Continued fractions are quite awesome. Gospers monograph, which is also mentioned in the article, is quite readable: https://perl.plover.com/classes/cftalk/INFO/gosper.txt

One thing to watch out for is that when you compute floor(sqrt(2) * sqrt(2)), the computation will never finish. The reason is that continued fractions only compute approximations, so the approximation will only tell you that the result is very close to 2, but it might still be slightly above or below 2. That is, unless you implement some logic to deal with repeating coefficients, but then you will have the same problem with 1/pi times pi, for example.