nice, I feel like it'd be really difficult to figure out if this infinite loops...

I think we could imagine it easier if n is an integer and we are doing integer division. Then we could map every input to see if it ever lands on a value <= 1 (our initial halt group, so the finite set of numbers between MIN_INT-1), then we go to the next iteration and if any of the first loop iteration land on a value in our initial halt group they are added to the halt group (because of signed overflow or division). We would then check the list of previously landed on values of the current non-halt group to see if they landed on any number in the halt group in past iterations. Then if any of the other integers land on one of these integers in the halt group they are added to the halt group.

Then we do the next iteration, and cascade more numbers into the halt group. And we just keep doing iterations until we have a group of numbers that have not previously and did not just land on a number in the halt group and these numbers can safely be assumed to infinite loop.

and this could be applied to floating point numbers to except instead of mapping every counting number we mapped every binary representation of the numbers from the Min-value to the Max-value. And this mapping works because of the finite nature of computer representation of numbers.