| // 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 |
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| // (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 <climits> |
| #include <cstdarg> |
| |
| #include <double-conversion/bignum.h> |
| #include <double-conversion/cached-powers.h> |
| #include <double-conversion/ieee.h> |
| #include <double-conversion/strtod.h> |
| |
| namespace double_conversion { |
| |
| // 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 = UINT64_2PART_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 CutToMaxSignificantDigits(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); |
| } |
| |
| |
| // Trims the buffer and cuts it to at most kMaxSignificantDecimalDigits. |
| // If possible the input-buffer is reused, but if the buffer needs to be |
| // modified (due to cutting), then the input needs to be copied into the |
| // buffer_copy_space. |
| static void TrimAndCut(Vector<const char> buffer, int exponent, |
| char* buffer_copy_space, int space_size, |
| Vector<const char>* trimmed, int* updated_exponent) { |
| Vector<const char> left_trimmed = TrimLeadingZeros(buffer); |
| Vector<const char> right_trimmed = TrimTrailingZeros(left_trimmed); |
| exponent += left_trimmed.length() - right_trimmed.length(); |
| if (right_trimmed.length() > kMaxSignificantDecimalDigits) { |
| (void) space_size; // Mark variable as used. |
| ASSERT(space_size >= kMaxSignificantDecimalDigits); |
| CutToMaxSignificantDigits(right_trimmed, exponent, |
| buffer_copy_space, updated_exponent); |
| *trimmed = Vector<const char>(buffer_copy_space, |
| kMaxSignificantDecimalDigits); |
| } else { |
| *trimmed = right_trimmed; |
| *updated_exponent = exponent; |
| } |
| } |
| |
| |
| // 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(DOUBLE_CONVERSION_CORRECT_DOUBLE_OPERATIONS) |
| // 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 uses 64 bits and is therefore accurate. |
| // Note that the ARM simulator is compiled for 32bits. It therefore exhibits |
| // the same problem. |
| return false; |
| #else |
| 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; |
| #endif |
| } |
| |
| |
| // 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(UINT64_2PART_C(0xa0000000, 00000000), -60); |
| case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57); |
| case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54); |
| case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50); |
| case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47); |
| case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44); |
| case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40); |
| default: |
| UNREACHABLE(); |
| } |
| } |
| |
| |
| // 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; |
| uint64_t 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 |
| // - -1 if buffer*10^exponent < diy_fp. |
| // - 0 if buffer*10^exponent == diy_fp. |
| // - +1 if buffer*10^exponent > diy_fp. |
| // Preconditions: |
| // buffer.length() + exponent <= kMaxDecimalPower + 1 |
| // buffer.length() + exponent > kMinDecimalPower |
| // buffer.length() <= kMaxDecimalSignificantDigits |
| static int CompareBufferWithDiyFp(Vector<const char> buffer, |
| int exponent, |
| DiyFp diy_fp) { |
| 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 buffer_bignum; |
| Bignum diy_fp_bignum; |
| buffer_bignum.AssignDecimalString(buffer); |
| diy_fp_bignum.AssignUInt64(diy_fp.f()); |
| if (exponent >= 0) { |
| buffer_bignum.MultiplyByPowerOfTen(exponent); |
| } else { |
| diy_fp_bignum.MultiplyByPowerOfTen(-exponent); |
| } |
| if (diy_fp.e() > 0) { |
| diy_fp_bignum.ShiftLeft(diy_fp.e()); |
| } else { |
| buffer_bignum.ShiftLeft(-diy_fp.e()); |
| } |
| return Bignum::Compare(buffer_bignum, diy_fp_bignum); |
| } |
| |
| |
| // Returns true if the guess is the correct double. |
| // Returns false, when guess is either correct or the next-lower double. |
| static bool ComputeGuess(Vector<const char> trimmed, int exponent, |
| double* guess) { |
| if (trimmed.length() == 0) { |
| *guess = 0.0; |
| return true; |
| } |
| if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) { |
| *guess = Double::Infinity(); |
| return true; |
| } |
| if (exponent + trimmed.length() <= kMinDecimalPower) { |
| *guess = 0.0; |
| return true; |
| } |
| |
| if (DoubleStrtod(trimmed, exponent, guess) || |
| DiyFpStrtod(trimmed, exponent, guess)) { |
| return true; |
| } |
| if (*guess == Double::Infinity()) { |
| return true; |
| } |
| return false; |
| } |
| |
| double Strtod(Vector<const char> buffer, int exponent) { |
| char copy_buffer[kMaxSignificantDecimalDigits]; |
| Vector<const char> trimmed; |
| int updated_exponent; |
| TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits, |
| &trimmed, &updated_exponent); |
| exponent = updated_exponent; |
| |
| double guess; |
| bool is_correct = ComputeGuess(trimmed, exponent, &guess); |
| if (is_correct) return guess; |
| |
| DiyFp upper_boundary = Double(guess).UpperBoundary(); |
| int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_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(); |
| } |
| } |
| |
| static float SanitizedDoubletof(double d) { |
| ASSERT(d >= 0.0); |
| // ASAN has a sanitize check that disallows casting doubles to floats if |
| // they are too big. |
| // https://clang.llvm.org/docs/UndefinedBehaviorSanitizer.html#available-checks |
| // The behavior should be covered by IEEE 754, but some projects use this |
| // flag, so work around it. |
| float max_finite = 3.4028234663852885981170418348451692544e+38; |
| // The half-way point between the max-finite and infinity value. |
| // Since infinity has an even significand everything equal or greater than |
| // this value should become infinity. |
| double half_max_finite_infinity = |
| 3.40282356779733661637539395458142568448e+38; |
| if (d >= max_finite) { |
| if (d >= half_max_finite_infinity) { |
| return Single::Infinity(); |
| } else { |
| return max_finite; |
| } |
| } else { |
| return static_cast<float>(d); |
| } |
| } |
| |
| float Strtof(Vector<const char> buffer, int exponent) { |
| char copy_buffer[kMaxSignificantDecimalDigits]; |
| Vector<const char> trimmed; |
| int updated_exponent; |
| TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits, |
| &trimmed, &updated_exponent); |
| exponent = updated_exponent; |
| |
| double double_guess; |
| bool is_correct = ComputeGuess(trimmed, exponent, &double_guess); |
| |
| float float_guess = SanitizedDoubletof(double_guess); |
| if (float_guess == double_guess) { |
| // This shortcut triggers for integer values. |
| return float_guess; |
| } |
| |
| // We must catch double-rounding. Say the double has been rounded up, and is |
| // now a boundary of a float, and rounds up again. This is why we have to |
| // look at previous too. |
| // Example (in decimal numbers): |
| // input: 12349 |
| // high-precision (4 digits): 1235 |
| // low-precision (3 digits): |
| // when read from input: 123 |
| // when rounded from high precision: 124. |
| // To do this we simply look at the neigbors of the correct result and see |
| // if they would round to the same float. If the guess is not correct we have |
| // to look at four values (since two different doubles could be the correct |
| // double). |
| |
| double double_next = Double(double_guess).NextDouble(); |
| double double_previous = Double(double_guess).PreviousDouble(); |
| |
| float f1 = SanitizedDoubletof(double_previous); |
| float f2 = float_guess; |
| float f3 = SanitizedDoubletof(double_next); |
| float f4; |
| if (is_correct) { |
| f4 = f3; |
| } else { |
| double double_next2 = Double(double_next).NextDouble(); |
| f4 = SanitizedDoubletof(double_next2); |
| } |
| (void) f2; // Mark variable as used. |
| ASSERT(f1 <= f2 && f2 <= f3 && f3 <= f4); |
| |
| // If the guess doesn't lie near a single-precision boundary we can simply |
| // return its float-value. |
| if (f1 == f4) { |
| return float_guess; |
| } |
| |
| ASSERT((f1 != f2 && f2 == f3 && f3 == f4) || |
| (f1 == f2 && f2 != f3 && f3 == f4) || |
| (f1 == f2 && f2 == f3 && f3 != f4)); |
| |
| // guess and next are the two possible candidates (in the same way that |
| // double_guess was the lower candidate for a double-precision guess). |
| float guess = f1; |
| float next = f4; |
| DiyFp upper_boundary; |
| if (guess == 0.0f) { |
| float min_float = 1e-45f; |
| upper_boundary = Double(static_cast<double>(min_float) / 2).AsDiyFp(); |
| } else { |
| upper_boundary = Single(guess).UpperBoundary(); |
| } |
| int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary); |
| if (comparison < 0) { |
| return guess; |
| } else if (comparison > 0) { |
| return next; |
| } else if ((Single(guess).Significand() & 1) == 0) { |
| // Round towards even. |
| return guess; |
| } else { |
| return next; |
| } |
| } |
| |
| } // namespace double_conversion |