Swaps the vanilla running forward with bow drawn for the vanilla sprinting forward with bow drawn animation.

Basically I got sick of running around like a monkey, looking to find mods that fix it there were a few to choose from, none of which I really liked, one added a load of other animations I found frankly disturbing and the others had a fairly serious excess of arm pumping.

Finally I found "Running with your Bow Replacer by Linique/bethesda" on skyrim nexus:

http://skyrim.nexusmods.com/mods/12426

While it seemed to improve on the diagonal movement with a drawn bow it still had the excessive arm pumping that's not to my personal taste so I simply swapped out the vanilla running animation and the one from that mod and overwrote it with the vanilla sprinting animation.

In short I would personally recommend installing "Running with your Bow Replacer by Linique/bethesda"...

http://skyrim.nexusmods.com/mods/12426

...then installing this on top overwriting the single animation for running.

See the video I'll link on youtube, it shows in sequence the vanilla animations in use then only this mod installed alone and finally my recommended setup of this mod installed on top of and overwriting the single animation from "Running with your Bow Replacer by Linique/bethesda".

http://skyrim.nexusmods.com/mods/12426