There are some good books for people who'd like to learn more about how and why various bow types work. Archery - the Technical side is a classic from the '40's by renowned American physicists, articles by Tim Baker in the Traditional Bowyer's Bible-series are another, easier to get source that's largerly based on the first mentioned work.
There is no question that the Asiatic composite bow is a superior weapon to an English longbow, in terms of cast per pound. A 60-pound composite will shoot an arrow with up to 50 % more KE than a similar-weight longbow. A 120 lbs. war longbow will of course be more powerful than a typical Turkish flight composite you can see in archery books and museums (and what Saxton Pope based his theories about the composites' poor performance with heavy, English arrows on) , but not a war composite bow. Very few war composites have survived, but those few indicate that many had just as incredible poundages as the European war bows. Pound for pound, composites are more effective, and they pack their punch into a much more compact package than selfbows; try lugging a 80", wrist-thick piece of timber in a tight spot, then try a 48" composite. Longbows are much more cost effective: a skilled bowyer could produce four war bows from a single good (seasoned) yew trunk in a couple of days, the composite bowyer would need dozens of hours of work per bow, spread over months or even years, and harder-to-get materials.
The Mighty Longbow was nothing new in Europe in medieval times, as many authors would have it. The Vikings and the Germans, among others, used man-tall, very heavy yew longbows already in the Iron Age. What was exceptional in Medieval Europe was the Brits' systematic and successful endeavour to produce large numbers of achers who could effectively use 100+lbs. bows in combat (the average modern woodbow shooter has all he can take in a 70lbs. stick).
Almost all of the old and all of the modern distance shooting records have been made using flight arrows, not war or even target arrows. Flight arrows use very low mass and thin, headless, rear-balanced shafting to reach those distances. The record bows, themselves, are typically made fast at the cost of dependability, a tradeoff no warrior would make. The real-life, combat distances achievable with either longbows or composites are much inferior to what you can read on the 'Net.
The English Longbow Standard Arrow (a standardized Medieval war arrow) distance record, made with an almost 200-pound ELB, and shot by probably the world's greatest modern war bow shooter, Simon Stanley, is 273 yards. So much for the "400 yard Longbow shots".
I have no knowledge of any verified tests made with a war composite and appropriate war arrows. Based on the performance of other types of composite gear , the extreme range of Asiatic war composites (there were many types, used with many types of combat ammunition) would probably be somewhere between 250 and 400 yards.