introsort

In 1997, he developed the sorting algorithm called Introsort (also known as introspective sort).

No mention of falling back to another algorithm, and no mention of introsort.

That's normally called "introsort".

Experiments were done comparing an optimized version of spreadsort to the highly-optimized C++ , implemented with introsort.

If it is known in advance that heapsort is going to be necessary, using it directly will be faster than waiting for introsort to switch to it.

In this sense, mergesort, heapsort, and introsort are asymptotically optimal in terms of the number of comparisons they must perform, although this metric neglects other operations.

David Musser's well-known introsort achieves practical performance comparable to quicksort while preserving O(n log n) worst-case behavior by creating a hybrid of quicksort and heapsort.

The GNU Standard C++ library, for example, uses a hybrid sorting algorithm: introsort is performed first, to a maximum depth given by 2xlog n, where n is the number of elements, followed by an insertion sort on the result.