[Agema]

Considering the people of the medieval and ancient era managed to do stuff as complex as build Hagia Sophia, make Attic pottery and craft intricate jewellery, I think we can expect the average medieval bowyer to be able to make a pretty reliable bow. I suspect a trained archer could also adapt to a new bow of the same basic design pretty quickly.

I saw some program where a modern guy did horse archery and could fairly reliably hit a conventional archery target at 30 yards or so whilst the horse was moving. Also, if you consider an archer has to hit a block of infantry/cavalry however much wide and maybe 4-10 ranks deep, he's got a fair bit of room for inaccuracy.