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 // Copyright 2012 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/numbers/strtod.h" #include #include #include "src/common/globals.h" #include "src/numbers/bignum.h" #include "src/numbers/cached-powers.h" #include "src/numbers/double.h" #include "src/utils/utils.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 = 0xFFFF'FFFF'FFFF'FFFF; // clang-format off 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 }; // clang-format on static const int kExactPowersOfTenSize = arraysize(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 TrimLeadingZeros(Vector buffer) { for (int i = 0; i < buffer.length(); i++) { if (buffer[i] != '0') { return buffer.SubVector(i, buffer.length()); } } return Vector(buffer.begin(), 0); } static Vector TrimTrailingZeros(Vector buffer) { for (int i = buffer.length() - 1; i >= 0; --i) { if (buffer[i] != '0') { return buffer.SubVector(0, i + 1); } } return Vector(buffer.begin(), 0); } static void TrimToMaxSignificantDigits(Vector 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'. DCHECK_NE(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 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'; DCHECK(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 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 trimmed, int exponent, double* result) { #if (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. USE(exact_powers_of_ten); USE(kMaxExactDoubleIntegerDecimalDigits); USE(kExactPowersOfTenSize); 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(ReadUint64(trimmed, &read_digits)); DCHECK(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(ReadUint64(trimmed, &read_digits)); DCHECK(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(ReadUint64(trimmed, &read_digits)); DCHECK(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) { DCHECK_LT(0, exponent); DCHECK_LT(exponent, PowersOfTenCache::kDecimalExponentDistance); // Simply hardcode the remaining powers for the given decimal exponent // distance. DCHECK_EQ(PowersOfTenCache::kDecimalExponentDistance, 8); switch (exponent) { case 1: return DiyFp(0xA000'0000'0000'0000, -60); case 2: return DiyFp(0xC800'0000'0000'0000, -57); case 3: return DiyFp(0xFA00'0000'0000'0000, -54); case 4: return DiyFp(0x9C40'0000'0000'0000, -50); case 5: return DiyFp(0xC350'0000'0000'0000, -47); case 6: return DiyFp(0xF424'0000'0000'0000, -44); case 7: return DiyFp(0x9896'8000'0000'0000, -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 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; int64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2); int old_e = input.e(); input.Normalize(); error <<= old_e - input.e(); DCHECK_LE(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. DCHECK_EQ(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. DCHECK_EQ(DiyFp::kSignificandSize, 64); DCHECK_LT(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 buffer, int exponent, double guess) { if (guess == V8_INFINITY) { return guess; } DiyFp upper_boundary = Double(guess).UpperBoundary(); DCHECK(buffer.length() + exponent <= kMaxDecimalPower + 1); DCHECK_GT(buffer.length() + exponent, kMinDecimalPower); DCHECK_LE(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... DCHECK_LT((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 buffer, int exponent) { Vector left_trimmed = TrimLeadingZeros(buffer); Vector 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(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 internal } // namespace v8