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// Copyright 2010 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 "cached-powers.h"
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namespace double_conversion {
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// 2^53 = 9007199254740992.
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// Any integer with at most 15 decimal digits will hence fit into a double
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// (which has a 53bit significand) without loss of precision.
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static const int kMaxExactDoubleIntegerDecimalDigits = 15;
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// 2^64 = 18446744073709551616 > 10^19
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static const int kMaxUint64DecimalDigits = 19;
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// Max double: 1.7976931348623157 x 10^308
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// Min non-zero double: 4.9406564584124654 x 10^-324
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// Any x >= 10^309 is interpreted as +infinity.
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// Any x <= 10^-324 is interpreted as 0.
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// Note that 2.5e-324 (despite being smaller than the min double) will be read
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// as non-zero (equal to the min non-zero double).
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static const int kMaxDecimalPower = 309;
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static const int kMinDecimalPower = -324;
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// 2^64 = 18446744073709551616
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static const uint64_t kMaxUint64 = UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF);
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static const double exact_powers_of_ten[] = {
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10000000000.0, // 10^10
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1000000000000000000.0,
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10000000000000000000.0,
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100000000000000000000.0, // 10^20
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1000000000000000000000.0,
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// 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
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10000000000000000000000.0
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static const int kExactPowersOfTenSize = ARRAY_SIZE(exact_powers_of_ten);
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// Maximum number of significant digits in the decimal representation.
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// In fact the value is 772 (see conversions.cc), but to give us some margin
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// we round up to 780.
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static const int kMaxSignificantDecimalDigits = 780;
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static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
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for (int i = 0; i < buffer.length(); i++) {
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if (buffer[i] != '0') {
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return buffer.SubVector(i, buffer.length());
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return Vector<const char>(buffer.start(), 0);
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static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
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for (int i = buffer.length() - 1; i >= 0; --i) {
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if (buffer[i] != '0') {
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return buffer.SubVector(0, i + 1);
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return Vector<const char>(buffer.start(), 0);
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static void CutToMaxSignificantDigits(Vector<const char> buffer,
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char* significant_buffer,
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int* significant_exponent) {
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for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
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significant_buffer[i] = buffer[i];
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// The input buffer has been trimmed. Therefore the last digit must be
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// different from '0'.
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ASSERT(buffer[buffer.length() - 1] != '0');
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// Set the last digit to be non-zero. This is sufficient to guarantee
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significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
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*significant_exponent =
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exponent + (buffer.length() - kMaxSignificantDecimalDigits);
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// Trims the buffer and cuts it to at most kMaxSignificantDecimalDigits.
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// If possible the input-buffer is reused, but if the buffer needs to be
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// modified (due to cutting), then the input needs to be copied into the
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// buffer_copy_space.
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static void TrimAndCut(Vector<const char> buffer, int exponent,
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char* buffer_copy_space, int space_size,
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Vector<const char>* trimmed, int* updated_exponent) {
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Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
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Vector<const char> right_trimmed = TrimTrailingZeros(left_trimmed);
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exponent += left_trimmed.length() - right_trimmed.length();
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if (right_trimmed.length() > kMaxSignificantDecimalDigits) {
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ASSERT(space_size >= kMaxSignificantDecimalDigits);
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CutToMaxSignificantDigits(right_trimmed, exponent,
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buffer_copy_space, updated_exponent);
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*trimmed = Vector<const char>(buffer_copy_space,
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kMaxSignificantDecimalDigits);
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*trimmed = right_trimmed;
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*updated_exponent = exponent;
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// Reads digits from the buffer and converts them to a uint64.
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// Reads in as many digits as fit into a uint64.
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// When the string starts with "1844674407370955161" no further digit is read.
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// Since 2^64 = 18446744073709551616 it would still be possible read another
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// digit if it was less or equal than 6, but this would complicate the code.
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static uint64_t ReadUint64(Vector<const char> buffer,
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int* number_of_read_digits) {
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while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
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int digit = buffer[i++] - '0';
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ASSERT(0 <= digit && digit <= 9);
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result = 10 * result + digit;
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*number_of_read_digits = i;
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// Reads a DiyFp from the buffer.
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// The returned DiyFp is not necessarily normalized.
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// If remaining_decimals is zero then the returned DiyFp is accurate.
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// Otherwise it has been rounded and has error of at most 1/2 ulp.
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static void ReadDiyFp(Vector<const char> buffer,
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int* remaining_decimals) {
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uint64_t significand = ReadUint64(buffer, &read_digits);
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if (buffer.length() == read_digits) {
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*result = DiyFp(significand, 0);
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*remaining_decimals = 0;
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// Round the significand.
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if (buffer[read_digits] >= '5') {
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// Compute the binary exponent.
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*result = DiyFp(significand, exponent);
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*remaining_decimals = buffer.length() - read_digits;
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static bool DoubleStrtod(Vector<const char> trimmed,
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#if !defined(DOUBLE_CONVERSION_CORRECT_DOUBLE_OPERATIONS)
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// On x86 the floating-point stack can be 64 or 80 bits wide. If it is
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// 80 bits wide (as is the case on Linux) then double-rounding occurs and the
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// result is not accurate.
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// We know that Windows32 uses 64 bits and is therefore accurate.
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// Note that the ARM simulator is compiled for 32bits. It therefore exhibits
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if (trimmed.length() <= kMaxExactDoubleIntegerDecimalDigits) {
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// The trimmed input fits into a double.
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// If the 10^exponent (resp. 10^-exponent) fits into a double too then we
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// can compute the result-double simply by multiplying (resp. dividing) the
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// This is possible because IEEE guarantees that floating-point operations
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// return the best possible approximation.
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if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
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// 10^-exponent fits into a double.
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*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
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ASSERT(read_digits == trimmed.length());
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*result /= exact_powers_of_ten[-exponent];
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if (0 <= exponent && exponent < kExactPowersOfTenSize) {
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// 10^exponent fits into a double.
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*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
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ASSERT(read_digits == trimmed.length());
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*result *= exact_powers_of_ten[exponent];
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int remaining_digits =
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kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
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if ((0 <= exponent) &&
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(exponent - remaining_digits < kExactPowersOfTenSize)) {
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// The trimmed string was short and we can multiply it with
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// 10^remaining_digits. As a result the remaining exponent now fits
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// into a double too.
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*result = static_cast<double>(ReadUint64(trimmed, &read_digits));
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ASSERT(read_digits == trimmed.length());
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*result *= exact_powers_of_ten[remaining_digits];
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*result *= exact_powers_of_ten[exponent - remaining_digits];
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// Returns 10^exponent as an exact DiyFp.
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// The given exponent must be in the range [1; kDecimalExponentDistance[.
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static DiyFp AdjustmentPowerOfTen(int exponent) {
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ASSERT(0 < exponent);
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ASSERT(exponent < PowersOfTenCache::kDecimalExponentDistance);
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// Simply hardcode the remaining powers for the given decimal exponent
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ASSERT(PowersOfTenCache::kDecimalExponentDistance == 8);
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case 1: return DiyFp(UINT64_2PART_C(0xa0000000, 00000000), -60);
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case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57);
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case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54);
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case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50);
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case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47);
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case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44);
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case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40);
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// If the function returns true then the result is the correct double.
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// Otherwise it is either the correct double or the double that is just below
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// the correct double.
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static bool DiyFpStrtod(Vector<const char> buffer,
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int remaining_decimals;
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ReadDiyFp(buffer, &input, &remaining_decimals);
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// Since we may have dropped some digits the input is not accurate.
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// If remaining_decimals is different than 0 than the error is at most
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// .5 ulp (unit in the last place).
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// We don't want to deal with fractions and therefore keep a common
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const int kDenominatorLog = 3;
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const int kDenominator = 1 << kDenominatorLog;
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// Move the remaining decimals into the exponent.
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exponent += remaining_decimals;
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int error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
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int old_e = input.e();
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error <<= old_e - input.e();
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ASSERT(exponent <= PowersOfTenCache::kMaxDecimalExponent);
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if (exponent < PowersOfTenCache::kMinDecimalExponent) {
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int cached_decimal_exponent;
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PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
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&cached_decimal_exponent);
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if (cached_decimal_exponent != exponent) {
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int adjustment_exponent = exponent - cached_decimal_exponent;
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DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
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input.Multiply(adjustment_power);
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if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
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// The product of input with the adjustment power fits into a 64 bit
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ASSERT(DiyFp::kSignificandSize == 64);
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// The adjustment power is exact. There is hence only an error of 0.5.
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error += kDenominator / 2;
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input.Multiply(cached_power);
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// The error introduced by a multiplication of a*b equals
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// error_a + error_b + error_a*error_b/2^64 + 0.5
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// Substituting a with 'input' and b with 'cached_power' we have
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// error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
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// error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
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int error_b = kDenominator / 2;
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int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
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int fixed_error = kDenominator / 2;
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error += error_b + error_ab + fixed_error;
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error <<= old_e - input.e();
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// See if the double's significand changes if we add/subtract the error.
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int order_of_magnitude = DiyFp::kSignificandSize + input.e();
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int effective_significand_size =
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Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
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int precision_digits_count =
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DiyFp::kSignificandSize - effective_significand_size;
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if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
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// This can only happen for very small denormals. In this case the
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// half-way multiplied by the denominator exceeds the range of an uint64.
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// Simply shift everything to the right.
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int shift_amount = (precision_digits_count + kDenominatorLog) -
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DiyFp::kSignificandSize + 1;
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input.set_f(input.f() >> shift_amount);
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input.set_e(input.e() + shift_amount);
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// We add 1 for the lost precision of error, and kDenominator for
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// the lost precision of input.f().
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error = (error >> shift_amount) + 1 + kDenominator;
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precision_digits_count -= shift_amount;
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// We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
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ASSERT(DiyFp::kSignificandSize == 64);
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ASSERT(precision_digits_count < 64);
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uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
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uint64_t precision_bits = input.f() & precision_bits_mask;
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uint64_t half_way = one64 << (precision_digits_count - 1);
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precision_bits *= kDenominator;
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half_way *= kDenominator;
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DiyFp rounded_input(input.f() >> precision_digits_count,
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input.e() + precision_digits_count);
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if (precision_bits >= half_way + error) {
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rounded_input.set_f(rounded_input.f() + 1);
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// If the last_bits are too close to the half-way case than we are too
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// inaccurate and round down. In this case we return false so that we can
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// fall back to a more precise algorithm.
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*result = Double(rounded_input).value();
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if (half_way - error < precision_bits && precision_bits < half_way + error) {
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// Too imprecise. The caller will have to fall back to a slower version.
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// However the returned number is guaranteed to be either the correct
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// double, or the next-lower double.
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// - -1 if buffer*10^exponent < diy_fp.
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// - 0 if buffer*10^exponent == diy_fp.
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// - +1 if buffer*10^exponent > diy_fp.
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// buffer.length() + exponent <= kMaxDecimalPower + 1
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// buffer.length() + exponent > kMinDecimalPower
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// buffer.length() <= kMaxDecimalSignificantDigits
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static int CompareBufferWithDiyFp(Vector<const char> buffer,
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ASSERT(buffer.length() + exponent <= kMaxDecimalPower + 1);
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ASSERT(buffer.length() + exponent > kMinDecimalPower);
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ASSERT(buffer.length() <= kMaxSignificantDecimalDigits);
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// Make sure that the Bignum will be able to hold all our numbers.
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// Our Bignum implementation has a separate field for exponents. Shifts will
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// consume at most one bigit (< 64 bits).
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// ln(10) == 3.3219...
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ASSERT(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
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Bignum buffer_bignum;
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Bignum diy_fp_bignum;
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buffer_bignum.AssignDecimalString(buffer);
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diy_fp_bignum.AssignUInt64(diy_fp.f());
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buffer_bignum.MultiplyByPowerOfTen(exponent);
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diy_fp_bignum.MultiplyByPowerOfTen(-exponent);
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if (diy_fp.e() > 0) {
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diy_fp_bignum.ShiftLeft(diy_fp.e());
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buffer_bignum.ShiftLeft(-diy_fp.e());
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return Bignum::Compare(buffer_bignum, diy_fp_bignum);
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// Returns true if the guess is the correct double.
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// Returns false, when guess is either correct or the next-lower double.
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static bool ComputeGuess(Vector<const char> trimmed, int exponent,
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if (trimmed.length() == 0) {
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if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) {
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*guess = Double::Infinity();
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if (exponent + trimmed.length() <= kMinDecimalPower) {
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if (DoubleStrtod(trimmed, exponent, guess) ||
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DiyFpStrtod(trimmed, exponent, guess)) {
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if (*guess == Double::Infinity()) {
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double Strtod(Vector<const char> buffer, int exponent) {
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char copy_buffer[kMaxSignificantDecimalDigits];
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Vector<const char> trimmed;
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int updated_exponent;
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TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
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&trimmed, &updated_exponent);
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exponent = updated_exponent;
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bool is_correct = ComputeGuess(trimmed, exponent, &guess);
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if (is_correct) return guess;
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DiyFp upper_boundary = Double(guess).UpperBoundary();
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int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
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if (comparison < 0) {
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} else if (comparison > 0) {
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return Double(guess).NextDouble();
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} else if ((Double(guess).Significand() & 1) == 0) {
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// Round towards even.
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return Double(guess).NextDouble();
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float Strtof(Vector<const char> buffer, int exponent) {
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char copy_buffer[kMaxSignificantDecimalDigits];
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Vector<const char> trimmed;
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int updated_exponent;
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TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
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&trimmed, &updated_exponent);
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exponent = updated_exponent;
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bool is_correct = ComputeGuess(trimmed, exponent, &double_guess);
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float float_guess = static_cast<float>(double_guess);
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if (float_guess == double_guess) {
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// This shortcut triggers for integer values.
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// We must catch double-rounding. Say the double has been rounded up, and is
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// now a boundary of a float, and rounds up again. This is why we have to
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// look at previous too.
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// Example (in decimal numbers):
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// high-precision (4 digits): 1235
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// low-precision (3 digits):
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// when read from input: 123
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// when rounded from high precision: 124.
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// To do this we simply look at the neigbors of the correct result and see
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// if they would round to the same float. If the guess is not correct we have
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// to look at four values (since two different doubles could be the correct
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double double_next = Double(double_guess).NextDouble();
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double double_previous = Double(double_guess).PreviousDouble();
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float f1 = static_cast<float>(double_previous);
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float f2 = float_guess;
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float f3 = static_cast<float>(double_next);
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double double_next2 = Double(double_next).NextDouble();
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f4 = static_cast<float>(double_next2);
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assert(f1 <= f2 && f2 <= f3 && f3 <= f4);
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// If the guess doesn't lie near a single-precision boundary we can simply
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// return its float-value.
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assert((f1 != f2 && f2 == f3 && f3 == f4) ||
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(f1 == f2 && f2 != f3 && f3 == f4) ||
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(f1 == f2 && f2 == f3 && f3 != f4));
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// guess and next are the two possible canditates (in the same way that
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// double_guess was the lower candidate for a double-precision guess).
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DiyFp upper_boundary;
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float min_float = 1e-45f;
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upper_boundary = Double(static_cast<double>(min_float) / 2).AsDiyFp();
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upper_boundary = Single(guess).UpperBoundary();
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int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
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if (comparison < 0) {
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} else if (comparison > 0) {
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} else if ((Single(guess).Significand() & 1) == 0) {
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// Round towards even.
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} // namespace double_conversion
added
removed
mozilla/mfbt/double-conversion/strtod.cc
