Many methods are available to detect silent errors in high-performance computing (HPC) applications. Each method comes with a cost, a recall (fraction of all errors that are actually detected, i.e., false negatives), and a precision (fraction of true errors amongst all detected errors, i.e., false positives). The main contribution of this paper is to characterize the optimal computing pattern for an application: which detector(s) to use, how many detectors of each type to use, together with the length of the work segment that precedes each of them. We first prove that detectors with imperfect precisions offer limited usefulness. Then we focus on detectors with perfect precision , and we conduct a comprehensive complexity analysis of this optimization problem, showing NP-completeness and designing an FPTAS (Fully Polynomial-Time Approximation Scheme). On the practical side, we provide a greedy algorithm, whose performance is shown to be close to the optimal for a realistic set of evaluation scenarios. Extensive simulations illustrate the usefulness of detectors with false negatives, which are available at a lower cost than the guaranteed detectors.