constructing $\mathbf R$ via binary search

Question

Using the function $$f(x) = \begin{cases}\frac{x}{4} &\text{if }|x|\le 2\\[0.3em] \frac{|x| - 1}{x} & \text{otherwise}\end{cases}$$ one can map $\mathbf Q$ into the interval $[-1, 1]$ in a way amenable to binary search.

Suppose $x$ is the real number we want to represent as a binary search sequence. For the first term of the sequence, record 1 if $x$ is positive, $-1$ if $x$ is negative, 0 if $x$ is exactly 0: i.e., start in the middle of $[-1, 1]$, and figure out whether you need to move left or right. Then ask if $|x| > 2$: if so, then is $|x| > 4$? what about $|x| > 8$? keep doubling until the number is bounded above, and then subtract: is $|x| > 6$? keep narrowing the search down until it terminates... or doesn't.

This is in no way a standard way to record real numbers, so I prepared some examples.

$0 = (0)$

$1 = (1, -1, 0)$

$-1 = (-1, 1, 0)$

$2 = (1, 0)$

$27 = (1, 1, 1, 1, 1, -1, 1, -1, 1, 0)$

$40 = (1, 1, 1, 1, 1, 1, -1, -1, 0)$

$67 = (1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 0)$

$\pi = (1, 1, -1, 1, -1, -1, 1,\dots)$

Question: how do I add and multiply these sequences in a term-wise manner?