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Example: Computing Squares

Assume that multiplication is expensive. Consider the problem of computing squares of succesive integers.

for i := 0 to 99 do
  x[i] := i*i;

next := 0;
delta := 1;
for i := 0 to 99 do
     x[i]  := next;
     next  := next + delta;
     delta := delta + 2
The second version has more code, but does no multiplication.

This form of computation has a long history; it was the basis of Babbage's Difference Engine.