Re: Dr. Dobbs Dev. Update 1/5 July 94 & Schneier

[Alan Barrett <barrett@daisy.ee.und.ac.za>]

> You can do better than a bit-serial loop -- though not down to
> one instruction!  There are a lot of very cool approaches, only
> one of which I remember.

Bit counting was discussed in great detail in comp.lang.c in October
1990.  I saved an excellent summary by Chris Torek, which I can post if
there is interest.  It includes a program to test 17 different methods
of bit counting, and a table of results from six machine/compiler
combinations.

In 5 of the 6 tested environments, the fastest method for counting the
1's in a 32-bit word turned out to be some variant of a table lookup
(but not always the same variant).  In 1 of the 6 tested environments,
the fastest code was the following, which is similar to that posted here
by Eli Brandt:

/*
 * Explanation:
 * First we add 32 1-bit fields to get 16 2-bit fields.
 * Each 2-bit field is one of 00, 01, or 10 (binary).
 * We then add all the two-bit fields to get 8 4-bit fields.
 * These are all one of 0000, 0001, 0010, 0011, or 0100.
 *
 * Now we can do something different, becuase for the first
 * time the value in each k-bit field (k now being 4) is small
 * enough that adding two k-bit fields results in a value that
 * still fits in the k-bit field.  The result is four 4-bit
 * fields containing one of {0000,0001,...,0111,1000} and four
 * more 4-bit fields containing junk (sums that are uninteresting).
 * Pictorially:
 *	    n = 0aaa0bbb0ccc0ddd0eee0fff0ggg0hhh
 *	 n>>4 = 00000aaa0bbb0ccc0ddd0eee0fff0ggg
 *	  sum = 0aaaWWWWiiiiXXXXjjjjYYYYkkkkZZZZ
 * where W, X, Y, and Z are the interesting sums (each at most 1000,
 * or 8 decimal).  Masking with 0x0f0f0f0f extracts these.
 *
 * Now we can change tactics yet again, because now we have:
 *	    n = 0000WWWW0000XXXX0000YYYY0000ZZZZ
 *	 n>>8 = 000000000000WWWW0000XXXX0000YYYY
 * so	  sum = 0000WWWW000ppppp000qqqqq000rrrrr
 * where p and r are the interesting sums (and each is at most
 * 10000, or 16 decimal).  The sum `q' is junk, like i, j, and
 * k above; but it is not necessarry to discard it this time.
 * One more fold, this time by sixteen bits, gives
 *	    n = 0000WWWW000ppppp000qqqqq000rrrrr
 *	n>>16 = 00000000000000000000WWWW000ppppp
 * so	  sum = 0000WWWW000ppppp000sssss00tttttt
 * where s is at most 11000 and t is it most 100000 (32 decimal).
 *
 * Now we have t = r+p = (Z+Y)+(X+W) = ((h+g)+(f+e))+((d+c)+(b+a)),
 * or in other words, t is the number of bits set in the original
 * 32-bit longword.  So all we have to do is return the low byte
 * (or low 6 bits, but `low byte' is typically just as easy if not
 * easier).
 *
 * This technique is also applicable to 64 and 128 bit words, but
 * 256 bit or larger word sizes require at least one more masking
 * step.
 */
int
tG_sumbits(n)
	register unsigned long n;
{

	n = (n & 0x55555555) + ((n >> 1) & 0x55555555);
	n = (n & 0x33333333) + ((n >> 2) & 0x33333333);
	n = (n + (n >> 4)) & 0x0f0f0f0f;
	n += n >> 8;
	n += n >> 16;
	return (n & 0xff);
}

--apb (Alan Barrett)