This is interesting. I think a number could be of low complexity in terms of approximating it to within smaller and smaller $\epsilon $, while of high complexity in terms of finding its binary representation. Just imagine that its binary representation has extremely long stretches of 0s (and/or 1s) so the number is unusually close to dyadic rationals in a sense.
You may find something about this in Chapter 7 of Weihrauch's book Computable Analysis.