Problem 1: I'm forced to make too many simplifications to my model to make it easy enough to compute. This introduces serious errors.

Solution: Avoid simplifications that introduce serious errors by using Mathematica's...

...powerful symbolic algebra.

Mathematica's own symbolic algebra can perform mathematical expressions with hundreds or thousands of terms without flubbing a single minus sign, and once the calculations have been done, it's easy to use the results, interactively or under programmed control, over and over. And that means the user doesn't have to do the math by hand (where an error may be introduced) and then code the result into the computer (where yet another error may be introduced).

...high-precision numeric calculations.

This is not meant to slight Mathematica's prodigious numerical capabilities.

Mathematica's fast, high-precision numerical calculations let you avoid the kind of errors that fixed-precision calculations can lead to--such as the one that clobbered the Ariane 5, a seven-billion-dollar rocket recently destroyed by a numerical overflow--a calculation that wasn't performed with sufficient numerical precision.