recharge Formula: How to Calculate Total Recharge Time using Varying Recharge Buffs

[Bopper]

Update: Additional Content is provided in the comments below that includes derivations of this formula.

 

I presented this earlier in an unrelated forum post but I felt it was worth sharing to an audience looking for this type of information (or at the very least it makes it easier for me to find this formula when I need it again down the road 😁).

 

Have you ever tried to calculate how long it takes for a power to recharge when your recharge buffs are not constant? Perhaps you have Hasten that isn't perma'd, leaving you with 120 seconds of +70% Recharge bonus then an unknown amount of time until Hasten is off cooldown. Perhaps you had a Force Feedback Proc that triggers a +100% Recharge bonus for 5 seconds...so how does that impact your total recharge time of a power? Perhaps you cast Ageless Destiny and it provides a cascade of Recharge bonuses that diminish over time... how to you factor that in?

 

The good news, there is a VERY simple formula that you can use that incorporates every recharge buff you have. Here it is:

 

 

 

Examples:

The above formula might be difficult to decipher, but I'll try to explain. Any recharge buffs that are permanent, or temporary buffs that last at least as long as the Total Recharge Time, are added together and go into the denominator of the formula. So, let's say we have 95% recharge enhancement slotted into the power and we have 80% global recharge from outside sources (set IO bonuses, LotG +Rech, etc). This gives us a total of 275% recharge in the power (100% from base, 175% from boosts), thus the denominator becomes:

 

 

 

As for the numerator, that is simply the Base Recharge of the power minus the sum of the products of each temporary buff's duration and amount. So to continue our example, let's assume the power we are analyzing is Hasten which has a base recharge of 450 seconds (X) and provides a 70% recharge buff (B1) for 120 seconds (T1). How long does it take for Hasten to recharge? Plug it in.

 

 

 

So we are not quite perma-hasten, but at least we know we will have 13.1 seconds of Hasten being inactive. Now lets assume we used a power during this period that has the Force Feedback +Recharge proc in it. How much would we reduce the total recharge time if we got Force Feedback to proc once? That would be a 100% recharge buff (B2) that lasts for 5 seconds (T2). Let's plug it in.

 

 

 

As we can see, the single proc was able to reduce out total recharge time by an additional 1.8 seconds. This may not seem like much, but consider it like this: how much extra permanent recharge would we need to achieve the same performance as one Force Feedback proc?

 

 

 

I went on a slight tangent there, but it's interesting to see that just a single FF proc can have that much of an effect on our equivalent permanent recharge boost. Now that we've gone this far, let's expand our problem solving capabilities ever more. With no FF procs, we had a recharge of 133.1 seconds. How many FF procs would we need to get Hasten down to 120 seconds or less (perma-Hasten)? To solve this, we will need to make a small tweak to the formula. Since we are assuming we are achieving perma-Hasten, we can no longer treat Hasten's recharge buff as temporary. Instead we treat the buff as permanent and it will go into the denominator of our formula. Also, since we don't know how many FF procs we need, we incorporate a variable that we will solve for (N):

 

 

 

From the results we now know that we would need the FF to proc at least 8 times to achieve Perma-Hasten. So let's finish this example off by answering if we did get 8 FF procs within a 120 second window, what would Hasten's final recharge time be?

 

 

 

Potential Oops:

Now, let's say we didn't know Hasten would become perma'd with 8 FF procs and we had used the original formula (thus treating Hasten as a temporary buff and using its values in the numerator). What could we have done to know we made a mistake? Simple, whenever you calculate the Total Recharge Time of a power and it results in being less than a temporary buff's duration, that buff needs to be treated as a permanent buff and you need to re-do the calculations accordingly. In our example, we would have calculated T = 118.5 seconds, which is less than 120 seconds and ultimately a wrong answer (although in this case, it would have been close).

 

Destiny Example:

Alright, that is a lot but hopefully everyone is starting to understand how to use the formula to calculate these recharge times. Let's now look at another example of Hasten where it gets paired with another 120 second duration power in the T4 Ageless Destiny. This power provides a 40% recharge buff for 10 seconds, a 10% recharge buff for 30 seconds, a 10% recharge buff for 60 seconds, plus a 10% recharge buff for 120 seconds (they all stack). Let's see if Hasten, with the T4 Ageless Destiny, and the same permanent boosts we've been using throughout (95% enhance, 80% global) would be enough to make Hasten permanent.

 

 

 

In this example, we still would not achieve perma-Hasten. We could solve for what amount of permanent recharge boost would be needed to achieve a total recharge time of 120 seconds:

 

 

 

So close.  If we had another 9.167% recharge (284.167% total), we would be able to achieve perma-Hasten through the use of Hasten and T4 Ageless Destiny.

 

So there you have it. A simple single formula for solving a power's recharge time. I hope it was helpful. Bookmark this info somewhere because I guarantee you will use if at some point while building up a character in a Hero Designer tool (e.g. Mid's Reborn) and you will click on a buff and not trust the recharge numbers it spits out.

 

Edited October 25, 2019 by Bopper

[EmmySky]

My Aunt wrote calculus textbooks.  I can barely add to 20 without taking off my shoes.  How about I out-source all my math needs to you?

 

Honestly, its all Greek to me, but kudos on finding a way to incorporate so many variables into a 'simple' equation you can use.  (Emphasis on you because, well, I will be calling on folks like you with better brains if I need anything this complex.)  

[MetaVileTerror]

I do hope you were being intentionally facetious with that all-cap'd "very" in the third paragraph . . . 

[roleki]

I dunno if I would consider a formula to be any kind of 'simple' if I don't even have the fonts on my computer to type it.

 

But I am glad YOU have them!

[golstat2003]

Or I could just check the Monitoring function in game that does it for you already.

[Bopper]

4 minutes ago, golstat2003 said:

Or I could just check the Monitoring function in game that does it for you already.

I am unaware of a monitoring function in game that factors in changing recharge boost levels. I know the countdown timer calculates the remaining time using the current recharge boost levels, but that is not what I'm doing here.

 

2 hours ago, roleki said:

I dunno if I would consider a formula to be any kind of 'simple' if I don't even have the fonts on my computer to type it.

 

But I am glad YOU have them!

I'm concerned. What do you mean you dont have the fonts? Are you unable to access the four mathematical functions used (plus, minus, multiply, divide)? Perhaps you're talking about the sigma summation symbols I used (looks like a triangular "E")? If so, I would hope my breaking it out would have explained it, or even possibly the many examples I showed would have explained it. If not, let me know, I'll be happy to help you understand the formula.

 

4 hours ago, MetaVileTerror said:

I do hope you were being intentionally facetious with that all-cap'd "very" in the third paragraph . . . 

Nope. Very easy formula. We are merely using math concepts from 4th grade and below: addition, subtraction, multiplication, and division. However, my use of mathematical notations more commonly seen in high school and college might have scared people off. I will help people out and provide a simple excel formula that does all this in one line later (after work when I get on my personal laptop).

5 hours ago, EmmySky said:

My Aunt wrote calculus textbooks.  I can barely add to 20 without taking off my shoes.  How about I out-source all my math needs to you?

 

Honestly, its all Greek to me, but kudos on finding a way to incorporate so many variables into a 'simple' equation you can use.  (Emphasis on you because, well, I will be calling on folks like you with better brains if I need anything this complex.)  

It was my hope that this would be easy enough for everyone to grasp, but admittedly my experiences are biased. I'm used to teaching/writing for engineers and statisticians and my methods are probably not going to be grasped by everyone. That's a failure on my part for not delivering the material properly to my intended audience. I hoped my examples would flush that out, but maybe I missed something. I am glad you picked up on the fact that this formula takes all variables and simplifies them into a succinct function. I was going to show everyone the painstaking iterative method I used to do, but judging by the replies I'm glad I didn't. It was ugly lol.

[golstat2003]

3 hours ago, Bopper said:

I am unaware of a monitoring function in game that factors in changing recharge boost levels. I know the countdown timer calculates the remaining time using the current recharge boost levels, but that is not what I'm doing here.

 

I'm concerned. What do you mean you dont have the fonts? Are you unable to access the four mathematical functions used (plus, minus, multiply, divide)? Perhaps you're talking about the sigma summation symbols I used (looks like a triangular "E")? If so, I would hope my breaking it out would have explained it, or even possibly the many examples I showed would have explained it. If not, let me know, I'll be happy to help you understand the formula.

 

Nope. Very easy formula. We are merely using math concepts from 4th grade and below: addition, subtraction, multiplication, and division. However, my use of mathematical notations more commonly seen in high school and college might have scared people off. I will help people out and provide a simple excel formula that does all this in one line later (after work when I get on my personal laptop).

It was my hope that this would be easy enough for everyone to grasp, but admittedly my experiences are biased. I'm used to teaching/writing for engineers and statisticians and my methods are probably not going to be grasped by everyone. That's a failure on my part for not delivering the material properly to my intended audience. I hoped my examples would flush that out, but maybe I missed something. I am glad you picked up on the fact that this formula takes all variables and simplifies them into a succinct function. I was going to show everyone the painstaking iterative method I used to do, but judging by the replies I'm glad I didn't. It was ugly lol.

Yeah you need to keep stuff like that a bit more simple. I was able to follow it cause of my years of experience in IT. Not everyone has that. 😛

[Inverted]

Thank you!  That's exactly what I needed to figure out of my fort's mind link would be perma or not.

[Bopper]

2 minutes ago, Inverted said:

Thank you!  That's exactly what I needed to figure out of my fort's mind link would be perma or not.

You're welcome, I'm so glad it worked. 

41 minutes ago, golstat2003 said:

Yeah you need to keep stuff like that a bit more simple. I was able to follow it cause of my years of experience in IT. Not everyone has that. 😛

I would certainly love to make it clearer, I'm just stuck on how to better explain that all you're doing is adding up your total permanent buffs. Then you just add up the product of all your temporary buffs with their durations. I can maybe derive the formula, if it would help people see the steps. But who knows. I'm open to any suggestions. 

[quixoteprog]

I'm not sure if this is accurate.

 

I am only vaguely aware of the Force Feedback proc so forgive my ignorance.

 

I am assuming that the FF proc is a global recharge bonus, since you apply it to Hasten. But if that is the case then would not this be a differential function? Not a simple sum? Meaning that with each FF proc you will be bringing Hasten closer to perma, resulting in the percentage of time for which it is in effect being longer thus reducing the need for the FF bonus to make it perma.

 

Does that make sense?

 

So, Hasten lasts for 2 minutes and if I have its recharge down to 3 minutes it will be in effect for 67% of the time. Meaning it gives the equivalent of 47% (.7 X .67 )bonus that is perma.

 

If I reduce it's recharge by 10 seconds to 2:50 that means (.7 X .71) it is now the equivalent of a 49% perma bonus.

 

Each subsequent proc will increase the effect of any other non permanent bonus, obviously, like Accelerated Metabolism.

 

Am I missing something? I'll admit I did not read through all of your responses.   

Edited October 23, 2019 by quixoteprog

[roleki]

15 hours ago, Bopper said:

 

So close.  If we had another 9.167% recharge (284.167% total), we would be able to achieve perma-Hasten through the use of Hasten and T4 Ageless 

I know I joked earlier, but I do have a question:

 

On one of my characters, through slotting 5x LotG, having a smattering of +Recharge set bonuses, and taking Spiritual Radial Revamp, I am sitting at a solid 191.25 recharge with perma-hasten.   Is that expressed as 291.25 in your formula?  Otherwise, I am not sure how I have perma-hasten, because I am nowhere near 284.

 

ETA duh, nevermind, I forgot you're talking about getting perma hasten with just Ageless' decaying +Recharge.  Never mind!

Edited October 23, 2019 by roleki

[Bopper]

2 hours ago, quixoteprog said:

Does that make sense?

I follow what you're saying and you are highlighting the reason why this simple formula is needed. I also used to get crossed up in the details of small changes having non-linear effects.

 

2 hours ago, quixoteprog said:

But if that is the case then would not this be a differential function?

It is in fact an integration problem.

Luckily integration can be done piecewise since after all, integration is just a fancy way of doing addition. Also luckily our haste is piecewise scalar, so when we're calculating the area under the curve, it is merely a constant height multiplied by a width (the width of the piece where haste is constant).

 

Ultimately, that is what my formula is doing, but with some steps simplified. I will probably write up a derivation in the next couple days that will show exactly what I'm talking about and will hopefully put pictures to functions to make it all clear.

 

Edit: Authors note. I tend to use the term Haste interchangeably with Recharge, so I apologize if there is any confusion. I know some people will see the term Haste and think I'm talking about the power Hasten.

Edited October 23, 2019 by Bopper

[Bopper]

15 minutes ago, roleki said:

I am sitting at a solid 191.25 recharge with perma-hasten.   Is that expressed as 291.25 in your formula?

I know you answered your own question, but I thought I should clarify where I use the base 100%. In the denominator of the formula, the "1" is the base 100%.

[Fan Mail]

Homecoming should pay you for this.  Wait...no one is getting paid.  Um...thank you!?!

[Clave Dark 5]

... I was told there'd be no math.

 

(GJ on this!)

[DSorrow]

Looks good to me. Ran a few checks with the iterative method I've always used and this matches all my results while being much more simple and user friendly. Thumbs up!

[quixoteprog]

My use of the term "differential" may have led to a misunderstanding. I agree that an integral could be used if the proc were a continuous quantity, but that is obviously not the case. As you point out the problem is just a summation of a series. 

 

My point was that because the problem is recursive i.e. that the change caused by the proc. feeds back into the effect of powers like Hasten (and Accelerated Metabolism, etc.) you can't count Hasten as a "Permanent Recharge Buff".  (I would prefer the term "constant" instead of permanent, but that is just my math OCD kicking in 8-) Either way, since these are variable quantities I think the formula you are using will be slightly different.

 

Not being critical here. I think the equation is very helpful even as is and will be a good tool to calculate recharge bonuses in most cases and will even come close for procs. Just wanted to point out this twist in the problem that it looks to me like you might have missed. If I am wrong, then feel free to set me straight. Lord knows somebody should.

[Bopper]

18 minutes ago, quixoteprog said:

If I am wrong, then feel free to set me straight

I would like to prove to you that math is right*, but I'm not sure what you are doing on your end that gets you different recharge results. Can you provide an example, step by step?

 

*ok, one caveat I didn't cover, my math did not factor in the fact that sometimes you might hit the recharge cap (500%). I am working on a derivation, but I will need time to produce plots. Once I finish, I think it will help folks see how the math is done and hopefully makes things clearer. 

 

20 minutes ago, quixoteprog said:

I would prefer the term "constant" instead of permanent,

I actually prefer that too. I will likely use that term going forward. 

[Bopper]

7 hours ago, Clave Dark 5 said:

... I was told there'd be no math.

 

(GJ on this!)

With me...there will always be math. 😝

 

1 hour ago, DSorrow said:

Looks good to me. Ran a few checks with the iterative method I've always used and this matches all my results while being much more simple and user friendly. Thumbs up!

I'm very glad it is. I know I wasn't the only one using that iterative method, so you were exactly the type of audience I was trying to reach out to. 

[Bopper]

Recharge2.docx

 

Derivation:

Attached is the Word document of my derivation. For those of you who do not have Microsoft Word, I screenshot my derivation below:

 

 

 

 

 

 

 

 

 

Edited October 25, 2019 by Bopper