Given a decimal x, I want to test if x is within 10^-12 of a rational number with denominator 9999 or less. Obviously I could do it by looking at x, 2x, 3x, and so on, and seeing if any of these is sufficiently close to an integer. But is there a more efficient algorithm?
There is an algorithm called the continued fraction algorithm that will give you "best" rational approximations in a certain defined sense. You can stop when your denominator exceeds 9999 and then go back to the previous convergent and compare to see if it is close enough. Of course if the decimal is a small enough rational number the algorithm will terminate early.