// 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/fixed-dtoa.h" #include <stdint.h> #include <cmath> #include "src/base/logging.h" #include "src/base/numbers/double.h" namespace v8 { namespace base { // Represents a 128bit type. This class should be replaced by a native type on // platforms that support 128bit integers. class UInt128 { public: UInt128() : high_bits_(0), low_bits_(0) {} UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) {} void Multiply(uint32_t multiplicand) { uint64_t accumulator; accumulator = (low_bits_ & kMask32) * multiplicand; uint32_t part = static_cast<uint32_t>(accumulator & kMask32); accumulator >>= 32; accumulator = accumulator + (low_bits_ >> 32) * multiplicand; low_bits_ = (accumulator << 32) + part; accumulator >>= 32; accumulator = accumulator + (high_bits_ & kMask32) * multiplicand; part = static_cast<uint32_t>(accumulator & kMask32); accumulator >>= 32; accumulator = accumulator + (high_bits_ >> 32) * multiplicand; high_bits_ = (accumulator << 32) + part; DCHECK_EQ(accumulator >> 32, 0); } void Shift(int shift_amount) { DCHECK(-64 <= shift_amount && shift_amount <= 64); if (shift_amount == 0) { return; } else if (shift_amount == -64) { high_bits_ = low_bits_; low_bits_ = 0; } else if (shift_amount == 64) { low_bits_ = high_bits_; high_bits_ = 0; } else if (shift_amount <= 0) { high_bits_ <<= -shift_amount; high_bits_ += low_bits_ >> (64 + shift_amount); low_bits_ <<= -shift_amount; } else { low_bits_ >>= shift_amount; low_bits_ += high_bits_ << (64 - shift_amount); high_bits_ >>= shift_amount; } } // Modifies *this to *this MOD (2^power). // Returns *this DIV (2^power). int DivModPowerOf2(int power) { if (power >= 64) { int result = static_cast<int>(high_bits_ >> (power - 64)); high_bits_ -= static_cast<uint64_t>(result) << (power - 64); return result; } else { uint64_t part_low = low_bits_ >> power; uint64_t part_high = high_bits_ << (64 - power); int result = static_cast<int>(part_low + part_high); high_bits_ = 0; low_bits_ -= part_low << power; return result; } } bool IsZero() const { return high_bits_ == 0 && low_bits_ == 0; } int BitAt(int position) { if (position >= 64) { return static_cast<int>(high_bits_ >> (position - 64)) & 1; } else { return static_cast<int>(low_bits_ >> position) & 1; } } private: static const uint64_t kMask32 = 0xFFFFFFFF; // Value == (high_bits_ << 64) + low_bits_ uint64_t high_bits_; uint64_t low_bits_; }; static const int kDoubleSignificandSize = 53; // Includes the hidden bit. static void FillDigits32FixedLength(uint32_t number, int requested_length, Vector<char> buffer, int* length) { for (int i = requested_length - 1; i >= 0; --i) { buffer[(*length) + i] = '0' + number % 10; number /= 10; } *length += requested_length; } static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) { int number_length = 0; // We fill the digits in reverse order and exchange them afterwards. while (number != 0) { int digit = number % 10; number /= 10; buffer[(*length) + number_length] = '0' + digit; number_length++; } // Exchange the digits. int i = *length; int j = *length + number_length - 1; while (i < j) { char tmp = buffer[i]; buffer[i] = buffer[j]; buffer[j] = tmp; i++; j--; } *length += number_length; } static void FillDigits64FixedLength(uint64_t number, int requested_length, Vector<char> buffer, int* length) { const uint32_t kTen7 = 10000000; // For efficiency cut the number into 3 uint32_t parts, and print those. uint32_t part2 = static_cast<uint32_t>(number % kTen7); number /= kTen7; uint32_t part1 = static_cast<uint32_t>(number % kTen7); uint32_t part0 = static_cast<uint32_t>(number / kTen7); FillDigits32FixedLength(part0, 3, buffer, length); FillDigits32FixedLength(part1, 7, buffer, length); FillDigits32FixedLength(part2, 7, buffer, length); } static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) { const uint32_t kTen7 = 10000000; // For efficiency cut the number into 3 uint32_t parts, and print those. uint32_t part2 = static_cast<uint32_t>(number % kTen7); number /= kTen7; uint32_t part1 = static_cast<uint32_t>(number % kTen7); uint32_t part0 = static_cast<uint32_t>(number / kTen7); if (part0 != 0) { FillDigits32(part0, buffer, length); FillDigits32FixedLength(part1, 7, buffer, length); FillDigits32FixedLength(part2, 7, buffer, length); } else if (part1 != 0) { FillDigits32(part1, buffer, length); FillDigits32FixedLength(part2, 7, buffer, length); } else { FillDigits32(part2, buffer, length); } } static void DtoaRoundUp(Vector<char> buffer, int* length, int* decimal_point) { // An empty buffer represents 0. if (*length == 0) { buffer[0] = '1'; *decimal_point = 1; *length = 1; return; } // Round the last digit until we either have a digit that was not '9' or until // we reached the first digit. buffer[(*length) - 1]++; for (int i = (*length) - 1; i > 0; --i) { if (buffer[i] != '0' + 10) { return; } buffer[i] = '0'; buffer[i - 1]++; } // If the first digit is now '0' + 10, we would need to set it to '0' and add // a '1' in front. However we reach the first digit only if all following // digits had been '9' before rounding up. Now all trailing digits are '0' and // we simply switch the first digit to '1' and update the decimal-point // (indicating that the point is now one digit to the right). if (buffer[0] == '0' + 10) { buffer[0] = '1'; (*decimal_point)++; } } // The given fractionals number represents a fixed-point number with binary // point at bit (-exponent). // Preconditions: // -128 <= exponent <= 0. // 0 <= fractionals * 2^exponent < 1 // The buffer holds the result. // The function will round its result. During the rounding-process digits not // generated by this function might be updated, and the decimal-point variable // might be updated. If this function generates the digits 99 and the buffer // already contained "199" (thus yielding a buffer of "19999") then a // rounding-up will change the contents of the buffer to "20000". static void FillFractionals(uint64_t fractionals, int exponent, int fractional_count, Vector<char> buffer, int* length, int* decimal_point) { DCHECK(-128 <= exponent && exponent <= 0); // 'fractionals' is a fixed-point number, with binary point at bit // (-exponent). Inside the function the non-converted remainder of fractionals // is a fixed-point number, with binary point at bit 'point'. if (-exponent <= 64) { // One 64 bit number is sufficient. DCHECK_EQ(fractionals >> 56, 0); int point = -exponent; for (int i = 0; i < fractional_count; ++i) { if (fractionals == 0) break; // Instead of multiplying by 10 we multiply by 5 and adjust the point // location. This way the fractionals variable will not overflow. // Invariant at the beginning of the loop: fractionals < 2^point. // Initially we have: point <= 64 and fractionals < 2^56 // After each iteration the point is decremented by one. // Note that 5^3 = 125 < 128 = 2^7. // Therefore three iterations of this loop will not overflow fractionals // (even without the subtraction at the end of the loop body). At this // time point will satisfy point <= 61 and therefore fractionals < 2^point // and any further multiplication of fractionals by 5 will not overflow. fractionals *= 5; point--; int digit = static_cast<int>(fractionals >> point); buffer[*length] = '0' + digit; (*length)++; fractionals -= static_cast<uint64_t>(digit) << point; } // If the first bit after the point is set we have to round up. if (point > 0 && ((fractionals >> (point - 1)) & 1) == 1) { DtoaRoundUp(buffer, length, decimal_point); } } else { // We need 128 bits. DCHECK(64 < -exponent && -exponent <= 128); UInt128 fractionals128 = UInt128(fractionals, 0); fractionals128.Shift(-exponent - 64); int point = 128; for (int i = 0; i < fractional_count; ++i) { if (fractionals128.IsZero()) break; // As before: instead of multiplying by 10 we multiply by 5 and adjust the // point location. // This multiplication will not overflow for the same reasons as before. fractionals128.Multiply(5); point--; int digit = fractionals128.DivModPowerOf2(point); buffer[*length] = '0' + digit; (*length)++; } if (fractionals128.BitAt(point - 1) == 1) { DtoaRoundUp(buffer, length, decimal_point); } } } // Removes leading and trailing zeros. // If leading zeros are removed then the decimal point position is adjusted. static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) { while (*length > 0 && buffer[(*length) - 1] == '0') { (*length)--; } int first_non_zero = 0; while (first_non_zero < *length && buffer[first_non_zero] == '0') { first_non_zero++; } if (first_non_zero != 0) { for (int i = first_non_zero; i < *length; ++i) { buffer[i - first_non_zero] = buffer[i]; } *length -= first_non_zero; *decimal_point -= first_non_zero; } } bool FastFixedDtoa(double v, int fractional_count, Vector<char> buffer, int* length, int* decimal_point) { const uint32_t kMaxUInt32 = 0xFFFFFFFF; uint64_t significand = Double(v).Significand(); int exponent = Double(v).Exponent(); // v = significand * 2^exponent (with significand a 53bit integer). // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we // don't know how to compute the representation. 2^73 ~= 9.5*10^21. // If necessary this limit could probably be increased, but we don't need // more. if (exponent > 20) return false; if (fractional_count > 20) return false; *length = 0; // At most kDoubleSignificandSize bits of the significand are non-zero. // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero // bits: 0..11*..0xxx..53*..xx if (exponent + kDoubleSignificandSize > 64) { // The exponent must be > 11. // // We know that v = significand * 2^exponent. // And the exponent > 11. // We simplify the task by dividing v by 10^17. // The quotient delivers the first digits, and the remainder fits into a 64 // bit number. // Dividing by 10^17 is equivalent to dividing by 5^17*2^17. const uint64_t kFive17 = 0xB1'A2BC'2EC5; // 5^17 uint64_t divisor = kFive17; int divisor_power = 17; uint64_t dividend = significand; uint32_t quotient; uint64_t remainder; // Let v = f * 2^e with f == significand and e == exponent. // Then need q (quotient) and r (remainder) as follows: // v = q * 10^17 + r // f * 2^e = q * 10^17 + r // f * 2^e = q * 5^17 * 2^17 + r // If e > 17 then // f * 2^(e-17) = q * 5^17 + r/2^17 // else // f = q * 5^17 * 2^(17-e) + r/2^e if (exponent > divisor_power) { // We only allow exponents of up to 20 and therefore (17 - e) <= 3 dividend <<= exponent - divisor_power; quotient = static_cast<uint32_t>(dividend / divisor); remainder = (dividend % divisor) << divisor_power; } else { divisor <<= divisor_power - exponent; quotient = static_cast<uint32_t>(dividend / divisor); remainder = (dividend % divisor) << exponent; } FillDigits32(quotient, buffer, length); FillDigits64FixedLength(remainder, divisor_power, buffer, length); *decimal_point = *length; } else if (exponent >= 0) { // 0 <= exponent <= 11 significand <<= exponent; FillDigits64(significand, buffer, length); *decimal_point = *length; } else if (exponent > -kDoubleSignificandSize) { // We have to cut the number. uint64_t integrals = significand >> -exponent; uint64_t fractionals = significand - (integrals << -exponent); if (integrals > kMaxUInt32) { FillDigits64(integrals, buffer, length); } else { FillDigits32(static_cast<uint32_t>(integrals), buffer, length); } *decimal_point = *length; FillFractionals(fractionals, exponent, fractional_count, buffer, length, decimal_point); } else if (exponent < -128) { // This configuration (with at most 20 digits) means that all digits must be // 0. DCHECK_LE(fractional_count, 20); buffer[0] = '\0'; *length = 0; *decimal_point = -fractional_count; } else { *decimal_point = 0; FillFractionals(significand, exponent, fractional_count, buffer, length, decimal_point); } TrimZeros(buffer, length, decimal_point); buffer[*length] = '\0'; if ((*length) == 0) { // The string is empty and the decimal_point thus has no importance. Mimick // Gay's dtoa and and set it to -fractional_count. *decimal_point = -fractional_count; } return true; } } // namespace base } // namespace v8