to the actual v. That is, even if several
// digits might correctly yield 'v' when read again, the closest will be
// computed.
static bool Grisu3(double v, char** outptr, int* decimal_exponent) { … }

// The "counted" version of grisu3 (see above) only generates requested_digits
// number of digits. This version does not generate the shortest representation,
// and with enough requested digits 0.1 will at some point print as 0.9999999...
// Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
// therefore the rounding strategy for halfway cases is irrelevant.
static bool Grisu3Counted(double v, int requested_digits, Vector<char> buffer,
                          int* length, int* decimal_exponent) { … }

bool FastDtoa(double v, FastDtoaMode mode, int requested_digits,
              Vector<char> buffer, int* length, int* decimal_point) { … }

}  // namespace base
}  // namespace v8
chromium/v8/src/base/numbers/fast-dtoa.cc
