I wrote a paper on the computable reals with a page of (Gambit) Scheme code at the end that implements some of the examples. The code was organized for clarity of presentation (base 10, easiest algorithms, etc.), so it's not production quality, but there are references for people who are interested:
https://scholarship.claremont.edu/jhm/vol12/iss1/25/
I made sure that the typesetting didn't screw up the code, that you can just copy it to a Gambit REPL and everything works.
And Section 3 tells a funny story about me learning how to compute square roots by hand. (Perhaps only a mathematician could write that sentence!)
Brad