There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use (bi)homographic functions (Raney's approach was extended to bihomographic functions by Liardet and Stambul in 1998) but Raney's one use a binary encoding of continued fractions and transducers from automata theory. This makes Raney's approach attractive (for me).
It seems to me one is able to compute $n^{th}$-roots of some reals - i.e. $\sqrt[n]{a}$ with $a\in \mathbb{R}$ - following Gosper's way. Do you know if one can do so with Raney's transducers (see On continued fractions and finite automata) ?