I think the fundamental problems is that if you start a computation using N most significant bits and then incrementally add more bits, e.g. N+M bits total, then your first N bits might change as a result.

E.g. decimal example:

    1.00/1.00 = 1.00
    1.000/1.001 = 0.999000999000...

(adding one more bit changes the first bits of the outcome)