bubble sort with early exit is
for (size_t i = 0; i < n; i++) {
bool sorted = 1;
for (size_t j = 1; j < n - i; j++) {
if (a[j-1] > a[j]) {
item_t tmp = a[j];
a[j] = a[j-1];
a[j-1] = tmp;
sorted = 0;
}
}
if (sorted) break;
}
if you compile it https://godbolt.org/z/jdzov8cPn you can see that the inner loop goes from 10 instructions to 11. that doesn't guarantee that it's 9% slower (that depends a lot on the platform, and in particular on a big superscalar cpu it probably just uses a little more power but the same number of clocks) but it does tend in that direction. of course occasionally it will save you a pass or two over the array, which will tend to compensate, but it won't usually save you 9% of the passes
if some item is in position n and needs to be in position m, where m < n, it will require n - m passes of bubble sort to get there. a little thought suffices to show that, for randomly shuffled input, where m = 0 the item has to move about half the size of the array, where m = 1 it has to move about half the size of the array minus one half, where m = 2 it has to move about half the size of the array minus one; but it's the maximum of these three numbers which determines the number of passes after which all three of those positions will have the correct item, and that maximum is heavily skewed to be almost the entire size of the array. the chance that none of those three items is in the last 25% of the array originally is only .75*3 = 42%. and as the value of three increases, it gets worse. so in general early exit doesn't buy you much
as for modern implementations of bubble sort, they're all either didactic classroom exercises like this one or careless mistakes like the original freebsd code; because insertion sort always beats it, there aren't like a community of bubble sort implementors who have annual bubble sort conferences
Although I certainly agree that Bubble Sort is almost always a bad choice, it still is used in computer graphics, or other situations when you have only one or two misplaced pairs of elements. In this case it will outperform the other algorithms, in average case while maintainig functionally in the less than ideal situation.
Besides it is entirely irrelevant what are the purposes the algorithm is used today, didactic, deliberate or poor judgement, the de facto standard way of doing it today is to check for the early exit. Your argument about poor performance on the already sorted data is a straw man argument, your are stubbornly defending. Well ok, if you believe so...
it might help your intuition to think of it this way:
1. with random input, the first ten positions of output are pretty likely (65%) to have an item that was originally in the last 10% of the input, and almost certain (97%) to have an item that was originally in the last 30% of the input. so those ten items alone account for needing 95% of the passes you'd need without an early exit, minus about five, in 65% of cases.
2. the first 20 positions are pretty likely (64%) to have an item that was originally in the last 5% of the input, and almost certain (96%) to have an item that was originally in the last 15% of the input. so those 20 items account for needing 97% of the passes you'd need without an early exit, minus about ten, in 64% of cases.
3. the first 30 positions are pretty likely (64%) to have an item that was originally in the last 3.3% of the input, and almost certain (96%) to have an item that was originally in the last 10% of the input. so those 30 items account for needing 98.5% of the passes you'd need without an early exit, minus about 15, in 64% of cases.
of course, if you're only sorting 30 items, the first ten positions of output are actually a third of all the positions, so "minus five" is pretty significant. and probably if you're sorting more than about 30 items at a time you ought to use a linearithmic sort or a linear-time radix sort instead of a quadratic sort
similarly, it's reasonable to argue that complexity (in the sense of containing many parts, not in the sense of asymptotic computational work required) is subjective, and to disagree about it
but i think these are sort of moot points
insertion sort still beats bubble sort in the situations where you have only one or two misplaced pairs of elements, and generally replaces it as soon as somebody notices
as for 'the de facto standard way of doing it today', i suggest you argue with clrs, not with me
the wikipedia article has a cleverer early-exit version of bubble sort that keeps track of the last pair it had to swap; it will do 1.5n-1 comparisons in that case on average (because the swapped pair is on average in the middle of the array)
above i linked to working code on godbolt (modulo the extra unnecessary swaps on duplicate keys that you were kind enough to point out earlier); edit a comparison counter and a swap counter into it and you'll see
someone else might have written the original version, i can't remember any more, but i think it was probably me
it's true that, if your data is exactly sorted, or has a small number of items that are far too early (e.g., 43 3 4 5 6 7 8 9 10), you can modify bubble sort to notice this and exit early, at the expense of making it more complicated and even slower in the average case
however, this doesn't help if instead you have a small number of items that are far too late (e.g., 4 5 6 7 8 9 10 43 3), in which case bubble sort takes the same number of passes as it normally would, while insertion sort takes linear time. of course you can complicate bubble sort further into "shaker sort" to handle this, which makes it even more complicated but at least doesn't slow it down any further
but all of this is sort of futile because it's still half as fast as insertion sort in the case of unsorted input, and even without the tweaks, no simpler