CP-Algorithms Library
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tests/karatsuba.cpp
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- Last update: 2026-09-12 23:50:56+02:00
Depends on
cp-algo/math/common.hpp- cp-algo/math/karatsuba.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/number_theory/nimber.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/bit.hpp
- cp-algo/util/simd.hpp
Code
#include "cp-algo/math/karatsuba.hpp"
#include <random>
using namespace cp_algo;
using namespace cp_algo::math;
template<class T>
T coefficient(uint64_t x) {
if constexpr(std::is_same_v<T, nimber::f2_64>) {
T res{};
res.r = x;
return res;
} else if constexpr(modint_type<T>) {
return T(x % T::mod());
} else {
return T(x % 7);
}
}
template<class T>
void check(size_t n, size_t m, bool square = false) {
std::mt19937_64 rng(100 * n + m);
std::vector<T> a(n), b(m);
for(auto &x: a) {x = coefficient<T>(rng());}
for(auto &x: b) {x = coefficient<T>(rng());}
if(square) {b = a;}
big_vector<T> expected(n && m ? n + m - 1 : 0);
if(std::min(n, m) < 80) {
for(size_t i = 0; i < n; i++) {
for(size_t j = 0; j < m; j++) {expected[i + j] += a[i] * b[j];}
}
} else {
// Compare Toom interpolation against the original three-product recursion.
size_t N = std::bit_ceil(std::max(n, m));
auto x = a, y = b;
x.resize(N); y.resize(N);
expected.resize(2 * N);
with_bit_ceil(N, [&]<auto NN>() {_karatsuba<NN>(x, y, expected);});
expected.resize(n + m - 1);
}
auto actual = square ? karatsuba(a, a) : karatsuba(a, b);
assert(actual == expected);
}
template<class T>
void boundaries() {
for(size_t n: {0, 1, 2, 3, 7, 8, 9, 15, 16, 17, 33, 65}) {
for(size_t m: {0, 1, 3, 8, 17, 65}) {check<T>(n, m);}
}
for(size_t n: {4095, 4096, 4097, 8192, 8193}) {
check<T>(n, n - 13);
check<T>(n, 17);
}
check<T>(8193, 8193, true);
}
int main() {
boundaries<modint<1000000007>>();
boundaries<nimber::f2_64>();
boundaries<modint<2147483647>>(); // Leaf sums must not overflow uint64_t.
check<modint<1>>(8193, 8193);
check<modint<2>>(8193, 8193);
check<modint<15>>(8193, 8193); // Noninvertible interpolation denominators.
check<modint<49>>(8193, 8193); // Composite, but all denominators invertible.
check<int64_t>(65, 65);
for(int modulus: {1000000007, 49, 37, 15, 1000000007}) {
dynamic_modint<>::with_mod(modulus, [] {check<dynamic_modint<>>(8193, 8179);});
}
check<modint<2305843009213693951LL>>(4097, 4097);
std::cout << "Karatsuba and Toom-4 properties passed\n";
}
#line 1 "cp-algo/math/karatsuba.hpp"
#line 1 "cp-algo/number_theory/nimber.hpp"
#include <array>
#include <bit>
#include <cstdint>
#include <immintrin.h>
// Ensure PCLMULQDQ is available at compile time
#if defined(__PCLMUL__)
static constexpr bool CP_ALGO_HAS_PCLMUL = true;
#else
static constexpr bool CP_ALGO_HAS_PCLMUL = false;
#endif
static_assert(CP_ALGO_HAS_PCLMUL,
"PCLMULQDQ intrinsics not available. Enable it with '-mpclmul' or add '#pragma GCC target(\"pclmul\")' or compile with '-march=native' on supported CPUs.");
namespace cp_algo::math::nimber {
inline constexpr std::array<uint64_t, 64> BASIS_COL = {
0x0000000000000001ull, 0x5211145c804b6109ull, 0x7c8bc2cad259879full, 0x565854b4c60c1e0bull,
0x4068acf7104c20c3ull, 0x662d2bd0f2739155ull, 0x7a90c83701fa8323ull, 0x21cfa750247e8755ull,
0x67d1044e545abf47ull, 0x4d9d3b5a8568f839ull, 0x567a9d7331b6b3c6ull, 0x1ca54bfdd6d1ae59ull,
0x454fa483275db25cull, 0x6766df6fec4e9d44ull, 0x35cb621cec1fe7f9ull, 0x4c606d3e52faf263ull,
0x57640dc825a57954ull, 0x7aca87838b7f6315ull, 0x6d53c884ebf2b0edull, 0x3721d998bb50164bull,
0x7aa7c62fd6cd53abull, 0x47cbb2c51f7c040full, 0x132063b7f5e42489ull, 0x0c1b36c8b2993f8aull,
0x60119ecff680497aull, 0x5175da444cc11791ull, 0x5792ff4554765b09ull, 0x0c9fdb8a01334e82ull,
0x2be0a763a68a4725ull, 0x3c2dc8260ad051f6ull, 0x6c4c9fed8816bb9cull, 0x630062753ffaf766ull,
0x7b37d31b5d519225ull, 0x2364f7f79705691cull, 0x453eb8a83e2fec71ull, 0x7c0121b37e828666ull,
0x59190d3250e66011ull, 0x103207f9dda18caeull, 0x28233dce01c69b76ull, 0x4fa519899227a5e7ull,
0x4567ba46ee7bc6cdull, 0x0a284773d021afd5ull, 0x63894079bbe3a824ull, 0x11013c7fdfaaa5c2ull,
0x1aa984f18574f3b0ull, 0x0cbaba126fd0c4dbull, 0x0b8797719e6dc725ull, 0x4a2845680aefaa72ull,
0x536d2535f6934e15ull, 0x01db7a57effcd689ull, 0x7e1ed0ad01e2a5adull, 0x0aedc9b3cee826f6ull,
0x7ba716eccf9f68e1ull, 0x5d5e23bc0f3dc38full, 0x0b5f2a3b88674d83ull, 0x2de9bafc2f00f8d4ull,
0x3b56712ad419c7e0ull, 0x3ab4be8c30c19253ull, 0x2708522ffaa654b0ull, 0x2b8bca57bf643598ull,
0x588825d1a5fa8e1cull, 0x86adf8bf4d45962full, 0x51b4c15d8719dd73ull, 0xe4a2b3b59783d0aaull
};
inline constexpr std::array<uint64_t, 64> INV_COL = {
0x0000000000000001ull, 0x19c9369f278adc02ull, 0xa181e7d66f5ff795ull, 0x5db84357ce785d09ull,
0xa0bae2f9d2430cc8ull, 0xb7ea5a9705b771c0ull, 0xba4f3cd82801769dull, 0x4886cde01b8241d0ull,
0x0a6f43f2aaf612edull, 0xebd0142f98030a32ull, 0xa81f89cda43f3792ull, 0xe99aec6b66ccb814ull,
0xa69d1ff025fc2f82ull, 0x48a81132d25db068ull, 0x4a900f9dcaa9644full, 0xe5ce4ea88259972aull,
0xf7094c336029f04cull, 0xe191dde287bc9c6bull, 0xaacaff12bff239b8ull, 0x49bc5212be1bc1caull,
0xfe57defb454446cfull, 0xa1dffcf944bdf6a7ull, 0xb9f1bdb5cee941eeull, 0x12e5e889275c22deull,
0x5bcb6b117b77eeedull, 0x03eb1ab59d05ae4bull, 0x02a25d7076ddd386ull, 0x53164a606c612245ull,
0xebb33f5822f66059ull, 0xe9be765f5747b93eull, 0x552a78df373a354full, 0xbcf5ac65f31fb8bfull,
0xe411e728becdc77bull, 0xf35c26d7b57cdca6ull, 0x4499da83de4ca5f7ull, 0x40ab25bdca4ae226ull,
0xee004b6f1dff7218ull, 0x0d122da9821c5b41ull, 0x51fbfcb058120efeull, 0xa148b1fa84905b22ull,
0xbb8ed3e647604d8dull, 0xe2d93fef2472776full, 0x4c17a2541a10e6b5ull, 0x1d879e08903708e7ull,
0x0fbe7d0d1934da90ull, 0x5bf977d9c6f61d30ull, 0x06832fc918260412ull, 0x0fe22e843ebf73e3ull,
0x4d7ef4e4fa28d60dull, 0x402250d979afbed5ull, 0x067902b8c8ca2d4full, 0xf38d113fe1d6bb16ull,
0x414f0248b02b5b7dull, 0xf041922915824ce9ull, 0x11a72fb5e30c93d9ull, 0x12e54f4d63102aeeull,
0xbc46ac14b3141c6cull, 0x1f172b3c16c645bbull, 0x584b492ed4e8fa6cull, 0x00a852e9a32cc133ull,
0xa180861bce00a45eull, 0xa194b6bcb4645fb9ull, 0x4509002ad808a4fbull, 0xc5172a0055602f69ull
};
template <const auto& COLS>
consteval auto make_byte_tables() {
std::array<std::array<uint64_t, 1 << 8>, 8> T{};
for (int pos = 0; pos < 8; pos++) {
for (int col = 0; col < 8; col++) {
for (int mask = 0; mask < (1 << col); mask++) {
T[pos][mask | (1 << col)] = T[pos][mask] ^ COLS[pos * 8 + col];
}
}
}
return T;
}
inline constexpr auto INV_BYTE = make_byte_tables<INV_COL>();
inline constexpr auto BASIS_BYTE = make_byte_tables<BASIS_COL>();
[[gnu::always_inline]]
inline uint64_t nim_to_poly(uint64_t x) {
auto xb = std::bit_cast<std::array<uint8_t, 8>>(x);
return INV_BYTE[0][xb[0]] ^ INV_BYTE[1][xb[1]]
^ INV_BYTE[2][xb[2]] ^ INV_BYTE[3][xb[3]]
^ INV_BYTE[4][xb[4]] ^ INV_BYTE[5][xb[5]]
^ INV_BYTE[6][xb[6]] ^ INV_BYTE[7][xb[7]];
}
[[gnu::always_inline]]
inline uint64_t poly_to_nim(uint64_t c) {
auto cb = std::bit_cast<std::array<uint8_t, 8>>(c);
return BASIS_BYTE[0][cb[0]] ^ BASIS_BYTE[1][cb[1]]
^ BASIS_BYTE[2][cb[2]] ^ BASIS_BYTE[3][cb[3]]
^ BASIS_BYTE[4][cb[4]] ^ BASIS_BYTE[5][cb[5]]
^ BASIS_BYTE[6][cb[6]] ^ BASIS_BYTE[7][cb[7]];
}
// Carryless multiply over GF(2) using PCLMULQDQ
[[gnu::always_inline]]
inline __m128i clmul(int64_t a, int64_t b) {
return _mm_clmulepi64_si128(__m128i{a, 0}, __m128i{b, 0}, 0);
}
// Reduction table for high bits overflow
inline constexpr std::array<uint64_t, 16> RED_OVER = [] {
std::array<uint64_t, 16> red{};
for (int q = 0; q < 16; ++q) {
uint64_t o = q ^ (q >> 1) ^ (q >> 3);
red[q] = o ^ (o << 1) ^ (o << 3) ^ (o << 4);
}
return red;
}();
// Reduce modulo x^64 + x^4 + x^3 + x + 1
[[gnu::always_inline]]
inline uint64_t reduce_mod(__m128i v) {
uint64_t h = v[1];
return v[0] ^ h ^ (h << 1) ^ (h << 3) ^ (h << 4) ^ RED_OVER[h >> 60];
}
[[gnu::always_inline]]
inline uint64_t f2_64_product(uint64_t a, uint64_t b) {
return reduce_mod(clmul(a, b));
}
// Public nimber product via isomorphism (no recursion, no Gauss at runtime)
[[gnu::always_inline]]
inline uint64_t nim_product(uint64_t a, uint64_t b) {
return poly_to_nim(f2_64_product(
nim_to_poly(a),
nim_to_poly(b)
));
}
struct f2_64 {
uint64_t r;
operator uint64_t() const {return r;}
f2_64() = default;
f2_64& operator+=(const f2_64 &other) {
r ^= other.r;
return *this;
}
f2_64& operator-=(const f2_64 &other) {
r ^= other.r;
return *this;
}
f2_64& operator *=(const f2_64 &other) {
r = f2_64_product(r, other.r);
return *this;
}
f2_64 operator*(const f2_64 &other) const {return f2_64(*this) *= other;}
f2_64 operator+(const f2_64 &other) const {return f2_64(*this) += other;}
f2_64 operator-(const f2_64 &other) const {return f2_64(*this) -= other;}
};
}
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#line 5 "cp-algo/math/common.hpp"
#include <cassert>
#line 7 "cp-algo/math/common.hpp"
#include <vector>
#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 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static Int remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
constexpr modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator - () const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator /= (const modint &t) {
return to_modint() *= t.inv();
}
modint& operator *= (const modint &t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator += (const modint &t) {
r += t.r; r = std::min(r, r - remod());
return to_modint();
}
modint& operator -= (const modint &t) {
r -= t.r; r = std::min(r, r + remod());
return to_modint();
}
modint operator + (const modint &t) const {return modint(to_modint()) += t;}
modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
// Why <=> doesn't work?..
auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static uint64_t modmod8() {return uint64_t(8 * modmod());}
void add_unsafe(UInt t) {r += t;}
void pseudonormalize() {r = std::min(r, r - modmod8());}
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) {r = rr;}
UInt getr_direct() const {return r;}
protected:
UInt r;
private:
constexpr modint& to_modint() {return static_cast<modint&>(*this);}
constexpr modint const& to_modint() const {return static_cast<modint const&>(*this);}
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() {return m;}
static constexpr Base::UInt remod() {return m;}
auto getr() const {return Base::r;}
};
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Base::UInt m_reduce(Base::UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return typename Base::UInt(ab % mod());
} else {
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator *= (const dynamic_modint &t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() {return m;}
static Int remod() {return 2 * m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() {switch_mod(prev);}
} _;
switch_mod(tmp);
return callback();
}
private:
static thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#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::popcount(Align) == 1, "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;
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/util/bit.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\")")
#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);
}
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 8 "cp-algo/util/bit.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"avx2,bmi,bmi2,lzcnt,popcnt\")")
#else
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"bmi,bmi2,lzcnt,popcnt\")")
#endif
#define CP_ALGO_BIT_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_BIT_OPS_TARGET
CP_ALGO_BIT_PRAGMA_PUSH
namespace cp_algo {
template<typename Uint>
constexpr size_t bit_width = sizeof(Uint) * 8;
// n < 64
uint64_t mask(size_t n) {
return (1ULL << n) - 1;
}
size_t order_of_bit(auto x, size_t k) {
return k ? std::popcount(x << (bit_width<decltype(x)> - k)) : 0;
}
inline size_t kth_set_bit(uint64_t x, size_t k) {
return std::countr_zero(_pdep_u64(1ULL << k, x));
}
template<int fl = 0>
void with_bit_floor(size_t n, auto &&callback) {
if constexpr (fl >= 63) {
return;
} else if (n >> (fl + 1)) {
with_bit_floor<fl + 1>(n, callback);
} else {
callback.template operator()<1ULL << fl>();
}
}
void with_bit_ceil(size_t n, auto &&callback) {
with_bit_floor(n, [&]<size_t N>() {
if(N == n) {
callback.template operator()<N>();
} else {
callback.template operator()<N << 1>();
}
});
}
inline uint32_t read_bits(char const* p) {
return _mm256_movemask_epi8(__m256i(vector_cast<u8x32 const>(p[0]) + (127 - '0')));
}
inline uint64_t read_bits64(char const* p) {
return read_bits(p) | (uint64_t(read_bits(p + 32)) << 32);
}
inline void write_bits(char *p, uint32_t bits) {
static constexpr u8x32 shuffler = {
0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 3, 3, 3, 3, 3, 3
};
auto shuffled = u8x32(_mm256_shuffle_epi8(__m256i() + bits, __m256i(shuffler)));
static constexpr u8x32 mask = {
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128
};
for(int z = 0; z < 32; z++) {
p[z] = shuffled[z] & mask[z] ? '1' : '0';
}
}
inline void write_bits64(char *p, uint64_t bits) {
write_bits(p, uint32_t(bits));
write_bits(p + 32, uint32_t(bits >> 32));
}
}
#pragma GCC pop_options
#line 10 "cp-algo/math/karatsuba.hpp"
#include <span>
namespace cp_algo::math {
constexpr size_t NN = 8;
template<auto N>
void base_conv(auto &&_a, auto &&_b, auto &&_c) {
auto a = &_a[0];
auto b = &_b[0];
auto c = &_c[0];
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
c[i + j] += a[i] * b[j];
}
}
}
// Optimized base case for F2_64: uses 256-bit VPCLMULQDQ
// Computes 4 products per iteration
template<size_t N>
[[gnu::target("avx2,vpclmulqdq")]]
void base_conv_f2_64(auto &&a, auto &&b, auto &&c) {
if constexpr (N % 2) {
static_assert(N < 2);
base_conv<N>(a, b, c);
return;
}
alignas(32) __m128i pr0[2 * N] = {};
alignas(32) __m128i pr1[2 * N] = {};
for (size_t i = 0; i + 1 < N; i += 2) {
auto va = (__m256i)u64x4{a[i], 0, a[i + 1], 0};
for (size_t j = 0; j + 1 < N; j += 2) {
auto vb = (__m256i)u64x4{b[j], b[j + 1], b[j], b[j + 1]};
(__m256i&)pr0[i + j] ^= _mm256_clmulepi64_epi128(va, vb, 0);
(__m256i&)pr1[i + j] ^= _mm256_clmulepi64_epi128(va, vb, 16);
}
}
c[0].r = nimber::reduce_mod(pr0[0]);
for (size_t i = 1; i < 2 * N - 1; i++) {
c[i].r ^= nimber::reduce_mod(pr0[i] ^ pr1[i - 1]);
}
}
template<auto N>
void base_conv_modint(auto &&a, auto &&b, auto &&c) {
using base = std::decay_t<decltype(a[0])>;
uint64_t largest = base::mod() - 1;
if(largest && largest > UINT64_MAX / N / largest) {
base_conv<N>(a, b, c);
return;
}
if constexpr (N % 4) {
static_assert(N < 4);
base_conv<N>(a, b, c);
return;
}
alignas(32) uint64_t pr0[2 * N] = {}, pr1[2 * N] = {};
alignas(32) uint64_t pr2[2 * N] = {}, pr3[2 * N] = {};
for (size_t i = 0; i < N; i += 4) {
auto va0 = __m256i() + a[i].getr();
auto va1 = __m256i() + a[i + 1].getr();
auto va2 = __m256i() + a[i + 2].getr();
auto va3 = __m256i() + a[i + 3].getr();
size_t j = 0;
for (; j + 3 < N; j += 4) {
auto vb = (__m256i)u64x4{
b[j].getr(), b[j + 1].getr(), b[j + 2].getr(), b[j + 3].getr()
};
(__m256i&)pr0[i + j] += _mm256_mul_epu32(va0, vb);
(__m256i&)pr1[i + j] += _mm256_mul_epu32(va1, vb);
(__m256i&)pr2[i + j] += _mm256_mul_epu32(va2, vb);
(__m256i&)pr3[i + j] += _mm256_mul_epu32(va3, vb);
}
}
for (size_t i = 0; i < 2 * N - 1; i++) {
if (i > 0) {
pr2[i] += pr3[i - 1];
pr1[i] += pr2[i - 1];
pr0[i] += pr1[i - 1];
}
c[i].setr((typename base::UInt)(pr0[i] % base::mod()));
}
}
// Generic Karatsuba multiplication algorithm for polynomials
// N is the input length and must be a power of two.
template<auto N>
void _karatsuba(auto &&a, auto &&b, auto &&c) {
[[gnu::assume(N <= 1<<19)]];
using base = std::decay_t<decltype(a[0])>;
if constexpr (N <= NN) {
if constexpr (std::is_same_v<base, nimber::f2_64>) {
base_conv_f2_64<N>(a, b, c);
} else if constexpr (modint_type<base>) {
base_conv_modint<N>(a, b, c);
} else {
base_conv<N>(a, b, c);
}
} else {
constexpr auto h = N / 2;
auto a0 = &a[0], a1 = a0 + h, b0 = &b[0], b1 = b0 + h;
auto c0 = &c[0], c1 = c0 + h, c2 = c0 + 2 * h;
_karatsuba<h>(a0, b0, c0);
_karatsuba<h>(a1, b1, c2);
static big_vector<base> buf(4 * h);
auto f = &buf[0];
auto sum_a = f + 2 * h, sum_b = f + 3 * h;
for (size_t i = 0; i < h; i++) {
sum_a[i] = a0[i] + a1[i];
sum_b[i] = b0[i] + b1[i];
}
memset(f, 0, sizeof(base) * 2 * h);
_karatsuba<h>(sum_a, sum_b, f);
for(size_t i = 0; i < h; i++) {
auto A = c0[i], &B = c1[i], &C = c2[i], D = c2[i + h];
auto BC = B - C;
B = BC + f[i] - A;
C = f[i + h] - D - BC;
}
}
}
namespace karatsuba_detail {
template<class T>
T constant(uint64_t x) {
if constexpr(std::is_same_v<T, nimber::f2_64>) {
T res{};
res.r = x;
return res;
} else {
return T(x);
}
}
template<class T>
T inverse(T a) {
if constexpr(std::is_same_v<T, nimber::f2_64>) {
return bpow(a, UINT64_MAX - 1, constant<T>(1));
} else {
// Euclid also supports composite moduli coprime to 2, 3, and 5.
auto x = a.getr(), y = typename T::UInt(T::mod());
typename T::Int2 u = 1, v = 0;
while(y) {
auto q = x / y;
x -= q * y;
std::swap(x, y);
u -= q * v;
std::swap(u, v);
}
assert(x == 1);
return T(u);
}
}
template<class T>
auto const& interpolation() {
auto build = [] {
std::array<std::array<T, 14>, 7> a{};
// Values at 0, 1, 2, 3, 4, 5, and infinity.
for(size_t i = 0; i < 6; i++) {
T x = constant<T>(i), p = constant<T>(1);
for(size_t j = 0; j < 7; j++) {a[i][j] = p; p *= x;}
}
a[6][6] = constant<T>(1);
for(size_t i = 0; i < 7; i++) {a[i][i + 7] = constant<T>(1);}
for(size_t i = 0; i < 7; i++) {
size_t pivot = i;
while(pivot < 7 && a[pivot][i] == T{}) {pivot++;}
assert(pivot < 7);
std::swap(a[i], a[pivot]);
auto inv = inverse(a[i][i]);
for(auto &x: a[i]) {x *= inv;}
for(size_t j = 0; j < 7; j++) {
if(j == i) {continue;}
auto q = a[j][i];
for(size_t k = 0; k < 14; k++) {a[j][k] -= q * a[i][k];}
}
}
std::array<std::array<T, 7>, 7> res{};
for(size_t i = 0; i < 7; i++) {
for(size_t j = 0; j < 7; j++) {res[i][j] = a[i][j + 7];}
}
return res;
};
static thread_local auto matrix = build();
if constexpr(modint_type<T>) {
static thread_local auto modulus = T::mod();
if(modulus != T::mod()) {matrix = build(); modulus = T::mod();}
}
return matrix;
}
// Modular evaluation points: 0, 1, -1, 2, -2, 3, infinity.
template<class T>
T evaluate_modular(T a0, T a1, T a2, T a3, size_t k) {
if(k <= 2) {
auto even = a0 + a2, odd = a1 + a3;
return k == 1 ? even + odd : even - odd;
}
if(k == 5) {return ((a3 * T(3) + a2) * T(3) + a1) * T(3) + a0;}
a2 += a2; a2 += a2;
a3 += a3; a3 += a3;
auto odd = a1 + a3;
odd += odd;
return k == 3 ? (a0 + a2) + odd : (a0 + a2) - odd;
}
template<class T>
void interpolate_modular(T const* p, T* c, size_t h) {
auto i2 = inverse(T(2)), i3 = inverse(T(3)), i4 = i2 * i2;
auto i5 = inverse(T(5)), i8 = i4 * i2, i12 = i4 * i3;
for(size_t i = 0; i < 2 * h - 1; i++) {
auto c0 = p[i], c6 = p[12*h + i];
auto p1 = p[2*h + i], m1 = p[4*h + i];
auto p2 = p[6*h + i], m2 = p[8*h + i], p3 = p[10*h + i];
// Separate even and odd powers before solving the small systems.
auto e1 = (p1 + m1) * i2 - c0 - c6;
auto e2 = (p2 + m2) * i2 - c0 - T(64) * c6;
auto c4 = (e2 - T(4) * e1) * i12, c2 = e1 - c4;
auto o1 = (p1 - m1) * i2, o2 = (p2 - m2) * i4;
auto o3 = (p3 - c0 - T(9) * c2 - T(81) * c4 - T(729) * c6) * i3;
auto d2 = (o2 - o1) * i3, d3 = (o3 - o1) * i8;
auto c5 = (d3 - d2) * i5, c3 = d2 - T(5) * c5, c1 = o1 - c3 - c5;
c[i] += c0; c[h + i] += c1; c[2*h + i] += c2;
c[3*h + i] += c3; c[4*h + i] += c4;
c[5*h + i] += c5; c[6*h + i] += c6;
}
}
// Toom-4 uses seven products for four blocks; Karatsuba handles the leaves.
template<size_t N, class T>
void mul(T const* a, T const* b, T* c) {
if constexpr(N > (1 << 19)) {
__builtin_unreachable(); // Same size bound as _karatsuba.
} else if constexpr(N <= 4096) {
std::fill_n(c, 2 * N, T{});
_karatsuba<N>(a, b, c);
} else {
constexpr size_t h = N / 4;
static big_vector<T> products(14 * h), av(h), bv(h);
mul<h>(a, b, products.data());
mul<h>(a + 3 * h, b + 3 * h, products.data() + 12 * h);
for(size_t k = 1; k < 6; k++) {
auto point = constant<T>(k);
for(size_t i = 0; i < h; i++) {
if constexpr(modint_type<T>) {
av[i] = evaluate_modular(a[i], a[i + h], a[i + 2*h], a[i + 3*h], k);
bv[i] = evaluate_modular(b[i], b[i + h], b[i + 2*h], b[i + 3*h], k);
} else {
av[i] = ((a[i + 3*h] * point + a[i + 2*h]) * point + a[i + h]) * point + a[i];
bv[i] = ((b[i + 3*h] * point + b[i + 2*h]) * point + b[i + h]) * point + b[i];
}
}
mul<h>(av.data(), bv.data(), products.data() + 2 * k * h);
}
std::fill_n(c, 2 * N, T{});
if constexpr(modint_type<T>) {
interpolate_modular(products.data(), c, h);
} else {
auto const& matrix = interpolation<T>();
for(size_t j = 0; j < 7; j++) {
for(size_t k = 0; k < 7; k++) {
auto x = matrix[j][k];
if(x == T{}) {continue;}
if(x == constant<T>(1)) {
for(size_t i = 0; i < 2*h - 1; i++) {c[j*h + i] += products[k*2*h + i];}
} else {
for(size_t i = 0; i < 2*h - 1; i++) {c[j*h + i] += x * products[k*2*h + i];}
}
}
}
}
}
}
}
// Large field inputs use Toom-4 above the Karatsuba recursion.
// Runtime wrapper that deduces N at compile time.
// Resizes inputs to the next power of 2 and result to n + m - 1
auto karatsuba(auto &a, auto &b) {
using base = std::decay_t<decltype(a[0])>;
auto n = std::size(a);
auto m = std::size(b);
if(!n || !m) {return big_vector<base>{};}
auto N = std::bit_ceil(std::max(n, m));
a.resize(N);
b.resize(N);
// The recursion reads the zero coefficient at index 2*N-1 when joining halves.
big_vector<base> c(2 * N);
with_bit_ceil(N, [&]<auto NN>() {
if constexpr(std::is_same_v<base, nimber::f2_64>) {
karatsuba_detail::mul<NN>(std::data(a), std::data(b), c.data());
return;
} else if constexpr(modint_type<base>) {
if(base::mod() > 5 && base::mod() % 2 && base::mod() % 3 && base::mod() % 5) {
karatsuba_detail::mul<NN>(std::data(a), std::data(b), c.data());
return;
}
}
_karatsuba<NN>(a, b, c);
});
c.resize(n + m - 1);
return c;
}
}
#line 2 "tests/karatsuba.cpp"
#include <random>
using namespace cp_algo;
using namespace cp_algo::math;
template<class T>
T coefficient(uint64_t x) {
if constexpr(std::is_same_v<T, nimber::f2_64>) {
T res{};
res.r = x;
return res;
} else if constexpr(modint_type<T>) {
return T(x % T::mod());
} else {
return T(x % 7);
}
}
template<class T>
void check(size_t n, size_t m, bool square = false) {
std::mt19937_64 rng(100 * n + m);
std::vector<T> a(n), b(m);
for(auto &x: a) {x = coefficient<T>(rng());}
for(auto &x: b) {x = coefficient<T>(rng());}
if(square) {b = a;}
big_vector<T> expected(n && m ? n + m - 1 : 0);
if(std::min(n, m) < 80) {
for(size_t i = 0; i < n; i++) {
for(size_t j = 0; j < m; j++) {expected[i + j] += a[i] * b[j];}
}
} else {
// Compare Toom interpolation against the original three-product recursion.
size_t N = std::bit_ceil(std::max(n, m));
auto x = a, y = b;
x.resize(N); y.resize(N);
expected.resize(2 * N);
with_bit_ceil(N, [&]<auto NN>() {_karatsuba<NN>(x, y, expected);});
expected.resize(n + m - 1);
}
auto actual = square ? karatsuba(a, a) : karatsuba(a, b);
assert(actual == expected);
}
template<class T>
void boundaries() {
for(size_t n: {0, 1, 2, 3, 7, 8, 9, 15, 16, 17, 33, 65}) {
for(size_t m: {0, 1, 3, 8, 17, 65}) {check<T>(n, m);}
}
for(size_t n: {4095, 4096, 4097, 8192, 8193}) {
check<T>(n, n - 13);
check<T>(n, 17);
}
check<T>(8193, 8193, true);
}
int main() {
boundaries<modint<1000000007>>();
boundaries<nimber::f2_64>();
boundaries<modint<2147483647>>(); // Leaf sums must not overflow uint64_t.
check<modint<1>>(8193, 8193);
check<modint<2>>(8193, 8193);
check<modint<15>>(8193, 8193); // Noninvertible interpolation denominators.
check<modint<49>>(8193, 8193); // Composite, but all denominators invertible.
check<int64_t>(65, 65);
for(int modulus: {1000000007, 49, 37, 15, 1000000007}) {
dynamic_modint<>::with_mod(modulus, [] {check<dynamic_modint<>>(8193, 8179);});
}
check<modint<2305843009213693951LL>>(4097, 4097);
std::cout << "Karatsuba and Toom-4 properties passed\n";
}
#line 1 "cp-algo/math/karatsuba.hpp"
#line 1 "cp-algo/number_theory/nimber.hpp"
#include <array>
#include <bit>
#include <cstdint>
#include <immintrin.h>
#if defined(__PCLMUL__)
static constexpr bool CP_ALGO_HAS_PCLMUL=true;
#else
static constexpr bool CP_ALGO_HAS_PCLMUL=false;
#endif
static_assert(CP_ALGO_HAS_PCLMUL,"PCLMULQDQ intrinsics not available. Enable it with '-mpclmul' or add '#pragma GCC target(\"pclmul\")' or compile with '-march=native' on supported CPUs.");namespace cp_algo::math::nimber{inline constexpr std::array<uint64_t,64>BASIS_COL={0x0000000000000001ull,0x5211145c804b6109ull,0x7c8bc2cad259879full,0x565854b4c60c1e0bull,0x4068acf7104c20c3ull,0x662d2bd0f2739155ull,0x7a90c83701fa8323ull,0x21cfa750247e8755ull,0x67d1044e545abf47ull,0x4d9d3b5a8568f839ull,0x567a9d7331b6b3c6ull,0x1ca54bfdd6d1ae59ull,0x454fa483275db25cull,0x6766df6fec4e9d44ull,0x35cb621cec1fe7f9ull,0x4c606d3e52faf263ull,0x57640dc825a57954ull,0x7aca87838b7f6315ull,0x6d53c884ebf2b0edull,0x3721d998bb50164bull,0x7aa7c62fd6cd53abull,0x47cbb2c51f7c040full,0x132063b7f5e42489ull,0x0c1b36c8b2993f8aull,0x60119ecff680497aull,0x5175da444cc11791ull,0x5792ff4554765b09ull,0x0c9fdb8a01334e82ull,0x2be0a763a68a4725ull,0x3c2dc8260ad051f6ull,0x6c4c9fed8816bb9cull,0x630062753ffaf766ull,0x7b37d31b5d519225ull,0x2364f7f79705691cull,0x453eb8a83e2fec71ull,0x7c0121b37e828666ull,0x59190d3250e66011ull,0x103207f9dda18caeull,0x28233dce01c69b76ull,0x4fa519899227a5e7ull,0x4567ba46ee7bc6cdull,0x0a284773d021afd5ull,0x63894079bbe3a824ull,0x11013c7fdfaaa5c2ull,0x1aa984f18574f3b0ull,0x0cbaba126fd0c4dbull,0x0b8797719e6dc725ull,0x4a2845680aefaa72ull,0x536d2535f6934e15ull,0x01db7a57effcd689ull,0x7e1ed0ad01e2a5adull,0x0aedc9b3cee826f6ull,0x7ba716eccf9f68e1ull,0x5d5e23bc0f3dc38full,0x0b5f2a3b88674d83ull,0x2de9bafc2f00f8d4ull,0x3b56712ad419c7e0ull,0x3ab4be8c30c19253ull,0x2708522ffaa654b0ull,0x2b8bca57bf643598ull,0x588825d1a5fa8e1cull,0x86adf8bf4d45962full,0x51b4c15d8719dd73ull,0xe4a2b3b59783d0aaull};inline constexpr std::array<uint64_t,64>INV_COL={0x0000000000000001ull,0x19c9369f278adc02ull,0xa181e7d66f5ff795ull,0x5db84357ce785d09ull,0xa0bae2f9d2430cc8ull,0xb7ea5a9705b771c0ull,0xba4f3cd82801769dull,0x4886cde01b8241d0ull,0x0a6f43f2aaf612edull,0xebd0142f98030a32ull,0xa81f89cda43f3792ull,0xe99aec6b66ccb814ull,0xa69d1ff025fc2f82ull,0x48a81132d25db068ull,0x4a900f9dcaa9644full,0xe5ce4ea88259972aull,0xf7094c336029f04cull,0xe191dde287bc9c6bull,0xaacaff12bff239b8ull,0x49bc5212be1bc1caull,0xfe57defb454446cfull,0xa1dffcf944bdf6a7ull,0xb9f1bdb5cee941eeull,0x12e5e889275c22deull,0x5bcb6b117b77eeedull,0x03eb1ab59d05ae4bull,0x02a25d7076ddd386ull,0x53164a606c612245ull,0xebb33f5822f66059ull,0xe9be765f5747b93eull,0x552a78df373a354full,0xbcf5ac65f31fb8bfull,0xe411e728becdc77bull,0xf35c26d7b57cdca6ull,0x4499da83de4ca5f7ull,0x40ab25bdca4ae226ull,0xee004b6f1dff7218ull,0x0d122da9821c5b41ull,0x51fbfcb058120efeull,0xa148b1fa84905b22ull,0xbb8ed3e647604d8dull,0xe2d93fef2472776full,0x4c17a2541a10e6b5ull,0x1d879e08903708e7ull,0x0fbe7d0d1934da90ull,0x5bf977d9c6f61d30ull,0x06832fc918260412ull,0x0fe22e843ebf73e3ull,0x4d7ef4e4fa28d60dull,0x402250d979afbed5ull,0x067902b8c8ca2d4full,0xf38d113fe1d6bb16ull,0x414f0248b02b5b7dull,0xf041922915824ce9ull,0x11a72fb5e30c93d9ull,0x12e54f4d63102aeeull,0xbc46ac14b3141c6cull,0x1f172b3c16c645bbull,0x584b492ed4e8fa6cull,0x00a852e9a32cc133ull,0xa180861bce00a45eull,0xa194b6bcb4645fb9ull,0x4509002ad808a4fbull,0xc5172a0055602f69ull};template<const auto&COLS>consteval auto make_byte_tables(){std::array<std::array<uint64_t,1<<8>,8>T{};for(int pos=0;pos<8;pos++){for(int col=0;col<8;col++){for(int mask=0;mask<(1<<col);mask++){T[pos][mask|(1<<col)]=T[pos][mask]^COLS[pos*8+col];}}}return T;}inline constexpr auto INV_BYTE=make_byte_tables<INV_COL>();inline constexpr auto BASIS_BYTE=make_byte_tables<BASIS_COL>();[[gnu::always_inline]]inline uint64_t nim_to_poly(uint64_t x){auto xb=std::bit_cast<std::array<uint8_t,8>>(x);return INV_BYTE[0][xb[0]]^INV_BYTE[1][xb[1]]^INV_BYTE[2][xb[2]]^INV_BYTE[3][xb[3]]^INV_BYTE[4][xb[4]]^INV_BYTE[5][xb[5]]^INV_BYTE[6][xb[6]]^INV_BYTE[7][xb[7]];}[[gnu::always_inline]]inline uint64_t poly_to_nim(uint64_t c){auto cb=std::bit_cast<std::array<uint8_t,8>>(c);return BASIS_BYTE[0][cb[0]]^BASIS_BYTE[1][cb[1]]^BASIS_BYTE[2][cb[2]]^BASIS_BYTE[3][cb[3]]^BASIS_BYTE[4][cb[4]]^BASIS_BYTE[5][cb[5]]^BASIS_BYTE[6][cb[6]]^BASIS_BYTE[7][cb[7]];}[[gnu::always_inline]]inline __m128i clmul(int64_t a,int64_t b){return _mm_clmulepi64_si128(__m128i{a,0},__m128i{b,0},0);}inline constexpr std::array<uint64_t,16>RED_OVER=[]{std::array<uint64_t,16>red{};for(int q=0;q<16;++q){uint64_t o=q^(q>>1)^(q>>3);red[q]=o^(o<<1)^(o<<3)^(o<<4);}return red;}();[[gnu::always_inline]]inline uint64_t reduce_mod(__m128i v){uint64_t h=v[1];return v[0]^h^(h<<1)^(h<<3)^(h<<4)^RED_OVER[h>>60];}[[gnu::always_inline]]inline uint64_t f2_64_product(uint64_t a,uint64_t b){return reduce_mod(clmul(a,b));}[[gnu::always_inline]]inline uint64_t nim_product(uint64_t a,uint64_t b){return poly_to_nim(f2_64_product(nim_to_poly(a),nim_to_poly(b)));}struct f2_64{uint64_t r;operator uint64_t()const{return r;}f2_64()=default;f2_64&operator+=(const f2_64&other){r^=other.r;return*this;}f2_64&operator-=(const f2_64&other){r^=other.r;return*this;}f2_64&operator*=(const f2_64&other){r=f2_64_product(r,other.r);return*this;}f2_64 operator*(const f2_64&other)const{return f2_64(*this)*=other;}f2_64 operator+(const f2_64&other)const{return f2_64(*this)+=other;}f2_64 operator-(const f2_64&other)const{return f2_64(*this)-=other;}};}
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#line 5 "cp-algo/math/common.hpp"
#include <cassert>
#line 7 "cp-algo/math/common.hpp"
#include <vector>
#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 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{template<typename modint,typename _Int>struct modint_base{using Int=_Int;using UInt=std::make_unsigned_t<Int>;static constexpr size_t bits=sizeof(Int)*8;using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>;using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr modint_base(Int2 rr){to_modint().setr(UInt((rr+modmod())%mod()));}constexpr modint inv()const{return bpow(to_modint(),mod()-2);}modint operator-()const{modint neg;neg.r=std::min(-r,remod()-r);return neg;}modint&operator/=(const modint&t){return to_modint()*=t.inv();}modint&operator*=(const modint&t){r=UInt(UInt2(r)*t.r%mod());return to_modint();}modint&operator+=(const modint&t){r+=t.r;r=std::min(r,r-remod());return to_modint();}modint&operator-=(const modint&t){r-=t.r;r=std::min(r,r+remod());return to_modint();}modint operator+(const modint&t)const{return modint(to_modint())+=t;}modint operator-(const modint&t)const{return modint(to_modint())-=t;}modint operator*(const modint&t)const{return modint(to_modint())*=t;}modint operator/(const modint&t)const{return modint(to_modint())/=t;}auto operator==(const modint&t)const{return to_modint().getr()==t.getr();}auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();}auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();}auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();}auto operator<(const modint&t)const{return to_modint().getr()<t.getr();}auto operator>(const modint&t)const{return to_modint().getr()>t.getr();}Int rem()const{UInt R=to_modint().getr();return R-(R>(UInt)mod()/2)*mod();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}static uint64_t modmod8(){return uint64_t(8*modmod());}void add_unsafe(UInt t){r+=t;}void pseudonormalize(){r=std::min(r,r-modmod8());}modint const&normalize(){if(r>=(UInt)mod()){r%=mod();}return to_modint();}void setr_direct(UInt rr){r=rr;}UInt getr_direct()const{return r;}protected:UInt r;private:constexpr modint&to_modint(){return static_cast<modint&>(*this);}constexpr modint const&to_modint()const{return static_cast<modint const&>(*this);}};template<typename modint>concept modint_type=std::is_base_of_v<modint_base<modint,typename modint::Int>,modint>;template<modint_type modint>decltype(std::cin)&operator>>(decltype(std::cin)&in,modint&x){typename modint::UInt r;auto&res=in>>r;x.setr(r);return res;}template<modint_type modint>decltype(std::cout)&operator<<(decltype(std::cout)&out,modint const&x){return out<<x.getr();}template<auto m>struct modint:modint_base<modint<m>,decltype(m)>{using Base=modint_base<modint<m>,decltype(m)>;using Base::Base;static constexpr Base::Int mod(){return m;}static constexpr Base::UInt remod(){return m;}auto getr()const{return Base::r;}};template<typename Int=int>struct dynamic_modint:modint_base<dynamic_modint<Int>,Int>{using Base=modint_base<dynamic_modint<Int>,Int>;using Base::Base;static Base::UInt m_reduce(Base::UInt2 ab){if(mod()%2==0)[[unlikely]]{return typename Base::UInt(ab%mod());}else{typename Base::UInt2 m=typename Base::UInt(ab)*imod();return typename Base::UInt((ab+m*mod())>>Base::bits);}}static Base::UInt m_transform(Base::UInt a){if(mod()%2==0)[[unlikely]]{return a;}else{return m_reduce(a*pw128());}}dynamic_modint&operator*=(const dynamic_modint&t){Base::r=m_reduce(typename Base::UInt2(Base::r)*t.r);return*this;}void setr(Base::UInt rr){Base::r=m_transform(rr);}Base::UInt getr()const{typename Base::UInt res=m_reduce(Base::r);return std::min(res,res-mod());}static Int mod(){return m;}static Int remod(){return 2*m;}static Base::UInt imod(){return im;}static Base::UInt2 pw128(){return r2;}static void switch_mod(Int nm){m=nm;im=m%2?inv2(-m):0;r2=static_cast<Base::UInt>(static_cast<Base::UInt2>(-1)%m+1);}auto static with_mod(Int tmp,auto callback){struct scoped{Int prev=mod();~scoped(){switch_mod(prev);}}_;switch_mod(tmp);return callback();}private:static thread_local Int m;static thread_local Base::UInt im,r2;};template<typename Int>Int thread_local dynamic_modint<Int>::m=1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im=-1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2=0;}
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#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::popcount(Align)==1,"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;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/util/bit.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\")")
#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 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 8 "cp-algo/util/bit.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"avx2,bmi,bmi2,lzcnt,popcnt\")")
#else
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"bmi,bmi2,lzcnt,popcnt\")")
#endif
#define CP_ALGO_BIT_PRAGMA_PUSH _Pragma("GCC push_options") CP_ALGO_BIT_OPS_TARGET
CP_ALGO_BIT_PRAGMA_PUSH
namespace cp_algo{template<typename Uint>constexpr size_t bit_width=sizeof(Uint)*8;uint64_t mask(size_t n){return(1ULL<<n)-1;}size_t order_of_bit(auto x,size_t k){return k?std::popcount(x<<(bit_width<decltype(x)>-k)):0;}inline size_t kth_set_bit(uint64_t x,size_t k){return std::countr_zero(_pdep_u64(1ULL<<k,x));}template<int fl=0>void with_bit_floor(size_t n,auto&&callback){if constexpr(fl>=63){return;}else if(n>>(fl+1)){with_bit_floor<fl+1>(n,callback);}else{callback.template operator()<1ULL<<fl>();}}void with_bit_ceil(size_t n,auto&&callback){with_bit_floor(n,[&]<size_t N>(){if(N==n){callback.template operator()<N>();}else{callback.template operator()<N<<1>();}});}inline uint32_t read_bits(char const*p){return _mm256_movemask_epi8(__m256i(vector_cast<u8x32 const>(p[0])+(127-'0')));}inline uint64_t read_bits64(char const*p){return read_bits(p)|(uint64_t(read_bits(p+32))<<32);}inline void write_bits(char*p,uint32_t bits){static constexpr u8x32 shuffler={0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3};auto shuffled=u8x32(_mm256_shuffle_epi8(__m256i()+bits,__m256i(shuffler)));static constexpr u8x32 mask={1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128};for(int z=0;z<32;z++){p[z]=shuffled[z]&mask[z]?'1':'0';}}inline void write_bits64(char*p,uint64_t bits){write_bits(p,uint32_t(bits));write_bits(p+32,uint32_t(bits>>32));}}
#pragma GCC pop_options
#line 10 "cp-algo/math/karatsuba.hpp"
#include <span>
namespace cp_algo::math{constexpr size_t NN=8;template<auto N>void base_conv(auto&&_a,auto&&_b,auto&&_c){auto a=&_a[0];auto b=&_b[0];auto c=&_c[0];for(size_t i=0;i<N;i++){for(size_t j=0;j<N;j++){c[i+j]+=a[i]*b[j];}}}template<size_t N>[[gnu::target("avx2,vpclmulqdq")]]void base_conv_f2_64(auto&&a,auto&&b,auto&&c){if constexpr(N%2){static_assert(N<2);base_conv<N>(a,b,c);return;}alignas(32)__m128i pr0[2*N]={};alignas(32)__m128i pr1[2*N]={};for(size_t i=0;i+1<N;i+=2){auto va=(__m256i)u64x4{a[i],0,a[i+1],0};for(size_t j=0;j+1<N;j+=2){auto vb=(__m256i)u64x4{b[j],b[j+1],b[j],b[j+1]};(__m256i&)pr0[i+j]^=_mm256_clmulepi64_epi128(va,vb,0);(__m256i&)pr1[i+j]^=_mm256_clmulepi64_epi128(va,vb,16);}}c[0].r=nimber::reduce_mod(pr0[0]);for(size_t i=1;i<2*N-1;i++){c[i].r^=nimber::reduce_mod(pr0[i]^pr1[i-1]);}}template<auto N>void base_conv_modint(auto&&a,auto&&b,auto&&c){using base=std::decay_t<decltype(a[0])>;uint64_t largest=base::mod()-1;if(largest&&largest>UINT64_MAX/N/largest){base_conv<N>(a,b,c);return;}if constexpr(N%4){static_assert(N<4);base_conv<N>(a,b,c);return;}alignas(32)uint64_t pr0[2*N]={},pr1[2*N]={};alignas(32)uint64_t pr2[2*N]={},pr3[2*N]={};for(size_t i=0;i<N;i+=4){auto va0=__m256i()+a[i].getr();auto va1=__m256i()+a[i+1].getr();auto va2=__m256i()+a[i+2].getr();auto va3=__m256i()+a[i+3].getr();size_t j=0;for(;j+3<N;j+=4){auto vb=(__m256i)u64x4{b[j].getr(),b[j+1].getr(),b[j+2].getr(),b[j+3].getr()};(__m256i&)pr0[i+j]+=_mm256_mul_epu32(va0,vb);(__m256i&)pr1[i+j]+=_mm256_mul_epu32(va1,vb);(__m256i&)pr2[i+j]+=_mm256_mul_epu32(va2,vb);(__m256i&)pr3[i+j]+=_mm256_mul_epu32(va3,vb);}}for(size_t i=0;i<2*N-1;i++){if(i>0){pr2[i]+=pr3[i-1];pr1[i]+=pr2[i-1];pr0[i]+=pr1[i-1];}c[i].setr((typename base::UInt)(pr0[i]%base::mod()));}}template<auto N>void _karatsuba(auto&&a,auto&&b,auto&&c){[[gnu::assume(N<=1<<19)]];using base=std::decay_t<decltype(a[0])>;if constexpr(N<=NN){if constexpr(std::is_same_v<base,nimber::f2_64>){base_conv_f2_64<N>(a,b,c);}else if constexpr(modint_type<base>){base_conv_modint<N>(a,b,c);}else{base_conv<N>(a,b,c);}}else{constexpr auto h=N/2;auto a0=&a[0],a1=a0+h,b0=&b[0],b1=b0+h;auto c0=&c[0],c1=c0+h,c2=c0+2*h;_karatsuba<h>(a0,b0,c0);_karatsuba<h>(a1,b1,c2);static big_vector<base>buf(4*h);auto f=&buf[0];auto sum_a=f+2*h,sum_b=f+3*h;for(size_t i=0;i<h;i++){sum_a[i]=a0[i]+a1[i];sum_b[i]=b0[i]+b1[i];}memset(f,0,sizeof(base)*2*h);_karatsuba<h>(sum_a,sum_b,f);for(size_t i=0;i<h;i++){auto A=c0[i],&B=c1[i],&C=c2[i],D=c2[i+h];auto BC=B-C;B=BC+f[i]-A;C=f[i+h]-D-BC;}}}namespace karatsuba_detail{template<class T>T constant(uint64_t x){if constexpr(std::is_same_v<T,nimber::f2_64>){T res{};res.r=x;return res;}else{return T(x);}}template<class T>T inverse(T a){if constexpr(std::is_same_v<T,nimber::f2_64>){return bpow(a,UINT64_MAX-1,constant<T>(1));}else{auto x=a.getr(),y=typename T::UInt(T::mod());typename T::Int2 u=1,v=0;while(y){auto q=x/y;x-=q*y;std::swap(x,y);u-=q*v;std::swap(u,v);}assert(x==1);return T(u);}}template<class T>auto const&interpolation(){auto build=[]{std::array<std::array<T,14>,7>a{};for(size_t i=0;i<6;i++){T x=constant<T>(i),p=constant<T>(1);for(size_t j=0;j<7;j++){a[i][j]=p;p*=x;}}a[6][6]=constant<T>(1);for(size_t i=0;i<7;i++){a[i][i+7]=constant<T>(1);}for(size_t i=0;i<7;i++){size_t pivot=i;while(pivot<7&&a[pivot][i]==T{}){pivot++;}assert(pivot<7);std::swap(a[i],a[pivot]);auto inv=inverse(a[i][i]);for(auto&x:a[i]){x*=inv;}for(size_t j=0;j<7;j++){if(j==i){continue;}auto q=a[j][i];for(size_t k=0;k<14;k++){a[j][k]-=q*a[i][k];}}}std::array<std::array<T,7>,7>res{};for(size_t i=0;i<7;i++){for(size_t j=0;j<7;j++){res[i][j]=a[i][j+7];}}return res;};static thread_local auto matrix=build();if constexpr(modint_type<T>){static thread_local auto modulus=T::mod();if(modulus!=T::mod()){matrix=build();modulus=T::mod();}}return matrix;}template<class T>T evaluate_modular(T a0,T a1,T a2,T a3,size_t k){if(k<=2){auto even=a0+a2,odd=a1+a3;return k==1?even+odd:even-odd;}if(k==5){return((a3*T(3)+a2)*T(3)+a1)*T(3)+a0;}a2+=a2;a2+=a2;a3+=a3;a3+=a3;auto odd=a1+a3;odd+=odd;return k==3?(a0+a2)+odd:(a0+a2)-odd;}template<class T>void interpolate_modular(T const*p,T*c,size_t h){auto i2=inverse(T(2)),i3=inverse(T(3)),i4=i2*i2;auto i5=inverse(T(5)),i8=i4*i2,i12=i4*i3;for(size_t i=0;i<2*h-1;i++){auto c0=p[i],c6=p[12*h+i];auto p1=p[2*h+i],m1=p[4*h+i];auto p2=p[6*h+i],m2=p[8*h+i],p3=p[10*h+i];auto e1=(p1+m1)*i2-c0-c6;auto e2=(p2+m2)*i2-c0-T(64)*c6;auto c4=(e2-T(4)*e1)*i12,c2=e1-c4;auto o1=(p1-m1)*i2,o2=(p2-m2)*i4;auto o3=(p3-c0-T(9)*c2-T(81)*c4-T(729)*c6)*i3;auto d2=(o2-o1)*i3,d3=(o3-o1)*i8;auto c5=(d3-d2)*i5,c3=d2-T(5)*c5,c1=o1-c3-c5;c[i]+=c0;c[h+i]+=c1;c[2*h+i]+=c2;c[3*h+i]+=c3;c[4*h+i]+=c4;c[5*h+i]+=c5;c[6*h+i]+=c6;}}template<size_t N,class T>void mul(T const*a,T const*b,T*c){if constexpr(N>(1<<19)){__builtin_unreachable();}else if constexpr(N<=4096){std::fill_n(c,2*N,T{});_karatsuba<N>(a,b,c);}else{constexpr size_t h=N/4;static big_vector<T>products(14*h),av(h),bv(h);mul<h>(a,b,products.data());mul<h>(a+3*h,b+3*h,products.data()+12*h);for(size_t k=1;k<6;k++){auto point=constant<T>(k);for(size_t i=0;i<h;i++){if constexpr(modint_type<T>){av[i]=evaluate_modular(a[i],a[i+h],a[i+2*h],a[i+3*h],k);bv[i]=evaluate_modular(b[i],b[i+h],b[i+2*h],b[i+3*h],k);}else{av[i]=((a[i+3*h]*point+a[i+2*h])*point+a[i+h])*point+a[i];bv[i]=((b[i+3*h]*point+b[i+2*h])*point+b[i+h])*point+b[i];}}mul<h>(av.data(),bv.data(),products.data()+2*k*h);}std::fill_n(c,2*N,T{});if constexpr(modint_type<T>){interpolate_modular(products.data(),c,h);}else{auto const&matrix=interpolation<T>();for(size_t j=0;j<7;j++){for(size_t k=0;k<7;k++){auto x=matrix[j][k];if(x==T{}){continue;}if(x==constant<T>(1)){for(size_t i=0;i<2*h-1;i++){c[j*h+i]+=products[k*2*h+i];}}else{for(size_t i=0;i<2*h-1;i++){c[j*h+i]+=x*products[k*2*h+i];}}}}}}}}auto karatsuba(auto&a,auto&b){using base=std::decay_t<decltype(a[0])>;auto n=std::size(a);auto m=std::size(b);if(!n||!m){return big_vector<base>{};}auto N=std::bit_ceil(std::max(n,m));a.resize(N);b.resize(N);big_vector<base>c(2*N);with_bit_ceil(N,[&]<auto NN>(){if constexpr(std::is_same_v<base,nimber::f2_64>){karatsuba_detail::mul<NN>(std::data(a),std::data(b),c.data());return;}else if constexpr(modint_type<base>){if(base::mod()>5&&base::mod()%2&&base::mod()%3&&base::mod()%5){karatsuba_detail::mul<NN>(std::data(a),std::data(b),c.data());return;}}_karatsuba<NN>(a,b,c);});c.resize(n+m-1);return c;}}
#line 2 "tests/karatsuba.cpp"
#include <random>
using namespace cp_algo;using namespace cp_algo::math;template<class T>T coefficient(uint64_t x){if constexpr(std::is_same_v<T,nimber::f2_64>){T res{};res.r=x;return res;}else if constexpr(modint_type<T>){return T(x%T::mod());}else{return T(x%7);}}template<class T>void check(size_t n,size_t m,bool square=false){std::mt19937_64 rng(100*n+m);std::vector<T>a(n),b(m);for(auto&x:a){x=coefficient<T>(rng());}for(auto&x:b){x=coefficient<T>(rng());}if(square){b=a;}big_vector<T>expected(n&&m?n+m-1:0);if(std::min(n,m)<80){for(size_t i=0;i<n;i++){for(size_t j=0;j<m;j++){expected[i+j]+=a[i]*b[j];}}}else{size_t N=std::bit_ceil(std::max(n,m));auto x=a,y=b;x.resize(N);y.resize(N);expected.resize(2*N);with_bit_ceil(N,[&]<auto NN>(){_karatsuba<NN>(x,y,expected);});expected.resize(n+m-1);}auto actual=square?karatsuba(a,a):karatsuba(a,b);assert(actual==expected);}template<class T>void boundaries(){for(size_t n:{0,1,2,3,7,8,9,15,16,17,33,65}){for(size_t m:{0,1,3,8,17,65}){check<T>(n,m);}}for(size_t n:{4095,4096,4097,8192,8193}){check<T>(n,n-13);check<T>(n,17);}check<T>(8193,8193,true);}int main(){boundaries<modint<1000000007>>();boundaries<nimber::f2_64>();boundaries<modint<2147483647>>();check<modint<1>>(8193,8193);check<modint<2>>(8193,8193);check<modint<15>>(8193,8193);check<modint<49>>(8193,8193);check<int64_t>(65,65);for(int modulus:{1000000007,49,37,15,1000000007}){dynamic_modint<>::with_mod(modulus,[]{check<dynamic_modint<>>(8193,8179);});}check<modint<2305843009213693951LL>>(4097,4097);std::cout<<"Karatsuba and Toom-4 properties passed\n";}