6.4.4 Rational Numbers

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/*

Perform operations on rational numbers internally represented as two integers: a numerator and a
denominator. The template integer type must support streamed input/output, comparisons, and
arithmetic operations.

- `Rational<Int>(n)` constructs a rational number with numerator `n` and denominator $1$.
- `Rational<Int>(n, d)` constructs a rational number with numerator `n` and denominator `d`.
- `operator>>` inputs a rational number using the next integer from the stream as the numerator and
  1 as the denominator.
- `operator<<` outputs the rational number as a string consisting of possibly a minus sign followed
  by the numerator, followed by a slash, followed by the denominator.
- `to_string()`, `to_llong()`, `to_double()`, and `to_ldouble()` return the rational converted to an
  `std::string`, `int64_t`, `double`, and `long double` respectively with potential truncation or
  approximation for the primitive types. Equivalent conversions are also available through explicit
  casts to `int`, `long long`, `double`, and `long double`.
- `to_arithmetic<T>()` is the general form behind the three numeric conversions above, returning the
  rational as any arithmetic type `T` readable from a stream. Integral `T` truncates toward zero.
- `abs()`, `floor()`, and `ceil()` return the absolute value, floor, and ceiling.
- Operators `<`, `>`, `<=`, `>=`, `==`, `!=`, `+`, `-`, `*`, `/`, `%`, `++`, `--`, `+=`, `-=`, `*=`,
  `/=`, and `%=` are defined analogous to those on numerical primitives. The comparisons and binary
  arithmetic are hidden friends, so a raw integer operand works on either side.

Overflow warning: Internal operations do not check for overflow. Comparisons and arithmetic
cross-multiply numerators and denominators, so instantiate with a wider integer type (such as
`__int128`, or even `BigInt`) if the values may grow large.

Time Complexity:
- O(log s) operations on `Int` per call to the constructor `Rational(n, d)`, where $s = |n| + |d|$,
  spent reducing to lowest terms.
- O(1) operations on `Int` per call to everything else.

Space Complexity:
- Two `Int` values for storage of the rational number.
- O(1) auxiliary `Int` values for all operations.

*/

#include <cassert>
#include <cstdint>
#include <istream>
#include <ostream>
#include <sstream>
#include <string>

template<typename Int>
class Rational {
  Int num, den;

 public:
  Rational() : num(0), den(1) {}
  Rational(const Int &n) : num(n), den(1) {}

  Rational(const Int &n, const Int &d) : num(n), den(d) {
    assert(den != 0);
    if (den < 0) {
      num = -num;
      den = -den;
    }
    Int a(num < 0 ? -num : num), b(den), tmp;
    while (a != 0 && b != 0) {
      tmp = a % b;
      a = b;
      b = tmp;
    }
    Int gcd = (b == 0) ? a : b;
    num /= gcd;
    den /= gcd;
  }

  friend std::istream &operator>>(std::istream &in, Rational &r) {
    in >> r.num;
    r.den = 1;
    return in;
  }

  friend std::ostream &operator<<(std::ostream &out, const Rational &r) {
    out << r.num << "/" << r.den;
    return out;
  }

  std::string to_string() const {
    std::stringstream ss;
    ss << num << "/" << den;
    return ss.str();
  }

  // Round-trips through a stringstream so that any Int supporting streamed I/O (e.g. a big-integer
  // type) converts, even without a direct cast to the target type. Integral T truncates.
  template<typename T>
  T to_arithmetic() const {
    std::stringstream ss;
    ss << num << " " << den;
    T n, d;
    ss >> n >> d;
    return n / d;
  }

  int64_t to_llong() const { return to_arithmetic<int64_t>(); }
  double to_double() const { return to_arithmetic<double>(); }
  long double to_ldouble() const { return to_arithmetic<long double>(); }

  explicit operator int() const { return static_cast<int>(to_llong()); }
  explicit operator long long() const { return static_cast<long long>(to_llong()); }
  explicit operator double() const { return to_double(); }
  explicit operator long double() const { return to_ldouble(); }

  // The comparison and binary arithmetic operators are hidden friends, so a raw integer operand on
  // either side converts through the implicit constructor.
  friend bool operator<(const Rational &a, const Rational &b) {
    return a.num * b.den < b.num * a.den;
  }

  friend bool operator==(const Rational &a, const Rational &b) {
    return a.num == b.num && a.den == b.den;
  }

  // clang-format off
  friend bool operator>(const Rational &a, const Rational &b) { return b < a; }
  friend bool operator<=(const Rational &a, const Rational &b) { return !(b < a); }
  friend bool operator>=(const Rational &a, const Rational &b) { return !(a < b); }
  friend bool operator!=(const Rational &a, const Rational &b) { return !(a == b); }
  // clang-format on

  Rational abs() const { return Rational(num < 0 ? -num : num, den); }
  friend Rational abs(const Rational &r) { return r.abs(); }
  Int floor() const { return num < 0 ? -((-num + den - 1) / den) : num / den; }
  Int ceil() const { return num < 0 ? -(-num / den) : (num + den - 1) / den; }

  friend Rational operator+(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den + b.num * a.den, a.den * b.den);
  }

  friend Rational operator-(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den - b.num * a.den, a.den * b.den);
  }

  friend Rational operator*(const Rational &a, const Rational &b) {
    return Rational(a.num * b.num, a.den * b.den);
  }

  friend Rational operator/(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den, a.den * b.num);
  }

  friend Rational operator%(const Rational &a, const Rational &b) {
    return a - b * Rational(a.num * b.den / (b.num * a.den), 1);
  }

  // clang-format off
  Rational operator-() const { return Rational(-num, den); }
  Rational operator++(int) { Rational t(*this); operator++(); return t; }
  Rational operator--(int) { Rational t(*this); operator--(); return t; }
  Rational &operator++() { *this = *this + 1; return *this; }
  Rational &operator--() { *this = *this - 1; return *this; }
  Rational &operator+=(const Rational &r) { *this = *this + r; return *this; }
  Rational &operator-=(const Rational &r) { *this = *this - r; return *this; }
  Rational &operator*=(const Rational &r) { *this = *this * r; return *this; }
  Rational &operator/=(const Rational &r) { *this = *this / r; return *this; }
  Rational &operator%=(const Rational &r) { *this = *this % r; return *this; }
  // clang-format on
};

/*** Example Usage ***/

#include <cmath>
using namespace std;

bool EQ(double a, double b) {
  return fabs(a - b) < 1e-9;
}

int main() {
  using Rational = Rational<int64_t>;
  assert(Rational(-21, 1) % 2 == -1);
  Rational r(Rational(-53, 10) % Rational(-17, 10));
  assert(EQ(r.to_ldouble(), fmod(-5.3, -1.7)));
  assert(r.to_string() == "-1/5");
  assert(static_cast<int>(Rational(7, 2)) == 3);
  assert(static_cast<long long>(Rational(7, 2)) == 3LL);
  assert(EQ(static_cast<double>(Rational(1, 2)), 0.5));

  // Raw integers work on either side of comparisons and arithmetic.
  assert(2 + Rational(1, 2) == Rational(5, 2));
  assert(1 < Rational(3, 2) && Rational(3, 2) < 2);
  assert(2 % Rational(3, 4) == Rational(1, 2));

  // Construction reduces the fraction and moves any sign onto the numerator.
  assert(Rational(2, 4).to_string() == "1/2");
  assert(Rational(1, -2).to_string() == "-1/2");
  assert(Rational(5).to_string() == "5/1");

  // Arithmetic between rationals reduces the result too.
  assert(Rational(1, 2) + Rational(1, 3) == Rational(5, 6));
  assert(Rational(1, 2) - Rational(1, 3) == Rational(1, 6));
  assert(Rational(2, 3) * Rational(3, 4) == Rational(1, 2));
  assert(Rational(1, 2) / Rational(3, 4) == Rational(2, 3));

  // Rounding takes a different branch on each sign, and is exact on whole numbers.
  assert(Rational(7, 2).floor() == 3 && Rational(7, 2).ceil() == 4);
  assert(Rational(-7, 2).floor() == -4 && Rational(-7, 2).ceil() == -3);
  assert(Rational(4, 2).floor() == 2 && Rational(4, 2).ceil() == 2);
  assert(Rational(-7, 2).abs() == Rational(7, 2) && abs(Rational(-7, 2)) == Rational(7, 2));

  // The named conversions and the general form behind them.
  assert(Rational(7, 2).to_llong() == 3 && EQ(Rational(7, 2).to_double(), 3.5));
  assert(Rational(7, 2).to_arithmetic<int>() == 3);

  // Stream input reads one integer, and output prints the reduced fraction.
  Rational streamed;
  istringstream in("7");
  in >> streamed;
  assert(streamed == 7);
  ostringstream out;
  out << Rational(-3, 6);
  assert(out.str() == "-1/2");
  return 0;
}