Note that "Attack Routine" does not mean a level 14 fighter can attack twice when an opponent flees. A level 14 fighter has two "attack routines" per round, each "attack routine" being a single attack (or grapples, etc.) (See DMG pages 62-3).Here is the relevant rule, from Gygax's DMG p 70:
Breaking Off From Melee:At such times as any creature decides, it can break off the engagement and flee the melee. To do, however, allows the opponent a free attack or attack routine. This attack is calculated as if it were a rear attack upon a stunned opponent.
Claw+Claw+Bite for an animal is an attack routine. (See the dragon entries, which have text like "The attack of a gold dragon can be a claw/claw/bite routine...")
A hasted animal could use their attack routine twice - but not then do a double claw/claw/bite routine against a fleeing foe.
Cheers,
Merric