At a fundamental level most physical equations are time reversible. In this paper we propose an integrator that preserves this property at the discrete computational level. Our simulations can be run forward and backwards and trace the same path exactly bitwise. We achieve this by implementing theoretically reversible integrators using a mix of fixed and floating point arithmetic. Our main application is in efficiently implementing the reverse step in the adjoint method used in optimization. Our integrator has applications in differential simulations and machine learning (backpropagation).
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