<linux@horizon.com> wrote:

This is an extremely well-known technique.  You can see a similar version that
uses a multiply for the last few steps at
http://graphics.stanford.edu/~seander/bithacks.html#CountBitsSetParallel whch
refers to "Software Optimization Guide for AMD Athlon 64 and Opteron
Processors"
http://www.amd.com/us-en/assets/content_type/white_papers_and_tech_docs/25112.PDF

It's section 8.6, "Efficient Implementation of Population-Count Function in
32-bit Mode", pages 179-180.

It uses the name that I am more familiar with, "popcount" (population count),
although "Hamming weight" also makes sense.

Anyway, the proof of correctness proceeds as follows:

	b = a - ((a >> 1) & 0x55555555);
	c = (b & 0x33333333) + ((b >> 2) & 0x33333333);
	d = (c + (c >> 4)) & 0x0f0f0f0f;
#if SLOW_MULTIPLY
	e = d + (d >> 8)
	f = e + (e >> 16);
	return f & 63;
#else
	/* Useful if multiply takes at most 4 cycles */
	return (d * 0x01010101) >> 24;
#endif

The input value a can be th...