Re Algorithms

Yes. One might want to approximate an irrational interval with a ratio
at a certain prime or odd limit, even if there are closer approximations
with a higher limit. CF's will probably work, but the Brun algorithm is
easier (for me, at least) and shows the semiconvergents as well. I've
programmed it and can evaluate each convergent and semiconvergent that
appear and can select whichever ratio I want.

I've programmed CF's too and sometimes use them

As for brute force, that works well too and I've programmed versions of
it as well.

All these methods have their uses and it's sometimes fun to watch
various solutions emerge on the screen.

--John