This paper presents an algorithm for the efficient approximation of the saddle-extremum persistence diagram of a scalar field. Vidal et al. introduced recently a fast algorithm for such an approximation (by interrupting a progressive computation framework). However, no theoretical guarantee was provided regarding its approximation quality. In this work, we revisit the progressive framework of Vidal et al. and we introduce in contrast a novel approximation algorithm, with a user controlled approximation error, specifically, on the Bottleneck distance to the exact solution. Our approach is based on a hierarchical representation of the input data, and relies on local simplifications of the scalar field to accelerate the computation, while maintaining a controlled bound on the output error. The locality of our approach enables further speedups thanks to shared memory parallelism. Experiments conducted on real life datasets show that for a mild error tolerance (5% relative Bottleneck distance), our approach improves runtime performance by 18% on average (and up to 48% on large, noisy datasets) in comparison to standard, exact, publicly available implementations. In addition to the strong guarantees on its approximation error, we show that our algorithm also provides in practice outputs which are on average 5 times more accurate (in terms of the L2-Wasserstein distance) than a naive approximation baseline. We illustrate the utility of our approach for interactive data exploration and we document visualization strategies for conveying the uncertainty related to our approximations.