So why don't we just make the data bus larger so that these polynomial computers can use bigger numbers and thus be more accurate? Why have data paths in the largest computers been stuck at 56 bits for many years now? Because, if we use the same polynomial technology to build a computer with a wider data bus, the size of the truth table that must be solved in hardware expands at an exponential rate.
This is true in the polynomial world, for integer and floating point multiplication and division. It is true in the residue world for integer residue division, but it is not true for residue rational point division nor is it true for residue addition, subtraction, or multiplication. Therefore, it is practical to build a rational residue computer with a wider data bus in order to provide much more accuracy, and still be produce as many or even more instructions per second.
A residue rational computer offers a means to achieve rapid rational computations. Using an integer residue division algorithm, it can do finite precision rational arithmetic approximately as fast, as floating point arithmetic, and with quite a bit more accuracy.
Floating point arithmetic is useful because it provides a form of approximate arithmetic that takes advantage of characteristics of polynomial number systems. A direct residue analog to polynomial floating point arithmetic is possible and one has indeed been proposed by Alan Huang when he was with Stanford University. The difficulty of Huang's system was that he was unable to propose an efficient division algorithm or solutions to the problems of overflow detection and magnitude comparison.
A system of residue rational arithmetic has been proposed that appears to solve these problems. The concepts involved are moderately complex, and provide an engineering and programming challenge, but it could be well worth the effort required to overcome these hurdles.
Floating point arithmetic is a subset of rational arithmetic. Any quantity that can be expressed in floating point arithmetic can be exactly expressed in rational arithmetic, but many rational arithmetic numbers cannot be exactly expressed in floating point. Rational arithmetic is more accurate than floating point, therefore, if a computer could do rational arithmetic as fast as floating point, it would be a better computer. I make this point again because many people seem to believe that rational arithmetic is somehow radically different from, and inferior to floating point. That may be true if you consider polynomial arithmetic the only possible basis for computer design, but the situation changes radically when you if you widen your scope to include using both residue and polynomial floating-point in the same computer.