I was talking to a few people about this on #haskell, and it was suggested I ask here. I should say that I'm playing around here; don't mistake this for an urgent request or a serious problem. Suppose I wanted to implement automatic differentiation of simple functions on real numbers; then I'd take the operations from Num, Fractional, and Floating, and define how to perform them on pairs of values and their differentials, and then I'd write a differentiate function... but finding an appropriate type for that function seems to be a challenge. We have: 1. Differentiating a function of the most general type (Num a => a -> a) should produce a result of type (Num a => a -> a). 2. Differentiating a function of the more specific type (Fractional a => a -> a) should produce a result of that type (Fractional a => a -> a). 3. Differentiating a function of the most specific type (Floating a => a -> a) should produce a result of type (Floating a => a -> a). 4. BUT, differentiating a function which is of a more specific type than (Floating a => a -> a) is not, in general, possible. So differentiate should have type A a => (forall b. A b => b -> b) -> a -> a, but ONLY if the type class A is a superclass of Floating. Two partial solutions are: I can just define the differentiate function for Floating; but that means if I differentiate (\x -> x + 1), the result is a function only on floating point numbers, which is less than desirable. Or, I can define several functions: say, diffNum, diffFractional, and diffFloating... all of which have precisely the same implementation, but different types and require copy/paste to make them work. Any thoughts? For reference, here's the code I kludged together. (Again, I'm only playing around... so I wrote this very quickly and may have gotten some things wrong; don't use my code without checking it first! In particular, I know that this code produces derivative functions whose domain is too large.)
data AD a = AD a a deriving Eq
instance Show a => Show (AD a) where show (AD x e) = show x ++ " + " ++ show e ++ " eps"
instance Num a => Num (AD a) where (AD x e) + (AD y f) = AD (x + y) (e + f) (AD x e) - (AD y f) = AD (x - y) (e - f) (AD x e) * (AD y f) = AD (x * y) (e * y + x * f) negate (AD x e) = AD (negate x) (negate e) abs (AD 0 _) = error "not differentiable: |0|" abs (AD x e) = AD (abs x) (e * signum x) signum (AD 0 e) = error "not differentiable: signum(0)" signum (AD x e) = AD (signum x) 0 fromInteger i = AD (fromInteger i) 0
instance Fractional a => Fractional (AD a) where (AD x e) / (AD y f) = AD (x / y) ((e * y - x * f) / (y * y)) recip (AD x e) = AD (1 / x) ((-e) / (x * x)) fromRational x = AD (fromRational x) 0
instance Floating a => Floating (AD a) where pi = AD pi 0 exp (AD x e) = AD (exp x) (e * exp x) sqrt (AD x e) = AD (sqrt x) (e / (2 * sqrt x)) log (AD x e) = AD (log x) (e / x) (AD x e) ** (AD y f) = AD (x ** y) (e * y * (x ** (y-1)) + f * (x ** y) * log x) sin (AD x e) = AD (sin x) (e * cos x) cos (AD x e) = AD (cos x) (-e * sin x) asin (AD x e) = AD (asin x) (e / sqrt (1 - x ** 2)) acos (AD x e) = AD (acos x) (-e / sqrt (1 - x ** 2)) atan (AD x e) = AD (atan x) (e / (1 + x ** 2)) sinh (AD x e) = AD (sinh x) (e * cosh x) cosh (AD x e) = AD (cosh x) (e * sinh x) asinh (AD x e) = AD (asinh x) (e / sqrt (x^2 + 1)) acosh (AD x e) = AD (acosh x) (e / sqrt (x^2 - 1)) atanh (AD x e) = AD (atanh x) (e / (1 - x^2))
diffNum :: Num b => (forall a. Num a => a -> a) -> b -> b diffFractional :: Fractional b => (forall a. Fractional a => a -> a) -> b -> b diffFloating :: Floating b => (forall a. Floating a => a -> a) -> b -> b
diffNum f x = let AD y dy = f (AD x 1) in dy diffFractional f x = let AD y dy = f (AD x 1) in dy diffFloating f x = let AD y dy = f (AD x 1) in dy
-- Chris Smith