... steps¹

One thing that needs to be stated is that we are repeatedly assigning values to variables. In the interests of efficiency, any algorithm that deals with multi-precision arithmetic should create new variables (allocate memory) and put values in them infrequently. One way to do this is to decide in advance (by a priori calculations) how many variables we will need and how large they can get. Then we keep track of value variable ``names'' by means of a symmetric matrix (the number of variables is likely to be much smaller than the numbers we are dealing with).

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... enough²

However, note that x $\mapsto$ x² - 2 is in fact a powering map in disguise. If we put x = y + 1/y then this is the same as y $\mapsto$ y²

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