Let me elucidate this vague question with an example. Consider for example the following Gauss sum of roots of unity
$N=(e^{2\pi i/5} + e^{2\pi i/4} e^{4\pi i/5} + e^{4\pi i/4} e^{8\pi i/5} + e^{6\pi i/4} e^{16\pi i/5})^4$.
We know from Galois theory that this is expression is in $\mathbb{Z}[i]$. We could devise an algorithm to determine what this Gaussian integer is:
Approximate $e^{2\pi ij/5}$ via a decimal expansion to $m$ decimal places and use this approximation to compute $N$ while keeping the $4$-th roots of unity as formal symbols. Then round the coefficients to the nearest integers. As $m\rightarrow \infty$, we have that the decimal expansions should approach the correct integers, but
Question: If we are trying to determine a runtime for this algorithm, it must include $m$. Is there an effectively computable $m$ such that we can guarantee that the 'rounding' step returns the correct integers? I am questioning this in general, not just for this specific example: can there be guarantees in such algorithms?
For example, wolfram alpha uses a similar approximation, and returns -14.9999999999999999999999999999999999999999999999999999999999999 + 20.0000000000000000000000000000000000000000000000000000000000000 i. I want to say that this guarantees the expression is $-15+20i$, but is this true?