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We introduce ReachNN*, a tool for reachability analysis of neural-network controlled systems (NNCSs). The theoretical foundation of ReachNN* is the use of Bernstein polynomials to approximate any Lipschitz-continuous neural-network controller with different types of activation functions, with provable approximation error bounds. In addition, the sampling-based error bound estimation in ReachNN* is amenable to GPU-based parallel computing. For further improvement in runtime and error bound estimation, ReachNN* also features optional controller re-synthesis via a technique called verification-aware knowledge distillation (KD) to reduce the Lipschitz constant of the neural-network controller. Experiment results across a set of benchmarks show to efficiency improvement over the previous prototype. Moreover, KD enables proof of reachability of NNCSs whose verification results were previously …