verify/math/generalized_floor_sum.test.cpp

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Last update: 2026-10-06 01:57:24+09:00
Problem: https://judge.yosupo.jp/problem/aplusb

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include "../../math/generalized_floor_sum.hpp"
#include "../../math/modint.hpp"
#include "../../math/number_theory.hpp"

#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"

using Mint = m1une::math::modint998244353;

long long floor_div(long long numerator, long long denominator) {
    long long quotient = numerator / denominator;
    if (numerator % denominator < 0) --quotient;
    return quotient;
}

template <class T, int MaxPower, int MaxFloorPower>
auto naive(long long n, long long mod, long long a, long long b) {
    m1une::math::GeneralizedFloorSumTable<
        T,
        MaxPower,
        MaxFloorPower
    > result{};
    for (long long x = 0; x < n; ++x) {
        long long y = floor_div(a * x + b, mod);
        T x_power = T(1);
        for (int p = 0; p <= MaxPower; ++p) {
            T y_power = T(1);
            for (int q = 0; q <= MaxFloorPower; ++q) {
                result[p][q] += x_power * y_power;
                y_power *= T(y);
            }
            x_power *= T(x);
        }
    }
    return result;
}

void test_fixed_cases() {
    auto count_only =
        m1une::math::generalized_floor_sum_table<Mint, 0, 0>(12, 1, -5, -8);
    assert(count_only[0][0] == Mint(12));

    auto empty =
        m1une::math::generalized_floor_sum_table<Mint, 3, 3>(0, 7, 2, 3);
    for (const auto& row : empty) {
        for (Mint value : row) assert(value == Mint(0));
    }

    auto actual =
        m1une::math::generalized_floor_sum_table<Mint, 4, 4>(20, 11, -7, -9);
    auto expected = naive<Mint, 4, 4>(20, 11, -7, -9);
    assert(actual == expected);

    Mint moment =
        m1une::math::generalized_floor_sum<Mint, 3, 2>(20, 11, -7, -9);
    assert(moment == expected[3][2]);

    auto unsigned_actual =
        m1une::math::generalized_floor_sum_table<std::uint64_t, 3, 3>(
            30,
            17,
            23,
            -41
        );
    auto unsigned_expected = naive<std::uint64_t, 3, 3>(30, 17, 23, -41);
    assert(unsigned_actual == unsigned_expected);

    long long ordinary = m1une::math::floor_sum(1000000, 998244353, 123456, -789);
    Mint generalized =
        m1une::math::generalized_floor_sum<Mint, 0, 1>(
            1000000,
            998244353,
            123456,
            -789
        );
    assert(generalized == Mint(ordinary));
}

void test_randomized_against_naive() {
    std::uint64_t state = 0x6a09e667f3bcc909ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int test = 0; test < 5000; ++test) {
        long long n = static_cast<long long>(random() % 25);
        long long mod = 1 + static_cast<long long>(random() % 20);
        long long a = static_cast<long long>(random() % 101) - 50;
        long long b = static_cast<long long>(random() % 101) - 50;

        auto actual =
            m1une::math::generalized_floor_sum_table<Mint, 4, 4>(
                n,
                mod,
                a,
                b
            );
        auto expected = naive<Mint, 4, 4>(n, mod, a, b);
        assert(actual == expected);
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_fixed_cases();
    test_randomized_against_naive();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}

#line 1 "verify/math/generalized_floor_sum.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#line 1 "math/generalized_floor_sum.hpp"



#include <array>
#include <cassert>
#include <cstdint>
#include <type_traits>
#include <utility>

namespace m1une {
namespace math {

template <class T, int MaxPower, int MaxFloorPower>
using GeneralizedFloorSumTable =
    std::array<std::array<T, MaxFloorPower + 1>, MaxPower + 1>;

namespace generalized_floor_sum_detail {

using SignedWide = __int128_t;
using UnsignedWide = __uint128_t;

template <class T>
T from_wide(SignedWide value) {
    bool negative = value < 0;
    UnsignedWide magnitude;
    if (negative) {
        magnitude = static_cast<UnsignedWide>(-(value + 1));
        ++magnitude;
    } else {
        magnitude = static_cast<UnsignedWide>(value);
    }

    T result = T();
    T binary_place = T(1);
    while (magnitude > 0) {
        if ((magnitude & 1) != 0) result += binary_place;
        magnitude >>= 1;
        if (magnitude > 0) binary_place += binary_place;
    }
    return negative ? T() - result : result;
}

inline SignedWide floor_div(SignedWide numerator, SignedWide denominator) {
    assert(denominator > 0);
    SignedWide quotient = numerator / denominator;
    if (numerator % denominator < 0) --quotient;
    return quotient;
}

template <class T, int MaxPower, int MaxFloorPower>
class MomentMonoid {
   public:
    using Table = GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower>;

    struct Data {
        Table sums{};
        T delta_x = T();
        T delta_y = T();
    };

    static constexpr int MaximumDegree =
        MaxPower > MaxFloorPower ? MaxPower : MaxFloorPower;

    MomentMonoid() {
        binomial_[0][0] = T(1);
        for (int degree = 0; degree < MaximumDegree; ++degree) {
            for (int index = 0; index <= degree; ++index) {
                binomial_[degree + 1][index] += binomial_[degree][index];
                binomial_[degree + 1][index + 1] +=
                    binomial_[degree][index];
            }
        }
    }

    const T& binomial(int n, int k) const {
        assert(0 <= k && k <= n && n <= MaximumDegree);
        return binomial_[n][k];
    }

    Data unit() const {
        return Data();
    }

    Data x_step() const {
        Data result;
        result.sums[0][0] = T(1);
        result.delta_x = T(1);
        return result;
    }

    Data y_step() const {
        Data result;
        result.delta_y = T(1);
        return result;
    }

    Data concatenate(Data left, Data right) const {
        std::array<T, MaxPower + 1> x_powers{};
        std::array<T, MaxFloorPower + 1> y_powers{};
        x_powers[0] = T(1);
        y_powers[0] = T(1);
        for (int power = 0; power < MaxPower; ++power) {
            x_powers[power + 1] = x_powers[power] * left.delta_x;
        }
        for (int power = 0; power < MaxFloorPower; ++power) {
            y_powers[power + 1] = y_powers[power] * left.delta_y;
        }

        // Shift the y-coordinate of every sampled x-step in the right path.
        for (int x_power = 0; x_power <= MaxPower; ++x_power) {
            for (int old_power = MaxFloorPower; old_power >= 0; --old_power) {
                T source = right.sums[x_power][old_power];
                for (int new_power = old_power + 1;
                     new_power <= MaxFloorPower;
                     ++new_power) {
                    right.sums[x_power][new_power] +=
                        binomial_[new_power][old_power] *
                        y_powers[new_power - old_power] * source;
                }
            }
        }

        // Shift x, then append all samples from the right path.
        for (int y_power = 0; y_power <= MaxFloorPower; ++y_power) {
            for (int old_power = 0; old_power <= MaxPower; ++old_power) {
                T source = right.sums[old_power][y_power];
                for (int new_power = old_power;
                     new_power <= MaxPower;
                     ++new_power) {
                    left.sums[new_power][y_power] +=
                        binomial_[new_power][old_power] *
                        x_powers[new_power - old_power] * source;
                }
            }
        }

        left.delta_x += right.delta_x;
        left.delta_y += right.delta_y;
        return left;
    }

   private:
    std::array<std::array<T, MaximumDegree + 1>, MaximumDegree + 1>
        binomial_{};
};

template <class Monoid>
typename Monoid::Data monoid_power(
    const Monoid& monoid,
    typename Monoid::Data base,
    UnsignedWide exponent
) {
    typename Monoid::Data result = monoid.unit();
    while (exponent > 0) {
        if ((exponent & 1) != 0) {
            result = monoid.concatenate(std::move(result), base);
        }
        exponent >>= 1;
        if (exponent > 0) {
            base = monoid.concatenate(base, base);
        }
    }
    return result;
}

template <class Monoid>
typename Monoid::Data floor_path_product(
    const Monoid& monoid,
    UnsignedWide n,
    UnsignedWide a,
    UnsignedWide b,
    UnsignedWide modulus
) {
    assert(modulus > 0);
    UnsignedWide height = (a * n + b) / modulus;
    typename Monoid::Data x = monoid.x_step();
    typename Monoid::Data y = monoid.y_step();
    typename Monoid::Data prefix = monoid.unit();
    typename Monoid::Data suffix = monoid.unit();

    while (true) {
        UnsignedWide slope_quotient = a / modulus;
        UnsignedWide intercept_quotient = b / modulus;
        a %= modulus;
        b %= modulus;

        x = monoid.concatenate(
            std::move(x),
            monoid_power(monoid, y, slope_quotient)
        );
        prefix = monoid.concatenate(
            std::move(prefix),
            monoid_power(monoid, y, intercept_quotient)
        );
        height -= slope_quotient * n + intercept_quotient;
        if (height == 0) break;

        assert(a > 0);
        UnsignedWide boundary =
            (modulus * height - b - 1) / a + 1;
        suffix = monoid.concatenate(
            y,
            monoid.concatenate(
                monoid_power(monoid, x, n - boundary),
                std::move(suffix)
            )
        );
        b = modulus - b - 1 + a;
        n = height - 1;
        height = boundary;
        std::swap(modulus, a);
        std::swap(x, y);
    }

    x = monoid_power(monoid, x, n);
    return monoid.concatenate(
        monoid.concatenate(std::move(prefix), std::move(x)),
        std::move(suffix)
    );
}

template <class T, int MaxPower, int MaxFloorPower>
GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower>
nonnegative_slope_table(
    const MomentMonoid<T, MaxPower, MaxFloorPower>& monoid,
    SignedWide n,
    SignedWide modulus,
    SignedWide a,
    SignedWide b
) {
    assert(n >= 0 && modulus > 0 && a >= 0);
    SignedWide y_offset = floor_div(b, modulus);
    SignedWide normalized_b = b - y_offset * modulus;

    auto path = floor_path_product(
        monoid,
        static_cast<UnsignedWide>(n),
        static_cast<UnsignedWide>(a),
        static_cast<UnsignedWide>(normalized_b),
        static_cast<UnsignedWide>(modulus)
    );

    std::array<T, MaxFloorPower + 1> offset_powers{};
    offset_powers[0] = T(1);
    T offset = from_wide<T>(y_offset);
    for (int power = 0; power < MaxFloorPower; ++power) {
        offset_powers[power + 1] = offset_powers[power] * offset;
    }

    GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower> result{};
    for (int x_power = 0; x_power <= MaxPower; ++x_power) {
        for (int y_power = 0; y_power <= MaxFloorPower; ++y_power) {
            for (int inner_power = 0;
                 inner_power <= y_power;
                 ++inner_power) {
                result[x_power][y_power] +=
                    monoid.binomial(y_power, inner_power) *
                    offset_powers[y_power - inner_power] *
                    path.sums[x_power][inner_power];
            }
        }
    }
    return result;
}

}  // namespace generalized_floor_sum_detail

// Returns every sum of x^p * floor((a*x+b)/mod)^q for 0 <= x < n,
// 0 <= p <= MaxPower, and 0 <= q <= MaxFloorPower.
template <class T, int MaxPower, int MaxFloorPower, class I>
GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower>
generalized_floor_sum_table(I n, I mod, I a, I b) {
    static_assert(MaxPower >= 0 && MaxFloorPower >= 0);
    static_assert(
        std::is_integral_v<I> && std::is_signed_v<I> && sizeof(I) <= 8,
        "generalized_floor_sum_table requires signed integer arguments"
    );
    assert(n >= 0);
    assert(mod > 0);

    namespace detail = generalized_floor_sum_detail;
    using Monoid = detail::MomentMonoid<T, MaxPower, MaxFloorPower>;
    static const Monoid monoid;

    detail::SignedWide wide_n = n;
    detail::SignedWide wide_mod = mod;
    detail::SignedWide wide_a = a;
    detail::SignedWide wide_b = b;
    if (wide_n == 0) {
        return GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower>();
    }
    if (wide_a >= 0) {
        return detail::nonnegative_slope_table(
            monoid,
            wide_n,
            wide_mod,
            wide_a,
            wide_b
        );
    }

    // Substitute x = n - 1 - t to make the slope nonnegative.
    auto reflected = detail::nonnegative_slope_table(
        monoid,
        wide_n,
        wide_mod,
        -wide_a,
        wide_a * (wide_n - 1) + wide_b
    );
    std::array<T, MaxPower + 1> offset_powers{};
    offset_powers[0] = T(1);
    T offset = detail::from_wide<T>(wide_n - 1);
    for (int power = 0; power < MaxPower; ++power) {
        offset_powers[power + 1] = offset_powers[power] * offset;
    }

    GeneralizedFloorSumTable<T, MaxPower, MaxFloorPower> result{};
    for (int x_power = 0; x_power <= MaxPower; ++x_power) {
        for (int y_power = 0; y_power <= MaxFloorPower; ++y_power) {
            for (int inner_power = 0;
                 inner_power <= x_power;
                 ++inner_power) {
                T coefficient =
                    monoid.binomial(x_power, inner_power) *
                    offset_powers[x_power - inner_power];
                if ((inner_power & 1) != 0) coefficient = T() - coefficient;
                result[x_power][y_power] +=
                    coefficient * reflected[inner_power][y_power];
            }
        }
    }
    return result;
}

template <class T, int Power, int FloorPower, class I>
T generalized_floor_sum(I n, I mod, I a, I b) {
    return generalized_floor_sum_table<T, Power, FloorPower>(n, mod, a, b)
        [Power][FloorPower];
}

}  // namespace math
}  // namespace m1une


#line 1 "math/modint.hpp"



#line 6 "math/modint.hpp"
#include <iostream>
#line 9 "math/modint.hpp"

namespace m1une {
namespace math {

template <uint32_t Modulus>
struct ModInt {
    static_assert(0 < Modulus, "Modulus must be positive");

   private:
    uint32_t _v;

   public:
    static constexpr uint32_t mod() {
        return Modulus;
    }

    static constexpr ModInt raw(uint32_t v) noexcept {
        ModInt x;
        x._v = v;
        return x;
    }

    constexpr ModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    constexpr ModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(Modulus);
            if (x < 0) x += Modulus;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % Modulus);
        }
    }

    constexpr uint32_t val() const noexcept {
        return _v;
    }

    constexpr ModInt& operator++() noexcept {
        _v++;
        if (_v == Modulus) _v = 0;
        return *this;
    }

    constexpr ModInt& operator--() noexcept {
        if (_v == 0) _v = Modulus;
        _v--;
        return *this;
    }

    constexpr ModInt operator++(int) noexcept {
        ModInt res = *this;
        ++*this;
        return res;
    }

    constexpr ModInt operator--(int) noexcept {
        ModInt res = *this;
        --*this;
        return res;
    }

    constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= Modulus) _v -= Modulus;
        return *this;
    }

    constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= Modulus) _v += Modulus;
        return *this;
    }

    constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
        uint64_t z = _v;
        z *= rhs._v;
        _v = static_cast<uint32_t>(z % Modulus);
        return *this;
    }

    constexpr ModInt& operator/=(const ModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    constexpr ModInt operator+(const ModInt& rhs) const noexcept {
        return ModInt(*this) += rhs;
    }
    constexpr ModInt operator-(const ModInt& rhs) const noexcept {
        return ModInt(*this) -= rhs;
    }
    constexpr ModInt operator*(const ModInt& rhs) const noexcept {
        return ModInt(*this) *= rhs;
    }
    constexpr ModInt operator/(const ModInt& rhs) const noexcept {
        return ModInt(*this) /= rhs;
    }

    constexpr bool operator==(const ModInt& rhs) const noexcept {
        return _v == rhs._v;
    }
    constexpr bool operator!=(const ModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    constexpr ModInt pow(long long n) const noexcept {
        ModInt res = raw(1 % Modulus);
        ModInt x = n < 0 ? inv() : *this;
        uint64_t exponent = n < 0 ? uint64_t(-(n + 1)) + 1 : uint64_t(n);
        while (exponent > 0) {
            if (exponent & 1) res *= x;
            x *= x;
            exponent >>= 1;
        }
        return res;
    }

    constexpr ModInt inv() const noexcept {
        int64_t a = _v, b = Modulus, u = 1, v = 0;
        while (b) {
            int64_t t = a / b;
            a -= t * b;
            std::swap(a, b);
            u -= t * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= Modulus;
        if (u < 0) u += Modulus;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const ModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, ModInt& rhs) {
        long long v;
        is >> v;
        rhs = ModInt(v);
        return is;
    }
};

using modint998244353 = ModInt<998244353>;
using modint1000000007 = ModInt<1000000007>;

template <int Id = 0>
struct DynamicModInt {
   private:
    uint32_t _v;
    inline static uint32_t _mod = 1;

   public:
    static uint32_t mod() noexcept {
        return _mod;
    }

    static void set_mod(uint32_t modulus) noexcept {
        assert(modulus > 0);
        assert(modulus <= uint32_t(1) << 31);
        _mod = modulus;
    }

    static DynamicModInt raw(uint32_t v) noexcept {
        assert(v < _mod);
        DynamicModInt x;
        x._v = v;
        return x;
    }

    DynamicModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    DynamicModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(_mod);
            if (x < 0) x += _mod;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % _mod);
        }
    }

    uint32_t val() const noexcept {
        return _v;
    }

    DynamicModInt& operator++() noexcept {
        _v++;
        if (_v == _mod) _v = 0;
        return *this;
    }

    DynamicModInt& operator--() noexcept {
        if (_v == 0) _v = _mod;
        _v--;
        return *this;
    }

    DynamicModInt operator++(int) noexcept {
        DynamicModInt result = *this;
        ++*this;
        return result;
    }

    DynamicModInt operator--(int) noexcept {
        DynamicModInt result = *this;
        --*this;
        return result;
    }

    DynamicModInt& operator+=(const DynamicModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= _mod) _v -= _mod;
        return *this;
    }

    DynamicModInt& operator-=(const DynamicModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= _mod) _v += _mod;
        return *this;
    }

    DynamicModInt& operator*=(const DynamicModInt& rhs) noexcept {
        _v = static_cast<uint32_t>(uint64_t(_v) * rhs._v % _mod);
        return *this;
    }

    DynamicModInt& operator/=(const DynamicModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    DynamicModInt operator+(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) += rhs;
    }

    DynamicModInt operator-(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) -= rhs;
    }

    DynamicModInt operator*(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) *= rhs;
    }

    DynamicModInt operator/(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) /= rhs;
    }

    bool operator==(const DynamicModInt& rhs) const noexcept {
        return _v == rhs._v;
    }

    bool operator!=(const DynamicModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    DynamicModInt pow(long long exponent) const noexcept {
        DynamicModInt result = raw(1 % _mod);
        DynamicModInt base = exponent < 0 ? inv() : *this;
        uint64_t magnitude =
            exponent < 0 ? uint64_t(-(exponent + 1)) + 1 : uint64_t(exponent);
        while (magnitude > 0) {
            if (magnitude & 1) result *= base;
            base *= base;
            magnitude >>= 1;
        }
        return result;
    }

    DynamicModInt inv() const noexcept {
        int64_t a = _v, b = _mod, u = 1, v = 0;
        while (b) {
            int64_t quotient = a / b;
            a -= quotient * b;
            std::swap(a, b);
            u -= quotient * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= _mod;
        if (u < 0) u += _mod;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const DynamicModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, DynamicModInt& rhs) {
        long long value;
        is >> value;
        rhs = DynamicModInt(value);
        return is;
    }
};

}  // namespace math
}  // namespace m1une


#line 1 "math/number_theory.hpp"



#line 6 "math/number_theory.hpp"
#include <limits>
#include <tuple>
#line 9 "math/number_theory.hpp"
#include <vector>

namespace m1une {
namespace math {

namespace internal {

inline long long safe_mod(long long x, long long mod) {
    x %= mod;
    if (x < 0) x += mod;
    return x;
}

inline unsigned __int128 floor_sum_unsigned(unsigned long long n, unsigned long long mod, unsigned long long a,
                                            unsigned long long b) {
    unsigned __int128 answer = 0;
    while (true) {
        if (a >= mod) {
            answer += static_cast<unsigned __int128>(n) * (n - 1) / 2 * (a / mod);
            a %= mod;
        }
        if (b >= mod) {
            answer += static_cast<unsigned __int128>(n) * (b / mod);
            b %= mod;
        }

        const unsigned __int128 y_max = static_cast<unsigned __int128>(a) * n + b;
        if (y_max < mod) break;
        n = static_cast<unsigned long long>(y_max / mod);
        b = static_cast<unsigned long long>(y_max % mod);
        unsigned long long tmp = mod;
        mod = a;
        a = tmp;
    }
    return answer;
}

}  // namespace internal

// Returns (g, x, y), where g = gcd(a, b) is nonnegative and
// a * x + b * y = g. Returns (0, 0, 0) when a = b = 0.
inline std::tuple<long long, long long, long long> extended_gcd(long long a,
                                                               long long b) {
    using i128 = __int128;
    if (a == 0 && b == 0) return {0, 0, 0};

    i128 old_remainder = a;
    i128 remainder = b;
    if (old_remainder < 0) old_remainder = -old_remainder;
    if (remainder < 0) remainder = -remainder;
    i128 old_x = 1;
    i128 x = 0;
    i128 old_y = 0;
    i128 y = 1;

    while (remainder != 0) {
        i128 quotient = old_remainder / remainder;

        i128 next = old_remainder - quotient * remainder;
        old_remainder = remainder;
        remainder = next;

        next = old_x - quotient * x;
        old_x = x;
        x = next;

        next = old_y - quotient * y;
        old_y = y;
        y = next;
    }

    if (a < 0) old_x = -old_x;
    if (b < 0) old_y = -old_y;

#ifndef NDEBUG
    const i128 minimum = std::numeric_limits<long long>::min();
    const i128 maximum = std::numeric_limits<long long>::max();
    assert(old_remainder <= maximum);
    assert(minimum <= old_x && old_x <= maximum);
    assert(minimum <= old_y && old_y <= maximum);
#endif
    return {static_cast<long long>(old_remainder), static_cast<long long>(old_x),
            static_cast<long long>(old_y)};
}

inline long long pow_mod(long long x, unsigned long long exponent, long long mod) {
    assert(mod >= 1);
    if (mod == 1) return 0;

    unsigned long long base = static_cast<unsigned long long>(internal::safe_mod(x, mod));
    unsigned long long result = 1;
    const unsigned long long unsigned_mod = static_cast<unsigned long long>(mod);
    while (exponent > 0) {
        if (exponent & 1) {
            result = static_cast<unsigned long long>(static_cast<unsigned __int128>(result) * base % unsigned_mod);
        }
        base = static_cast<unsigned long long>(static_cast<unsigned __int128>(base) * base % unsigned_mod);
        exponent >>= 1;
    }
    return static_cast<long long>(result);
}

// Returns gcd(a, mod) and x such that a * x is congruent to gcd(a, mod)
// modulo mod. The returned x is in [0, mod / gcd(a, mod)).
inline std::pair<long long, long long> inv_gcd(long long a, long long mod) {
    assert(mod >= 1);
    a = internal::safe_mod(a, mod);
    if (a == 0) return {mod, 0};

    long long s = mod;
    long long t = a;
    long long m0 = 0;
    long long m1 = 1;
    while (t > 0) {
        const long long quotient = s / t;
        s -= t * quotient;
        m0 -= m1 * quotient;

        long long tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += mod / s;
    return {s, m0};
}

inline long long inv_mod(long long x, long long mod) {
    const auto result = inv_gcd(x, mod);
    assert(result.first == 1);
    return result.second;
}

// Returns the smallest nonnegative solution and the least common multiple of
// the moduli. Returns {0, 0} when the system is inconsistent.
inline std::pair<long long, long long> crt(const std::vector<long long>& remainders,
                                           const std::vector<long long>& moduli) {
    assert(remainders.size() == moduli.size());

    long long r0 = 0;
    long long m0 = 1;
    for (int i = 0; i < int(remainders.size()); i++) {
        assert(moduli[i] >= 1);
        long long r1 = internal::safe_mod(remainders[i], moduli[i]);
        long long m1 = moduli[i];

        if (m0 < m1) {
            long long tmp = r0;
            r0 = r1;
            r1 = tmp;
            tmp = m0;
            m0 = m1;
            m1 = tmp;
        }
        if (m0 % m1 == 0) {
            if (r0 % m1 != r1) return {0, 0};
            continue;
        }

        const auto inverse = inv_gcd(m0, m1);
        const long long gcd = inverse.first;
        const long long reduced_modulus = m1 / gcd;
        const __int128 difference = static_cast<__int128>(r1) - r0;
        if (difference % gcd != 0) return {0, 0};

        __int128 multiplier = difference / gcd % reduced_modulus;
        multiplier = multiplier * inverse.second % reduced_modulus;
        if (multiplier < 0) multiplier += reduced_modulus;

        const __int128 new_modulus = static_cast<__int128>(m0) * reduced_modulus;
        assert(new_modulus <= std::numeric_limits<long long>::max());
        __int128 new_remainder = static_cast<__int128>(r0) + multiplier * m0;
        new_remainder %= new_modulus;
        if (new_remainder < 0) new_remainder += new_modulus;
        r0 = static_cast<long long>(new_remainder);
        m0 = static_cast<long long>(new_modulus);
    }
    return {r0, m0};
}

// Returns sum_{i=0}^{n-1} floor((a * i + b) / mod).
inline long long floor_sum(long long n, long long mod, long long a, long long b) {
    assert(n >= 0);
    assert(mod >= 1);

    const long long normalized_a = internal::safe_mod(a, mod);
    const long long normalized_b = internal::safe_mod(b, mod);
    __int128 answer = (static_cast<__int128>(a) - normalized_a) / mod * n * (n - 1) / 2;
    answer += (static_cast<__int128>(b) - normalized_b) / mod * n;
    answer += internal::floor_sum_unsigned(static_cast<unsigned long long>(n), static_cast<unsigned long long>(mod),
                                           static_cast<unsigned long long>(normalized_a),
                                           static_cast<unsigned long long>(normalized_b));

    assert(answer >= std::numeric_limits<long long>::min());
    assert(answer <= std::numeric_limits<long long>::max());
    return static_cast<long long>(answer);
}

}  // namespace math
}  // namespace m1une


#line 6 "verify/math/generalized_floor_sum.test.cpp"

#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 10 "verify/math/generalized_floor_sum.test.cpp"

using Mint = m1une::math::modint998244353;

long long floor_div(long long numerator, long long denominator) {
    long long quotient = numerator / denominator;
    if (numerator % denominator < 0) --quotient;
    return quotient;
}

template <class T, int MaxPower, int MaxFloorPower>
auto naive(long long n, long long mod, long long a, long long b) {
    m1une::math::GeneralizedFloorSumTable<
        T,
        MaxPower,
        MaxFloorPower
    > result{};
    for (long long x = 0; x < n; ++x) {
        long long y = floor_div(a * x + b, mod);
        T x_power = T(1);
        for (int p = 0; p <= MaxPower; ++p) {
            T y_power = T(1);
            for (int q = 0; q <= MaxFloorPower; ++q) {
                result[p][q] += x_power * y_power;
                y_power *= T(y);
            }
            x_power *= T(x);
        }
    }
    return result;
}

void test_fixed_cases() {
    auto count_only =
        m1une::math::generalized_floor_sum_table<Mint, 0, 0>(12, 1, -5, -8);
    assert(count_only[0][0] == Mint(12));

    auto empty =
        m1une::math::generalized_floor_sum_table<Mint, 3, 3>(0, 7, 2, 3);
    for (const auto& row : empty) {
        for (Mint value : row) assert(value == Mint(0));
    }

    auto actual =
        m1une::math::generalized_floor_sum_table<Mint, 4, 4>(20, 11, -7, -9);
    auto expected = naive<Mint, 4, 4>(20, 11, -7, -9);
    assert(actual == expected);

    Mint moment =
        m1une::math::generalized_floor_sum<Mint, 3, 2>(20, 11, -7, -9);
    assert(moment == expected[3][2]);

    auto unsigned_actual =
        m1une::math::generalized_floor_sum_table<std::uint64_t, 3, 3>(
            30,
            17,
            23,
            -41
        );
    auto unsigned_expected = naive<std::uint64_t, 3, 3>(30, 17, 23, -41);
    assert(unsigned_actual == unsigned_expected);

    long long ordinary = m1une::math::floor_sum(1000000, 998244353, 123456, -789);
    Mint generalized =
        m1une::math::generalized_floor_sum<Mint, 0, 1>(
            1000000,
            998244353,
            123456,
            -789
        );
    assert(generalized == Mint(ordinary));
}

void test_randomized_against_naive() {
    std::uint64_t state = 0x6a09e667f3bcc909ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int test = 0; test < 5000; ++test) {
        long long n = static_cast<long long>(random() % 25);
        long long mod = 1 + static_cast<long long>(random() % 20);
        long long a = static_cast<long long>(random() % 101) - 50;
        long long b = static_cast<long long>(random() % 101) - 50;

        auto actual =
            m1une::math::generalized_floor_sum_table<Mint, 4, 4>(
                n,
                mod,
                a,
                b
            );
        auto expected = naive<Mint, 4, 4>(n, mod, a, b);
        assert(actual == expected);
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_fixed_cases();
    test_randomized_against_naive();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

verify/math/generalized_floor_sum.test.cpp

Depends on

Code