Is there a shorter way to express this haskell idiom?

Question

Every so often I have a function of type a -> b and a function of type b -> b -> c and I would like a function of type a -> a -> c. For example if I wanted to check the second element of each two tuples were equal

snd :: (a , b) -> b
(==) :: Eq a => a -> a -> Bool

And I want something of type

Eq b => (a, b) -> (c, b) -> Bool

The best way I have found to do this so far is

f g z=(.g).z.g

Which is ok, but it feels unneccessarily complex for something that is this simple. The pointfree representation is even worse:

flip=<<((.).).(.).flip(.)

Additionally it feels like this idiom should be representable with some sort of abstraction (it feels a lot like (<*>)). I attempted to find a replacement by abstracting our functions to functors, the result is not very enlightening

flip=<<(fmap.).fmap.flip fmap ::
  Functor f => f a -> (a -> a -> b) -> f (f b)

Is there a shorter way to express this idea? Particularly the pointfree version?

Answer 1

I think eta-reduction is severely overrated. I see a lot more attempts at eta-reduction in the wild than I think is reasonable.

So it is in this case: eta-reducing this function is not helpful, I think. It's much more readable and elegant in its full form:

f :: (a -> a -> c) -> (a -> b) -> b -> b -> c
f g h a b = g (h a) (h b)

---

On a related note, such function exists in the standard libraries. It's called on, and the idea is to use it in infix form, so that it almost reads like English:

eqAmounts = (==) `on` amount
cmpNames = compare `on` name