verify/math/gcd_of_gaussian_integers.test.cpp

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

Depends on

• Gaussian Integer (math/gaussian_integer.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

#include "../../math/gaussian_integer.hpp"
#include "../../utilities/fast_io.hpp"

#include <cassert>
#include <cstdint>
#include <limits>
#include <tuple>

namespace {

using Gaussian = m1une::math::GaussianInteger<long long>;

constexpr bool compile_time_tests() {
    constexpr Gaussian first(5, 7);
    constexpr Gaussian second(2, -1);
    static_assert(first + second == Gaussian(7, 6));
    static_assert(first - second == Gaussian(3, 8));
    static_assert(first * second == Gaussian(17, 9));
    static_assert(first.conjugate() == Gaussian(5, -7));
    static_assert(first.norm() == 74);
    static_assert(Gaussian(0, 1).is_unit());
    static_assert(!Gaussian(1, 1).is_unit());

    constexpr auto division = first.divmod(second);
    static_assert(division.first * second + division.second == first);
    static_assert(division.second.norm() < second.norm());

    static_assert(Gaussian(2, 3).normalized() == Gaussian(2, 3));
    static_assert(Gaussian(2, -3).normalized() == Gaussian(3, 2));
    static_assert(Gaussian(-2, -3).normalized() == Gaussian(2, 3));
    static_assert(Gaussian(-2, 3).normalized() == Gaussian(3, 2));
    static_assert(Gaussian(0, 4).normalized() == Gaussian(4, 0));
    static_assert(Gaussian(0, -4).normalized() == Gaussian(4, 0));
    return true;
}

static_assert(compile_time_tests());

void test_division() {
    const long long minimum = std::numeric_limits<long long>::min();
    const Gaussian extreme(minimum, minimum);
    assert(extreme.norm() == (__uint128_t(1) << 127));
    const auto extreme_division = Gaussian(minimum, 0).divmod(Gaussian(1, 0));
    assert(extreme_division.first == Gaussian(minimum, 0));
    assert(extreme_division.second.is_zero());

    for (long long first_real = -12; first_real <= 12; first_real++) {
        for (long long first_imag = -12; first_imag <= 12; first_imag++) {
            const Gaussian first(first_real, first_imag);
            for (long long second_real = -5; second_real <= 5; second_real++) {
                for (long long second_imag = -5; second_imag <= 5; second_imag++) {
                    const Gaussian second(second_real, second_imag);
                    if (second.is_zero()) continue;
                    const auto [quotient, remainder] = first.divmod(second);
                    assert(quotient * second + remainder == first);
                    assert(remainder.norm() < second.norm());
                    assert(first / second == quotient);
                    assert(first % second == remainder);
                    assert(m1une::math::gaussian_divides(second, first) ==
                           remainder.is_zero());
                }
            }
        }
    }
}

void test_gcd() {
    std::uint64_t state = UINT64_C(0x84a71c39d5e602fb);
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    const Gaussian units[4] = {
        Gaussian(1, 0),
        Gaussian(0, 1),
        Gaussian(-1, 0),
        Gaussian(0, -1)
    };

    for (int trial = 0; trial < 10000; trial++) {
        const Gaussian first(
            static_cast<long long>(random() % 2000001) - 1000000,
            static_cast<long long>(random() % 2000001) - 1000000
        );
        const Gaussian second(
            static_cast<long long>(random() % 2000001) - 1000000,
            static_cast<long long>(random() % 2000001) - 1000000
        );

        const Gaussian gcd = m1une::math::gaussian_gcd(first, second);
        assert(gcd.is_zero() || (0 < gcd.real && 0 <= gcd.imag));
        assert(m1une::math::gaussian_divides(gcd, first));
        assert(m1une::math::gaussian_divides(gcd, second));

        const auto [extended_gcd, first_coefficient, second_coefficient] =
            m1une::math::extended_gaussian_gcd(first, second);
        assert(extended_gcd == gcd);
        assert(first * first_coefficient + second * second_coefficient == gcd);

        const Gaussian first_unit = units[random() % 4];
        const Gaussian second_unit = units[random() % 4];
        assert(m1une::math::gaussian_gcd(
                   first * first_unit,
                   second * second_unit
               ) == gcd);
        assert(m1une::math::gaussian_associates(gcd, gcd * first_unit));

        if (trial < 1000) {
            for (long long real = -5; real <= 5; real++) {
                for (long long imag = -5; imag <= 5; imag++) {
                    const Gaussian divisor(real, imag);
                    if (m1une::math::gaussian_divides(divisor, first) &&
                        m1une::math::gaussian_divides(divisor, second)) {
                        assert(m1une::math::gaussian_divides(divisor, gcd));
                    }
                }
            }
        }
    }

    const Gaussian zero;
    assert(m1une::math::gaussian_gcd(zero, zero) == zero);
    assert(m1une::math::gaussian_divides(zero, zero));
    assert(!m1une::math::gaussian_divides(zero, Gaussian(1)));
}

}  // namespace

int main() {
    test_division();
    test_gcd();

    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;
    int test_count;
    fast_input >> test_count;
    while (test_count--) {
        long long first_real, first_imag, second_real, second_imag;
        fast_input >> first_real >> first_imag >> second_real >> second_imag;
        const Gaussian gcd = m1une::math::gaussian_gcd(
            Gaussian(first_real, first_imag),
            Gaussian(second_real, second_imag)
        );
        fast_output << gcd.real << ' ' << gcd.imag << '\n';
    }
}

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

#line 1 "math/gaussian_integer.hpp"



#include <cassert>
#include <concepts>
#include <cstdint>
#include <limits>
#include <tuple>
#include <utility>

namespace m1une {
namespace math {

template <std::signed_integral T = long long>
struct GaussianInteger {
    static_assert(sizeof(T) <= sizeof(long long));

    using value_type = T;
    using norm_type = __uint128_t;

    T real;
    T imag;

   private:
    using wide_type = __int128_t;

    struct SignedMagnitude {
        norm_type magnitude;
        bool negative;
    };

    static constexpr norm_type magnitude(T value) {
        const wide_type wide = value;
        if (wide < 0) {
            return static_cast<norm_type>(-(wide + 1)) + 1;
        }
        return static_cast<norm_type>(wide);
    }

    static constexpr SignedMagnitude signed_value(T value) {
        return SignedMagnitude{magnitude(value), value < 0};
    }

    static constexpr SignedMagnitude negate(SignedMagnitude value) {
        if (value.magnitude != 0) value.negative = !value.negative;
        return value;
    }

    static constexpr SignedMagnitude add_signed(
        SignedMagnitude first,
        SignedMagnitude second
    ) {
        if (first.negative == second.negative) {
            return SignedMagnitude{
                first.magnitude + second.magnitude,
                first.negative
            };
        }
        if (first.magnitude < second.magnitude) {
            return SignedMagnitude{
                second.magnitude - first.magnitude,
                second.negative
            };
        }
        return SignedMagnitude{
            first.magnitude - second.magnitude,
            first.magnitude == second.magnitude ? false : first.negative
        };
    }

    static constexpr SignedMagnitude product(T first, T second) {
        const norm_type result = magnitude(first) * magnitude(second);
        return SignedMagnitude{
            result,
            result != 0 && ((first < 0) != (second < 0))
        };
    }

    static constexpr T narrow(SignedMagnitude value) {
        const norm_type maximum =
            static_cast<norm_type>(std::numeric_limits<T>::max());
        if (!value.negative) {
            assert(value.magnitude <= maximum);
            return static_cast<T>(value.magnitude);
        }

        assert(value.magnitude <= maximum + 1);
        if (value.magnitude == maximum + 1) {
            return std::numeric_limits<T>::min();
        }
        return static_cast<T>(-static_cast<wide_type>(value.magnitude));
    }

    static constexpr std::pair<SignedMagnitude, SignedMagnitude>
    product_components(
        const GaussianInteger& first,
        const GaussianInteger& second
    ) {
        const SignedMagnitude product_real = add_signed(
            product(first.real, second.real),
            negate(product(first.imag, second.imag))
        );
        const SignedMagnitude product_imag = add_signed(
            product(first.real, second.imag),
            product(first.imag, second.real)
        );
        return {product_real, product_imag};
    }

    static constexpr T round_ratio(
        SignedMagnitude numerator,
        norm_type denominator
    ) {
        assert(denominator != 0);
        norm_type quotient = numerator.magnitude / denominator;
        const norm_type remainder = numerator.magnitude % denominator;
        if (remainder + remainder >= denominator) quotient++;
        return narrow(SignedMagnitude{quotient, numerator.negative});
    }

   public:
    constexpr GaussianInteger() : real(0), imag(0) {}

    constexpr GaussianInteger(T real_) : real(real_), imag(0) {}

    constexpr GaussianInteger(T real_, T imag_)
        : real(real_), imag(imag_) {}

    constexpr bool is_zero() const {
        return real == 0 && imag == 0;
    }

    constexpr bool is_unit() const {
        return norm() == 1;
    }

    constexpr norm_type norm() const {
        const norm_type real_magnitude = magnitude(real);
        const norm_type imag_magnitude = magnitude(imag);
        return real_magnitude * real_magnitude +
               imag_magnitude * imag_magnitude;
    }

    constexpr GaussianInteger conjugate() const {
        return GaussianInteger(real, narrow(negate(signed_value(imag))));
    }

    constexpr GaussianInteger normalizing_unit() const {
        if (is_zero() || (0 < real && 0 <= imag)) {
            return GaussianInteger(1, 0);
        }
        if (0 < real) return GaussianInteger(0, 1);
        if (real < 0 && imag <= 0) return GaussianInteger(-1, 0);
        if (real < 0) return GaussianInteger(0, -1);
        if (0 < imag) return GaussianInteger(0, -1);
        return GaussianInteger(0, 1);
    }

    constexpr GaussianInteger normalized() const {
        return normalizing_unit() * *this;
    }

    constexpr std::pair<GaussianInteger, GaussianInteger> divmod(
        const GaussianInteger& divisor
    ) const {
        const norm_type divisor_norm = divisor.norm();
        assert(divisor_norm != 0);

        const SignedMagnitude numerator_real = add_signed(
            product(real, divisor.real),
            product(imag, divisor.imag)
        );
        const SignedMagnitude numerator_imag = add_signed(
            product(imag, divisor.real),
            negate(product(real, divisor.imag))
        );
        const GaussianInteger quotient(
            round_ratio(numerator_real, divisor_norm),
            round_ratio(numerator_imag, divisor_norm)
        );

        const auto product = product_components(quotient, divisor);
        const GaussianInteger remainder(
            narrow(add_signed(signed_value(real), negate(product.first))),
            narrow(add_signed(signed_value(imag), negate(product.second)))
        );
        assert(remainder.norm() < divisor_norm);
        return {quotient, remainder};
    }

    constexpr GaussianInteger operator+() const {
        return *this;
    }

    constexpr GaussianInteger operator-() const {
        return GaussianInteger(
            narrow(negate(signed_value(real))),
            narrow(negate(signed_value(imag)))
        );
    }

    constexpr GaussianInteger& operator+=(const GaussianInteger& other) {
        real = narrow(add_signed(signed_value(real), signed_value(other.real)));
        imag = narrow(add_signed(signed_value(imag), signed_value(other.imag)));
        return *this;
    }

    constexpr GaussianInteger& operator-=(const GaussianInteger& other) {
        real = narrow(add_signed(
            signed_value(real),
            negate(signed_value(other.real))
        ));
        imag = narrow(add_signed(
            signed_value(imag),
            negate(signed_value(other.imag))
        ));
        return *this;
    }

    constexpr GaussianInteger& operator*=(const GaussianInteger& other) {
        const auto result = product_components(*this, other);
        real = narrow(result.first);
        imag = narrow(result.second);
        return *this;
    }

    constexpr GaussianInteger& operator/=(const GaussianInteger& other) {
        *this = divmod(other).first;
        return *this;
    }

    constexpr GaussianInteger& operator%=(const GaussianInteger& other) {
        *this = divmod(other).second;
        return *this;
    }

    friend constexpr GaussianInteger operator+(
        GaussianInteger left,
        const GaussianInteger& right
    ) {
        return left += right;
    }

    friend constexpr GaussianInteger operator-(
        GaussianInteger left,
        const GaussianInteger& right
    ) {
        return left -= right;
    }

    friend constexpr GaussianInteger operator*(
        GaussianInteger left,
        const GaussianInteger& right
    ) {
        return left *= right;
    }

    friend constexpr GaussianInteger operator/(
        GaussianInteger left,
        const GaussianInteger& right
    ) {
        return left /= right;
    }

    friend constexpr GaussianInteger operator%(
        GaussianInteger left,
        const GaussianInteger& right
    ) {
        return left %= right;
    }

    friend constexpr bool operator==(
        const GaussianInteger& first,
        const GaussianInteger& second
    ) = default;
};

template <std::signed_integral T>
constexpr bool gaussian_divides(
    const GaussianInteger<T>& divisor,
    const GaussianInteger<T>& value
) {
    if (divisor.is_zero()) return value.is_zero();
    return (value % divisor).is_zero();
}

template <std::signed_integral T>
constexpr bool gaussian_associates(
    const GaussianInteger<T>& first,
    const GaussianInteger<T>& second
) {
    return first.normalized() == second.normalized();
}

template <std::signed_integral T>
constexpr GaussianInteger<T> gaussian_gcd(
    GaussianInteger<T> first,
    GaussianInteger<T> second
) {
    while (!second.is_zero()) {
        first %= second;
        std::swap(first, second);
    }
    return first.normalized();
}

template <std::signed_integral T>
constexpr std::tuple<
    GaussianInteger<T>,
    GaussianInteger<T>,
    GaussianInteger<T>
> extended_gaussian_gcd(
    GaussianInteger<T> first,
    GaussianInteger<T> second
) {
    using G = GaussianInteger<T>;
    G old_remainder = first;
    G remainder = second;
    G old_first_coefficient(1);
    G first_coefficient(0);
    G old_second_coefficient(0);
    G second_coefficient(1);

    while (!remainder.is_zero()) {
        const G quotient = old_remainder / remainder;

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

        next = old_first_coefficient - quotient * first_coefficient;
        old_first_coefficient = first_coefficient;
        first_coefficient = next;

        next = old_second_coefficient - quotient * second_coefficient;
        old_second_coefficient = second_coefficient;
        second_coefficient = next;
    }

    const G unit = old_remainder.normalizing_unit();
    return {
        unit * old_remainder,
        unit * old_first_coefficient,
        unit * old_second_coefficient
    };
}

}  // namespace math
}  // namespace m1une


#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#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>
#include <type_traits>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#include <vector>

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 5 "verify/math/gcd_of_gaussian_integers.test.cpp"

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

namespace {

using Gaussian = m1une::math::GaussianInteger<long long>;

constexpr bool compile_time_tests() {
    constexpr Gaussian first(5, 7);
    constexpr Gaussian second(2, -1);
    static_assert(first + second == Gaussian(7, 6));
    static_assert(first - second == Gaussian(3, 8));
    static_assert(first * second == Gaussian(17, 9));
    static_assert(first.conjugate() == Gaussian(5, -7));
    static_assert(first.norm() == 74);
    static_assert(Gaussian(0, 1).is_unit());
    static_assert(!Gaussian(1, 1).is_unit());

    constexpr auto division = first.divmod(second);
    static_assert(division.first * second + division.second == first);
    static_assert(division.second.norm() < second.norm());

    static_assert(Gaussian(2, 3).normalized() == Gaussian(2, 3));
    static_assert(Gaussian(2, -3).normalized() == Gaussian(3, 2));
    static_assert(Gaussian(-2, -3).normalized() == Gaussian(2, 3));
    static_assert(Gaussian(-2, 3).normalized() == Gaussian(3, 2));
    static_assert(Gaussian(0, 4).normalized() == Gaussian(4, 0));
    static_assert(Gaussian(0, -4).normalized() == Gaussian(4, 0));
    return true;
}

static_assert(compile_time_tests());

void test_division() {
    const long long minimum = std::numeric_limits<long long>::min();
    const Gaussian extreme(minimum, minimum);
    assert(extreme.norm() == (__uint128_t(1) << 127));
    const auto extreme_division = Gaussian(minimum, 0).divmod(Gaussian(1, 0));
    assert(extreme_division.first == Gaussian(minimum, 0));
    assert(extreme_division.second.is_zero());

    for (long long first_real = -12; first_real <= 12; first_real++) {
        for (long long first_imag = -12; first_imag <= 12; first_imag++) {
            const Gaussian first(first_real, first_imag);
            for (long long second_real = -5; second_real <= 5; second_real++) {
                for (long long second_imag = -5; second_imag <= 5; second_imag++) {
                    const Gaussian second(second_real, second_imag);
                    if (second.is_zero()) continue;
                    const auto [quotient, remainder] = first.divmod(second);
                    assert(quotient * second + remainder == first);
                    assert(remainder.norm() < second.norm());
                    assert(first / second == quotient);
                    assert(first % second == remainder);
                    assert(m1une::math::gaussian_divides(second, first) ==
                           remainder.is_zero());
                }
            }
        }
    }
}

void test_gcd() {
    std::uint64_t state = UINT64_C(0x84a71c39d5e602fb);
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    const Gaussian units[4] = {
        Gaussian(1, 0),
        Gaussian(0, 1),
        Gaussian(-1, 0),
        Gaussian(0, -1)
    };

    for (int trial = 0; trial < 10000; trial++) {
        const Gaussian first(
            static_cast<long long>(random() % 2000001) - 1000000,
            static_cast<long long>(random() % 2000001) - 1000000
        );
        const Gaussian second(
            static_cast<long long>(random() % 2000001) - 1000000,
            static_cast<long long>(random() % 2000001) - 1000000
        );

        const Gaussian gcd = m1une::math::gaussian_gcd(first, second);
        assert(gcd.is_zero() || (0 < gcd.real && 0 <= gcd.imag));
        assert(m1une::math::gaussian_divides(gcd, first));
        assert(m1une::math::gaussian_divides(gcd, second));

        const auto [extended_gcd, first_coefficient, second_coefficient] =
            m1une::math::extended_gaussian_gcd(first, second);
        assert(extended_gcd == gcd);
        assert(first * first_coefficient + second * second_coefficient == gcd);

        const Gaussian first_unit = units[random() % 4];
        const Gaussian second_unit = units[random() % 4];
        assert(m1une::math::gaussian_gcd(
                   first * first_unit,
                   second * second_unit
               ) == gcd);
        assert(m1une::math::gaussian_associates(gcd, gcd * first_unit));

        if (trial < 1000) {
            for (long long real = -5; real <= 5; real++) {
                for (long long imag = -5; imag <= 5; imag++) {
                    const Gaussian divisor(real, imag);
                    if (m1une::math::gaussian_divides(divisor, first) &&
                        m1une::math::gaussian_divides(divisor, second)) {
                        assert(m1une::math::gaussian_divides(divisor, gcd));
                    }
                }
            }
        }
    }

    const Gaussian zero;
    assert(m1une::math::gaussian_gcd(zero, zero) == zero);
    assert(m1une::math::gaussian_divides(zero, zero));
    assert(!m1une::math::gaussian_divides(zero, Gaussian(1)));
}

}  // namespace

int main() {
    test_division();
    test_gcd();

    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;
    int test_count;
    fast_input >> test_count;
    while (test_count--) {
        long long first_real, first_imag, second_real, second_imag;
        fast_input >> first_real >> first_imag >> second_real >> second_imag;
        const Gaussian gcd = m1une::math::gaussian_gcd(
            Gaussian(first_real, first_imag),
            Gaussian(second_real, second_imag)
        );
        fast_output << gcd.real << ' ' << gcd.imag << '\n';
    }
}

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