Sequences of operations

  One interesting consideration is the following. Suppose we wish to compute

$$(\pi+(1+(1+(1+ \dots (1+1)\dots))))$$

Using signed binary (mantissa, exponent) representation we would require two digits of $\pi$ to obtain one digit of output, although many more digits of the later numerals representing the number one. Suppose on the other hand we compute:

$$(1+(1+(1+(1+ \dots (1+\pi)\dots))))$$

Suppose we had 100 additions, we may require 101 digits from $\pi$, although the result of the computation is the same. This implies that applying the operations sensibly, and computing expressions in a sort of tree form would be sensible. For example, although both results are the same,

(((a+a)+(a+a)) + ((a+a)+(a+a))) + (((a+a)+(a+a)) + ((a+a)+(a+a)))

requires less digits of a than

(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+(a+a)))))))))))))))

The first example requires only four digits of a to obtain one digit of output, whereas the second requires sixteen. In general if the second form requires n digits of a, one would expect the first form to require $\lceil ln(x) \rceil$ digits.