Replies: 476
PTS 2.3.x Changes
- Ability tooltip value
- List of skill coefficients for PTS 2.3.2
- Resistance
- Changed how armour focus was being applied
- Critical modifier
- Major Force is multiplicative
- New Section: Miscellaneous equations
- Champion Point System: Warrior
- Made recommendations in favour of Hardy and Elemental Defender over Armour Focus and Spell Shield
- Champion Point System: Mage
- Changed the function to optimise for ideal CP distribution. I could not think of a way to provide ideal CP distribution over a wide crit chance or crit modifier and have only provided two examples.
- Precise and Nirnhoned
- Changed from a PvP setting to a PvE setting
Acknowledgement
A special thank you to addon developers for allowing me to understand the mechanics in the game in particular @Atropos for FTC, @SpellBuilder for LUI, @Kith for Srendarr, @Coolmodi for MitigationPercent and TemplarExecute and all the other developers for the bunch of addons I use daily in ESO. I would also like to extend my gratitude to the community for the kind words and valuable discussion about ESO game mechanics
Introduction
Welcome to my update for the PTS 2.2.3 including most equations from my previous post but with a bit more organisation and colour to help guide you through it!
This post is divided into two main sections: Fundamental Equations and Application of Equations. Fundamental Equations covers the majority of calculations in the game while Application of Equations uses the derived equations to draw conclusions regarding what trait or mundus to use. While my initial focus was on Magicka Sorcerers, the focus has spread a bit and is probably of interest to most Magicka based classes. Note that a large number of equations can be applied to Stamina builds as well by simply substituting for the relevant stamina analogue. I am lacking healing related equations and I may delve into that in the future. Also include, at the end of this post, are spreadsheets that implements the Fundamental Equations as well as Application of Equations. The equations and values use are valid for the Orsinium PTS so some of it will be incorrect in the current live version 2.1.x. The spreadsheets can be used to determine the relative strengths of different sets. The spreadsheets are view only to prevent tampering but please feel free to make a copy for your own calculations. If you do use the spreadsheets I would appreciate feedback in terms of accuracy or ease of use.
Fundamental equations
Stat pool
[spoiler]
The Base value at V16 is 8744 for Health and 7958 for Magicka and Stamina. Attribute points is the number of points spent in Health, Magicka or Stamina multiplied by 122 for Health and 111 for Magicka and Stamina. CPI is a cumulative percentage increase due to points spent in the corresponding constellation and can be calculated as follow
The Lord Mundus gives 1452 Health. The Mage and Tower Mundus gives 1320 Magicka and Stamina, respectively. Divines is the sum of divines. For example 4 pieces of green equipment with divines (4.5% each) means Divines = 1.18. Limited testing on the PTS suggest that Divines is rounded to 2 decimal places.
Example
I have a V16 Breton Sorcerer on the PTS. My gear gives me 7924 Magicka. This includes enchantments and gear bonuses. I have 64 points in Magicka, thus Attribute Points is 7104. I have 100 Champion Points in the Mage constellation which means that CPI is 1.134. I am using the Crown Fortifying Meal which gives 3617 Magicka. Skills is 1.31 (8% Bound Aegis, 5% Inner light, 2% Magicka Controller, 10% Gift of Magnus, 6% Undaunted Mettle). Putting this all into the formula, 
My in-game Magicka pool in the PTS is 38899.
[/spoiler]
Spell damage
[spoiler]
The Apprentice Mundus provides 166 Spell damage at V16.
[/spoiler]
Ability tooltip value
[spoiler]
The tooltip value for the majority of skill conforms to 
where a and b are coefficients. b is typically around 10.5 and a varies for each skill. The range of a is typically between 0.02 to 0.2. It is quite challenging to get extremely accurate values for a even with plane fitting over a large data range. We can fix b without much loss in the fitting for a thus we can introduce to concept of effective pool
This allows a fast evaluation method for different builds with varying Magicka and Spell Damage.
A technically more accurate estimate can be obtained by using
[img]https://i.gyazo.com/eb82a847e05085af973286c03deaa144.png[/img]
However the first formula presented in this section usually provides sufficient accuracy and will be used for the remainder of this post.
Some skills notably Hardened Ward and Annulment scale of only Magicka. In some cases the coefficient a is modified by Champion Points, at the time of writing, Thaumathurge appears to increase the tooltip for all abilities even non-DoTs. Elemental Expert is correctly being applied to tooltips. In the remainder of this post, it will be assumed that tooltip refers to the base tooltip unmodified by Champion Points.
A list of skill coefficients for the PTS 2.3.2 can be found at
https://docs.google.com/spreadsheets/d/1YN8YWDpi1-d4CfoagRy1F9ath2w2nb-TniL4MjdJdz4/edit?usp=sharing
[/spoiler]
Recovery
[spoiler]
Base Magicka and Stamina recovery at V16 is 514. Base Health recovery at V16 is 309. The Atronach mundus provides 198 Magicka recovery at V16. Other skills include Magicka Aid [Support Passive]. Skills include Magicka Controller [Mages’ Guild Passive], Major Intellect, Recovery [Light Armour Passive], and Spellcharge [Altmer Passive].
Example
I’m using a V16 Altmer with 765 Magicka Recovery from Gear and have the Atronach mundus but no divines pieces equipped. I have 30 points in Arcanist (10.8%). I have two Support skills slotted with 2 points in Magicka Aid so Other skills is 1.2. I am using a Crown Refreshing Drink which provides 361 Magicka Recovery. Skills is 1.55, 6% from Magicka Controller, 20% from Major Intellect, 20% from Recovery and 9% from Spellcharge. This yields 
My in-game Magicka Recovery in the PTS is 3602.
[/spoiler]
Spell Cost
[spoiler]
The Base Cost of a spell is the tooltip cost value, without any points in Magician and without any equipment or skills that provide either a percentage or flat cost reduction, divided by 1.1625. For instance, the tooltip cost value for Fire Rune is 3654. The Base Cost for Fire Rune is then 3143. Flat Cost Reduction is typically in the form of enchantments on jewellery and % Cost Reduction comes from skills and passives. Note that the 2 piece Molag Kena is a 33% cost increase when activated.
Example
I am calculating the cost to cast Fire Rune. The Base Cost is 3141 at V16. I have 70 points in Magician (12.9%) and 3 legendary reduce spell cost enchantment (203 each). %Cost Reduction is 0.33, 15% from Evocation [Light Armour Passive], 15% from Mage Adept [Mages’ Guild Passive] and 3% from Magicka Mastery [Breton Passive]. The cost for casting Fire Rune is 
This matches the in-game cost for Fire Rune in the PTS.
If I was to activate the 2 piece bonus of Molag Kena the cost for Fire Rune will be
The in-game cost for Fire Rune on the PTS is 2475.
[/spoiler]
Spell Critical
[spoiler]
Simply add up all sources that increase Spell Critical. 219 Spell Critical rating is equivalent to 1% Spell Critical
[/spoiler]
Critical modifier
[spoiler]
This formula has been updated due to changes in [2.2.4]
where Fl is the floor function, Rd is the round function and Elfborn_Real is similar to the tooltip value of Elfborn but when it is used no unexpected rounding errors are found. 
Rd(x, 2) rounds a number to 2 decimal places and Fl(x, 0.01) truncates a number at the 2nd decimal places. Here are examples of both functions in action, Rd(23.458,2) = 23.46, Fl(23.458,0.01) = 23.45. Skills tested were the Piercing Spear passive and Trap Beast (Minor Force). Aggressive Horn (Major Force) is multiplicative. Thanks to @Beltan3 and @hofawd with some help in getting this formula down.
By the way, if you don’t mind some error in your calculation, a simpler formula is
Due to rounding errors, Elfborn still suffers from jump. Any points in between jumps do not increase your critical modifier. @Erraln has kindly listed all the jump points in this thread but for convenience, I’ll put them here as well. The Elfborn jump points are at
1,2,4,7,9,12,15,18,22,26,29,33,38,42,46,51,56,61,66,71,76,81,87,92, and 98.
[/spoiler]
Resistance
[spoiler]
Your spell resistance can be calculated with the following formula
where Gear is the sum of tooltip armour values, Resolve is a Heavy Armour Passive, Defending is a weapon trait. Other includes Breton Spell Resistance Passive, Balanced Warrior Passive (Templars), Major Ward, Spell Warding and Spell Shield CP.
Similarly, physical resistance is calculated as follows
The Lady mundus provides 1980 Physical Resistance at V16 and is put into the variable Other. The Reinforced trait increases the tooltip armour value and will be included when calculating Gear. The Shield Expert passive under the Steed increases the tooltip value of the shield thus is also included in Gear.
Example
I have 5 pieces of heavy armour, 1 medium and 1 light. The sum of all my tooltip armour values, Gear, is 16666. Resolves grants 1811 resistance and Spell Warding grants 363 Spell Resistance. I have a legendary defending weapon equipped (6%). My set bonus for physical resistance is 5805. I have 100 points in Heavy Armour Focus (5281). I also have the Spell Resistance (Breton passive) and Balanced Warrior (Templar passive) passives. With Major Resolve and Ward active, I estimate my physical and spell resistance to be
My actual physical and spell resistance are 35951 and 31828, respectively.
[/spoiler]
Critical resistance
[spoiler]
Critical resistance is not needed in PvE since monsters do not do critical damage. An enemies’ critical modifier can be reduced by equipping gear with the Impenetrable trait or by spending points in the Resilient champion point sign. Every percent in Resilient decreases the enemies’ critical modifier by the same amount and every 250 points of critical resistance reduces an enemies’ critical modifier by 3.5%
Example
If you are PvP’ing against an enemy with a critical modifier of 0.5 and you have 500 critical resistance (2 legendary equipment) and 48 points (15%) in Resilient, then
[/spoiler]
Miscellaneous equations
[spoiler]
Bash Cost
[img]https://i.gyazo.com/df6a388d7b0849c273bd2de731188241.png[/img]
Bash cost reduction comes from the Shield-Play enchantment. The effect of the enchantment should be multiplied by 1.1625 to get the Bash Cost Reduction. Bashing Focus has no impact on Bash Cost.
Block Cost
[img]https://i.gyazo.com/5c3326507587f91cd6724475364ebb4a.png[/img]
Skills tested were Absorb Magicka, Bracing and Fortress.
Block Mitigation
[img]https://i.gyazo.com/d6858e946d35e1646cbe8bd5c530b3ed.png[/img]
Skills tested were Spear Wall, Deflect Bolts, Footman, Absorb Magicka and Sword and Board. There are some tooltip inconsistencies with Spear Wall and Deflect bolts but the error is not large.
Dodge Cost
[img]https://i.gyazo.com/e034dc3dbb05a10bb5f9a62ad67a291c.png[/img]
Heavy Attack Resource Return
[img]https://i.gyazo.com/fbe3ed43b9ce802b660b65112baf6577.png[/img]
Cycle of Life is a Restoration Staff passive. I only tested this briefly but for some reason Lightning staves was returning ~12.6% additional resources.
Sneak Cost
[img]https://i.gyazo.com/e1501f82ea655f9b1cf3b7625ef40eed.png[/img]
Sprint Cost
[img]https://i.gyazo.com/0f7ec517b0b987d8ab9d95a36465464e.png[/img]
[/spoiler]
Damage calculation
Base damage
[spoiler]
The base damage formula is 
where
[img]https://i.gyazo.com/b7a7c91d99b14abfe8c893ee2f089f23.png[/img]
Attacker Bonus includes relevant champion point signs, Minor Berserk [Combat Prayer] and Elemental Talent [Altmer Passive]. These stack multiplicatively. Defender bonus refers to relevant champion point signs. Battle Spirit can be included by simply multiplying by 0.5. In 2.2.3, the impact of Thaumathurge is included in the tooltip but the impact of Elemental Expert is not included in the tooltip, so be wary you’re not double counting Thaumathurge. Resistance is the relevant physical or spell resistance. Penetration is the sum of percentage based penetration. For magicka builds, this is either 18% for a legendary nirnhoned weapon or 28% when using a legendary nirnhoned weapon and casting a Destruction Staff spell due to the Penetrating Magic passive. Examples of Flat Penetration are Concentration for Light Armour users, Spell Erosion and Piercing. The base Flat Penetration is 100. Veteran rank 16 corresponds to level 66. For PvE, most mobs have a level of 50. Examples of Armour Debuff are Major and Minor Fracture and Breach, 5 piece bonus of Night Mother’s Gaze and Glyphs of Crushing. The resistance of some bosses in 2.1.x can be found at [2.1] Project Resistance.
PvE Example
A V16 Breton cast Crystal Fragments on Slimecraw (Wayrest Normal Dungeon). Slimecraw’s spell resistance is 18200.The tooltip value without any points in Elemental Expert is 7832 (Refer to the section Ability tooltip value if you’re confused). I have 75 points in Elemental Expert (20.4%) and 25 points in Spell Erosion (1492 Flat Spell Penetration). My Flat Spell Penetration is 100[base]+4884[Concentration]+1492[Spell Erosion] = 6476. I’m using an epic sharpened staff (12% penetration). I have combat prayer active. Given these values
[img]https://i.gyazo.com/06d9e9727846e66a62317b084ddd9d5f.png[/img]
and
[img]https://i.gyazo.com/a0e99a5bb9395f2af926752aa732fb77.png[/img]
The actual damage recorded by CLS is 8243.
[/spoiler]
Average damage
[spoiler]
The average damage when taking into account critical damage is 
[/spoiler]
Healing calculation
Base Healing
[spoiler]
The base healing formula for a healer using a variety of sources of Healing Done and Healing Taken and Healing Received is
The Tooltip value is increased by Restoration Master, Soul Siphoner, Major Mending and the Ritual Mundus. These add additively. Healing Done was tested with Blessed. Healing Taken was tested with Tormentor and Leeching sets. Healing Received was tested to be additive with Quick Recovery [Champion Point], Rapid Mending [Heavy Armour Passive], Minor Vitality [Swallow Soul & Coagulating Blood], Burning Heart [Draconic Power Passive] and Quick to Mend [Argonian Passive].
Example
A V16 Argonian cast Healing Springs. This character has 12% Healing Taken from the Tormentor and Leeching Sets and has 100 points in Blessed (25% Healing Done) and 100 points in Quick Recovery (16% Healing Received). In addition, this character is wearing 7 Heavy armour pieces (7% Healing Received), has the Minor Vitality buff, has a Draconic Power ability active and has 3 points in the Quick to Mend passive. This tooltip value includes the bonuses from Restoration Master, Major Mending and the Ritual Mundus.
The recorded in-game healing is 3264.
[/spoiler]
Application of equations
Champion Point System
Warrior
[spoiler]
Most players will be looking to spend points in either Hardy, Elemental Defender or Thick-Skinned. Sorcerers will also be looking to spend points into Bastion. Not much can be said for Armour Focus or Spell Shield as a 100 points provides only 5281 resistance which corresponds to 8% mitigation at V16 which corresponds to 20 points in Hardy or Elemental Defender. In addition, Armour Focus and Spell Shield are affected by percentage penetration and are not taken into account when a damage shield is used.
[/spoiler]
Thief
[spoiler]
For Magicka builds, the decision is on how to spread points between Magician and Arcanist. Essentially, we’re trying to optimise the function
[img]https://i.gyazo.com/c214533a81871daf4e80c8016e8182bf.png[/img]
This function represents the average magicka gain per second. Since this spreadsheet is rather complicated, I recommend selecting your Race and Class and then filling in all the grey filled cells. Most of them should be reasonably easy to understand and I’ve made comments for most of them. Ensure that the Final Magicka Recovery and Final Spell Cost matches your ingame values to make sure all inputs are correct. The two inputs that require further explanation are Average Base Spell Cost and Cast per second. Both can be determined from a parse. I’ll provide an example in the future. But for now assume that I casted 10 Funnel Healths and 1 Crippling Grasp over 20 seconds. Look up the Base Spell Cost of both spell which are 1233 and 2923 respectively. The Average Base Spell Cost can be determine from Input – Spell Cost or from the Skill Data spreadsheet. So my Average Base Spell Cost is (1233*10+2923)/11=1387. Cast per second is 11/20 = 0.55. Look over to the right and find Recovery 0, Cost Reduction 0 to determine the ideal CP distribution.
If you wanted to mess around with the number of jewellery to enchant remove the impact of jewellery from Magicka Recovery – Gear and Spell Cost – Gear (Flat CR) then just look at the corresponding output line. The spreadsheet can be found at
https://docs.google.com/spreadsheets/d/1Yp90QzJnsL5PNqSgKqDGv9IpYzJajG_tnlslFNPOf9g/edit?usp=sharing
[/spoiler]
Mage
[spoiler]
For a Magicka damage dealer, there are five signs of interest, Elemental Expert, Thaumathurge, Spell Erosion, Elfborn and Staff Expert.
The mechanics of all these except Staff Expert have been discussed in the sections on Critical Modifier and Base Damage. Staff expert increases the damage of light and medium attacks that is your weaving damage. Light and medium staff attacks are also increased by Elemental Expert.
In order to determine the optimal Champion Point distribution, we first need to consider the ratio of Elemental, DoT and Staff attacks. Then we can optimise the following function
[img]https://i.gyazo.com/34fbbb2fa63e4600fc57d33892aaed2d.png[/img]
One way to approach this is to enumerate all possible CP distributions and then calculate the function. However, this is quite a challenging task since with 167 Mage CP and 5 CP stars to consider the total possible of combinations is 30507895 (166C4). Although, some of these can be eliminated through some insight on the relative strengths of the CP stars. While this can be done, I have decided to make the equation a bit more accessible with the loss of a tiny amount of accuracy. I’ve included a spreadsheet that will do this
https://docs.google.com/spreadsheets/d/1Zp9v1Vp4Z7X6zfDfcxTwyAnejv-tEC5LujbXYBiVMDk/edit?usp=sharing
On the first sheet you’ll see a range of inputs including
- Number of Champion Points
- Critical Chance
- Critical Modifier
- Defined as Critical Damage/Non-Critical Damage – 1
- Target Resistance
- Target Level
- Percentage Penetration
- Flat Penetration
- Elemental Ratio
- DoT Ratio
- Staff Ratio
It will output the ideal CP distribution in Elemental Expert, Thaumathurge, Staff Expert, Elfborn and Spell Erosion. It does not take into account the Elfborn jump points in the calculation so if it suggest a non-Elfborn jump point it will also display the nearest Elfborn jump points. Also all CP passives are ignored. This means that there may be situations where it will not recommend putting 30 points into the Apprentice first.
[b]How it works?[/b]
It starts by assuming you have 0 in all 5 CP stars. It then calculates the following function
[img]https://i.gyazo.com/34fbbb2fa63e4600fc57d33892aaed2d.png[/img]
for an increase of 1 point in each star. It selects the optimum distribution then keeps going until it reaches your stated number of Champion Points. I had to use a continuous equation to model how the CP stars vary with points spent. The equation used can be seen in Sheet 3. Because a continuous equation was used some deviation from my previous discrete optimisation will be present. In addition, I could not include the impact of Elfborn jump points thus jump points are suggested at the end of the optimisation. While the previous discrete optimisation is probably better to model the jump points, I hope that this method of presenting the ideal mage CP distribution for magicka builds will be easier to use thus increasing it’s accuracy as you can put in your own relevant values instead of looking for the closest table match.
[/spoiler]
Precise and Nirnhoned
[spoiler]
Recall that the average damage equation is
where
For the purpose of this analysis, it is possible to ignore the Tooltip, Attacker Bonus and Defender Bonus. But for completeness, I’ll combine it all into a coefficient, k, that I will show later that it can be eliminated for this assessment. This discussion assumes a PvE situation thus the level is set to 50.
It is clear that we can separate Mitigation into two parts, one containing the ‘Base’, Mit_Base, and the other the ‘Penetration’, Mit_Pen, component
The average damage with Nirnhoned, Dmg_N, is then
In comparison the average damage with Precise, Dmg_P, is
To decide which is preferable, we will subtract the average damage with Precise, Dmg_P, from the average damage with Nirnhoned, Dmg_N,
The coefficient k is always positive so we can ignore it when determining if the equation is positive, that is favouring Nirnhoned, or negative, that is favouring Precise. For simplicity, I’ll assume that 0 points are placed in Spell Erosion. Flat Penetration is assumed to be 4984. Then for select values of Resistance we can create graphs showing when Nirnhoned or Precise is better. In the graphs below, blue means Precise is better and red implies Nirnhoned is better.
[IMG]https://i.gyazo.com/429c82339723021055eb179cb0fa93ae.png[/IMG]
[IMG]https://i.gyazo.com/d39e27248c398d937e9eee7960f1e8d9.png[/IMG]
[IMG]https://i.gyazo.com/80ddf9a2d92ad070da7ac0aff13e2444.png[/IMG]
[IMG]https://i.gyazo.com/0a3d2f9415c14dc93c01864d4cddc3d5.png[/IMG]
Below ~8000 resistance, precise is always better and above ~14000 resistance nirnhoned is always superior. Also having a high critical chance favours Nirnhoned while having a high critical modifier favours Precise.
To help with your decision between Nirnhoned and Precise, I have included an additional calculator in my latest spreadsheet. It uses the critical chance, critical modifier, points in spell erosion, focus, and monster resistance to determine which is preferred.
Alternatively the equation below can be used to determine the threshold resistance for Precise/Nirnhoned
[img]https://i.gyazo.com/1d335d27a1c33c8296e385c059534c3e.png[/img]
[/spoiler]
Divines and Infused
[spoiler]
The current meta advice suggest using Infused on large pieces (Head, Chest, Legs and Shield) and Divines on small pieces (Shoulders, Waist, Hands and Feet). There are no other viable traits for optimising damage. However, with the buffs to the Thief and Shadow mundus stones this advice is called into question. In this section, I will derive the conditions where Infused or Divines should be used. It is quite laborious mathematics and this section has been implemented in my spreadsheet. The equations here are to explain how the calculation is done.
The amount of magicka gained from using Infused on a legendary large piece instead of Divines, Inf, is
173 is the difference between in enchantment of a V16 legendary Magicka enchant on an Infused large piece compared to a non-Infused large piece. CPI and Skills were defined in Stat Pool.
The average damage when using an Infused piece is
The coefficient, k, in this section is
Dmg_Inf can be separated into two parts, Dmg_Base and Dmg’_Inf. The former is the damage component from not using any trait and the latter is the bonus damage coming from using infused
The average damage increase when using Divines depends on the mundus stone. For completeness, I’ll analyse the Apprentice, Mage, Thief and Shadow mundus stones.
Apprentice
The average damage increase when using Divines with the Apprentice mundus is
Skill_SD refers to abilities and buffs that increase spell damage, notably Minor and Major Sorcery and the Expert Mage passive.
Again this can be separated into two components, Dmg_Base and Dmg_Divine^App’
Mage
The average damage increase when using Divines with the Mage mundus is
Skills refers to abilities and passives that increase maximum Magicka and was defined in more detail in <b>Stat Pool</b>.
Again this can be separated into two components, Dmg_Base and Dmg_Divine^Mage’
Thief
The average damage increase when using Divines with the Thief mundus is
Again this can be separated into two components, Dmg_Base and Dmg_Divine^Thief’
Shadow
The average damage increase when using Divines with the Shadow mundus is
I’m using a simplified form of the Critical Modifier equation here but in my implementation of the equation in my spreadsheet I use the more exact form of the Critical Modifier. Skill_S refers to skills that increase the critical modifier such as Piercing Spear [Templar passive], Hemorrhage [Nightblade passive], Trap Beast [Minor Force] and Aggressive Warhorn [Major Force].
This can be separated into two components, Dmg_Base and Dmg_Divine^Shadow’
The way to decide between Infused and Divines is then to evaluate Dmg_Inf’ - Dmg_Divine’ with the corresponding mundus stone. Since several variables need to be taken into account, I’ve simply implemented my calculation into my spreadsheets. On a personal note, in the majority of calculations that I have performed Divines with Thief or Shadow outperforms Infused. However, if the Apprentice or Mageis used then Infused on large pieces is preferred.
[/spoiler]
Mundus stone: Apprentice, Mage, Thief and Shadow
[spoiler]
There are four mundus stones of interest to optimising Magicka based damage dealers. They are the Apprentice, Mage, Thief and Shadow. I’ll begin with the average damage equation for each mundus. Astute readers might notice a striking similarity with the section <b>Divines and Infused</b>.
Apprentice
In this section, the coefficient, k, is defined to be
Skill_SD refers to abilities and buffs that increase spell damage, notably Minor and Major Sorcery and the Expert Mage passive.
Dmg_App can be separated into two components, Dmg_Base and Dmg_App’. The former is the damage component without any mundus active and the latter is the damage from using the Apprentice mundus.
Mage
Skills refers to abilities and passives that increase maximum Magicka and was defined in more detail in Stat Pool. Separating into a base and Mage component yields
Thief
Once again, this can be separated into a base and Thief component.
Shadow
I’m using a simplified form of the Critical Modifier equation here but in my implementation of the equation in my spreadsheet I use the more exact form of the Critical Modifier. Skill_S refers to skills that increase the critical modifier such as Piercing Spear [Templar passive], Hemorrhage [Nightblade passive], Trap Beast [Minor Force] and Aggressive Warhorn [Major Force].
This can be separated into two components, Dmg_Base and Dmg_Shadow’
From this, we can easily conclude that the Apprentice mundus is preferred to the Mage mundus in nearly all cases for increasing damage since
where
At V16 the Apprentice provides 166 Spell Damage, <i>b</i> is approximately 10.5 and Skill_SD is typically 1.2 due to the Major Sorcery buff. The Mage gives 1320 Magicka at V16 and Skills is typically around 1.31 for a Sorcerer and is lower for Templars. Putting this into the equation, we obtain
Thus the Apprentice is preferred. The Mage mundus is sometimes preferred due to increasing Magicka pool for stronger shields and higher pet damage as these scale solely off Magicka.
For the comparison between the Apprentice and Thief there are no easy simplifications and one is left to evaluate
There are many variables and no obvious simplifications thus I have simply implemented the laborious calculation in my spreadsheet. Similarly the comparison between Apprentice and Shadow is very involved and is implemented in the spreadsheet.
While the equations presented for the Thief are a crude approximation due to the complicated rounding in the more accurate formula for Critical Modifier, it is possible to make a rough comparison between the Thief and Shadow mundus stones.
Thus the Thief is better if
This is equivalent to
In the Orsinium PTS, the Shadow mundus increases Critical Modifier by 12% and the Thief increases Critical Chance by 11%. Putting this values in, we get the following inequality
which means that Thief is better than Shadow if your Critical Modifier is at least ~10% greater than your Critical Chance.
[/spoiler]
Percentage Penetration and Spell Damage Equivalence
[spoiler]
Since Maelstrom weapons cannot come in Nirnhoned, it is natural to ask how does a non-set Nirnhoned destruction staff compare to a Sharpened Maelstrom destruction staff.
To evaluate this, let us consider T1 and T2, where T1 is the base damage with S1 extra spell damage and T2 be the base damage with no extra spell damage but 4% extra penetration. Then
where M is Max Magicka, S0 is the base spell damage, S1 is the extra spell damage for T1 and
Note that any penetration that is common to T1 and T2 can be seen as just a reduction in the Resistance.
We then proceed to solve T1-T2=0 for S1
We can rearrange this to get
Now we have to put some typical endgame values, I’ll let M=43487, S_0=3764 and Mit=0.19 (17k boss resistance, 14% penetration, 4984 Focus, 0 Spell Erosion). Mit’=0.04*0.34
Thus for the stats assumed a Sharpened Maelstrom Destruction Staff is better than a non-set Nirnhoned Destruction Staff as the 4% additional penetration is equivalent to 133 Spell Damage which is less than the Maelstrom enchantment of 189 Spell Damage
[/spoiler]
Julianos and Twice-Born Star
[spoiler]
Due to changes to Elfborn and skills that increase critical damage, TBS is no longer optimal from a DPS point of view. In the spreadsheets below, I have introduced a new metric called the Combined Metric. This metric was introduced because staff attacks scale differently from abilities. For most abilities 10.5 Max Magicka ~ 1 Spell Damage but for staff attacks 40 Max Magicka ~ 1 Spell Damage. To obtain the Combined Metric, I assume approximately 15% of total dps comes from Heavy (Medium Attacks) and 85% comes from abilities, then the weighted average of the Ability and Attack metric results in the Combined Metric.
We see then that for the Combined Metric on staffs, Law of Julianos is better by about 1.5%. Previously, my calculations showed that Law of Julianos was ~0.5% better than TBS but that was without taking into account the higher Attack Metric. On the dual wield bar, we should use the Ability Metric since no weaves are used and in this situation Julianos is better by around 0.5%.
If you were to replace one magicka enchantment on a large piece in favour of health so that the Health with Julianos and Twice-Born Star are comparable then on the staff bar Julianos is better by 0.1% but worse by 0.1% on the dual wield bar. To help put all these percentage differences into context, my rough calculations suggest that not having Divines on one piece (maybe you have been unlucky and have a bank full of Well-Fitted Molag Kena shoulders/helm) equates to a loss of ~0.5%
I’ve heard of people saying that Twice-Born Star is better with lower CP but I have yet to see extensive calculations that demonstrate this. Using the spreadsheets below, I varied the amount of CP by adjusting both the number of points invested into the Mage Tree and assumed that the first 100 points will be put into Elemental Expert followed by all points (up to 66) into Elfborn. While this CP distribution is not absolutely optimal, it is reasonably close. In this case, even with 100 Mage CP (300 total CP) Julianos is better than Twice-Born Star on the staff bar (0.3%). On the Dual-Wield bar, Julianos outperforms Twice-Born Star at 129 Mage CP (387 total CP)
Note: Ignore the Magicka Recovery and Spell Cost boxes. I was too lazy to move them away before I took the pictures.
Twice-Born Star Staff
Twice-Born Star Dual Wield
Law of Julianos Staff
Law of Julianos Dual Wield
[/spoiler]
Damage distribution
[spoiler]
Damage calculation in ESO is rather straightforward you either crit or don’t crit. This leads to a binomial distribution. If you repeat the event (of using an ability) a sufficient number of times you can approximate the binomial distribution with a normal distribution with the following statistics
[img]https://i.gyazo.com/61662e6f049959bb0e8c03c16d6fdd8f.png[/img]
Since stats can already be estimated along with tooltip values and base damage numbers, it is possible to create total damage normal distributions (or DPS distributions) for different item sets.
So if I were to redo my section on Julianos and Twice-Born Star, I could use the calculated stats to calculate the tooltip value of Force Pulse. Here is a summary of the data required to create the normal distributions
[img]https://i.gyazo.com/16c57a35d6614f38ce30632d1c2c0422.png[/img]
and the distributions themsevles
[img]https://i.gyazo.com/a9d73ccaaf67a357e58fb3df5de03813.png[/img]
It’s rather interesting because initially I thought that while the average of Julianos was greater than Twice-Born Star, I felt that the higher crit modifier of Twice-Born Star would mean that there would be a good chance of it doing more damage than Julianos if crits went your way. However, in the case of 300 Force Pulses which is 100 cast or about 100 seconds of just using Force Pulse, not only is the average damage of Julianos greater than Twice-Born Star but the maximum damage for Twice-Born Star is rather similar to Julianos. It is still true that the SD of Julianos is smaller than the SD of Twice-Born Star. Even if Force Pulse were casted only 10 times (30 counts), you would expect similar’ish distribution.
[img]https://i.gyazo.com/cb97e0c6dc2ecffa63712efa850fd112.png[/img]
From here it’s straightforward to calculate the difference of the distributions
[img]https://i.gyazo.com/ab8617aa20061ebb7d63ac69b28fa6f4.png[/img]
We can setup damage thresholds and calculate the chance that a single test will be incorrect, inconclusive or correct. Someone a bit more motivated than me could then calculate the minimum sample size to get a reasonable estimate of the distribution and the chance of Type 1 or Type 2 errors.
To show that the calculated curves are reasonably close to the simulated curves, I’ll start of by showing the situation where you cast Force Pulse a 100 times so 300 data points and your recorded the total damage. If you repeated it a 100 times you’ll get something like the green curve below
[img]https://i.gyazo.com/b317eb4a2af6aa009819d6cf1f6ddbed.png[/img]
It doesn’t match the calculated curve (blue) that well because repeating it 100 times is a bit low. If you repeated it 10000 times, things get much better
[img]https://i.gyazo.com/b1e30417dea3d345ece323b7d56dcd0b.png[/img]
For me this means that I’m better off just calculating the mean and basing my conclusions off that or calculating the damage distributions (now that I’ve done the maths once and have a few scripts that are easily editable) than spending time in game casting Force Pulse on a boss repeatedly to get determine which set is better. I’ll still keep testing in game to ensure the base equations are accurate though.
[/spoiler]
Scathing Mage and Law of Julianos
[spoiler]
In the Thieves Guild update, the Scathing Mage set was buffed. In this post, I’ll compare Scathing Mage to Law of Julianos.
[img]https://i.gyazo.com/bb063b468347e95e7047caabd7109e95.png[/img]
[img]https://i.gyazo.com/23bb51c96f2fb73ab8ec479737736122.png[/img]
There are several approaches in determining when Scathing Mage is preferred over Julianos. I suggest looking at the number of attacks required to proc Scathing Mage. From there we can estimate the uptime of Scathing Mage and then it is a straightforward comparison to Julianos. I’ll show some example calculations based on my sorcerer.
The proc chance per attack with Scathing Mage is quite simplistically
[img] https://i.gyazo.com/39bd55e4010aa3de560a75e0ca2b5338.png/img
Assuming that all attacks are independent we can model the resulting distribution of required attacks to proc Scathing Mage with a geometric distribution. In order to demonstrate that the geometric distribution is a suitable model I did some in-game testing. I counted the number of hits required to proc Scathing Mage. I only did 50 trials but I was reasonably convinced by the results. I trial here is defined as the number of hits needed to proc Scathing Mage. The image bellow summarises my in-game testing as well as the theoretical model. I was reasonably convinced with validity of the model despite the low number of trials.
[img]https://i.gyazo.com/dc2fb0638eeea04db760d3d713f1e828.png[/img]
The median of this distribution is
[img] https://i.gyazo.com/abb9cc082e2dd040ff366d470e080985.png/img
The median corresponds to the point where 50% of the time you’ll need less than X attacks to proc Scathing Mage and equivalently 50% of the time you’ll need more than X attacks to proc Scathing Mage. From this, it is reasonably straightforward to calculate the median amount of time required to proc Scathing Mage
[img] https://i.gyazo.com/2fce4dc69097fa60da63c88df4a279cf.png/img
From this the uptime of Scathing Mage is expected to be
[img] https://i.gyazo.com/2d0b1cb207399236d4f207555a772ccd.png/img
where there is an implicit assumption that the internal cooldown of Scathing Mage is the duration of the proc, which is based on personal testing during the IC PTS. The effective spell damage of Scathing Mage is then
[img] https://i.gyazo.com/329dffa8d5bd76270c9a6437147e8f54.png/img
Or to simplify it in a single equation
[img] https://i.gyazo.com/b49f7d5d2c2df70fc2e9c9f642313d56.png/img
Let us consider a magicka Sorcerer using the Thief mundus (6 pieces of Legendary divines) and with a precise staff. The spell critical of this character is 71.2% or 74.2% with Minor Prophecy active. I timed myself doing 50 Force Pulse/LA weaves and could do it in 1.16 seconds per weave or equivalently 3.45 attacks per second. Since this is probably close to the upper limit of attacks per second, I would estimate that the upper limit of the Scathing Mage 5 piece bonus for the magicka Sorcerer in question is
[img]https://i.gyazo.com/b2280c04400c4d3902a0ec4cb1469035.png[/img]
A more practical approach to estimating SM SD equivalence is by looking at a parse and determining the number of non-DoT attacks per second. In the example parse below, I estimate the number of non-Dot attacks to be 84 (Force Pulse + Crystal Fragments + Light Attack) which means that the number of non-DoT attacks in this example is 2.37 (parse duration 35.5s).
[img] https://i.gyazo.com/0fcffd520f31a5d7cf527e0d7b8fea5e.png/img
[img] https://i.gyazo.com/1fadf00f2cab5bc35f05b7815c7943ff.png [/img]
This is 97 SD greater than the 5 piece bonus of Julianos. Since this character has 43486 Max Magicka and 3189 Spell Damage, the 80 spell damage corresponds to an average tooltip damage increase of ~1.7%
[img] https://i.gyazo.com/ac81b09e0c86ff5b02e226815f968e9d.png/img
So based on the example parse above, if I were to use Scathing Mage instead of Julianos I would expect a DPS increase of around 425 (1.7%*25000). There is a slight simplification here since Light Attacks depend more strongly on Spell Damage than abilities so in fact my Light Attacks will do more than 1.7% damage with Scathing Mage.
However most Sorcerers rely quite a bit on Overload so the SM SD equivalence during Overload needs to evaluated. The magicka Sorcerer is using Nirnhoned swords (Spell Critical of 64.2%) for Overload and does 0.862 non-Dot attacks per second
[img] https://i.gyazo.com/53fdb3d0debd9e510b9ecb5b668fba95.png/img
[img]https://i.gyazo.com/7fce65bb137f736cf6541a23b0e618d3.png[/img]
This is 38 SD less than the 5 piece of Julianos and corresponds to an average damage loss of ~0.6% during Overload or around 200 DPS (0.6%*32000).
[img]https://embed.gyazo.com/cbbcdc7c53a45ecd4f84acd046128486.png[/img]
During Overload, I use about 18 Ultimate per second. It takes 6 seconds of non-Overload time to generate 18 Ultimate. Thus on average Scathing Mage provides an increase of 336 DPS.
[img] https://i.gyazo.com/cbbcdc7c53a45ecd4f84acd046128486.png/img
This is based on the assumption that you Overload all things equally.
To Precise or not?
An additional question that arises when using Scathing Mage is whether a Precise or Sharpened/Nirnhoned weapon should be used.
Percentage penetration can be converted into equivalent spell damage with the following equation
[img] https://i.gyazo.com/3ada96f9efdc3420d21eb367c5db50ae.png/img
where the variables Mit_pen and Mit_Base are defined as
[img] https://i.gyazo.com/7b6269eff8d8e32e404b97b768d552e6.png/img
For simplicity, b is assumed to be 10.5. Using some typical values, 14% penetration (Sharpened) is worth about
[img] https://i.gyazo.com/be902ff2a43440be00d60f24d023360a.png/img
Additional Spell Critical from Precise can be converted into equivalent spell damage with the following equation
[img] https://i.gyazo.com/041cfc55fa27b0f4d18543a469916e0b.png/img
Again using typical values, 7% Spell Critical (Precise) is worth about
[img] https://i.gyazo.com/904a16e266123aa639b78339a64efa80.png/img
This means that for mobs with 13k resistance which is a typical value for bosses debuffed with Major Breach, Sharpened/Nirnhoned is preferred over Precise. However, going with Precise will increase the uptime of Scathing Mage and the question is whether the increased uptime will compensate for the difference. Using my first parse with Force Pulse, the spell damage equivalence of Scathing Mage is expected to be 418 with a Sharpened Staff
[img] https://i.gyazo.com/56f87973fa67c101399fe90a5e4728b2.png/img
While the spell damage equivalence with Precise is
[img] https://i.gyazo.com/03f64160421f05f6216c5df3ec104611.png/img
Thus the average Spell Damage gain from using a Precise weapon with Scathing Mage compared to Sharpened is 9 base spell damage which is increased to around 11 by Major Sorcery and Expert Mage. However, the spell damage equivalent difference between Sharpened and Precise on a PvE monster with 13000 resistance is 93 which is larger thus a Sharpened weapon is still preferred even when Scathing Mage is used.
[/spoiler]
Molag Kena and Nerien’eth
[spoiler]
In this post, I’ll demonstrate how to compare Molag Kena to Nerien’eth on a single target.
[img]https://i.gyazo.com/51745b00e9d25ae967e6af384c879ab7.png[/img][img]https://i.gyazo.com/0ec8951c920931204c9e9750e772835d.png[/img]
The image for Nerien’eth was taken without CP. The tooltip is increased by both Elemental Expert and Thaumathurge.
My Max Magicka and Spell Damage with Nerien’eth is 43486 and 3189, respectively. The spell damage is with Major Sorcery and 3 Sorcerer abilities slotted. Molag Kena provides 129 Spell Damage and an additional 516 Spell Damage when proc’ed. I estimate a maximum uptime of Molag Kena at 86% (6/7 as you cannot re-proc it when it is active). This means that the maximum Spell Damage from Molag Kena is
[img] https://i.gyazo.com/c8737a1ef76049810d16fc2e67e1c6fb.png/img
This is increased to 722 Spell Damage after taking into account Major Sorcery and 3 Sorcerer abilities (6% increased Spell Damage from Expert Mage passive). Based on my Max Magicka and Spell Damage, this corresponds to an increase of ability tooltips by ~9.8%
[img] https://i.gyazo.com/406434a5b7484e6175c0be7729bb8379.png/img
So we can roughly say that we would expect a DPS increase of ~9.8% with Molag Kena. This is an approximation since Light Attacks depend more on Spell Damage than abilities so will be increased by more than 9.8%.
There are two ways to determine the contribution of Nerien’eth. One way is to look at a parse. In the example parse below, we can estimate the DPS increase of Nerien’eth by dividing the total damage contribution of Lich Crystal by the total damage dealt less the damage done by Lich Crystal. For the example parse below it turns out to be ~9.8%, similar to the estimated upper bound DPS increase with Molag Kena.
[img]https://embed.gyazo.com/1fadf00f2cab5bc35f05b7815c7943ff.png[/img]
[img] https://i.gyazo.com/03728b91b444b5a5aee5326f5eab3d35.png/img
The method above does depend on the RNG or a particular parse so a theoretical approach would be beneficial. The DPS contribution of Nerien’eth can be modelled by first modelling the number of attacks required proc Nerien’eth with a geometric distribution and then we will be able to obtain the uptime of Nerien’eth and finally estimate the DPS from the uptime.
I first tested the validity of a geometric distribution with in-game testing. I counted the number of attacks required to proc Nerien’eth and made the following image
[img] https://i.gyazo.com/3993ed88b8b263545bf487f4200b291b.png/img
The median number of attacks required to proc Nerien’eth is
[img] https://i.gyazo.com/733b8efb9fb4b271cf5bf692f6f4f31a.png/img
which means that the median amount of time required to proc Nerien’eth is
[img]https://i.gyazo.com/f6f27bb264fb434cbb5d49b071628c70.png[/img]
Since you cannot have two Lich crystals at once, the internal cooldown of Nerien’eth is estimated to be 3 seconds. Thus, I would expect one Lich crystal every
[img] https://i.gyazo.com/2bd5faede6423ce4dd0eaeb01494eb52.png/img
From my parse above, I did 84 non-DoT attacks over 35.5 seconds which means the number of non-DoT attacks that could proc Nerien’eth is 2.37. So I would expect one Lich crystal every
[img] https://i.gyazo.com/ebb9410f4990265279431636928ff6ab.png/img
The average damage for each Lich Crystal is
[img] https://i.gyazo.com/9d760160a21b4057fa19f5f9d31708ea.png/img
Let’s say I have 100 in Elemental Exper (25%) and 2 in Thaumathurge (1.6%) and my spell critical is 71.2% and my critical modifier 0.62 then the average damage for each of my Lich Crystal is
[img] https://i.gyazo.com/3e7495c85f1c7b6043961c0b981bd5df.png/img
So my expected DPS with Nerien’eth is 2319 (13385/5.77) which was reasonably close to the example parse. Based on this, I conclude that Nerien’eth is comparable to Molag Kena for the rotation shown in the parse.
Additional note: I considered making a comparison between Molag Kena, Nerien’eth and Skoria but there is something very peculiar about Skoria. It appears that DoTs must be running for some period of time before it can actually proc. I counted the number of Puncturing Sweep hits required to proc Skoria and found that for the first few hits Skoria will never proc. This is shown in the image below. A similar result can be obtained when using Elemental Blockade (data not shown)
[img] https://i.gyazo.com/077b32051fa6c58cda73e308f6a3c0eb.png/img
[/spoiler]
Character SpreadSheets
[spoiler]
Only made them for Sorcerers, Templar and Nightblades at the moment.Feel free to make a copy for your own calculations.
https://docs.google.com/spreadsheets/d/1e2M7ZU9ZKBxCNhDmvfS4GKPs_Nx7YLUny9vBmPaUmgA/edit?usp=sharing
@Beltan3 has made an awesome calculator. The post is at
http://tamrielfoundry.com/topic/stats-calculator/
and the calculator itself can be found at
[/spoiler]
Other useful spreadsheets (I’ve listed them at various points in the post but sometimes I forget where they are too :/ )
TG Mage CP Distribution
https://docs.google.com/spreadsheets/d/1Zp9v1Vp4Z7X6zfDfcxTwyAnejv-tEC5LujbXYBiVMDk/edit?usp=sharing
TG Skill Coefficients
https://docs.google.com/spreadsheets/d/1YN8YWDpi1-d4CfoagRy1F9ath2w2nb-TniL4MjdJdz4/edit?usp=sharing
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This topic was modified 6 days, 16 hours ago by
Asayre.
