ARS: when to update?

An email I got today from Heng Zhou wondered about the validity of the above form of the ARS algorithm. As printed in our book Monte Carlo Statistical Methods. The worry is that in the original version of the algorithm the envelope of the log-concave target f(.) is only updated for rejected values. My reply to the question is that there is no difference in the versions towards returning a value simulated from f, since changing the envelope between simulations does not modify the accept-reject nature of the algorithm. There is no issue of dependence between the simulations of this adaptive accept-reject method, all simulations remain independent. The question is rather one about efficiency, namely does it pay to update the envelope(s) when accepting a new value and I think it does because the costly part is the computation of f(x), rather than the call to the piecewise-exponential envelope. Correct me if I am wrong!

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