Probably a lot of people know about the tricks in HAKMEM for counting the number of nonzero bits in a machine word. And that in practice it's probably best to avoid multiplication, and either repeatedly remove the lowest nonzero bit with x &= x-1 (if few nonzeros are expected, as in bitboard chess programs) or do a table lookup on longer blocks of bits.

So what follows is not likely to be especially useful.

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