The normalized weapon damage for instant attacks is:
normalized_damage = base_weapon_damage + (X * Attack Power / 14)
where X is:
1.7 for daggers
2.4 for other one-handed weapons
3.3 for two-handed weapons
2.8 for ranged weapons
weapon speed for non-normalized attacks (see below)
Prior to patch 1.8, the weapon damage used for instant attacks was:
damage = base_weapon_damage + (weapon_speed * Attack Power / 14)
Note that ...
base_weapon_damage + weapon_speed * Attack Power / 14
is simply the formula for weapon (white) hit damage, as 14 AP = 1 DPS. The nomalized formula treats your weapon as if its swing time was a certain normal value for its type (1.7 sec for daggers, 3.3 sec for two-handed weapons, etc).
Nearly all instant attacks based upon weapon damage are affected by normalization; only a few are not.