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:heavy_check_mark: verify/math/fps/floating_point_convolution.test.cpp

Depends on

Code

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

#include "../../../math/fps/floating_point_convolution.hpp"

#include <algorithm>
#include <cassert>
#include <cmath>
#include <complex>
#include <cstdint>
#include "../../../utilities/fast_io.hpp"
#include <vector>

namespace {

template <class T>
std::vector<T> naive(
    const std::vector<T>& first,
    const std::vector<T>& second
) {
    if (first.empty() || second.empty()) return {};
    std::vector<T> result(first.size() + second.size() - 1);
    for (std::size_t i = 0; i < first.size(); ++i) {
        for (std::size_t j = 0; j < second.size(); ++j) {
            result[i + j] += first[i] * second[j];
        }
    }
    return result;
}

void test_fixed() {
    std::vector<double> first = {1.5, 2.0};
    std::vector<double> second = {3.0, -1.0};
    auto result = m1une::fps::convolution_fft(first, second);
    assert(std::fabs(result[0] - 4.5) < 1e-12);
    assert(std::fabs(result[1] - 4.5) < 1e-12);
    assert(std::fabs(result[2] + 2.0) < 1e-12);

    std::vector<std::complex<double>> complex_first;
    complex_first.emplace_back(1, 2);
    complex_first.emplace_back(3, -1);
    std::vector<std::complex<double>> complex_second;
    complex_second.emplace_back(-2, 1);
    complex_second.emplace_back(0, 4);
    auto complex_result =
        m1une::fps::convolution_fft(complex_first, complex_second);
    auto complex_expected = naive(complex_first, complex_second);
    for (std::size_t index = 0; index < complex_result.size(); ++index) {
        assert(std::abs(complex_result[index] - complex_expected[index]) < 1e-11);
    }

    std::vector<long long> integers = {1, -2, 3, 4};
    assert(
        m1une::fps::convolution_round(integers, integers)
        == naive(integers, integers)
    );
    assert(
        m1une::fps::convolution_fft(
            std::vector<double>(),
            std::vector<double>{1}
        ).empty()
    );
}

void test_randomized() {
    std::uint64_t state = 1409;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; ++trial) {
        int first_size = int(random() % 100);
        int second_size = int(random() % 100);
        std::vector<long double> first(first_size);
        std::vector<long double> second(second_size);
        for (long double& value : first) {
            value =
                (static_cast<long long>(random() % 2001) - 1000) / 100.0L;
        }
        for (long double& value : second) {
            value =
                (static_cast<long long>(random() % 2001) - 1000) / 100.0L;
        }
        auto actual = m1une::fps::convolution_fft(first, second);
        auto expected = naive(first, second);
        for (std::size_t index = 0; index < actual.size(); ++index) {
            long double magnitude = 0;
            for (std::size_t first_index = 0; first_index < first.size(); ++first_index) {
                if (index < first_index) continue;
                std::size_t second_index = index - first_index;
                if (second_index < second.size()) {
                    magnitude += std::fabs(
                        first[first_index] * second[second_index]
                    );
                }
            }
            [[maybe_unused]] long double tolerance =
                1e-12L * std::max(1.0L, magnitude);
            assert(std::fabs(actual[index] - expected[index]) <= tolerance);
        }

        std::vector<long long> integer_first(first_size);
        std::vector<long long> integer_second(second_size);
        for (long long& value : integer_first) {
            value = static_cast<long long>(random() % 2001) - 1000;
        }
        for (long long& value : integer_second) {
            value = static_cast<long long>(random() % 2001) - 1000;
        }
        assert(
            m1une::fps::convolution_round(integer_first, integer_second)
            == naive(integer_first, integer_second)
        );
    }
}

}  // namespace

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

    test_fixed();
    test_randomized();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
#line 1 "verify/math/fps/floating_point_convolution.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#line 1 "math/fps/floating_point_convolution.hpp"



#include <algorithm>
#include <bit>
#include <cmath>
#include <complex>
#include <concepts>
#include <numbers>
#include <type_traits>
#include <vector>

namespace m1une {
namespace fps {

namespace floating_point_convolution_detail {

template <std::floating_point Real>
void fft(std::vector<std::complex<Real>>& values, bool inverse) {
    int size = int(values.size());
    for (int index = 1, reversed = 0; index < size; ++index) {
        int bit = size >> 1;
        while (reversed & bit) {
            reversed ^= bit;
            bit >>= 1;
        }
        reversed ^= bit;
        if (index < reversed) std::swap(values[index], values[reversed]);
    }

    for (int length = 2; length <= size; length <<= 1) {
        Real angle = Real(2) * std::numbers::pi_v<Real> / Real(length);
        if (inverse) angle = -angle;
        std::complex<Real> step(std::cos(angle), std::sin(angle));
        int half = length >> 1;
        for (int offset = 0; offset < size; offset += length) {
            std::complex<Real> root(1, 0);
            for (int index = 0; index < half; ++index) {
                std::complex<Real> even = values[offset + index];
                std::complex<Real> odd = values[offset + index + half] * root;
                values[offset + index] = even + odd;
                values[offset + index + half] = even - odd;
                root *= step;
            }
        }
    }

    if (inverse) {
        Real inverse_size = Real(1) / Real(size);
        for (auto& value : values) value *= inverse_size;
    }
}

template <std::floating_point Real>
std::vector<std::complex<Real>> complex_convolution(const std::vector<std::complex<Real>>& first,
                                                    const std::vector<std::complex<Real>>& second) {
    if (first.empty() || second.empty()) return {};
    std::size_t result_size = first.size() + second.size() - 1;
    std::size_t size = std::bit_ceil(result_size);
    std::vector<std::complex<Real>> transformed_first(size);
    std::vector<std::complex<Real>> transformed_second(size);
    std::copy(first.begin(), first.end(), transformed_first.begin());
    std::copy(second.begin(), second.end(), transformed_second.begin());

    fft(transformed_first, false);
    fft(transformed_second, false);
    for (std::size_t index = 0; index < size; ++index) {
        transformed_first[index] *= transformed_second[index];
    }
    fft(transformed_first, true);
    transformed_first.resize(result_size);
    return transformed_first;
}

}  // namespace floating_point_convolution_detail

// Convolution of complex floating-point coefficients.
template <std::floating_point Real>
std::vector<std::complex<Real>> convolution_fft(const std::vector<std::complex<Real>>& first,
                                                const std::vector<std::complex<Real>>& second) {
    return floating_point_convolution_detail::complex_convolution(first, second);
}

// Convolution of real floating-point coefficients.
template <std::floating_point Real>
std::vector<Real> convolution_fft(const std::vector<Real>& first, const std::vector<Real>& second) {
    if (first.empty() || second.empty()) return {};
    std::vector<std::complex<Real>> complex_first(first.size());
    std::vector<std::complex<Real>> complex_second(second.size());
    for (std::size_t index = 0; index < first.size(); ++index) {
        complex_first[index] = std::complex<Real>(first[index], 0);
    }
    for (std::size_t index = 0; index < second.size(); ++index) {
        complex_second[index] = std::complex<Real>(second[index], 0);
    }
    auto result = floating_point_convolution_detail::complex_convolution(complex_first, complex_second);
    std::vector<Real> real_result(result.size());
    for (std::size_t index = 0; index < result.size(); ++index) {
        real_result[index] = result[index].real();
    }
    return real_result;
}

// Uses long-double FFT and rounds each coefficient to the nearest integer.
template <std::integral Integer>
std::vector<Integer> convolution_round(const std::vector<Integer>& first, const std::vector<Integer>& second) {
    if (first.empty() || second.empty()) return {};
    std::vector<long double> real_first(first.begin(), first.end());
    std::vector<long double> real_second(second.begin(), second.end());
    std::vector<long double> real_result = convolution_fft(real_first, real_second);
    std::vector<Integer> result(real_result.size());
    for (std::size_t index = 0; index < result.size(); ++index) {
        result[index] = static_cast<Integer>(std::round(real_result[index]));
    }
    return result;
}

}  // namespace fps
}  // namespace m1une


#line 4 "verify/math/fps/floating_point_convolution.test.cpp"

#line 6 "verify/math/fps/floating_point_convolution.test.cpp"
#include <cassert>
#line 9 "verify/math/fps/floating_point_convolution.test.cpp"
#include <cstdint>
#line 1 "utilities/fast_io.hpp"



#line 5 "utilities/fast_io.hpp"
#include <array>
#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 17 "utilities/fast_io.hpp"
#include <utility>
#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 12 "verify/math/fps/floating_point_convolution.test.cpp"

namespace {

template <class T>
std::vector<T> naive(
    const std::vector<T>& first,
    const std::vector<T>& second
) {
    if (first.empty() || second.empty()) return {};
    std::vector<T> result(first.size() + second.size() - 1);
    for (std::size_t i = 0; i < first.size(); ++i) {
        for (std::size_t j = 0; j < second.size(); ++j) {
            result[i + j] += first[i] * second[j];
        }
    }
    return result;
}

void test_fixed() {
    std::vector<double> first = {1.5, 2.0};
    std::vector<double> second = {3.0, -1.0};
    auto result = m1une::fps::convolution_fft(first, second);
    assert(std::fabs(result[0] - 4.5) < 1e-12);
    assert(std::fabs(result[1] - 4.5) < 1e-12);
    assert(std::fabs(result[2] + 2.0) < 1e-12);

    std::vector<std::complex<double>> complex_first;
    complex_first.emplace_back(1, 2);
    complex_first.emplace_back(3, -1);
    std::vector<std::complex<double>> complex_second;
    complex_second.emplace_back(-2, 1);
    complex_second.emplace_back(0, 4);
    auto complex_result =
        m1une::fps::convolution_fft(complex_first, complex_second);
    auto complex_expected = naive(complex_first, complex_second);
    for (std::size_t index = 0; index < complex_result.size(); ++index) {
        assert(std::abs(complex_result[index] - complex_expected[index]) < 1e-11);
    }

    std::vector<long long> integers = {1, -2, 3, 4};
    assert(
        m1une::fps::convolution_round(integers, integers)
        == naive(integers, integers)
    );
    assert(
        m1une::fps::convolution_fft(
            std::vector<double>(),
            std::vector<double>{1}
        ).empty()
    );
}

void test_randomized() {
    std::uint64_t state = 1409;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; ++trial) {
        int first_size = int(random() % 100);
        int second_size = int(random() % 100);
        std::vector<long double> first(first_size);
        std::vector<long double> second(second_size);
        for (long double& value : first) {
            value =
                (static_cast<long long>(random() % 2001) - 1000) / 100.0L;
        }
        for (long double& value : second) {
            value =
                (static_cast<long long>(random() % 2001) - 1000) / 100.0L;
        }
        auto actual = m1une::fps::convolution_fft(first, second);
        auto expected = naive(first, second);
        for (std::size_t index = 0; index < actual.size(); ++index) {
            long double magnitude = 0;
            for (std::size_t first_index = 0; first_index < first.size(); ++first_index) {
                if (index < first_index) continue;
                std::size_t second_index = index - first_index;
                if (second_index < second.size()) {
                    magnitude += std::fabs(
                        first[first_index] * second[second_index]
                    );
                }
            }
            [[maybe_unused]] long double tolerance =
                1e-12L * std::max(1.0L, magnitude);
            assert(std::fabs(actual[index] - expected[index]) <= tolerance);
        }

        std::vector<long long> integer_first(first_size);
        std::vector<long long> integer_second(second_size);
        for (long long& value : integer_first) {
            value = static_cast<long long>(random() % 2001) - 1000;
        }
        for (long long& value : integer_second) {
            value = static_cast<long long>(random() % 2001) - 1000;
        }
        assert(
            m1une::fps::convolution_round(integer_first, integer_second)
            == naive(integer_first, integer_second)
        );
    }
}

}  // namespace

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

    test_fixed();
    test_randomized();

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