m1une's library

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

View on GitHub

:heavy_check_mark: verify/math/stern_brocot_tree.test.cpp

Depends on

Code

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

#include "../../math/stern_brocot_tree.hpp"

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

namespace {

using Fraction = m1une::math::Rational<long long>;
using Direction = m1une::math::SternBrocotDirection;
using Path = m1une::math::SternBrocotPath;

std::vector<Direction> expand(const Path& path) {
    std::vector<Direction> result;
    for (const auto& run : path.runs) {
        for (uint64_t i = 0; i < run.count; i++) result.push_back(run.direction);
    }
    return result;
}

void test_features() {
    Path root = m1une::math::stern_brocot_path(1LL, 1LL);
    assert(root.empty());
    assert(root.depth() == 0);
    assert(m1une::math::stern_brocot_decode<long long>(root) == Fraction(1, 1));

    Path path = m1une::math::stern_brocot_path(5LL, 3LL);
    assert(path.depth() == 3);
    assert(path.runs.size() == 3);
    m1une::math::SternBrocotRun right_once{Direction::Right, 1};
    m1une::math::SternBrocotRun left_once{Direction::Left, 1};
    assert(path.runs[0] == right_once);
    assert(path.runs[1] == left_once);
    assert(path.runs[2] == right_once);
    assert(m1une::math::stern_brocot_decode<long long>(path) == Fraction(5, 3));
    assert(m1une::math::stern_brocot_parent(5LL, 3LL) == Fraction(3, 2));
    assert(m1une::math::stern_brocot_ancestor(5LL, 3LL, 2) == Fraction(2, 1));
    assert(m1une::math::stern_brocot_ancestor(5LL, 3LL, 3) == Fraction(1, 1));

    assert(
        m1une::math::stern_brocot_move(1LL, 1LL, Direction::Left, 4)
        == Fraction(1, 5)
    );
    assert(
        m1une::math::stern_brocot_move(1LL, 1LL, Direction::Right, 4)
        == Fraction(5, 1)
    );
    assert(m1une::math::stern_brocot_lca(5LL, 3LL, 7LL, 4LL) == Fraction(5, 3));
    assert(m1une::math::stern_brocot_lca(2LL, 5LL, 5LL, 2LL) == Fraction(1, 1));

    auto bounds = m1une::math::stern_brocot_bounds<long long>(path);
    std::pair<long long, long long> expected_left(3, 2);
    std::pair<long long, long long> expected_right(2, 1);
    assert(bounds.left == expected_left);
    assert(bounds.right == expected_right);
}

void test_exhaustive() {
    std::vector<Fraction> fractions;
    std::vector<Path> paths;
    for (long long numerator = 1; numerator <= 100; numerator++) {
        for (long long denominator = 1; denominator <= 100; denominator++) {
            if (std::gcd(numerator, denominator) != 1) continue;
            Path path = m1une::math::stern_brocot_path(numerator, denominator);
            Fraction decoded = m1une::math::stern_brocot_decode<long long>(path);
            assert(decoded == Fraction(numerator, denominator));
            assert(path.depth() == m1une::math::stern_brocot_depth(numerator, denominator));

            auto bounds = m1une::math::stern_brocot_bounds<long long>(path);
            auto [left_numerator, left_denominator] = bounds.left;
            auto [right_numerator, right_denominator] = bounds.right;
            assert(
                __int128(numerator) * left_denominator
                - __int128(left_numerator) * denominator
                == 1
            );
            assert(
                __int128(right_numerator) * denominator
                - __int128(numerator) * right_denominator
                == 1
            );

            auto left_child = m1une::math::stern_brocot_move(
                numerator,
                denominator,
                Direction::Left
            );
            auto right_child = m1une::math::stern_brocot_move(
                numerator,
                denominator,
                Direction::Right
            );
            assert(
                left_child
                == Fraction(
                    numerator + left_numerator,
                    denominator + left_denominator
                )
            );
            assert(
                right_child
                == Fraction(
                    numerator + right_numerator,
                    denominator + right_denominator
                )
            );
            assert(
                m1une::math::stern_brocot_parent(
                    left_child.numerator(),
                    left_child.denominator()
                )
                == Fraction(numerator, denominator)
            );
            assert(
                m1une::math::stern_brocot_parent(
                    right_child.numerator(),
                    right_child.denominator()
                )
                == Fraction(numerator, denominator)
            );

            fractions.emplace_back(numerator, denominator);
            paths.push_back(path);
        }
    }

    for (int first = 0; first < int(paths.size()); first += 17) {
        for (int second = 0; second < int(paths.size()); second += 23) {
            std::vector<Direction> first_expanded = expand(paths[first]);
            std::vector<Direction> second_expanded = expand(paths[second]);
            int common = 0;
            while (
                common < int(first_expanded.size()) &&
                common < int(second_expanded.size()) &&
                first_expanded[common] == second_expanded[common]
            ) {
                common++;
            }
            Path expected;
            for (int i = 0; i < common; i++) expected.push(first_expanded[i]);
            Path actual = m1une::math::stern_brocot_lca_path(paths[first], paths[second]);
            assert(actual == expected);
            assert(
                m1une::math::stern_brocot_lca(
                    fractions[first].numerator(),
                    fractions[first].denominator(),
                    fractions[second].numerator(),
                    fractions[second].denominator()
                )
                == m1une::math::stern_brocot_decode<long long>(expected)
            );
        }
    }
}

}  // namespace

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

    test_features();
    test_exhaustive();

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

#line 1 "math/stern_brocot_tree.hpp"



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

#line 1 "math/rational.hpp"



#include <algorithm>
#line 6 "math/rational.hpp"
#include <cmath>
#include <compare>
#line 9 "math/rational.hpp"
#include <iostream>
#line 11 "math/rational.hpp"
#include <sstream>
#include <string>
#include <type_traits>
#line 15 "math/rational.hpp"

namespace m1une {
namespace math {

namespace rational_detail {

template <class T>
concept IntegerLike =
    std::signed_integral<T> ||
    (!std::integral<T> && std::copyable<T> && requires(T first, T second) {
        T(0);
        T(1);
        { -first } -> std::same_as<T>;
        { first + second } -> std::same_as<T>;
        { first - second } -> std::same_as<T>;
        { first * second } -> std::same_as<T>;
        { first / second } -> std::same_as<T>;
        { first % second } -> std::same_as<T>;
        { first += second } -> std::same_as<T&>;
        { first -= second } -> std::same_as<T&>;
        { first /= second } -> std::same_as<T&>;
        { first == second } -> std::convertible_to<bool>;
        { first < second } -> std::convertible_to<bool>;
    });

}  // namespace rational_detail

template <rational_detail::IntegerLike T = long long>
struct Rational {
    static_assert(!std::signed_integral<T> || sizeof(T) <= sizeof(long long));

   private:
    static constexpr bool BUILTIN_INTEGER = std::signed_integral<T>;
    using Wide = std::conditional_t<BUILTIN_INTEGER, __int128_t, T>;
    using Magnitude = std::conditional_t<BUILTIN_INTEGER, __uint128_t, T>;

    T _numerator;
    T _denominator;

    static constexpr Magnitude magnitude(Wide value) {
        if constexpr (BUILTIN_INTEGER) {
            if (value < 0) {
                return static_cast<Magnitude>(-(value + 1)) + 1;
            }
            return static_cast<Magnitude>(value);
        } else {
            return value < 0 ? -value : value;
        }
    }

    static constexpr Magnitude gcd(Magnitude first, Magnitude second) {
        while (second != 0) {
            Magnitude remainder = first % second;
            first = second;
            second = remainder;
        }
        return first;
    }

    static constexpr T narrow(Wide value) {
        if constexpr (BUILTIN_INTEGER) {
            assert(Wide(std::numeric_limits<T>::min()) <= value);
            assert(value <= Wide(std::numeric_limits<T>::max()));
            return static_cast<T>(value);
        } else {
            return value;
        }
    }

    constexpr void assign_normalized(Wide numerator, Wide denominator) {
        assert(denominator != 0);
        if (numerator == 0) {
            _numerator = 0;
            _denominator = 1;
            return;
        }

        Magnitude divisor = gcd(magnitude(numerator), magnitude(denominator));
        numerator /= static_cast<Wide>(divisor);
        denominator /= static_cast<Wide>(divisor);
        if (denominator < 0) {
            numerator = -numerator;
            denominator = -denominator;
        }
        _numerator = narrow(numerator);
        _denominator = narrow(denominator);
    }

    static constexpr Rational from_wide(Wide numerator, Wide denominator) {
        Rational result;
        result.assign_normalized(numerator, denominator);
        return result;
    }

    static std::pair<long double, long long> decimal_scientific(const T& value) {
        std::ostringstream output;
        output << value;
        const std::string text = output.str();
        std::size_t begin = 0;
        int sign = 1;
        if (!text.empty() && (text[0] == '-' || text[0] == '+')) {
            if (text[0] == '-') sign = -1;
            begin = 1;
        }
        while (begin < text.size() && text[begin] == '0') ++begin;
        if (begin == text.size()) return std::make_pair(0.0L, 0LL);

        constexpr int DIGITS = std::numeric_limits<long double>::digits10 + 1;
        const std::size_t used = std::min<std::size_t>(DIGITS, text.size() - begin);
        long double significand = 0;
        for (std::size_t i = 0; i < used; ++i) {
            assert('0' <= text[begin + i] && text[begin + i] <= '9');
            significand = significand * 10 + (text[begin + i] - '0');
        }
        for (std::size_t i = 1; i < used; ++i) significand /= 10;
        const long long exponent = static_cast<long long>(text.size() - begin - 1);
        return std::make_pair(sign * significand, exponent);
    }

   public:
    constexpr Rational() : _numerator(0), _denominator(1) {}

    constexpr Rational(T integer) : _numerator(integer), _denominator(1) {}

    template <std::integral U>
        requires std::constructible_from<T, U> &&
                 (!std::same_as<std::remove_cv_t<U>, T>)
    constexpr Rational(U integer) : Rational(T(integer)) {}

    constexpr Rational(T numerator, T denominator) {
        assign_normalized(Wide(numerator), Wide(denominator));
    }

    constexpr T numerator() const {
        return _numerator;
    }

    constexpr T denominator() const {
        return _denominator;
    }

    constexpr bool is_integer() const {
        return _denominator == 1;
    }

    constexpr int sign() const {
        return (_numerator > 0) - (_numerator < 0);
    }

    constexpr Rational reciprocal() const {
        assert(_numerator != 0);
        return from_wide(Wide(_denominator), Wide(_numerator));
    }

    constexpr Rational abs() const {
        return _numerator < 0 ? -*this : *this;
    }

    constexpr long double to_long_double() const
        requires requires(const T& value) { static_cast<long double>(value); }
    {
        return static_cast<long double>(_numerator) / static_cast<long double>(_denominator);
    }

    long double to_long_double() const
        requires(!requires(const T& value) { static_cast<long double>(value); })
    {
        const auto [numerator, numerator_exponent] = decimal_scientific(_numerator);
        const auto [denominator, denominator_exponent] = decimal_scientific(_denominator);
        return numerator / denominator *
               std::pow(10.0L, numerator_exponent - denominator_exponent);
    }

    template <std::floating_point F>
    explicit constexpr operator F() const
        requires requires(const T& value) { static_cast<long double>(value); }
    {
        return static_cast<F>(to_long_double());
    }

    template <std::floating_point F>
    explicit operator F() const
        requires(!requires(const T& value) { static_cast<long double>(value); })
    {
        return static_cast<F>(to_long_double());
    }

    constexpr T trunc() const {
        return _numerator / _denominator;
    }

    constexpr T floor() const {
        T quotient = _numerator / _denominator;
        if (_numerator < 0 && _numerator % _denominator != 0) quotient -= T(1);
        return quotient;
    }

    constexpr T ceil() const {
        T quotient = _numerator / _denominator;
        if (0 < _numerator && _numerator % _denominator != 0) quotient += T(1);
        return quotient;
    }

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

    constexpr Rational operator-() const {
        return from_wide(-Wide(_numerator), Wide(_denominator));
    }

    constexpr Rational& operator+=(const Rational& other) {
        Magnitude common =
            gcd(static_cast<Magnitude>(_denominator), static_cast<Magnitude>(other._denominator));
        Wide left_scale = Wide(other._denominator) / static_cast<Wide>(common);
        Wide right_scale = Wide(_denominator) / static_cast<Wide>(common);
        Wide numerator =
            Wide(_numerator) * left_scale + Wide(other._numerator) * right_scale;

        // With both operands already reduced, every factor shared by the new
        // numerator and denominator must divide `common`.  Restricting the
        // second gcd to that value avoids a full-size gcd against the product
        // of both denominators, which is especially important for BigInt.
        Magnitude reduction = common == Magnitude(1)
                                  ? Magnitude(1)
                                  : gcd(magnitude(numerator), common);
        if (reduction != Magnitude(1)) {
            numerator /= static_cast<Wide>(reduction);
        }
        Wide remaining_denominator = Wide(other._denominator);
        if (reduction != Magnitude(1)) {
            remaining_denominator /= static_cast<Wide>(reduction);
        }
        _numerator = narrow(numerator);
        _denominator = narrow(right_scale * remaining_denominator);
        return *this;
    }

    constexpr Rational& operator-=(const Rational& other) {
        return *this += -other;
    }

    constexpr Rational& operator*=(const Rational& other) {
        Magnitude first_gcd = gcd(magnitude(Wide(_numerator)), static_cast<Magnitude>(other._denominator));
        Magnitude second_gcd = gcd(magnitude(Wide(other._numerator)), static_cast<Magnitude>(_denominator));
        assign_normalized((Wide(_numerator) / static_cast<Wide>(first_gcd)) *
                              (Wide(other._numerator) / static_cast<Wide>(second_gcd)),
                          (Wide(_denominator) / static_cast<Wide>(second_gcd)) *
                              (Wide(other._denominator) / static_cast<Wide>(first_gcd)));
        return *this;
    }

    constexpr Rational& operator/=(const Rational& other) {
        return *this *= other.reciprocal();
    }

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

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

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

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

    friend constexpr bool operator==(const Rational& left, const Rational& right) {
        return left._numerator == right._numerator && left._denominator == right._denominator;
    }

    friend constexpr std::strong_ordering operator<=>(const Rational& left, const Rational& right) {
        Wide first = Wide(left._numerator) * Wide(right._denominator);
        Wide second = Wide(right._numerator) * Wide(left._denominator);
        if (first < second) return std::strong_ordering::less;
        if (second < first) return std::strong_ordering::greater;
        return std::strong_ordering::equal;
    }

    friend std::ostream& operator<<(std::ostream& output, const Rational& value) {
        output << value._numerator;
        if (value._denominator != 1) {
            output << '/' << value._denominator;
        }
        return output;
    }

    friend std::istream& operator>>(std::istream& input, Rational& value) {
        std::string token;
        if (!(input >> token)) return input;

        std::size_t slash = token.find('/');
        if (slash != std::string::npos && token.find('/', slash + 1) != std::string::npos) {
            input.setstate(std::ios::failbit);
            return input;
        }

        T numerator = 0;
        T denominator = 1;
        std::istringstream numerator_input(token.substr(0, slash));
        if (!(numerator_input >> numerator) || numerator_input.peek() != std::char_traits<char>::eof()) {
            input.setstate(std::ios::failbit);
            return input;
        }
        if (slash != std::string::npos) {
            std::istringstream denominator_input(token.substr(slash + 1));
            if (!(denominator_input >> denominator) ||
                denominator_input.peek() != std::char_traits<char>::eof()) {
                input.setstate(std::ios::failbit);
                return input;
            }
        }
        value = Rational(numerator, denominator);
        return input;
    }
};

template <rational_detail::IntegerLike T>
constexpr Rational<T> abs(const Rational<T>& value) {
    return value.abs();
}

}  // namespace math
}  // namespace m1une

namespace std {

// Integer/rational common types already follow from implicit integer
// construction. Mixing a floating scalar explicitly chooses approximation.
template <m1une::math::rational_detail::IntegerLike T, floating_point F>
struct common_type<m1une::math::Rational<T>, F> {
    using type = long double;
};

template <floating_point F, m1une::math::rational_detail::IntegerLike T>
struct common_type<F, m1une::math::Rational<T>> {
    using type = long double;
};

}  // namespace std


#line 12 "math/stern_brocot_tree.hpp"

namespace m1une {
namespace math {

enum class SternBrocotDirection {
    Left,
    Right,
};

struct SternBrocotRun {
    SternBrocotDirection direction;
    uint64_t count;

    friend bool operator==(const SternBrocotRun&, const SternBrocotRun&) = default;
};

struct SternBrocotPath {
    std::vector<SternBrocotRun> runs;

    bool empty() const {
        return runs.empty();
    }

    uint64_t depth() const {
        uint64_t result = 0;
        for (const SternBrocotRun& run : runs) {
            assert(run.count <= std::numeric_limits<uint64_t>::max() - result);
            result += run.count;
        }
        return result;
    }

    void push(SternBrocotDirection direction, uint64_t count = 1) {
        if (count == 0) return;
        if (!runs.empty() && runs.back().direction == direction) {
            assert(count <= std::numeric_limits<uint64_t>::max() - runs.back().count);
            runs.back().count += count;
        } else {
            runs.push_back(SternBrocotRun{direction, count});
        }
    }

    bool move_up(uint64_t count = 1) {
        if (depth() < count) return false;
        while (count > 0) {
            SternBrocotRun& run = runs.back();
            uint64_t removed = run.count < count ? run.count : count;
            run.count -= removed;
            count -= removed;
            if (run.count == 0) runs.pop_back();
        }
        return true;
    }

    SternBrocotPath ancestor(uint64_t count) const {
        SternBrocotPath result = *this;
        [[maybe_unused]] bool valid = result.move_up(count);
        assert(valid);
        return result;
    }

    friend bool operator==(const SternBrocotPath&, const SternBrocotPath&) = default;
};

template <std::signed_integral T = long long>
struct SternBrocotBounds {
    std::pair<T, T> left;
    std::pair<T, T> right;
};

template <std::signed_integral T>
SternBrocotPath stern_brocot_path(T numerator, T denominator) {
    assert(0 < numerator);
    assert(0 < denominator);
    SternBrocotPath result;
    while (numerator != denominator) {
        if (numerator < denominator) {
            T count = (denominator - 1) / numerator;
            result.push(SternBrocotDirection::Left, uint64_t(count));
            denominator -= count * numerator;
        } else {
            T count = (numerator - 1) / denominator;
            result.push(SternBrocotDirection::Right, uint64_t(count));
            numerator -= count * denominator;
        }
    }
    assert(numerator == 1);
    return result;
}

template <std::signed_integral T = long long>
Rational<T> stern_brocot_decode(const SternBrocotPath& path) {
    using Wide = __int128_t;
    Wide left_numerator = 0;
    Wide left_denominator = 1;
    Wide right_numerator = 1;
    Wide right_denominator = 0;
    Wide numerator = 1;
    Wide denominator = 1;
    [[maybe_unused]] const Wide maximum = std::numeric_limits<T>::max();
    for (const SternBrocotRun& run : path.runs) {
        assert(run.count > 0);
        assert(Wide(run.count) <= maximum);
        Wide count = run.count;
        if (run.direction == SternBrocotDirection::Left) {
            right_numerator = numerator + (count - 1) * left_numerator;
            right_denominator = denominator + (count - 1) * left_denominator;
            numerator += count * left_numerator;
            denominator += count * left_denominator;
        } else {
            left_numerator = numerator + (count - 1) * right_numerator;
            left_denominator = denominator + (count - 1) * right_denominator;
            numerator += count * right_numerator;
            denominator += count * right_denominator;
        }
        assert(numerator <= maximum);
        assert(denominator <= maximum);
        assert(left_numerator <= maximum);
        assert(left_denominator <= maximum);
        assert(right_numerator <= maximum);
        assert(right_denominator <= maximum);
    }
    return Rational<T>(T(numerator), T(denominator));
}

template <std::signed_integral T>
uint64_t stern_brocot_depth(T numerator, T denominator) {
    return stern_brocot_path(numerator, denominator).depth();
}

inline SternBrocotPath stern_brocot_lca_path(
    const SternBrocotPath& first,
    const SternBrocotPath& second
) {
    SternBrocotPath result;
    int limit = int(first.runs.size() < second.runs.size() ? first.runs.size() : second.runs.size());
    for (int i = 0; i < limit; i++) {
        if (first.runs[i].direction != second.runs[i].direction) break;
        uint64_t common =
            first.runs[i].count < second.runs[i].count
                ? first.runs[i].count
                : second.runs[i].count;
        result.push(first.runs[i].direction, common);
        if (first.runs[i].count != second.runs[i].count) break;
    }
    return result;
}

template <std::signed_integral T>
Rational<T> stern_brocot_lca(
    T first_numerator,
    T first_denominator,
    T second_numerator,
    T second_denominator
) {
    SternBrocotPath path = stern_brocot_lca_path(
        stern_brocot_path(first_numerator, first_denominator),
        stern_brocot_path(second_numerator, second_denominator)
    );
    return stern_brocot_decode<T>(path);
}

template <std::signed_integral T>
Rational<T> stern_brocot_ancestor(T numerator, T denominator, uint64_t up) {
    SternBrocotPath path = stern_brocot_path(numerator, denominator);
    [[maybe_unused]] bool valid = path.move_up(up);
    assert(valid);
    return stern_brocot_decode<T>(path);
}

template <std::signed_integral T>
Rational<T> stern_brocot_parent(T numerator, T denominator) {
    return stern_brocot_ancestor(numerator, denominator, 1);
}

template <std::signed_integral T>
Rational<T> stern_brocot_move(
    T numerator,
    T denominator,
    SternBrocotDirection direction,
    uint64_t count = 1
) {
    SternBrocotPath path = stern_brocot_path(numerator, denominator);
    path.push(direction, count);
    return stern_brocot_decode<T>(path);
}

template <std::signed_integral T = long long>
SternBrocotBounds<T> stern_brocot_bounds(const SternBrocotPath& path) {
    using Wide = __int128_t;
    Wide left_numerator = 0;
    Wide left_denominator = 1;
    Wide right_numerator = 1;
    Wide right_denominator = 0;
    Wide numerator = 1;
    Wide denominator = 1;
    [[maybe_unused]] const Wide maximum = std::numeric_limits<T>::max();

    for (const SternBrocotRun& run : path.runs) {
        assert(run.count > 0);
        assert(Wide(run.count) <= maximum);
        Wide count = run.count;
        if (run.direction == SternBrocotDirection::Left) {
            right_numerator = numerator + (count - 1) * left_numerator;
            right_denominator = denominator + (count - 1) * left_denominator;
            numerator += count * left_numerator;
            denominator += count * left_denominator;
        } else {
            left_numerator = numerator + (count - 1) * right_numerator;
            left_denominator = denominator + (count - 1) * right_denominator;
            numerator += count * right_numerator;
            denominator += count * right_denominator;
        }
        assert(numerator <= maximum);
        assert(denominator <= maximum);
        assert(left_numerator <= maximum);
        assert(left_denominator <= maximum);
        assert(right_numerator <= maximum);
        assert(right_denominator <= maximum);
    }
    SternBrocotBounds<T> result;
    result.left = {T(left_numerator), T(left_denominator)};
    result.right = {T(right_numerator), T(right_denominator)};
    return result;
}

}  // namespace math
}  // namespace m1une


#line 4 "verify/math/stern_brocot_tree.test.cpp"

#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>
#line 15 "utilities/fast_io.hpp"
#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 8 "verify/math/stern_brocot_tree.test.cpp"
#include <numeric>
#line 11 "verify/math/stern_brocot_tree.test.cpp"

namespace {

using Fraction = m1une::math::Rational<long long>;
using Direction = m1une::math::SternBrocotDirection;
using Path = m1une::math::SternBrocotPath;

std::vector<Direction> expand(const Path& path) {
    std::vector<Direction> result;
    for (const auto& run : path.runs) {
        for (uint64_t i = 0; i < run.count; i++) result.push_back(run.direction);
    }
    return result;
}

void test_features() {
    Path root = m1une::math::stern_brocot_path(1LL, 1LL);
    assert(root.empty());
    assert(root.depth() == 0);
    assert(m1une::math::stern_brocot_decode<long long>(root) == Fraction(1, 1));

    Path path = m1une::math::stern_brocot_path(5LL, 3LL);
    assert(path.depth() == 3);
    assert(path.runs.size() == 3);
    m1une::math::SternBrocotRun right_once{Direction::Right, 1};
    m1une::math::SternBrocotRun left_once{Direction::Left, 1};
    assert(path.runs[0] == right_once);
    assert(path.runs[1] == left_once);
    assert(path.runs[2] == right_once);
    assert(m1une::math::stern_brocot_decode<long long>(path) == Fraction(5, 3));
    assert(m1une::math::stern_brocot_parent(5LL, 3LL) == Fraction(3, 2));
    assert(m1une::math::stern_brocot_ancestor(5LL, 3LL, 2) == Fraction(2, 1));
    assert(m1une::math::stern_brocot_ancestor(5LL, 3LL, 3) == Fraction(1, 1));

    assert(
        m1une::math::stern_brocot_move(1LL, 1LL, Direction::Left, 4)
        == Fraction(1, 5)
    );
    assert(
        m1une::math::stern_brocot_move(1LL, 1LL, Direction::Right, 4)
        == Fraction(5, 1)
    );
    assert(m1une::math::stern_brocot_lca(5LL, 3LL, 7LL, 4LL) == Fraction(5, 3));
    assert(m1une::math::stern_brocot_lca(2LL, 5LL, 5LL, 2LL) == Fraction(1, 1));

    auto bounds = m1une::math::stern_brocot_bounds<long long>(path);
    std::pair<long long, long long> expected_left(3, 2);
    std::pair<long long, long long> expected_right(2, 1);
    assert(bounds.left == expected_left);
    assert(bounds.right == expected_right);
}

void test_exhaustive() {
    std::vector<Fraction> fractions;
    std::vector<Path> paths;
    for (long long numerator = 1; numerator <= 100; numerator++) {
        for (long long denominator = 1; denominator <= 100; denominator++) {
            if (std::gcd(numerator, denominator) != 1) continue;
            Path path = m1une::math::stern_brocot_path(numerator, denominator);
            Fraction decoded = m1une::math::stern_brocot_decode<long long>(path);
            assert(decoded == Fraction(numerator, denominator));
            assert(path.depth() == m1une::math::stern_brocot_depth(numerator, denominator));

            auto bounds = m1une::math::stern_brocot_bounds<long long>(path);
            auto [left_numerator, left_denominator] = bounds.left;
            auto [right_numerator, right_denominator] = bounds.right;
            assert(
                __int128(numerator) * left_denominator
                - __int128(left_numerator) * denominator
                == 1
            );
            assert(
                __int128(right_numerator) * denominator
                - __int128(numerator) * right_denominator
                == 1
            );

            auto left_child = m1une::math::stern_brocot_move(
                numerator,
                denominator,
                Direction::Left
            );
            auto right_child = m1une::math::stern_brocot_move(
                numerator,
                denominator,
                Direction::Right
            );
            assert(
                left_child
                == Fraction(
                    numerator + left_numerator,
                    denominator + left_denominator
                )
            );
            assert(
                right_child
                == Fraction(
                    numerator + right_numerator,
                    denominator + right_denominator
                )
            );
            assert(
                m1une::math::stern_brocot_parent(
                    left_child.numerator(),
                    left_child.denominator()
                )
                == Fraction(numerator, denominator)
            );
            assert(
                m1une::math::stern_brocot_parent(
                    right_child.numerator(),
                    right_child.denominator()
                )
                == Fraction(numerator, denominator)
            );

            fractions.emplace_back(numerator, denominator);
            paths.push_back(path);
        }
    }

    for (int first = 0; first < int(paths.size()); first += 17) {
        for (int second = 0; second < int(paths.size()); second += 23) {
            std::vector<Direction> first_expanded = expand(paths[first]);
            std::vector<Direction> second_expanded = expand(paths[second]);
            int common = 0;
            while (
                common < int(first_expanded.size()) &&
                common < int(second_expanded.size()) &&
                first_expanded[common] == second_expanded[common]
            ) {
                common++;
            }
            Path expected;
            for (int i = 0; i < common; i++) expected.push(first_expanded[i]);
            Path actual = m1une::math::stern_brocot_lca_path(paths[first], paths[second]);
            assert(actual == expected);
            assert(
                m1une::math::stern_brocot_lca(
                    fractions[first].numerator(),
                    fractions[first].denominator(),
                    fractions[second].numerator(),
                    fractions[second].denominator()
                )
                == m1une::math::stern_brocot_decode<long long>(expected)
            );
        }
    }
}

}  // namespace

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

    test_features();
    test_exhaustive();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
Back to top page