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//===-- A class to manipulate wide integers. --------------------*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_SRC___SUPPORT_UINT_H
#define LLVM_LIBC_SRC___SUPPORT_UINT_H
#include "src/__support/CPP/array.h"
#include "src/__support/CPP/bit.h" // countl_zero
#include "src/__support/CPP/limits.h"
#include "src/__support/CPP/optional.h"
#include "src/__support/CPP/type_traits.h"
#include "src/__support/macros/attributes.h" // LIBC_INLINE
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/compiler.h" // LIBC_COMPILER_IS_CLANG
#include "src/__support/macros/properties/types.h" // LIBC_TYPES_HAS_INT128, LIBC_TYPES_HAS_INT64
#include "src/__support/math_extras.h" // add_with_carry, sub_with_borrow
#include "src/__support/number_pair.h"
#include <stddef.h> // For size_t
#include <stdint.h>
namespace LIBC_NAMESPACE {
namespace multiword {
// A type trait mapping unsigned integers to their half-width unsigned
// counterparts.
template <typename T> struct half_width;
template <> struct half_width<uint16_t> : cpp::type_identity<uint8_t> {};
template <> struct half_width<uint32_t> : cpp::type_identity<uint16_t> {};
#ifdef LIBC_TYPES_HAS_INT64
template <> struct half_width<uint64_t> : cpp::type_identity<uint32_t> {};
#ifdef LIBC_TYPES_HAS_INT128
template <> struct half_width<__uint128_t> : cpp::type_identity<uint64_t> {};
#endif // LIBC_TYPES_HAS_INT128
#endif // LIBC_TYPES_HAS_INT64
template <typename T> using half_width_t = typename half_width<T>::type;
// An array of two elements that can be used in multiword operations.
template <typename T> struct DoubleWide final : cpp::array<T, 2> {
using UP = cpp::array<T, 2>;
using UP::UP;
LIBC_INLINE constexpr DoubleWide(T lo, T hi) : UP({lo, hi}) {}
};
// Converts an unsigned value into a DoubleWide<half_width_t<T>>.
template <typename T> LIBC_INLINE constexpr auto split(T value) {
static_assert(cpp::is_unsigned_v<T>);
using half_type = half_width_t<T>;
return DoubleWide<half_type>(
half_type(value),
half_type(value >> cpp::numeric_limits<half_type>::digits));
}
// The low part of a DoubleWide value.
template <typename T> LIBC_INLINE constexpr T lo(const DoubleWide<T> &value) {
return value[0];
}
// The high part of a DoubleWide value.
template <typename T> LIBC_INLINE constexpr T hi(const DoubleWide<T> &value) {
return value[1];
}
// The low part of an unsigned value.
template <typename T> LIBC_INLINE constexpr half_width_t<T> lo(T value) {
return lo(split(value));
}
// The high part of an unsigned value.
template <typename T> LIBC_INLINE constexpr half_width_t<T> hi(T value) {
return hi(split(value));
}
// Returns 'a' times 'b' in a DoubleWide<word>. Cannot overflow by construction.
template <typename word>
LIBC_INLINE constexpr DoubleWide<word> mul2(word a, word b) {
if constexpr (cpp::is_same_v<word, uint8_t>) {
return split<uint16_t>(uint16_t(a) * uint16_t(b));
} else if constexpr (cpp::is_same_v<word, uint16_t>) {
return split<uint32_t>(uint32_t(a) * uint32_t(b));
}
#ifdef LIBC_TYPES_HAS_INT64
else if constexpr (cpp::is_same_v<word, uint32_t>) {
return split<uint64_t>(uint64_t(a) * uint64_t(b));
}
#endif
#ifdef LIBC_TYPES_HAS_INT128
else if constexpr (cpp::is_same_v<word, uint64_t>) {
return split<__uint128_t>(__uint128_t(a) * __uint128_t(b));
}
#endif
else {
using half_word = half_width_t<word>;
const auto shiftl = [](word value) -> word {
return value << cpp::numeric_limits<half_word>::digits;
};
const auto shiftr = [](word value) -> word {
return value >> cpp::numeric_limits<half_word>::digits;
};
// Here we do a one digit multiplication where 'a' and 'b' are of type
// word. We split 'a' and 'b' into half words and perform the classic long
// multiplication with 'a' and 'b' being two-digit numbers.
// a a_hi a_lo
// x b => x b_hi b_lo
// ---- -----------
// c result
// We convert 'lo' and 'hi' from 'half_word' to 'word' so multiplication
// doesn't overflow.
const word a_lo = lo(a);
const word b_lo = lo(b);
const word a_hi = hi(a);
const word b_hi = hi(b);
const word step1 = b_lo * a_lo; // no overflow;
const word step2 = b_lo * a_hi; // no overflow;
const word step3 = b_hi * a_lo; // no overflow;
const word step4 = b_hi * a_hi; // no overflow;
word lo_digit = step1;
word hi_digit = step4;
const word no_carry = 0;
word carry;
word _; // unused carry variable.
lo_digit = add_with_carry<word>(lo_digit, shiftl(step2), no_carry, carry);
hi_digit = add_with_carry<word>(hi_digit, shiftr(step2), carry, _);
lo_digit = add_with_carry<word>(lo_digit, shiftl(step3), no_carry, carry);
hi_digit = add_with_carry<word>(hi_digit, shiftr(step3), carry, _);
return DoubleWide<word>(lo_digit, hi_digit);
}
}
// In-place 'dst op= rhs' with operation with carry propagation. Returns carry.
template <typename Function, typename word, size_t N, size_t M>
LIBC_INLINE constexpr word inplace_binop(Function op_with_carry,
cpp::array<word, N> &dst,
const cpp::array<word, M> &rhs) {
static_assert(N >= M);
word carry_out = 0;
for (size_t i = 0; i < N; ++i) {
const bool has_rhs_value = i < M;
const word rhs_value = has_rhs_value ? rhs[i] : 0;
const word carry_in = carry_out;
dst[i] = op_with_carry(dst[i], rhs_value, carry_in, carry_out);
// stop early when rhs is over and no carry is to be propagated.
if (!has_rhs_value && carry_out == 0)
break;
}
return carry_out;
}
// In-place addition. Returns carry.
template <typename word, size_t N, size_t M>
LIBC_INLINE constexpr word add_with_carry(cpp::array<word, N> &dst,
const cpp::array<word, M> &rhs) {
return inplace_binop(LIBC_NAMESPACE::add_with_carry<word>, dst, rhs);
}
// In-place subtraction. Returns borrow.
template <typename word, size_t N, size_t M>
LIBC_INLINE constexpr word sub_with_borrow(cpp::array<word, N> &dst,
const cpp::array<word, M> &rhs) {
return inplace_binop(LIBC_NAMESPACE::sub_with_borrow<word>, dst, rhs);
}
// In-place multiply-add. Returns carry.
// i.e., 'dst += b * c'
template <typename word, size_t N>
LIBC_INLINE constexpr word mul_add_with_carry(cpp::array<word, N> &dst, word b,
word c) {
return add_with_carry(dst, mul2(b, c));
}
// An array of two elements serving as an accumulator during multiword
// computations.
template <typename T> struct Accumulator final : cpp::array<T, 2> {
using UP = cpp::array<T, 2>;
LIBC_INLINE constexpr Accumulator() : UP({0, 0}) {}
LIBC_INLINE constexpr T advance(T carry_in) {
auto result = UP::front();
UP::front() = UP::back();
UP::back() = carry_in;
return result;
}
LIBC_INLINE constexpr T sum() const { return UP::front(); }
LIBC_INLINE constexpr T carry() const { return UP::back(); }
};
// In-place multiplication by a single word. Returns carry.
template <typename word, size_t N>
LIBC_INLINE constexpr word scalar_multiply_with_carry(cpp::array<word, N> &dst,
word x) {
Accumulator<word> acc;
for (auto &val : dst) {
const word carry = mul_add_with_carry(acc, val, x);
val = acc.advance(carry);
}
return acc.carry();
}
// Multiplication of 'lhs' by 'rhs' into 'dst'. Returns carry.
// This function is safe to use for signed numbers.
// https://stackoverflow.com/a/20793834
// https://pages.cs.wisc.edu/%7Emarkhill/cs354/Fall2008/beyond354/int.mult.html
template <typename word, size_t O, size_t M, size_t N>
LIBC_INLINE constexpr word multiply_with_carry(cpp::array<word, O> &dst,
const cpp::array<word, M> &lhs,
const cpp::array<word, N> &rhs) {
static_assert(O >= M + N);
Accumulator<word> acc;
for (size_t i = 0; i < O; ++i) {
const size_t lower_idx = i < N ? 0 : i - N + 1;
const size_t upper_idx = i < M ? i : M - 1;
word carry = 0;
for (size_t j = lower_idx; j <= upper_idx; ++j)
carry += mul_add_with_carry(acc, lhs[j], rhs[i - j]);
dst[i] = acc.advance(carry);
}
return acc.carry();
}
template <typename word, size_t N>
LIBC_INLINE constexpr void quick_mul_hi(cpp::array<word, N> &dst,
const cpp::array<word, N> &lhs,
const cpp::array<word, N> &rhs) {
Accumulator<word> acc;
word carry = 0;
// First round of accumulation for those at N - 1 in the full product.
for (size_t i = 0; i < N; ++i)
carry += mul_add_with_carry(acc, lhs[i], rhs[N - 1 - i]);
for (size_t i = N; i < 2 * N - 1; ++i) {
acc.advance(carry);
carry = 0;
for (size_t j = i - N + 1; j < N; ++j)
carry += mul_add_with_carry(acc, lhs[j], rhs[i - j]);
dst[i - N] = acc.sum();
}
dst.back() = acc.carry();
}
template <typename word, size_t N>
LIBC_INLINE constexpr bool is_negative(cpp::array<word, N> &array) {
using signed_word = cpp::make_signed_t<word>;
return cpp::bit_cast<signed_word>(array.back()) < 0;
}
// An enum for the shift function below.
enum Direction { LEFT, RIGHT };
// A bitwise shift on an array of elements.
// 'offset' must be less than TOTAL_BITS (i.e., sizeof(word) * CHAR_BIT * N)
// otherwise the behavior is undefined.
template <Direction direction, bool is_signed, typename word, size_t N>
LIBC_INLINE constexpr cpp::array<word, N> shift(cpp::array<word, N> array,
size_t offset) {
static_assert(direction == LEFT || direction == RIGHT);
constexpr size_t WORD_BITS = cpp::numeric_limits<word>::digits;
#ifdef LIBC_TYPES_HAS_INT128
constexpr size_t TOTAL_BITS = N * WORD_BITS;
if constexpr (TOTAL_BITS == 128) {
using type = cpp::conditional_t<is_signed, __int128_t, __uint128_t>;
auto tmp = cpp::bit_cast<type>(array);
if constexpr (direction == LEFT)
tmp <<= offset;
else
tmp >>= offset;
return cpp::bit_cast<cpp::array<word, N>>(tmp);
}
#endif
if (LIBC_UNLIKELY(offset == 0))
return array;
const bool is_neg = is_signed && is_negative(array);
constexpr auto at = [](size_t index) -> int {
// reverse iteration when direction == LEFT.
if constexpr (direction == LEFT)
return int(N) - int(index) - 1;
return int(index);
};
const auto safe_get_at = [&](size_t index) -> word {
// return appropriate value when accessing out of bound elements.
const int i = at(index);
if (i < 0)
return 0;
if (i >= int(N))
return is_neg ? -1 : 0;
return array[i];
};
const size_t index_offset = offset / WORD_BITS;
const size_t bit_offset = offset % WORD_BITS;
#ifdef LIBC_COMPILER_IS_CLANG
__builtin_assume(index_offset < N);
#endif
cpp::array<word, N> out = {};
for (size_t index = 0; index < N; ++index) {
const word part1 = safe_get_at(index + index_offset);
const word part2 = safe_get_at(index + index_offset + 1);
word &dst = out[at(index)];
if (bit_offset == 0)
dst = part1; // no crosstalk between parts.
else if constexpr (direction == LEFT)
dst = (part1 << bit_offset) | (part2 >> (WORD_BITS - bit_offset));
else
dst = (part1 >> bit_offset) | (part2 << (WORD_BITS - bit_offset));
}
return out;
}
#define DECLARE_COUNTBIT(NAME, INDEX_EXPR) \
template <typename word, size_t N> \
LIBC_INLINE constexpr int NAME(const cpp::array<word, N> &val) { \
int bit_count = 0; \
for (size_t i = 0; i < N; ++i) { \
const int word_count = cpp::NAME<word>(val[INDEX_EXPR]); \
bit_count += word_count; \
if (word_count != cpp::numeric_limits<word>::digits) \
break; \
} \
return bit_count; \
}
DECLARE_COUNTBIT(countr_zero, i) // iterating forward
DECLARE_COUNTBIT(countr_one, i) // iterating forward
DECLARE_COUNTBIT(countl_zero, N - i - 1) // iterating backward
DECLARE_COUNTBIT(countl_one, N - i - 1) // iterating backward
} // namespace multiword
template <size_t Bits, bool Signed, typename WordType = uint64_t>
struct BigInt {
private:
static_assert(cpp::is_integral_v<WordType> && cpp::is_unsigned_v<WordType>,
"WordType must be unsigned integer.");
struct Division {
BigInt quotient;
BigInt remainder;
};
public:
using word_type = WordType;
using unsigned_type = BigInt<Bits, false, word_type>;
using signed_type = BigInt<Bits, true, word_type>;
LIBC_INLINE_VAR static constexpr bool SIGNED = Signed;
LIBC_INLINE_VAR static constexpr size_t BITS = Bits;
LIBC_INLINE_VAR
static constexpr size_t WORD_SIZE = sizeof(WordType) * CHAR_BIT;
static_assert(Bits > 0 && Bits % WORD_SIZE == 0,
"Number of bits in BigInt should be a multiple of WORD_SIZE.");
LIBC_INLINE_VAR static constexpr size_t WORD_COUNT = Bits / WORD_SIZE;
cpp::array<WordType, WORD_COUNT> val{}; // zero initialized.
LIBC_INLINE constexpr BigInt() = default;
LIBC_INLINE constexpr BigInt(const BigInt &other) = default;
template <size_t OtherBits, bool OtherSigned>
LIBC_INLINE constexpr BigInt(
const BigInt<OtherBits, OtherSigned, WordType> &other) {
if (OtherBits >= Bits) { // truncate
for (size_t i = 0; i < WORD_COUNT; ++i)
val[i] = other[i];
} else { // zero or sign extend
size_t i = 0;
for (; i < OtherBits / WORD_SIZE; ++i)
val[i] = other[i];
extend(i, Signed && other.is_neg());
}
}
// Construct a BigInt from a C array.
template <size_t N> LIBC_INLINE constexpr BigInt(const WordType (&nums)[N]) {
static_assert(N == WORD_COUNT);
for (size_t i = 0; i < WORD_COUNT; ++i)
val[i] = nums[i];
}
LIBC_INLINE constexpr explicit BigInt(
const cpp::array<WordType, WORD_COUNT> &words) {
val = words;
}
// Initialize the first word to |v| and the rest to 0.
template <typename T, typename = cpp::enable_if_t<cpp::is_integral_v<T>>>
LIBC_INLINE constexpr BigInt(T v) {
constexpr size_t T_SIZE = sizeof(T) * CHAR_BIT;
const bool is_neg = Signed && (v < 0);
for (size_t i = 0; i < WORD_COUNT; ++i) {
if (v == 0) {
extend(i, is_neg);
return;
}
val[i] = static_cast<WordType>(v);
if constexpr (T_SIZE > WORD_SIZE)
v >>= WORD_SIZE;
else
v = 0;
}
}
LIBC_INLINE constexpr BigInt &operator=(const BigInt &other) = default;
// constants
LIBC_INLINE static constexpr BigInt zero() { return BigInt(); }
LIBC_INLINE static constexpr BigInt one() { return BigInt(1); }
LIBC_INLINE static constexpr BigInt all_ones() { return ~zero(); }
LIBC_INLINE static constexpr BigInt min() {
BigInt out;
if constexpr (SIGNED)
out.set_msb();
return out;
}
LIBC_INLINE static constexpr BigInt max() {
BigInt out = all_ones();
if constexpr (SIGNED)
out.clear_msb();
return out;
}
// TODO: Reuse the Sign type.
LIBC_INLINE constexpr bool is_neg() const { return SIGNED && get_msb(); }
template <typename T> LIBC_INLINE constexpr explicit operator T() const {
return to<T>();
}
template <typename T>
LIBC_INLINE constexpr cpp::enable_if_t<
cpp::is_integral_v<T> && !cpp::is_same_v<T, bool>, T>
to() const {
constexpr size_t T_SIZE = sizeof(T) * CHAR_BIT;
T lo = static_cast<T>(val[0]);
if constexpr (T_SIZE <= WORD_SIZE)
return lo;
constexpr size_t MAX_COUNT =
T_SIZE > Bits ? WORD_COUNT : T_SIZE / WORD_SIZE;
for (size_t i = 1; i < MAX_COUNT; ++i)
lo += static_cast<T>(val[i]) << (WORD_SIZE * i);
if constexpr (Signed && (T_SIZE > Bits)) {
// Extend sign for negative numbers.
constexpr T MASK = (~T(0) << Bits);
if (is_neg())
lo |= MASK;
}
return lo;
}
LIBC_INLINE constexpr explicit operator bool() const { return !is_zero(); }
LIBC_INLINE constexpr bool is_zero() const {
for (auto part : val)
if (part != 0)
return false;
return true;
}
// Add 'rhs' to this number and store the result in this number.
// Returns the carry value produced by the addition operation.
LIBC_INLINE constexpr WordType add_overflow(const BigInt &rhs) {
return multiword::add_with_carry(val, rhs.val);
}
LIBC_INLINE constexpr BigInt operator+(const BigInt &other) const {
BigInt result = *this;
result.add_overflow(other);
return result;
}
// This will only apply when initializing a variable from constant values, so
// it will always use the constexpr version of add_with_carry.
LIBC_INLINE constexpr BigInt operator+(BigInt &&other) const {
// We use addition commutativity to reuse 'other' and prevent allocation.
other.add_overflow(*this); // Returned carry value is ignored.
return other;
}
LIBC_INLINE constexpr BigInt &operator+=(const BigInt &other) {
add_overflow(other); // Returned carry value is ignored.
return *this;
}
// Subtract 'rhs' to this number and store the result in this number.
// Returns the carry value produced by the subtraction operation.
LIBC_INLINE constexpr WordType sub_overflow(const BigInt &rhs) {
return multiword::sub_with_borrow(val, rhs.val);
}
LIBC_INLINE constexpr BigInt operator-(const BigInt &other) const {
BigInt result = *this;
result.sub_overflow(other); // Returned carry value is ignored.
return result;
}
LIBC_INLINE constexpr BigInt operator-(BigInt &&other) const {
BigInt result = *this;
result.sub_overflow(other); // Returned carry value is ignored.
return result;
}
LIBC_INLINE constexpr BigInt &operator-=(const BigInt &other) {
// TODO(lntue): Set overflow flag / errno when carry is true.
sub_overflow(other); // Returned carry value is ignored.
return *this;
}
// Multiply this number with x and store the result in this number.
LIBC_INLINE constexpr WordType mul(WordType x) {
return multiword::scalar_multiply_with_carry(val, x);
}
// Return the full product.
template <size_t OtherBits>
LIBC_INLINE constexpr auto
ful_mul(const BigInt<OtherBits, Signed, WordType> &other) const {
BigInt<Bits + OtherBits, Signed, WordType> result;
multiword::multiply_with_carry(result.val, val, other.val);
return result;
}
LIBC_INLINE constexpr BigInt operator*(const BigInt &other) const {
// Perform full mul and truncate.
return BigInt(ful_mul(other));
}
// Fast hi part of the full product. The normal product `operator*` returns
// `Bits` least significant bits of the full product, while this function will
// approximate `Bits` most significant bits of the full product with errors
// bounded by:
// 0 <= (a.full_mul(b) >> Bits) - a.quick_mul_hi(b)) <= WORD_COUNT - 1.
//
// An example usage of this is to quickly (but less accurately) compute the
// product of (normalized) mantissas of floating point numbers:
// (mant_1, mant_2) -> quick_mul_hi -> normalize leading bit
// is much more efficient than:
// (mant_1, mant_2) -> ful_mul -> normalize leading bit
// -> convert back to same Bits width by shifting/rounding,
// especially for higher precisions.
//
// Performance summary:
// Number of 64-bit x 64-bit -> 128-bit multiplications performed.
// Bits WORD_COUNT ful_mul quick_mul_hi Error bound
// 128 2 4 3 1
// 196 3 9 6 2
// 256 4 16 10 3
// 512 8 64 36 7
LIBC_INLINE constexpr BigInt quick_mul_hi(const BigInt &other) const {
BigInt result;
multiword::quick_mul_hi(result.val, val, other.val);
return result;
}
// BigInt(x).pow_n(n) computes x ^ n.
// Note 0 ^ 0 == 1.
LIBC_INLINE constexpr void pow_n(uint64_t power) {
static_assert(!Signed);
BigInt result = one();
BigInt cur_power = *this;
while (power > 0) {
if ((power % 2) > 0)
result *= cur_power;
power >>= 1;
cur_power *= cur_power;
}
*this = result;
}
// Performs inplace signed / unsigned division. Returns remainder if not
// dividing by zero.
// For signed numbers it behaves like C++ signed integer division.
// That is by truncating the fractionnal part
// https://stackoverflow.com/a/3602857
LIBC_INLINE constexpr cpp::optional<BigInt> div(const BigInt &divider) {
if (LIBC_UNLIKELY(divider.is_zero()))
return cpp::nullopt;
if (LIBC_UNLIKELY(divider == BigInt::one()))
return BigInt::zero();
Division result;
if constexpr (SIGNED)
result = divide_signed(*this, divider);
else
result = divide_unsigned(*this, divider);
*this = result.quotient;
return result.remainder;
}
// Efficiently perform BigInt / (x * 2^e), where x is a half-word-size
// unsigned integer, and return the remainder. The main idea is as follow:
// Let q = y / (x * 2^e) be the quotient, and
// r = y % (x * 2^e) be the remainder.
// First, notice that:
// r % (2^e) = y % (2^e),
// so we just need to focus on all the bits of y that is >= 2^e.
// To speed up the shift-and-add steps, we only use x as the divisor, and
// performing 32-bit shiftings instead of bit-by-bit shiftings.
// Since the remainder of each division step < x < 2^(WORD_SIZE / 2), the
// computation of each step is now properly contained within WordType.
// And finally we perform some extra alignment steps for the remaining bits.
LIBC_INLINE constexpr cpp::optional<BigInt>
div_uint_half_times_pow_2(multiword::half_width_t<WordType> x, size_t e) {
BigInt remainder;
if (x == 0)
return cpp::nullopt;
if (e >= Bits) {
remainder = *this;
*this = BigInt<Bits, false, WordType>();
return remainder;
}
BigInt quotient;
WordType x_word = static_cast<WordType>(x);
constexpr size_t LOG2_WORD_SIZE = cpp::bit_width(WORD_SIZE) - 1;
constexpr size_t HALF_WORD_SIZE = WORD_SIZE >> 1;
constexpr WordType HALF_MASK = ((WordType(1) << HALF_WORD_SIZE) - 1);
// lower = smallest multiple of WORD_SIZE that is >= e.
size_t lower = ((e >> LOG2_WORD_SIZE) + ((e & (WORD_SIZE - 1)) != 0))
<< LOG2_WORD_SIZE;
// lower_pos is the index of the closest WORD_SIZE-bit chunk >= 2^e.
size_t lower_pos = lower / WORD_SIZE;
// Keep track of current remainder mod x * 2^(32*i)
WordType rem = 0;
// pos is the index of the current 64-bit chunk that we are processing.
size_t pos = WORD_COUNT;
// TODO: look into if constexpr(Bits > 256) skip leading zeroes.
for (size_t q_pos = WORD_COUNT - lower_pos; q_pos > 0; --q_pos) {
// q_pos is 1 + the index of the current WORD_SIZE-bit chunk of the
// quotient being processed. Performing the division / modulus with
// divisor:
// x * 2^(WORD_SIZE*q_pos - WORD_SIZE/2),
// i.e. using the upper (WORD_SIZE/2)-bit of the current WORD_SIZE-bit
// chunk.
rem <<= HALF_WORD_SIZE;
rem += val[--pos] >> HALF_WORD_SIZE;
WordType q_tmp = rem / x_word;
rem %= x_word;
// Performing the division / modulus with divisor:
// x * 2^(WORD_SIZE*(q_pos - 1)),
// i.e. using the lower (WORD_SIZE/2)-bit of the current WORD_SIZE-bit
// chunk.
rem <<= HALF_WORD_SIZE;
rem += val[pos] & HALF_MASK;
quotient.val[q_pos - 1] = (q_tmp << HALF_WORD_SIZE) + rem / x_word;
rem %= x_word;
}
// So far, what we have is:
// quotient = y / (x * 2^lower), and
// rem = (y % (x * 2^lower)) / 2^lower.
// If (lower > e), we will need to perform an extra adjustment of the
// quotient and remainder, namely:
// y / (x * 2^e) = [ y / (x * 2^lower) ] * 2^(lower - e) +
// + (rem * 2^(lower - e)) / x
// (y % (x * 2^e)) / 2^e = (rem * 2^(lower - e)) % x
size_t last_shift = lower - e;
if (last_shift > 0) {
// quotient * 2^(lower - e)
quotient <<= last_shift;
WordType q_tmp = 0;
WordType d = val[--pos];
if (last_shift >= HALF_WORD_SIZE) {
// The shifting (rem * 2^(lower - e)) might overflow WordTyoe, so we
// perform a HALF_WORD_SIZE-bit shift first.
rem <<= HALF_WORD_SIZE;
rem += d >> HALF_WORD_SIZE;
d &= HALF_MASK;
q_tmp = rem / x_word;
rem %= x_word;
last_shift -= HALF_WORD_SIZE;
} else {
// Only use the upper HALF_WORD_SIZE-bit of the current WORD_SIZE-bit
// chunk.
d >>= HALF_WORD_SIZE;
}
if (last_shift > 0) {
rem <<= HALF_WORD_SIZE;
rem += d;
q_tmp <<= last_shift;
x_word <<= HALF_WORD_SIZE - last_shift;
q_tmp += rem / x_word;
rem %= x_word;
}
quotient.val[0] += q_tmp;
if (lower - e <= HALF_WORD_SIZE) {
// The remainder rem * 2^(lower - e) might overflow to the higher
// WORD_SIZE-bit chunk.
if (pos < WORD_COUNT - 1) {
remainder[pos + 1] = rem >> HALF_WORD_SIZE;
}
remainder[pos] = (rem << HALF_WORD_SIZE) + (val[pos] & HALF_MASK);
} else {
remainder[pos] = rem;
}
} else {
remainder[pos] = rem;
}
// Set the remaining lower bits of the remainder.
for (; pos > 0; --pos) {
remainder[pos - 1] = val[pos - 1];
}
*this = quotient;
return remainder;
}
LIBC_INLINE constexpr BigInt operator/(const BigInt &other) const {
BigInt result(*this);
result.div(other);
return result;
}
LIBC_INLINE constexpr BigInt &operator/=(const BigInt &other) {
div(other);
return *this;
}
LIBC_INLINE constexpr BigInt operator%(const BigInt &other) const {
BigInt result(*this);
return *result.div(other);
}
LIBC_INLINE constexpr BigInt &operator*=(const BigInt &other) {
*this = *this * other;
return *this;
}
LIBC_INLINE constexpr BigInt &operator<<=(size_t s) {
val = multiword::shift<multiword::LEFT, SIGNED>(val, s);
return *this;
}
LIBC_INLINE constexpr BigInt operator<<(size_t s) const {
return BigInt(multiword::shift<multiword::LEFT, SIGNED>(val, s));
}
LIBC_INLINE constexpr BigInt &operator>>=(size_t s) {
val = multiword::shift<multiword::RIGHT, SIGNED>(val, s);
return *this;
}
LIBC_INLINE constexpr BigInt operator>>(size_t s) const {
return BigInt(multiword::shift<multiword::RIGHT, SIGNED>(val, s));
}
#define DEFINE_BINOP(OP) \
LIBC_INLINE friend constexpr BigInt operator OP(const BigInt &lhs, \
const BigInt &rhs) { \
BigInt result; \
for (size_t i = 0; i < WORD_COUNT; ++i) \
result[i] = lhs[i] OP rhs[i]; \
return result; \
} \
LIBC_INLINE friend constexpr BigInt operator OP##=(BigInt &lhs, \
const BigInt &rhs) { \
for (size_t i = 0; i < WORD_COUNT; ++i) \
lhs[i] OP## = rhs[i]; \
return lhs; \
}
DEFINE_BINOP(&) // & and &=
DEFINE_BINOP(|) // | and |=
DEFINE_BINOP(^) // ^ and ^=
#undef DEFINE_BINOP
LIBC_INLINE constexpr BigInt operator~() const {
BigInt result;
for (size_t i = 0; i < WORD_COUNT; ++i)
result[i] = ~val[i];
return result;
}
LIBC_INLINE constexpr BigInt operator-() const {
BigInt result(*this);
result.negate();
return result;
}
LIBC_INLINE friend constexpr bool operator==(const BigInt &lhs,
const BigInt &rhs) {
for (size_t i = 0; i < WORD_COUNT; ++i)
if (lhs.val[i] != rhs.val[i])
return false;
return true;
}
LIBC_INLINE friend constexpr bool operator!=(const BigInt &lhs,
const BigInt &rhs) {
return !(lhs == rhs);
}
LIBC_INLINE friend constexpr bool operator>(const BigInt &lhs,
const BigInt &rhs) {
return cmp(lhs, rhs) > 0;
}
LIBC_INLINE friend constexpr bool operator>=(const BigInt &lhs,
const BigInt &rhs) {
return cmp(lhs, rhs) >= 0;
}
LIBC_INLINE friend constexpr bool operator<(const BigInt &lhs,
const BigInt &rhs) {
return cmp(lhs, rhs) < 0;
}
LIBC_INLINE friend constexpr bool operator<=(const BigInt &lhs,
const BigInt &rhs) {
return cmp(lhs, rhs) <= 0;
}
LIBC_INLINE constexpr BigInt &operator++() {
increment();
return *this;
}
LIBC_INLINE constexpr BigInt operator++(int) {
BigInt oldval(*this);
increment();
return oldval;
}
LIBC_INLINE constexpr BigInt &operator--() {
decrement();
return *this;
}
LIBC_INLINE constexpr BigInt operator--(int) {
BigInt oldval(*this);
decrement();
return oldval;
}
// Return the i-th word of the number.
LIBC_INLINE constexpr const WordType &operator[](size_t i) const {
return val[i];
}
// Return the i-th word of the number.
LIBC_INLINE constexpr WordType &operator[](size_t i) { return val[i]; }
private:
LIBC_INLINE friend constexpr int cmp(const BigInt &lhs, const BigInt &rhs) {
constexpr auto compare = [](WordType a, WordType b) {
return a == b ? 0 : a > b ? 1 : -1;
};
if constexpr (Signed) {
const bool lhs_is_neg = lhs.is_neg();
const bool rhs_is_neg = rhs.is_neg();
if (lhs_is_neg != rhs_is_neg)
return rhs_is_neg ? 1 : -1;
}
for (size_t i = WORD_COUNT; i-- > 0;)
if (auto cmp = compare(lhs[i], rhs[i]); cmp != 0)
return cmp;
return 0;
}
LIBC_INLINE constexpr void bitwise_not() {
for (auto &part : val)
part = ~part;
}
LIBC_INLINE constexpr void negate() {
bitwise_not();
increment();
}
LIBC_INLINE constexpr void increment() {
multiword::add_with_carry(val, cpp::array<WordType, 1>{1});
}
LIBC_INLINE constexpr void decrement() {
multiword::add_with_carry(val, cpp::array<WordType, 1>{1});
}
LIBC_INLINE constexpr void extend(size_t index, bool is_neg) {
const WordType value = is_neg ? cpp::numeric_limits<WordType>::max()
: cpp::numeric_limits<WordType>::min();
for (size_t i = index; i < WORD_COUNT; ++i)
val[i] = value;
}
LIBC_INLINE constexpr bool get_msb() const {
return val.back() >> (WORD_SIZE - 1);
}
LIBC_INLINE constexpr void set_msb() {
val.back() |= mask_leading_ones<WordType, 1>();
}
LIBC_INLINE constexpr void clear_msb() {
val.back() &= mask_trailing_ones<WordType, WORD_SIZE - 1>();
}
LIBC_INLINE constexpr void set_bit(size_t i) {
const size_t word_index = i / WORD_SIZE;
val[word_index] |= WordType(1) << (i % WORD_SIZE);
}
LIBC_INLINE constexpr static Division divide_unsigned(const BigInt &dividend,
const BigInt &divider) {
BigInt remainder = dividend;
BigInt quotient;
if (remainder >= divider) {
BigInt subtractor = divider;
int cur_bit = multiword::countl_zero(subtractor.val) -
multiword::countl_zero(remainder.val);
subtractor <<= cur_bit;
for (; cur_bit >= 0 && remainder > 0; --cur_bit, subtractor >>= 1) {
if (remainder < subtractor)
continue;
remainder -= subtractor;
quotient.set_bit(cur_bit);
}
}
return Division{quotient, remainder};
}
LIBC_INLINE constexpr static Division divide_signed(const BigInt &dividend,
const BigInt &divider) {
// Special case because it is not possible to negate the min value of a
// signed integer.
if (dividend == min() && divider == min())
return Division{one(), zero()};
// 1. Convert the dividend and divisor to unsigned representation.
unsigned_type udividend(dividend);
unsigned_type udivider(divider);
// 2. Negate the dividend if it's negative, and similarly for the divisor.
const bool dividend_is_neg = dividend.is_neg();
const bool divider_is_neg = divider.is_neg();
if (dividend_is_neg)
udividend.negate();
if (divider_is_neg)
udivider.negate();
// 3. Use unsigned multiword division algorithm.
const auto unsigned_result = divide_unsigned(udividend, udivider);
// 4. Convert the quotient and remainder to signed representation.
Division result;
result.quotient = signed_type(unsigned_result.quotient);
result.remainder = signed_type(unsigned_result.remainder);
// 5. Negate the quotient if the dividend and divisor had opposite signs.
if (dividend_is_neg != divider_is_neg)
result.quotient.negate();
// 6. Negate the remainder if the dividend was negative.
if (dividend_is_neg)
result.remainder.negate();
return result;
}
friend signed_type;
friend unsigned_type;
};
namespace internal {
// We default BigInt's WordType to 'uint64_t' or 'uint32_t' depending on type
// availability.
template <size_t Bits>
struct WordTypeSelector : cpp::type_identity<
#ifdef LIBC_TYPES_HAS_INT64
uint64_t
#else
uint32_t
#endif // LIBC_TYPES_HAS_INT64
> {
};
// Except if we request 32 bits explicitly.
template <> struct WordTypeSelector<32> : cpp::type_identity<uint32_t> {};
template <size_t Bits>
using WordTypeSelectorT = typename WordTypeSelector<Bits>::type;
} // namespace internal
template <size_t Bits>
using UInt = BigInt<Bits, false, internal::WordTypeSelectorT<Bits>>;
template <size_t Bits>
using Int = BigInt<Bits, true, internal::WordTypeSelectorT<Bits>>;
// Provides limits of U/Int<128>.
template <> class cpp::numeric_limits<UInt<128>> {
public:
LIBC_INLINE static constexpr UInt<128> max() { return UInt<128>::max(); }
LIBC_INLINE static constexpr UInt<128> min() { return UInt<128>::min(); }
// Meant to match std::numeric_limits interface.
// NOLINTNEXTLINE(readability-identifier-naming)
LIBC_INLINE_VAR static constexpr int digits = 128;
};
template <> class cpp::numeric_limits<Int<128>> {
public:
LIBC_INLINE static constexpr Int<128> max() { return Int<128>::max(); }
LIBC_INLINE static constexpr Int<128> min() { return Int<128>::min(); }
// Meant to match std::numeric_limits interface.
// NOLINTNEXTLINE(readability-identifier-naming)
LIBC_INLINE_VAR static constexpr int digits = 128;
};
// type traits to determine whether a T is a BigInt.
template <typename T> struct is_big_int : cpp::false_type {};
template <size_t Bits, bool Signed, typename T>
struct is_big_int<BigInt<Bits, Signed, T>> : cpp::true_type {};
template <class T>
LIBC_INLINE_VAR constexpr bool is_big_int_v = is_big_int<T>::value;
// extensions of type traits to include BigInt
// is_integral_or_big_int
template <typename T>
struct is_integral_or_big_int
: cpp::bool_constant<(cpp::is_integral_v<T> || is_big_int_v<T>)> {};
template <typename T>
LIBC_INLINE_VAR constexpr bool is_integral_or_big_int_v =
is_integral_or_big_int<T>::value;
// make_big_int_unsigned
template <typename T> struct make_big_int_unsigned;
template <size_t Bits, bool Signed, typename T>
struct make_big_int_unsigned<BigInt<Bits, Signed, T>>
: cpp::type_identity<BigInt<Bits, false, T>> {};
template <typename T>
using make_big_int_unsigned_t = typename make_big_int_unsigned<T>::type;
// make_big_int_signed
template <typename T> struct make_big_int_signed;
template <size_t Bits, bool Signed, typename T>
struct make_big_int_signed<BigInt<Bits, Signed, T>>
: cpp::type_identity<BigInt<Bits, true, T>> {};
template <typename T>
using make_big_int_signed_t = typename make_big_int_signed<T>::type;
// make_integral_or_big_int_unsigned
template <typename T, class = void> struct make_integral_or_big_int_unsigned;
template <typename T>
struct make_integral_or_big_int_unsigned<
T, cpp::enable_if_t<cpp::is_integral_v<T>>> : cpp::make_unsigned<T> {};
template <typename T>
struct make_integral_or_big_int_unsigned<T, cpp::enable_if_t<is_big_int_v<T>>>
: make_big_int_unsigned<T> {};
template <typename T>
using make_integral_or_big_int_unsigned_t =
typename make_integral_or_big_int_unsigned<T>::type;
// make_integral_or_big_int_signed
template <typename T, class = void> struct make_integral_or_big_int_signed;
template <typename T>
struct make_integral_or_big_int_signed<T,
cpp::enable_if_t<cpp::is_integral_v<T>>>
: cpp::make_signed<T> {};
template <typename T>
struct make_integral_or_big_int_signed<T, cpp::enable_if_t<is_big_int_v<T>>>
: make_big_int_signed<T> {};
template <typename T>
using make_integral_or_big_int_signed_t =
typename make_integral_or_big_int_signed<T>::type;
namespace cpp {
// Specialization of cpp::bit_cast ('bit.h') from T to BigInt.
template <typename To, typename From>
LIBC_INLINE constexpr cpp::enable_if_t<
(sizeof(To) == sizeof(From)) && cpp::is_trivially_copyable<To>::value &&
cpp::is_trivially_copyable<From>::value && is_big_int<To>::value,
To>
bit_cast(const From &from) {
To out;
using Storage = decltype(out.val);
out.val = cpp::bit_cast<Storage>(from);
return out;
}
// Specialization of cpp::bit_cast ('bit.h') from BigInt to T.
template <typename To, size_t Bits>
LIBC_INLINE constexpr cpp::enable_if_t<
sizeof(To) == sizeof(UInt<Bits>) &&
cpp::is_trivially_constructible<To>::value &&
cpp::is_trivially_copyable<To>::value &&
cpp::is_trivially_copyable<UInt<Bits>>::value,
To>
bit_cast(const UInt<Bits> &from) {
return cpp::bit_cast<To>(from.val);
}
// Specialization of cpp::popcount ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
popcount(T value) {
int bits = 0;
for (auto word : value.val)
if (word)
bits += popcount(word);
return bits;
}
// Specialization of cpp::has_single_bit ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, bool>
has_single_bit(T value) {
int bits = 0;
for (auto word : value.val) {
if (word == 0)
continue;
bits += popcount(word);
if (bits > 1)
return false;
}
return bits == 1;
}
// Specialization of cpp::countr_zero ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
countr_zero(const T &value) {
return multiword::countr_zero(value.val);
}
// Specialization of cpp::countl_zero ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
countl_zero(const T &value) {
return multiword::countl_zero(value.val);
}
// Specialization of cpp::countl_one ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
countl_one(T value) {
return multiword::countl_one(value.val);
}
// Specialization of cpp::countr_one ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
countr_one(T value) {
return multiword::countr_one(value.val);
}
// Specialization of cpp::bit_width ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
bit_width(T value) {
return cpp::numeric_limits<T>::digits - cpp::countl_zero(value);
}
// Forward-declare rotr so that rotl can use it.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
rotr(T value, int rotate);
// Specialization of cpp::rotl ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
rotl(T value, int rotate) {
constexpr unsigned N = cpp::numeric_limits<T>::digits;
rotate = rotate % N;
if (!rotate)
return value;
if (rotate < 0)
return cpp::rotr<T>(value, -rotate);
return (value << rotate) | (value >> (N - rotate));
}
// Specialization of cpp::rotr ('bit.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
rotr(T value, int rotate) {
constexpr unsigned N = cpp::numeric_limits<T>::digits;
rotate = rotate % N;
if (!rotate)
return value;
if (rotate < 0)
return cpp::rotl<T>(value, -rotate);
return (value >> rotate) | (value << (N - rotate));
}
} // namespace cpp
// Specialization of mask_trailing_ones ('math_extras.h') for BigInt.
template <typename T, size_t count>
LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
mask_trailing_ones() {
static_assert(!T::SIGNED && count <= T::BITS);
if (count == T::BITS)
return T::all_ones();
constexpr size_t QUOTIENT = count / T::WORD_SIZE;
constexpr size_t REMAINDER = count % T::WORD_SIZE;
T out; // zero initialized
for (size_t i = 0; i <= QUOTIENT; ++i)
out[i] = i < QUOTIENT
? -1
: mask_trailing_ones<typename T::word_type, REMAINDER>();
return out;
}
// Specialization of mask_leading_ones ('math_extras.h') for BigInt.
template <typename T, size_t count>
LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T> mask_leading_ones() {
static_assert(!T::SIGNED && count <= T::BITS);
if (count == T::BITS)
return T::all_ones();
constexpr size_t QUOTIENT = (T::BITS - count - 1U) / T::WORD_SIZE;
constexpr size_t REMAINDER = count % T::WORD_SIZE;
T out; // zero initialized
for (size_t i = QUOTIENT; i < T::WORD_COUNT; ++i)
out[i] = i > QUOTIENT
? -1
: mask_leading_ones<typename T::word_type, REMAINDER>();
return out;
}
// Specialization of mask_trailing_zeros ('math_extras.h') for BigInt.
template <typename T, size_t count>
LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
mask_trailing_zeros() {
return mask_leading_ones<T, T::BITS - count>();
}
// Specialization of mask_leading_zeros ('math_extras.h') for BigInt.
template <typename T, size_t count>
LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, T>
mask_leading_zeros() {
return mask_trailing_ones<T, T::BITS - count>();
}
// Specialization of count_zeros ('math_extras.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
count_zeros(T value) {
return cpp::popcount(~value);
}
// Specialization of first_leading_zero ('math_extras.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
first_leading_zero(T value) {
return value == cpp::numeric_limits<T>::max() ? 0
: cpp::countl_one(value) + 1;
}
// Specialization of first_leading_one ('math_extras.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
first_leading_one(T value) {
return first_leading_zero(~value);
}
// Specialization of first_trailing_zero ('math_extras.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
first_trailing_zero(T value) {
return value == cpp::numeric_limits<T>::max() ? 0
: cpp::countr_zero(~value) + 1;
}
// Specialization of first_trailing_one ('math_extras.h') for BigInt.
template <typename T>
[[nodiscard]] LIBC_INLINE constexpr cpp::enable_if_t<is_big_int_v<T>, int>
first_trailing_one(T value) {
return value == cpp::numeric_limits<T>::max() ? 0
: cpp::countr_zero(value) + 1;
}
} // namespace LIBC_NAMESPACE
#endif // LLVM_LIBC_SRC___SUPPORT_UINT_H