State of the art implementations of the exponential function rely on interpolation tables, Taylor expansions or IEEE manipulations containing a small fraction of integer operations. Unfortunately, none of these methods is able to maximize the profit of vectorization and at the same time, provide sufficient accuracy. Indeed, many applications such as solving PDEs, simulations of neuronal networks, Fourier transforms and many more involved a large quantity of exponentials that have to be computed efficiently. In this paper we device and demonstrate the usefulness of a novel formulation to compute the exponential employing only floating point operations, with a flexible accuracy ranging from a few digits up to the full machine precision. Using the presented algorithm we can compute exponentials of large vectors, in any application setting, maximizing the performance gains of the vectorization units available to modern processors. This immediately results in a speedup for all applications.