CP-Algorithms Library
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tests/bigint.cpp
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- Last update: 2026-10-16 14:16:08+02:00
Depends on
cp-algo/math/bigint.hpp- cp-algo/math/common.hpp
- cp-algo/math/cvector.hpp
- cp-algo/math/fft_simple.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/simd.hpp
Code
#include "cp-algo/math/bigint.hpp"
#include <random>
using namespace cp_algo;
using namespace cp_algo::math;
template<base_v radix>
void check() {
using B = bigint<radix>;
std::mt19937_64 rng(71823);
auto value = [](B const& a) {
__int128 result = 0;
for(auto digit: a.digits | std::views::reverse) {result = result * B::Base + digit;}
return a.negative ? -result : result;
};
std::vector<B> cases{B(0), B(1), B(-1), B({0, 1}, false), B({0, 1}, true)};
auto zero = B(0);
zero.negate(); // A negative zero still compares equal to zero.
cases.push_back(zero);
for(int i = 0; i < 100; i++) {
cases.emplace_back(big_basic_string<uint64_t>{rng() % B::Base, rng() % B::Base}, rng() % 2);
}
for(auto const& a: cases) {
for(auto const& b: cases) {
assert((a <=> b) == (value(a) <=> value(b)));
}
for(size_t n = 0; n <= 3; n++) {
__int128 power = 1;
// Two limbs fit in __int128; larger truncations leave the input unchanged.
for(size_t i = 0; i < std::min(n, size_t(2)); i++) {power *= B::Base;}
assert(value(a.take(n)) == (n >= 2 ? value(a) : value(a) % power));
assert(value(a.drop(n)) == (n >= 2 ? 0 : value(a) / power));
}
}
for(int64_t x: {-1000000, -1, 0, 1, 1000000}) {
B a(x);
assert(value(a.pad(1)) == __int128(x) * B::Base);
assert(a.pad(1).drop(1).digits == a.digits);
}
}
int main() {
check<x10>();
check<x16>();
std::cout << "Bigint ordering and slicing passed independent integer checks\n";
}
#line 1 "cp-algo/math/bigint.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#include <bit>
#include <iostream>
#include <forward_list>
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template<typename T, size_t Align = 32>
class big_alloc {
static_assert(Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::has_single_bit(Align), "Align must be a power of two");
public:
using value_type = T;
template<class U>
struct rebind {
using other = big_alloc<U, Align>;
};
constexpr bool operator==(const big_alloc&) const = default;
big_alloc() noexcept = default;
template<typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded, PROT_READ | PROT_WRITE, MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if(!p) {
return;
}
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
munmap(p, padded);
return;
}
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T>
using big_vector = std::vector<T, big_alloc<T>>;
template<typename T>
using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T>
using big_deque = std::deque<T, big_alloc<T>>;
template<typename T>
using big_stack = std::stack<T, big_deque<T>>;
template<typename T>
using big_queue = std::queue<T, big_deque<T>>;
template<typename T>
using big_priority_queue = std::priority_queue<T, big_vector<T>>;
template<typename T>
using big_forward_list = std::forward_list<T, big_alloc<T>>;
using big_string = big_basic_string<char>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
template<typename T, typename Compare = std::less<T>>
using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
template<typename T, typename Compare = std::less<T>>
using big_set = std::set<T, Compare, big_alloc<T>>;
}
#line 1 "cp-algo/math/fft_simple.hpp"
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random {
std::mt19937_64 gen(
std::chrono::steady_clock::now().time_since_epoch().count()
);
uint64_t rng() {
return gen();
}
}
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#line 8 "cp-algo/math/common.hpp"
#include <algorithm>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
// Nonnegative 64-bit exponents, with an associative operation and its identity.
// Windows >1 precompute odd powers only when that saves operations.
template<int window = 1>
auto bpow(auto const& x, auto n, auto const& one, auto op) {
static_assert(window >= 1 && window <= 6);
if constexpr(window > 1) {
if(n == 0) {return one;}
int bits = std::bit_width(uint64_t(n));
auto low_bit = [&](int high) {
int low = std::max(0, high - window + 1);
while(!((n >> low) & 1)) {low++;}
return low;
};
int first = low_bit(bits - 1);
int cost = (1 << (window - 1)) + first;
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {j--;}
else {cost++; j = low_bit(j) - 1;}
}
// Do not pay for the table when binary powering uses fewer operations.
if(cost >= bits + std::popcount(uint64_t(n)) - 2) {return bpow<1>(x, n, one, op);}
using T = std::decay_t<decltype(x)>;
std::vector<T> odd;
odd.reserve(1 << (window - 1));
odd.push_back(x);
auto square = op(x, x);
while(odd.size() < size_t(1 << (window - 1))) {odd.push_back(op(odd.back(), square));}
auto ans = odd[(n >> first) / 2];
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {ans = op(ans, ans); j--;}
else {
int low = low_bit(j), length = j - low + 1;
auto digit = (n >> low) & ((1u << length) - 1);
for(int i = 0; i < length; i++) {ans = op(ans, ans);}
ans = op(ans, odd[digit / 2]);
j = low - 1;
}
}
return ans;
} else {
if (n == 0) {
return one;
}
auto ans = x;
for(int j = std::bit_width<uint64_t>(n) - 2; ~j; j--) {
ans = op(ans, ans);
if((n >> j) & 1) {
ans = op(ans, x);
}
}
return ans;
}
}
template<int window = 1>
auto bpow(auto x, auto n, auto ans) {
return bpow<window>(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using u64x8 = simd<uint64_t, 8>;
using u32x16 = simd<uint32_t, 16>;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using u16x16 = simd<uint16_t, 16>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using u16x8 = simd<uint16_t, 8>;
using u16x4 = simd<uint16_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using u8x16 = simd<uint8_t, 16>;
using u8x8 = simd<uint8_t, 8>;
using u8x4 = simd<uint8_t, 4>;
using dx4 = simd<double, 4>;
inline dx4 abs(dx4 a) {
return dx4{
std::abs(a[0]),
std::abs(a[1]),
std::abs(a[2]),
std::abs(a[3])
};
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
auto x_ninv = u64x4(u32x8(low32(x)) * imod);
x += x_ninv * uint64_t(mod);
#endif
return swap_bytes(x);
}
// x - mod where x >= mod, for x in [0, 2 mod). Unsigned, so it holds up to mod < 2^31;
// zero upper halves of 64-bit lanes stay zero.
inline u32x8 reduce_once(u32x8 x, uint32_t mod) {
auto y = x - mod;
return x < y ? x : y;
}
inline u64x4 reduce_once(u64x4 x, uint32_t mod) {
return u64x4(reduce_once(u32x8(x), mod));
}
inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
return montgomery_reduce(x * y, mod, imod);
#endif
}
inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
inline bool is_aligned(const auto *p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
// Custom implementation, since std::complex is UB on non-floating types
template<typename T>
struct complex {
using value_type = T;
T x, y;
inline constexpr complex(): x(), y() {}
inline constexpr complex(T const& x): x(x), y() {}
inline constexpr complex(T const& x, T const& y): x(x), y(y) {}
inline complex& operator *= (T const& t) {x *= t; y *= t; return *this;}
inline complex& operator /= (T const& t) {x /= t; y /= t; return *this;}
inline complex operator * (T const& t) const {return complex(*this) *= t;}
inline complex operator / (T const& t) const {return complex(*this) /= t;}
inline complex& operator += (complex const& t) {x += t.x; y += t.y; return *this;}
inline complex& operator -= (complex const& t) {x -= t.x; y -= t.y; return *this;}
inline complex operator * (complex const& t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
inline complex operator / (complex const& t) const {return *this * t.conj() / t.norm();}
inline complex operator + (complex const& t) const {return complex(*this) += t;}
inline complex operator - (complex const& t) const {return complex(*this) -= t;}
inline complex& operator *= (complex const& t) {return *this = *this * t;}
inline complex& operator /= (complex const& t) {return *this = *this / t;}
inline complex operator - () const {return {-x, -y};}
inline complex conj() const {return {x, -y};}
inline T norm() const {return x * x + y * y;}
inline T abs() const {return std::sqrt(norm());}
inline T const real() const {return x;}
inline T const imag() const {return y;}
inline T& real() {return x;}
inline T& imag() {return y;}
inline static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
inline auto operator <=> (complex const& t) const = default;
};
template<typename T> inline complex<T> conj(complex<T> const& x) {return x.conj();}
template<typename T> inline T norm(complex<T> const& x) {return x.norm();}
template<typename T> inline T abs(complex<T> const& x) {return x.abs();}
template<typename T> inline T& real(complex<T> &x) {return x.real();}
template<typename T> inline T& imag(complex<T> &x) {return x.imag();}
template<typename T> inline T const real(complex<T> const& x) {return x.real();}
template<typename T> inline T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
inline constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
inline std::ostream& operator << (std::ostream &out, complex<T> const& x) {
return out << x.real() << ' ' << x.imag();
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo {
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string, double> checkpoints;
double last;
#endif
template<bool final = false>
void checkpoint([[maybe_unused]] auto const& _msg) {
#ifdef CP_ALGO_CHECKPOINT
big_string msg = _msg;
double now = (double)clock() / CLOCKS_PER_SEC;
double delta = now - last;
last = now;
if(msg.size() && !final) {
checkpoints[msg] += delta;
}
if(final) {
for(auto const& [key, value] : checkpoints) {
std::cerr << key << ": " << value * 1000 << " ms\n";
}
std::cerr << "Total: " << now * 1000 << " ms\n";
}
#endif
}
template<bool final = false>
void checkpoint() {
checkpoint<final>("");
}
}
#line 7 "cp-algo/math/cvector.hpp"
#include <immintrin.h>
#include <numbers>
#include <cstring>
#include <ranges>
#line 13 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace stdx = std::experimental;
namespace cp_algo::math::fft {
static constexpr size_t flen = 4;
using ftype = double;
using vftype = dx4;
using point = complex<ftype>;
using vpoint = complex<vftype>;
static constexpr vftype vz = {};
vpoint vi(vpoint const& r) {
return {-imag(r), real(r)};
}
// Spectrum storage skips zero-filling on resize; every user writes before reading.
template<class T>
struct spectrum_alloc: big_alloc<T> {
using big_alloc<T>::big_alloc;
template<class U>
struct rebind {
using other = spectrum_alloc<U>;
};
template<class U>
requires(std::is_same_v<U, vpoint>)
void construct(U*) noexcept {}
};
using spectrum_vector = std::vector<vpoint, spectrum_alloc<vpoint>>;
struct cvector {
spectrum_vector r;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
r.assign(n / flen, vpoint{});
prepare_roots(n / 16);
checkpoint("cvector create");
}
vpoint& at(size_t k) { return r[k / flen]; }
vpoint at(size_t k) const { return r[k / flen]; }
template<class pt = point>
inline void set(size_t k, pt const& t) {
if constexpr(std::is_same_v<pt, point>) {
real(r[k / flen])[k % flen] = real(t);
imag(r[k / flen])[k % flen] = imag(t);
} else {
at(k) = t;
}
}
template<class pt = point>
inline pt get(size_t k) const {
if constexpr(std::is_same_v<pt, point>) {
return {real(r[k / flen])[k % flen], imag(r[k / flen])[k % flen]};
} else {
return at(k);
}
}
size_t size() const {
return flen * r.size();
}
static constexpr size_t eval_arg(size_t n) {
if(n < pre_evals) {
return eval_args[n];
} else {
return eval_arg(n / 2) | (n & 1) << (std::bit_width(n) - 1);
}
}
static constexpr point eval_point(size_t n) {
if(n % 2) {
return -eval_point(n - 1);
} else if(n % 4) {
return eval_point(n - 2) * point(0, 1);
} else if(n / 4 < pre_evals) {
return evalp[n / 4];
} else if(n / 4 - pre_evals < extra.size()) {
return extra[n / 4 - pre_evals];
} else {
return polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(n) * (ftype)eval_arg(n));
}
}
static constexpr std::array<point, 32> roots = []() {
std::array<point, 32> res;
for(size_t i = 2; i < 32; i++) {
res[i] = polar<ftype>(1., std::numbers::pi / (1ull << (i - 2)));
}
return res;
}();
static constexpr point root(size_t n) {
return roots[std::bit_width(n)];
}
template<int step>
static void exec_on_eval(size_t n, size_t k, auto &&callback) {
callback(k, root(4 * step * n) * eval_point(step * k));
}
template<int step>
static void exec_on_evals(size_t n, auto &&callback) {
point factor = root(4 * step * n);
if constexpr(step == 1 || step == 2 || step == 4) {
prepare_roots((step * n + 3) / 4);
size_t i = 0;
if constexpr(step == 1) {
for(; i + 4 <= n; i += 4) {
size_t k = i / 4;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point v = factor * e;
callback(i, v);
callback(i + 1, -v);
point iv(-imag(v), real(v));
callback(i + 2, iv);
callback(i + 3, -iv);
}
} else if constexpr(step == 2) {
for(; i + 2 <= n; i += 2) {
size_t k = i / 2;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point v = factor * e;
callback(i, v);
callback(i + 1, point(-imag(v), real(v)));
}
}
for(; i < n; i++) {
size_t index = step * i, k = index / 4;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
if(index & 2) {
e = e * point(0, 1);
}
if(index & 1) {
e = -e;
}
callback(i, factor * e);
}
} else {
for(size_t i = 0; i < n; i++) {
callback(i, factor * eval_point(step * i));
}
}
}
static void do_dot_iter(point rt, vpoint& Bv, vpoint const& Av, vpoint& res) {
res += Av * Bv;
real(Bv) = rotate_right(real(Bv));
imag(Bv) = rotate_right(imag(Bv));
auto x = real(Bv)[0], y = imag(Bv)[0];
real(Bv)[0] = x * real(rt) - y * imag(rt);
imag(Bv)[0] = x * imag(rt) + y * real(rt);
}
template<size_t fixed = 0>
void dot(cvector const& t) {
size_t n = fixed ? fixed : this->size();
exec_on_evals<1>(n / flen, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = at(k);
auto Cv = t.at(k);
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto Cr = Cv * vrt;
vpoint res = vz;
auto iter = [&]<int i> [[gnu::always_inline]] () {
auto wrap = [&](vftype original, vftype rotated) {
if constexpr(i == 0) {
return original;
} else {
return __builtin_shufflevector(rotated, original, 4 - i, 5 - i, 6 - i, 7 - i);
}
};
vpoint Cw = {wrap(real(Cv), real(Cr)), wrap(imag(Cv), imag(Cr))};
vpoint Av = {vz + Ax[i], vz + Ay[i]};
return Av * Cw;
};
auto p0 = iter.template operator()<0>(), p1 = iter.template operator()<1>();
auto p2 = iter.template operator()<2>(), p3 = iter.template operator()<3>();
res = (p0 + p1) + (p2 + p3);
set(k, res);
});
checkpoint("dot");
}
// normalize=false leaves the inverse-transform scale for the caller.
template<bool partial = true, bool normalize = true, size_t fixed = 0>
void ifft() {
size_t n = fixed ? fixed : size();
if constexpr(!partial) {
prepare_roots(n / 4);
point pi(0, 1);
exec_on_evals<4>(n / 4, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 4;
point v1 = conj(rt);
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1);
auto C = get(k + 2);
auto D = get(k + 3);
set(k, (A + B) + (C + D));
set(k + 2, ((A + B) - (C + D)) * v2);
set(k + 1, ((A - B) - pi * (C - D)) * v1);
set(k + 3, ((A - B) + pi * (C - D)) * v3);
});
}
bool parity = std::countr_zero(n) % 2;
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 2 * flen;
vpoint cvrt = {vz + real(rt), vz - imag(rt)};
auto B = at(k) - at(k + flen);
at(k) += at(k + flen);
at(k + flen) = B * cvrt;
});
}
transform<true, fixed>(n, parity);
checkpoint("ifft");
if constexpr(normalize) {
auto scale = vz + ftype(partial ? flen : 1) / ftype(n);
for(size_t k = 0; k < n; k += flen) {
set(k, get<vpoint>(k) * scale);
}
}
}
template<bool partial = true, size_t fixed = 0>
void fft() {
size_t n = fixed ? fixed : size();
prepare_roots(n / (partial ? 16 : 4));
bool parity = std::countr_zero(n) % 2;
transform<false, fixed>(n, parity);
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 2 * flen;
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto t = at(k + flen) * vrt;
at(k + flen) = at(k) - t;
at(k) += t;
});
}
if constexpr(!partial) {
prepare_roots(n / 4);
point pi(0, 1);
exec_on_evals<4>(n / 4, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 4;
point v1 = rt;
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1) * v1;
auto C = get(k + 2) * v2;
auto D = get(k + 3) * v3;
set(k, (A + C) + (B + D));
set(k + 1, (A + C) - (B + D));
set(k + 2, (A - C) + pi * (B - D));
set(k + 3, (A - C) - pi * (B - D));
});
}
checkpoint("fft");
}
static std::array<vpoint, 4> transpose(std::array<vpoint, 4> const& a) {
auto half = [](auto part) {
auto a0 = __m256d(part(0)), a1 = __m256d(part(1)), a2 = __m256d(part(2)), a3 = __m256d(part(3));
auto t0 = _mm256_unpacklo_pd(a0, a1), t1 = _mm256_unpackhi_pd(a0, a1), t2 = _mm256_unpacklo_pd(a2, a3),
t3 = _mm256_unpackhi_pd(a2, a3);
return std::array<vftype, 4>{
vftype(_mm256_permute2f128_pd(t0, t2, 0x20)), vftype(_mm256_permute2f128_pd(t1, t3, 0x20)),
vftype(_mm256_permute2f128_pd(t0, t2, 0x31)), vftype(_mm256_permute2f128_pd(t1, t3, 0x31))};
};
auto re = half([&](int k) { return real(a[k]); }), im = half([&](int k) { return imag(a[k]); });
return {vpoint{re[0], im[0]}, vpoint{re[1], im[1]}, vpoint{re[2], im[2]}, vpoint{re[3], im[3]}};
}
// Multiply groups of 16 points as polynomials modulo x^16 - rt: one radix-4 level on
// each operand, a lane-parallel 4-coefficient Karatsuba product, one inverse level.
// The weights rt, rt^2, rt^3 of a tile of groups are computed four groups per SIMD
// operation ahead of the tile, so the product loop only broadcasts them from memory.
static constexpr size_t dot_tile = 64;
template<size_t fixed, bool square = false>
void dot_fused16(cvector const& t, size_t offset, size_t length) {
constexpr size_t T = dot_tile;
const size_t n = fixed ? fixed : size();
point factor = root(n);
auto fr = _mm256_set1_pd(real(factor)), fi = _mm256_set1_pd(imag(factor));
alignas(32) double tw[6][T];
const bool ahead = offset + length < n;
auto cmadd = [] [[gnu::always_inline]] (vpoint x, vpoint y, vpoint c) {
auto re = _mm256_fmadd_pd(__m256d(real(x)), __m256d(real(y)),
_mm256_fnmadd_pd(__m256d(imag(x)), __m256d(imag(y)), __m256d(real(c))));
auto im = _mm256_fmadd_pd(__m256d(real(x)), __m256d(imag(y)),
_mm256_fmadd_pd(__m256d(imag(x)), __m256d(real(y)), __m256d(imag(c))));
return vpoint{vftype(re), vftype(im)};
};
auto cmcross = [] [[gnu::always_inline]] (vpoint x, vpoint y, vpoint p, vpoint q) {
auto re = _mm256_sub_pd(_mm256_fmsub_pd(__m256d(real(x)), __m256d(real(y)), __m256d(real(p))),
_mm256_fmadd_pd(__m256d(imag(x)), __m256d(imag(y)), __m256d(real(q))));
auto im = _mm256_add_pd(_mm256_fmsub_pd(__m256d(real(x)), __m256d(imag(y)), __m256d(imag(p))),
_mm256_fmsub_pd(__m256d(imag(x)), __m256d(real(y)), __m256d(imag(q))));
return vpoint{vftype(re), vftype(im)};
};
// z * conj(v)
auto mulconj = [] [[gnu::always_inline]] (vpoint z, vpoint v) {
auto re = _mm256_fmadd_pd(__m256d(real(z)), __m256d(real(v)),
_mm256_mul_pd(__m256d(imag(z)), __m256d(imag(v))));
auto im = _mm256_fmsub_pd(__m256d(imag(z)), __m256d(real(v)),
_mm256_mul_pd(__m256d(real(z)), __m256d(imag(v))));
return vpoint{vftype(re), vftype(im)};
};
for(size_t tile = offset; tile < offset + length; tile += 16 * T) {
size_t k0 = tile / 16;
auto const* e = reinterpret_cast<double const*>(k0 < pre_evals ? evalp.data() + k0
: extra.data() + (k0 - pre_evals));
for(size_t g = 0; g < T; g += 4) {
auto x = _mm256_loadu_pd(e + 2 * g), y = _mm256_loadu_pd(e + 2 * g + 4);
auto er = _mm256_permute4x64_pd(_mm256_unpacklo_pd(x, y), 0xD8),
ei = _mm256_permute4x64_pd(_mm256_unpackhi_pd(x, y), 0xD8);
auto r1 = _mm256_fmsub_pd(fr, er, _mm256_mul_pd(fi, ei)),
i1 = _mm256_fmadd_pd(fr, ei, _mm256_mul_pd(fi, er));
auto r2 = _mm256_fmsub_pd(r1, r1, _mm256_mul_pd(i1, i1)),
i2 = _mm256_fmadd_pd(r1, i1, _mm256_mul_pd(i1, r1));
auto r3 = _mm256_fmsub_pd(r1, r2, _mm256_mul_pd(i1, i2)),
i3 = _mm256_fmadd_pd(r1, i2, _mm256_mul_pd(i1, r2));
_mm256_store_pd(tw[0] + g, r1);
_mm256_store_pd(tw[1] + g, i1);
_mm256_store_pd(tw[2] + g, r2);
_mm256_store_pd(tw[3] + g, i2);
_mm256_store_pd(tw[4] + g, r3);
_mm256_store_pd(tw[5] + g, i3);
}
for(size_t g = 0; g < T; g++) {
size_t pos = tile + 16 * g;
if(ahead) {
// Pull the next block of both operands towards the cache while this one is multiplied.
auto const* pa = reinterpret_cast<char const*>(r.data() + (pos + length) / flen);
auto const* pb = reinterpret_cast<char const*>(t.r.data() + (pos + length) / flen);
for(size_t c = 0; c < 4; c++) {
_mm_prefetch(pa + 64 * c, _MM_HINT_T2);
_mm_prefetch(pb + 64 * c, _MM_HINT_T2);
}
}
auto bc = [&] [[gnu::always_inline]] (size_t c) {
return vftype(_mm256_broadcast_sd(tw[c] + g));
};
vpoint v1 = {bc(0), bc(1)}, v2 = {bc(2), bc(3)}, v3 = {bc(4), bc(5)};
auto forward = [&] [[gnu::always_inline]] (cvector const& a) {
auto A = a.at(pos), B = a.at(pos + 4) * v1, C = a.at(pos + 8) * v2, D = a.at(pos + 12) * v3;
return std::array<vpoint, 4>{(A + C) + (B + D), (A + C) - (B + D), (A - C) + vi(B - D),
(A - C) - vi(B - D)};
};
auto a = transpose(forward(*this));
auto b = [&] {
if constexpr(square) {
return a;
} else {
return transpose(forward(t));
}
}();
// The four residues are taken modulo x^4 - rt * {1, -1, i, -i}.
const auto flip_re = _mm256_set_pd(0., -0., -0., 0.), flip_im = _mm256_set_pd(-0., 0., -0., 0.);
vpoint w = {
vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(real(v1)), __m256d(imag(v1)), 0b1100), flip_re)),
vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(imag(v1)), __m256d(real(v1)), 0b1100), flip_im))};
auto mul = [&] [[gnu::always_inline]] (vpoint a0, vpoint a1, vpoint b0, vpoint b1) {
auto p = a0 * b0, q = a1 * b1;
return std::array<vpoint, 2>{cmadd(w, q, p), cmcross(a0 + a1, b0 + b1, p, q)};
};
auto p = mul(a[0], a[2], b[0], b[2]), q = mul(a[1], a[3], b[1], b[3]);
auto m = mul(a[0] + a[1], a[2] + a[3], b[0] + b[1], b[2] + b[3]);
auto c = transpose({cmadd(w, q[1], p[0]), (m[0] - p[0]) - q[0], p[1] + q[0], (m[1] - p[1]) - q[1]});
auto A = c[0], B = c[1], C = c[2], D = c[3];
at(pos) = (A + B) + (C + D);
at(pos + 8) = mulconj((A + B) - (C + D), v2);
at(pos + 4) = mulconj((A - B) - vi(C - D), v1);
at(pos + 12) = mulconj((A - B) + vi(C - D), v3);
}
}
}
// A product in two steps, this <- this * t modulo x^n - i up to the factor n / flen:
// forward() on both operands, then multiply(t), which may be *this for a square.
// The forward transform stops at groups of 16 points and dot_fused16 multiplies those,
// which saves a pass over each operand and one over the result. That needs powers of
// four, so a length 2 * 4^k takes its radix-two level at the top, modulo
// x^(n/2) -+ sqrt(i); short lengths take the complete transform and the 4-point product.
void forward() {
if(!fused_leaves()) {
return fft();
}
size_t n = size(), part = fused_part();
if(part < n) {
vpoint vrt = {vz + real(roots[4]), vz + imag(roots[4])};
for(size_t k = 0; k < part; k += flen) {
auto t = at(k + part) * vrt;
at(k + part) = at(k) - t;
at(k) += t;
}
}
for(size_t offset = 0; offset < n; offset += part) {
transform<false, 0, 0, true>(n, false, offset, part);
}
checkpoint("fft");
}
void multiply(cvector const& t) {
if(!fused_leaves()) {
dot(t);
return ifft<true, false>();
}
size_t n = size(), part = fused_part();
dot_fused16<0>(t, 0, n);
checkpoint("dot");
for(size_t offset = 0; offset < n; offset += part) {
transform<true, 0, 0, true>(n, false, offset, part);
}
if(part < n) {
vpoint cvrt = {vz + real(roots[4]), vz - imag(roots[4])};
for(size_t k = 0; k < part; k += flen) {
auto t = at(k) - at(k + part);
at(k) += at(k + part);
at(k + part) = t * cvrt;
}
}
checkpoint("ifft");
}
// Radix-64 out-of-cache pass for n = 2^24, run as two tiled radix-8 stages.
// With Fuse=1 (forward only) the first stage lifts u32 residues to Gaussian
// coordinates on load, so the spectrum buffer is written once and never re-read
// before the in-cache block phase. The index-keyed noise makes both branches see
// the same representatives.
struct fuse_args {
const uint32_t* src = nullptr;
size_t count = 0;
uint64_t seed = 0;
double a = 0, b = 0, a_over_p = 0, b_over_p = 0;
};
static constexpr size_t sweep_tile = 256;
template<bool inverse, int Mode, size_t Tile, int Fuse = 0, bool Neg = false>
void sweep8(fuse_args const& fa = {}) {
constexpr size_t n = 1 << 24;
static const std::array<std::array<point, 7>, 9> weights = []() {
std::array<std::array<point, 7>, 9> table;
for(size_t count: {size_t(1), size_t(8)}) {
for(size_t k = 0; k < count; k++) {
size_t rev = 0, x = k;
for(size_t c = count; c > 1; c >>= 1) {
rev = (rev << 1) | (x & 1);
x >>= 1;
}
long double angle = std::numbers::pi_v<long double> * (1 + 4 * rev) / (16 * count);
if(k & 1) {
angle -= std::numbers::pi_v<long double> / 4;
}
for(size_t j = 1; j < 8; j++) {
long double a = j * angle;
if constexpr(Mode == 0) {
table[(count - 1) / 7 + k][j - 1] = {double(cosl(a)), double(sinl(a))};
} else {
table[(count - 1) / 7 + k][j - 1] = {double(sinl(a)), double(tanl(a / 2))};
}
}
}
}
return table;
}();
auto lift = [&] [[gnu::always_inline]] (size_t idx) -> vpoint {
i32x4 bits{};
if(idx + 4 <= fa.count) {
std::memcpy(&bits, fa.src + idx, sizeof(bits));
} else if(idx < fa.count) {
for(size_t j = 0; j < fa.count - idx; j++) {
bits[j] = int32_t(fa.src[idx + j]);
}
} else {
return vpoint{vz, vz};
}
auto x = __builtin_convertvector(bits, vftype);
u32x4 h =
u32x4{uint32_t(idx), uint32_t(idx + 1), uint32_t(idx + 2), uint32_t(idx + 3)} ^ uint32_t(fa.seed);
h *= 0x9E3779B1u;
h ^= h >> 15;
h *= 0x85EBCA77u;
h ^= h >> 13;
h *= 0xC2B2AE3Du;
h ^= h >> 16;
auto noise = __builtin_convertvector(i32x4(h), vftype) * 0x1p-32;
auto q = round(x * fa.a_over_p + noise), t = round(x * fa.b_over_p + noise);
auto re = x - q * fa.a - t * fa.b, im = t * fa.a - q * fa.b;
return vpoint{re, Neg ? -im : im};
};
auto stage = [&]<bool top> [[gnu::always_inline]] (size_t offset, size_t length, size_t begin, size_t end) {
size_t step = length / 8, k = offset / length;
auto const& w = weights[(n / length - 1) / 7 + k];
auto rot = [&]<size_t J> [[gnu::always_inline]] (vpoint z) {
auto c = w[J - 1];
if constexpr(Mode == 0) {
return z * vpoint{vz + real(c), inverse ? vz - imag(c) : vz + imag(c)};
} else {
auto s = inverse ? vz - real(c) : vz + real(c), t = inverse ? vz - imag(c) : vz + imag(c);
auto x = vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(imag(z)), __m256d(real(z))));
auto y = vftype(_mm256_fmadd_pd(__m256d(s), __m256d(x), __m256d(imag(z))));
return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(y), __m256d(x))), y};
}
};
auto add = [] [[gnu::always_inline]] (vpoint a, vpoint b) {
if constexpr(Mode != 2) {
return a + b;
} else {
return vpoint{vftype(_mm256_fmadd_pd(__m256d(real(a)), _mm256_set1_pd(1), __m256d(real(b)))),
vftype(_mm256_fmadd_pd(__m256d(imag(a)), _mm256_set1_pd(1), __m256d(imag(b))))};
}
};
auto sub = [] [[gnu::always_inline]] (vpoint a, vpoint b) {
if constexpr(Mode != 2) {
return a - b;
} else {
return vpoint{vftype(_mm256_fmsub_pd(__m256d(real(a)), _mm256_set1_pd(1), __m256d(real(b)))),
vftype(_mm256_fmsub_pd(__m256d(imag(a)), _mm256_set1_pd(1), __m256d(imag(b))))};
}
};
auto d4 = [] [[gnu::always_inline]] (vpoint a, vpoint b, vpoint c, vpoint d) {
auto s = a + c, t = a - c, u = b + d, v = vi(b - d);
if constexpr(inverse) {
return std::array<vpoint, 4>{s + u, t - v, s - u, t + v};
} else {
return std::array<vpoint, 4>{s + u, t + v, s - u, t - v};
}
};
constexpr double q = 0.707106781186547524400844362104849039;
auto r1 = [] [[gnu::always_inline]] (vpoint z) {
if constexpr(inverse) {
return vpoint{(real(z) + imag(z)) * q, (imag(z) - real(z)) * q};
} else {
return vpoint{(real(z) - imag(z)) * q, (real(z) + imag(z)) * q};
}
};
auto r3 = [] [[gnu::always_inline]] (vpoint z) {
if constexpr(inverse) {
return vpoint{(imag(z) - real(z)) * q, (-imag(z) - real(z)) * q};
} else {
return vpoint{(-real(z) - imag(z)) * q, (real(z) - imag(z)) * q};
}
};
std::array<vpoint*, 8> input, output;
for(size_t j = 0; j < 8; j++) {
input[j] = output[j] = r.data() + (offset + begin + j * step) / flen;
}
if(k & 1) {
constexpr std::array<size_t, 8> perm = {7, 6, 4, 5, 0, 1, 2, 3};
for(size_t j = 0; j < 8; j++) {
if constexpr(inverse) {
input[j] = output[perm[j]];
} else {
output[j] = input[perm[j]];
}
}
}
constexpr bool fused_in = top && Fuse == 1 && !inverse;
for(size_t j = 0; j < (end - begin) / flen; j++) {
size_t base = offset + begin + j * flen;
auto in = [&]<size_t S> [[gnu::always_inline]] () {
if constexpr(fused_in) {
return lift(base + S * step);
} else {
return input[S][j];
}
};
if constexpr(!inverse) {
auto E =
d4(in.template operator()<0>(), rot.template operator()<2>(in.template operator()<2>()),
rot.template operator()<4>(in.template operator()<4>()),
rot.template operator()<6>(in.template operator()<6>()));
auto O = d4(rot.template operator()<1>(in.template operator()<1>()),
rot.template operator()<3>(in.template operator()<3>()),
rot.template operator()<5>(in.template operator()<5>()),
rot.template operator()<7>(in.template operator()<7>()));
auto a = O[0], b = r1(O[1]), c = vi(O[2]), d = r3(O[3]);
output[0][j] = add(E[0], a);
output[1][j] = sub(E[0], a);
output[2][j] = add(E[2], c);
output[3][j] = sub(E[2], c);
output[4][j] = add(E[1], b);
output[5][j] = sub(E[1], b);
output[6][j] = add(E[3], d);
output[7][j] = sub(E[3], d);
} else {
auto E = d4(add(input[0][j], input[1][j]), add(input[4][j], input[5][j]),
add(input[2][j], input[3][j]), add(input[6][j], input[7][j]));
auto O = d4(sub(input[0][j], input[1][j]), r1(sub(input[4][j], input[5][j])),
-vi(sub(input[2][j], input[3][j])), r3(sub(input[6][j], input[7][j])));
output[0][j] = E[0];
output[1][j] = rot.template operator()<1>(O[0]);
output[2][j] = rot.template operator()<2>(E[1]);
output[3][j] = rot.template operator()<3>(O[1]);
output[4][j] = rot.template operator()<4>(E[2]);
output[5][j] = rot.template operator()<5>(O[2]);
output[6][j] = rot.template operator()<6>(E[3]);
output[7][j] = rot.template operator()<7>(O[3]);
}
}
};
constexpr size_t h = n / 64;
for(size_t j = 0; j < h; j += Tile) {
size_t end = std::min(h, j + Tile);
auto first = [&]() {
for(size_t t = 0; t < 8; t++) {
stage.template operator()<true>(0, n, j + t * h, end + t * h);
}
};
auto second = [&]() {
for(size_t k = 0; k < 8; k++) {
stage.template operator()<false>(k * n / 8, n / 8, j, end);
}
};
if constexpr(inverse) {
second();
first();
} else {
first();
second();
}
}
}
// Product for n = 2^24: the top three stages split each input into 64 independent
// blocks. Both forward transforms read the u32 inputs directly; each block then
// completes its two forward transforms, the product, and the inverse before the
// final three inverse stages combine the results.
// Fusing the Gaussian lift into the first pass saves a write and a read of each
// spectrum, but makes that pass compute-bound on the judge (measured slower there),
// so it stays opt-in; by default both spectra are filled first and swept in place.
static constexpr bool fuse_forward = false;
template<bool Neg, bool Fused = fuse_forward, bool Square = false>
void cache_product(cvector& b, fuse_args const& fa = {}, fuse_args const& fb = {}) {
constexpr size_t n = 1 << 24, block = 1 << 18;
prepare_roots(n / 16);
prepare_shear_roots();
if constexpr(Fused) {
sweep8<false, 2, sweep_tile, 1, Neg>(fa);
if constexpr(!Square) {
b.sweep8<false, 2, sweep_tile, 1, Neg>(fb);
}
} else {
sweep8<false, 2, sweep_tile>();
if constexpr(!Square) {
b.sweep8<false, 2, sweep_tile>();
}
}
checkpoint("sweep forward");
for(size_t offset = 0; offset < n; offset += block) {
transform<false, n, block, true>(n, false, offset, block);
if constexpr(!Square) {
b.transform<false, n, block, true>(n, false, offset, block);
}
dot_fused16<n, Square>(b, offset, block);
transform<true, n, block, true>(n, false, offset, block);
}
checkpoint("blocks");
sweep8<true, 2, sweep_tile>();
checkpoint("sweep inverse");
}
static constexpr size_t pre_evals = 1 << 16;
static const std::array<size_t, pre_evals> eval_args;
static const std::array<point, pre_evals> evalp;
private:
// The power of four that the fused product works on: all of n = 4^k, half of n = 2 * 4^k.
size_t fused_part() const {
return size() >> (std::countr_zero(size()) % 2);
}
bool fused_leaves() const {
return fused_part() >= 16 * dot_tile;
}
// Tile two radix-four stages together before descending into each child.
template<bool inverse, size_t fixed = 0, size_t range_fixed = 0, bool omit16 = false>
void transform(size_t input_n, bool parity, size_t range_offset = 0, size_t range_length = 0) {
if constexpr(range_fixed) {
range_length = range_fixed;
}
const size_t n = fixed ? fixed : input_n;
if constexpr(!range_fixed) {
prepare_roots(n / 16);
if constexpr(fixed == (1 << 24)) {
prepare_shear_roots();
}
}
size_t log_n = std::countr_zero(n);
auto butterfly = [&] [[gnu::always_inline]] (size_t offset, size_t length, size_t begin, size_t end) {
if constexpr(omit16) {
if(length == 16) {
return;
}
}
size_t step = length / 4, log_length = std::countr_zero(length), k = offset >> log_length;
auto *p0 = r.data() + (offset + begin) / flen, *p1 = r.data() + (offset + begin + step) / flen,
*p2 = r.data() + (offset + begin + 2 * step) / flen,
*p3 = r.data() + (offset + begin + 3 * step) / flen;
auto run = [&]<bool shear> [[gnu::always_inline]] () {
vpoint v1, v2, v3;
vftype t1{}, t2{}, t3{};
if constexpr(shear && fixed == (1 << 24)) {
auto const& c = shear_roots[((n >> log_length) - 1) / 3 + k];
v1 = {vz, inverse ? vz - real(c[0]) : vz + real(c[0])};
v2 = {vz, inverse ? vz - real(c[1]) : vz + real(c[1])};
v3 = {vz, inverse ? vz - real(c[2]) : vz + real(c[2])};
t1 = inverse ? vz - imag(c[0]) : vz + imag(c[0]);
t2 = inverse ? vz - imag(c[1]) : vz + imag(c[1]);
t3 = inverse ? vz - imag(c[2]) : vz + imag(c[2]);
} else {
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point rt = roots[log_n + 5 - log_length] * e;
v1 = {vz + real(rt), inverse ? vz - imag(rt) : vz + imag(rt)};
if constexpr(shear) {
if(k & 1) {
if constexpr(inverse) {
v1 = vi(v1);
} else {
v1 = -vi(v1);
}
}
}
v2 = v1 * v1;
v3 = v1 * v2;
if constexpr(shear) {
t1 = imag(v1) / (vz + 1.0 + real(v1));
t2 = imag(v2) / (vz + 1.0 + real(v2));
t3 = imag(v3) / (vz + 1.0 + real(v3));
}
}
auto rotate = [] [[gnu::always_inline]] (vpoint z, vpoint v, vftype t) {
if constexpr(!shear) {
return z * v;
} else {
auto x = vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(imag(z)), __m256d(real(z))));
auto y = vftype(_mm256_fmadd_pd(__m256d(imag(v)), __m256d(x), __m256d(imag(z))));
return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(y), __m256d(x))), y};
}
};
auto *i0 = p0, *i1 = p1, *i2 = p2, *i3 = p3, *o0 = p0, *o1 = p1, *o2 = p2, *o3 = p3;
if constexpr(shear) {
if(k & 1) {
if constexpr(inverse) {
i0 = p3;
i1 = p2;
i2 = p0;
i3 = p1;
} else {
o0 = p3;
o1 = p2;
o2 = p0;
o3 = p1;
}
}
}
for(size_t j = 0; j < (end - begin) / flen; j++) {
auto A = i0[j], B = i1[j], C = i2[j], D = i3[j];
if constexpr(inverse) {
o0[j] = (A + B) + (C + D);
o2[j] = rotate((A + B) - (C + D), v2, t2);
o1[j] = rotate((A - B) - vi(C - D), v1, t1);
o3[j] = rotate((A - B) + vi(C - D), v3, t3);
} else {
B = rotate(B, v1, t1);
C = rotate(C, v2, t2);
D = rotate(D, v3, t3);
o0[j] = (A + C) + (B + D);
o1[j] = (A + C) - (B + D);
o2[j] = (A - C) + vi(B - D);
o3[j] = (A - C) - vi(B - D);
}
}
};
if(length >= shear_min<fixed>) {
run.template operator()<true>();
} else {
run.template operator()<false>();
}
};
auto recurse = [&](this auto const& self, size_t offset, size_t length) -> void {
if(length < 4 * flen) {
return;
}
if(length >= (1 << 15)) {
size_t step = length / 16;
if constexpr(inverse) {
for(size_t t = 0; t < 16; t++) {
self(offset + t * step, step);
}
}
for(size_t j = 0; j < step; j += 256) {
size_t end = std::min(step, j + 256);
if constexpr(inverse) {
for(size_t t = 0; t < 4; t++) {
butterfly(offset + t * length / 4, length / 4, j, end);
}
for(size_t t = 0; t < 4; t++) {
butterfly(offset, length, j + t * step, end + t * step);
}
} else {
for(size_t t = 0; t < 4; t++) {
butterfly(offset, length, j + t * step, end + t * step);
}
for(size_t t = 0; t < 4; t++) {
butterfly(offset + t * length / 4, length / 4, j, end);
}
}
}
if constexpr(!inverse) {
for(size_t t = 0; t < 16; t++) {
self(offset + t * step, step);
}
}
} else if(length >= (size_t(1) << (6 + parity))) {
auto finish = [&]<bool par>() {
constexpr size_t chunk = size_t(1) << (6 + par);
constexpr size_t bottom = size_t(1) << (4 + par);
auto small = [&]<size_t L> [[gnu::always_inline]] (this auto const& self, size_t pos) -> void {
if constexpr(inverse && L > bottom) {
for(size_t q = 0; q < 4; q++) {
self.template operator()<L / 4>(pos + q * (L / 4));
}
}
butterfly(pos, L, 0, L / 4);
if constexpr(!inverse && L > bottom) {
for(size_t q = 0; q < 4; q++) {
self.template operator()<L / 4>(pos + q * (L / 4));
}
}
};
for(size_t leaf = offset; leaf < offset + length; leaf += chunk) {
if constexpr(inverse) {
small.template operator()<chunk>(leaf);
size_t level =
std::min<size_t>(std::countr_one(leaf + chunk - 1), std::countr_zero(length));
for(size_t lvl = 6 + 2 + par; lvl <= level; lvl += 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
} else {
size_t level = std::min<size_t>(std::countr_zero(n + leaf), std::countr_zero(length));
level -= level % 2 != par;
for(size_t lvl = level; lvl >= 6 + 2 + par; lvl -= 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
small.template operator()<chunk>(leaf);
}
}
};
if(parity) {
finish.template operator()<true>();
} else {
finish.template operator()<false>();
}
} else {
if constexpr(inverse) {
for(size_t leaf = offset + 3 * flen; leaf < offset + length; leaf += 4 * flen) {
size_t level = std::min<size_t>(std::countr_one(leaf + 3), std::countr_zero(length));
for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
}
} else {
for(size_t leaf = offset; leaf < offset + length; leaf += 4 * flen) {
size_t level = std::min<size_t>(std::countr_zero(n + leaf), std::countr_zero(length));
level -= level % 2 != parity;
for(size_t lvl = level; lvl >= 4; lvl -= 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
}
}
}
};
// Radix two is performed separately at the leaves.
recurse(range_offset, range_length ? range_length : n);
}
// Shortest butterfly that uses the shear rotation; the fixed 2^24 transform reads its
// shear roots from a table, so they are worth using down to the last in-block level.
template<size_t fixed>
static constexpr size_t shear_min = fixed == (1 << 24) ? 64 : 256;
static big_vector<std::array<point, 3>> shear_roots;
static void prepare_shear_roots() {
constexpr size_t n = 1 << 24;
if(!shear_roots.empty()) {
return;
}
shear_roots.resize((n / 16 - 1) / 3, std::array<point, 3>{});
for(size_t len = n; len >= shear_min<n>; len /= 4) {
size_t count = n / len, base = (count - 1) / 3;
point factor = roots[29 - std::countr_zero(len)];
for(size_t k = 0; k < count; k++) {
point rt = factor * (k < pre_evals ? evalp[k] : extra[k - pre_evals]);
vpoint v1 = {vz + real(rt), vz + imag(rt)};
if(k & 1) {
v1 = -vi(v1);
}
vpoint v2 = v1 * v1, v3 = v1 * v2;
vftype t1 = imag(v1) / (vz + 1.0 + real(v1)), t2 = imag(v2) / (vz + 1.0 + real(v2)),
t3 = imag(v3) / (vz + 1.0 + real(v3));
shear_roots[base + k] = {point(imag(v1)[0], t1[0]), point(imag(v2)[0], t2[0]),
point(imag(v3)[0], t3[0])};
}
}
}
static big_vector<point> extra;
// Keep the usual table small; cache additional roots for large transforms.
static void prepare_roots(size_t n) {
if(n <= pre_evals + extra.size()) {
return;
}
size_t old = extra.size();
extra.resize(std::bit_ceil(n) - pre_evals);
static const std::array<point, 256> coarse = []() {
std::array<point, 256> out;
for(size_t i = 0; i < 256; i++) {
out[i] = polar<ftype>(1., std::numbers::pi * double((eval_args[256 + i] - 1) / 2) / 512.0);
}
return out;
}();
for(size_t h = pre_evals + old; h < pre_evals + extra.size(); h *= 2) {
for(size_t i = h; i < 2 * h; i += 256) {
point fine = polar<ftype>(1., std::numbers::pi / double(4 * h) * double(eval_arg(4 * i)));
for(size_t j = 0; j < 256; j++) {
extra[i + j - pre_evals] = coarse[j] * fine;
}
}
}
}
};
big_vector<std::array<point, 3>> cvector::shear_roots;
big_vector<point> cvector::extra;
const std::array<size_t, cvector::pre_evals> cvector::eval_args = []() {
std::array<size_t, pre_evals> res = {};
for(size_t i = 1; i < pre_evals; i++) {
res[i] = res[i >> 1] | (i & 1) << (std::bit_width(i) - 1);
}
return res;
}();
const std::array<point, cvector::pre_evals> cvector::evalp = []() {
std::array<point, pre_evals> res = {};
res[0] = 1;
for(size_t n = 1; n < pre_evals; n++) {
res[n] = polar<ftype>(1., std::numbers::pi * ftype(eval_args[n]) / ftype(4 * std::bit_floor(n)));
}
return res;
}();
}
#pragma GCC pop_options
#line 6 "cp-algo/math/fft_simple.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
struct dft_simple {
cp_algo::math::fft::cvector cv;
dft_simple(auto const& a, size_t n): cv(n) {
for(size_t i = 0; i < std::min(std::size(a), n); i++) {
real(cv.at(i))[i % 4] = ftype(a[i]);
imag(cv.at(i))[i % 4] = ftype(i + n < std::size(a) ? a[i + n] : 0);
}
checkpoint("dft64 init");
cv.fft();
}
void dot(dft_simple const& t) {
cv.dot(t.cv);
}
void recover_mod(auto &res, size_t k) {
cv.ifft();
size_t n = cv.size();
for(size_t i = 0; i < std::min(k, n); i++) {
res[i] = llround(real(cv.get(i)));
}
for(size_t i = n; i < k; i++) {
res[i] = llround(imag(cv.get(i - n)));
}
cp_algo::checkpoint("recover mod");
}
};
// Multiplies a and b, assuming perfect precision and no overflow
void conv_simple(auto& a, auto const& b) {
if (empty(a) || empty(b)) {
a.clear();
return;
}
size_t n = a.size(), m = b.size();
size_t N = std::max(flen, std::bit_ceil(n + m - 1) / 2);
dft_simple A(a, N), B(b, N);
A.dot(B);
a.resize(n + m - 1);
A.recover_mod(a, n + m - 1);
}
}
#pragma GCC pop_options
#line 5 "cp-algo/math/bigint.hpp"
#include <bits/stdc++.h>
namespace cp_algo::math {
enum base_v {
x10 = uint64_t(1e16),
x16 = uint64_t(1ull << 60)
};
template<base_v base = x10>
struct bigint {
static constexpr uint64_t Base = uint64_t(base);
static constexpr uint16_t digit_length = base == x10 ? 16 : 15;
static constexpr uint16_t sub_base = base == x10 ? 10 : 16;
static constexpr uint32_t meta_base = base == x10 ? uint32_t(1e4) : uint32_t(1 << 15);
big_basic_string<uint64_t> digits;
bool negative;
auto operator <=> (bigint const& other) const {
auto sign = [](bigint const& x) { return x.digits.empty() ? 0 : x.negative ? -1 : 1; };
if(auto cmp = sign(*this) <=> sign(other); cmp != 0) {
return cmp;
}
auto cmp = digits.size() <=> other.digits.size();
if(cmp == 0) {
cmp = std::lexicographical_compare_three_way(
digits.rbegin(), digits.rend(), other.digits.rbegin(), other.digits.rend());
}
return !digits.empty() && negative ? 0 <=> cmp : cmp;
}
bigint() {}
bigint(big_basic_string<uint64_t> d, bool neg): digits(std::move(d)), negative(neg) {
normalize();
}
bigint& pad_inplace(size_t to_add) {
digits.insert(0, to_add, 0);
return normalize();
}
bigint& drop_inplace(size_t to_drop) {
digits.erase(0, std::min(to_drop, size(digits)));
return normalize();
}
bigint& take_inplace(size_t to_keep) {
digits.erase(std::min(to_keep, size(digits)), std::string::npos);
return normalize();
}
bigint& top_inplace(size_t to_keep) {
if (to_keep >= size(digits)) {
return pad_inplace(to_keep - size(digits));
} else {
return drop_inplace(size(digits) - to_keep);
}
}
bigint pad(size_t to_add) const {
return bigint{big_basic_string<uint64_t>(to_add, 0) + digits, negative};
}
bigint drop(size_t to_drop) const {
return bigint{digits.substr(std::min(to_drop, size(digits))), negative};
}
bigint take(size_t to_keep) const {
return bigint{digits.substr(0, std::min(to_keep, size(digits))), negative};
}
bigint top(size_t to_keep) const {
if (to_keep >= size(digits)) {
return pad(to_keep - size(digits));
} else {
return drop(size(digits) - to_keep);
}
}
bigint& normalize() {
while (!empty(digits) && digits.back() == 0) {
digits.pop_back();
}
if (digits.empty()) {
negative = false;
}
return *this;
}
bigint& negate() {
negative ^= 1;
return *this;
}
bigint operator -() {
return bigint(*this).negate();
}
bigint& operator -= (const bigint& other) {
if (negative != other.negative) {
return (negate() += other).negate().normalize();
}
digits.resize(std::max(size(digits), size(other.digits)));
bool carry = false;
auto d_ptr = std::assume_aligned<32>(digits.data());
auto o_ptr = std::assume_aligned<32>(other.digits.data());
size_t N = size(other.digits);
size_t i = 0;
for (; i < N; i++) {
d_ptr[i] -= o_ptr[i] + carry;
carry = d_ptr[i] >= base;
d_ptr[i] += carry ? uint64_t(base) : 0;
}
if (carry) {
N = size(digits);
for (; i < N && d_ptr[i] == 0; i++) {
d_ptr[i] = base - 1;
}
if (i < N) {
d_ptr[i]--;
} else {
// Two's complement: flip all digits then add 1
for (i = 0; i < N; i++) {
d_ptr[i] = base - d_ptr[i] - 1;
}
bool carry = true;
for (i = 0; i < N && carry; i++) {
d_ptr[i]++;
carry = d_ptr[i] >= base;
d_ptr[i] -= carry * base;
}
negate();
}
}
return normalize();
}
bigint& operator += (const bigint& other) {
if (negative != other.negative) {
return (negate() -= other).negate().normalize();
}
digits.resize(std::max(size(digits), size(other.digits)));
bool carry = false;
auto d_ptr = std::assume_aligned<32>(digits.data());
auto o_ptr = std::assume_aligned<32>(other.digits.data());
size_t N = size(other.digits);
size_t i = 0;
for (; i < N; i++) {
d_ptr[i] += o_ptr[i] + carry;
carry = d_ptr[i] >= base;
d_ptr[i] -= carry ? uint64_t(base) : 0;
}
if (carry) {
N = size(digits);
for (; i < N && d_ptr[i] == uint64_t(base) - 1; i++) {
d_ptr[i] = 0;
}
if (i < N) {
d_ptr[i]++;
} else {
digits.push_back(1);
}
}
return *this;
}
bigint(int64_t x) {
negative = x < 0;
x = negative ? -x : x;
digits = x ? big_basic_string<uint64_t>{uint64_t(x)} : big_basic_string<uint64_t>{};
}
bigint(std::span<char> s): negative(false) {
if (size(s) < digit_length) {
int64_t val = 0;
std::from_chars(s.data(), s.data() + size(s), val, sub_base);
*this = bigint(val);
return;
}
if (!empty(s) && s[0] == '-') {
negative = true;
s = s.subspan(1);
}
size_t len = size(s);
assert(len > 0);
size_t num_digits = (len + digit_length - 1) / digit_length;
digits.resize(num_digits);
size_t i = len;
for (size_t j = 0; j < num_digits - 1; j++) {
std::from_chars(s.data() + i - digit_length, s.data() + i, digits[j], sub_base);
i -= digit_length;
}
std::from_chars(s.data(), s.data() + i, digits.back(), sub_base);
normalize();
}
bigint operator + (const bigint& other) const {
return bigint(*this) += other;
}
bigint operator - (const bigint& other) const {
return bigint(*this) -= other;
}
void to_metabase() {
auto N = ssize(digits);
digits.resize(4 * N);
for (auto i = N - 1; i >= 0; i--) {
uint64_t val = digits[i];
digits[4 * i] = val % meta_base;
val /= meta_base;
digits[4 * i + 1] = val % meta_base;
val /= meta_base;
digits[4 * i + 2] = val % meta_base;
val /= meta_base;
digits[4 * i + 3] = val;
}
}
void from_metabase() {
auto N = (ssize(digits) + 3) / 4;
digits.resize(4 * N);
uint64_t carry = 0;
for (int i = 0; i < N; i++) {
__uint128_t val = digits[4 * i + 3];
val = val * meta_base + digits[4 * i + 2];
val = val * meta_base + digits[4 * i + 1];
val = val * meta_base + digits[4 * i];
val += carry;
digits[i] = uint64_t(val % base);
carry = uint64_t(val / base);
}
digits.resize(N);
while (carry) {
digits.push_back(carry % base);
carry /= base;
}
}
bigint& operator *= (int64_t other) {
if (other < 0) {
negative ^= 1;
other = -other;
}
if (other == 0) {
return *this = bigint(0);
} else if (other == 1) {
return *this;
}
uint64_t carry = 0;
for (auto &d: digits) {
__uint128_t val = __uint128_t(d) * other + carry;
d = uint64_t(val % base);
carry = uint64_t(val / base);
}
if (carry) {
digits.push_back(carry % base);
carry /= base;
}
return *this;
}
bigint operator * (int64_t other) const {
return bigint(*this) *= other;
}
friend bigint operator * (int64_t lhs, const bigint& rhs) {
return bigint(rhs) *= lhs;
}
bigint& mul_inplace(auto &&other) {
negative ^= other.negative;
auto n = size(digits), m = size(other.digits);
if (n < m) {
std::swap(n, m);
std::swap(digits, other.digits);
}
if (m <= 1) {
return *this *= int64_t(m == 0 ? 0 : other.digits[0]);
}
// Small m: use schoolbook long multiplication in base `Base`
// Threshold chosen empirically to avoid FFT overhead on small sizes
constexpr size_t SMALL_M_THRESHOLD = 32;
if (m <= SMALL_M_THRESHOLD) {
big_basic_string<uint64_t> res;
res.assign(n + m, 0);
for (size_t i = 0; i < n; i++) {
if (digits[i] == 0) continue;
uint64_t carry = 0;
for (size_t j = 0; j < m; j++) {
__uint128_t cur = res[i + j]
+ (__uint128_t)digits[i] * other.digits[j]
+ carry;
res[i + j] = uint64_t(cur % Base);
carry = uint64_t(cur / Base);
}
size_t k = i + m;
if (carry) {
uint64_t cur = res[k] + carry;
res[k] = cur % Base;
carry = cur / Base;
k++;
}
}
digits = std::move(res);
return normalize();
}
to_metabase();
other.to_metabase();
fft::conv_simple(digits, other.digits);
from_metabase();
return normalize();
}
bigint& operator *= (bigint const& other) {
return mul_inplace(bigint(other));
}
bigint operator * (const bigint& other) const {
return bigint(*this).mul_inplace(bigint(other));
}
};
template<base_v base>
decltype(std::cin)& operator >> (decltype(std::cin) &in, cp_algo::math::bigint<base> &x) {
std::string s;
in >> s;
x = {s};
return in;
}
template<base_v base, bool fill = true>
auto& print_digit(auto &out, uint64_t d) {
char buf[16];
auto [ptr, ec] = std::to_chars(buf, buf + sizeof(buf), d, bigint<base>::sub_base);
if constexpr (base == x16) {
std::ranges::transform(buf, buf, toupper);
}
auto len = ptr - buf;
if constexpr (fill) {
out << std::string(bigint<base>::digit_length - len, '0');
}
return out << std::string_view(buf, len);
}
template<bool fill_all = false, base_v base>
auto& print_bigint(auto &out, cp_algo::math::bigint<base> const& x) {
if (x.negative) {
out << '-';
}
if (empty(x.digits)) {
return print_digit<base, fill_all>(out, 0);
}
print_digit<base, fill_all>(out, x.digits.back());
for (auto d: x.digits | std::views::reverse | std::views::drop(1)) {
print_digit<base, true>(out, d);
}
return out;
}
template<base_v base>
decltype(std::cout)& operator << (decltype(std::cout) &out, cp_algo::math::bigint<base> const& x) {
return print_bigint(out, x);
}
}
#line 3 "tests/bigint.cpp"
using namespace cp_algo;
using namespace cp_algo::math;
template<base_v radix>
void check() {
using B = bigint<radix>;
std::mt19937_64 rng(71823);
auto value = [](B const& a) {
__int128 result = 0;
for(auto digit: a.digits | std::views::reverse) {result = result * B::Base + digit;}
return a.negative ? -result : result;
};
std::vector<B> cases{B(0), B(1), B(-1), B({0, 1}, false), B({0, 1}, true)};
auto zero = B(0);
zero.negate(); // A negative zero still compares equal to zero.
cases.push_back(zero);
for(int i = 0; i < 100; i++) {
cases.emplace_back(big_basic_string<uint64_t>{rng() % B::Base, rng() % B::Base}, rng() % 2);
}
for(auto const& a: cases) {
for(auto const& b: cases) {
assert((a <=> b) == (value(a) <=> value(b)));
}
for(size_t n = 0; n <= 3; n++) {
__int128 power = 1;
// Two limbs fit in __int128; larger truncations leave the input unchanged.
for(size_t i = 0; i < std::min(n, size_t(2)); i++) {power *= B::Base;}
assert(value(a.take(n)) == (n >= 2 ? value(a) : value(a) % power));
assert(value(a.drop(n)) == (n >= 2 ? 0 : value(a) / power));
}
}
for(int64_t x: {-1000000, -1, 0, 1, 1000000}) {
B a(x);
assert(value(a.pad(1)) == __int128(x) * B::Base);
assert(a.pad(1).drop(1).digits == a.digits);
}
}
int main() {
check<x10>();
check<x16>();
std::cout << "Bigint ordering and slicing passed independent integer checks\n";
}
#line 1 "cp-algo/math/bigint.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#include <bit>
#include <iostream>
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{ template<typename T,size_t Align=32> class big_alloc{ static_assert(Align>=alignof(void*),"Align must be at least pointer-size"); static_assert(std::has_single_bit(Align),"Align must be a power of two"); public: using value_type=T; template<class U> struct rebind{ using other=big_alloc<U,Align>; }; constexpr bool operator==(const big_alloc&)const=default; big_alloc()noexcept=default; template<typename U,std::size_t A> big_alloc(const big_alloc<U,A>&)noexcept{} [[nodiscard]]T*allocate(std::size_t n){ std::size_t padded=round_up(n*sizeof(T)); std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){ void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0); madvise(raw,padded,MADV_HUGEPAGE); return static_cast<T*>(raw); }
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align))); } void deallocate(T*p,std::size_t n)noexcept{ if(!p){ return; } std::size_t padded=round_up(n*sizeof(T)); std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){ munmap(p,padded); return; }
#endif
::operator delete(p,padded,std::align_val_t(align)); } private: static constexpr std::size_t MEGABYTE=1<<20; static constexpr std::size_t round_up(std::size_t x)noexcept{ return(x+Align-1)/Align*Align; } }; template<typename T> using big_vector=std::vector<T,big_alloc<T>>; template<typename T> using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>; template<typename T> using big_deque=std::deque<T,big_alloc<T>>; template<typename T> using big_stack=std::stack<T,big_deque<T>>; template<typename T> using big_queue=std::queue<T,big_deque<T>>; template<typename T> using big_priority_queue=std::priority_queue<T,big_vector<T>>; template<typename T> using big_forward_list=std::forward_list<T,big_alloc<T>>; using big_string=big_basic_string<char>; template<typename Key,typename Value,typename Compare=std::less<Key>> using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>; template<typename T,typename Compare=std::less<T>> using big_multiset=std::multiset<T,Compare,big_alloc<T>>; template<typename T,typename Compare=std::less<T>> using big_set=std::set<T,Compare,big_alloc<T>>; }
#line 1 "cp-algo/math/fft_simple.hpp"
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random{ std::mt19937_64 gen( std::chrono::steady_clock::now().time_since_epoch().count() ); uint64_t rng(){ return gen(); } }
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#line 8 "cp-algo/math/common.hpp"
#include <algorithm>
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64; template<int window=1> auto bpow(auto const&x,auto n,auto const&one,auto op){ static_assert(window>=1&&window<=6); if constexpr(window>1){ if(n==0){return one;} int bits=std::bit_width(uint64_t(n)); auto low_bit= [&](int high){ int low=std::max(0,high-window+1); while(!((n>>low)&1)){low++;} return low; }; int first=low_bit(bits-1); int cost=(1<<(window-1))+first; for(int j=first-1;j>=0;){ if(!((n>>j)&1)){j--;} else{cost++;j=low_bit(j)-1;} } if(cost>=bits+std::popcount(uint64_t(n))-2){return bpow<1>(x,n,one,op);} using T=std::decay_t<decltype(x)>; std::vector<T>odd; odd.reserve(1<<(window-1)); odd.push_back(x); auto square=op(x,x); while(odd.size()<size_t(1<<(window-1))){odd.push_back(op(odd.back(),square));} auto ans=odd[(n>>first)/2]; for(int j=first-1;j>=0;){ if(!((n>>j)&1)){ans=op(ans,ans);j--;} else{ int low=low_bit(j),length=j-low+1; auto digit=(n>>low)&((1u<<length)-1); for(int i=0;i<length;i++){ans=op(ans,ans);} ans=op(ans,odd[digit/2]); j=low-1; } } return ans; }else{ if(n==0){ return one; } auto ans=x; for(int j=std::bit_width<uint64_t>(n)-2;~j;j--){ ans=op(ans,ans); if((n>>j)&1){ ans=op(ans,x); } } return ans; } } template<int window=1> auto bpow(auto x,auto n,auto ans){ return bpow<window>(x,n,ans,std::multiplies{}); } template<typename T> T bpow(T const&x,auto n){ return bpow(x,n,T(1)); } inline constexpr auto inv2(auto x){ assert(x%2); std::make_unsigned_t<decltype(x)>y=1; while(y*x!=1){ y*=2-x*y; } return y; } }
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH _Pragma("GCC push_options") CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{ template<typename T,size_t len> using simd[[gnu::vector_size(len*sizeof(T))]] =T; using u64x8=simd<uint64_t,8>; using u32x16=simd<uint32_t,16>; using i64x4=simd<int64_t,4>; using u64x4=simd<uint64_t,4>; using u32x8=simd<uint32_t,8>; using u16x16=simd<uint16_t,16>; using i32x4=simd<int32_t,4>; using u32x4=simd<uint32_t,4>; using u16x8=simd<uint16_t,8>; using u16x4=simd<uint16_t,4>; using i16x4=simd<int16_t,4>; using u8x32=simd<uint8_t,32>; using u8x16=simd<uint8_t,16>; using u8x8=simd<uint8_t,8>; using u8x4=simd<uint8_t,4>; using dx4=simd<double,4>; inline dx4 abs(dx4 a){ return dx4{ std::abs(a[0]), std::abs(a[1]), std::abs(a[2]), std::abs(a[3]) }; } static constexpr dx4 magic=dx4()+(3ULL<<51); inline i64x4 lround(dx4 x){ return i64x4(x+magic)-i64x4(magic); } inline dx4 to_double(i64x4 x){ return dx4(x+i64x4(magic))-magic; } inline dx4 round(dx4 a){ return dx4{ std::nearbyint(a[0]), std::nearbyint(a[1]), std::nearbyint(a[2]), std::nearbyint(a[3]) }; } inline u64x4 low32(u64x4 x){ return x&uint32_t(-1); } inline auto swap_bytes(auto x){ return decltype(x)(__builtin_shufflevector(u32x8(x),u32x8(x),1,0,3,2,5,4,7,6)); } inline u64x4 montgomery_reduce(u64x4 x,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
auto x_ninv=u64x4(_mm256_mul_epu32(__m256i(x),__m256i()+imod)); x+=u64x4(_mm256_mul_epu32(__m256i(x_ninv),__m256i()+mod));
#else
auto x_ninv=u64x4(u32x8(low32(x))*imod); x+=x_ninv*uint64_t(mod);
#endif
return swap_bytes(x); } inline u32x8 reduce_once(u32x8 x,uint32_t mod){ auto y=x-mod; return x<y?x:y; } inline u64x4 reduce_once(u64x4 x,uint32_t mod){ return u64x4(reduce_once(u32x8(x),mod)); } inline u64x4 montgomery_mul(u64x4 x,u64x4 y,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x),__m256i(y))),mod,imod);
#else
return montgomery_reduce(x*y,mod,imod);
#endif
} inline u32x8 montgomery_mul(u32x8 x,u32x8 y,uint32_t mod,uint32_t imod){ return u32x8(montgomery_mul(u64x4(x),u64x4(y),mod,imod))| u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)),u64x4(swap_bytes(y)),mod,imod))); } inline dx4 rotate_right(dx4 x){ static constexpr u64x4 shuffler={3,0,1,2}; return __builtin_shuffle(x,shuffler); } template<std::size_t Align=32> inline bool is_aligned(const auto*p)noexcept{ return(reinterpret_cast<std::uintptr_t>(p)%Align)==0; } template<class Target> inline Target&vector_cast(auto&&p){ return*reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p)); } }
#pragma GCC pop_options
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{ template<typename T> struct complex{ using value_type=T; T x,y; inline constexpr complex():x(),y(){} inline constexpr complex(T const&x):x(x),y(){} inline constexpr complex(T const&x,T const&y):x(x),y(y){} inline complex&operator*=(T const&t){x*=t;y*=t;return*this;} inline complex&operator/=(T const&t){x/=t;y/=t;return*this;} inline complex operator*(T const&t)const{return complex(*this)*=t;} inline complex operator/(T const&t)const{return complex(*this)/=t;} inline complex&operator+=(complex const&t){x+=t.x;y+=t.y;return*this;} inline complex&operator-=(complex const&t){x-=t.x;y-=t.y;return*this;} inline complex operator*(complex const&t)const{return{x*t.x-y*t.y,x*t.y+y*t.x};} inline complex operator/(complex const&t)const{return*this*t.conj()/t.norm();} inline complex operator+(complex const&t)const{return complex(*this)+=t;} inline complex operator-(complex const&t)const{return complex(*this)-=t;} inline complex&operator*=(complex const&t){return*this= *this*t;} inline complex&operator/=(complex const&t){return*this= *this/t;} inline complex operator-()const{return{-x,-y};} inline complex conj()const{return{x,-y};} inline T norm()const{return x*x+y*y;} inline T abs()const{return std::sqrt(norm());} inline T const real()const{return x;} inline T const imag()const{return y;} inline T&real(){return x;} inline T&imag(){return y;} inline static constexpr complex polar(T r,T theta){return{T(r*cos(theta)),T(r*sin(theta))};} inline auto operator<=>(complex const&t)const=default; }; template<typename T>inline complex<T>conj(complex<T>const&x){return x.conj();} template<typename T>inline T norm(complex<T>const&x){return x.norm();} template<typename T>inline T abs(complex<T>const&x){return x.abs();} template<typename T>inline T&real(complex<T> &x){return x.real();} template<typename T>inline T&imag(complex<T> &x){return x.imag();} template<typename T>inline T const real(complex<T>const&x){return x.real();} template<typename T>inline T const imag(complex<T>const&x){return x.imag();} template<typename T> inline constexpr complex<T>polar(T r,T theta){ return complex<T>::polar(r,theta); } template<typename T> inline std::ostream&operator<<(std::ostream&out,complex<T>const&x){ return out<<x.real()<<' '<<x.imag(); } }
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo{
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string,double>checkpoints; double last;
#endif
template<bool final=false> void checkpoint([[maybe_unused]]auto const&_msg){
#ifdef CP_ALGO_CHECKPOINT
big_string msg=_msg; double now=(double)clock()/CLOCKS_PER_SEC; double delta=now-last; last=now; if(msg.size()&& !final){ checkpoints[msg] +=delta; } if(final){ for(auto const& [key,value] :checkpoints){ std::cerr<<key<<": "<<value*1000<<" ms\n"; } std::cerr<<"Total: "<<now*1000<<" ms\n"; }
#endif
} template<bool final=false> void checkpoint(){ checkpoint<final>(""); } }
#line 7 "cp-algo/math/cvector.hpp"
#include <immintrin.h>
#include <numbers>
#include <cstring>
#include <ranges>
#line 13 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace stdx=std::experimental; namespace cp_algo::math::fft{ static constexpr size_t flen=4; using ftype=double; using vftype=dx4; using point=complex<ftype>; using vpoint=complex<vftype>; static constexpr vftype vz={}; vpoint vi(vpoint const&r){ return{-imag(r),real(r)}; } template<class T> struct spectrum_alloc:big_alloc<T>{ using big_alloc<T>::big_alloc; template<class U> struct rebind{ using other=spectrum_alloc<U>; }; template<class U> requires(std::is_same_v<U,vpoint>) void construct(U*)noexcept{} }; using spectrum_vector=std::vector<vpoint,spectrum_alloc<vpoint>>; struct cvector{ spectrum_vector r; cvector(size_t n){ n=std::max(flen,std::bit_ceil(n)); r.assign(n/flen,vpoint{}); prepare_roots(n/16); checkpoint("cvector create"); } vpoint&at(size_t k){return r[k/flen];} vpoint at(size_t k)const{return r[k/flen];} template<class pt=point> inline void set(size_t k,pt const&t){ if constexpr(std::is_same_v<pt,point>){ real(r[k/flen])[k%flen] =real(t); imag(r[k/flen])[k%flen] =imag(t); }else{ at(k)=t; } } template<class pt=point> inline pt get(size_t k)const{ if constexpr(std::is_same_v<pt,point>){ return{real(r[k/flen])[k%flen],imag(r[k/flen])[k%flen]}; }else{ return at(k); } } size_t size()const{ return flen*r.size(); } static constexpr size_t eval_arg(size_t n){ if(n<pre_evals){ return eval_args[n]; }else{ return eval_arg(n/2)|(n&1)<<(std::bit_width(n)-1); } } static constexpr point eval_point(size_t n){ if(n%2){ return-eval_point(n-1); }else if(n%4){ return eval_point(n-2)*point(0,1); }else if(n/4<pre_evals){ return evalp[n/4]; }else if(n/4-pre_evals<extra.size()){ return extra[n/4-pre_evals]; }else{ return polar<ftype>(1.,std::numbers::pi/(ftype)std::bit_floor(n)*(ftype)eval_arg(n)); } } static constexpr std::array<point,32>roots= [](){ std::array<point,32>res; for(size_t i=2;i<32;i++){ res[i] =polar<ftype>(1.,std::numbers::pi/(1ull<<(i-2))); } return res; }(); static constexpr point root(size_t n){ return roots[std::bit_width(n)]; } template<int step> static void exec_on_eval(size_t n,size_t k,auto&&callback){ callback(k,root(4*step*n)*eval_point(step*k)); } template<int step> static void exec_on_evals(size_t n,auto&&callback){ point factor=root(4*step*n); if constexpr(step==1||step==2||step==4){ prepare_roots((step*n+3)/4); size_t i=0; if constexpr(step==1){ for(;i+4<=n;i+=4){ size_t k=i/4; point e=k<pre_evals?evalp[k] :extra[k-pre_evals]; point v=factor*e; callback(i,v); callback(i+1,-v); point iv(-imag(v),real(v)); callback(i+2,iv); callback(i+3,-iv); } }else if constexpr(step==2){ for(;i+2<=n;i+=2){ size_t k=i/2; point e=k<pre_evals?evalp[k] :extra[k-pre_evals]; point v=factor*e; callback(i,v); callback(i+1,point(-imag(v),real(v))); } } for(;i<n;i++){ size_t index=step*i,k=index/4; point e=k<pre_evals?evalp[k] :extra[k-pre_evals]; if(index&2){ e=e*point(0,1); } if(index&1){ e= -e; } callback(i,factor*e); } }else{ for(size_t i=0;i<n;i++){ callback(i,factor*eval_point(step*i)); } } } static void do_dot_iter(point rt,vpoint&Bv,vpoint const&Av,vpoint&res){ res+=Av*Bv; real(Bv)=rotate_right(real(Bv)); imag(Bv)=rotate_right(imag(Bv)); auto x=real(Bv)[0],y=imag(Bv)[0]; real(Bv)[0] =x*real(rt)-y*imag(rt); imag(Bv)[0] =x*imag(rt)+y*real(rt); } template<size_t fixed=0> void dot(cvector const&t){ size_t n=fixed?fixed:this->size(); exec_on_evals<1>(n/flen,[&] [[gnu::always_inline]](size_t k,point rt){ k*=flen; auto[Ax,Ay] =at(k); auto Cv=t.at(k); vpoint vrt={vz+real(rt),vz+imag(rt)}; auto Cr=Cv*vrt; vpoint res=vz; auto iter= [&]<int i> [[gnu::always_inline]](){ auto wrap= [&](vftype original,vftype rotated){ if constexpr(i==0){ return original; }else{ return __builtin_shufflevector(rotated,original,4-i,5-i,6-i,7-i); } }; vpoint Cw={wrap(real(Cv),real(Cr)),wrap(imag(Cv),imag(Cr))}; vpoint Av={vz+Ax[i],vz+Ay[i]}; return Av*Cw; }; auto p0=iter.template operator()<0>(),p1=iter.template operator()<1>(); auto p2=iter.template operator()<2>(),p3=iter.template operator()<3>(); res=(p0+p1)+(p2+p3); set(k,res); }); checkpoint("dot"); } template<bool partial=true,bool normalize=true,size_t fixed=0> void ifft(){ size_t n=fixed?fixed:size(); if constexpr(!partial){ prepare_roots(n/4); point pi(0,1); exec_on_evals<4>(n/4,[&] [[gnu::always_inline]](size_t k,point rt){ k*=4; point v1=conj(rt); point v2=v1*v1; point v3=v1*v2; auto A=get(k); auto B=get(k+1); auto C=get(k+2); auto D=get(k+3); set(k,(A+B)+(C+D)); set(k+2,((A+B)-(C+D))*v2); set(k+1,((A-B)-pi*(C-D))*v1); set(k+3,((A-B)+pi*(C-D))*v3); }); } bool parity=std::countr_zero(n)%2; if(parity){ exec_on_evals<2>(n/(2*flen),[&] [[gnu::always_inline]](size_t k,point rt){ k*=2*flen; vpoint cvrt={vz+real(rt),vz-imag(rt)}; auto B=at(k)-at(k+flen); at(k)+=at(k+flen); at(k+flen)=B*cvrt; }); } transform<true,fixed>(n,parity); checkpoint("ifft"); if constexpr(normalize){ auto scale=vz+ftype(partial?flen:1)/ftype(n); for(size_t k=0;k<n;k+=flen){ set(k,get<vpoint>(k)*scale); } } } template<bool partial=true,size_t fixed=0> void fft(){ size_t n=fixed?fixed:size(); prepare_roots(n/(partial?16:4)); bool parity=std::countr_zero(n)%2; transform<false,fixed>(n,parity); if(parity){ exec_on_evals<2>(n/(2*flen),[&] [[gnu::always_inline]](size_t k,point rt){ k*=2*flen; vpoint vrt={vz+real(rt),vz+imag(rt)}; auto t=at(k+flen)*vrt; at(k+flen)=at(k)-t; at(k)+=t; }); } if constexpr(!partial){ prepare_roots(n/4); point pi(0,1); exec_on_evals<4>(n/4,[&] [[gnu::always_inline]](size_t k,point rt){ k*=4; point v1=rt; point v2=v1*v1; point v3=v1*v2; auto A=get(k); auto B=get(k+1)*v1; auto C=get(k+2)*v2; auto D=get(k+3)*v3; set(k,(A+C)+(B+D)); set(k+1,(A+C)-(B+D)); set(k+2,(A-C)+pi*(B-D)); set(k+3,(A-C)-pi*(B-D)); }); } checkpoint("fft"); } static std::array<vpoint,4>transpose(std::array<vpoint,4>const&a){ auto half= [](auto part){ auto a0=__m256d(part(0)),a1=__m256d(part(1)),a2=__m256d(part(2)),a3=__m256d(part(3)); auto t0=_mm256_unpacklo_pd(a0,a1),t1=_mm256_unpackhi_pd(a0,a1),t2=_mm256_unpacklo_pd(a2,a3), t3=_mm256_unpackhi_pd(a2,a3); return std::array<vftype,4>{ vftype(_mm256_permute2f128_pd(t0,t2,0x20)),vftype(_mm256_permute2f128_pd(t1,t3,0x20)), vftype(_mm256_permute2f128_pd(t0,t2,0x31)),vftype(_mm256_permute2f128_pd(t1,t3,0x31))}; }; auto re=half([&](int k){return real(a[k]);}),im=half([&](int k){return imag(a[k]);}); return{vpoint{re[0],im[0]},vpoint{re[1],im[1]},vpoint{re[2],im[2]},vpoint{re[3],im[3]}}; } static constexpr size_t dot_tile=64; template<size_t fixed,bool square=false> void dot_fused16(cvector const&t,size_t offset,size_t length){ constexpr size_t T=dot_tile; const size_t n=fixed?fixed:size(); point factor=root(n); auto fr=_mm256_set1_pd(real(factor)),fi=_mm256_set1_pd(imag(factor)); alignas(32)double tw[6][T]; const bool ahead=offset+length<n; auto cmadd= [] [[gnu::always_inline]](vpoint x,vpoint y,vpoint c){ auto re=_mm256_fmadd_pd(__m256d(real(x)),__m256d(real(y)), _mm256_fnmadd_pd(__m256d(imag(x)),__m256d(imag(y)),__m256d(real(c)))); auto im=_mm256_fmadd_pd(__m256d(real(x)),__m256d(imag(y)), _mm256_fmadd_pd(__m256d(imag(x)),__m256d(real(y)),__m256d(imag(c)))); return vpoint{vftype(re),vftype(im)}; }; auto cmcross= [] [[gnu::always_inline]](vpoint x,vpoint y,vpoint p,vpoint q){ auto re=_mm256_sub_pd(_mm256_fmsub_pd(__m256d(real(x)),__m256d(real(y)),__m256d(real(p))), _mm256_fmadd_pd(__m256d(imag(x)),__m256d(imag(y)),__m256d(real(q)))); auto im=_mm256_add_pd(_mm256_fmsub_pd(__m256d(real(x)),__m256d(imag(y)),__m256d(imag(p))), _mm256_fmsub_pd(__m256d(imag(x)),__m256d(real(y)),__m256d(imag(q)))); return vpoint{vftype(re),vftype(im)}; }; auto mulconj= [] [[gnu::always_inline]](vpoint z,vpoint v){ auto re=_mm256_fmadd_pd(__m256d(real(z)),__m256d(real(v)), _mm256_mul_pd(__m256d(imag(z)),__m256d(imag(v)))); auto im=_mm256_fmsub_pd(__m256d(imag(z)),__m256d(real(v)), _mm256_mul_pd(__m256d(real(z)),__m256d(imag(v)))); return vpoint{vftype(re),vftype(im)}; }; for(size_t tile=offset;tile<offset+length;tile+=16*T){ size_t k0=tile/16; auto const*e=reinterpret_cast<double const*>(k0<pre_evals?evalp.data()+k0 :extra.data()+(k0-pre_evals)); for(size_t g=0;g<T;g+=4){ auto x=_mm256_loadu_pd(e+2*g),y=_mm256_loadu_pd(e+2*g+4); auto er=_mm256_permute4x64_pd(_mm256_unpacklo_pd(x,y),0xD8), ei=_mm256_permute4x64_pd(_mm256_unpackhi_pd(x,y),0xD8); auto r1=_mm256_fmsub_pd(fr,er,_mm256_mul_pd(fi,ei)), i1=_mm256_fmadd_pd(fr,ei,_mm256_mul_pd(fi,er)); auto r2=_mm256_fmsub_pd(r1,r1,_mm256_mul_pd(i1,i1)), i2=_mm256_fmadd_pd(r1,i1,_mm256_mul_pd(i1,r1)); auto r3=_mm256_fmsub_pd(r1,r2,_mm256_mul_pd(i1,i2)), i3=_mm256_fmadd_pd(r1,i2,_mm256_mul_pd(i1,r2)); _mm256_store_pd(tw[0] +g,r1); _mm256_store_pd(tw[1] +g,i1); _mm256_store_pd(tw[2] +g,r2); _mm256_store_pd(tw[3] +g,i2); _mm256_store_pd(tw[4] +g,r3); _mm256_store_pd(tw[5] +g,i3); } for(size_t g=0;g<T;g++){ size_t pos=tile+16*g; if(ahead){ auto const*pa=reinterpret_cast<char const*>(r.data()+(pos+length)/flen); auto const*pb=reinterpret_cast<char const*>(t.r.data()+(pos+length)/flen); for(size_t c=0;c<4;c++){ _mm_prefetch(pa+64*c,_MM_HINT_T2); _mm_prefetch(pb+64*c,_MM_HINT_T2); } } auto bc= [&] [[gnu::always_inline]](size_t c){ return vftype(_mm256_broadcast_sd(tw[c] +g)); }; vpoint v1={bc(0),bc(1)},v2={bc(2),bc(3)},v3={bc(4),bc(5)}; auto forward= [&] [[gnu::always_inline]](cvector const&a){ auto A=a.at(pos),B=a.at(pos+4)*v1,C=a.at(pos+8)*v2,D=a.at(pos+12)*v3; return std::array<vpoint,4>{(A+C)+(B+D),(A+C)-(B+D),(A-C)+vi(B-D), (A-C)-vi(B-D)}; }; auto a=transpose(forward(*this)); auto b= [&]{ if constexpr(square){ return a; }else{ return transpose(forward(t)); } }(); const auto flip_re=_mm256_set_pd(0.,-0.,-0.,0.),flip_im=_mm256_set_pd(-0.,0.,-0.,0.); vpoint w={ vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(real(v1)),__m256d(imag(v1)),0b1100),flip_re)), vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(imag(v1)),__m256d(real(v1)),0b1100),flip_im))}; auto mul= [&] [[gnu::always_inline]](vpoint a0,vpoint a1,vpoint b0,vpoint b1){ auto p=a0*b0,q=a1*b1; return std::array<vpoint,2>{cmadd(w,q,p),cmcross(a0+a1,b0+b1,p,q)}; }; auto p=mul(a[0],a[2],b[0],b[2]),q=mul(a[1],a[3],b[1],b[3]); auto m=mul(a[0] +a[1],a[2] +a[3],b[0] +b[1],b[2] +b[3]); auto c=transpose({cmadd(w,q[1],p[0]),(m[0] -p[0])-q[0],p[1] +q[0],(m[1] -p[1])-q[1]}); auto A=c[0],B=c[1],C=c[2],D=c[3]; at(pos)=(A+B)+(C+D); at(pos+8)=mulconj((A+B)-(C+D),v2); at(pos+4)=mulconj((A-B)-vi(C-D),v1); at(pos+12)=mulconj((A-B)+vi(C-D),v3); } } } void forward(){ if(!fused_leaves()){ return fft(); } size_t n=size(),part=fused_part(); if(part<n){ vpoint vrt={vz+real(roots[4]),vz+imag(roots[4])}; for(size_t k=0;k<part;k+=flen){ auto t=at(k+part)*vrt; at(k+part)=at(k)-t; at(k)+=t; } } for(size_t offset=0;offset<n;offset+=part){ transform<false,0,0,true>(n,false,offset,part); } checkpoint("fft"); } void multiply(cvector const&t){ if(!fused_leaves()){ dot(t); return ifft<true,false>(); } size_t n=size(),part=fused_part(); dot_fused16<0>(t,0,n); checkpoint("dot"); for(size_t offset=0;offset<n;offset+=part){ transform<true,0,0,true>(n,false,offset,part); } if(part<n){ vpoint cvrt={vz+real(roots[4]),vz-imag(roots[4])}; for(size_t k=0;k<part;k+=flen){ auto t=at(k)-at(k+part); at(k)+=at(k+part); at(k+part)=t*cvrt; } } checkpoint("ifft"); } struct fuse_args{ const uint32_t*src=nullptr; size_t count=0; uint64_t seed=0; double a=0,b=0,a_over_p=0,b_over_p=0; }; static constexpr size_t sweep_tile=256; template<bool inverse,int Mode,size_t Tile,int Fuse=0,bool Neg=false> void sweep8(fuse_args const&fa={}){ constexpr size_t n=1<<24; static const std::array<std::array<point,7>,9>weights= [](){ std::array<std::array<point,7>,9>table; for(size_t count:{size_t(1),size_t(8)}){ for(size_t k=0;k<count;k++){ size_t rev=0,x=k; for(size_t c=count;c>1;c>>=1){ rev=(rev<<1)|(x&1); x>>=1; } long double angle=std::numbers::pi_v<long double> *(1+4*rev)/(16*count); if(k&1){ angle-=std::numbers::pi_v<long double> /4; } for(size_t j=1;j<8;j++){ long double a=j*angle; if constexpr(Mode==0){ table[(count-1)/7+k][j-1] ={double(cosl(a)),double(sinl(a))}; }else{ table[(count-1)/7+k][j-1] ={double(sinl(a)),double(tanl(a/2))}; } } } } return table; }(); auto lift= [&] [[gnu::always_inline]](size_t idx)->vpoint{ i32x4 bits{}; if(idx+4<=fa.count){ std::memcpy(&bits,fa.src+idx,sizeof(bits)); }else if(idx<fa.count){ for(size_t j=0;j<fa.count-idx;j++){ bits[j] =int32_t(fa.src[idx+j]); } }else{ return vpoint{vz,vz}; } auto x=__builtin_convertvector(bits,vftype); u32x4 h= u32x4{uint32_t(idx),uint32_t(idx+1),uint32_t(idx+2),uint32_t(idx+3)}^uint32_t(fa.seed); h*=0x9E3779B1u; h^=h>>15; h*=0x85EBCA77u; h^=h>>13; h*=0xC2B2AE3Du; h^=h>>16; auto noise=__builtin_convertvector(i32x4(h),vftype)*0x1p-32; auto q=round(x*fa.a_over_p+noise),t=round(x*fa.b_over_p+noise); auto re=x-q*fa.a-t*fa.b,im=t*fa.a-q*fa.b; return vpoint{re,Neg?-im:im}; }; auto stage= [&]<bool top> [[gnu::always_inline]](size_t offset,size_t length,size_t begin,size_t end){ size_t step=length/8,k=offset/length; auto const&w=weights[(n/length-1)/7+k]; auto rot= [&]<size_t J> [[gnu::always_inline]](vpoint z){ auto c=w[J-1]; if constexpr(Mode==0){ return z*vpoint{vz+real(c),inverse?vz-imag(c):vz+imag(c)}; }else{ auto s=inverse?vz-real(c):vz+real(c),t=inverse?vz-imag(c):vz+imag(c); auto x=vftype(_mm256_fnmadd_pd(__m256d(t),__m256d(imag(z)),__m256d(real(z)))); auto y=vftype(_mm256_fmadd_pd(__m256d(s),__m256d(x),__m256d(imag(z)))); return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t),__m256d(y),__m256d(x))),y}; } }; auto add= [] [[gnu::always_inline]](vpoint a,vpoint b){ if constexpr(Mode!=2){ return a+b; }else{ return vpoint{vftype(_mm256_fmadd_pd(__m256d(real(a)),_mm256_set1_pd(1),__m256d(real(b)))), vftype(_mm256_fmadd_pd(__m256d(imag(a)),_mm256_set1_pd(1),__m256d(imag(b))))}; } }; auto sub= [] [[gnu::always_inline]](vpoint a,vpoint b){ if constexpr(Mode!=2){ return a-b; }else{ return vpoint{vftype(_mm256_fmsub_pd(__m256d(real(a)),_mm256_set1_pd(1),__m256d(real(b)))), vftype(_mm256_fmsub_pd(__m256d(imag(a)),_mm256_set1_pd(1),__m256d(imag(b))))}; } }; auto d4= [] [[gnu::always_inline]](vpoint a,vpoint b,vpoint c,vpoint d){ auto s=a+c,t=a-c,u=b+d,v=vi(b-d); if constexpr(inverse){ return std::array<vpoint,4>{s+u,t-v,s-u,t+v}; }else{ return std::array<vpoint,4>{s+u,t+v,s-u,t-v}; } }; constexpr double q=0.707106781186547524400844362104849039; auto r1= [] [[gnu::always_inline]](vpoint z){ if constexpr(inverse){ return vpoint{(real(z)+imag(z))*q,(imag(z)-real(z))*q}; }else{ return vpoint{(real(z)-imag(z))*q,(real(z)+imag(z))*q}; } }; auto r3= [] [[gnu::always_inline]](vpoint z){ if constexpr(inverse){ return vpoint{(imag(z)-real(z))*q,(-imag(z)-real(z))*q}; }else{ return vpoint{(-real(z)-imag(z))*q,(real(z)-imag(z))*q}; } }; std::array<vpoint*,8>input,output; for(size_t j=0;j<8;j++){ input[j] =output[j] =r.data()+(offset+begin+j*step)/flen; } if(k&1){ constexpr std::array<size_t,8>perm={7,6,4,5,0,1,2,3}; for(size_t j=0;j<8;j++){ if constexpr(inverse){ input[j] =output[perm[j]]; }else{ output[j] =input[perm[j]]; } } } constexpr bool fused_in=top&&Fuse==1&& !inverse; for(size_t j=0;j<(end-begin)/flen;j++){ size_t base=offset+begin+j*flen; auto in= [&]<size_t S> [[gnu::always_inline]](){ if constexpr(fused_in){ return lift(base+S*step); }else{ return input[S][j]; } }; if constexpr(!inverse){ auto E= d4(in.template operator()<0>(),rot.template operator()<2>(in.template operator()<2>()), rot.template operator()<4>(in.template operator()<4>()), rot.template operator()<6>(in.template operator()<6>())); auto O=d4(rot.template operator()<1>(in.template operator()<1>()), rot.template operator()<3>(in.template operator()<3>()), rot.template operator()<5>(in.template operator()<5>()), rot.template operator()<7>(in.template operator()<7>())); auto a=O[0],b=r1(O[1]),c=vi(O[2]),d=r3(O[3]); output[0][j] =add(E[0],a); output[1][j] =sub(E[0],a); output[2][j] =add(E[2],c); output[3][j] =sub(E[2],c); output[4][j] =add(E[1],b); output[5][j] =sub(E[1],b); output[6][j] =add(E[3],d); output[7][j] =sub(E[3],d); }else{ auto E=d4(add(input[0][j],input[1][j]),add(input[4][j],input[5][j]), add(input[2][j],input[3][j]),add(input[6][j],input[7][j])); auto O=d4(sub(input[0][j],input[1][j]),r1(sub(input[4][j],input[5][j])), -vi(sub(input[2][j],input[3][j])),r3(sub(input[6][j],input[7][j]))); output[0][j] =E[0]; output[1][j] =rot.template operator()<1>(O[0]); output[2][j] =rot.template operator()<2>(E[1]); output[3][j] =rot.template operator()<3>(O[1]); output[4][j] =rot.template operator()<4>(E[2]); output[5][j] =rot.template operator()<5>(O[2]); output[6][j] =rot.template operator()<6>(E[3]); output[7][j] =rot.template operator()<7>(O[3]); } } }; constexpr size_t h=n/64; for(size_t j=0;j<h;j+=Tile){ size_t end=std::min(h,j+Tile); auto first= [&](){ for(size_t t=0;t<8;t++){ stage.template operator()<true>(0,n,j+t*h,end+t*h); } }; auto second= [&](){ for(size_t k=0;k<8;k++){ stage.template operator()<false>(k*n/8,n/8,j,end); } }; if constexpr(inverse){ second(); first(); }else{ first(); second(); } } } static constexpr bool fuse_forward=false; template<bool Neg,bool Fused=fuse_forward,bool Square=false> void cache_product(cvector&b,fuse_args const&fa={},fuse_args const&fb={}){ constexpr size_t n=1<<24,block=1<<18; prepare_roots(n/16); prepare_shear_roots(); if constexpr(Fused){ sweep8<false,2,sweep_tile,1,Neg>(fa); if constexpr(!Square){ b.sweep8<false,2,sweep_tile,1,Neg>(fb); } }else{ sweep8<false,2,sweep_tile>(); if constexpr(!Square){ b.sweep8<false,2,sweep_tile>(); } } checkpoint("sweep forward"); for(size_t offset=0;offset<n;offset+=block){ transform<false,n,block,true>(n,false,offset,block); if constexpr(!Square){ b.transform<false,n,block,true>(n,false,offset,block); } dot_fused16<n,Square>(b,offset,block); transform<true,n,block,true>(n,false,offset,block); } checkpoint("blocks"); sweep8<true,2,sweep_tile>(); checkpoint("sweep inverse"); } static constexpr size_t pre_evals=1<<16; static const std::array<size_t,pre_evals>eval_args; static const std::array<point,pre_evals>evalp; private: size_t fused_part()const{ return size()>>(std::countr_zero(size())%2); } bool fused_leaves()const{ return fused_part()>=16*dot_tile; } template<bool inverse,size_t fixed=0,size_t range_fixed=0,bool omit16=false> void transform(size_t input_n,bool parity,size_t range_offset=0,size_t range_length=0){ if constexpr(range_fixed){ range_length=range_fixed; } const size_t n=fixed?fixed:input_n; if constexpr(!range_fixed){ prepare_roots(n/16); if constexpr(fixed==(1<<24)){ prepare_shear_roots(); } } size_t log_n=std::countr_zero(n); auto butterfly= [&] [[gnu::always_inline]](size_t offset,size_t length,size_t begin,size_t end){ if constexpr(omit16){ if(length==16){ return; } } size_t step=length/4,log_length=std::countr_zero(length),k=offset>>log_length; auto*p0=r.data()+(offset+begin)/flen,*p1=r.data()+(offset+begin+step)/flen, *p2=r.data()+(offset+begin+2*step)/flen, *p3=r.data()+(offset+begin+3*step)/flen; auto run= [&]<bool shear> [[gnu::always_inline]](){ vpoint v1,v2,v3; vftype t1{},t2{},t3{}; if constexpr(shear&&fixed==(1<<24)){ auto const&c=shear_roots[((n>>log_length)-1)/3+k]; v1={vz,inverse?vz-real(c[0]):vz+real(c[0])}; v2={vz,inverse?vz-real(c[1]):vz+real(c[1])}; v3={vz,inverse?vz-real(c[2]):vz+real(c[2])}; t1=inverse?vz-imag(c[0]):vz+imag(c[0]); t2=inverse?vz-imag(c[1]):vz+imag(c[1]); t3=inverse?vz-imag(c[2]):vz+imag(c[2]); }else{ point e=k<pre_evals?evalp[k] :extra[k-pre_evals]; point rt=roots[log_n+5-log_length] *e; v1={vz+real(rt),inverse?vz-imag(rt):vz+imag(rt)}; if constexpr(shear){ if(k&1){ if constexpr(inverse){ v1=vi(v1); }else{ v1= -vi(v1); } } } v2=v1*v1; v3=v1*v2; if constexpr(shear){ t1=imag(v1)/(vz+1.0+real(v1)); t2=imag(v2)/(vz+1.0+real(v2)); t3=imag(v3)/(vz+1.0+real(v3)); } } auto rotate= [] [[gnu::always_inline]](vpoint z,vpoint v,vftype t){ if constexpr(!shear){ return z*v; }else{ auto x=vftype(_mm256_fnmadd_pd(__m256d(t),__m256d(imag(z)),__m256d(real(z)))); auto y=vftype(_mm256_fmadd_pd(__m256d(imag(v)),__m256d(x),__m256d(imag(z)))); return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t),__m256d(y),__m256d(x))),y}; } }; auto*i0=p0,*i1=p1,*i2=p2,*i3=p3,*o0=p0,*o1=p1,*o2=p2,*o3=p3; if constexpr(shear){ if(k&1){ if constexpr(inverse){ i0=p3; i1=p2; i2=p0; i3=p1; }else{ o0=p3; o1=p2; o2=p0; o3=p1; } } } for(size_t j=0;j<(end-begin)/flen;j++){ auto A=i0[j],B=i1[j],C=i2[j],D=i3[j]; if constexpr(inverse){ o0[j] =(A+B)+(C+D); o2[j] =rotate((A+B)-(C+D),v2,t2); o1[j] =rotate((A-B)-vi(C-D),v1,t1); o3[j] =rotate((A-B)+vi(C-D),v3,t3); }else{ B=rotate(B,v1,t1); C=rotate(C,v2,t2); D=rotate(D,v3,t3); o0[j] =(A+C)+(B+D); o1[j] =(A+C)-(B+D); o2[j] =(A-C)+vi(B-D); o3[j] =(A-C)-vi(B-D); } } }; if(length>=shear_min<fixed>){ run.template operator()<true>(); }else{ run.template operator()<false>(); } }; auto recurse= [&](this auto const&self,size_t offset,size_t length)->void{ if(length<4*flen){ return; } if(length>=(1<<15)){ size_t step=length/16; if constexpr(inverse){ for(size_t t=0;t<16;t++){ self(offset+t*step,step); } } for(size_t j=0;j<step;j+=256){ size_t end=std::min(step,j+256); if constexpr(inverse){ for(size_t t=0;t<4;t++){ butterfly(offset+t*length/4,length/4,j,end); } for(size_t t=0;t<4;t++){ butterfly(offset,length,j+t*step,end+t*step); } }else{ for(size_t t=0;t<4;t++){ butterfly(offset,length,j+t*step,end+t*step); } for(size_t t=0;t<4;t++){ butterfly(offset+t*length/4,length/4,j,end); } } } if constexpr(!inverse){ for(size_t t=0;t<16;t++){ self(offset+t*step,step); } } }else if(length>=(size_t(1)<<(6+parity))){ auto finish= [&]<bool par>(){ constexpr size_t chunk=size_t(1)<<(6+par); constexpr size_t bottom=size_t(1)<<(4+par); auto small= [&]<size_t L> [[gnu::always_inline]](this auto const&self,size_t pos)->void{ if constexpr(inverse&&L>bottom){ for(size_t q=0;q<4;q++){ self.template operator()<L/4>(pos+q*(L/4)); } } butterfly(pos,L,0,L/4); if constexpr(!inverse&&L>bottom){ for(size_t q=0;q<4;q++){ self.template operator()<L/4>(pos+q*(L/4)); } } }; for(size_t leaf=offset;leaf<offset+length;leaf+=chunk){ if constexpr(inverse){ small.template operator()<chunk>(leaf); size_t level= std::min<size_t>(std::countr_one(leaf+chunk-1),std::countr_zero(length)); for(size_t lvl=6+2+par;lvl<=level;lvl+=2){ size_t len=size_t(1)<<lvl; butterfly(leaf& ~(len-1),len,0,len/4); } }else{ size_t level=std::min<size_t>(std::countr_zero(n+leaf),std::countr_zero(length)); level-=level%2!=par; for(size_t lvl=level;lvl>=6+2+par;lvl-=2){ size_t len=size_t(1)<<lvl; butterfly(leaf& ~(len-1),len,0,len/4); } small.template operator()<chunk>(leaf); } } }; if(parity){ finish.template operator()<true>(); }else{ finish.template operator()<false>(); } }else{ if constexpr(inverse){ for(size_t leaf=offset+3*flen;leaf<offset+length;leaf+=4*flen){ size_t level=std::min<size_t>(std::countr_one(leaf+3),std::countr_zero(length)); for(size_t lvl=4+parity;lvl<=level;lvl+=2){ size_t len=size_t(1)<<lvl; butterfly(leaf& ~(len-1),len,0,len/4); } } }else{ for(size_t leaf=offset;leaf<offset+length;leaf+=4*flen){ size_t level=std::min<size_t>(std::countr_zero(n+leaf),std::countr_zero(length)); level-=level%2!=parity; for(size_t lvl=level;lvl>=4;lvl-=2){ size_t len=size_t(1)<<lvl; butterfly(leaf& ~(len-1),len,0,len/4); } } } } }; recurse(range_offset,range_length?range_length:n); } template<size_t fixed> static constexpr size_t shear_min=fixed==(1<<24)?64:256; static big_vector<std::array<point,3>>shear_roots; static void prepare_shear_roots(){ constexpr size_t n=1<<24; if(!shear_roots.empty()){ return; } shear_roots.resize((n/16-1)/3,std::array<point,3>{}); for(size_t len=n;len>=shear_min<n>;len/=4){ size_t count=n/len,base=(count-1)/3; point factor=roots[29-std::countr_zero(len)]; for(size_t k=0;k<count;k++){ point rt=factor*(k<pre_evals?evalp[k] :extra[k-pre_evals]); vpoint v1={vz+real(rt),vz+imag(rt)}; if(k&1){ v1= -vi(v1); } vpoint v2=v1*v1,v3=v1*v2; vftype t1=imag(v1)/(vz+1.0+real(v1)),t2=imag(v2)/(vz+1.0+real(v2)), t3=imag(v3)/(vz+1.0+real(v3)); shear_roots[base+k] ={point(imag(v1)[0],t1[0]),point(imag(v2)[0],t2[0]), point(imag(v3)[0],t3[0])}; } } } static big_vector<point>extra; static void prepare_roots(size_t n){ if(n<=pre_evals+extra.size()){ return; } size_t old=extra.size(); extra.resize(std::bit_ceil(n)-pre_evals); static const std::array<point,256>coarse= [](){ std::array<point,256>out; for(size_t i=0;i<256;i++){ out[i] =polar<ftype>(1.,std::numbers::pi*double((eval_args[256+i] -1)/2)/512.0); } return out; }(); for(size_t h=pre_evals+old;h<pre_evals+extra.size();h*=2){ for(size_t i=h;i<2*h;i+=256){ point fine=polar<ftype>(1.,std::numbers::pi/double(4*h)*double(eval_arg(4*i))); for(size_t j=0;j<256;j++){ extra[i+j-pre_evals] =coarse[j] *fine; } } } } }; big_vector<std::array<point,3>>cvector::shear_roots; big_vector<point>cvector::extra; const std::array<size_t,cvector::pre_evals>cvector::eval_args= [](){ std::array<size_t,pre_evals>res={}; for(size_t i=1;i<pre_evals;i++){ res[i] =res[i>>1] |(i&1)<<(std::bit_width(i)-1); } return res; }(); const std::array<point,cvector::pre_evals>cvector::evalp= [](){ std::array<point,pre_evals>res={}; res[0] =1; for(size_t n=1;n<pre_evals;n++){ res[n] =polar<ftype>(1.,std::numbers::pi*ftype(eval_args[n])/ftype(4*std::bit_floor(n))); } return res; }(); }
#pragma GCC pop_options
#line 6 "cp-algo/math/fft_simple.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft{ struct dft_simple{ cp_algo::math::fft::cvector cv; dft_simple(auto const&a,size_t n):cv(n){ for(size_t i=0;i<std::min(std::size(a),n);i++){ real(cv.at(i))[i%4] =ftype(a[i]); imag(cv.at(i))[i%4] =ftype(i+n<std::size(a)?a[i+n] :0); } checkpoint("dft64 init"); cv.fft(); } void dot(dft_simple const&t){ cv.dot(t.cv); } void recover_mod(auto&res,size_t k){ cv.ifft(); size_t n=cv.size(); for(size_t i=0;i<std::min(k,n);i++){ res[i] =llround(real(cv.get(i))); } for(size_t i=n;i<k;i++){ res[i] =llround(imag(cv.get(i-n))); } cp_algo::checkpoint("recover mod"); } }; void conv_simple(auto&a,auto const&b){ if(empty(a)||empty(b)){ a.clear(); return; } size_t n=a.size(),m=b.size(); size_t N=std::max(flen,std::bit_ceil(n+m-1)/2); dft_simple A(a,N),B(b,N); A.dot(B); a.resize(n+m-1); A.recover_mod(a,n+m-1); } }
#pragma GCC pop_options
#line 5 "cp-algo/math/bigint.hpp"
#include <bits/stdc++.h>
namespace cp_algo::math{ enum base_v{ x10=uint64_t(1e16), x16=uint64_t(1ull<<60) }; template<base_v base=x10> struct bigint{ static constexpr uint64_t Base=uint64_t(base); static constexpr uint16_t digit_length=base==x10?16:15; static constexpr uint16_t sub_base=base==x10?10:16; static constexpr uint32_t meta_base=base==x10?uint32_t(1e4):uint32_t(1<<15); big_basic_string<uint64_t>digits; bool negative; auto operator<=>(bigint const&other)const{ auto sign= [](bigint const&x){return x.digits.empty()?0:x.negative?-1:1;}; if(auto cmp=sign(*this)<=>sign(other);cmp!=0){ return cmp; } auto cmp=digits.size()<=>other.digits.size(); if(cmp==0){ cmp=std::lexicographical_compare_three_way( digits.rbegin(),digits.rend(),other.digits.rbegin(),other.digits.rend()); } return!digits.empty()&&negative?0<=>cmp:cmp; } bigint(){} bigint(big_basic_string<uint64_t>d,bool neg):digits(std::move(d)),negative(neg){ normalize(); } bigint&pad_inplace(size_t to_add){ digits.insert(0,to_add,0); return normalize(); } bigint&drop_inplace(size_t to_drop){ digits.erase(0,std::min(to_drop,size(digits))); return normalize(); } bigint&take_inplace(size_t to_keep){ digits.erase(std::min(to_keep,size(digits)),std::string::npos); return normalize(); } bigint&top_inplace(size_t to_keep){ if(to_keep>=size(digits)){ return pad_inplace(to_keep-size(digits)); }else{ return drop_inplace(size(digits)-to_keep); } } bigint pad(size_t to_add)const{ return bigint{big_basic_string<uint64_t>(to_add,0)+digits,negative}; } bigint drop(size_t to_drop)const{ return bigint{digits.substr(std::min(to_drop,size(digits))),negative}; } bigint take(size_t to_keep)const{ return bigint{digits.substr(0,std::min(to_keep,size(digits))),negative}; } bigint top(size_t to_keep)const{ if(to_keep>=size(digits)){ return pad(to_keep-size(digits)); }else{ return drop(size(digits)-to_keep); } } bigint&normalize(){ while(!empty(digits)&&digits.back()==0){ digits.pop_back(); } if(digits.empty()){ negative=false; } return*this; } bigint&negate(){ negative^=1; return*this; } bigint operator-(){ return bigint(*this).negate(); } bigint&operator-=(const bigint&other){ if(negative!=other.negative){ return(negate()+=other).negate().normalize(); } digits.resize(std::max(size(digits),size(other.digits))); bool carry=false; auto d_ptr=std::assume_aligned<32>(digits.data()); auto o_ptr=std::assume_aligned<32>(other.digits.data()); size_t N=size(other.digits); size_t i=0; for(;i<N;i++){ d_ptr[i] -=o_ptr[i] +carry; carry=d_ptr[i] >=base; d_ptr[i] +=carry?uint64_t(base):0; } if(carry){ N=size(digits); for(;i<N&&d_ptr[i] ==0;i++){ d_ptr[i] =base-1; } if(i<N){ d_ptr[i]--; }else{ for(i=0;i<N;i++){ d_ptr[i] =base-d_ptr[i] -1; } bool carry=true; for(i=0;i<N&&carry;i++){ d_ptr[i]++; carry=d_ptr[i] >=base; d_ptr[i] -=carry*base; } negate(); } } return normalize(); } bigint&operator+=(const bigint&other){ if(negative!=other.negative){ return(negate()-=other).negate().normalize(); } digits.resize(std::max(size(digits),size(other.digits))); bool carry=false; auto d_ptr=std::assume_aligned<32>(digits.data()); auto o_ptr=std::assume_aligned<32>(other.digits.data()); size_t N=size(other.digits); size_t i=0; for(;i<N;i++){ d_ptr[i] +=o_ptr[i] +carry; carry=d_ptr[i] >=base; d_ptr[i] -=carry?uint64_t(base):0; } if(carry){ N=size(digits); for(;i<N&&d_ptr[i] ==uint64_t(base)-1;i++){ d_ptr[i] =0; } if(i<N){ d_ptr[i]++; }else{ digits.push_back(1); } } return*this; } bigint(int64_t x){ negative=x<0; x=negative?-x:x; digits=x?big_basic_string<uint64_t>{uint64_t(x)}:big_basic_string<uint64_t>{}; } bigint(std::span<char>s):negative(false){ if(size(s)<digit_length){ int64_t val=0; std::from_chars(s.data(),s.data()+size(s),val,sub_base); *this=bigint(val); return; } if(!empty(s)&&s[0] =='-'){ negative=true; s=s.subspan(1); } size_t len=size(s); assert(len>0); size_t num_digits=(len+digit_length-1)/digit_length; digits.resize(num_digits); size_t i=len; for(size_t j=0;j<num_digits-1;j++){ std::from_chars(s.data()+i-digit_length,s.data()+i,digits[j],sub_base); i-=digit_length; } std::from_chars(s.data(),s.data()+i,digits.back(),sub_base); normalize(); } bigint operator+(const bigint&other)const{ return bigint(*this)+=other; } bigint operator-(const bigint&other)const{ return bigint(*this)-=other; } void to_metabase(){ auto N=ssize(digits); digits.resize(4*N); for(auto i=N-1;i>=0;i--){ uint64_t val=digits[i]; digits[4*i] =val%meta_base; val/=meta_base; digits[4*i+1] =val%meta_base; val/=meta_base; digits[4*i+2] =val%meta_base; val/=meta_base; digits[4*i+3] =val; } } void from_metabase(){ auto N=(ssize(digits)+3)/4; digits.resize(4*N); uint64_t carry=0; for(int i=0;i<N;i++){ __uint128_t val=digits[4*i+3]; val=val*meta_base+digits[4*i+2]; val=val*meta_base+digits[4*i+1]; val=val*meta_base+digits[4*i]; val+=carry; digits[i] =uint64_t(val%base); carry=uint64_t(val/base); } digits.resize(N); while(carry){ digits.push_back(carry%base); carry/=base; } } bigint&operator*=(int64_t other){ if(other<0){ negative^=1; other= -other; } if(other==0){ return*this=bigint(0); }else if(other==1){ return*this; } uint64_t carry=0; for(auto&d:digits){ __uint128_t val=__uint128_t(d)*other+carry; d=uint64_t(val%base); carry=uint64_t(val/base); } if(carry){ digits.push_back(carry%base); carry/=base; } return*this; } bigint operator*(int64_t other)const{ return bigint(*this)*=other; } friend bigint operator*(int64_t lhs,const bigint&rhs){ return bigint(rhs)*=lhs; } bigint&mul_inplace(auto&&other){ negative^=other.negative; auto n=size(digits),m=size(other.digits); if(n<m){ std::swap(n,m); std::swap(digits,other.digits); } if(m<=1){ return*this*=int64_t(m==0?0:other.digits[0]); } constexpr size_t SMALL_M_THRESHOLD=32; if(m<=SMALL_M_THRESHOLD){ big_basic_string<uint64_t>res; res.assign(n+m,0); for(size_t i=0;i<n;i++){ if(digits[i] ==0)continue; uint64_t carry=0; for(size_t j=0;j<m;j++){ __uint128_t cur=res[i+j] +(__uint128_t)digits[i] *other.digits[j] +carry; res[i+j] =uint64_t(cur%Base); carry=uint64_t(cur/Base); } size_t k=i+m; if(carry){ uint64_t cur=res[k] +carry; res[k] =cur%Base; carry=cur/Base; k++; } } digits=std::move(res); return normalize(); } to_metabase(); other.to_metabase(); fft::conv_simple(digits,other.digits); from_metabase(); return normalize(); } bigint&operator*=(bigint const&other){ return mul_inplace(bigint(other)); } bigint operator*(const bigint&other)const{ return bigint(*this).mul_inplace(bigint(other)); } }; template<base_v base> decltype(std::cin)&operator>>(decltype(std::cin)&in,cp_algo::math::bigint<base> &x){ std::string s; in>>s; x={s}; return in; } template<base_v base,bool fill=true> auto&print_digit(auto&out,uint64_t d){ char buf[16]; auto[ptr,ec] =std::to_chars(buf,buf+sizeof(buf),d,bigint<base>::sub_base); if constexpr(base==x16){ std::ranges::transform(buf,buf,toupper); } auto len=ptr-buf; if constexpr(fill){ out<<std::string(bigint<base>::digit_length-len,'0'); } return out<<std::string_view(buf,len); } template<bool fill_all=false,base_v base> auto&print_bigint(auto&out,cp_algo::math::bigint<base>const&x){ if(x.negative){ out<<'-'; } if(empty(x.digits)){ return print_digit<base,fill_all>(out,0); } print_digit<base,fill_all>(out,x.digits.back()); for(auto d:x.digits|std::views::reverse|std::views::drop(1)){ print_digit<base,true>(out,d); } return out; } template<base_v base> decltype(std::cout)&operator<<(decltype(std::cout)&out,cp_algo::math::bigint<base>const&x){ return print_bigint(out,x); } }
#line 3 "tests/bigint.cpp"
using namespace cp_algo; using namespace cp_algo::math; template<base_v radix> void check(){ using B=bigint<radix>; std::mt19937_64 rng(71823); auto value= [](B const&a){ __int128 result=0; for(auto digit:a.digits|std::views::reverse){result=result*B::Base+digit;} return a.negative?-result:result; }; std::vector<B>cases{B(0),B(1),B(-1),B({0,1},false),B({0,1},true)}; auto zero=B(0); zero.negate(); cases.push_back(zero); for(int i=0;i<100;i++){ cases.emplace_back(big_basic_string<uint64_t>{rng()%B::Base,rng()%B::Base},rng()%2); } for(auto const&a:cases){ for(auto const&b:cases){ assert((a<=>b)==(value(a)<=>value(b))); } for(size_t n=0;n<=3;n++){ __int128 power=1; for(size_t i=0;i<std::min(n,size_t(2));i++){power*=B::Base;} assert(value(a.take(n))==(n>=2?value(a):value(a)%power)); assert(value(a.drop(n))==(n>=2?0:value(a)/power)); } } for(int64_t x:{-1000000,-1,0,1,1000000}){ B a(x); assert(value(a.pad(1))==__int128(x)*B::Base); assert(a.pad(1).drop(1).digits==a.digits); } } int main(){ check<x10>(); check<x16>(); std::cout<<"Bigint ordering and slicing passed independent integer checks\n"; }