dual.fs

type ^T Dual
  when ^T: (static member (+): ^T * ^T -> ^T)
  and ^T: (static member (-): ^T * ^T -> ^T)
  and ^T: (static member (*): ^T * ^T -> ^T)
  and ^T: (static member (/): ^T * ^T -> ^T)
  (real: ^T, imag: ^T) =
    member inline this.x = real
    member inline this.y = imag

    static member inline (+) (lhs: Dual< ^T>, rhs: Dual< ^T>) =
      Dual< ^T>(lhs.x + rhs.x, lhs.y + rhs.y)
    
    static member inline (-) (lhs: Dual< ^T>, rhs: Dual< ^T>) =
      Dual< ^T>(lhs.x - rhs.x, lhs.y - rhs.y)

    static member inline (*) (lhs: Dual< ^T>, rhs: Dual< ^T>) =
      Dual< ^T>(lhs.x * rhs.x, lhs.x * rhs.y + lhs.y * rhs.x)

    static member inline (/) (lhs: Dual< ^T>, rhs: Dual< ^T>) =
      Dual< ^T>(
        lhs.x / rhs.x, 
        (lhs.y * rhs.x + lhs.x * rhs.y) / (rhs.x * rhs.x))
  
    static member inline (*) (lhs: Dual< ^T>, rhs: ^T) =
      Dual< ^T>(lhs.x * rhs, lhs.y * rhs)

    static member inline (*) (lhs: ^T, rhs: Dual< ^T>) =
      Dual< ^T>(lhs * rhs.x, lhs * rhs.y)

    static member inline (+) (lhs: Dual< ^T>, rhs: ^T) =
      Dual< ^T>(lhs.x + rhs, lhs.y)

    static member inline (-) (lhs: ^T, rhs: Dual< ^T>) =
      Dual< ^T>(lhs - rhs.x, rhs.y)


let inline dual x = 
  let y = LanguagePrimitives.GenericOne
  Dual (x, y) 

// implicitly (x: int) -> int ???
let inline F x =  3 * x * x + 2 * x

let x = dual 2

//error FS0001: This expression was expected to have type
//    'int Dual'    
//but here has type
//    'int'    
let result : Dual<int> = F x