I'm glad I finally found an alg that someone likes for speed! However, before the fact that I started looking at Kåre's algorithms in detail, I didn't realize that he actually had that algorithm in his 3 Flip text document. It was just surrounded by beginning and ending outer layer turns.
However, at the same time that I found that alg, I tested alternate but related paths. Here's the best one I found that uses L turns as well and could possibly be even faster than that alg.