An interesting way to get the popcount, or the number of bits set in an integer is via a SWAR (SIMD Within A Register) algorithm. While its performance is very good (16 instructions when compiled with gcc -O3), why the algorithm works is somewhat opaque:

int swar(uint32_t i) {
  i = i - ((i >> 1) & 0x55555555);
  i = (i & 0x33333333) + ((i >> 2) & 0x33333333);
  return (((i + (i >> 4)) & 0x0F0F0F0F) * 0x01010101) >> 24;
}

For the time being, we assert that the above function is the same as the more verbose one below. Note how we distribute integer addition over bitwise & in (C) – this isn’t true in general and will be justified later.

int swar_expanded(uint32_t i) {
  uint32_t j = (i >> 1) & 0x55555555;
  i = i - j; // (A)
  i = (i & 0x33333333) + ((i >> 2) & 0x33333333); // (B)
  i = (i & 0x0F0F0F0F) + ((i >> 4) & 0x0F0F0F0F); // (C)
  i = i * 0x01010101; // (D)
  return i >> 24;
}

(For the rest of this post, β(n) is the number of ones in the binary representation of n (so swar(n) calculates β(n) for a 32 bit n); and ⟨ a b c … ⟩ denotes the binary number that has a as its more significant bit, b as its second most significant bit and so on.)

Notice how a 32 bit integer can been seen as a sequence of 32 1 bit integers, each with the property that the integer contains the number of high bits in itself? Statement (A) of the algorithm expands this condition to hold for two bits. If i is ⟨ b0 b1 b2 … ⟩, then this is how j lines up with it:

i = ⟨ b0 b1 b2 b3 b4 b5 b6 b7 b8 ... ⟩
j = ⟨  0 b0  0 b2  0 b4  0 b6  0 ... ⟩

Since ⟨ b0 b1 ⟩ ≥ ⟨ b0 ⟩, carry from one 2-bit tuple can’t “spill over” to a higher 2-bit tuple. This means bits 2k and (2k + 1) of (i - j) is determined by corresponding bits of i and the transform can be completely described by this table:

T(⟨ 0 0 ⟩) = ⟨ 0 0 ⟩
T(⟨ 0 1 ⟩) = ⟨ 0 1 ⟩
T(⟨ 1 0 ⟩) = ⟨ 0 1 ⟩
T(⟨ 1 1 ⟩) = ⟨ 1 0 ⟩

Do you see the pattern above? T(⟨a b⟩) is β(⟨a b⟩)! If i was ⟨ b0 b1 b2 … ⟩, (i - j) is ⟨ c0 c1 c2 … ⟩ such that ⟨ c(2k) c(2k + 1) ⟩ is β(⟨ b(2k) b(2k + 1) ⟩).

Statement (B) expands this condition further to hold for 4 bit sequences, as can be seen from how (i & 0x33333333) and ((i >> 2) & 0x33333333) line up:

    0  0 c2 c3  0  0 c6 c7  0  0 ...
 +       c0 c1  0  0 c4 c5  0  0 c8 c9 ...
 = d0 d1 d2 d3 ...

In the above example, the first four bits had ⟨ c0 c1 ⟩ + ⟨ c2 c3 ⟩ ones (because of (A)); which is exactly equal to ⟨ d0 d1 d2 d3 ⟩. Statement (C) expands this condition to hold for groups of 8 bits in exactly the same way.

In (D), multiplying by 0x01010101 sets the most significant byte of the result to the sum of the all the bytes (assuming there are no inter-byte carries). This is exactly what we were trying to compute — the total number of high bits in the input! We right shift by 24 to get to the MSB and return it.

The optimized version combines (C) to ((i + (i >> 4)) & 0x0F0F0F0F) = (E), so that we do one bitwise & instead of two. While (a + b) & c is not always the same as (a & c + b & c), they are equal in this case. To “prove” this informally, we see that for i = ⟨ x0 x1 x2 … ⟩ (C) is

    0  0  0  0 x4 x5 x6 x7 ...
 +  0  0  0  0 x0 x1 x2 x3 ...

and (E) is

   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 ...
 +  0  0  0  0 x0 x1 x2 x3 x4 x5 ...
 &  0  0  0  0  1  1  1  1  0  0 ...

and that for (C) and (E) to be different, an overflowed bit should have changed bit 7. But there won’t be an overflow past, say, bit 12 since ⟨ x12 x13 x14 x15 ⟩ + ⟨ x8 x9 x10 x11 ⟩ ≤ 8 and hence can fit in four bits.

As a concept check, I suggest implementing a 64 bit version. Here is my solution:

int pop_cnt_64(uint64_t i) {
  i = i - ((i >> 1) & 0x5555555555555555);
  i = (i & 0x3333333333333333) + ((i >> 2) & 0x3333333333333333);
  return (((i + (i >> 4)) & 0x0F0F0F0F0F0F0F0F) *
          0x0101010101010101) >> 56;
}

This is probably slightly slower than the 32 bit version since the integer literals now need to be moved into registers explicitly before they can be used (they’re too large to be embedded as immediates). Note that the patterns for the integer literals remained the same. How will they change for a (hypothetical) pop_cnt_512?