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Weapon Procs and Triggered Weapon Effects
One of the main reasons you want to avoid losing regular swings is the effect this loss has on procs and triggers. With less swings, these often powerful effects are less likely to be activated. This is because the chance for a weapon to trigger a proc is calculated as a percentage chance for an unhasted weapon of that weapon's particular delay.
Procs have descriptions like "Procs 3.2 times per minute." What this means is that a short delay weapon (1.2 seconds for example) has a 6.4% chance to proc on any given swing while a slower weapon (4 seconds for example) has a 21.33% chance to proc on any given swing. That in itself is not a point in favour of slow weapons as they will still proc the same number of times over any given time-frame. However, the fast weapon loses far more swings due to auto-attack queueing than the slow weapon as a proportion of the total number of swings.
The proportion of lost attacks is more than simply the factor of its higher weapon speed. Because it is so much harder to squeeze spells and combat arts in between fast swings you are more likely to "double queue" auto attacks and lose more auto attacks more often. Thus, a proc effect that would normally be 3 procs per minute can easily be cut down to 2 procs per minute or worse.
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What About Casting Times?
This is somewhat simple. The longer the casting time of the spell or combat art being used the more likely you are going to mess up the timing of auto attacks and miss out on one of them. Instant attacks still have an animation time, so that needs to be kept in mind as well. Remember that one auto attack can be queued, so your timing does not have to be perfect. The problem only really arises when it is time for a new auto attack to happen when one is already queued. You cannot have more than one auto attack queued at the same time.
This guide explains the mechanics behind weapon speeds in EverQuest 2.