Wanted: Efficient Algorithm to Square an Integer Number

[Thomas Okken]

Many years ago, I wrote a program to visualize the Mandelbrot set on my Macintosh. This was the original model, with the 7.83 MHz 68000 CPU.

To get as much speed as possible, I coded the iteration using fixed-point math, in assembly language. This required, among other things, calculating a couple of 32-bit multiplications, and since the 68000 could only multiply 16-bit numbers, this required performing four multiplications and combining their results: a_low * b_low + (a_low * b_high + a_high * b_low) << 16 + a_high * b_high << 32.

When squaring a number, you can save one multiplication here, since a_low * b_high and a_high * b_low are the same in this case. The difference is going to be even more pronounced if you're implementing 32-bit multiplication on an 8-bit CPU, and I imagine similar optimizations come into play when implementing it in hardware.