Frac —— 分数四则运算
=======================

    .. code-block:: CPP

        template<class T>
        struct Frac {
            T num;
            T den;
            Frac(T num_, T den_) : num(num_), den(den_) {
                if (den < 0) {
                    den = -den;
                    num = -num;
                }
            }
            Frac() : Frac(0, 1) {}
            Frac(T num_) : Frac(num_, 1) {}
            explicit operator double() const {
                return 1. * num / den;
            }
            Frac &operator+=(const Frac &rhs) {
                num = num * rhs.den + rhs.num * den;
                den *= rhs.den;
                return *this;
            }
            Frac &operator-=(const Frac &rhs) {
                num = num * rhs.den - rhs.num * den;
                den *= rhs.den;
                return *this;
            }
            Frac &operator*=(const Frac &rhs) {
                num *= rhs.num;
                den *= rhs.den;
                return *this;
            }
            Frac &operator/=(const Frac &rhs) {
                num *= rhs.den;
                den *= rhs.num;
                if (den < 0) {
                    num = -num;
                    den = -den;
                }
                return *this;
            }
            friend Frac operator+(Frac lhs, const Frac &rhs) {
                return lhs += rhs;
            }
            friend Frac operator-(Frac lhs, const Frac &rhs) {
                return lhs -= rhs;
            }
            friend Frac operator*(Frac lhs, const Frac &rhs) {
                return lhs *= rhs;
            }
            friend Frac operator/(Frac lhs, const Frac &rhs) {
                return lhs /= rhs;
            }
            friend Frac operator-(const Frac &a) {
                return Frac(-a.num, a.den);
            }
            friend bool operator==(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den == rhs.num * lhs.den;
            }
            friend bool operator!=(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den != rhs.num * lhs.den;
            }
            friend bool operator<(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den < rhs.num * lhs.den;
            }
            friend bool operator>(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den > rhs.num * lhs.den;
            }
            friend bool operator<=(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den <= rhs.num * lhs.den;
            }
            friend bool operator>=(const Frac &lhs, const Frac &rhs) {
                return lhs.num * rhs.den >= rhs.num * lhs.den;
            }
            friend std::ostream &operator<<(std::ostream &os, Frac x) {
                T g = std::gcd(x.num, x.den);
                if (x.den == g) {
                    return os << x.num / g;
                } else {
                    return os << x.num / g << "/" << x.den / g;
                }
            }
        };