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DEFINITIONS
This source file includes following definitions.
- TrimLeadingZeros
- TrimTrailingZeros
- TrimToMaxSignificantDigits
- ReadUint64
- ReadDiyFp
- DoubleStrtod
- AdjustmentPowerOfTen
- DiyFpStrtod
- BignumStrtod
- Strtod
// Copyright 2012 the V8 project authors. All rights reserved.
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following
// disclaimer in the documentation and/or other materials provided
// with the distribution.
// * Neither the name of Google Inc. nor the names of its
// contributors may be used to endorse or promote products derived
// from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
#include <stdarg.h>
#include <math.h>
#include "globals.h"
#include "utils.h"
#include "strtod.h"
#include "bignum.h"
#include "cached-powers.h"
#include "double.h"
namespace v8 {
namespace internal {
// 2^53 = 9007199254740992.
// Any integer with at most 15 decimal digits will hence fit into a double
// (which has a 53bit significand) without loss of precision.
static const int kMaxExactDoubleIntegerDecimalDigits = 15;
// 2^64 = 18446744073709551616 > 10^19
static const int kMaxUint64DecimalDigits = 19;
// Max double: 1.7976931348623157 x 10^308
// Min non-zero double: 4.9406564584124654 x 10^-324
// Any x >= 10^309 is interpreted as +infinity.
// Any x <= 10^-324 is interpreted as 0.
// Note that 2.5e-324 (despite being smaller than the min double) will be read
// as non-zero (equal to the min non-zero double).
static const int kMaxDecimalPower = 309;
static const int kMinDecimalPower = -324;
// 2^64 = 18446744073709551616
static const uint64_t kMaxUint64 = V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF);
static const double exact_powers_of_ten[] = {
1.0, // 10^0
10.0,
100.0,
1000.0,
10000.0,
100000.0,
1000000.0,
10000000.0,
100000000.0,
1000000000.0,
10000000000.0, // 10^10
100000000000.0,
1000000000000.0,
10000000000000.0,
100000000000000.0,
1000000000000000.0,
10000000000000000.0,
100000000000000000.0,
1000000000000000000.0,
10000000000000000000.0,
100000000000000000000.0, // 10^20
1000000000000000000000.0,
// 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
10000000000000000000000.0
};
static const int kExactPowersOfTenSize = ARRAY_SIZE(exact_powers_of_ten);
// Maximum number of significant digits in the decimal representation.
// In fact the value is 772 (see conversions.cc), but to give us some margin
// we round up to 780.
static const int kMaxSignificantDecimalDigits = 780;
static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
for (int i = 0; i < buffer.length(); i++) {
if (buffer[i] != '0') {
return buffer.SubVector(i, buffer.length());
}
}
return Vector<const char>(buffer.start(), 0);
}
static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
for (int i = buffer.length() - 1; i >= 0; --i) {
if (buffer[i] != '0') {
return buffer.SubVector(0, i + 1);
}
}
return Vector<const char>(buffer.start(), 0);
}
static void TrimToMaxSignificantDigits(Vector<const char> buffer,
int exponent,
char* significant_buffer,
int* significant_exponent) {
for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
significant_buffer[i] = buffer[i];
}
// The input buffer has been trimmed. Therefore the last digit must be
// different from '0'.
ASSERT(buffer[buffer.length() - 1] != '0');
// Set the last digit to be non-zero. This is sufficient to guarantee
// correct rounding.
significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
*significant_exponent =
exponent + (buffer.length() - kMaxSignificantDecimalDigits);
}
// Reads digits from the buffer and converts them to a uint64.
// Reads in as many digits as fit into a uint64.
// When the string starts with "1844674407370955161" no further digit is read.
// Since 2^64 = 18446744073709551616 it would still be possible read another
// digit if it was less or equal than 6, but this would complicate the code.
static uint64_t ReadUint64(Vector<const char> buffer,
int* number_of_read_digits) {
uint64_t result = 0;
int i = 0;
while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
int digit = buffer[i++] - '0';
ASSERT(0 <= digit && digit <= 9);
result = 10 * result + digit;
}
*number_of_read_digits = i;
return result;
}
// Reads a DiyFp from the buffer.
// The returned DiyFp is not necessarily normalized.
// If remaining_decimals is zero then the returned DiyFp is accurate.
// Otherwise it has been rounded and has error of at most 1/2 ulp.
static void ReadDiyFp(Vector<const char> buffer,
DiyFp* result,
int* remaining_decimals) {
int read_digits;
uint64_t significand = ReadUint64(buffer, &read_digits);
if (buffer.length() == read_digits) {
*result = DiyFp(significand, 0);
*remaining_decimals = 0;
} else {
// Round the significand.
if (buffer[read_digits] >= '5') {
significand++;
}
// Compute the binary exponent.
int exponent = 0;
*result = DiyFp(significand, exponent);
*remaining_decimals = buffer.length() - read_digits;
}
}
static bool DoubleStrtod(Vector<const char> trimmed,
int exponent,
double* result) {
#if (defined(V8_TARGET_ARCH_IA32) || defined(USE_SIMULATOR)) \
&& !defined(_MSC_VER)
// On x86 the floating-point stack can be 64 or 80 bits wide. If it is
// 80 bits wide (as is the case on Linux) then double-rounding occurs and the
// result is not accurate.
// We know that Windows32 with MSVC, unlike with MinGW32, uses 64 bits and is
// therefore accurate.
// Note that the ARM and MIPS simulators are compiled for 32bits. They
// therefore exhibit the same problem.
return false;
#endif
if (trimmed.length() <= kMaxExactDoubleIntegerDecimalDigits) {
int read_digits;
// The trimmed input fits into a double.
// If the 10^exponent (resp. 10^-exponent) fits into a double too then we
// can compute the result-double simply by multiplying (resp. dividing) the
// two numbers.
// This is possible because IEEE guarantees that floating-point operations
// return the best possible approximation.
if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
// 10^-exponent fits into a double.
*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
ASSERT(read_digits == trimmed.length());
*result /= exact_powers_of_ten[-exponent];
return true;
}
if (0 <= exponent && exponent < kExactPowersOfTenSize) {
// 10^exponent fits into a double.
*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
ASSERT(read_digits == trimmed.length());
*result *= exact_powers_of_ten[exponent];
return true;
}
int remaining_digits =
kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
if ((0 <= exponent) &&
(exponent - remaining_digits < kExactPowersOfTenSize)) {
// The trimmed string was short and we can multiply it with
// 10^remaining_digits. As a result the remaining exponent now fits
// into a double too.
*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
ASSERT(read_digits == trimmed.length());
*result *= exact_powers_of_ten[remaining_digits];
*result *= exact_powers_of_ten[exponent - remaining_digits];
return true;
}
}
return false;
}
// Returns 10^exponent as an exact DiyFp.
// The given exponent must be in the range [1; kDecimalExponentDistance[.
static DiyFp AdjustmentPowerOfTen(int exponent) {
ASSERT(0 < exponent);
ASSERT(exponent < PowersOfTenCache::kDecimalExponentDistance);
// Simply hardcode the remaining powers for the given decimal exponent
// distance.
ASSERT(PowersOfTenCache::kDecimalExponentDistance == 8);
switch (exponent) {
case 1: return DiyFp(V8_2PART_UINT64_C(0xa0000000, 00000000), -60);
case 2: return DiyFp(V8_2PART_UINT64_C(0xc8000000, 00000000), -57);
case 3: return DiyFp(V8_2PART_UINT64_C(0xfa000000, 00000000), -54);
case 4: return DiyFp(V8_2PART_UINT64_C(0x9c400000, 00000000), -50);
case 5: return DiyFp(V8_2PART_UINT64_C(0xc3500000, 00000000), -47);
case 6: return DiyFp(V8_2PART_UINT64_C(0xf4240000, 00000000), -44);
case 7: return DiyFp(V8_2PART_UINT64_C(0x98968000, 00000000), -40);
default:
UNREACHABLE();
return DiyFp(0, 0);
}
}
// If the function returns true then the result is the correct double.
// Otherwise it is either the correct double or the double that is just below
// the correct double.
static bool DiyFpStrtod(Vector<const char> buffer,
int exponent,
double* result) {
DiyFp input;
int remaining_decimals;
ReadDiyFp(buffer, &input, &remaining_decimals);
// Since we may have dropped some digits the input is not accurate.
// If remaining_decimals is different than 0 than the error is at most
// .5 ulp (unit in the last place).
// We don't want to deal with fractions and therefore keep a common
// denominator.
const int kDenominatorLog = 3;
const int kDenominator = 1 << kDenominatorLog;
// Move the remaining decimals into the exponent.
exponent += remaining_decimals;
int error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
int old_e = input.e();
input.Normalize();
error <<= old_e - input.e();
ASSERT(exponent <= PowersOfTenCache::kMaxDecimalExponent);
if (exponent < PowersOfTenCache::kMinDecimalExponent) {
*result = 0.0;
return true;
}
DiyFp cached_power;
int cached_decimal_exponent;
PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
&cached_power,
&cached_decimal_exponent);
if (cached_decimal_exponent != exponent) {
int adjustment_exponent = exponent - cached_decimal_exponent;
DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
input.Multiply(adjustment_power);
if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
// The product of input with the adjustment power fits into a 64 bit
// integer.
ASSERT(DiyFp::kSignificandSize == 64);
} else {
// The adjustment power is exact. There is hence only an error of 0.5.
error += kDenominator / 2;
}
}
input.Multiply(cached_power);
// The error introduced by a multiplication of a*b equals
// error_a + error_b + error_a*error_b/2^64 + 0.5
// Substituting a with 'input' and b with 'cached_power' we have
// error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
// error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
int error_b = kDenominator / 2;
int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
int fixed_error = kDenominator / 2;
error += error_b + error_ab + fixed_error;
old_e = input.e();
input.Normalize();
error <<= old_e - input.e();
// See if the double's significand changes if we add/subtract the error.
int order_of_magnitude = DiyFp::kSignificandSize + input.e();
int effective_significand_size =
Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
int precision_digits_count =
DiyFp::kSignificandSize - effective_significand_size;
if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
// This can only happen for very small denormals. In this case the
// half-way multiplied by the denominator exceeds the range of an uint64.
// Simply shift everything to the right.
int shift_amount = (precision_digits_count + kDenominatorLog) -
DiyFp::kSignificandSize + 1;
input.set_f(input.f() >> shift_amount);
input.set_e(input.e() + shift_amount);
// We add 1 for the lost precision of error, and kDenominator for
// the lost precision of input.f().
error = (error >> shift_amount) + 1 + kDenominator;
precision_digits_count -= shift_amount;
}
// We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
ASSERT(DiyFp::kSignificandSize == 64);
ASSERT(precision_digits_count < 64);
uint64_t one64 = 1;
uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
uint64_t precision_bits = input.f() & precision_bits_mask;
uint64_t half_way = one64 << (precision_digits_count - 1);
precision_bits *= kDenominator;
half_way *= kDenominator;
DiyFp rounded_input(input.f() >> precision_digits_count,
input.e() + precision_digits_count);
if (precision_bits >= half_way + error) {
rounded_input.set_f(rounded_input.f() + 1);
}
// If the last_bits are too close to the half-way case than we are too
// inaccurate and round down. In this case we return false so that we can
// fall back to a more precise algorithm.
*result = Double(rounded_input).value();
if (half_way - error < precision_bits && precision_bits < half_way + error) {
// Too imprecise. The caller will have to fall back to a slower version.
// However the returned number is guaranteed to be either the correct
// double, or the next-lower double.
return false;
} else {
return true;
}
}
// Returns the correct double for the buffer*10^exponent.
// The variable guess should be a close guess that is either the correct double
// or its lower neighbor (the nearest double less than the correct one).
// Preconditions:
// buffer.length() + exponent <= kMaxDecimalPower + 1
// buffer.length() + exponent > kMinDecimalPower
// buffer.length() <= kMaxDecimalSignificantDigits
static double BignumStrtod(Vector<const char> buffer,
int exponent,
double guess) {
if (guess == V8_INFINITY) {
return guess;
}
DiyFp upper_boundary = Double(guess).UpperBoundary();
ASSERT(buffer.length() + exponent <= kMaxDecimalPower + 1);
ASSERT(buffer.length() + exponent > kMinDecimalPower);
ASSERT(buffer.length() <= kMaxSignificantDecimalDigits);
// Make sure that the Bignum will be able to hold all our numbers.
// Our Bignum implementation has a separate field for exponents. Shifts will
// consume at most one bigit (< 64 bits).
// ln(10) == 3.3219...
ASSERT(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
Bignum input;
Bignum boundary;
input.AssignDecimalString(buffer);
boundary.AssignUInt64(upper_boundary.f());
if (exponent >= 0) {
input.MultiplyByPowerOfTen(exponent);
} else {
boundary.MultiplyByPowerOfTen(-exponent);
}
if (upper_boundary.e() > 0) {
boundary.ShiftLeft(upper_boundary.e());
} else {
input.ShiftLeft(-upper_boundary.e());
}
int comparison = Bignum::Compare(input, boundary);
if (comparison < 0) {
return guess;
} else if (comparison > 0) {
return Double(guess).NextDouble();
} else if ((Double(guess).Significand() & 1) == 0) {
// Round towards even.
return guess;
} else {
return Double(guess).NextDouble();
}
}
double Strtod(Vector<const char> buffer, int exponent) {
Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
Vector<const char> trimmed = TrimTrailingZeros(left_trimmed);
exponent += left_trimmed.length() - trimmed.length();
if (trimmed.length() == 0) return 0.0;
if (trimmed.length() > kMaxSignificantDecimalDigits) {
char significant_buffer[kMaxSignificantDecimalDigits];
int significant_exponent;
TrimToMaxSignificantDigits(trimmed, exponent,
significant_buffer, &significant_exponent);
return Strtod(Vector<const char>(significant_buffer,
kMaxSignificantDecimalDigits),
significant_exponent);
}
if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) return V8_INFINITY;
if (exponent + trimmed.length() <= kMinDecimalPower) return 0.0;
double guess;
if (DoubleStrtod(trimmed, exponent, &guess) ||
DiyFpStrtod(trimmed, exponent, &guess)) {
return guess;
}
return BignumStrtod(trimmed, exponent, guess);
}
} } // namespace v8::internal