Tifa's CP Library

:heavy_check_mark: tsh_fps (src/code/poly/tsh_fps.hpp)

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#ifndef TIFALIBS_POLY_TSH_FPS
#define TIFALIBS_POLY_TSH_FPS

#include "../comb/gen_fact.hpp"
#include "../comb/gen_ifact.hpp"
#include "poly.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
CEXP poly<mint, ccore> tsh_fps(poly<mint, ccore> CR f, mint c, vecu64 CR fact, vecu64 CR ifact) {
  u32 n = f.size();
  if (n == 1) return f;
  poly<mint, ccore> s = f, p(f.size());
  flt_ (u32, i, 0, n) p[n - i - 1] = f[i] * fact[i];
  {
    mint _ = 1;
    for (u32 i = 0; i < n; ++i, _ *= c) s[i] = _ * ifact[i];
  }
  (p *= s).resize(n);
  flt_ (u32, i, 0, n) s[n - i - 1] = p[i] * ifact[n - i - 1];
  return s;
}
template <class mint, class ccore>
CEXP poly<mint, ccore> tsh_fps(poly<mint, ccore> CR f, mint c) {
  u32 n = f.size();
  if (n == 1) return f;
  return tsh_fps<mint, ccore>(f, c, gen_fact(n, mint::mod()), gen_ifact(n, mint::mod()));
}

}  // namespace tifa_libs::math

#endif
#line 1 "src/code/poly/tsh_fps.hpp"



#line 1 "src/code/comb/gen_fact.hpp"



#line 1 "src/code/math/mul_mod.hpp"



#line 1 "src/code/math/safe_mod.hpp"



#line 1 "src/code/util/traits.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/stdc++.h>

#define CEXP constexpr
#define TPN typename
#define CR const&

#define cT_(...) std::conditional_t<sizeof(__VA_ARGS__) <= sizeof(size_t), __VA_ARGS__, __VA_ARGS__ CR>
#define fle_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i <= i##e; ++i)
#define flt_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i < i##e; ++i)

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;

using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;

using f32 = float;
using f64 = double;
using f128 = long double;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
template <class T>
using vec = std::vector<T>;
template <class T>
using vvec = vec<vec<T>>;
template <class T>
using v3ec = vec<vvec<T>>;
template <class U, class T>
using vecp = vec<std::pair<U, T>>;
template <class U, class T>
using vvecp = vvec<std::pair<U, T>>;
template <class T>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<T>>;

template <class T, class C = std::less<T>>
using pq = std::priority_queue<T, vec<T>, C>;
template <class T>
using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

using namespace std::literals;

CEXP i8 operator""_i8(unsigned long long x) { return (i8)x; }
CEXP i16 operator""_i16(unsigned long long x) { return (i16)x; }
CEXP i32 operator""_i32(unsigned long long x) { return (i32)x; }
CEXP i64 operator""_i64(unsigned long long x) { return (i64)x; }
CEXP isz operator""_iz(unsigned long long x) { return (isz)x; }

CEXP u8 operator""_u8(unsigned long long x) { return (u8)x; }
CEXP u16 operator""_u16(unsigned long long x) { return (u16)x; }
CEXP u32 operator""_u32(unsigned long long x) { return (u32)x; }
CEXP u64 operator""_u64(unsigned long long x) { return (u64)x; }
CEXP usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};
inline const auto fn_is0 = [](auto x) { return x == 0; };

// std::sqrt(std::numeric_limits<FP>::epsilon())
template <std::floating_point FP>
CEXP inline FP eps_v = FP(1e-8L);
using std::numbers::pi_v;

namespace tifa_libs {
using std::min, std::max, std::swap;
template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }
}  // namespace tifa_libs


#line 5 "src/code/util/traits.hpp"

namespace tifa_libs {

template <class T>
concept iterable_c = requires(T v) {
  { v.begin() } -> std::same_as<TPN T::iterator>;
  { v.end() } -> std::same_as<TPN T::iterator>;
};

template <class T>
concept container_c = iterable_c<T> && !std::derived_from<T, std::basic_string<TPN T::value_type>> && !std::derived_from<T, std::basic_string_view<TPN T::value_type>>;

template <class T>
CEXP bool is_char_v = std::is_same_v<T, char> || std::is_same_v<T, signed char> || std::is_same_v<T, unsigned char>;
template <class T>
concept char_c = is_char_v<T>;

template <class T>
CEXP bool is_s128_v = std::is_same_v<T, __int128_t> || std::is_same_v<T, __int128>;
template <class T>
concept s128_c = is_s128_v<T>;

template <class T>
CEXP bool is_u128_v = std::is_same_v<T, __uint128_t> || std::is_same_v<T, unsigned __int128>;
template <class T>
concept u128_c = is_u128_v<T>;

template <class T>
CEXP bool is_i128_v = is_s128_v<T> || is_u128_v<T>;
template <class T>
concept i128_c = is_u128_v<T>;

template <class T>
CEXP bool is_int_v = std::is_integral_v<T> || is_i128_v<T>;
template <class T>
concept int_c = is_int_v<T>;

template <class T>
CEXP bool is_sint_v = is_s128_v<T> || (is_int_v<T> && std::is_signed_v<T>);
template <class T>
concept sint_c = is_sint_v<T>;

template <class T>
CEXP bool is_uint_v = is_u128_v<T> || (is_int_v<T> && std::is_unsigned_v<T>);
template <class T>
concept uint_c = is_uint_v<T>;

template <class T>
concept mint_c = requires(T x) {
  { x.mod() } -> uint_c;
  { x.val() } -> uint_c;
};

template <class T>
concept dft_c = requires(T x, vec<TPN T::data_t> v, u32 n) {
  { x.size() } -> std::same_as<u32>;
  x.bzr(n);
  x.dif(v, n);
  x.dit(v, n);
};

template <class T>
concept ntt_c = dft_c<T> && requires(T x) {
  T::max_size;
  T::G;
};

template <class T>
CEXP bool is_arithm_v = std::is_arithmetic_v<T> || is_int_v<T>;
template <class T>
concept arithm_c = is_arithm_v<T>;

template <class T>
struct to_sint : std::make_signed<T> {};
template <>
struct to_sint<u128> {
  using type = u128;
};
template <>
struct to_sint<i128> {
  using type = u128;
};
template <class T>
using to_sint_t = TPN to_sint<T>::type;

template <class T>
struct to_uint : std::make_unsigned<T> {};
template <>
struct to_uint<u128> {
  using type = u128;
};
template <>
struct to_uint<i128> {
  using type = u128;
};
template <class T>
using to_uint_t = TPN to_uint<T>::type;

}  // namespace tifa_libs


#line 5 "src/code/math/safe_mod.hpp"

namespace tifa_libs::math {

template <sint_c T>
CEXP T safe_mod(T x, to_uint_t<T> mod) { return ((x %= (T)mod) < 0 ? x + (T)mod : x); }

}  // namespace tifa_libs::math


#line 5 "src/code/math/mul_mod.hpp"

namespace tifa_libs::math {

CEXP i64 mul_mod_s(i64 a, i64 b, u64 mod) {
  if (std::bit_width((u64)abs(a)) + std::bit_width((u64)abs(b)) < 64) return safe_mod(a * b % (i64)mod, mod);
  return safe_mod((i64)((i128)a * b % mod), mod);
}
CEXP u64 mul_mod_u(u64 a, u64 b, u64 mod) {
  if (std::bit_width(a) + std::bit_width(b) <= 64) return a * b % mod;
  return (u64)((u128)a * b % mod);
}

}  // namespace tifa_libs::math


#line 5 "src/code/comb/gen_fact.hpp"

namespace tifa_libs::math {

// i! from i=0..n-1
CEXP vecu64 gen_fact(u32 n, u64 mod) {
  if (n == 0) return {};
  if (n == 1) return {1};
  vecu64 ans(n);
  ans[0] = ans[1] = 1;
  flt_ (u32, i, 2, n) ans[i] = mul_mod_u(ans[i - 1], i, mod);
  return ans;
}
// i! from i=0..n-1
template <class mint>
CEXP vec<mint> gen_fact(u32 n) {
  vec<mint> ans(n);
  auto _ = gen_fact(n, mint::mod());
  flt_ (u32, i, 0, n) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/gen_ifact.hpp"



#line 1 "src/code/comb/gen_inv.hpp"



#line 5 "src/code/comb/gen_inv.hpp"

namespace tifa_libs::math {

// i^{-1} from i=0..n-1
CEXP vecu64 gen_inv(u32 n, u64 mod) {
  if (n == 0) return {};
  if (n == 1) return {1};
  vecu64 ans(n);
  ans[0] = ans[1] = 1;
  flt_ (u32, i, 2, n) ans[i] = mul_mod_u(mod - mod / i, ans[mod % i], mod);
  return ans;
}
// i^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_inv(u32 n) {
  vec<mint> ans(n);
  auto _ = gen_inv(n, mint::mod());
  flt_ (u32, i, 0, n) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 6 "src/code/comb/gen_ifact.hpp"

namespace tifa_libs::math {

// (i!)^{-1} from i=0..n-1
CEXP vecu64 gen_ifact(u32 n, u64 mod, vecu64 inv) {
  flt_ (u32, i, 2, n) inv[i] = mul_mod_u(inv[i], inv[i - 1], mod);
  return inv;
}
// (i!)^{-1} from i=0..n-1
CEXP vecu64 gen_ifact(u32 n, u64 mod) { return gen_ifact(n, mod, gen_inv(n, mod)); }
// (i!)^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_ifact(u32 n, vec<mint> inv) {
  flt_ (u32, i, 2, n) inv[i] *= inv[i - 1];
  return inv;
}
// (i!)^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_ifact(u32 n) { return gen_ifact(n, gen_inv<mint>(n)); }

}  // namespace tifa_libs::math


#line 1 "src/code/poly/poly.hpp"



#line 5 "src/code/poly/poly.hpp"

namespace tifa_libs::math {

// clang-format off
enum ccore_t { ct_FFT, ct_3NTT, ct_NTT };
// clang-format on

template <class mint, class ccore>
requires requires(ccore cc, vec<mint> l, vec<mint> CR r, u32 sz) {
  { ccore::ct_cat } -> std::same_as<ccore_t CR>;
  cc.conv(l, r);
  cc.conv(l, r, sz);
}
class poly {
  vec<mint> d;

 public:
  using value_type = mint;
  using data_type = vec<value_type>;
  using ccore_type = ccore;
  static inline ccore_type conv_core;

  explicit CEXP poly(u32 sz = 1, cT_(value_type) val = value_type{}) : d(sz, val) {}
  CEXP poly(TPN data_type::const_iterator begin, TPN data_type::const_iterator end) : d(begin, end) {}
  CEXP poly(std::initializer_list<value_type> v) : d(v) {}
  template <class T>
  explicit CEXP poly(vec<T> CR v) : d(v) {}

  friend CEXP std::istream &operator>>(std::istream &is, poly &poly) {
    for (auto &val : poly.d) is >> val;
    return is;
  }
  friend CEXP std::ostream &operator<<(std::ostream &os, poly CR poly) {
    if (!poly.size()) return os;
    for (u32 i = 1; i < poly.size(); ++i) os << poly[i - 1] << ' ';
    return os << poly.d.back();
  }

  CEXP u32 size() const { return (u32)d.size(); }
  CEXP bool empty() const {
    for (auto &&i : d)
      if (i != 0) return 0;
    return 1;
  }
  CEXP data_type &data() { return d; }
  CEXP data_type CR data() const { return d; }

  CEXP value_type &operator[](u32 x) { return d[x]; }
  CEXP value_type CR operator[](u32 x) const { return d[x]; }
  CEXP value_type operator()(value_type x) const {
    value_type ans = 0;
    for (u32 i = size() - 1; ~i; --i) ans = ans * x + d[i];
    return ans;
  }

  template <class F>
  requires requires(F f, u32 idx, mint &val) { f(idx, val); }
  CEXP void apply_range(u32 l, u32 r, F &&f) {
    assert(l < r && r <= size());
    flt_ (u32, i, l, r) f(i, d[i]);
  }
  template <class F>
  CEXP void apply(F &&f) { apply_range(0, size(), std::forward<F>(f)); }
  CEXP void resize(u32 size) { d.resize(size); }
  CEXP poly pre(u32 size) const {
    poly _ = *this;
    _.resize(size);
    return _;
  }
  CEXP void strip() {
    auto it = std::find_if(d.rbegin(), d.rend(), [](cT_(mint) x) { return x.val() != 0; });
    d.resize(usz(d.rend() - it));
    if (d.empty()) d.push_back(value_type(0));
  }
  friend poly stripped(poly p) {
    p.strip();
    return p;
  }
  CEXP void reverse(u32 n = 0) { std::reverse(d.begin(), d.begin() + (n ? n : size())); }
  CEXP void conv(poly CR r, u32 ans_size = 0) { conv_core.conv(d, r.d, ans_size); }

  CEXP poly operator-() const {
    poly ret = *this;
    ret.apply([](u32, auto &v) { v = -v; });
    return ret;
  }

  friend CEXP poly operator+(poly p, value_type c) {
    p[0] += c;
    return p;
  }
  friend CEXP poly operator+(value_type c, poly CR p) { return p + c; }
  friend CEXP poly operator-(poly p, value_type c) {
    p[0] -= c;
    return p;
  }
  friend CEXP poly operator-(value_type c, poly CR p) { return p - c; }

  CEXP poly &operator*=(value_type c) {
    apply([&c](u32, auto &v) { v *= c; });
    return *this;
  }
  friend CEXP poly operator*(poly p, value_type c) { return p *= c; }
  friend CEXP poly operator*(value_type c, poly p) { return p *= c; }

  CEXP poly &operator+=(poly CR r) {
    if (!r.size()) return *this;
    resize(max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v += r[i]; });
    return *this;
  }
  friend CEXP poly operator+(poly l, poly CR r) { return l += r; }

  CEXP poly &operator-=(poly CR r) {
    if (!r.size()) return *this;
    resize(max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v -= r[i]; });
    return *this;
  }
  friend CEXP poly operator-(poly l, poly CR r) { return l -= r; }

  CEXP poly &operator*=(poly CR r) {
    if (!r.size()) {
      resize(1);
      d[0] = 0;
      return *this;
    }
    conv(r);
    return *this;
  }
  friend CEXP poly operator*(poly l, poly CR r) { return l *= r; }

  CEXP auto operator<=>(poly CR r) const { return stripped(*this).d <=> stripped(r).d; }
  CEXP bool operator==(poly CR r) const { return stripped(*this).d == stripped(r).d; }
};

}  // namespace tifa_libs::math


#line 7 "src/code/poly/tsh_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
CEXP poly<mint, ccore> tsh_fps(poly<mint, ccore> CR f, mint c, vecu64 CR fact, vecu64 CR ifact) {
  u32 n = f.size();
  if (n == 1) return f;
  poly<mint, ccore> s = f, p(f.size());
  flt_ (u32, i, 0, n) p[n - i - 1] = f[i] * fact[i];
  {
    mint _ = 1;
    for (u32 i = 0; i < n; ++i, _ *= c) s[i] = _ * ifact[i];
  }
  (p *= s).resize(n);
  flt_ (u32, i, 0, n) s[n - i - 1] = p[i] * ifact[n - i - 1];
  return s;
}
template <class mint, class ccore>
CEXP poly<mint, ccore> tsh_fps(poly<mint, ccore> CR f, mint c) {
  u32 n = f.size();
  if (n == 1) return f;
  return tsh_fps<mint, ccore>(f, c, gen_fact(n, mint::mod()), gen_ifact(n, mint::mod()));
}

}  // namespace tifa_libs::math


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