HP calcs are really not that accurate..

[ttw]

There are lots of convergent interval algorithms. While intervals do use conservative rounding estimates, there are two common operations that shrink intervals. The first is the obvious division by a constant. (A,B)/2 becomes (A/2, B/2) with A/2 being rounded down and B/2 being rounded up. The new interval is smaller than the original (not twice as small though.)

The second, and not as obvious shrinking algorithm is the interval Newton's method. One starts with an interval (A,B) which is guaranteed to cover the answer; next the interval Newton is applied (it's not just applying Newton's method with interval rounding to the endpoints; there is some other stuff that can be done but I don't remember all of it.) One gets a new interval (C,D) that also includes the solution. This interval may actually be larger than the input interval. (Magic Manipulation Alert) As both (A,B) and (C,D) include the desired point, the intersection of the intervals must also include the solution. So the interval (Max(A,C), Min(B,D)) (no rounding involved) must also contain the solution. In one dimension, this method always converges (unlike ordinary Newton) and provides a convergent sequence of intervals bracketing the solution. The convergence isn't guaranteed in several dimensions though.