It depends on the type of composite bow, and of course on the details of the individual bows.

If we compare bows of the same draw weight and draw length (and the same brace height), the composite bow has two key advantages:

1. The composite bow is a reflex-recurve bow, and has a convex force-draw curve, while a straight bow like a longbow has a slightly concave force-draw curve. This means that the composite bow stores more energy than the longbow (for the same draw weight and draw length).

2. A "generic" composite bow is smaller than a longbow, and has lighter limbs due to that smaller size. The lighter limbs mean that less kinetic energy is needed to move the limbs of the bow at a given speed, and that the composite bow is more efficient for arrows of a given weight.

Either of these mean that a given arrow is shot faster by the composite bow, provided more range, and better anti-armour performance, and potentially more accuracy. Both together mean an even bigger improvement.

However, such a composite would rarely shoot similar arrows to a longbow,. Since the longbow is big, with relatively heavy limbs, it is used with heavier arrows (which provides better efficiency, and better anti-armour performance, at the cost of range and arrow speed). Longbow arrows tend to be about 80-100g, while the "generic" composite bow is usually used with arrows of about 30g.

There are some composite bows as big and heavy as a typical Medieval longbow, such as the Manchu bow. Like the longbow, such bows are used with heavy arrows. Manchu war arrows are often about 125g; they are longer than longbow arrows since the bow is also a very long draw bow. Such a bow doesn't have the second advantage of lighter limbs above, but still retains the advantage of a convex force-draw curve (and the longer draw length means that even more energy is stored and delivered to the arrow for a given draw weight).

Composite construction is important for achieving compact light limbs while retaining the same draw length as a longer self-bow. The sinew and horn used in the composite bow can be stretched and compressed further than wood, and a shorter composite limb can be be bent into a tighter curve than a wooden limb, even with the pre-straining that reflex involves. Composite construction also makes it fairly simple to make the bow strongly reflexed and recurved, but it isn't necessary - for example, the Japanese yumi (laminated but not composite) is a reflex-recurve bow, and also a very long draw bow. The yumi pays a price for achieving that very long draw without composite construction: it's even longer and heavier than a European longbow.

While the composite bow is, in principle, better-performing than a longbow, the longbow has a major advantage: it's much cheaper and quicker to make. In Europe c. 1500, it seems that a gun might cost more than a hundred times as much as a longbow, while composite bows were typically more expensive than guns (which was an important reason for the Muscovite army to switch from the (composite) bow to the gun). (This doesn't mean that 100 archers could be equipped and supported for the cost of 1 gunner - they were paid similarly, and arrows were about the same price as a bow, and bows didn't last as long as guns.)

Further reading:

Measurements of force-draw curves and efficiency vs arrow mass for replicas of Turkish composite bows:

• https://www.atarn.org/islamic/Performance/Performance_of_Turkish_bows.htm

Some simple theory of why the efficiency depends on arrow mass:

• https://arxiv.org/abs/1101.1677