After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers $\alpha$, $\beta$, $\gamma$, $ \delta$ and the task is to find successively good approximations for rational numbers $q_i$ in
$q_1\alpha+q_2\beta+q_3\gamma+q_4\delta\approx0$
Basically it's like continued fraction approximations, but for more irrational numbers.
What is such an algorithm called?
It's called an integer relation algorithm. In addition to LLL mentioned in the comments, there's also PSLQ and others. The Wikipedia article on integer relation algorithms will get you started on understanding these algorithms.