The solutions to the problems of overflow detection and magnitude comparison are easiest to explain. The speed with which polynomial calculations can be executed is heavily dependant on their resolution. A low resolution calculation in floating point using a small mantissa and a small exponent can be executed very rapidly. If it is small enough, it can be executed in parallel with a residue calculation in about the same amount of time. The small integer range floating point calculation produces results that are useless in terms of the accuracy required for scientific calculations, but they can, in most cases, tell us whether overflow occurred, and can be used for the purposes of magnitude comparison. It is necessary to occasionally "refresh" the accuracy of the floating point numbers by translating them from the residue, but this needs to be done rarely enough to make the system practical. This approach costs us a little hardware overhead, and some time overhead too, but it solves the problem of magnitude comparison and overflow detection.