Fast calculate hamming distance in C

Question

I read the Wikipedia article on Hamming Weight and noticed something interesting:

It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string. In this binary case, it is also called the population count, popcount or sideways sum.

[emphasis mine]

So something occurred to me. Could I calculate the Hamming Distance between two strings by XORing them and then taking the Hamming Weight (POPCOUNT) of the resulting string?

Something along the lines of this (using gcc intrinsics):

#include <stdint.h>

int hammingDistance (uint64_t x, uint64_t y) {
        uint64_t res = x ^ y;
        return __builtin_popcountll (res);
}

---

Now, as for why I would want to do this, well, on some platforms, yes, this would just translate to gcc emitting a call to a function that calculates popcount. For instance, on x64 without popcnt, gcc spits out (Godbolt's GCC Online):

hammingDistance:
    sub rsp, 8
    xor rdi, rsi
    call    __popcountdi2
    add rsp, 8
    ret

OTOH, if you have a platform that supports POPCOUNT, like x64 models including nehalem and after (which have POPCNT), you get (Godbolt's GCC Online):

hammingDistance:
    xor rdi, rsi
    popcnt  rax, rdi
    ret

which should be waaay faster, especially once inlined.

---

But back to the original question. Can you take the Hamming Weight of the XOR of two strings to find their Hamming Distance? ie:

HD = HW (x xor y)

Answer 1

Hamming distance between two equal length strings, x and y, is defined to be the number of positions where they differ. In the case of x and y being bitstrings, x^y is a string with 1s in exactly the positions they differ. Thus, HammingDistance(x,y) = Number of 1s in x^y, for bitstrings. Also, HammingWeight(x) = number of 1s in x for a bitstring x. Thus, your first claim, HammingDistance(x,y) = HammingWeight(x^y) is true for bitstrings. Having established that, it is clear that your implementation is correct.