At some point you _do_ have to use floating point (the whole point of numeric computations is to do things you can't do symbolically), however, this project might enable judicous use of symbolic manipulation in coordination with floating point. Holding off the computation until further in the program might have advantages: Automatic reordering of computations to help preserve precision, automatic symbolic manipulation to cancel out large terms, and perhaps automatic calculation of the amount of precision needed (eg if calculating "a < b", generate the digits of a and b until they differ). The programmer could specify exactly when symbolic manipulation is desired vs numeric.

The quadratic equation case he shows is pretty good. Say you want to solve ax^2 + bx + c = 0. In C you might write "(-b + sqrt(b * b - 4 * a * c))/(2 * a)" which has all the precision problems he described. However, in his library that code would be converted to an AST, then symbolically simplified given the variable values, and _then_ evaluted numerically. The intermediate symbolic step would be able to automatically account for the edge cases, eg if c was 0, so that the numeric step wouldn't fail. In this case, the symbolic solver would recognize that (+, (sqrt, (-, ( *, b, b), 0)), b) is exactly zero before the floating point computation occurs.

I often write scientific programs with the issues he describes, and I have also wished for something along these lines but kind of suspect it is impossible. I usually have to play in mathematica/on paper a bit before finding an algorithm that takes care of the numeric issues (yes, even when using multiprecision), and while it would be nice for it to happen automatically, my impression is that the algorithms I find on paper are nontrivial for a computer to come up with. In some cases, however, the computer could have done it for me. But I have a feeling that in the case precision becomes an issue, even if this library gets me 90% of the way, I would still want total manual control of how the calculations occur. But I still want him to try in case it works!