It's fine if the poster above me wishes to change languages. Anyone who chooses to stick with Haskell should get comfortable with the notion that functions can diverge, and accept that this is fully consistent with the Haskell type system. The return types may seem unhelpful, but they are accurate.
Haskell can't express the difference, but languages with effects, like Koka and F*, can!
The first effect is called div in Koka and the second is called exn, section 2.2 here
https://www.microsoft.com/en-us/research/wp-content/uploads/...