We present a new elementary algorithm for computing $M(x) = \sum_{n \leq x} \mu(n),$ where $\mu(n)$ is the M"{o}bius function. Our algorithm takes \[\begin{aligned} \mathrm{time} \ \ O_\epsilon\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+\epsilon} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right)\end{aligned},\] which improves on existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias-Odlyzko with computations based on the integrals of $\zeta(s)$ that only takes time $O(x^{1/2 + \epsilon})$, our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analyzing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup. This talk is based on joint work with Harald Andr'{e}s Helfgott.