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An optimisation where a function of some systematically changing variable is calculated more efficiently by using previous values of the function. In a procedural language this would apply to an expression involving a loop variable and in a declarative language it would apply to the argument of a recursive function. E.g.
f x = ... (2**x) ... (f (x+1)) ...
f x = f' x (2**x) where f ' x z = ... z ... (f' (x+1) 2*z) ...
Here the expensive operation (2**x) has been replaced by the cheaper 2*z in the recursive function f'. This maintains the invariant that z = 2**x for any call to f'.