CP-Algorithms Library
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cp-algo/math/factorials.hpp
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- Last update: 2026-01-05 01:05:39+01:00
- Include:
#include "cp-algo/math/factorials.hpp"
Depends on
- cp-algo/math/combinatorics.hpp
- cp-algo/math/common.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/bump_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/simd.hpp
Verified with
- Many Factorials (verify/combi/many_facts.test.cpp)
- Many Factorials (dynamic Mod) (verify/combi/many_facts_dynamic.test.cpp)
Code
#ifndef CP_ALGO_MATH_FACTORIALS_HPP
#define CP_ALGO_MATH_FACTORIALS_HPP
#include "../util/checkpoint.hpp"
#include "../util/bump_alloc.hpp"
#include "../util/simd.hpp"
#include "../math/combinatorics.hpp"
#include "../number_theory/modint.hpp"
#include <ranges>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
template<bool use_bump_alloc = false, int maxn = -1>
auto facts(auto const& args) {
static_assert(!use_bump_alloc || maxn > 0, "maxn must be set if use_bump_alloc is true");
constexpr int max_mod = 1'000'000'000;
constexpr int accum = 4;
constexpr int simd_size = 8;
constexpr int block = 1 << 18;
constexpr int subblock = block / simd_size;
using base = std::decay_t<decltype(args[0])>;
static_assert(modint_type<base>, "Base type must be a modint type");
using T = std::array<int, 2>;
using alloc = std::conditional_t<use_bump_alloc,
bump_alloc<T, 30 * maxn>,
big_alloc<T>>;
std::basic_string<T, std::char_traits<T>, alloc> odd_args_per_block[max_mod / subblock];
std::basic_string<T, std::char_traits<T>, alloc> reg_args_per_block[max_mod / subblock];
constexpr int limit_reg = max_mod / 64;
int limit_odd = 0;
big_vector<base> res(size(args), 1);
const int mod = base::mod();
const int imod = -math::inv2(mod);
for(auto [i, xy]: std::views::zip(args, res) | std::views::enumerate) {
auto [x, y] = xy;
int t = x.getr();
if(t >= mod / 2) {
t = mod - t - 1;
y = t % 2 ? 1 : mod-1;
}
auto pw = 32ull * (t + 1);
while(t > limit_reg) {
limit_odd = std::max(limit_odd, (t - 1) / 2);
odd_args_per_block[(t - 1) / 2 / subblock].push_back({int(i), (t - 1) / 2});
t /= 2;
pw += t;
}
reg_args_per_block[t / subblock].push_back({int(i), t});
y *= pow_fixed<base, 2>(int(pw % (mod - 1)));
}
checkpoint("init");
base bi2x32 = pow_fixed<base, 2>(32).inv();
auto process = [&](int limit, auto &args_per_block, auto step, auto &&proj) {
base fact = 1;
for(int b = 0; b <= limit; b += accum * block) {
u32x8 cur[accum];
static std::array<u32x8, subblock> prods[accum];
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j] = uint32_t(b + z * block + j * subblock);
cur[z][j] = proj(cur[z][j]);
prods[z][0][j] = cur[z][j] + !cur[z][j];
prods[z][0][j] = uint32_t(uint64_t(prods[z][0][j]) * bi2x32.getr() % mod);
#pragma GCC diagnostic pop
}
}
for(int i = 1; i < block / simd_size; i++) {
for(int z = 0; z < accum; z++) {
cur[z] += step;
prods[z][i] = montgomery_mul(prods[z][i - 1], cur[z], mod, imod);
}
}
checkpoint("inner loop");
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
int bl = b + z * block + j * subblock;
for(auto [i, x]: args_per_block[bl / subblock]) {
res[i] *= fact * prods[z][x - bl][j];
}
fact *= base(prods[z].back()[j]);
}
}
checkpoint("mul ans");
}
};
process(limit_reg, reg_args_per_block, 1, std::identity{});
process(limit_odd, odd_args_per_block, 2, [](uint32_t x) {return 2 * x + 1;});
auto invs = bulk_invs<base>(res);
for(auto [i, x]: res | std::views::enumerate) {
if (args[i] >= mod / 2) {
x = invs[i];
}
}
checkpoint("inv ans");
return res;
}
}
#pragma GCC pop_options
#endif // CP_ALGO_MATH_FACTORIALS_HPP
#line 1 "cp-algo/math/factorials.hpp"
#line 1 "cp-algo/util/checkpoint.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 <iostream>
#include <generator>
#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);
madvise(raw, padded, MADV_POPULATE_WRITE);
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>>;
template<typename Ref, typename V = void>
using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}
// Deduction guide to make elements_of with big_generator default to big_alloc
namespace std::ranges {
template<typename Ref, typename V>
elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}
#line 5 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#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 1 "cp-algo/util/bump_alloc.hpp"
#line 5 "cp-algo/util/bump_alloc.hpp"
namespace cp_algo {
template<class T, size_t max_len>
struct bump_alloc {
static char* buf;
static size_t buf_ind;
using value_type = T;
template <class U> struct rebind { using other = bump_alloc<U, max_len>; };
constexpr bool operator==(const bump_alloc&) const = default;
constexpr bool operator!=(const bump_alloc&) const = default;
bump_alloc() = default;
template<class U> bump_alloc(const U&) {}
T* allocate(size_t n) {
buf_ind -= n * sizeof(T);
buf_ind &= 0 - alignof(T);
return (T*)(buf + buf_ind);
}
void deallocate(T*, size_t) {}
};
template<class T, size_t max_len>
char* bump_alloc<T, max_len>::buf = big_alloc<char>().allocate(max_len * sizeof(T));
template<class T, size_t max_len>
size_t bump_alloc<T, max_len>::buf_ind = max_len * sizeof(T);
}
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#include <cstdint>
#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 1 "cp-algo/math/combinatorics.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#line 5 "cp-algo/math/common.hpp"
#include <cassert>
#include <bit>
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
auto bpow(auto const& x, auto n, auto const& one, auto op) {
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;
}
auto bpow(auto x, auto n, auto ans) {
return bpow(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 6 "cp-algo/math/combinatorics.hpp"
#include <ranges>
namespace cp_algo::math {
// fact/rfact/small_inv are caching
// Beware of usage with dynamic mod
template<typename T>
T fact(auto n) {
static big_vector<T> F(maxn);
static bool init = false;
if(!init) {
F[0] = T(1);
for(int i = 1; i < maxn; i++) {
F[i] = F[i - 1] * T(i);
}
init = true;
}
return F[n];
}
// Only works for modint types
template<typename T>
T rfact(auto n) {
static big_vector<T> F(maxn);
static bool init = false;
if(!init) {
int t = (int)std::min<int64_t>(T::mod(), maxn) - 1;
F[t] = T(1) / fact<T>(t);
for(int i = t - 1; i >= 0; i--) {
F[i] = F[i + 1] * T(i + 1);
}
init = true;
}
return F[n];
}
template<typename T, int base>
T pow_fixed(int n) {
static big_vector<T> prec_low(1 << 16);
static big_vector<T> prec_high(1 << 16);
static bool init = false;
if(!init) {
init = true;
prec_low[0] = prec_high[0] = T(1);
T step_low = T(base);
T step_high = bpow(T(base), 1 << 16);
for(int i = 1; i < (1 << 16); i++) {
prec_low[i] = prec_low[i - 1] * step_low;
prec_high[i] = prec_high[i - 1] * step_high;
}
}
return prec_low[n & 0xFFFF] * prec_high[n >> 16];
}
template<typename T>
big_vector<T> bulk_invs(auto const& args) {
big_vector<T> res(std::size(args), args[0]);
for(size_t i = 1; i < std::size(args); i++) {
res[i] = res[i - 1] * args[i];
}
auto all_invs = T(1) / res.back();
for(size_t i = std::size(args) - 1; i > 0; i--) {
res[i] = all_invs * res[i - 1];
all_invs *= args[i];
}
res[0] = all_invs;
return res;
}
template<typename T>
T small_inv(auto n) {
static auto F = bulk_invs<T>(std::views::iota(1, maxn));
return F[n - 1];
}
template<typename T>
T binom_large(T n, auto r) {
assert(r < maxn);
T ans = 1;
for(decltype(r) i = 0; i < r; i++) {
ans = ans * T(n - i) * small_inv<T>(i + 1);
}
return ans;
}
template<typename T>
T binom(auto n, auto r) {
if(r < 0 || r > n) {
return T(0);
} else if(n >= maxn) {
return binom_large(T(n), r);
} else {
return fact<T>(n) * rfact<T>(r) * rfact<T>(n - r);
}
}
}
#line 1 "cp-algo/number_theory/modint.hpp"
#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 9 "cp-algo/math/factorials.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
template<bool use_bump_alloc = false, int maxn = -1>
auto facts(auto const& args) {
static_assert(!use_bump_alloc || maxn > 0, "maxn must be set if use_bump_alloc is true");
constexpr int max_mod = 1'000'000'000;
constexpr int accum = 4;
constexpr int simd_size = 8;
constexpr int block = 1 << 18;
constexpr int subblock = block / simd_size;
using base = std::decay_t<decltype(args[0])>;
static_assert(modint_type<base>, "Base type must be a modint type");
using T = std::array<int, 2>;
using alloc = std::conditional_t<use_bump_alloc,
bump_alloc<T, 30 * maxn>,
big_alloc<T>>;
std::basic_string<T, std::char_traits<T>, alloc> odd_args_per_block[max_mod / subblock];
std::basic_string<T, std::char_traits<T>, alloc> reg_args_per_block[max_mod / subblock];
constexpr int limit_reg = max_mod / 64;
int limit_odd = 0;
big_vector<base> res(size(args), 1);
const int mod = base::mod();
const int imod = -math::inv2(mod);
for(auto [i, xy]: std::views::zip(args, res) | std::views::enumerate) {
auto [x, y] = xy;
int t = x.getr();
if(t >= mod / 2) {
t = mod - t - 1;
y = t % 2 ? 1 : mod-1;
}
auto pw = 32ull * (t + 1);
while(t > limit_reg) {
limit_odd = std::max(limit_odd, (t - 1) / 2);
odd_args_per_block[(t - 1) / 2 / subblock].push_back({int(i), (t - 1) / 2});
t /= 2;
pw += t;
}
reg_args_per_block[t / subblock].push_back({int(i), t});
y *= pow_fixed<base, 2>(int(pw % (mod - 1)));
}
checkpoint("init");
base bi2x32 = pow_fixed<base, 2>(32).inv();
auto process = [&](int limit, auto &args_per_block, auto step, auto &&proj) {
base fact = 1;
for(int b = 0; b <= limit; b += accum * block) {
u32x8 cur[accum];
static std::array<u32x8, subblock> prods[accum];
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j] = uint32_t(b + z * block + j * subblock);
cur[z][j] = proj(cur[z][j]);
prods[z][0][j] = cur[z][j] + !cur[z][j];
prods[z][0][j] = uint32_t(uint64_t(prods[z][0][j]) * bi2x32.getr() % mod);
#pragma GCC diagnostic pop
}
}
for(int i = 1; i < block / simd_size; i++) {
for(int z = 0; z < accum; z++) {
cur[z] += step;
prods[z][i] = montgomery_mul(prods[z][i - 1], cur[z], mod, imod);
}
}
checkpoint("inner loop");
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
int bl = b + z * block + j * subblock;
for(auto [i, x]: args_per_block[bl / subblock]) {
res[i] *= fact * prods[z][x - bl][j];
}
fact *= base(prods[z].back()[j]);
}
}
checkpoint("mul ans");
}
};
process(limit_reg, reg_args_per_block, 1, std::identity{});
process(limit_odd, odd_args_per_block, 2, [](uint32_t x) {return 2 * x + 1;});
auto invs = bulk_invs<base>(res);
for(auto [i, x]: res | std::views::enumerate) {
if (args[i] >= mod / 2) {
x = invs[i];
}
}
checkpoint("inv ans");
return res;
}
}
#pragma GCC pop_options
#ifndef CP_ALGO_MATH_FACTORIALS_HPP
#define CP_ALGO_MATH_FACTORIALS_HPP
#include "../util/checkpoint.hpp"
#include "../util/bump_alloc.hpp"
#include "../util/simd.hpp"
#include "../math/combinatorics.hpp"
#include "../number_theory/modint.hpp"
#include <ranges>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<bool use_bump_alloc=false,int maxn=-1>auto facts(auto const&args){static_assert(!use_bump_alloc||maxn>0,"maxn must be set if use_bump_alloc is true");constexpr int max_mod=1'000'000'000;constexpr int accum=4;constexpr int simd_size=8;constexpr int block=1<<18;constexpr int subblock=block/simd_size;using base=std::decay_t<decltype(args[0])>;static_assert(modint_type<base>,"Base type must be a modint type");using T=std::array<int,2>;using alloc=std::conditional_t<use_bump_alloc,bump_alloc<T,30*maxn>,big_alloc<T>>;std::basic_string<T,std::char_traits<T>,alloc>odd_args_per_block[max_mod/subblock];std::basic_string<T,std::char_traits<T>,alloc>reg_args_per_block[max_mod/subblock];constexpr int limit_reg=max_mod/64;int limit_odd=0;big_vector<base>res(size(args),1);const int mod=base::mod();const int imod=-math::inv2(mod);for(auto[i,xy]:std::views::zip(args,res)|std::views::enumerate){auto[x,y]=xy;int t=x.getr();if(t>=mod/2){t=mod-t-1;y=t%2?1:mod-1;}auto pw=32ull*(t+1);while(t>limit_reg){limit_odd=std::max(limit_odd,(t-1)/2);odd_args_per_block[(t-1)/2/subblock].push_back({int(i),(t-1)/2});t/=2;pw+=t;}reg_args_per_block[t/subblock].push_back({int(i),t});y*=pow_fixed<base,2>(int(pw%(mod-1)));}checkpoint("init");base bi2x32=pow_fixed<base,2>(32).inv();auto process=[&](int limit,auto&args_per_block,auto step,auto&&proj){base fact=1;for(int b=0;b<=limit;b+=accum*block){u32x8 cur[accum];static std::array<u32x8,subblock>prods[accum];for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j]=uint32_t(b+z*block+j*subblock);cur[z][j]=proj(cur[z][j]);prods[z][0][j]=cur[z][j]+!cur[z][j];prods[z][0][j]=uint32_t(uint64_t(prods[z][0][j])*bi2x32.getr()%mod);
#pragma GCC diagnostic pop
}}for(int i=1;i<block/simd_size;i++){for(int z=0;z<accum;z++){cur[z]+=step;prods[z][i]=montgomery_mul(prods[z][i-1],cur[z],mod,imod);}}checkpoint("inner loop");for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){int bl=b+z*block+j*subblock;for(auto[i,x]:args_per_block[bl/subblock]){res[i]*=fact*prods[z][x-bl][j];}fact*=base(prods[z].back()[j]);}}checkpoint("mul ans");}};process(limit_reg,reg_args_per_block,1,std::identity{});process(limit_odd,odd_args_per_block,2,[](uint32_t x){return 2*x+1;});auto invs=bulk_invs<base>(res);for(auto[i,x]:res|std::views::enumerate){if(args[i]>=mod/2){x=invs[i];}}checkpoint("inv ans");return res;}}
#pragma GCC pop_options
#endif
#line 1 "cp-algo/math/factorials.hpp"
#line 1 "cp-algo/util/checkpoint.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 <iostream>
#include <generator>
#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);madvise(raw,padded,MADV_POPULATE_WRITE);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>>;template<typename Ref,typename V=void>using big_generator=std::generator<Ref,V,big_alloc<std::byte>>;}namespace std::ranges{template<typename Ref,typename V>elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>;}
#line 5 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#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 1 "cp-algo/util/bump_alloc.hpp"
#line 5 "cp-algo/util/bump_alloc.hpp"
namespace cp_algo{template<class T,size_t max_len>struct bump_alloc{static char*buf;static size_t buf_ind;using value_type=T;template<class U>struct rebind{using other=bump_alloc<U,max_len>;};constexpr bool operator==(const bump_alloc&)const=default;constexpr bool operator!=(const bump_alloc&)const=default;bump_alloc()=default;template<class U>bump_alloc(const U&){}T*allocate(size_t n){buf_ind-=n*sizeof(T);buf_ind&=0-alignof(T);return(T*)(buf+buf_ind);}void deallocate(T*,size_t){}};template<class T,size_t max_len>char*bump_alloc<T,max_len>::buf=big_alloc<char>().allocate(max_len*sizeof(T));template<class T,size_t max_len>size_t bump_alloc<T,max_len>::buf_ind=max_len*sizeof(T);}
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#include <cstdint>
#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 1 "cp-algo/math/combinatorics.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#line 5 "cp-algo/math/common.hpp"
#include <cassert>
#include <bit>
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;auto bpow(auto const&x,auto n,auto const&one,auto op){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;}auto bpow(auto x,auto n,auto ans){return bpow(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 6 "cp-algo/math/combinatorics.hpp"
#include <ranges>
namespace cp_algo::math{template<typename T>T fact(auto n){static big_vector<T>F(maxn);static bool init=false;if(!init){F[0]=T(1);for(int i=1;i<maxn;i++){F[i]=F[i-1]*T(i);}init=true;}return F[n];}template<typename T>T rfact(auto n){static big_vector<T>F(maxn);static bool init=false;if(!init){int t=(int)std::min<int64_t>(T::mod(),maxn)-1;F[t]=T(1)/fact<T>(t);for(int i=t-1;i>=0;i--){F[i]=F[i+1]*T(i+1);}init=true;}return F[n];}template<typename T,int base>T pow_fixed(int n){static big_vector<T>prec_low(1<<16);static big_vector<T>prec_high(1<<16);static bool init=false;if(!init){init=true;prec_low[0]=prec_high[0]=T(1);T step_low=T(base);T step_high=bpow(T(base),1<<16);for(int i=1;i<(1<<16);i++){prec_low[i]=prec_low[i-1]*step_low;prec_high[i]=prec_high[i-1]*step_high;}}return prec_low[n&0xFFFF]*prec_high[n>>16];}template<typename T>big_vector<T>bulk_invs(auto const&args){big_vector<T>res(std::size(args),args[0]);for(size_t i=1;i<std::size(args);i++){res[i]=res[i-1]*args[i];}auto all_invs=T(1)/res.back();for(size_t i=std::size(args)-1;i>0;i--){res[i]=all_invs*res[i-1];all_invs*=args[i];}res[0]=all_invs;return res;}template<typename T>T small_inv(auto n){static auto F=bulk_invs<T>(std::views::iota(1,maxn));return F[n-1];}template<typename T>T binom_large(T n,auto r){assert(r<maxn);T ans=1;for(decltype(r)i=0;i<r;i++){ans=ans*T(n-i)*small_inv<T>(i+1);}return ans;}template<typename T>T binom(auto n,auto r){if(r<0||r>n){return T(0);}else if(n>=maxn){return binom_large(T(n),r);}else{return fact<T>(n)*rfact<T>(r)*rfact<T>(n-r);}}}
#line 1 "cp-algo/number_theory/modint.hpp"
#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 9 "cp-algo/math/factorials.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<bool use_bump_alloc=false,int maxn=-1>auto facts(auto const&args){static_assert(!use_bump_alloc||maxn>0,"maxn must be set if use_bump_alloc is true");constexpr int max_mod=1'000'000'000;constexpr int accum=4;constexpr int simd_size=8;constexpr int block=1<<18;constexpr int subblock=block/simd_size;using base=std::decay_t<decltype(args[0])>;static_assert(modint_type<base>,"Base type must be a modint type");using T=std::array<int,2>;using alloc=std::conditional_t<use_bump_alloc,bump_alloc<T,30*maxn>,big_alloc<T>>;std::basic_string<T,std::char_traits<T>,alloc>odd_args_per_block[max_mod/subblock];std::basic_string<T,std::char_traits<T>,alloc>reg_args_per_block[max_mod/subblock];constexpr int limit_reg=max_mod/64;int limit_odd=0;big_vector<base>res(size(args),1);const int mod=base::mod();const int imod=-math::inv2(mod);for(auto[i,xy]:std::views::zip(args,res)|std::views::enumerate){auto[x,y]=xy;int t=x.getr();if(t>=mod/2){t=mod-t-1;y=t%2?1:mod-1;}auto pw=32ull*(t+1);while(t>limit_reg){limit_odd=std::max(limit_odd,(t-1)/2);odd_args_per_block[(t-1)/2/subblock].push_back({int(i),(t-1)/2});t/=2;pw+=t;}reg_args_per_block[t/subblock].push_back({int(i),t});y*=pow_fixed<base,2>(int(pw%(mod-1)));}checkpoint("init");base bi2x32=pow_fixed<base,2>(32).inv();auto process=[&](int limit,auto&args_per_block,auto step,auto&&proj){base fact=1;for(int b=0;b<=limit;b+=accum*block){u32x8 cur[accum];static std::array<u32x8,subblock>prods[accum];for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j]=uint32_t(b+z*block+j*subblock);cur[z][j]=proj(cur[z][j]);prods[z][0][j]=cur[z][j]+!cur[z][j];prods[z][0][j]=uint32_t(uint64_t(prods[z][0][j])*bi2x32.getr()%mod);
#pragma GCC diagnostic pop
}}for(int i=1;i<block/simd_size;i++){for(int z=0;z<accum;z++){cur[z]+=step;prods[z][i]=montgomery_mul(prods[z][i-1],cur[z],mod,imod);}}checkpoint("inner loop");for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){int bl=b+z*block+j*subblock;for(auto[i,x]:args_per_block[bl/subblock]){res[i]*=fact*prods[z][x-bl][j];}fact*=base(prods[z].back()[j]);}}checkpoint("mul ans");}};process(limit_reg,reg_args_per_block,1,std::identity{});process(limit_odd,odd_args_per_block,2,[](uint32_t x){return 2*x+1;});auto invs=bulk_invs<base>(res);for(auto[i,x]:res|std::views::enumerate){if(args[i]>=mod/2){x=invs[i];}}checkpoint("inv ans");return res;}}
#pragma GCC pop_options