Warrior's Charge for TP generation

A typical way to use Warrior's Charge

When I had the one perfunctory merit for Warrior's Charge—not like there was anything compelling in Group 2—I reserved it exclusively for weapon skills instead of TP gain. For an ability that can be used on demand when available, this is a typical application especially for "zerging," when you can't really ensure that the potential TP gain from the guaranteed double attack will let you squeeze out another WS in the 45 seconds of Mighty Strikes. But to be honest, I liked the big numbers for Steel Cyclone.

A more efficient way to use Warrior's Charge?

Of course, if you're doing some long-term activity, like meriting, it is theoretically more efficient (that word again...) in the "long run" to use Warrior's Charge in the auto-attack phase than for weapon skills. By "long run," I'm basically referring to TP-burning, where the benefit of the average increase in TP gain from Warrior's Charge actually and most ideally manifests in higher average WS frequency per unit time.

Would you actually want to waste 22 merits to reduce the recast time to 5 minutes, though? Perhaps it would help to quantify the effective increase in double attack rate from using Warrior's Charge every five minutes.

Expressing the effect of Warrior's Charge as a rate of double attack

First, a preliminary observation. While Warrior's Charge confers an absolute increase of one double attack per use, its relative contribution to long-run rate of damage decreases with increasing delay reduction.

For example, if there are few attack rounds in the time period between uses of Warrior's Charge, then the contribution of Warrior's Charge is relatively large. But if there are many attack rounds in that window, then the contribution of Warrior's Charge is relatively small.

With that in mind, it is necessary to make an explicit statement about the amount of delay reduction present before determining the effective double attack rate with ideal use of Warrior's Charge. I acknowledge that the effect of Warrior's Charge is "discrete," and not continually present, but the rate of double attack is just an expected value (average) anyway, a mathematical conceit, so it's natural to account for the effect of Warrior's Charge in a weighted average, which is the "effective" double attack rate in the long run (to be shown later).

As an example, suppose that you have 5/5 Warrior's Charge, for a five-minute recast. The average DA rate for Warrior's Charge, one double attack every five minutes, is equivalent to 2.8% DA given 504 delay, and 2.296% DA given 504 delay and 18% haste. These values are calculated as follows:

\[\frac{1\ \mbox{DA}}{5\ \cancel{\mbox{min}}}\cdot\frac{1\ \mbox{min}}{60\ \mbox{s}}\cdot\frac{1\ \mbox{s}}{60\ \mbox{delay}}\cdot\frac{504\ \mbox{delay}}{1\ \mbox{round}}\cdot100\% =2.8\%\ \frac{\mbox{DA}}{\mbox{round}}\]
\[\frac{1\ \mbox{DA}}{5\ \mbox{min}}\cdot\frac{1\ \mbox{min}}{60\ \mbox{s}}\cdot\frac{1\ \mbox{s}}{60\ \mbox{delay}}\cdot\frac{504(.82)\ \mbox{delay}}{1\ \mbox{round}}\cdot100\% =2.296\%\ \frac{\mbox{DA}}{\mbox{round}}\]
Again, the length of the interval between attacks, which is affected by delay reduction, determines the relative contribution of Warrior's Charge on a per-round basis.

That makes sense, but how does Warrior's Charge actually affect the effective double attack rate?

The effective double attack rate, which takes into account the contribution of Warrior's Charge, is not the sum of the base DA rate and the contrived DA rate from Warrior's Charge. It's a weighted average based on how often WC takes effect, ideally as often as possible, or once every five minutes. That may be a confusing statement, so I provide further corny explanation.

Approaching this question from a probabilistic point of view, the above rates can be treated as the unconditional "probabilities" that Warrior's Charge takes effect in one attack round. There is nothing probabilistic about how often Warrior's Charge is used, but I am using probability language for the sake of explaining how the effective double attack rate is calculated.

If you are familiar with the phrase "percent of the time" appended to a number, perhaps this explanation will actually make some sense. Probability statements are often colloquially expressed in terms of "percent of the time." Given 0% delay reduction, Warrior's Charge "takes effect 2.8 percent of the time." Given 18% delay reduction, Warrior's Charge "takes effect 2.296 percent of the time," and so on.

Given that Warrior's Charge has just been used, the "probability" of a double attack in the subsequent attack round is 1; otherwise, it is whatever your DA rate normally is. Therefore, the "effective" double attack rate is just a weighted average, an application of the law of total probability treating the DA rates as probabilities. As a probability statement, the effective double attack rate is

\[P(\mbox{DA}) = P(\mbox{DA} \mid \mbox{WC})P(\mbox{WC}) + P(\mbox{DA} \mid \overline{\mbox{WC}})P(\overline{\mbox{WC}})\]
Note that this expression is valid in extreme hypothetical cases. If your delay reduction approaches 100%, the relative effect of Warrior's Charge tends to 0. Therefore, your effective DA rate (which is a semi-probability, so to speak) cannot exceed 1. If you never use Warrior's Charge, then your effective DA rate is just your base DA rate.

To give an explicit example, suppose my DA rate from Warrior's Charge is 2.296% (shown above) and my base DA rate (before Warrior's Charge) is 19%. Then, my effective DA rate is

\[P(\mbox{DA}) = 1(.02296) + (.19)(1-.02296) = .2085976\]
This effective double attack rate, which accounts for the discrete contributions of Warrior's Charge, can then be used to estimate the average number of rounds to 100 TP or the average number of hits in a weapon skill.

Showing that Warrior's Charge is better (in the "long run") for TP gain than for weapon skills

Certainly, if you don't find occasion to use Warrior's Charge for meaningful TP gain, you might as well use it for weapon skills. No one ever said anything about not using one's discretion and judgment.

Still, it is easy to argue that you get more out of Warrior's Charge for TP spamming without doing arithmetic. In the auto-attack phase, on average the extra DA increases TP gain, leading to higher WS frequency, and also contributes to auto-attack damage. For weapon skills, the extra DA merely gives a slightly higher average TP return and slightly higher WS damage.

Numbers provide a nice summary, however, so I present the results of some number-crunching for TP spamming with

• sufficient Store TP for a "6-hit" setup (5 hits to 100 TP given sufficient TP return from the previous weapon skill)
  
• a 3-hit weapon skill (like Raging Rush or King's Justice)
  
• 18% delay reduction
• 5/5 Warrior's Charge (so 2.296% DA rate from WC)
  
• 19% base double attack rate (so effective DA rate of 20.86%)
  
• 95% hit rate
• 106 "base" damage for TP (average pDIF of 1)
• 159 "base" damage for WS (average pDIF of 1)
  

(All numbers in the following table are averages.)

Comparison of average damage per second for a "cycle" of auto-attack and WS damage (5 hits to 100 TP)

Use of Warrior's Charge	Rounds	TP Hits	WS Hits	Time (s)	DPS
5/5 Warrior's Charge only for TP gain	4.497	5.164	3.211	30.979	34.149
5/5 Warrior's Charge only for WS ("e-penis")	4.557	5.151	3.228	31.388	33.752
No Warrior's Charge	4.557	5.151	3.211	31.388	33.662

As expected, the e-penis approach is slightly better than nothing, but worse than the "optimal" approach, provided you actually have opportunities to use Warrior's Charge for TP generation. 22 merit points into Warrior's Charge only gets you up to a 1.446% improvement in theoretical, long-run rate of damage, which itself is an inefficient use of merit points compared to other warrior-specific options such as Group 1 double attack rate.

Moreover, as shown earlier, Warrior's Charge becomes relatively less effective for increasing WS frequency the more delay reduction you have, as it provides only a static increase in TP gain while being unaffected by delay reduction.

By this point, it should be easy to accept using Warrior's Charge for increasing WS frequency where applicable, but let's return to the 45-second Mighty Strikes "zerg." Warrior's Charge is relatively less effective with increasing haste (which is desirable for zerging), but the key word is "relatively." In that small time frame, you should still be better off using Warrior's Charge for the extra TP to get another WS off than to tack on another hit to a weapon skill.