AD Reflection - How does it work?

Question

I have seen the ad package and i understand how it does automatic differentiation by providing a different instance of the class Floating and then implementing the rules of derivatives.

But in the example

Prelude Debug.SimpleReflect Numeric.AD> diff atanh x
recip (1 - x * x) * 1

We see that it can represent functions as ASTs and show them as a string with variable names.

I wonder how they did that, because when i write:

f :: Floating a => a -> a
f x = x^2

No matter what instance I provide, i will get a function f :: Something -> Something and not a representation like f :: AST, or f :: String

The instance cannot "know" what the parameters are.

How they are able to do it ?

Answer 1

It has nothing to do with the AD package, actually, and everything to do with the x in diff atanh x.

To see this, let's define our own AST type

data AST = AST :+ AST
         | AST :* AST
         | AST :- AST
         | Negate AST
         | Abs AST
         | Signum AST
         | FromInteger Integer
         | Variable String

We can define a Num instance for this type

instance Num (AST) where
  (+) = (:+)
  (*) = (:*)
  (-) = (:-)
  negate = Negate
  abs = Abs
  signum = Signum
  fromInteger = FromInteger

And a Show instance

instance Show (AST) where
  showsPrec p (a :+ b) = showParen (p > 6) (showsPrec 6 a . showString " + " . showsPrec 6 b)
  showsPrec p (a :* b) = showParen (p > 7) (showsPrec 7 a . showString " * " . showsPrec 7 b)
  showsPrec p (a :- b) = showParen (p > 6) (showsPrec 6 a . showString " - " . showsPrec 7 b)
  showsPrec p (Negate a) = showParen (p >= 10) (showString "negate " . showsPrec 10 a)
  showsPrec p (Abs a) = showParen (p >= 10) (showString "abs " . showsPrec 10 a)
  showsPrec p (Signum a) = showParen (p >= 10) (showString "signum " . showsPrec 10 a)
  showsPrec p (FromInteger n) = showsPrec p n
  showsPrec _ (Variable v) = showString v

So now if we define a function:

f :: Num a => a -> a
f a = a ^ 2

and an AST variable:

x :: AST
x = Variable "x"

We can run the function to produce either integer values or AST values:

λ f 5
25
λ f x
x * x

If we wanted to be able to use our AST type with your function f :: Floating a => a -> a; f x = x^2, we'd need to extend its definition to allow us to implement Floating (AST).