No? Or maybe I'm missing something. If the goal is to be able to bound the computation of f, you can:

1) compute f with interval arithmetic

2) compute f normally and f' with interval arithmetic

3) compute f rounding towards zero, compute f' from f rounded towards infinity, and round f' up (if f positive) or round f' down (if f negative).

In all 3 cases you can use what you computed to figure out bounds on f, (1) is direct, the other two need extra work.