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// Copyright 2010 the V8 project authors. All rights reserved.
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above
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// copyright notice, this list of conditions and the following
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// disclaimer in the documentation and/or other materials provided
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// with the distribution.
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// * Neither the name of Google Inc. nor the names of its
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// contributors may be used to endorse or promote products derived
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// from this software without specific prior written permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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namespace double_conversion {
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: bigits_(bigits_buffer_, kBigitCapacity), used_digits_(0), exponent_(0) {
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for (int i = 0; i < kBigitCapacity; ++i) {
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static int BitSize(S value) {
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return 8 * sizeof(value);
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// Guaranteed to lie in one Bigit.
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void Bignum::AssignUInt16(uint16_t value) {
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ASSERT(kBigitSize >= BitSize(value));
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if (value == 0) return;
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void Bignum::AssignUInt64(uint64_t value) {
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const int kUInt64Size = 64;
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if (value == 0) return;
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int needed_bigits = kUInt64Size / kBigitSize + 1;
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EnsureCapacity(needed_bigits);
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for (int i = 0; i < needed_bigits; ++i) {
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bigits_[i] = value & kBigitMask;
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value = value >> kBigitSize;
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used_digits_ = needed_bigits;
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void Bignum::AssignBignum(const Bignum& other) {
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exponent_ = other.exponent_;
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for (int i = 0; i < other.used_digits_; ++i) {
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bigits_[i] = other.bigits_[i];
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// Clear the excess digits (if there were any).
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for (int i = other.used_digits_; i < used_digits_; ++i) {
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used_digits_ = other.used_digits_;
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static uint64_t ReadUInt64(Vector<const char> buffer,
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for (int i = from; i < from + digits_to_read; ++i) {
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int digit = buffer[i] - '0';
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ASSERT(0 <= digit && digit <= 9);
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result = result * 10 + digit;
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void Bignum::AssignDecimalString(Vector<const char> value) {
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// 2^64 = 18446744073709551616 > 10^19
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const int kMaxUint64DecimalDigits = 19;
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int length = value.length();
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// Let's just say that each digit needs 4 bits.
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while (length >= kMaxUint64DecimalDigits) {
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uint64_t digits = ReadUInt64(value, pos, kMaxUint64DecimalDigits);
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pos += kMaxUint64DecimalDigits;
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length -= kMaxUint64DecimalDigits;
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MultiplyByPowerOfTen(kMaxUint64DecimalDigits);
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uint64_t digits = ReadUInt64(value, pos, length);
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MultiplyByPowerOfTen(length);
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static int HexCharValue(char c) {
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if ('0' <= c && c <= '9') return c - '0';
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if ('a' <= c && c <= 'f') return 10 + c - 'a';
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if ('A' <= c && c <= 'F') return 10 + c - 'A';
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return 0; // To make compiler happy.
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void Bignum::AssignHexString(Vector<const char> value) {
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int length = value.length();
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int needed_bigits = length * 4 / kBigitSize + 1;
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EnsureCapacity(needed_bigits);
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int string_index = length - 1;
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for (int i = 0; i < needed_bigits - 1; ++i) {
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// These bigits are guaranteed to be "full".
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Chunk current_bigit = 0;
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for (int j = 0; j < kBigitSize / 4; j++) {
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current_bigit += HexCharValue(value[string_index--]) << (j * 4);
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bigits_[i] = current_bigit;
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used_digits_ = needed_bigits - 1;
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Chunk most_significant_bigit = 0; // Could be = 0;
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for (int j = 0; j <= string_index; ++j) {
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most_significant_bigit <<= 4;
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most_significant_bigit += HexCharValue(value[j]);
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if (most_significant_bigit != 0) {
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bigits_[used_digits_] = most_significant_bigit;
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void Bignum::AddUInt64(uint64_t operand) {
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if (operand == 0) return;
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other.AssignUInt64(operand);
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void Bignum::AddBignum(const Bignum& other) {
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ASSERT(other.IsClamped());
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// If this has a greater exponent than other append zero-bigits to this.
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// After this call exponent_ <= other.exponent_.
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// There are two possibilities:
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// aaaaaaaaaaa 0000 (where the 0s represent a's exponent)
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// In both cases we might need a carry bigit.
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EnsureCapacity(1 + Max(BigitLength(), other.BigitLength()) - exponent_);
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int bigit_pos = other.exponent_ - exponent_;
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ASSERT(bigit_pos >= 0);
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for (int i = 0; i < other.used_digits_; ++i) {
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Chunk sum = bigits_[bigit_pos] + other.bigits_[i] + carry;
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bigits_[bigit_pos] = sum & kBigitMask;
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carry = sum >> kBigitSize;
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Chunk sum = bigits_[bigit_pos] + carry;
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bigits_[bigit_pos] = sum & kBigitMask;
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carry = sum >> kBigitSize;
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used_digits_ = Max(bigit_pos, used_digits_);
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void Bignum::SubtractBignum(const Bignum& other) {
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ASSERT(other.IsClamped());
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// We require this to be bigger than other.
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ASSERT(LessEqual(other, *this));
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int offset = other.exponent_ - exponent_;
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for (i = 0; i < other.used_digits_; ++i) {
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ASSERT((borrow == 0) || (borrow == 1));
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Chunk difference = bigits_[i + offset] - other.bigits_[i] - borrow;
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bigits_[i + offset] = difference & kBigitMask;
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borrow = difference >> (kChunkSize - 1);
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while (borrow != 0) {
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Chunk difference = bigits_[i + offset] - borrow;
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bigits_[i + offset] = difference & kBigitMask;
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borrow = difference >> (kChunkSize - 1);
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void Bignum::ShiftLeft(int shift_amount) {
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if (used_digits_ == 0) return;
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exponent_ += shift_amount / kBigitSize;
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int local_shift = shift_amount % kBigitSize;
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EnsureCapacity(used_digits_ + 1);
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BigitsShiftLeft(local_shift);
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void Bignum::MultiplyByUInt32(uint32_t factor) {
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if (factor == 1) return;
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if (used_digits_ == 0) return;
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// The product of a bigit with the factor is of size kBigitSize + 32.
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// Assert that this number + 1 (for the carry) fits into double chunk.
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ASSERT(kDoubleChunkSize >= kBigitSize + 32 + 1);
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DoubleChunk carry = 0;
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for (int i = 0; i < used_digits_; ++i) {
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DoubleChunk product = static_cast<DoubleChunk>(factor) * bigits_[i] + carry;
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bigits_[i] = static_cast<Chunk>(product & kBigitMask);
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carry = (product >> kBigitSize);
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EnsureCapacity(used_digits_ + 1);
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bigits_[used_digits_] = carry & kBigitMask;
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carry >>= kBigitSize;
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void Bignum::MultiplyByUInt64(uint64_t factor) {
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if (factor == 1) return;
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ASSERT(kBigitSize < 32);
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uint64_t low = factor & 0xFFFFFFFF;
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uint64_t high = factor >> 32;
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for (int i = 0; i < used_digits_; ++i) {
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uint64_t product_low = low * bigits_[i];
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uint64_t product_high = high * bigits_[i];
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uint64_t tmp = (carry & kBigitMask) + product_low;
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bigits_[i] = tmp & kBigitMask;
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carry = (carry >> kBigitSize) + (tmp >> kBigitSize) +
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(product_high << (32 - kBigitSize));
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EnsureCapacity(used_digits_ + 1);
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bigits_[used_digits_] = carry & kBigitMask;
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carry >>= kBigitSize;
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void Bignum::MultiplyByPowerOfTen(int exponent) {
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const uint64_t kFive27 = UINT64_2PART_C(0x6765c793, fa10079d);
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const uint16_t kFive1 = 5;
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const uint16_t kFive2 = kFive1 * 5;
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const uint16_t kFive3 = kFive2 * 5;
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const uint16_t kFive4 = kFive3 * 5;
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const uint16_t kFive5 = kFive4 * 5;
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const uint16_t kFive6 = kFive5 * 5;
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const uint32_t kFive7 = kFive6 * 5;
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const uint32_t kFive8 = kFive7 * 5;
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const uint32_t kFive9 = kFive8 * 5;
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const uint32_t kFive10 = kFive9 * 5;
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const uint32_t kFive11 = kFive10 * 5;
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const uint32_t kFive12 = kFive11 * 5;
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const uint32_t kFive13 = kFive12 * 5;
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const uint32_t kFive1_to_12[] =
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{ kFive1, kFive2, kFive3, kFive4, kFive5, kFive6,
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kFive7, kFive8, kFive9, kFive10, kFive11, kFive12 };
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ASSERT(exponent >= 0);
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if (exponent == 0) return;
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if (used_digits_ == 0) return;
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// We shift by exponent at the end just before returning.
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int remaining_exponent = exponent;
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while (remaining_exponent >= 27) {
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MultiplyByUInt64(kFive27);
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remaining_exponent -= 27;
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while (remaining_exponent >= 13) {
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MultiplyByUInt32(kFive13);
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remaining_exponent -= 13;
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if (remaining_exponent > 0) {
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MultiplyByUInt32(kFive1_to_12[remaining_exponent - 1]);
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void Bignum::Square() {
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int product_length = 2 * used_digits_;
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EnsureCapacity(product_length);
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// Comba multiplication: compute each column separately.
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// Example: r = a2a1a0 * b2b1b0.
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// 10 * (a1b0 + a0b1) +
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// 100 * (a2b0 + a1b1 + a0b2) +
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// 1000 * (a2b1 + a1b2) +
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// In the worst case we have to accumulate nb-digits products of digit*digit.
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// Assert that the additional number of bits in a DoubleChunk are enough to
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// sum up used_digits of Bigit*Bigit.
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if ((1 << (2 * (kChunkSize - kBigitSize))) <= used_digits_) {
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DoubleChunk accumulator = 0;
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// First shift the digits so we don't overwrite them.
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int copy_offset = used_digits_;
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for (int i = 0; i < used_digits_; ++i) {
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bigits_[copy_offset + i] = bigits_[i];
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// We have two loops to avoid some 'if's in the loop.
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for (int i = 0; i < used_digits_; ++i) {
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// Process temporary digit i with power i.
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// The sum of the two indices must be equal to i.
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int bigit_index1 = i;
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int bigit_index2 = 0;
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// Sum all of the sub-products.
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while (bigit_index1 >= 0) {
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Chunk chunk1 = bigits_[copy_offset + bigit_index1];
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Chunk chunk2 = bigits_[copy_offset + bigit_index2];
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accumulator += static_cast<DoubleChunk>(chunk1) * chunk2;
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bigits_[i] = static_cast<Chunk>(accumulator) & kBigitMask;
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accumulator >>= kBigitSize;
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for (int i = used_digits_; i < product_length; ++i) {
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int bigit_index1 = used_digits_ - 1;
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int bigit_index2 = i - bigit_index1;
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// Invariant: sum of both indices is again equal to i.
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// Inner loop runs 0 times on last iteration, emptying accumulator.
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while (bigit_index2 < used_digits_) {
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Chunk chunk1 = bigits_[copy_offset + bigit_index1];
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Chunk chunk2 = bigits_[copy_offset + bigit_index2];
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accumulator += static_cast<DoubleChunk>(chunk1) * chunk2;
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// The overwritten bigits_[i] will never be read in further loop iterations,
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// because bigit_index1 and bigit_index2 are always greater
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// than i - used_digits_.
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bigits_[i] = static_cast<Chunk>(accumulator) & kBigitMask;
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accumulator >>= kBigitSize;
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// Since the result was guaranteed to lie inside the number the
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// accumulator must be 0 now.
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ASSERT(accumulator == 0);
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// Don't forget to update the used_digits and the exponent.
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used_digits_ = product_length;
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void Bignum::AssignPowerUInt16(uint16_t base, int power_exponent) {
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ASSERT(power_exponent >= 0);
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if (power_exponent == 0) {
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// We expect base to be in range 2-32, and most often to be 10.
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// It does not make much sense to implement different algorithms for counting
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while ((base & 1) == 0) {
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while (tmp_base != 0) {
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int final_size = bit_size * power_exponent;
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// 1 extra bigit for the shifting, and one for rounded final_size.
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EnsureCapacity(final_size / kBigitSize + 2);
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// Left to Right exponentiation.
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while (power_exponent >= mask) mask <<= 1;
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// The mask is now pointing to the bit above the most significant 1-bit of
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// Get rid of first 1-bit;
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uint64_t this_value = base;
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bool delayed_multipliciation = false;
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const uint64_t max_32bits = 0xFFFFFFFF;
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while (mask != 0 && this_value <= max_32bits) {
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this_value = this_value * this_value;
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// Verify that there is enough space in this_value to perform the
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// multiplication. The first bit_size bits must be 0.
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if ((power_exponent & mask) != 0) {
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uint64_t base_bits_mask =
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~((static_cast<uint64_t>(1) << (64 - bit_size)) - 1);
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bool high_bits_zero = (this_value & base_bits_mask) == 0;
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if (high_bits_zero) {
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delayed_multipliciation = true;
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AssignUInt64(this_value);
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if (delayed_multipliciation) {
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MultiplyByUInt32(base);
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// Now do the same thing as a bignum.
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if ((power_exponent & mask) != 0) {
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MultiplyByUInt32(base);
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// And finally add the saved shifts.
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ShiftLeft(shifts * power_exponent);
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// Precondition: this/other < 16bit.
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uint16_t Bignum::DivideModuloIntBignum(const Bignum& other) {
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ASSERT(other.IsClamped());
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ASSERT(other.used_digits_ > 0);
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// Easy case: if we have less digits than the divisor than the result is 0.
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// Note: this handles the case where this == 0, too.
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if (BigitLength() < other.BigitLength()) {
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// Start by removing multiples of 'other' until both numbers have the same
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while (BigitLength() > other.BigitLength()) {
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// This naive approach is extremely inefficient if the this divided other
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// might be big. This function is implemented for doubleToString where
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// the result should be small (less than 10).
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ASSERT(other.bigits_[other.used_digits_ - 1] >= ((1 << kBigitSize) / 16));
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// Remove the multiples of the first digit.
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// Example this = 23 and other equals 9. -> Remove 2 multiples.
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result += bigits_[used_digits_ - 1];
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SubtractTimes(other, bigits_[used_digits_ - 1]);
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ASSERT(BigitLength() == other.BigitLength());
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// Both bignums are at the same length now.
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// Since other has more than 0 digits we know that the access to
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// bigits_[used_digits_ - 1] is safe.
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Chunk this_bigit = bigits_[used_digits_ - 1];
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Chunk other_bigit = other.bigits_[other.used_digits_ - 1];
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if (other.used_digits_ == 1) {
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// Shortcut for easy (and common) case.
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int quotient = this_bigit / other_bigit;
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bigits_[used_digits_ - 1] = this_bigit - other_bigit * quotient;
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int division_estimate = this_bigit / (other_bigit + 1);
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result += division_estimate;
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SubtractTimes(other, division_estimate);
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if (other_bigit * (division_estimate + 1) > this_bigit) {
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// No need to even try to subtract. Even if other's remaining digits were 0
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// another subtraction would be too much.
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while (LessEqual(other, *this)) {
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SubtractBignum(other);
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static int SizeInHexChars(S number) {
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while (number != 0) {
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static char HexCharOfValue(int value) {
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ASSERT(0 <= value && value <= 16);
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if (value < 10) return value + '0';
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return value - 10 + 'A';
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bool Bignum::ToHexString(char* buffer, int buffer_size) const {
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// Each bigit must be printable as separate hex-character.
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ASSERT(kBigitSize % 4 == 0);
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const int kHexCharsPerBigit = kBigitSize / 4;
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if (used_digits_ == 0) {
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if (buffer_size < 2) return false;
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// We add 1 for the terminating '\0' character.
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int needed_chars = (BigitLength() - 1) * kHexCharsPerBigit +
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SizeInHexChars(bigits_[used_digits_ - 1]) + 1;
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if (needed_chars > buffer_size) return false;
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int string_index = needed_chars - 1;
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buffer[string_index--] = '\0';
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for (int i = 0; i < exponent_; ++i) {
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for (int j = 0; j < kHexCharsPerBigit; ++j) {
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buffer[string_index--] = '0';
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for (int i = 0; i < used_digits_ - 1; ++i) {
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Chunk current_bigit = bigits_[i];
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for (int j = 0; j < kHexCharsPerBigit; ++j) {
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buffer[string_index--] = HexCharOfValue(current_bigit & 0xF);
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// And finally the last bigit.
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Chunk most_significant_bigit = bigits_[used_digits_ - 1];
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while (most_significant_bigit != 0) {
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buffer[string_index--] = HexCharOfValue(most_significant_bigit & 0xF);
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most_significant_bigit >>= 4;
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Bignum::Chunk Bignum::BigitAt(int index) const {
609
if (index >= BigitLength()) return 0;
610
if (index < exponent_) return 0;
611
return bigits_[index - exponent_];
615
int Bignum::Compare(const Bignum& a, const Bignum& b) {
616
ASSERT(a.IsClamped());
617
ASSERT(b.IsClamped());
618
int bigit_length_a = a.BigitLength();
619
int bigit_length_b = b.BigitLength();
620
if (bigit_length_a < bigit_length_b) return -1;
621
if (bigit_length_a > bigit_length_b) return +1;
622
for (int i = bigit_length_a - 1; i >= Min(a.exponent_, b.exponent_); --i) {
623
Chunk bigit_a = a.BigitAt(i);
624
Chunk bigit_b = b.BigitAt(i);
625
if (bigit_a < bigit_b) return -1;
626
if (bigit_a > bigit_b) return +1;
627
// Otherwise they are equal up to this digit. Try the next digit.
633
int Bignum::PlusCompare(const Bignum& a, const Bignum& b, const Bignum& c) {
634
ASSERT(a.IsClamped());
635
ASSERT(b.IsClamped());
636
ASSERT(c.IsClamped());
637
if (a.BigitLength() < b.BigitLength()) {
638
return PlusCompare(b, a, c);
640
if (a.BigitLength() + 1 < c.BigitLength()) return -1;
641
if (a.BigitLength() > c.BigitLength()) return +1;
642
// The exponent encodes 0-bigits. So if there are more 0-digits in 'a' than
643
// 'b' has digits, then the bigit-length of 'a'+'b' must be equal to the one
645
if (a.exponent_ >= b.BigitLength() && a.BigitLength() < c.BigitLength()) {
650
// Starting at min_exponent all digits are == 0. So no need to compare them.
651
int min_exponent = Min(Min(a.exponent_, b.exponent_), c.exponent_);
652
for (int i = c.BigitLength() - 1; i >= min_exponent; --i) {
653
Chunk chunk_a = a.BigitAt(i);
654
Chunk chunk_b = b.BigitAt(i);
655
Chunk chunk_c = c.BigitAt(i);
656
Chunk sum = chunk_a + chunk_b;
657
if (sum > chunk_c + borrow) {
660
borrow = chunk_c + borrow - sum;
661
if (borrow > 1) return -1;
662
borrow <<= kBigitSize;
665
if (borrow == 0) return 0;
670
void Bignum::Clamp() {
671
while (used_digits_ > 0 && bigits_[used_digits_ - 1] == 0) {
674
if (used_digits_ == 0) {
681
bool Bignum::IsClamped() const {
682
return used_digits_ == 0 || bigits_[used_digits_ - 1] != 0;
686
void Bignum::Zero() {
687
for (int i = 0; i < used_digits_; ++i) {
695
void Bignum::Align(const Bignum& other) {
696
if (exponent_ > other.exponent_) {
697
// If "X" represents a "hidden" digit (by the exponent) then we are in the
698
// following case (a == this, b == other):
699
// a: aaaaaaXXXX or a: aaaaaXXX
700
// b: bbbbbbX b: bbbbbbbbXX
701
// We replace some of the hidden digits (X) of a with 0 digits.
702
// a: aaaaaa000X or a: aaaaa0XX
703
int zero_digits = exponent_ - other.exponent_;
704
EnsureCapacity(used_digits_ + zero_digits);
705
for (int i = used_digits_ - 1; i >= 0; --i) {
706
bigits_[i + zero_digits] = bigits_[i];
708
for (int i = 0; i < zero_digits; ++i) {
711
used_digits_ += zero_digits;
712
exponent_ -= zero_digits;
713
ASSERT(used_digits_ >= 0);
714
ASSERT(exponent_ >= 0);
719
void Bignum::BigitsShiftLeft(int shift_amount) {
720
ASSERT(shift_amount < kBigitSize);
721
ASSERT(shift_amount >= 0);
723
for (int i = 0; i < used_digits_; ++i) {
724
Chunk new_carry = bigits_[i] >> (kBigitSize - shift_amount);
725
bigits_[i] = ((bigits_[i] << shift_amount) + carry) & kBigitMask;
729
bigits_[used_digits_] = carry;
735
void Bignum::SubtractTimes(const Bignum& other, int factor) {
736
ASSERT(exponent_ <= other.exponent_);
738
for (int i = 0; i < factor; ++i) {
739
SubtractBignum(other);
744
int exponent_diff = other.exponent_ - exponent_;
745
for (int i = 0; i < other.used_digits_; ++i) {
746
DoubleChunk product = static_cast<DoubleChunk>(factor) * other.bigits_[i];
747
DoubleChunk remove = borrow + product;
748
Chunk difference = bigits_[i + exponent_diff] - (remove & kBigitMask);
749
bigits_[i + exponent_diff] = difference & kBigitMask;
750
borrow = static_cast<Chunk>((difference >> (kChunkSize - 1)) +
751
(remove >> kBigitSize));
753
for (int i = other.used_digits_ + exponent_diff; i < used_digits_; ++i) {
754
if (borrow == 0) return;
755
Chunk difference = bigits_[i] - borrow;
756
bigits_[i] = difference & kBigitMask;
757
borrow = difference >> (kChunkSize - 1);
764
} // namespace double_conversion
added
removed
mozilla/mfbt/double-conversion/bignum.cc
