Time for Prof. Angus to weigh in, here. comparing the efficiency of the two is like comparing apples and oranges. Both were equally efficient. Also, you need to differentiate war bows from general service hunting bows. We know that Welsh longbowmen could fire between 6-10 arrows per minute with the war bow. We know they were lethal at 400 paces from battlefield accounts. We know that they could pierce 4 inches of oak. Many of you may not know that yew in England after the first quarter of the 1300's was too spongy for decent bows. That's why a "stave" tax was imposed on wine imported from the continent. Prior to about 1305, the weather was warm and dry enough in England that their wine was better than French wine, but with the onset of the "little ice age", that changed. The hayday of the longbow was during the 100 years' war. By the time of the battle of Crecy (1346), longbows were made from yew imported from Portugal, Spain, even France! Jeremy makes a telling point about humidity, hence the need to stop using English yew.
if you define "efficiency" as firing a faster arrow further and greater rate of fire, you'll have to give it to the Asiatic composite bow every time. Any questions of rate of fire can easily be answered by watching Lucas Novotny of Saluki Bows. Efficiency in warfare I would also give to the composite bow. Bear in mind that troops would not have armor sufficient to ward of an arrow-only those of higher rank! the grunts of the day wore-at best-hardened leather and a quilted undergarment. Either bow would be effective against that. You'll find it's how they were employed in warfare that differentiates the bows, and here, it's apples etc. Longbows were fired from a standing platform, composites were fired from horseback at a full gallop. history has battles where the yeoman archer did not prevail: Genghis and the boys never lost a single one!