bignum.cc

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// Copyright 2010 the V8 project authors. All rights reserved.
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#include <algorithm>
#include <cstring>
 
#include "bignum.h"
#include "utils.h"
 
namespace double_conversion {
 
Bignum::Chunk& Bignum::RawBigit(const int index) {
  DOUBLE_CONVERSION_ASSERT(static_cast<unsigned>(index) < kBigitCapacity);
  return bigits_buffer_[index];
}
 
 
const Bignum::Chunk& Bignum::RawBigit(const int index) const {
  DOUBLE_CONVERSION_ASSERT(static_cast<unsigned>(index) < kBigitCapacity);
  return bigits_buffer_[index];
}
 
 
template<typename S>
static int BitSize(const S value) {
  (void) value;  // Mark variable as used.
  return 8 * sizeof(value);
}
 
// Guaranteed to lie in one Bigit.
void Bignum::AssignUInt16(const uint16_t value) {
  DOUBLE_CONVERSION_ASSERT(kBigitSize >= BitSize(value));
  Zero();
  if (value > 0) {
    RawBigit(0) = value;
    used_bigits_ = 1;
  }
}
 
 
void Bignum::AssignUInt64(uint64_t value) {
  Zero();
  for(int i = 0; value > 0; ++i) {
    RawBigit(i) = value & kBigitMask;
    value >>= kBigitSize;
    ++used_bigits_;
  }
}
 
 
void Bignum::AssignBignum(const Bignum& other) {
  exponent_ = other.exponent_;
  for (int i = 0; i < other.used_bigits_; ++i) {
    RawBigit(i) = other.RawBigit(i);
  }
  used_bigits_ = other.used_bigits_;
}
 
 
static uint64_t ReadUInt64(const Vector<const char> buffer,
                           const int from,
                           const int digits_to_read) {
  uint64_t result = 0;
  for (int i = from; i < from + digits_to_read; ++i) {
    const int digit = buffer[i] - '0';
    DOUBLE_CONVERSION_ASSERT(0 <= digit && digit <= 9);
    result = result * 10 + digit;
  }
  return result;
}
 
 
void Bignum::AssignDecimalString(const Vector<const char> value) {
  // 2^64 = 18446744073709551616 > 10^19
  static const int kMaxUint64DecimalDigits = 19;
  Zero();
  int length = value.length();
  unsigned pos = 0;
  // Let's just say that each digit needs 4 bits.
  while (length >= kMaxUint64DecimalDigits) {
    const uint64_t digits = ReadUInt64(value, pos, kMaxUint64DecimalDigits);
    pos += kMaxUint64DecimalDigits;
    length -= kMaxUint64DecimalDigits;
    MultiplyByPowerOfTen(kMaxUint64DecimalDigits);
    AddUInt64(digits);
  }
  const uint64_t digits = ReadUInt64(value, pos, length);
  MultiplyByPowerOfTen(length);
  AddUInt64(digits);
  Clamp();
}
 
 
static uint64_t HexCharValue(const int c) {
  if ('0' <= c && c <= '9') {
    return c - '0';
  }
  if ('a' <= c && c <= 'f') {
    return 10 + c - 'a';
  }
  DOUBLE_CONVERSION_ASSERT('A' <= c && c <= 'F');
  return 10 + c - 'A';
}
 
 
// Unlike AssignDecimalString(), this function is "only" used
// for unit-tests and therefore not performance critical.
void Bignum::AssignHexString(Vector<const char> value) {
  Zero();
  // Required capacity could be reduced by ignoring leading zeros.
  EnsureCapacity(((value.length() * 4) + kBigitSize - 1) / kBigitSize);
  DOUBLE_CONVERSION_ASSERT(sizeof(uint64_t) * 8 >= kBigitSize + 4);  // TODO: static_assert
  // Accumulates converted hex digits until at least kBigitSize bits.
  // Works with non-factor-of-four kBigitSizes.
  uint64_t tmp = 0;
  for (int cnt = 0; !value.is_empty(); value.pop_back()) {
    tmp |= (HexCharValue(value.last()) << cnt);
    if ((cnt += 4) >= kBigitSize) {
      RawBigit(used_bigits_++) = (tmp & kBigitMask);
      cnt -= kBigitSize;
      tmp >>= kBigitSize;
    }
  }
  if (tmp > 0) {
    DOUBLE_CONVERSION_ASSERT(tmp <= kBigitMask);
    RawBigit(used_bigits_++) = static_cast<Bignum::Chunk>(tmp & kBigitMask);
  }
  Clamp();
}
 
 
void Bignum::AddUInt64(const uint64_t operand) {
  if (operand == 0) {
    return;
  }
  Bignum other;
  other.AssignUInt64(operand);
  AddBignum(other);
}
 
 
void Bignum::AddBignum(const Bignum& other) {
  DOUBLE_CONVERSION_ASSERT(IsClamped());
  DOUBLE_CONVERSION_ASSERT(other.IsClamped());
 
  // If this has a greater exponent than other append zero-bigits to this.
  // After this call exponent_ <= other.exponent_.
  Align(other);
 
  // There are two possibilities:
  //   aaaaaaaaaaa 0000  (where the 0s represent a's exponent)
  //     bbbbb 00000000
  //   ----------------
  //   ccccccccccc 0000
  // or
  //    aaaaaaaaaa 0000
  //  bbbbbbbbb 0000000
  //  -----------------
  //  cccccccccccc 0000
  // In both cases we might need a carry bigit.
 
  EnsureCapacity(1 + (std::max)(BigitLength(), other.BigitLength()) - exponent_);
  Chunk carry = 0;
  int bigit_pos = other.exponent_ - exponent_;
  DOUBLE_CONVERSION_ASSERT(bigit_pos >= 0);
  for (int i = used_bigits_; i < bigit_pos; ++i) {
    RawBigit(i) = 0;
  }
  for (int i = 0; i < other.used_bigits_; ++i) {
    const Chunk my = (bigit_pos < used_bigits_) ? RawBigit(bigit_pos) : 0;
    const Chunk sum = my + other.RawBigit(i) + carry;
    RawBigit(bigit_pos) = sum & kBigitMask;
    carry = sum >> kBigitSize;
    ++bigit_pos;
  }
  while (carry != 0) {
    const Chunk my = (bigit_pos < used_bigits_) ? RawBigit(bigit_pos) : 0;
    const Chunk sum = my + carry;
    RawBigit(bigit_pos) = sum & kBigitMask;
    carry = sum >> kBigitSize;
    ++bigit_pos;
  }
  used_bigits_ = static_cast<int16_t>(std::max(bigit_pos, static_cast<int>(used_bigits_)));
  DOUBLE_CONVERSION_ASSERT(IsClamped());
}
 
 
void Bignum::SubtractBignum(const Bignum& other) {
  DOUBLE_CONVERSION_ASSERT(IsClamped());
  DOUBLE_CONVERSION_ASSERT(other.IsClamped());
  // We require this to be bigger than other.
  DOUBLE_CONVERSION_ASSERT(LessEqual(other, *this));
 
  Align(other);
 
  const int offset = other.exponent_ - exponent_;
  Chunk borrow = 0;
  int i;
  for (i = 0; i < other.used_bigits_; ++i) {
    DOUBLE_CONVERSION_ASSERT((borrow == 0) || (borrow == 1));
    const Chunk difference = RawBigit(i + offset) - other.RawBigit(i) - borrow;
    RawBigit(i + offset) = difference & kBigitMask;
    borrow = difference >> (kChunkSize - 1);
  }
  while (borrow != 0) {
    const Chunk difference = RawBigit(i + offset) - borrow;
    RawBigit(i + offset) = difference & kBigitMask;
    borrow = difference >> (kChunkSize - 1);
    ++i;
  }
  Clamp();
}
 
 
void Bignum::ShiftLeft(const int shift_amount) {
  if (used_bigits_ == 0) {
    return;
  }
  exponent_ += static_cast<int16_t>(shift_amount / kBigitSize);
  const int local_shift = shift_amount % kBigitSize;
  EnsureCapacity(used_bigits_ + 1);
  BigitsShiftLeft(local_shift);
}
 
 
void Bignum::MultiplyByUInt32(const uint32_t factor) {
  if (factor == 1) {
    return;
  }
  if (factor == 0) {
    Zero();
    return;
  }
  if (used_bigits_ == 0) {
    return;
  }
  // The product of a bigit with the factor is of size kBigitSize + 32.
  // Assert that this number + 1 (for the carry) fits into double chunk.
  DOUBLE_CONVERSION_ASSERT(kDoubleChunkSize >= kBigitSize + 32 + 1);
  DoubleChunk carry = 0;
  for (int i = 0; i < used_bigits_; ++i) {
    const DoubleChunk product = static_cast<DoubleChunk>(factor) * RawBigit(i) + carry;
    RawBigit(i) = static_cast<Chunk>(product & kBigitMask);
    carry = (product >> kBigitSize);
  }
  while (carry != 0) {
    EnsureCapacity(used_bigits_ + 1);
    RawBigit(used_bigits_) = carry & kBigitMask;
    used_bigits_++;
    carry >>= kBigitSize;
  }
}
 
 
void Bignum::MultiplyByUInt64(const uint64_t factor) {
  if (factor == 1) {
    return;
  }
  if (factor == 0) {
    Zero();
    return;
  }
  if (used_bigits_ == 0) {
    return;
  }
  DOUBLE_CONVERSION_ASSERT(kBigitSize < 32);
  uint64_t carry = 0;
  const uint64_t low = factor & 0xFFFFFFFF;
  const uint64_t high = factor >> 32;
  for (int i = 0; i < used_bigits_; ++i) {
    const uint64_t product_low = low * RawBigit(i);
    const uint64_t product_high = high * RawBigit(i);
    const uint64_t tmp = (carry & kBigitMask) + product_low;
    RawBigit(i) = tmp & kBigitMask;
    carry = (carry >> kBigitSize) + (tmp >> kBigitSize) +
        (product_high << (32 - kBigitSize));
  }
  while (carry != 0) {
    EnsureCapacity(used_bigits_ + 1);
    RawBigit(used_bigits_) = carry & kBigitMask;
    used_bigits_++;
    carry >>= kBigitSize;
  }
}
 
 
void Bignum::MultiplyByPowerOfTen(const int exponent) {
  static const uint64_t kFive27 = DOUBLE_CONVERSION_UINT64_2PART_C(0x6765c793, fa10079d);
  static const uint16_t kFive1 = 5;
  static const uint16_t kFive2 = kFive1 * 5;
  static const uint16_t kFive3 = kFive2 * 5;
  static const uint16_t kFive4 = kFive3 * 5;
  static const uint16_t kFive5 = kFive4 * 5;
  static const uint16_t kFive6 = kFive5 * 5;
  static const uint32_t kFive7 = kFive6 * 5;
  static const uint32_t kFive8 = kFive7 * 5;
  static const uint32_t kFive9 = kFive8 * 5;
  static const uint32_t kFive10 = kFive9 * 5;
  static const uint32_t kFive11 = kFive10 * 5;
  static const uint32_t kFive12 = kFive11 * 5;
  static const uint32_t kFive13 = kFive12 * 5;
  static const uint32_t kFive1_to_12[] =
      { kFive1, kFive2, kFive3, kFive4, kFive5, kFive6,
        kFive7, kFive8, kFive9, kFive10, kFive11, kFive12 };
 
  DOUBLE_CONVERSION_ASSERT(exponent >= 0);
 
  if (exponent == 0) {
    return;
  }
  if (used_bigits_ == 0) {
    return;
  }
  // We shift by exponent at the end just before returning.
  int remaining_exponent = exponent;
  while (remaining_exponent >= 27) {
    MultiplyByUInt64(kFive27);
    remaining_exponent -= 27;
  }
  while (remaining_exponent >= 13) {
    MultiplyByUInt32(kFive13);
    remaining_exponent -= 13;
  }
  if (remaining_exponent > 0) {
    MultiplyByUInt32(kFive1_to_12[remaining_exponent - 1]);
  }
  ShiftLeft(exponent);
}
 
 
void Bignum::Square() {
  DOUBLE_CONVERSION_ASSERT(IsClamped());
  const int product_length = 2 * used_bigits_;
  EnsureCapacity(product_length);
 
  // Comba multiplication: compute each column separately.
  // Example: r = a2a1a0 * b2b1b0.
  //    r =  1    * a0b0 +
  //        10    * (a1b0 + a0b1) +
  //        100   * (a2b0 + a1b1 + a0b2) +
  //        1000  * (a2b1 + a1b2) +
  //        10000 * a2b2
  //
  // In the worst case we have to accumulate nb-digits products of digit*digit.
  //
  // Assert that the additional number of bits in a DoubleChunk are enough to
  // sum up used_digits of Bigit*Bigit.
  if ((1 << (2 * (kChunkSize - kBigitSize))) <= used_bigits_) {
    DOUBLE_CONVERSION_UNIMPLEMENTED();
  }
  DoubleChunk accumulator = 0;
  // First shift the digits so we don't overwrite them.
  const int copy_offset = used_bigits_;
  for (int i = 0; i < used_bigits_; ++i) {
    RawBigit(copy_offset + i) = RawBigit(i);
  }
  // We have two loops to avoid some 'if's in the loop.
  for (int i = 0; i < used_bigits_; ++i) {
    // Process temporary digit i with power i.
    // The sum of the two indices must be equal to i.
    int bigit_index1 = i;
    int bigit_index2 = 0;
    // Sum all of the sub-products.
    while (bigit_index1 >= 0) {
      const Chunk chunk1 = RawBigit(copy_offset + bigit_index1);
      const Chunk chunk2 = RawBigit(copy_offset + bigit_index2);
      accumulator += static_cast<DoubleChunk>(chunk1) * chunk2;
      bigit_index1--;
      bigit_index2++;
    }
    RawBigit(i) = static_cast<Chunk>(accumulator) & kBigitMask;
    accumulator >>= kBigitSize;
  }
  for (int i = used_bigits_; i < product_length; ++i) {
    int bigit_index1 = used_bigits_ - 1;
    int bigit_index2 = i - bigit_index1;
    // Invariant: sum of both indices is again equal to i.
    // Inner loop runs 0 times on last iteration, emptying accumulator.
    while (bigit_index2 < used_bigits_) {
      const Chunk chunk1 = RawBigit(copy_offset + bigit_index1);
      const Chunk chunk2 = RawBigit(copy_offset + bigit_index2);
      accumulator += static_cast<DoubleChunk>(chunk1) * chunk2;
      bigit_index1--;
      bigit_index2++;
    }
    // The overwritten RawBigit(i) will never be read in further loop iterations,
    // because bigit_index1 and bigit_index2 are always greater
    // than i - used_bigits_.
    RawBigit(i) = static_cast<Chunk>(accumulator) & kBigitMask;
    accumulator >>= kBigitSize;
  }
  // Since the result was guaranteed to lie inside the number the
  // accumulator must be 0 now.
  DOUBLE_CONVERSION_ASSERT(accumulator == 0);
 
  // Don't forget to update the used_digits and the exponent.
  used_bigits_ = static_cast<int16_t>(product_length);
  exponent_ *= 2;
  Clamp();
}
 
 
void Bignum::AssignPowerUInt16(uint16_t base, const int power_exponent) {
  DOUBLE_CONVERSION_ASSERT(base != 0);
  DOUBLE_CONVERSION_ASSERT(power_exponent >= 0);
  if (power_exponent == 0) {
    AssignUInt16(1);
    return;
  }
  Zero();
  int shifts = 0;
  // We expect base to be in range 2-32, and most often to be 10.
  // It does not make much sense to implement different algorithms for counting
  // the bits.
  while ((base & 1) == 0) {
    base >>= 1;
    shifts++;
  }
  int bit_size = 0;
  int tmp_base = base;
  while (tmp_base != 0) {
    tmp_base >>= 1;
    bit_size++;
  }
  const int final_size = bit_size * power_exponent;
  // 1 extra bigit for the shifting, and one for rounded final_size.
  EnsureCapacity(final_size / kBigitSize + 2);
 
  // Left to Right exponentiation.
  int mask = 1;
  while (power_exponent >= mask) mask <<= 1;
 
  // The mask is now pointing to the bit above the most significant 1-bit of
  // power_exponent.
  // Get rid of first 1-bit;
  mask >>= 2;
  uint64_t this_value = base;
 
  bool delayed_multiplication = false;
  const uint64_t max_32bits = 0xFFFFFFFF;
  while (mask != 0 && this_value <= max_32bits) {
    this_value = this_value * this_value;
    // Verify that there is enough space in this_value to perform the
    // multiplication.  The first bit_size bits must be 0.
    if ((power_exponent & mask) != 0) {
      DOUBLE_CONVERSION_ASSERT(bit_size > 0);
      const uint64_t base_bits_mask =
        ~((static_cast<uint64_t>(1) << (64 - bit_size)) - 1);
      const bool high_bits_zero = (this_value & base_bits_mask) == 0;
      if (high_bits_zero) {
        this_value *= base;
      } else {
        delayed_multiplication = true;
      }
    }
    mask >>= 1;
  }
  AssignUInt64(this_value);
  if (delayed_multiplication) {
    MultiplyByUInt32(base);
  }
 
  // Now do the same thing as a bignum.
  while (mask != 0) {
    Square();
    if ((power_exponent & mask) != 0) {
      MultiplyByUInt32(base);
    }
    mask >>= 1;
  }
 
  // And finally add the saved shifts.
  ShiftLeft(shifts * power_exponent);
}
 
 
// Precondition: this/other < 16bit.
uint16_t Bignum::DivideModuloIntBignum(const Bignum& other) {
  DOUBLE_CONVERSION_ASSERT(IsClamped());
  DOUBLE_CONVERSION_ASSERT(other.IsClamped());
  DOUBLE_CONVERSION_ASSERT(other.used_bigits_ > 0);
 
  // Easy case: if we have less digits than the divisor than the result is 0.
  // Note: this handles the case where this == 0, too.
  if (BigitLength() < other.BigitLength()) {
    return 0;
  }
 
  Align(other);
 
  uint16_t result = 0;
 
  // Start by removing multiples of 'other' until both numbers have the same
  // number of digits.
  while (BigitLength() > other.BigitLength()) {
    // This naive approach is extremely inefficient if `this` divided by other
    // is big. This function is implemented for doubleToString where
    // the result should be small (less than 10).
    DOUBLE_CONVERSION_ASSERT(other.RawBigit(other.used_bigits_ - 1) >= ((1 << kBigitSize) / 16));
    DOUBLE_CONVERSION_ASSERT(RawBigit(used_bigits_ - 1) < 0x10000);
    // Remove the multiples of the first digit.
    // Example this = 23 and other equals 9. -> Remove 2 multiples.
    result += static_cast<uint16_t>(RawBigit(used_bigits_ - 1));
    SubtractTimes(other, RawBigit(used_bigits_ - 1));
  }
 
  DOUBLE_CONVERSION_ASSERT(BigitLength() == other.BigitLength());
 
  // Both bignums are at the same length now.
  // Since other has more than 0 digits we know that the access to
  // RawBigit(used_bigits_ - 1) is safe.
  const Chunk this_bigit = RawBigit(used_bigits_ - 1);
  const Chunk other_bigit = other.RawBigit(other.used_bigits_ - 1);
 
  if (other.used_bigits_ == 1) {
    // Shortcut for easy (and common) case.
    int quotient = this_bigit / other_bigit;
    RawBigit(used_bigits_ - 1) = this_bigit - other_bigit * quotient;
    DOUBLE_CONVERSION_ASSERT(quotient < 0x10000);
    result += static_cast<uint16_t>(quotient);
    Clamp();
    return result;
  }
 
  const int division_estimate = this_bigit / (other_bigit + 1);
  DOUBLE_CONVERSION_ASSERT(division_estimate < 0x10000);
  result += static_cast<uint16_t>(division_estimate);
  SubtractTimes(other, division_estimate);
 
  if (other_bigit * (division_estimate + 1) > this_bigit) {
    // No need to even try to subtract. Even if other's remaining digits were 0
    // another subtraction would be too much.
    return result;
  }
 
  while (LessEqual(other, *this)) {
    SubtractBignum(other);
    result++;
  }
  return result;
}
 
 
template<typename S>
static int SizeInHexChars(S number) {
  DOUBLE_CONVERSION_ASSERT(number > 0);
  int result = 0;
  while (number != 0) {
    number >>= 4;
    result++;
  }
  return result;
}
 
 
static char HexCharOfValue(const int value) {
  DOUBLE_CONVERSION_ASSERT(0 <= value && value <= 16);
  if (value < 10) {
    return static_cast<char>(value + '0');
  }
  return static_cast<char>(value - 10 + 'A');
}
 
 
bool Bignum::ToHexString(char* buffer, const int buffer_size) const {
  DOUBLE_CONVERSION_ASSERT(IsClamped());
  // Each bigit must be printable as separate hex-character.
  DOUBLE_CONVERSION_ASSERT(kBigitSize % 4 == 0);
  static const int kHexCharsPerBigit = kBigitSize / 4;
 
  if (used_bigits_ == 0) {
    if (buffer_size < 2) {
      return false;
    }
    buffer[0] = '0';
    buffer[1] = '\0';
    return true;
  }
  // We add 1 for the terminating '\0' character.
  const int needed_chars = (BigitLength() - 1) * kHexCharsPerBigit +
    SizeInHexChars(RawBigit(used_bigits_ - 1)) + 1;
  if (needed_chars > buffer_size) {
    return false;
  }
  int string_index = needed_chars - 1;
  buffer[string_index--] = '\0';
  for (int i = 0; i < exponent_; ++i) {
    for (int j = 0; j < kHexCharsPerBigit; ++j) {
      buffer[string_index--] = '0';
    }
  }
  for (int i = 0; i < used_bigits_ - 1; ++i) {
    Chunk current_bigit = RawBigit(i);
    for (int j = 0; j < kHexCharsPerBigit; ++j) {
      buffer[string_index--] = HexCharOfValue(current_bigit & 0xF);
      current_bigit >>= 4;
    }
  }
  // And finally the last bigit.
  Chunk most_significant_bigit = RawBigit(used_bigits_ - 1);
  while (most_significant_bigit != 0) {
    buffer[string_index--] = HexCharOfValue(most_significant_bigit & 0xF);
    most_significant_bigit >>= 4;
  }
  return true;
}
 
 
Bignum::Chunk Bignum::BigitOrZero(const int index) const {
  if (index >= BigitLength()) {
    return 0;
  }
  if (index < exponent_) {
    return 0;
  }
  return RawBigit(index - exponent_);
}
 
 
int Bignum::Compare(const Bignum& a, const Bignum& b) {
  DOUBLE_CONVERSION_ASSERT(a.IsClamped());
  DOUBLE_CONVERSION_ASSERT(b.IsClamped());
  const int bigit_length_a = a.BigitLength();
  const int bigit_length_b = b.BigitLength();
  if (bigit_length_a < bigit_length_b) {
    return -1;
  }
  if (bigit_length_a > bigit_length_b) {
    return +1;
  }
  for (int i = bigit_length_a - 1; i >= (std::min)(a.exponent_, b.exponent_); --i) {
    const Chunk bigit_a = a.BigitOrZero(i);
    const Chunk bigit_b = b.BigitOrZero(i);
    if (bigit_a < bigit_b) {
      return -1;
    }
    if (bigit_a > bigit_b) {
      return +1;
    }
    // Otherwise they are equal up to this digit. Try the next digit.
  }
  return 0;
}
 
 
int Bignum::PlusCompare(const Bignum& a, const Bignum& b, const Bignum& c) {
  DOUBLE_CONVERSION_ASSERT(a.IsClamped());
  DOUBLE_CONVERSION_ASSERT(b.IsClamped());
  DOUBLE_CONVERSION_ASSERT(c.IsClamped());
  if (a.BigitLength() < b.BigitLength()) {
    return PlusCompare(b, a, c);
  }
  if (a.BigitLength() + 1 < c.BigitLength()) {
    return -1;
  }
  if (a.BigitLength() > c.BigitLength()) {
    return +1;
  }
  // The exponent encodes 0-bigits. So if there are more 0-digits in 'a' than
  // 'b' has digits, then the bigit-length of 'a'+'b' must be equal to the one
  // of 'a'.
  if (a.exponent_ >= b.BigitLength() && a.BigitLength() < c.BigitLength()) {
    return -1;
  }
 
  Chunk borrow = 0;
  // Starting at min_exponent all digits are == 0. So no need to compare them.
  const int min_exponent = (std::min)((std::min)(a.exponent_, b.exponent_), c.exponent_);
  for (int i = c.BigitLength() - 1; i >= min_exponent; --i) {
    const Chunk chunk_a = a.BigitOrZero(i);
    const Chunk chunk_b = b.BigitOrZero(i);
    const Chunk chunk_c = c.BigitOrZero(i);
    const Chunk sum = chunk_a + chunk_b;
    if (sum > chunk_c + borrow) {
      return +1;
    } else {
      borrow = chunk_c + borrow - sum;
      if (borrow > 1) {
        return -1;
      }
      borrow <<= kBigitSize;
    }
  }
  if (borrow == 0) {
    return 0;
  }
  return -1;
}
 
 
void Bignum::Clamp() {
  while (used_bigits_ > 0 && RawBigit(used_bigits_ - 1) == 0) {
    used_bigits_--;
  }
  if (used_bigits_ == 0) {
    // Zero.
    exponent_ = 0;
  }
}
 
 
void Bignum::Align(const Bignum& other) {
  if (exponent_ > other.exponent_) {
    // If "X" represents a "hidden" bigit (by the exponent) then we are in the
    // following case (a == this, b == other):
    // a:  aaaaaaXXXX   or a:   aaaaaXXX
    // b:     bbbbbbX      b: bbbbbbbbXX
    // We replace some of the hidden digits (X) of a with 0 digits.
    // a:  aaaaaa000X   or a:   aaaaa0XX
    const int zero_bigits = exponent_ - other.exponent_;
    EnsureCapacity(used_bigits_ + zero_bigits);
    for (int i = used_bigits_ - 1; i >= 0; --i) {
      RawBigit(i + zero_bigits) = RawBigit(i);
    }
    for (int i = 0; i < zero_bigits; ++i) {
      RawBigit(i) = 0;
    }
    used_bigits_ += static_cast<int16_t>(zero_bigits);
    exponent_ -= static_cast<int16_t>(zero_bigits);
 
    DOUBLE_CONVERSION_ASSERT(used_bigits_ >= 0);
    DOUBLE_CONVERSION_ASSERT(exponent_ >= 0);
  }
}
 
 
void Bignum::BigitsShiftLeft(const int shift_amount) {
  DOUBLE_CONVERSION_ASSERT(shift_amount < kBigitSize);
  DOUBLE_CONVERSION_ASSERT(shift_amount >= 0);
  Chunk carry = 0;
  for (int i = 0; i < used_bigits_; ++i) {
    const Chunk new_carry = RawBigit(i) >> (kBigitSize - shift_amount);
    RawBigit(i) = ((RawBigit(i) << shift_amount) + carry) & kBigitMask;
    carry = new_carry;
  }
  if (carry != 0) {
    RawBigit(used_bigits_) = carry;
    used_bigits_++;
  }
}
 
 
void Bignum::SubtractTimes(const Bignum& other, const int factor) {
  DOUBLE_CONVERSION_ASSERT(exponent_ <= other.exponent_);
  if (factor < 3) {
    for (int i = 0; i < factor; ++i) {
      SubtractBignum(other);
    }
    return;
  }
  Chunk borrow = 0;
  const int exponent_diff = other.exponent_ - exponent_;
  for (int i = 0; i < other.used_bigits_; ++i) {
    const DoubleChunk product = static_cast<DoubleChunk>(factor) * other.RawBigit(i);
    const DoubleChunk remove = borrow + product;
    const Chunk difference = RawBigit(i + exponent_diff) - (remove & kBigitMask);
    RawBigit(i + exponent_diff) = difference & kBigitMask;
    borrow = static_cast<Chunk>((difference >> (kChunkSize - 1)) +
                                (remove >> kBigitSize));
  }
  for (int i = other.used_bigits_ + exponent_diff; i < used_bigits_; ++i) {
    if (borrow == 0) {
      return;
    }
    const Chunk difference = RawBigit(i) - borrow;
    RawBigit(i) = difference & kBigitMask;
    borrow = difference >> (kChunkSize - 1);
  }
  Clamp();
}
 
 
}  // namespace double_conversion