fixed-dtoa.cc

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// Copyright 2011 the V8 project authors. All rights reserved.
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#include <cmath>

#include "../include/v8stdint.h"
#include "checks.h"
#include "utils.h"

#include "double.h"
#include "fixed-dtoa.h"

namespace v8 {
namespace internal {

// 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;
    ASSERT((accumulator >> 32) == 0);
  }

  void Shift(int shift_amount) {
    ASSERT(-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 RoundUp(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) {
  ASSERT(-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.
    ASSERT(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 (((fractionals >> (point - 1)) & 1) == 1) {
      RoundUp(buffer, length, decimal_point);
    }
  } else {  // We need 128 bits.
    ASSERT(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) {
      RoundUp(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 = V8_2PART_UINT64_C(0xB1, A2BC2EC5);  // 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.
    ASSERT(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 v8::internal