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Fire vs Ice: Detailed DPS Analysis

#1
Comparing Long Term Ideal DPS of Fire and Ice
It's common knowledge that fire beats ice for DPS on long boss fights. But what do the numbers say about how big the difference is?

I will neglect additional damage spells—incinerate and frostbite—and only focus on the core damage skills. I also ignore the effects of lures/resistances, and assume zero evasions. Basically, I compare the DPS of fire and ice if every spell hit for 100% of its maximum potential damage.

What does "Long Term Ideal DPS" mean?
"Long Term Ideal" are implies these assumptions.
1. You are fighting a boss that has a huge amount of health (hence, long term).
2. The boss has zero fire and ice resistance; thus, lures are irrelevant. This is extremely unrealistic, but makes things much simpler (hence, ideal).
3. You have perfect casting. Your damage spells are continuously casted nonstop with no interruptions, energy hunger, or death. The main damage spell (fire bolt/ice shards) is prioritized over the secondary AoE spell (firestorm/ice blast) when both spells are ready at the same time.
4. There are no evades. If there were evades, they should affect both fire and ice equally anyway (because it's a long fight and there are many hits).

DPS simply means damage per second. That's what we're trying to compare here: rate of damage of fire vs rate of damage of ice.

Skip directly to Conclusions if you don't want to read the math.

The Original Formulas
The following maxed (level 50) skill formulas come from this skill formula index. They predict the maximum possible skill power based on your focus stat and fire or ice magic ability. I color coded them to easily separate fire skills from ice skills.

Level 50 Fire Attunement Damage Boost = 22.331*sqrt(Focus) + 10.044*sqrt(Fire Magic) + 307.16
Level 50 Fire Bolt Damage = 72.853*sqrt(Focus) + 21.611*sqrt(Fire Magic) + 264.84
Level 50 Firestorm Damage = 90.164*sqrt(Focus) + 25.515*sqrt(Fire Magic) + 365.43

Level 50 Ice Attunement Damage Boost = 33.548*sqrt(Focus) + 15.221*sqrt(Ice Magic) + 450.48
Level 50 Ice Shards Damage = 96.322*sqrt(Focus) + 30.47*sqrt(Ice Magic) + 702.43
Level 50 Ice Blast Damage = 117.97*sqrt(Focus) + 32.835*sqrt(Ice Magic) + 487.25

Adding Attunement Effects to Damage Spells
Let's assume that attunement adds the maximum amount of damage to each spell hit. Now, after adding the attunement formula to the damage spell formulas:

Fire Bolt Damage with Attune = 95.184*sqrt(Focus) + 31.655*sqrt(Fire Magic) + 572
Firestorm Damage with Attune = 112.495*sqrt(Focus) + 35.559*sqrt(Fire Magic) + 672.59

Ice Shards Damage with Attune = 129.87*sqrt(Focus) + 45.691*sqrt(Ice Magic) + 1152.91
Ice Blast Damage with Attune = 151.518*sqrt(Focus) + 48.056*sqrt(Ice Magic) + 937.73

Looking at Speed
An attunement lasts 3 minutes. In 3 minutes of ideal casting, a fire mage using both damage spells can cast 25 bolts and 10 storms. In 3 minutes of ideal casting, an ice mage using both damage spells can cast 12 shards and 8 blasts. Multiply each damage formula by their corresponding counts to get:

Total Fire Bolt Damage during Attune = 2379.6*sqrt(Focus) + 791.375*sqrt(Fire Magic) + 14300
Total Firestorm Damage during Attune = 1124.95*sqrt(Focus) + 355.59*sqrt(Fire Magic) + 6725.9

Total Ice Shards Damage during Attune = 1558.44*sqrt(Focus) + 548.292*sqrt(Ice Magic) + 13834.92
Total Ice Blast Damage during Attune = 1212.144*sqrt(Focus) + 384.448*sqrt(Ice Magic) + 7501.84

Combining the Damage Spells
Add fire bolt and firestorm for total fire mage damage in 3 minutes. Add ice shards and ice blast for total ice mage damage in 3 minutes.

Total Fire Mage Damage in 3 mins = 3504.55*sqrt(Focus) + 1146.965*sqrt(Fire Magic) + 21025.9

Total Ice Mage Damage in 3 mins = 2770.584*sqrt(Focus) + 932.74*sqrt(Ice Magic) + 21336.76

Average Long Term Ideal DPS
Divide by 180 to get damage per second.

Average Fire Mage Long Term Ideal DPS = 19.47*sqrt(Focus) + 6.372*sqrt(Fire Magic) + 116.8

Average Ice Mage Long Term Ideal DPS = 15.39*sqrt(Focus) + 5.182*sqrt(Ice Magic) + 118.5


Conclusions
Keep in mind that this guide assumes maxed out (level 50) attunes and damage spells. This is what we end up with:

Average Fire Mage Long Term Ideal DPS = 19.47*sqrt(Focus) + 6.372*sqrt(Fire Magic) + 116.8
Average Ice Mage Long Term Ideal DPS = 15.39*sqrt(Focus) + 5.182*sqrt(Ice Magic) + 118.5

So how much better is fire than ice on long fights? Plug in your own numbers and find out.

For me, my focus/fire magic would give me approximately 1100 average ideal DPS if I had maxed fire skills. If I had the same amount of ice magic and maxed ice skills, I would have about 900 average ideal DPS.

Re: Fire vs Ice: Detailed DPS Analysis

#2
The fact that each evasion on ice will decrease the DPS more harshly compared to an evasion on a fire mage is also something to note.
Edit: Actually, that is debatable since fire mage casts more spells therefore there's more chance of evasions...

Swan, absolutely love the content you put out. Just curious if you would be willing to do me a favour.

Compare the long term DPS of a hrung dark skull of fire bolt with a shadow mord grim book, except consider all the stat points (focus and vitality) gained from the book as just focus, if that makes sense.

Regardless, very nice post. Nice to see numbers to back up thoughts that have occurred to me for a while.
Image


A mage at heart... Fire mage that is.
Currently actively playing the game.

Re: Fire vs Ice: Detailed DPS Analysis

#3
I would think that 49 attunement, rather than 50, is more realistic since most mages will not dedicate more than two ring slots to attainment.

Armo wrote:The fact that each evasion on ice will decrease the DPS more harshly compared to an evasion on a fire mage is also something to note.
Edit: Actually, that is debatable since fire mage casts more spells therefore there's more chance of evasions...


But, since ice's damage per cast output is higher an evasion is more "costly" than for fire. So the frequency of evasions may be higher with the frequency of casts increasing, but the cost of those evasions is higher with a higher damage, lower frequency cast, such as ice.

And I agree...great post.
Xanadu - Mage 228+ - Forever - Rosmerta

Re: Fire vs Ice: Detailed DPS Analysis

#5
Suppose that you have an 80% chance of hitting a boss (20% chance for evasion), and equal fire and ice abilities.

You can do about twice as many bolts as shards over a period of time. After casting 10 bolts, you should only hit 8. Or after casting 5 shards, you should only hit 4.

In both cases, the percent damage reduction should be the same for both fire and ice. Remember that you're assuming a long boss fight, so there will be many hits. Therefore, evasions affect both fire and ice equally (in terms of %).

Armo wrote:The fact that each evasion on ice will decrease the DPS more harshly compared to an evasion on a fire mage is also something to note.
Edit: Actually, that is debatable since fire mage casts more spells therefore there's more chance of evasions...

Swan, absolutely love the content you put out. Just curious if you would be willing to do me a favour.

Compare the long term DPS of a hrung dark skull of fire bolt with a shadow mord grim book, except consider all the stat points (focus and vitality) gained from the book as just focus, if that makes sense.

Regardless, very nice post. Nice to see numbers to back up thoughts that have occurred to me for a while.

I would have to know your focus/fire magic amounts before equipping the shadow grim or dark skull. Also, should I ignore the 9 skill points from the dark skull?

Re: Fire vs Ice: Detailed DPS Analysis

#7
Armo wrote:The fact that each evasion on ice will decrease the DPS more harshly compared to an evasion on a fire mage is also something to note.
Edit: Actually, that is debatable since fire mage casts more spells therefore there's more chance of evasions...

Swan, absolutely love the content you put out. Just curious if you would be willing to do me a favour.

Compare the long term DPS of a hrung dark skull of fire bolt with a shadow mord grim book, except consider all the stat points (focus and vitality) gained from the book as just focus, if that makes sense.

Regardless, very nice post. Nice to see numbers to back up thoughts that have occurred to me for a while.

I would like to know that as well, i always wondered if mordy grims are even better than skulls.
#NerfMages #AvoidBalance #WhyPlayARogue #MeatShieldOnly #HealingSlavesOnly

OP dps warrior on Belenus, hot af melee druid on Nuada. #Elementals #Apex

Re: Fire vs Ice: Detailed DPS Analysis

#9
Armo wrote:Just curious if you would be willing to do me a favour.

Compare the long term DPS of a hrung dark skull of fire bolt with a shadow mord grim book, except consider all the stat points (focus and vitality) gained from the book as just focus, if that makes sense.

Armo wrote:3000 fire magic and 1470 focus before equipping the grim.

Armo's Favor

The Gear
Dark Skull of Fire Bolt — reduces recast of fire bolt by 20%, +9 fire bolt
Shadowheart Grimoire of Flame — 240 focus, 75 vitality, +400 fire magic

Assumptions
1. Treat the 240 focus and 75 vitality as just 315 focus.
2. Neglect the fact that skill evasions occur. I know this is an unrealistic assumption, but I simply can't quantify +400 fire magic (yet) because I haven't done any skill evasion testing.
3. Ignore the 9 extra skill points on the dark skull. Fire bolt should be maxed either way. Whether these 9 skill points make up for the 400 fire magic loss is up for debate.
4. Pretend that Armo has a level 50 attune (because that's what I have a formula for). Since we're comparing one weapon against another, this isn't an important detail.
5. Again, we assume that Armo does perfect casting, with no interruptions/deaths, and always hitting fire bolt and firestorm as soon as they cool down.
6. Also, we assume that it's a long boss fight, and resistances are zero (so lures are irrelevant).

Precise Fire Bolt Timing
After quite a bit of timing, I have determined the exact casting time (including the small delay) of a normal fire bolt to be 1.2 seconds. The recast/reload/cooldown time is 6 seconds. So, a normal fire bolt can be casted every 7.2 seconds.
A dark skull of fire bolt does -20% to the regular recast time of 6 seconds. That means shaving off 0.2*6 = 1.2 seconds per fire bolt. So, fire bolt would be possible to use every 6 seconds instead of 7.2 seconds.
Using perfect casting, I casted 25 fire bolts during a 3 minute attune. With a dark skull, I would have casted 30 in 3 minutes. With that in mind, let's start over with the basic formulas.

Fire Formulas
Level 50 Fire Attunement Damage Boost = 22.331*sqrt(Focus) + 10.044*sqrt(Fire Magic) + 307.16
Level 50 Fire Bolt Damage = 72.853*sqrt(Focus) + 21.611*sqrt(Fire Magic) + 264.84
Level 50 Firestorm Damage = 90.164*sqrt(Focus) + 25.515*sqrt(Fire Magic) + 365.43

Damage for Dark Skull
Plug in the provided numbers, 1470 focus and 3000 fire magic without the grimoire.
Dark Skull Fire Attunement Damage Boost = 1713.48
Dark Skull Fire Bolt Damage = 4241.75
Dark Skull Firestorm Damage = 5219.88


Attuned Damage for Dark Skull
Just adding it up.
Dark Skull Fire Bolt Damage with Attune = 5955.23
Dark Skull Firestorm Damage with Attune = 6933.36


Incorporating Speed Factor for Dark Skull
With the dark skull's -20% recast time effect, there can be 30 fire bolts and 10 firestorms per 3 minute attunement.
Dark Skull Armo's Damage in 3 mins = 30*5955.23 + 10*6933.36 = 247990.5

Final Dark Skull DPS
Divide by 180.
Dark Skull Armo's DPS = 1377.73

Damage for Shadowheart Grimoire
Now we have 1785 focus and 3400 fire magic with the grimoire.
Shadowheart Grimoire Fire Attunement Damage Boost = 1836.29
Shadowheart Grimoire Fire Bolt Damage = 4602.95
Shadowheart Grimoire Firestorm Damage = 5662.56


Attuned Damage for Shadowheart Grimoire
Just adding it up.
Shadowheart Grimoire Fire Bolt Damage with Attune = 6439.24
Shadowheart Grimoire Firestorm Damage with Attune = 7498.85


Incorporating Speed Factor for Shadowheart Grimoire
With normal cooldowns, there can be 25 fire bolts and 10 firestorms per 3 minute attunement.
Shadowheart Grimoire Armo's Damage in 3 mins = 25*6439.24 + 10*7498.85 = 235969.5

Final Shadowheart Grimoire DPS
Divide by 180.
Shadowheart Grimoire Armo's DPS = 1310.94


Conclusions
The ideal DPS for a dark skull of fire bolt is 1378. For a shadowheart grimoire of flame, ideal DPS is 1311.

The dark skull wins, but it's ideal DPS is only 5.1% higher than the grimoire's DPS. There are still other things to take into consideration.

My recommendation is to use the shadowheart grimoire. 5% isn't a very big DPS increase for the dark skull, while going from 3000 fire magic to 3400 fire magic is a 13% ability increase for the shadowheart grimoire. Thus, you have a better chance to hit your target. Also keep in mind that your fire lure will be more powerful due to the focus/fire magic boost.

Using the lure formula from the index, and 1470 focus and 3000 fire magic, a maxed lure with dark skull equipped reduces 1173 fire resistance. With the grimoire (1785 focus and 3400 fire magic), a maxed lure reduces 1260 fire resistance.

So, to sum it all up:

Dark Skull of Fire Bolt
1. 5.1% higher ideal DPS, or 67 more damage per second on zero resistance targets.
2. 9 extra skill points.

Shadowheart Grimoire of Flame
1. 13% higher fire magic, or 400 more fire magic. (better chance to hit target)
2. 7.4% higher fire lure, or 87 more resistance reduced.

My opinion: shadowheart grimoire of flame wins overall! Despite losing the ideal DPS battle by a small margin, it's other specs (better hit chance, more resistance reduction) make it the better choice for real DPS. But it's up to you to decide.

Re: Fire vs Ice: Detailed DPS Analysis

#10
Also remember that you won't always be spamming fire bolt and firestorm back to back. You might die; you might get interrupted; you might have to stop and lure, e shield, attune; you might want to wait a few seconds for sigils to kick in; you might even take time to cast a third skill like ice shards or incinerate. There are so many reasons why you wouldn't always be perfectly casting at full speed, and so many ways to delay.

When leveling, you might have to run to the next mob. This takes time, and your fire bolt might already have cooled down long ago.

If you aren't casting perfectly at full speed, the cool down effect of the dark skull is wasted. Your DPS would no longer be higher for the dark skull.

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