The formal safety analysis of automated driving vehicles poses unique challenges due to their dynamic operating conditions and significant complexity. This paper presents a case study of applying formal safety verification to adaptive cruise controllers. Unlike the majority of existing verification approaches in the automotive domain, which only analyze (potentially imperfect) controller models, employ simulation to find counter-examples or use online monitors for runtime verification, our method verifies controllers at code level by utilizing bounded model checking. Verification is performed against an invariant set derived from formal specifications and an analytical model of the required behavior. For neural network controllers, we propose a scalable three-step decomposition, which additionally uses a neural network verifier. We show that both traditionally implemented as well as neural network controllers are verified within minutes. The dual focus on formal safety and implementation verification provides a comprehensive framework applicable to similar cyber-physical systems.