// Copyright 2011 the V8 project authors. All rights reserved. // Use of this source code is governed by a BSD-style license that can be // found in the LICENSE file. #include "src/numbers/bignum.h" #include "src/utils/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) { DCHECK_GE(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] = static_cast<Chunk>(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; int to = from + digits_to_read; for (int i = from; i < to; ++i) { int digit = buffer[i] - '0'; DCHECK(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(); } 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) { DCHECK(IsClamped()); DCHECK(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_; DCHECK_GE(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_); DCHECK(IsClamped()); } void Bignum::SubtractBignum(const Bignum& other) { DCHECK(IsClamped()); DCHECK(other.IsClamped()); // We require this to be bigger than other. DCHECK(LessEqual(other, *this)); Align(other); int offset = other.exponent_ - exponent_; Chunk borrow = 0; int i; for (i = 0; i < other.used_digits_; ++i) { DCHECK((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. DCHECK_GE(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_] = static_cast<Chunk>(carry & kBigitMask); used_digits_++; carry >>= kBigitSize; } } void Bignum::MultiplyByUInt64(uint64_t factor) { if (factor == 1) return; if (factor == 0) { Zero(); return; } DCHECK_LT(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] = static_cast<Chunk>(tmp & kBigitMask); carry = (carry >> kBigitSize) + (tmp >> kBigitSize) + (product_high << (32 - kBigitSize)); } while (carry != 0) { EnsureCapacity(used_digits_ + 1); bigits_[used_digits_] = static_cast<Chunk>(carry & kBigitMask); used_digits_++; carry >>= kBigitSize; } } void Bignum::MultiplyByPowerOfTen(int exponent) { const uint64_t kFive27 = 0x6765'C793'FA10'079D; 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}; DCHECK_GE(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() { DCHECK(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. DCHECK_EQ(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) { DCHECK_NE(base, 0); DCHECK_GE(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) { DCHECK(IsClamped()); DCHECK(other.IsClamped()); DCHECK_GT(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). DCHECK(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]); } DCHECK(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) { DCHECK_GT(number, 0); int result = 0; while (number != 0) { number >>= 4; result++; } return result; } bool Bignum::ToHexString(char* buffer, int buffer_size) const { DCHECK(IsClamped()); // Each bigit must be printable as separate hex-character. DCHECK_EQ(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) { DCHECK(a.IsClamped()); DCHECK(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) { DCHECK(a.IsClamped()); DCHECK(b.IsClamped()); DCHECK(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; DCHECK_GE(used_digits_, 0); DCHECK_GE(exponent_, 0); } } void Bignum::BigitsShiftLeft(int shift_amount) { DCHECK_LT(shift_amount, kBigitSize); DCHECK_GE(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) { #ifdef DEBUG Bignum a, b; a.AssignBignum(*this); b.AssignBignum(other); b.MultiplyByUInt32(factor); a.SubtractBignum(b); #endif DCHECK(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] - static_cast<Chunk>(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(); DCHECK(Bignum::Equal(a, *this)); } } // namespace internal } // namespace v8