// Copyright 2018 The Abseil Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#ifndef ABSL_STRINGS_INTERNAL_CHARCONV_BIGINT_H_
#define ABSL_STRINGS_INTERNAL_CHARCONV_BIGINT_H_
#include <algorithm>
#include <cstdint>
#include <iostream>
#include <string>
#include "absl/strings/ascii.h"
#include "absl/strings/internal/charconv_parse.h"
#include "absl/strings/string_view.h"
namespace absl {
namespace strings_internal {
// The largest power that 5 that can be raised to, and still fit in a uint32_t.
constexpr int kMaxSmallPowerOfFive = 13;
// The largest power that 10 that can be raised to, and still fit in a uint32_t.
constexpr int kMaxSmallPowerOfTen = 9;
extern const uint32_t kFiveToNth[kMaxSmallPowerOfFive + 1];
extern const uint32_t kTenToNth[kMaxSmallPowerOfTen + 1];
// Large, fixed-width unsigned integer.
//
// Exact rounding for decimal-to-binary floating point conversion requires very
// large integer math, but a design goal of absl::from_chars is to avoid
// allocating memory. The integer precision needed for decimal-to-binary
// conversions is large but bounded, so a huge fixed-width integer class
// suffices.
//
// This is an intentionally limited big integer class. Only needed operations
// are implemented. All storage lives in an array data member, and all
// arithmetic is done in-place, to avoid requiring separate storage for operand
// and result.
//
// This is an internal class. Some methods live in the .cc file, and are
// instantiated only for the values of max_words we need.
template <int max_words>
class BigUnsigned {
public:
static_assert(max_words == 4 || max_words == 84,
"unsupported max_words value");
BigUnsigned() : size_(0), words_{} {}
explicit BigUnsigned(uint32_t v) : size_(v > 0 ? 1 : 0), words_{v} {}
explicit BigUnsigned(uint64_t v)
: size_(0),
words_{static_cast<uint32_t>(v & 0xffffffff),
static_cast<uint32_t>(v >> 32)} {
if (words_[1]) {
size_ = 2;
} else if (words_[0]) {
size_ = 1;
}
}
// Constructs a BigUnsigned from the given string_view containing a decimal
// value. If the input std::string is not a decimal integer, constructs a 0
// instead.
explicit BigUnsigned(absl::string_view sv) : size_(0), words_{} {
// Check for valid input, returning a 0 otherwise. This is reasonable
// behavior only because this constructor is for unit tests.
if (std::find_if_not(sv.begin(), sv.end(), ascii_isdigit) != sv.end() ||
sv.empty()) {
return;
}
int exponent_adjust =
ReadDigits(sv.data(), sv.data() + sv.size(), Digits10() + 1);
if (exponent_adjust > 0) {
MultiplyByTenToTheNth(exponent_adjust);
}
}
// Loads the mantissa value of a previously-parsed float.
//
// Returns the associated decimal exponent. The value of the parsed float is
// exactly *this * 10**exponent.
int ReadFloatMantissa(const ParsedFloat& fp, int significant_digits);
// Returns the number of decimal digits of precision this type provides. All
// numbers with this many decimal digits or fewer are representable by this
// type.
//
// Analagous to std::numeric_limits<BigUnsigned>::digits10.
static constexpr int Digits10() {
// 9975007/1035508 is very slightly less than log10(2**32).
return static_cast<uint64_t>(max_words) * 9975007 / 1035508;
}
// Shifts left by the given number of bits.
void ShiftLeft(int count) {
if (count > 0) {
const int word_shift = count / 32;
if (word_shift >= max_words) {
SetToZero();
return;
}
size_ = std::min(size_ + word_shift, max_words);
count %= 32;
if (count == 0) {
std::copy_backward(words_, words_ + size_ - word_shift, words_ + size_);
} else {
for (int i = std::min(size_, max_words - 1); i > word_shift; --i) {
words_[i] = (words_[i - word_shift] << count) |
(words_[i - word_shift - 1] >> (32 - count));
}
words_[word_shift] = words_[0] << count;
// Grow size_ if necessary.
if (size_ < max_words && words_[size_]) {
++size_;
}
}
std::fill(words_, words_ + word_shift, 0u);
}
}
// Multiplies by v in-place.
void MultiplyBy(uint32_t v) {
if (size_ == 0 || v == 1) {
return;
}
if (v == 0) {
SetToZero();
return;
}
const uint64_t factor = v;
uint64_t window = 0;
for (int i = 0; i < size_; ++i) {
window += factor * words_[i];
words_[i] = window & 0xffffffff;
window >>= 32;
}
// If carry bits remain and there's space for them, grow size_.
if (window && size_ < max_words) {
words_[size_] = window & 0xffffffff;
++size_;
}
}
void MultiplyBy(uint64_t v) {
uint32_t words[2];
words[0] = static_cast<uint32_t>(v);
words[1] = static_cast<uint32_t>(v >> 32);
if (words[1] == 0) {
MultiplyBy(words[0]);
} else {
MultiplyBy(2, words);
}
}
// Multiplies in place by 5 to the power of n. n must be non-negative.
void MultiplyByFiveToTheNth(int n) {
while (n >= kMaxSmallPowerOfFive) {
MultiplyBy(kFiveToNth[kMaxSmallPowerOfFive]);
n -= kMaxSmallPowerOfFive;
}
if (n > 0) {
MultiplyBy(kFiveToNth[n]);
}
}
// Multiplies in place by 10 to the power of n. n must be non-negative.
void MultiplyByTenToTheNth(int n) {
if (n > kMaxSmallPowerOfTen) {
// For large n, raise to a power of 5, then shift left by the same amount.
// (10**n == 5**n * 2**n.) This requires fewer multiplications overall.
MultiplyByFiveToTheNth(n);
ShiftLeft(n);
} else if (n > 0) {
// We can do this more quickly for very small N by using a single
// multiplication.
MultiplyBy(kTenToNth[n]);
}
}
// Returns the value of 5**n, for non-negative n. This implementation uses
// a lookup table, and is faster then seeding a BigUnsigned with 1 and calling
// MultiplyByFiveToTheNth().
static BigUnsigned FiveToTheNth(int n);
// Multiplies by another BigUnsigned, in-place.
template <int M>
void MultiplyBy(const BigUnsigned<M>& other) {
MultiplyBy(other.size(), other.words());
}
void SetToZero() {
std::fill(words_, words_ + size_, 0u);
size_ = 0;
}
// Returns the value of the nth word of this BigUnsigned. This is
// range-checked, and returns 0 on out-of-bounds accesses.
uint32_t GetWord(int index) const {
if (index < 0 || index >= size_) {
return 0;
}
return words_[index];
}
// Returns this integer as a decimal std::string. This is not used in the decimal-
// to-binary conversion; it is intended to aid in testing.
std::string ToString() const;
int size() const { return size_; }
const uint32_t* words() const { return words_; }
private:
// Reads the number between [begin, end), possibly containing a decimal point,
// into this BigUnsigned.
//
// Callers are required to ensure [begin, end) contains a valid number, with
// one or more decimal digits and at most one decimal point. This routine
// will behave unpredictably if these preconditions are not met.
//
// Only the first `significant_digits` digits are read. Digits beyond this
// limit are "sticky": If the final significant digit is 0 or 5, and if any
// dropped digit is nonzero, then that final significant digit is adjusted up
// to 1 or 6. This adjustment allows for precise rounding.
//
// Returns `exponent_adjustment`, a power-of-ten exponent adjustment to
// account for the decimal point and for dropped significant digits. After
// this function returns,
// actual_value_of_parsed_string ~= *this * 10**exponent_adjustment.
int ReadDigits(const char* begin, const char* end, int significant_digits);
// Performs a step of big integer multiplication. This computes the full
// (64-bit-wide) values that should be added at the given index (step), and
// adds to that location in-place.
//
// Because our math all occurs in place, we must multiply starting from the
// highest word working downward. (This is a bit more expensive due to the
// extra carries involved.)
//
// This must be called in steps, for each word to be calculated, starting from
// the high end and working down to 0. The first value of `step` should be
// `std::min(original_size + other.size_ - 2, max_words - 1)`.
// The reason for this expression is that multiplying the i'th word from one
// multiplicand and the j'th word of another multiplicand creates a
// two-word-wide value to be stored at the (i+j)'th element. The highest
// word indices we will access are `original_size - 1` from this object, and
// `other.size_ - 1` from our operand. Therefore,
// `original_size + other.size_ - 2` is the first step we should calculate,
// but limited on an upper bound by max_words.
// Working from high-to-low ensures that we do not overwrite the portions of
// the initial value of *this which are still needed for later steps.
//
// Once called with step == 0, *this contains the result of the
// multiplication.
//
// `original_size` is the size_ of *this before the first call to
// MultiplyStep(). `other_words` and `other_size` are the contents of our
// operand. `step` is the step to perform, as described above.
void MultiplyStep(int original_size, const uint32_t* other_words,
int other_size, int step);
void MultiplyBy(int other_size, const uint32_t* other_words) {
const int original_size = size_;
const int first_step =
std::min(original_size + other_size - 2, max_words - 1);
for (int step = first_step; step >= 0; --step) {
MultiplyStep(original_size, other_words, other_size, step);
}
}
// Adds a 32-bit value to the index'th word, with carry.
void AddWithCarry(int index, uint32_t value) {
if (value) {
while (index < max_words && value > 0) {
words_[index] += value;
// carry if we overflowed in this word:
if (value > words_[index]) {
value = 1;
++index;
} else {
value = 0;
}
}
size_ = std::min(max_words, std::max(index + 1, size_));
}
}
void AddWithCarry(int index, uint64_t value) {
if (value && index < max_words) {
uint32_t high = value >> 32;
uint32_t low = value & 0xffffffff;
words_[index] += low;
if (words_[index] < low) {
++high;
if (high == 0) {
// Carry from the low word caused our high word to overflow.
// Short circuit here to do the right thing.
AddWithCarry(index + 2, static_cast<uint32_t>(1));
return;
}
}
if (high > 0) {
AddWithCarry(index + 1, high);
} else {
// Normally 32-bit AddWithCarry() sets size_, but since we don't call
// it when `high` is 0, do it ourselves here.
size_ = std::min(max_words, std::max(index + 1, size_));
}
}
}
// Divide this in place by a constant divisor. Returns the remainder of the
// division.
template <uint32_t divisor>
uint32_t DivMod() {
uint64_t accumulator = 0;
for (int i = size_ - 1; i >= 0; --i) {
accumulator <<= 32;
accumulator += words_[i];
// accumulator / divisor will never overflow an int32_t in this loop
words_[i] = static_cast<uint32_t>(accumulator / divisor);
accumulator = accumulator % divisor;
}
while (size_ > 0 && words_[size_ - 1] == 0) {
--size_;
}
return static_cast<uint32_t>(accumulator);
}
// The number of elements in words_ that may carry significant values.
// All elements beyond this point are 0.
//
// When size_ is 0, this BigUnsigned stores the value 0.
// When size_ is nonzero, is *not* guaranteed that words_[size_ - 1] is
// nonzero. This can occur due to overflow truncation.
// In particular, x.size_ != y.size_ does *not* imply x != y.
int size_;
uint32_t words_[max_words];
};
// Compares two big integer instances.
//
// Returns -1 if lhs < rhs, 0 if lhs == rhs, and 1 if lhs > rhs.
template <int N, int M>
int Compare(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
int limit = std::max(lhs.size(), rhs.size());
for (int i = limit - 1; i >= 0; --i) {
const uint32_t lhs_word = lhs.GetWord(i);
const uint32_t rhs_word = rhs.GetWord(i);
if (lhs_word < rhs_word) {
return -1;
} else if (lhs_word > rhs_word) {
return 1;
}
}
return 0;
}
template <int N, int M>
bool operator==(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
int limit = std::max(lhs.size(), rhs.size());
for (int i = 0; i < limit; ++i) {
if (lhs.GetWord(i) != rhs.GetWord(i)) {
return false;
}
}
return true;
}
template <int N, int M>
bool operator!=(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
return !(lhs == rhs);
}
template <int N, int M>
bool operator<(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
return Compare(lhs, rhs) == -1;
}
template <int N, int M>
bool operator>(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
return rhs < lhs;
}
template <int N, int M>
bool operator<=(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
return !(rhs < lhs);
}
template <int N, int M>
bool operator>=(const BigUnsigned<N>& lhs, const BigUnsigned<M>& rhs) {
return !(lhs < rhs);
}
// Output operator for BigUnsigned, for testing purposes only.
template <int N>
std::ostream& operator<<(std::ostream& os, const BigUnsigned<N>& num) {
return os << num.ToString();
}
// Explicit instantiation declarations for the sizes of BigUnsigned that we
// are using.
//
// For now, the choices of 4 and 84 are arbitrary; 4 is a small value that is
// still bigger than an int128, and 84 is a large value we will want to use
// in the from_chars implementation.
//
// Comments justifying the use of 84 belong in the from_chars implementation,
// and will be added in a follow-up CL.
extern template class BigUnsigned<4>;
extern template class BigUnsigned<84>;
} // namespace strings_internal
} // namespace absl
#endif // ABSL_STRINGS_INTERNAL_CHARCONV_BIGINT_H_
