Given a 64-bit quantity, the simplest (and obvious) way to count the number of bits that are set in it is to simply walk through all the bits and add 1 to a running count. This is slow, as it requires 64 iterations, but is simple and easy to convince yourself that it is correct. For instance:

>>> popCountSlow 0x0123456789ABCDEF
32 :: SWord8

Faster version. This is essentially the same algorithm, except we go 8 bits at a time instead of one by one, by using a precomputed table of population-count values for each byte. This algorithm loops only 8 times, and hence is at least 8 times more efficient.

Look-up table, containing population counts for all possible 8-bit value, from 0 to 255. Note that we do not "hard-code" the values, but merely use the slow version to compute them.

States the correctness of faster population-count algorithm, with respect to the reference slow version. Turns out Z3's default solver is rather slow for this one, but there's a magic incantation to make it go fast. See https://github.com/Z3Prover/z3/issues/1150 for details.

>>> let cmd = "(check-sat-using (then (using-params ackermannize_bv :div0_ackermann_limit 1000000) simplify bit-blast sat))"
>>> proveWith z3{satCmd = cmd} fastPopCountIsCorrect
Q.E.D.

Not only we can prove that faster version is correct, but we can also automatically generate C code to compute population-counts for us. This action will generate all the C files that you will need, including a driver program for test purposes.

Below is the generated header file for popCountFast:

>>> genPopCountInC
== BEGIN: "Makefile" ================
# Makefile for popCount. Automatically generated by SBV. Do not edit!

# include any user-defined .mk file in the current directory.
-include *.mk

CC?=gcc
CCFLAGS?=-Wall -O3 -DNDEBUG -fomit-frame-pointer

all: popCount_driver

popCount.o: popCount.c popCount.h
	${CC} ${CCFLAGS} -c $< -o $@

popCount_driver.o: popCount_driver.c
	${CC} ${CCFLAGS} -c $< -o $@

popCount_driver: popCount.o popCount_driver.o
	${CC} ${CCFLAGS} $^ -o $@

clean:
	rm -f *.o

veryclean: clean
	rm -f popCount_driver
== END: "Makefile" ==================
== BEGIN: "popCount.h" ================
/* Header file for popCount. Automatically generated by SBV. Do not edit! */

#ifndef __popCount__HEADER_INCLUDED__
#define __popCount__HEADER_INCLUDED__

#include <stdio.h>
#include <stdlib.h>
#include <inttypes.h>
#include <stdint.h>
#include <stdbool.h>
#include <string.h>
#include <math.h>

/* The boolean type */
typedef bool SBool;

/* The float type */
typedef float SFloat;

/* The double type */
typedef double SDouble;

/* Unsigned bit-vectors */
typedef uint8_t  SWord8;
typedef uint16_t SWord16;
typedef uint32_t SWord32;
typedef uint64_t SWord64;

/* Signed bit-vectors */
typedef int8_t  SInt8;
typedef int16_t SInt16;
typedef int32_t SInt32;
typedef int64_t SInt64;

/* Entry point prototype: */
SWord8 popCount(const SWord64 x);

#endif /* __popCount__HEADER_INCLUDED__ */
== END: "popCount.h" ==================
== BEGIN: "popCount_driver.c" ================
/* Example driver program for popCount. */
/* Automatically generated by SBV. Edit as you see fit! */

#include <stdio.h>
#include "popCount.h"

int main(void)
{
  const SWord8 __result = popCount(0x1b02e143e4f0e0e5ULL);

  printf("popCount(0x1b02e143e4f0e0e5ULL) = %"PRIu8"\n", __result);

  return 0;
}
== END: "popCount_driver.c" ==================
== BEGIN: "popCount.c" ================
/* File: "popCount.c". Automatically generated by SBV. Do not edit! */

#include "popCount.h"

SWord8 popCount(const SWord64 x)
{
  const SWord64 s0 = x;
  static const SWord8 table0[] = {
      0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3,
      3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4,
      3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2,
      2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5,
      3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5,
      5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 1, 2, 2, 3,
      2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4,
      4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
      3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4,
      4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6,
      5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 4, 5,
      5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8
  };
  const SWord64 s11 = s0 & 0x00000000000000ffULL;
  const SWord8  s12 = table0[s11];
  const SWord64 s14 = s0 >> 8;
  const SWord64 s15 = 0x00000000000000ffULL & s14;
  const SWord8  s16 = table0[s15];
  const SWord8  s17 = s12 + s16;
  const SWord64 s18 = s14 >> 8;
  const SWord64 s19 = 0x00000000000000ffULL & s18;
  const SWord8  s20 = table0[s19];
  const SWord8  s21 = s17 + s20;
  const SWord64 s22 = s18 >> 8;
  const SWord64 s23 = 0x00000000000000ffULL & s22;
  const SWord8  s24 = table0[s23];
  const SWord8  s25 = s21 + s24;
  const SWord64 s26 = s22 >> 8;
  const SWord64 s27 = 0x00000000000000ffULL & s26;
  const SWord8  s28 = table0[s27];
  const SWord8  s29 = s25 + s28;
  const SWord64 s30 = s26 >> 8;
  const SWord64 s31 = 0x00000000000000ffULL & s30;
  const SWord8  s32 = table0[s31];
  const SWord8  s33 = s29 + s32;
  const SWord64 s34 = s30 >> 8;
  const SWord64 s35 = 0x00000000000000ffULL & s34;
  const SWord8  s36 = table0[s35];
  const SWord8  s37 = s33 + s36;
  const SWord64 s38 = s34 >> 8;
  const SWord64 s39 = 0x00000000000000ffULL & s38;
  const SWord8  s40 = table0[s39];
  const SWord8  s41 = s37 + s40;

  return s41;
}
== END: "popCount.c" ==================