Bignum implementation of Strtod.

src/bignum.cc

Index: src/bignum.cc
diff --git a/src/bignum.cc b/src/bignum.cc
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+// Copyright 2010 the V8 project authors. All rights reserved.
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are
+// met:
+//
+//     * Redistributions of source code must retain the above copyright
+//       notice, this list of conditions and the following disclaimer.
+//     * Redistributions in binary form must reproduce the above
+//       copyright notice, this list of conditions and the following
+//       disclaimer in the documentation and/or other materials provided
+//       with the distribution.
+//     * Neither the name of Google Inc. nor the names of its
+//       contributors may be used to endorse or promote products derived
+//       from this software without specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+#include "v8.h"
+
+#include "bignum.h"
+#include "utils.h"
+
+namespace v8 {
+namespace internal {
+
+Bignum::Bignum()
+    : 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) {
+  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();
+  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';
+  if ('A' <= c && c <= 'F') return 10 + c - 'A';
+  UNREACHABLE();
+  return 0;  // To make compiler happy.
+}
+
+
+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 = V8_2PART_UINT64_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_multipliciation = 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) {
+      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_multipliciation = true;
+      }
+    }
+    mask >>= 1;
+  }
+  AssignUInt64(this_value);
+  if (delayed_multipliciation) {
+    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 the this divided other
+    // might be 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));
+    // Remove the multiples of the first digit.
+    // Example this = 23 and other equals 9. -> Remove 2 multiples.
+    result += 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;
+    result += quotient;
+    Clamp();
+    return result;
+  }
+
+  int division_estimate = this_bigit / (other_bigit + 1);
+  result += 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 value + '0';
+  return 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);
+    ++i;
+  }
+  Clamp();
+}
+
+
+} }  // namespace v8::internal