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Various studies have shown that immune system inspired hypermutation operators can allow artificial immune systems (AIS) to be very efficient at escaping local optima of multimodal optimisation problems. However, this efficiency comes at the expense of considerably slower runtimes during the exploitation phase compared to standard evolutionary algorithms. We propose modifications to the traditional `hypermutations with mutation potential' (HMP) that allow them to be efficient at exploitation as well as maintaining their effective explorative characteristics. Rather than deterministically evaluating fitness after each bit-flip of a hypermutation, we sample the fitness function stochastically with a `parabolic' distribution which allows the `stop at first constructive mutation' (FCM) variant of HMP to reduce the linear amount of wasted function evaluations when no improvement is found to a constant. The stochastic distribution also allows the removal of the FCM mechanism altogether as originally desired in the design of the HMP operators. We rigorously prove the effectiveness of the proposed operators for all the benchmark functions where the performance of HMP is rigorously understood in the literature and validating the gained insights to show linear speed-ups for the identification of high quality approximate solutions to classical NP-Hard problems from combinatorial optimisation. We then show the superiority of the HMP operators to the traditional ones in an analysis of the complete standard Opt-IA AIS, where the stochastic evaluation scheme allows HMP and ageing operators to work in harmony. Through a comparative performance study of other `fast mutation' operators from the literature, we conclude that a power-law distribution for the parabolic evaluation scheme is the best compromise in black box scenarios where little problem knowledge is available.