I bet there is a specific optimization that recognizes common patterns for counting the number of set bits and replaces it with popcntq. There is no popcnt() function built into C, so everyone who needs it uses one of a few common implementations.
(I suspected it was this kind of thing, but I thought maybe it was something more generic.)
// step 1: Check if the loop-back branch is in desirable form.
// step 2: detect instructions corresponding to "x2 = x1 & (x1 - 1)"
// step 3: Check the recurrence of variable X
// step 4: Find the instruction which count the population: cnt2 = cnt1 + 1
It's very, very specific. But they allow that the loop has something more, see the resulting transformation at the end:
// After this step, this loop (conceptually) would look like following:
// newcnt = __builtin_ctpop(x);
// t = newcnt;
// if (x)
// do { cnt++; x &= x-1; t--) } while (t > 0);