Notes on higher-dimensional tarai functions

0. Introduction I. Takeuchi defined the following recursive function, called the tarai function, in [5]. t(x, y, z) = if x ≤ y then y else t(t(x− 1, y, z), t(y − 1, z, x), t(z − 1, x, y)) This function requires many recursive calls even for small x, y, and z, so it is used to see how effectively the programming language implementation handles recursive calls. In [3], J. McCarthy proved that this recursion terminates without call-by-need and t can be computed in the following way. t(x, y, z) = if x ≤ y then y else if y ≤ z then z else x