  Simulation-based verification algorithms can provide formal safety guarantees
for nonlinear and hybrid systems. The previous algorithms rely on user provided
model annotations called discrepancy function, which are crucial for computing
reachtubes from simulations. In this paper, we eliminate this requirement by
presenting an algorithm for computing piece-wise exponential discrepancy
functions. The algorithm relies on computing local convergence or divergence
rates of trajectories along a simulation using a coarse over-approximation of
the reach set and bounding the maximal eigenvalue of the Jacobian over this
over-approximation. The resulting discrepancy function preserves the soundness
and the relative completeness of the verification algorithm. We also provide a
coordinate transformation method to improve the local estimates for the
convergence or divergence rates in practical examples. We extend the method to
get the input-to-state discrepancy of nonlinear dynamical systems which can be
used for compositional analysis. Our experiments show that the approach is
effective in terms of running time for several benchmark problems, scales
reasonably to larger dimensional systems, and compares favorably with respect
to available tools for nonlinear models.
