Let's say I have some fraction $\frac{n}{m}$, which is fully reduced. how can I approximate its decimal expansion to a given accuracy?
Like $\frac{1}{7}$ is 0.143 if you want 3 decimal places of accuracy but 0.14285714 if you want 8 decimal places of accuracy.
Currently I use the following algorithm
Let be $a \in \{1,2,\ldots\}$ a specifier for accuracy.
Calculate:
$$
\begin{align}
p &= \lceil \log_{10}(m) \rceil + a \\\\
f &= \lfloor \frac{10^p}{m} \rfloor \\\\
v &= n \cdot f
\end{align}
$$
Then in $v$ insert the decimal comma at the correct place or add 0. with leading zeros.
Is this a good algorithm or are there improvements I could do?
You could do long division, see https://www.mathsisfun.com/long_division3.html for details.