Thrill of the Spreadsheet: Bestial Wrath in Legion

I touched on this last week, when talking about the datamined tier 19 set bonuses, but I wasn't happy with my approximation of Wild Call's effect on Bestial Wrath up-time. So, I spent a bit more time thinking about how to more accurately model the relationship between Dire Beast resets and Bestial Wrath. It's probably worth noting up front that nothing I'm going to talk about today will have the slightest effect on how you play your BM hunter in legion.  It's just for funsies.  If you find math/spreadsheets fun.  As such, I decided to include a really chill sitting-on-the-back-porch-drinking-coffee kind of song, just to set the mood.

Bestial Wrath and Dire Beast

If you haven't been following the changes coming in Legion, as it stands now, Bestial Wrath lasts for 10 seconds, and has a 90 second cooldown.  While this may at first glance seem like a fairly significant nerf from the current version of Bestial Wrath, which last 15 seconds, has a 60 second CD, and reduces the focus cost of all abilities by 50%, they've also added a mechanic to Dire Beast which reduces the CD of BW by 15 seconds each time you cast a Dire Beast, as well as increased the damage buff.

Dire Beast received some other changes as well; most notably, it will now be baseline, instead of a talent.  In addition, Dire Beast will have a 10 second CD, and will be replacing the current version of Cobra Shot as our only ability that generates focus for the hunter (our pets still have the passive Go for the Throat ability).  

What this means for our rotation is if we use Dire Beast on CD, we're looking a net CD of 30 seconds.  That is, if "x" is the number of seconds between casts of Bestial Wrath, and we BW and DB at the 0 second mark, by the 40 second mark we will cast our 5th Dire Beast: 90 sec [base cd] - 30 sec [elapsed time] - 4*15 [CD reduction from Dire beast casts] = 0 sec".  

Wild Call

Were that all we had to deal with, I probably wouldn't be writing about it in this blog. There is an additional mechanic being introduced, though, called Wild Call. Wild Call gives you a 20% chance to reset the cooldown of Dire Beast any time you crit.  

This introduces a couple new elements into our calculations.  For one, our crit percentage is going to be a factor in determining how often we cast Bestial Wrath.  The second part is more difficult to define, but essentially it's our "total chances to crit".  This is affected by a couple of things, but primarily, our haste / attack speed, which in turn of course affects our base focus regen, which also plays a large role in our total chances to crit.  

Instead of saying "Crit Percentage" and "Chances to Crit", from this point forward, I'm going to define "C" as the floating point of my crit percentage; that is, if I have 30% crit, C = 0.30.  I should not here that "C" will be different from what's shown on your character stat sheets depending on what target you're hitting.  Against raid bosses, there will be the level difference crit supression. For our purposes here, we'll use for our value of "C" your actual chance to crit.

I'm also going to use "T" as how often I have a chance to crit (and therefor a chance to proc wild call). "T" is then the inverse of my actions per second (APS).  As it currently works on the Legion Alpha, only Auto-Shot Crits can proc Wild Call.  However, due to the wording of the tool tip, I'm currently working under the assumption that this will eventually change, most likely not to include all abilities (like pet abilities), but at least all of the hunter shots. Regardless, as this ability is reiterated, we will just need to change our value "T", and the rest of the formula should remain the same.

Skoll, Hati and I, hanging out around the bonfire at Trueshot Lodge
after a rather long battle against the training dummies, trying to gather
enough data to definitively claim Wild Call was only proc'ing on
Auto-Shots.  Also, I found Skoll on my first trip to Northrend on the
Legion Alpha, so that's kind of cool.  This "caption" is getting quite
long...

With only auto-shots proc'ing Wild Call, T will equal "3/(1+H)" where H is our Haste percent in floating point. For example, if we have 15% haste, then we have "T = 3/1.15 = 2.6".  If additional abilities are able to proc Wild Call in later versions, then we'll just average those in.  That is, if Cobra Shot also can proc Wild Call, and my rotation uses 3 Cobra Shots every 6 seconds, then that would average out to an additional one chance to crit every 2 seconds; between auto-shots every 2.6 seconds and cobra shots every 2 seconds, that would mean that T=1.13. 

With known values for "T" and "C", we can put together a little calculator.  To determine how often we can expect to proc Wild Call, we'll use the duration that would give us an expected value of 1 Wild Call proc.  I'll define that duration as "W" (in seconds).  So "W = T/(C*0.2)",  that is to say, "C*0.2" is our chance to proc Wild Call on each hit, then "T/(C*0.2)" will give us the amount of time it will take until our EV is raised to 1.

Wild Call's affect on Bestial Wrath

We have Dire Beast's affect on Bestial Wrath already, from above:

0 = 90 - x + ((x%10 + 1) * 15);
 x = 30

What we're trying to accomplish is having the difference between the CD of Bestial Wrath  and the sum of the time elapsed and the CD reduction due to Dire Beast casts equal 0.  If Wild Call procs just gave us additional time reduced form the CD of BW, we could just include the product of the expected value of WC and the CD reduction in our equation there.  But instead, it resets the CD of Dire Beast, giving us the additional 15 seconds off the BW CD, but also pushing back when the next Dire Beast will happen.

Instead, what we need to do is represent the Dire Beast resets from Wild Call averaged out as a flat CD reduction to all Dire Beast CDs.  There are a couple of considerations we want to cover here, too.  For one, Wild Call doesn't reduce the CD of Dire Beast by a fixed amount; it reduced the CD by 0 - 9 seconds.  That is, if Wild Call procs between 0 and 1 second after you cast Dire Beast, then you'll have 9 seconds removed from the CD (since you'll be able to cast Dire Beast again on the next GCD); if instead, Wild Call procs between 9 and 10 seconds after you cast your last Dire beast, it won't reduce the CD of Dire Beast at all, since the next available time you'd be able to cast it, you were going to be casting it anyway.  Because Wild Call could proc anywhere between 0 and 9 seconds, we'll call that an average of a 4.5 second CD reduction.  

For eery duration "W" (as we defined above: "W = T/(C*0.2)"), we can expect to have a 4.5 second reduction in the CD of Dire Beast.  "(W-4.5)/W" will then give us the percent of our original Dire Beast CD we should experience (again, on average), and if we multiply that by our base Dire Beast CD, "10*(W-4.5)/W", we'll get the average Dire Beast duration.  We can then substitute that for the "10" that appeared in the original formula for determining the Bestial Wrath CD, to find what our average CD of BW should be:

90 = x + ((x/D + 1) * 15);
x = 75D/(15+D)

From here, we can start looking at how our Attack Speed / Haste and our Crit rating affect Bestial Wrath up-time.  In the table below, I plugged in a few different values for both T (how often we have a chance to proc Wild Hunt) and C (our crit percentage).  

T =	2.60	2.60	2.60	2.00	2.00	2.00	1.13	1.13	1.13
C =	20%	30%	40%	20%	30%	40%	20%	30%	40%
Avg DB CD =	9.31	8.96	8.62	9.10	8.65	8.20	8.41	7.61	6.81
Avg BW CD =	28.72	28.05	27.36	28.32	27.43	26.51	26.94	25.24	23.43
BW Up-time =	34.8%	35.7%	36.5%	35.3%	36.5%	37.7%	37.1%	39.6%	42.7%

If you'd like to recreate this little calculator so you can play with the values T and C, I used roughly the following:

T =	___
C =	___%
Avg DB CD =	=10*((B1/(B2*0.2))-4.5)/(B1/(B2*0.2))
Avg BW CD =	=75*B3/(15+B3)
BW Up-time =	=10/B4

Problems and Actual Uses:

There are some fun little factoids that come out of this.  For instance, at 32.25% crit, you'll have an expected value of 1 Wild Call proc per Barrage (if it turns out that Barrage crits are able to proc Wild Call).  That doesn't mean you'll be gauranteed a proc, but on average, you should get one every Barrage. 

That, however, brings up a problem with our formula above.  Any time we have more than one chance to crit each second, or T < 1, then we'll run into GCD trouble.  With Barrage, for example, you'll never get more than one Dire Beast reset from Barrage, because you won't be able to use Dire Beast again until you're finished casting Barrage, meaning there will be no CD to reset, and of course, if you break your cast to cast a Dire Beast, you can't start the Barrage cast back up.  This would be true of anything happening more than once per second, like if you cobra shot and auto-shot during the same GCD (which is fairly common).  Overall, the value of both crit and higher attack speeds / haste will still continue to benefit you, but it won't be at a linear rate.  

There is one practical use for the above math, and that is determining whether or not it's going to be more valuable to use BW exactly on CD, or if you can save it until right before Dire Beast.  In the formula above, we're assuming a Dire Beast is used on the same GCD that Bestial Wrath is cast, directly following it, in fact.  If you remove the "+1" from "90 = x + ((x/D + 1) * 15)", you'll get the average CD of Bestial Wrath where you instead use BW directly after casting Dire Beast.  While playing in game, pretty much all of the time BW will be coming off CD due to casting a Dire Beast (which reduces the CD of BW by 15 seconds, as stated above).  So we can compare the predicted up-time of Bestial Wrath with different values for the time between casting BW and DB.  

Finding My People

If it's not clear from the 2000 words I just wrote about finding the EV of BW's CD; I perhaps have somewhat bizarre interests, at times.  I don't do it on purpose or anything, I just find myself wondering about things that I guess not everyone wonders about. As plenty of people make it clear that they have absolutely no interest in the things I find interesting, it always kind of catches me off guard when I find people who will not only put up with my interests, but explore them and encourage them. So, I wanted to take a minute to say thanks to all of you out there who kind of play the role of sounding board for me, and push me in the right directions when I get stuck.  

When I was writing the second line of this blog post: "I spent a bit more time thinking about how to more accurately model the relationship between Dire Beast resets and Bestial Wrath", I couldn't decide whether or not the phrase "more accurate" made sense. That is, is something either accurate or not, or are there degrees of accuracy? So I threw the question up on twitter, and actually got a lot of responses from people who had obviously thought about the difference between gradable and absolute adjectives.

Maybe that's a really small thing, but it struck me as pretty great that I've just kind of naturally found a group of people who enjoy thinking about things.  Left me feeling a bit sentimental. In one of my favorite lines ever written, by probably my favorite author of all time, Kurt Vonnegut responds to a question about why he continues to write, which he posed to himself in the book Time Quake:

Still and all, why bother?  Here's my answer: Many people need desperately to receive this message:
"I feel and think much as you do, care about many of the things you care about, although most people don't care about them. You are not alone."

Thanks for being so cool.