"A lot of common numerical operations result in irrationals, including all root, logarithmic, exponential, and trigonometric functions, and that would take an i

"A lot of common numerical operations result in irrationals, including all root, logarithmic, exponential, and trigonometric functions, and that would take an infinite amount of space to represent faithfully." Actually symbolic representation can help here - not just to represent the values, but help the final precision. As a trivial example 2/sqrt(2) can be reduced to sqrt(2) with no loss of precision from the division. This is a lot of hassle though, and not a perfect solution, but it helps solve rounding errors from irrational constants or functions... in a sense some of your intermediary values have infinite precision (but in reality you removed the operation with a clever optimisation/precision conserving trick).