#line 1 "verify/verify-unit-test/complex-fft.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
//
#line 2 "template/template.hpp"
using namespace std;
// intrinstic
#include <immintrin.h>
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
// utility
#line 1 "template/util.hpp"
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
template <typename T, typename U>
struct P : pair<T, U> {
template <typename... Args>
P(Args... args) : pair<T, U>(args...) {}
using pair<T, U>::first;
using pair<T, U>::second;
P &operator+=(const P &r) {
first += r.first;
second += r.second;
return *this;
}
P &operator-=(const P &r) {
first -= r.first;
second -= r.second;
return *this;
}
P &operator*=(const P &r) {
first *= r.first;
second *= r.second;
return *this;
}
template <typename S>
P &operator*=(const S &r) {
first *= r, second *= r;
return *this;
}
P operator+(const P &r) const { return P(*this) += r; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator*(const P &r) const { return P(*this) *= r; }
template <typename S>
P operator*(const S &r) const {
return P(*this) *= r;
}
P operator-() const { return P{-first, -second}; }
};
using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;
template <typename T>
int sz(const T &t) {
return t.size();
}
template <typename T, typename U>
inline bool amin(T &x, U y) {
return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
inline T Max(const vector<T> &v) {
return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
return accumulate(begin(v), end(v), 0LL);
}
template <typename T>
int lb(const vector<T> &v, const T &a) {
return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
return upper_bound(begin(v), end(v), a) - begin(v);
}
constexpr long long TEN(int n) {
long long ret = 1, x = 10;
for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
return ret;
}
template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
return make_pair(t, u);
}
template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
vector<T> ret(v.size() + 1);
if (rev) {
for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
} else {
for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
}
return ret;
};
template <typename T>
vector<T> mkuni(const vector<T> &v) {
vector<T> ret(v);
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());
return ret;
}
template <typename F>
vector<int> mkord(int N, F f) {
vector<int> ord(N);
iota(begin(ord), end(ord), 0);
sort(begin(ord), end(ord), f);
return ord;
}
template <typename T>
vector<int> mkinv(vector<T> &v) {
int max_val = *max_element(begin(v), end(v));
vector<int> inv(max_val + 1, -1);
for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
return inv;
}
vector<int> mkiota(int n) {
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
return ret;
}
template <typename T>
T mkrev(const T &v) {
T w{v};
reverse(begin(w), end(w));
return w;
}
template <typename T>
bool nxp(T &v) {
return next_permutation(begin(v), end(v));
}
// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
vector<vector<T>> ret;
vector<T> v;
auto dfs = [&](auto rc, int i) -> void {
if (i == (int)a.size()) {
ret.push_back(v);
return;
}
for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
};
dfs(dfs, 0);
return ret;
}
// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
T res = I;
for (; n; f(a = a * a), n >>= 1) {
if (n & 1) f(res = res * a);
}
return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I = T{1}) {
return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}
template <typename T>
T Rev(const T &v) {
T res = v;
reverse(begin(res), end(res));
return res;
}
template <typename T>
vector<T> Transpose(const vector<T> &v) {
using U = typename T::value_type;
if(v.empty()) return {};
int H = v.size(), W = v[0].size();
vector res(W, T(H, U{}));
for (int i = 0; i < H; i++) {
for (int j = 0; j < W; j++) {
res[j][i] = v[i][j];
}
}
return res;
}
template <typename T>
vector<T> Rotate(const vector<T> &v, int clockwise = true) {
using U = typename T::value_type;
int H = v.size(), W = v[0].size();
vector res(W, T(H, U{}));
for (int i = 0; i < H; i++) {
for (int j = 0; j < W; j++) {
if (clockwise) {
res[W - 1 - j][i] = v[i][j];
} else {
res[j][H - 1 - i] = v[i][j];
}
}
}
return res;
}
} // namespace Nyaan
#line 58 "template/template.hpp"
// bit operation
#line 1 "template/bitop.hpp"
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
return __builtin_popcountll(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} // namespace Nyaan
#line 61 "template/template.hpp"
// inout
#line 1 "template/inout.hpp"
namespace Nyaan {
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << p.first << " " << p.second;
return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
int s = (int)v.size();
for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
for (auto &x : v) is >> x;
return is;
}
istream &operator>>(istream &is, __int128_t &x) {
string S;
is >> S;
x = 0;
int flag = 0;
for (auto &c : S) {
if (c == '-') {
flag = true;
continue;
}
x *= 10;
x += c - '0';
}
if (flag) x = -x;
return is;
}
istream &operator>>(istream &is, __uint128_t &x) {
string S;
is >> S;
x = 0;
for (auto &c : S) {
x *= 10;
x += c - '0';
}
return is;
}
ostream &operator<<(ostream &os, __int128_t x) {
if (x == 0) return os << 0;
if (x < 0) os << '-', x = -x;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
if (x == 0) return os << 0;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
struct IoSetupNya {
IoSetupNya() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetupnya;
} // namespace Nyaan
#line 64 "template/template.hpp"
// debug
#line 1 "template/debug.hpp"
namespace DebugImpl {
template <typename U, typename = void>
struct is_specialize : false_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, typename U::iterator, void>::type>
: true_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, decltype(U::first), void>::type>
: true_type {};
template <typename U>
struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type {
};
void dump(const char& t) { cerr << t; }
void dump(const string& t) { cerr << t; }
void dump(const bool& t) { cerr << (t ? "true" : "false"); }
void dump(__int128_t t) {
if (t == 0) cerr << 0;
if (t < 0) cerr << '-', t = -t;
string S;
while (t) S.push_back('0' + t % 10), t /= 10;
reverse(begin(S), end(S));
cerr << S;
}
void dump(__uint128_t t) {
if (t == 0) cerr << 0;
string S;
while (t) S.push_back('0' + t % 10), t /= 10;
reverse(begin(S), end(S));
cerr << S;
}
template <typename U,
enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr>
void dump(const U& t) {
cerr << t;
}
template <typename T>
void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) {
string res;
if (t == Nyaan::inf) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::inf) res = "-inf";
}
if constexpr (sizeof(T) == 8) {
if (t == Nyaan::infLL) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::infLL) res = "-inf";
}
}
if (res.empty()) res = to_string(t);
cerr << res;
}
template <typename T, typename U>
void dump(const pair<T, U>&);
template <typename T>
void dump(const pair<T*, int>&);
template <typename T>
void dump(const T& t,
enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) {
cerr << "[ ";
for (auto it = t.begin(); it != t.end();) {
dump(*it);
cerr << (++it == t.end() ? "" : ", ");
}
cerr << " ]";
}
template <typename T, typename U>
void dump(const pair<T, U>& t) {
cerr << "( ";
dump(t.first);
cerr << ", ";
dump(t.second);
cerr << " )";
}
template <typename T>
void dump(const pair<T*, int>& t) {
cerr << "[ ";
for (int i = 0; i < t.second; i++) {
dump(t.first[i]);
cerr << (i == t.second - 1 ? "" : ", ");
}
cerr << " ]";
}
void trace() { cerr << endl; }
template <typename Head, typename... Tail>
void trace(Head&& head, Tail&&... tail) {
cerr << " ";
dump(head);
if (sizeof...(tail) != 0) cerr << ",";
trace(std::forward<Tail>(tail)...);
}
} // namespace DebugImpl
#ifdef NyaanDebug
#define trc(...) \
do { \
cerr << "## " << #__VA_ARGS__ << " = "; \
DebugImpl::trace(__VA_ARGS__); \
} while (0)
#else
#define trc(...) (void(0))
#endif
#ifdef NyaanLocal
#define trc2(...) \
do { \
cerr << "## " << #__VA_ARGS__ << " = "; \
DebugImpl::trace(__VA_ARGS__); \
} while (0)
#else
#define trc2(...) (void(0))
#endif
#line 67 "template/template.hpp"
// macro
#line 1 "template/macro.hpp"
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
int __VA_ARGS__; \
in(__VA_ARGS__)
#define inl(...) \
long long __VA_ARGS__; \
in(__VA_ARGS__)
#define ins(...) \
string __VA_ARGS__; \
in(__VA_ARGS__)
#define in2(s, t) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i]); \
}
#define in3(s, t, u) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i]); \
}
#define in4(s, t, u, v) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i], v[i]); \
}
#define die(...) \
do { \
Nyaan::out(__VA_ARGS__); \
return; \
} while (0)
#line 70 "template/template.hpp"
namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }
#line 4 "verify/verify-unit-test/complex-fft.test.cpp"
//
#line 2 "misc/rng.hpp"
#line 2 "internal/internal-seed.hpp"
#line 4 "internal/internal-seed.hpp"
using namespace std;
namespace internal {
unsigned long long non_deterministic_seed() {
unsigned long long m =
chrono::duration_cast<chrono::nanoseconds>(
chrono::high_resolution_clock::now().time_since_epoch())
.count();
m ^= 9845834732710364265uLL;
m ^= m << 24, m ^= m >> 31, m ^= m << 35;
return m;
}
unsigned long long deterministic_seed() { return 88172645463325252UL; }
// 64 bit の seed 値を生成 (手元では seed 固定)
// 連続で呼び出すと同じ値が何度も返ってくるので注意
// #define RANDOMIZED_SEED するとシードがランダムになる
unsigned long long seed() {
#if defined(NyaanLocal) && !defined(RANDOMIZED_SEED)
return deterministic_seed();
#else
return non_deterministic_seed();
#endif
}
} // namespace internal
#line 4 "misc/rng.hpp"
namespace my_rand {
using i64 = long long;
using u64 = unsigned long long;
// [0, 2^64 - 1)
u64 rng() {
static u64 _x = internal::seed();
return _x ^= _x << 7, _x ^= _x >> 9;
}
// [l, r]
i64 rng(i64 l, i64 r) {
assert(l <= r);
return l + rng() % u64(r - l + 1);
}
// [l, r)
i64 randint(i64 l, i64 r) {
assert(l < r);
return l + rng() % u64(r - l);
}
// choose n numbers from [l, r) without overlapping
vector<i64> randset(i64 l, i64 r, i64 n) {
assert(l <= r && n <= r - l);
unordered_set<i64> s;
for (i64 i = n; i; --i) {
i64 m = randint(l, r + 1 - i);
if (s.find(m) != s.end()) m = r - i;
s.insert(m);
}
vector<i64> ret;
for (auto& x : s) ret.push_back(x);
sort(begin(ret), end(ret));
return ret;
}
// [0.0, 1.0)
double rnd() { return rng() * 5.42101086242752217004e-20; }
// [l, r)
double rnd(double l, double r) {
assert(l < r);
return l + rnd() * (r - l);
}
template <typename T>
void randshf(vector<T>& v) {
int n = v.size();
for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]);
}
} // namespace my_rand
using my_rand::randint;
using my_rand::randset;
using my_rand::randshf;
using my_rand::rnd;
using my_rand::rng;
#line 2 "ntt/complex-fft.hpp"
namespace ArbitraryModConvolution {
template <typename T>
struct Cp {
T x, y;
constexpr Cp() : x(0), y(0) {}
constexpr Cp(T _x, T _y) : x(_x), y(_y) {}
constexpr inline Cp operator+(const Cp& c) const {
return Cp(x + c.x, y + c.y);
}
constexpr inline Cp operator-(const Cp& c) const {
return Cp(x - c.x, y - c.y);
}
constexpr inline Cp operator*(const Cp& c) const {
return Cp(x * c.x - y * c.y, x * c.y + y * c.x);
}
constexpr inline Cp operator-() const { return Cp(-x, -y); }
constexpr inline Cp conj() const { return Cp(x, -y); }
constexpr inline Cp rotl() const { return Cp(-y, x); }
friend ostream& operator<<(ostream& os, const Cp& c) {
os << "(" << c.x << ", " << c.y << ")" << endl;
return os;
}
};
using C = Cp<double>;
const long double PI = acosl(-1);
struct CooleyTukey {
static vector<C> w;
static void setw(int k) {
--k;
if ((int)w.size() >= (1 << k)) return;
w.resize(1 << k);
vector<Cp<long double>> base(k);
const long double arg = PI / (1 << k);
for (int i = 0, j = 1 << (k - 1); j; i++, j >>= 1) {
complex<long double> z = exp(complex<long double>(1i) * (arg * j));
base[i] = Cp<long double>{z.real(), z.imag()};
}
genw(0, k - 1, Cp<long double>{1, 0}, base);
}
static void genw(int i, int b, Cp<long double> z,
const vector<Cp<long double>>& base) {
if (b == -1) {
w[i].x = z.x, w[i].y = z.y;
} else {
genw(i, b - 1, z, base);
genw(i | (1 << b), b - 1, z * base[b], base);
}
}
static void fft(vector<C>& a, int k) {
if (k <= 0) return;
if (k == 1) {
C a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if (k & 1) {
int v = 1 << (k - 1);
for (int j = 0; j < v; ++j) {
C ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] = a[j] + ajv;
}
}
int u = 1 << (k & 1), v = 1 << (k - 2 - (k & 1));
while (v) {
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = v;
for (; j0 < je; ++j0, ++j1, ++j2, ++j3) {
C t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
C t0p2 = t0 + t2, t1p3 = t1 + t3;
C t0m2 = t0 - t2, t1m3 = (t1 - t3) * w[1];
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
// jh >= 1
for (int jh = 1; jh < u; ++jh) {
int j0 = jh * v * 4;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
C ww = w[jh];
C xx = w[jh << 1];
C wx = ww * xx;
for (; j0 < je; ++j0, ++j1, ++j2, ++j3) {
C t0 = a[j0], t1 = a[j1] * xx, t2 = a[j2] * ww, t3 = a[j3] * wx;
C t0p2 = t0 + t2, t1p3 = t1 + t3;
C t0m2 = t0 - t2, t1m3 = (t1 - t3) * w[1];
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
u <<= 2, v >>= 2;
}
}
static void ifft(vector<C>& a, int k) {
if ((int)a.size() <= 1) return;
if (k == 1) {
C a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
int u = 1 << (k - 2);
int v = 1;
while (u) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
C t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
C t0p1 = t0 + t1, t2p3 = t2 + t3;
C t0m1 = t0 - t1, t2m3 = (t2 - t3) * w[1].conj();
a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
}
}
// jh >= 1
for (int jh = 1; jh < u; ++jh) {
int j0 = (jh * v) << 2;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
C ww = w[jh].conj();
C xx = w[jh << 1].conj();
C yy = w[(jh << 1) + 1].conj();
for (; j0 < je; ++j0, ++j1, ++j2, ++j3) {
C t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
C t0p1 = t0 + t1, t2p3 = t2 + t3;
C t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
a[j1] = t0m1 + t2m3, a[j3] = (t0m1 - t2m3) * ww;
}
}
u >>= 2;
v <<= 2;
}
if (k & 1) {
u = 1 << (k - 1);
for (int j = 0; j < u; j++) {
C ajv = a[j] - a[j + u];
a[j] = a[j] + a[j + u];
a[j + u] = ajv;
}
}
}
static void fft_real(vector<C>& AL, vector<C>& AH, int k) {
fft(AL, k);
AH[0] = C{AL[0].y * 2.0, 0};
AL[0] = C{AL[0].x * 2.0, 0};
AH[1] = C{AL[1].y * 2.0, 0};
AL[1] = C{AL[1].x * 2.0, 0};
for (int i = 2, y = 2; y < (1 << k); y <<= 1) {
for (; i < 2 * y; i += 2) {
int j = i ^ (y - 1);
AH[i] = (AL[j].conj() - AL[i]).rotl();
AL[i] = (AL[j].conj() + AL[i]);
AH[j] = AH[i].conj();
AL[j] = AL[i].conj();
}
}
}
// naive convolution for int
template <typename T, enable_if_t<is_integral<T>::value, nullptr_t> = nullptr>
static vector<long long> multiply(const vector<T>& s, const vector<T>& t) {
int l = s.size() + t.size() - 1;
if (min(s.size(), t.size()) <= 40) {
vector<long long> u(l);
for (int i = 0; i < (int)s.size(); i++) {
for (int j = 0; j < (int)t.size(); j++) u[i + j] += 1LL * s[i] * t[j];
}
return u;
}
int k = 2, M = 4;
while (M < l) M <<= 1, ++k;
setw(k);
auto round = [](double x) -> long long {
return (long long)(x + (x > 0 ? 0.5 : -0.5));
};
vector<C> a(M);
for (int i = 0; i < (int)s.size(); i++) a[i].x = s[i];
for (int i = 0; i < (int)t.size(); i++) a[i].y = t[i];
fft(a, k);
a[0].y = 4.0 * a[0].x * a[0].y;
a[1].y = 4.0 * a[1].x * a[1].y;
a[0].x = a[1].x = 0.0;
for (int i = 2; i < M; i += 2) {
int c = 1 << (31 - __builtin_clz(i));
int j = i ^ (c - 1);
a[i] = (a[i] + a[j].conj()) * (a[i] - a[j].conj());
a[j] = -a[i].conj();
}
vector<C> b(M / 2);
for (int j = 0; j < M / 2; j++) {
C tmp1 = a[j * 2 + 0] + a[j * 2 + 1];
C tmp2 = (a[j * 2 + 0] - a[j * 2 + 1]) * w[j].conj();
b[j] = tmp1 + tmp2.rotl();
}
ifft(b, k - 1);
vector<long long> u(l);
for (int i = 0; i < l; i++) {
if (i & 1) {
u[i] = round(-b[i >> 1].x / (4.0 * M));
} else {
u[i] = round(b[i >> 1].y / (4.0 * M));
}
}
return u;
}
static vector<double> multiply(const vector<double>& s,
const vector<double>& t) {
int l = s.size() + t.size() - 1;
if (min(s.size(), t.size()) <= 40) {
vector<double> u(l);
for (int i = 0; i < (int)s.size(); i++) {
for (int j = 0; j < (int)t.size(); j++) u[i + j] += s[i] * t[j];
}
return u;
}
int k = 2, M = 4;
while (M < l) M <<= 1, ++k;
setw(k);
vector<C> a(M);
for (int i = 0; i < (int)s.size(); i++) a[i].x = s[i];
for (int i = 0; i < (int)t.size(); i++) a[i].y = t[i];
fft(a, k);
a[0].y = 4.0 * a[0].x * a[0].y;
a[1].y = 4.0 * a[1].x * a[1].y;
a[0].x = a[1].x = 0.0;
for (int i = 2; i < M; i += 2) {
int c = 1 << (31 - __builtin_clz(i));
int j = i ^ (c - 1);
a[i] = (a[i] + a[j].conj()) * (a[i] - a[j].conj());
a[j] = -a[i].conj();
}
vector<C> b(M / 2);
for (int j = 0; j < M / 2; j++) {
C tmp1 = a[j * 2 + 0] + a[j * 2 + 1];
C tmp2 = (a[j * 2 + 0] - a[j * 2 + 1]) * w[j].conj();
b[j] = tmp1 + tmp2.rotl();
}
ifft(b, k - 1);
vector<double> u(l);
for (int i = 0; i < l; i++) {
if (i & 1) {
u[i] = -b[i >> 1].x / (4.0 * M);
} else {
u[i] = b[i >> 1].y / (4.0 * M);
}
}
return u;
}
template <unsigned int MOD = -1u>
static conditional_t<MOD == -1u, vector<__uint128_t>, vector<int>>
multiply_15bit(const vector<int>& a, const vector<int>& b) {
using u64 = unsigned long long;
constexpr u64 B = 32000;
int l = a.size() + b.size() - 1;
int k = 2, M = 4;
while (M < l) M <<= 1, ++k;
setw(k);
auto round = [](double x) -> u64 { return u64(x + 0.5); };
vector<C> AL(M), AH(M), BL(M), BH(M);
for (int i = 0; i < (int)a.size(); i++) {
AL[i] = C{double(a[i] % B), double(a[i] / B)};
}
for (int i = 0; i < (int)b.size(); i++) {
BL[i] = C{double(b[i] % B), double(b[i] / B)};
}
fft_real(AL, AH, k);
fft_real(BL, BH, k);
for (int i = 0; i < M; i++) {
C tmp = AL[i] * BL[i] + (AH[i] * BH[i]).rotl();
BH[i] = AL[i] * BH[i] + (AH[i] * BL[i]).rotl();
BL[i] = tmp;
}
ifft(BL, k);
ifft(BH, k);
using return_type =
conditional_t<MOD + 1u == 0u, vector<__uint128_t>, vector<int>>;
return_type u(l);
double im = 1.0 / (4.0 * M);
for (int i = 0; i < l; i++) {
BL[i].x *= im, BL[i].y *= im;
BH[i].x *= im, BH[i].y *= im;
u64 s1 = round(BL[i].x);
u64 s2 = round(BH[i].x) + round(BH[i].y);
u64 s3 = round(BL[i].y);
if constexpr (MOD == -1u) {
u[i] += __uint128_t(s1);
u[i] += __uint128_t(s2) * B;
u[i] += __uint128_t(s3) * B * B;
} else {
u[i] += s1 % MOD;
u[i] += s2 % MOD * B % MOD;
if (u[i] >= MOD) u[i] -= MOD;
u[i] += s3 % MOD * (B * B % MOD) % MOD;
if (u[i] >= MOD) u[i] -= MOD;
}
}
return u;
}
};
vector<C> CooleyTukey::w;
} // namespace ArbitraryModConvolution
#line 7 "verify/verify-unit-test/complex-fft.test.cpp"
using namespace Nyaan;
vl naive(vi a, vi b) {
int n = sz(a), m = sz(b);
vl c(n + m - 1);
rep(i, n) rep(j, m) { c[i + j] += 1LL * a[i] * b[j]; }
return c;
}
vi naive_mod(vi a, vi b, int mod) {
int n = sz(a), m = sz(b);
vi c(n + m - 1);
rep(i, n) rep(j, m) {
ll x = c[i + j] + 1LL * a[i] * b[j];
c[i + j] = x % mod;
}
return c;
}
void is_same_negative(vi a, vi b) {
auto c1 = ArbitraryModConvolution::CooleyTukey::multiply(a, b);
auto c2 = naive(a, b);
if (c1 != c2) {
out(sz(a), sz(b));
rep(i, sz(c1)) {
if (c1[i] != c2[i]) out(i, c1[i], c2[i]);
}
exit(1);
}
}
void is_same_mod(vi a, vi b) {
auto c1 =
ArbitraryModConvolution::CooleyTukey::multiply_15bit<TEN(9) + 7>(a, b);
auto c2 = naive_mod(a, b, TEN(9) + 7);
if (c1 != c2) {
out(a);
out(b);
out(c1);
out(c2);
exit(1);
}
}
void test() {
// negative
rep(loop, 100) {
vi a(randint(1, TEN(3)));
vi b(randint(1, TEN(3)));
each(x, a) x = randint(-TEN(5), TEN(5));
each(x, b) x = randint(-TEN(5), TEN(5));
is_same_negative(a, b);
}
// mod
rep(loop, 100) {
vi a(randint(1, 100));
vi b(randint(1, 100));
each(x, a) x = randint(0, TEN(9) + 7);
each(x, b) x = randint(0, TEN(9) + 7);
is_same_mod(a, b);
}
}
void Nyaan::solve() {
test();
int a, b;
cin >> a >> b;
cout << (a + b) << endl;
}
verify/verify-unit-test/complex-fft.test.cpp