-- Automatic Differentiation

class Val a where val :: a -> Double

data Var = Var Double
instance Val Var where val (Var v) = v

data Fun = Fun Double Double
instance Show Fun where
    show (Fun v d) = show v ++ ", " ++ show d
instance Eq Fun where
    (==) (Fun av ad) (Fun bv bd) = av == bv && ad == bd
    (/=) (Fun av ad) (Fun bv bd) = av /= bv || ad /= bd
instance Num Fun where
    (+) (Fun av ad) (Fun bv bd) = Fun (av + bv) (ad + bd)
    (*) (Fun av ad) (Fun bv bd) = Fun (av * bv) (ad*bv + av*bd)
    (-) (Fun av ad) (Fun bv bd) = Fun (av - bv) (ad - bd)
instance Val Fun where val (Fun v d) = v
instance Fractional Fun where
    (/) (Fun av ad) (Fun bv bd) = Fun (av/bv) ((ad*bv - av*bd)/(bv*bv))
    fromRational r = Fun (fromRational r) 0.0
instance Floating Fun where
    exp   (Fun v d) = Fun (exp v) ((exp v) * d)
    log   (Fun v d) = Fun (log v) (d / v)
    sin   (Fun v d) = Fun (sin v) ((cos v) * d)
    cos   (Fun v d) = Fun (cos v) (-(sin v) * d)
    asin  (Fun v d) = Fun (asin v) (d / sqrt(1.0 - v**2.0))
    acos  (Fun v d) = Fun (acos v) (-d / sqrt(1.0 - v**2.0))
    atan  (Fun v d) = Fun (atan v) (d / (1.0 + v**2.0))
    sinh  (Fun v d) = Fun (sinh v) ((cosh v) * d)
    cosh  (Fun v d) = Fun (sinh v) ((sinh v) * d)
    asinh (Fun v d) = Fun (asinh v) (d / sqrt(1.0 + v**2.0))
    acosh (Fun v d) = Fun (acosh v)
                      (d / (sqrt(1.0 - v**2.0)*sqrt(1.0 + v**2.0)))
    atanh (Fun v d) = Fun (atan v) (d / (1.0 - v**2.0))

toFun (Var v) = Fun v 1.0
der (Fun _ d) = d

main =
    let x = toFun $ Var 3.141592653589793
    in do
        print $ (x**2.0) + 2.0*x + 3.0
        print $ 2.0*(exp (sin x)) + x
        print $ x/x
        print $ tan x

19.1527897082689, 8.28318530717959
5.14159265358979, -1.0
1.0, 0.0
-1.22460635382238e-16, 1.0