http://www.killingifrit.com/forums.php?m=posts&q=91713
Scroll down just over halfway and find the post by Ukiki. That seems to be the most accurate post I could find.
From what most have stated (most of them us the distance plugin too, it makes it easier, but almost too easy in my opinion, not sure if I'll use it or not when I start on rng again), the order from closest to farthest is: gun => xbow => bow. Roughly 10 meters is sweet spot for bow. To eyeball it, go to MG in Jeuno and go to SE corner. Your distance from the moogle is roughly 10m. The post about the MH is at the very bottom by Chiem. Hope this helps. ^^
Aegina, does using a macro with /equip range "whatever bow" not reset your tp? I know one time when I first started playing I went into equip window and re-equipped the weapon I was already using and it set my tp to 0. I dont know if this was a fluke, if macro'ing it prevents it, or if its been changed, but this was the problem I had.