At times I have computations which *should be* the same mathematically, but are carried out through a different sequence of specific computations.  One common example is in parallel frameworks where the order of computation is indeterminate because multiple workers/threads/processes are each calculating portions to aggregate.

Another related case is when I call some library to do an operation, but I did not write the library, nor do I understand its guts well.  For example, the tensor libraries used in neural networks that will calculate a loss function.  Occasionally I'd like to be able to replicate (within a tolerance) a computation the library performs using something more general like NumPy.  Having a few ulps difference is typical, but counts as validating the "same" answer.

Another occasion I encounter it is with data measurements.  Some sort of instrument collects measurements with a small jitter.  Two measurements that cannot be distinguished based on the precision of the instrument might nonetheless be stored as different floating point numbers.  In that case, I probably want to be able to tweak the tolerances for the specific case.