path: root/contrib/libs/double-conversion/bignum.cc

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#include "bignum.h" 
#include "utils.h" 
 
namespace double_conversion { 
 
Bignum::Bignum() 
    : bigits_buffer_(), bigits_(bigits_buffer_, kBigitCapacity), used_digits_(0), exponent_(0) {
  for (int i = 0; i < kBigitCapacity; ++i) { 
    bigits_[i] = 0; 
  } 
} 
 
 
template<typename S> 
static int BitSize(S value) { 
  (void) value;  // Mark variable as used. 
  return 8 * sizeof(value); 
} 
 
// Guaranteed to lie in one Bigit. 
void Bignum::AssignUInt16(uint16_t value) { 
  ASSERT(kBigitSize >= BitSize(value)); 
  Zero(); 
  if (value == 0) return; 
 
  EnsureCapacity(1); 
  bigits_[0] = value; 
  used_digits_ = 1; 
} 
 
 
void Bignum::AssignUInt64(uint64_t value) { 
  const int kUInt64Size = 64; 
 
  Zero(); 
  if (value == 0) return; 
 
  int needed_bigits = kUInt64Size / kBigitSize + 1; 
  EnsureCapacity(needed_bigits); 
  for (int i = 0; i < needed_bigits; ++i) { 
    bigits_[i] = value & kBigitMask; 
    value = value >> kBigitSize; 
  } 
  used_digits_ = needed_bigits; 
  Clamp(); 
} 
 
 
void Bignum::AssignBignum(const Bignum& other) { 
  exponent_ = other.exponent_; 
  for (int i = 0; i < other.used_digits_; ++i) { 
    bigits_[i] = other.bigits_[i]; 
  } 
  // Clear the excess digits (if there were any). 
  for (int i = other.used_digits_; i < used_digits_; ++i) { 
    bigits_[i] = 0; 
  } 
  used_digits_ = other.used_digits_; 
} 
 
 
static uint64_t ReadUInt64(Vector<const char> buffer, 
                           int from, 
                           int digits_to_read) { 
  uint64_t result = 0; 
  for (int i = from; i < from + digits_to_read; ++i) { 
    int digit = buffer[i] - '0'; 
    ASSERT(0 <= digit && digit <= 9); 
    result = result * 10 + digit; 
  } 
  return result; 
} 
 
 
void Bignum::AssignDecimalString(Vector<const char> value) { 
  // 2^64 = 18446744073709551616 > 10^19 
  const int kMaxUint64DecimalDigits = 19; 
  Zero(); 
  int length = value.length(); 
  unsigned int pos = 0; 
  // Let's just say that each digit needs 4 bits. 
  while (length >= kMaxUint64DecimalDigits) { 
    uint64_t digits = ReadUInt64(value, pos, kMaxUint64DecimalDigits); 
    pos += kMaxUint64DecimalDigits; 
    length -= kMaxUint64DecimalDigits; 
    MultiplyByPowerOfTen(kMaxUint64DecimalDigits); 
    AddUInt64(digits); 
  } 
  uint64_t digits = ReadUInt64(value, pos, length); 
  MultiplyByPowerOfTen(length); 
  AddUInt64(digits); 
  Clamp(); 
} 
 
 
static int HexCharValue(char c) { 
  if ('0' <= c && c <= '9') return c - '0'; 
  if ('a' <= c && c <= 'f') return 10 + c - 'a'; 
  ASSERT('A' <= c && c <= 'F'); 
  return 10 + c - 'A'; 
} 
 
 
void Bignum::AssignHexString(Vector<const char> value) { 
  Zero(); 
  int length = value.length(); 
 
  int needed_bigits = length * 4 / kBigitSize + 1; 
  EnsureCapacity(needed_bigits); 
  int string_index = length - 1; 
  for (int i = 0; i < needed_bigits - 1; ++i) { 
    // These bigits are guaranteed to be "full". 
    Chunk current_bigit = 0; 
    for (int j = 0; j < kBigitSize / 4; j++) { 
      current_bigit += HexCharValue(value[string_index--]) << (j * 4); 
    } 
    bigits_[i] = current_bigit; 
  } 
  used_digits_ = needed_bigits - 1; 
 
  Chunk most_significant_bigit = 0;  // Could be = 0; 
  for (int j = 0; j <= string_index; ++j) { 
    most_significant_bigit <<= 4; 
    most_significant_bigit += HexCharValue(value[j]); 
  } 
  if (most_significant_bigit != 0) { 
    bigits_[used_digits_] = most_significant_bigit; 
    used_digits_++; 
  } 
  Clamp(); 
} 
 
 
void Bignum::AddUInt64(uint64_t operand) { 
  if (operand == 0) return; 
  Bignum other; 
  other.AssignUInt64(operand); 
  AddBignum(other); 
} 
 
 
void Bignum::AddBignum(const Bignum& other) { 
  ASSERT(IsClamped()); 
  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 + Max(BigitLength(), other.BigitLength()) - exponent_); 
  Chunk carry = 0; 
  int bigit_pos = other.exponent_ - exponent_; 
  ASSERT(bigit_pos >= 0); 
  for (int i = 0; i < other.used_digits_; ++i) { 
    Chunk sum = bigits_[bigit_pos] + other.bigits_[i] + carry; 
    bigits_[bigit_pos] = sum & kBigitMask; 
    carry = sum >> kBigitSize; 
    bigit_pos++; 
  } 
 
  while (carry != 0) { 
    Chunk sum = bigits_[bigit_pos] + carry; 
    bigits_[bigit_pos] = sum & kBigitMask; 
    carry = sum >> kBigitSize; 
    bigit_pos++; 
  } 
  used_digits_ = Max(bigit_pos, used_digits_); 
  ASSERT(IsClamped()); 
} 
 
 
void Bignum::SubtractBignum(const Bignum& other) { 
  ASSERT(IsClamped()); 
  ASSERT(other.IsClamped()); 
  // We require this to be bigger than other. 
  ASSERT(LessEqual(other, *this)); 
 
  Align(other); 
 
  int offset = other.exponent_ - exponent_; 
  Chunk borrow = 0; 
  int i; 
  for (i = 0; i < other.used_digits_; ++i) { 
    ASSERT((borrow == 0) || (borrow == 1)); 
    Chunk difference = bigits_[i + offset] - other.bigits_[i] - borrow; 
    bigits_[i + offset] = difference & kBigitMask; 
    borrow = difference >> (kChunkSize - 1); 
  } 
  while (borrow != 0) { 
    Chunk difference = bigits_[i + offset] - borrow; 
    bigits_[i + offset] = difference & kBigitMask; 
    borrow = difference >> (kChunkSize - 1); 
    ++i; 
  } 
  Clamp(); 
} 
 
 
void Bignum::ShiftLeft(int shift_amount) { 
  if (used_digits_ == 0) return; 
  exponent_ += shift_amount / kBigitSize; 
  int local_shift = shift_amount % kBigitSize; 
  EnsureCapacity(used_digits_ + 1); 
  BigitsShiftLeft(local_shift); 
} 
 
 
void Bignum::MultiplyByUInt32(uint32_t factor) { 
  if (factor == 1) return; 
  if (factor == 0) { 
    Zero(); 
    return; 
  } 
  if (used_digits_ == 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. 
  ASSERT(kDoubleChunkSize >= kBigitSize + 32 + 1); 
  DoubleChunk carry = 0; 
  for (int i = 0; i < used_digits_; ++i) { 
    DoubleChunk product = static_cast<DoubleChunk>(factor) * bigits_[i] + carry; 
    bigits_[i] = static_cast<Chunk>(product & kBigitMask); 
    carry = (product >> kBigitSize); 
  } 
  while (carry != 0) { 
    EnsureCapacity(used_digits_ + 1); 
    bigits_[used_digits_] = carry & kBigitMask; 
    used_digits_++; 
    carry >>= kBigitSize; 
  } 
} 
 
 
void Bignum::MultiplyByUInt64(uint64_t factor) { 
  if (factor == 1) return; 
  if (factor == 0) { 
    Zero(); 
    return; 
  } 
  ASSERT(kBigitSize < 32); 
  uint64_t carry = 0; 
  uint64_t low = factor & 0xFFFFFFFF; 
  uint64_t high = factor >> 32; 
  for (int i = 0; i < used_digits_; ++i) { 
    uint64_t product_low = low * bigits_[i]; 
    uint64_t product_high = high * bigits_[i]; 
    uint64_t tmp = (carry & kBigitMask) + product_low; 
    bigits_[i] = tmp & kBigitMask; 
    carry = (carry >> kBigitSize) + (tmp >> kBigitSize) + 
        (product_high << (32 - kBigitSize)); 
  } 
  while (carry != 0) { 
    EnsureCapacity(used_digits_ + 1); 
    bigits_[used_digits_] = carry & kBigitMask; 
    used_digits_++; 
    carry >>= kBigitSize; 
  } 
} 
 
 
void Bignum::MultiplyByPowerOfTen(int exponent) { 
  const uint64_t kFive27 = UINT64_2PART_C(0x6765c793, fa10079d); 
  const uint16_t kFive1 = 5; 
  const uint16_t kFive2 = kFive1 * 5; 
  const uint16_t kFive3 = kFive2 * 5; 
  const uint16_t kFive4 = kFive3 * 5; 
  const uint16_t kFive5 = kFive4 * 5; 
  const uint16_t kFive6 = kFive5 * 5; 
  const uint32_t kFive7 = kFive6 * 5; 
  const uint32_t kFive8 = kFive7 * 5; 
  const uint32_t kFive9 = kFive8 * 5; 
  const uint32_t kFive10 = kFive9 * 5; 
  const uint32_t kFive11 = kFive10 * 5; 
  const uint32_t kFive12 = kFive11 * 5; 
  const uint32_t kFive13 = kFive12 * 5; 
  const uint32_t kFive1_to_12[] = 
      { kFive1, kFive2, kFive3, kFive4, kFive5, kFive6, 
        kFive7, kFive8, kFive9, kFive10, kFive11, kFive12 }; 
 
  ASSERT(exponent >= 0); 
  if (exponent == 0) return; 
  if (used_digits_ == 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() { 
  ASSERT(IsClamped()); 
  int product_length = 2 * used_digits_; 
  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_digits_) { 
    UNIMPLEMENTED(); 
  } 
  DoubleChunk accumulator = 0; 
  // First shift the digits so we don't overwrite them. 
  int copy_offset = used_digits_; 
  for (int i = 0; i < used_digits_; ++i) { 
    bigits_[copy_offset + i] = bigits_[i]; 
  } 
  // We have two loops to avoid some 'if's in the loop. 
  for (int i = 0; i < used_digits_; ++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) { 
      Chunk chunk1 = bigits_[copy_offset + bigit_index1]; 
      Chunk chunk2 = bigits_[copy_offset + bigit_index2]; 
      accumulator += static_cast<DoubleChunk>(chunk1) * chunk2; 
      bigit_index1--; 
      bigit_index2++; 
    } 
    bigits_[i] = static_cast<Chunk>(accumulator) & kBigitMask; 
    accumulator >>= kBigitSize; 
  } 
  for (int i = used_digits_; i < product_length; ++i) { 
    int bigit_index1 = used_digits_ - 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_digits_) { 
      Chunk chunk1 = bigits_[copy_offset + bigit_index1]; 
      Chunk chunk2 = bigits_[copy_offset + bigit_index2]; 
      accumulator += static_cast<DoubleChunk>(chunk1) * chunk2; 
      bigit_index1--; 
      bigit_index2++; 
    } 
    // The overwritten bigits_[i] will never be read in further loop iterations, 
    // because bigit_index1 and bigit_index2 are always greater 
    // than i - used_digits_. 
    bigits_[i] = static_cast<Chunk>(accumulator) & kBigitMask; 
    accumulator >>= kBigitSize; 
  } 
  // Since the result was guaranteed to lie inside the number the 
  // accumulator must be 0 now. 
  ASSERT(accumulator == 0); 
 
  // Don't forget to update the used_digits and the exponent. 
  used_digits_ = product_length; 
  exponent_ *= 2; 
  Clamp(); 
} 
 
 
void Bignum::AssignPowerUInt16(uint16_t base, int power_exponent) { 
  ASSERT(base != 0); 
  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++; 
  } 
  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) { 
      ASSERT(bit_size > 0);
      uint64_t base_bits_mask = 
          ~((static_cast<uint64_t>(1) << (64 - bit_size)) - 1); 
      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) { 
  ASSERT(IsClamped()); 
  ASSERT(other.IsClamped()); 
  ASSERT(other.used_digits_ > 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). 
    ASSERT(other.bigits_[other.used_digits_ - 1] >= ((1 << kBigitSize) / 16)); 
    ASSERT(bigits_[used_digits_ - 1] < 0x10000); 
    // Remove the multiples of the first digit. 
    // Example this = 23 and other equals 9. -> Remove 2 multiples. 
    result += static_cast<uint16_t>(bigits_[used_digits_ - 1]); 
    SubtractTimes(other, bigits_[used_digits_ - 1]); 
  } 
 
  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 
  // bigits_[used_digits_ - 1] is safe. 
  Chunk this_bigit = bigits_[used_digits_ - 1]; 
  Chunk other_bigit = other.bigits_[other.used_digits_ - 1]; 
 
  if (other.used_digits_ == 1) { 
    // Shortcut for easy (and common) case. 
    int quotient = this_bigit / other_bigit; 
    bigits_[used_digits_ - 1] = this_bigit - other_bigit * quotient; 
    ASSERT(quotient < 0x10000); 
    result += static_cast<uint16_t>(quotient); 
    Clamp(); 
    return result; 
  } 
 
  int division_estimate = this_bigit / (other_bigit + 1); 
  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) { 
  ASSERT(number > 0); 
  int result = 0; 
  while (number != 0) { 
    number >>= 4; 
    result++; 
  } 
  return result; 
} 
 
 
static char HexCharOfValue(int value) { 
  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, int buffer_size) const { 
  ASSERT(IsClamped()); 
  // Each bigit must be printable as separate hex-character. 
  ASSERT(kBigitSize % 4 == 0); 
  const int kHexCharsPerBigit = kBigitSize / 4; 
 
  if (used_digits_ == 0) { 
    if (buffer_size < 2) return false; 
    buffer[0] = '0'; 
    buffer[1] = '\0'; 
    return true; 
  } 
  // We add 1 for the terminating '\0' character. 
  int needed_chars = (BigitLength() - 1) * kHexCharsPerBigit + 
      SizeInHexChars(bigits_[used_digits_ - 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_digits_ - 1; ++i) { 
    Chunk current_bigit = bigits_[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 = bigits_[used_digits_ - 1]; 
  while (most_significant_bigit != 0) { 
    buffer[string_index--] = HexCharOfValue(most_significant_bigit & 0xF); 
    most_significant_bigit >>= 4; 
  } 
  return true; 
} 
 
 
Bignum::Chunk Bignum::BigitAt(int index) const { 
  if (index >= BigitLength()) return 0; 
  if (index < exponent_) return 0; 
  return bigits_[index - exponent_]; 
} 
 
 
int Bignum::Compare(const Bignum& a, const Bignum& b) { 
  ASSERT(a.IsClamped()); 
  ASSERT(b.IsClamped()); 
  int bigit_length_a = a.BigitLength(); 
  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 >= Min(a.exponent_, b.exponent_); --i) { 
    Chunk bigit_a = a.BigitAt(i); 
    Chunk bigit_b = b.BigitAt(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) { 
  ASSERT(a.IsClamped()); 
  ASSERT(b.IsClamped()); 
  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. 
  int min_exponent = Min(Min(a.exponent_, b.exponent_), c.exponent_); 
  for (int i = c.BigitLength() - 1; i >= min_exponent; --i) { 
    Chunk chunk_a = a.BigitAt(i); 
    Chunk chunk_b = b.BigitAt(i); 
    Chunk chunk_c = c.BigitAt(i); 
    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_digits_ > 0 && bigits_[used_digits_ - 1] == 0) { 
    used_digits_--; 
  } 
  if (used_digits_ == 0) { 
    // Zero. 
    exponent_ = 0; 
  } 
} 
 
 
bool Bignum::IsClamped() const { 
  return used_digits_ == 0 || bigits_[used_digits_ - 1] != 0; 
} 
 
 
void Bignum::Zero() { 
  for (int i = 0; i < used_digits_; ++i) { 
    bigits_[i] = 0; 
  } 
  used_digits_ = 0; 
  exponent_ = 0; 
} 
 
 
void Bignum::Align(const Bignum& other) { 
  if (exponent_ > other.exponent_) { 
    // If "X" represents a "hidden" digit (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 
    int zero_digits = exponent_ - other.exponent_; 
    EnsureCapacity(used_digits_ + zero_digits); 
    for (int i = used_digits_ - 1; i >= 0; --i) { 
      bigits_[i + zero_digits] = bigits_[i]; 
    } 
    for (int i = 0; i < zero_digits; ++i) { 
      bigits_[i] = 0; 
    } 
    used_digits_ += zero_digits; 
    exponent_ -= zero_digits; 
    ASSERT(used_digits_ >= 0); 
    ASSERT(exponent_ >= 0); 
  } 
} 
 
 
void Bignum::BigitsShiftLeft(int shift_amount) { 
  ASSERT(shift_amount < kBigitSize); 
  ASSERT(shift_amount >= 0); 
  Chunk carry = 0; 
  for (int i = 0; i < used_digits_; ++i) { 
    Chunk new_carry = bigits_[i] >> (kBigitSize - shift_amount); 
    bigits_[i] = ((bigits_[i] << shift_amount) + carry) & kBigitMask; 
    carry = new_carry; 
  } 
  if (carry != 0) { 
    bigits_[used_digits_] = carry; 
    used_digits_++; 
  } 
} 
 
 
void Bignum::SubtractTimes(const Bignum& other, int factor) { 
  ASSERT(exponent_ <= other.exponent_); 
  if (factor < 3) { 
    for (int i = 0; i < factor; ++i) { 
      SubtractBignum(other); 
    } 
    return; 
  } 
  Chunk borrow = 0; 
  int exponent_diff = other.exponent_ - exponent_; 
  for (int i = 0; i < other.used_digits_; ++i) { 
    DoubleChunk product = static_cast<DoubleChunk>(factor) * other.bigits_[i]; 
    DoubleChunk remove = borrow + product; 
    Chunk difference = bigits_[i + exponent_diff] - (remove & kBigitMask); 
    bigits_[i + exponent_diff] = difference & kBigitMask; 
    borrow = static_cast<Chunk>((difference >> (kChunkSize - 1)) + 
                                (remove >> kBigitSize)); 
  } 
  for (int i = other.used_digits_ + exponent_diff; i < used_digits_; ++i) { 
    if (borrow == 0) return; 
    Chunk difference = bigits_[i] - borrow; 
    bigits_[i] = difference & kBigitMask; 
    borrow = difference >> (kChunkSize - 1); 
  } 
  Clamp(); 
} 
 
 
}  // namespace double_conversion