Output negative digits

Negative digits are generated in the output if the numerator starts $(\overline{1}::\overline{1})$ or $(\overline{1}::0)$, which means that:

$$x \in \Big[-1,-\frac{1}{4}\Big]$$

The strategy here is identical to the one used when we output positive digits. We require three cases as before, and output either -1, $-\frac{1}{2}$, or $-\frac{1}{4}$. The derivations are not given here, but exactly the same method is used and the results are all of the same form with a few additions and subtractions swapped, and some minor sign changes present.