As an example, primality testing can now be done in polynomial time - a huge breakthrough for mathematicians - but in practice number sieves that give ~99.999% accuracy in constant time are used.

This reminds me of a recent post on http://lambda-the-ultimate.org where (twards the end) the author makes a comment on "Physics vs. Mathematics" (the thread it's attached to is a rather interesting debate on modern type systems). Math majors would rather have their toenails chewed off by chipmunks than use a theorum that wasn't completely true for every little edge case. But a Physics major (and Engineering majors, too) almost have to in order to get work done.

Orbits of planetary bodies comes to mind. Newton presented some simple equations that almost, but not quite, fit the data (he passed this off as God fiddling with the universe). Einstein gave a much more complex equation that perfectly fit the data. However, Newton's equation is still prefered by Physicists because it's easier to work with, and the minor differences in the outcome usually don't matter.

As programming is more or less an Engineering discipline, we can often get away with things that aren't mathmatically pure. I suspect that good programers know their math, but also know when to throw it away.