Guide to Choosing a Special Upgrade

[Justified]

TL:DR is at the bottom if you don't care for rhetoric or explanations.

Choosing a Special Upgrade

Implemented with G13 are new upgrades after gem upgrades after regular upgrades, the Special Upgrade! Unlike the earlier two upgrades, you don't need any prof to Special Upgrade your weapon, but instead will need to find an Upgrade Stone which drops from mobs and end chests in Theater Missions.

There are two different kinds of Upgrade Stones
Red (R-Type) - Increases your critical hit damage
Blue (S-Type) - Increases your attack power

Generally, the S-Type upgrades will be stronger than R-Type upgrades. This is because R-Type upgrades only help when you manage to land a critical hit, so it is greatly limited. On the other hand, S-Type upgrades will still gain damage when you land a critical hit, because it adds raw power. It requires high character stats to be able to use R-Type upgrades efficiently.

R-Type upgrades are as follows...


And S-Type upgrades...



Break-even Values

For the sake of simplification, we'll only look at maximum damage (including minimum damage and balance would add pages of algebra and not really change too much). Your damage is reflected by this formula...

Max Damage + ( Max Damage x Critical Bonus x Critical Chance )

If we change it to reflect S and R upgrades, and then set them equal to each other, we can find the damage in which both upgrades output the same amount of damage. If your damage is lower than this, S is better. Higher, and R is better.

S+Max + ( S+Max x Crit Bonus x Crit Chance ) = Max + ( Max x R+Crit Bonus x Crit Chance )

If we assume Rank 1 Critical Hit and 30% Critical Chance, the formula can be ultimately simplified to...

S x 1.45 = Max x R x 0.3
or
Max = ( S x 1.45 ) / ( R x 0.3 )

So, without further ado...

One-Handed Axe
G13 = 16 Max vs 23%
( 16 x 1.45 ) / ( 0.23 x 0.3 ) = 336 Max

G14 = 37 Max vs 53%
( 37 x 1.45 ) / ( 0.53 x 0.3 ) = 337 Max

One-Handed Weapons
Single-Wield
G13 = 13 Max vs 18%
( 13 x 1.45 ) / ( 0.18 x 0.3 ) = 349 Max

G14 = 31 Max vs 42%
( 31 x 1.45 ) / ( 0.42 x 0.3 ) = 357 Max

Dual-Wield*
G13 = 13 Max (x2) vs 18%
( 26 x 1.45 ) / ( 0.18 x 0.3 ) = 698 Max

G14 = 31 Max (x2) vs 42%
( 62 x 1.45 ) / ( 0.42 x 0.3 ) = 713 Max

* There is debate between whether or not R-Type upgrades stack (add together) or average (halved bonus) when dual-wielding. The break-even values listed here assume that they are averaged. In the case that they stack, the break-even values are exactly the same as single-wield.

Two-Handed Weapons / Bows
G13 = 21 Max vs 26%
( 21 x 1.45 ) / ( 0.26 x 0.3 ) = 390 Max

G14 = 48 Max vs 62%
( 48 x 1.45 ) / ( 0.62 x 0.3 ) = 375 Max

Cylinders
I'm not too familiar with alchemy formulas, so it's likely that the following is incorrect.
If any Alchemists out there would like to double-check this part, feel free to.

Tidal Wave (Water)
G13 = 15 Max vs 26%
( 15 x 1.45 ) / ( 0.26 x 0.3 ) = 279 damage per charge

G14 = 30 Max vs 62%
( 30 x 1.45 ) / ( 0.62 x 0.3 ) = 234 damage per charge

Volcano (Fire)
G13 = 9 Max vs 26%
( 9 x 1.45 ) / ( 0.26 x 0.3 ) = 167 damage per charge

G14 = 19 Max vs 62%
( 19 x 1.45 ) / ( 0.62 x 0.3 ) = 148 damage per charge

Wands and Staves
Again, due to being uncertain of the damage formula (the ambiguity of whether the 6 masteries and wand effects are added or multiplied), this section may be wrong. Assuming that the Special Upgrade is independent of all other multipliers and works on your grand total...

( Magic x S ) + ( Magic x S x Crit Bonus x Crit Chance ) = Magic + ( Magic x R x Crit Bonus x Crit Chance )

Since both upgrades are multipliers, these upgrades actually don't depend on your Magic. Magic is factored out from everything, and the equation simplifies to...

S x 1.45 = R x 0.45

Neither of these are variables, so instead we compare the two.

G13 = S 9% vs R 26%
( 0.09 x 1.45 ) = 13.05% S-Boost
( 0.26 x 0.45 ) = 11.70% R-Boost

G14 = S 21% vs R 62%
( 0.21 x 1.45 ) = 31.45% S-Boost
( 0.62 x 0.45 ) = 27.90% R-Boost

The S-Type upgrade will always deal more average damage. Although, for special cases like Ghasts, you may want the R-Type upgrade to bypass the massive defense rating.

And there you have it. This is a comparison on net average damage, so it'll only be accurate for 0 Defense mobs which you have maxed 30% Critical chance on. The more Defense a mob has, the better R-Type upgrades get (you don't need as much damage R). The less Critical chance you have, the better S-Type upgrades get (you need more damage for R). This is merely a guideline to help you make the best choice possible.


Things to Consider

Aside from Break-even values, you should keep other things in mind when choosing an upgrade. This is especially important if you are near these values and not greatly above or below them, since that means that the upgrades will be more or less equal in terms of average damage.

1. Transformations, Spotlights, and Personas favor the R-Type upgrade. (+1 R-Type)

In addition to food effects and magic music, there are many ways to greatly increase your damage beyond these break-even points. For some people, these weapons will only see action when you really want to pack a punch with your attacks. An R-Type critical hit with some or all of these bonuses applied will reach heights that S-Type weapons will never even dream of getting close to.

2. Consistent Damage may be more useful than Average Damage. (+1 S-Type)

The damage from S-Type upgrades may be more useful depending on what you're fighting. If you're close to a ohko, then you may want the damage boost from S-Type upgrades to get closer to (or over) the line. The ability to deal consistent ohkos can usually cut the time spent on doing anything in half. Being able to overkill a mob at a low rate isn't really that useful.

3. Epic Criticals are extremely useful in other situations. (+1 R-Type)

An example of where Criticals are useful is Peaca. The common method to killing Ghasts was to use Thunder which would always deal 1 damage (plus Int bonus of like 40) because of the extremely high defense rating. It was only on Criticals that Thunder would do over 100 damage.
Likewise, if you aren't closing the gap on consistent OHKOs, you might as well strive for larger criticals to deal with stronger mobs in harder areas. The higher numbers are pleasant to look at as well.

4. Critical Chance is a pain in the ass. (+1 S-Type)

It's sort of a Catch-22. To benefit from the large damage bonus of R-Type upgrades, you'll need high critical chance to bypass enemy protection. But to get that high critical chance, you'll have to sacrifice some of your damage enchants or upgrades. Transformations help (unless you're a lolPaladin) since it will both push you over the break-even point while adding a large chunk of critical, but otherwise this is a problem if you're not rebirthing at a low age. With S-Type, the criticals are still cool, but you don't really need them any more than normal.
Note that the blue spotlight in hardmode theater missions give a 30% (or was it 60%?) critical boost from the Will and Luck gains, as well as some Str/Int/Dex. The purple light has the same effect, but less of a boost and not as common.

5. The red glow is cooler than the blue one.

Srs. Normally I like blue, but since this is some odd aquamarine color and not a nice shade, the red one looks way better. It even matches the blood you'll be spilling.



And as promised, the TL:DR version.

Less is S, More for R.

One-Handed Axe
G13 = 336 Max
G14 = 337 Max

One-Handed Weapons
Single-Wield
G13 = 349 Max
G14 = 357 Max
Dual-Wield*
G13 = 698 Max
G14 = 713 Max
*Unverified

Two-Handed Weapons / Bows
G13 = 390 Max
G14 = 375 Max

Cylinders
Tidal Wave (Water)
G13 = 279 damage per charge
G14 = 234 damage per charge
Volcano (Fire)
G13 = 167 damage per charge
G14 = 148 damage per charge

Wands and Staves
G13 = 13.05% S-Boost vs 11.70% R-Boost
G14 = 31.45% S-Boost vs 27.90% R-Boost


Edit :
Hylianblade has stepped forth and explained his calculations, so I'm now including them as a reliable source.



These numbers take instability into consideration, thus Hylian's may be more accurate when including Minimum Damage and Balance. Mine are still accurate in terms of optimal damage, though it has a larger margin of error in practice.


Edit 2 :
2H Smash -vs- Dual Wield Smash

Because of the much higher bonuses that dual-wielding gets, it's commonly accepted that a 2H Windmill will never be able to outperform a DW Windmill, crit or not.

However, because 2H Weapons gain additional bonus for Smash, they have the potential to do greater damage. In particular, the gap is made more meaningful with the additional multipliers from R-Upgrades.
(This is assuming that R-Upgrades stack for Dual Wield, which is the popular opinion at this point)
The following is an evaluation of such (again, using only Max Damage in calculations).

Max Damage + ( Max Damage x Critical Bonus x Critical Chance ) x Smash Bonus

Include the Smash Bonus in the formula from before, assume 30% Critical Chance, and toss in the Critical Bonus from R-Upgrades, then simplify it to get...

2H Damage = 2H_Max x ( 6 + 1.8 x Critical Bonus )
DW Damage = DW_Max x ( 5 + 1.5 x Critical Bonus )

And set them equal to each other to find the breakeven point. Since we're considering different weapons here, the "Damage" variable is split into 2H Damage and DW Damage, and the resulting breakeven value is actually a ratio. This means that even at high or low damage ranges, the better choice greatly depends on what kind of weapons you're using, not just your stats.

G13 = 2H 26% vs DW 36%
9.168 x 2H_Max = 7.79 x DW_Max

2H Max = 84.97% DW_Max
DW Max = 117.69% 2H_Max


G14 = 2H 62% vs DW 84%
9.816 x 2H_Max = 8.51 x DW_Max

2H_Max = 86.7% DW_Max
DW_Max = 115.35% 2H_Max


What this means is, equip both your weapons and take a look at your final damage. (In the case of Dual Wield, equip one weapon and then add the total damage bonus from your second weapon to what you see in the character window). The ratios are actually the same, but both are calculated depending on which direction you want to calculate.

If, with your 2H Weapon, your damage is greater than 84.97% of your damage with Dual Wield, then Smashing with the 2H will result in more average damage. On the other hand, if your Dual Wield total is greater than 117.69% of your 2H damage, then Dual Wielding is better.

For example - if all weapons are Level 3 R-Type Upgraded...
300 Dual Wield Max = 255 2H Max
300 2H Max = 354 Dual Wield Max


Things to consider are, Dual Wielding is much more annoying since you'll need to Red Upgrade two weapons instead of just one. On the other hand, Dual Wield Windmill will outperform 2H Windmill by a large gap, so it is more situationally versatile.

[Zyrus]

Thanks justin. This helps alot. I came to the same conclusion but this confirms it.

[Adrianblade]

Sorry if I don't quite get it, but from what you've said R-type is better if you already have a high crit chance and if you can get the max of 30% on a mob after reducing crit because of said mobs prot. While S-type is better if you have lower crit and can't reach the max of 30%?
So that would mean for windmillers R-type is better because of windmill's ability to ignore prot and for smashers it would be better to go with S-type?

So I mean for example if I had a fully upgraded Glory Sword I might go for R-type since it already has a high crit; while for say a Dragon Fang (is it special upgradable?) S-type might be the way to go?


Ah wait no nvm, just reread that post again.

[Chihaya]

Nice Guide. Too bad I was going for S upgrades already since I duel wield, but it'll serve as a tool for future upgrades on my bow or something =)

[Pierce]

Thank you, I've had this discussion with Zid and Nixon in different threads and we reached the same conclusion, although we focused on wands only.

Edit: found it.

Here I compare Thunder damage between S-type and R-type upgraded wands, considering critical (30% of the time). As you can see, S-type is superior every time, with the exception of 5 criticals in a row... which is not so common, to say the least:

Damage upgrade:

a) Non crit

1.13x+1.13x... (5 times)= 5.65x

b) 1 crit

1.13x(2.5)+1.13x... (4 times)= 7.345x

c) 2 crits

1.13x(2.5)...(2 times)+1.13x...(3 times)= 9.04x

d) 3 crits

1.13x(2.5)...(3 times)+1.13x...(2times)=10.735x

e) 4 crits

1.13x(2.5)...(4 times)+1.13x= 13.56x

f) 5 crits (including last blow, 2x damage)

1.13x(2.5)...(6 times)=16.95x

g) Extra

Crits only on last blow: 10.17x

Critical upgrade

a) Non crit

b) 1x...(5 times)= 5x

1 crit

2.88x+4x= 6.88x

c) 2 crits

2.88x...(2 times)+3x= 8.76x

d) 3 crits

2.88x...(3 times)+2x= 10.64x

e) 4 crits

2.88x...(4 times)+1x= 12.52x

f) 5 crits

2.88x...(6 times)= 17.28x

g) Extra

Crits only on last blow: 9.76x

And this is what Zid calculated. As you can see, R-type upgrades are, on average, 5% weaker than S-type upgrades. (Divide averages and you'll get a quotient)

Going with what I know: Average Damage = [Max * Balance + Min * (1 - Balance) + (Critical Rate * Max * Total Critical Multiplier)]

Using pre-G13 r1 Ice Spear (200~300 damage) and 100% Magic Balance...

Lvl 4 S-Type Upgrade (+13% damage -> 226~339 final damage):

[339 * 100%] + [226 * (1 - 100%)] + [(30% * 339 * 150%)] =

339 + 0 + 152.55 =

492 damage on average

Lvl 4 R-Type Upgrade (+38% Critical damage):

[300 * 100%] + [200 * (1 - 100%)] + [(30% * 300 * 188%)] =

300 + 0 + 169.2 =

469 damage on average

This is for one charge of pre-G13 r1 Ice Spear, without considering the original damage upgrades on the wand/staff, damage from Intelligence, damage from Ice Magic Mastery, or damage from enchants.

I did the same calculation ignoring average damage and just focused on "damage"

S type: +13% Magic Damage.

1.13x+1.13x...(10 times)= 11.3x

Extra calc: crical hit occurs 30% of the time, then (I guess it compounds):

2.5x(1.13)+2.5x(1.13)+2.5x(1.13)+(1.13x...7times)= 16.385x

R type: +38% critical damage

2.88x+2.88x+2.88x+(1x...7 times)= 15.64x

You get the same quotient...

I think Int damage, masteries damage, enchants, etc. are irrelevant because they remain constant.

[Yanm]

*reads through guide*

I'm not fully understanding this. So, for the TL;DR part, you're saying if my damage with a 2hander is less than 390, S type is better and if it's higher, R type is better? The wand and magic part I understood thanks to gentrone's post earlier. After reading his post, I always thought it was, for multi hit skills, S type is better, but for single hit, R type is better. So.....even if I have 30% crit chance with smash on most normal HM Shadow Monsters, S type would still be better? Also, for dual-wielding with an ego, is S type still a better choice? Finally, wouldn't R type be better for a Titanic Crocodile Broad Axe with 201 max for Giant Full Swing?

[Justified]

*reads through guide*

I'm not fully understanding this. So, for the TL;DR part, you're saying if my damage with a 2hander is less than 390, S type is better and if it's higher, R type is better? The wand and magic part I understood thanks to gentrone's post earlier. After reading his post, I always thought it was, for multi hit skills, S type is better, but for single hit, R type is better. So.....even if I have 30% crit chance with smash on most normal HM Shadow Monsters, S type would still be better? Also, for dual-wielding with an ego, is S type still a better choice? Finally, wouldn't R type be better for a Titanic Crocodile Broad Axe with 201 max for Giant Full Swing?

The number of hits and skills used don't matter.
You don't need that much damage for R-Type upgrades to be better when you crit, but because you don't crit all of the time your average damage will usually be higher with an S-Type upgrade, which is applied all of the time including crits.

For Giant Full Swing, you get 98 Max from Trans, and 201 Max from the T/C BA. Since the breaking point for G13 is 390 Max, then you'd need 91 base Max damage from enchants and Str for R-Type to be better. Not too hard to get.

Just to illustrate it...
390 + 21 = 411 Max (Criticals 30% x 150% = 184.95) = 595.95 average damage
390 = 390 Max (Criticals 30% x 176% = 205.92) = 595.92 average damage
Giant Full Swing would multiply both by 900%, so they'd still be roughly equal.

[Jando]

Interesting. What path do Bow's follow?

[Fracture]

The number of hits and skills used don't matter.
You don't need that much damage for R-Type upgrades to be better when you crit, but because you don't crit all of the time your average damage will usually be higher with an S-Type upgrade, which is applied all of the time including crits.

For Giant Full Swing, you get 98 Max from Trans, and 201 Max from the T/C BA. Since the breaking point for G13 is 390 Max, then you'd need 91 base Max damage from enchants and Str for R-Type to be better. Not too hard to get.

Just to illustrate it...
390 + 21 = 411 Max (Criticals 30% x 150% = 184.95) = 595.95 average damage
390 = 390 Max (Criticals 30% x 176% = 205.92) = 595.92 average damage
Giant Full Swing would multiply both by 900%, so they'd still be roughly equal.

However, as Alchemy has set base damage for different attacks, this will matter for cylinders, meaning the breakeven point for it would need to be calculated on an attack-to-attack basis. Flame Burst, for instance, would be different from Heat Buster.

[Justified]

However, as Alchemy has set base damage for different attacks, this will matter for cylinders, meaning the breakeven point for it would need to be calculated on an attack-to-attack basis. Flame Burst, for instance, would be different from Heat Buster.

Not true. The breakeven point will remain the same, the only thing that changes is your base damage.

For Volcano Cylinders, the breaking point is 167 (assuming the damage formula was correct). This is true for all attacks, however your base damage will differ.
Flame Burst may be like 90 damage, while Heat Buster may be like 400. This means that R-Type is better if you want Heat Buster damage, while S-Type is better for Flame Burst.