On the rediscovery of Halley's iterative method for computing the zero of an analytic function

We show that Halley's basic sequence, resulting from accelerating the order of convergence of Newton's method, is the most efficient way of doing so in terms of usage of certain derivatives. This fact could explain why this process of accelerating the convergence of Newton's method is so frequently rediscovered. Then we present an algorithmic way of recognizing Halley's family and we apply this algorithm to examples of rediscoveries.