A lot of functions are polynomials.
f(x) = x3 + 5x2 + 2x + 4
If we write this in C, we get a formula something like this:
x*x*x + 5*x*x + 2*x + 4
Many operations. We could use pow() for x3, but that's not going to help; it's probably slower.
Consider instead a factoring:
((x + 5) * x + 2) * x + 4
This is significantly fewer operations.
Also, the factoring doesn't require as much analysis as it might look; this computation is easily written in a loop in C for an arbitrary array of coefficients.
int i;
double sum = 0;
for (i = deg; i >= 0; i--)
sum = sum * x + coeff[i];
Here coeff[i] is the coefficient of xi
(e.g. coeff[0] is the un-multiplied-by-x term)
and deg is the degree of the polynomial
(i.e. the coeff array has subscripts from 0 to deg inclusive).
This is, in fact, the standard way to evaluate polynomials. It also illustrates a general principle: that a little rearrangement of the way we think of something in mathematics can sometimes make something much easier to compute.