In model-based design of cyber-physical systems, such as switched mixed-signal circuits or software-controlled physical systems, it is common to develop a sequence of system models of different fidelity and complexity, each appropriate for a particular design or verification task. In such a sequence, one model is often derived from the other by a process of simplification or implementation. E.g. a Simulink model might be implemented on an embedded processor via automatic code generation. Three questions naturally present themselves: how do we quantify closeness between the two systems? How can we measure such closeness? If the original system satisfies some formal property, can we automatically infer what properties are then satisfied by the derived model? This paper addresses all three questions: we quantify the closeness between original and derived model via a distance measure between their outputs. We then propose two computational methods for approximating this closeness measure. Finally, we derive syntactical re-writing rules which, when applied to a Metric Temporal Logic specification satisfied by the original model, produce a formula satisfied by the derived model. We demonstrate the soundness of the theory with several experiments.