fileprivate func gcd(_ a: Int, _ b: Int) -> Int {
  return b == 0 ? a : gcd(b, a % b)
}

fileprivate func abs(_ n: Int) -> Int {
  return n > 0 ? n : -n
}

struct Ratio {
  let numerator: Int
  let denominator: Int

  init(_ numerator: Int, _ denominator: Int = 1) {
    var (numerator, denominator) = (numerator, denominator)

    let divisor = gcd(numerator, denominator)
    numerator /= divisor
    denominator /= divisor

    if denominator < 0 {
      (numerator, denominator) = (-numerator, -denominator)
    }

    self.numerator = numerator
    self.denominator = denominator
  }

  var asDouble: Double { return Double(numerator) / Double(denominator) }
  var asFloat: Float { return Float(numerator) / Float(denominator) }
  var abs: Ratio { return self.numerator < 0 ? -self : self }
}

extension Ratio: CustomDebugStringConvertible {
  var debugDescription: String {
    let denom = denominator == 1 ? "" : "/\(String(describing: denominator))"
    return "\(numerator)\(denom)"
  }
}

extension Ratio: ExpressibleByIntegerLiteral {
  init(integerLiteral value: IntegerLiteralType) {
    self.init(value)
  }
}

extension Ratio: Equatable {}
func ==(lhs: Ratio, rhs: Ratio) -> Bool {
  return lhs.numerator == rhs.numerator && lhs.denominator == rhs.denominator
}

extension Ratio: Comparable {}
func <(lhs: Ratio, rhs: Ratio) -> Bool {
  return (rhs.abs - lhs.abs).numerator > 0
}

func +(lhs: Ratio, rhs: Ratio) -> Ratio {
  return Ratio(
    lhs.numerator * rhs.denominator + rhs.numerator * lhs.denominator,
    lhs.denominator * rhs.denominator
  )
}

func +=(lhs: inout Ratio, rhs: Ratio) {
  lhs = lhs + rhs
}

prefix func -(r: Ratio) -> Ratio {
  return Ratio(-r.numerator, r.denominator)
}

func -(lhs: Ratio, rhs: Ratio) -> Ratio {
  return lhs + (-rhs)
}

func -=(lhs: inout Ratio, rhs: Ratio) {
  lhs = lhs - rhs
}

func *(lhs: Ratio, rhs: Ratio) -> Ratio {
  return Ratio(lhs.numerator * rhs.numerator,lhs.denominator * rhs.denominator)
}

func *=(lhs: inout Ratio, rhs: Ratio) {
  lhs = lhs * rhs
}

func /(lhs: Ratio, rhs: Ratio) -> Ratio {
  return lhs * Ratio(rhs.denominator, rhs.numerator)
}

func /=(lhs: inout Ratio, rhs: Ratio) {
  lhs = lhs / rhs
}