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Neural networks (NNs) playing the role of controllers have demonstrated impressive empirical performance on challenging control problems. However, the potential adoption of NN controllers in real-life applications has been significantly impeded by the growing concerns over the safety of these neural-network controlled systems (NNCSs). In this work, we present POLAR-Express, an efficient and precise formal reachability analysis tool for verifying the safety of NNCSs. POLAR-Express uses Taylor model arithmetic to propagate Taylor models (TMs) layer-by-layer across a neural network to compute an over-approximation of the neural network. It can be applied to analyze any feed-forward neural networks with continuous activation functions, such as ReLU, Sigmoid, and Tanh activation functions that cover the common benchmarks for NNCS reachability analysis. Compared with its earlier prototype POLAR, we …