double r2 = dx * dx + dy * dy;
double r3 = r2 * sqrt(r2);
Can the second line be replaced by something faster? Something that does not involve sqrt?
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I think another way to look at your question would be “how to calculate (or approximate) sqrt(n)“. From there your question would be trivial (n * sqrt(n)). Of course, you’d have to define how much error you could live with. Wikipedia gives you many options:
http://en.wikipedia.org/wiki/Methods_of_computing_square_roots