Designing an efficient yet accurate floating-point approximation of a mathematical function is an intricate and error-prone process. This warrants the use of formal methods, especially formal proof, to achieve some degree of confidence in the implementation. Unfortunately, the lack of automation or its poor interplay with the more manual parts of the proof makes it way too costly in practice. This article revisits the issue by proposing a methodology and some dedicated automation, and applies them to the use case of a faithful binary64 approximation of exponential. The peculiarity of this use case is that the target of the formal verification is not a simple modeling of an external code, it is an actual floating-point function defined in the logic of the Coq proof assistant, which is thus usable inside proofs once its correctness has been fully verified. This function presents all the attributes of a state-of-the-art implementation: bit-level manipulations, large tables of constants, obscure floating-point transformations, exceptional values, etc. This function has been integrated into the proof strategies of the CoqInterval library, bringing a 20x speedup with respect to the previous implementation.