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Synergies and Masteries Explained

Discussion in 'Sorceress' started by PhatTrumpet, Aug 8, 2006.

  1. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    Synergies and Masteries Explained

    Here it is folks, by mildly popular demand. I anxiously await your thoughts, comments, and criticism (hopefully constructive... go easy on me about the math part).


    Explanation of Synergies, Masteries, and the Interaction Between Them

    Table of Contents

    I. Introduction
    II. “Basic†Basics
    III. Interaction Basics
    IV. Optimal Skill Placement



    I. Introduction

    The effects of elemental masteries have been fairly well understood by the Diablo 2 community for quite some time now (except maybe Cold Mastery, but that’s another story). Each damage spell in the Sorceress’ arsenal would have a given base damage based on its current skill level (slvl), which was then enhanced by the percentage number listed in the skill description of that skill’s respective elemental mastery (Lightning Mastery, Fire Mastery). (Note: this same concept also applies to the Paladin’s Blessed Hammer, which is enhanced by the aura Concentration at half effectiveness.)

    With the advent of patch 1.10, however, came the introduction of synergies. Investing in lower-level, lesser-used skills now boosted the damage of many of the higher-level, commonly-used skills, and vice versa. No longer could a Sorceress simply dump 40 skill points into Nova and Lightning Mastery or Meteor and Fire Mastery, automatically becoming a bovine-slaughtering powerhouse. Sorceresses now needed to make an important decision with respect to their skill development plan: either specialize in one element, dealing impressive amounts of damage in one element while being relatively ineffective against monsters immune to that element; or specialize in two or three different elements, sacrificing huge damage for greater versatility. Items were also introduced in patch 1.10 that enhanced elemental damages beyond the effects of +skills gear (e.g. Eschuta’s Temper, Ormus’ Robes, and Rainbow Facet jewels).

    But how do all these damage modifiers work together? Are they multiplicative like Tiger Strike and Dragon Tail, or are they additive like sources of off-weapon enhanced damage? How should one go about placing skill points to maximize the damage of a particular spell throughout their Sorceress’ development? This guide attempts to answer these questions.


    II. “Basic†Basics

    You’ve probably heard it a million times by now: “synergies aren’t affected by +skills… only hard points count as synergies.†What people are trying to say is that if and only if you actually place an available skill point into a synergy skill do you gain exactly one point worth of synergy bonus. By placing one point in a synergy and getting +19 to all skills from gear you do not gain 20 times the synergy bonus; you only gain the one point of synergy bonus you actually invested. Similarly, if you’ve already maxed out a synergy (20 skill points invested) and you decide to put on a piece of gear that gives +1 to all skills, you do not gain any additional synergy bonus; you only benefit from the 20 points already invested. (Your spell damage will increase as a result of the additional +1 to all skills, but your synergy bonus will remain unchanged.) Synergy bonuses are only affected by hard points you actually invest into the synergy. If you want to gain the maximum benefit of a particular synergy, you must max out that synergy. Getting a particular synergy to slvl 20 through the use of +skills gear will not net the same result.

    Masteries, however, are just the opposite. You can invest one point into Fire or Lightning Mastery, equip gear that gives +19 to all skills, and reap the benefits of a level 20 Fire or Lightning Mastery. This is an important distinction that needs to be made and understood. This guide refers to total points invested in synergies and displayed slvl mastery because the former is completely independent of +skills gear while the latter is dependent on +skills gear.


    III. Interaction Basics

    Here is where most people’s questions will hopefully be answered.

    All synergy bonuses add together to form a total synergy bonus. This resulting total is applied to a particular skill’s base damage before any other modifiers. If we let U represent the base damage of the skill in question and X% represent the total synergy bonus, the synergy-modified damage, T, would be calculated as follows:

    T = U * [1 + (X% / 100)]

    The above calculated damage is now what the mastery bonus is applied to. The catch here is that any enhanced elemental damage you may have from gear also gets applied in this step. All elemental damage enhancements (from items like Eschuta’s Temper, Ormus’ Robes, Rainbow Facet jewels, etc.) are added directly to the displayed mastery bonus. This point warrants a bit of elaboration. Because elemental damage enhancements from gear get lumped together with the elemental mastery, the bigger the mastery is the less effective gear enhancements become (another proportionality issue). Something like Eschuta’s on a Lightning Sorceress will have relatively little effect on the final damage, whereas something like Death’s Fathom on a Cold Sorceress will boost her damage by monstrous amounts.

    Once all the mastery and gear bonuses are summed together, this resulting total is applied to the above calculated damage as follows, with S representing the fully-modified damage (or final damage) and Y% representing the sum of all gear and mastery bonuses:

    S = T * [1 + (Y% / 100)]

    Now, if we combine the two above formulas into one formula, we get the following:

    S = U * [1 + (X% / 100)] * [1 + (Y% / 100)]

    or,

    [final damage] = [base spell damage] * [total synergy bonus] * [total mastery and gear bonus]

    There are two key points to note here. The first point is that the only operation in the formula is multiplication, meaning that synergies and masteries are multiplicative. As synergies and masteries both increase as you level up, your damage will increase exponentially rather than linearly. The second point is that multiplication is a commutative operation, meaning that even though synergies are technically applied to the base spell damage before the sum of masteries and gear bonuses, the order in which either modifier is applied doesn’t matter as far as the final calculated damage is concerned. (You can verify this for yourself by plugging numbers into the final formula.)


    IV. Optimal Skill Placement

    This section addresses how to go about determining the appropriate allocation of skill points between synergies and mastery to attain maximum spell damage at every level of your Sorceress’ development. This discussion assumes that you’ve already invested 20 points in the spell you’re trying to maximize since that’s actually the first and most fundamental step to attaining maximum damage. (I seem to remember reading, possibly in the Paladin Mage Guide, that there’s a certain point with Fist of the Heavens at which additional points in synergies actually net you more damage than additional points in the skill itself, but as far as I know that’s not the case with any of the Sorceress’ skills.)

    Referring back to the final formula that was determined above, we see that we have four terms to contend with:

    [final damage]
    [base damage]
    [total synergy modifier]
    [total mastery and gear modifier]


    The final damage term is obviously what we’re trying to maximize here. Furthermore, because we’ve assumed that the skill itself is already maxed out, the base damage term is actually a constant and can be eliminated from the equation. Thus, the way to maximize the final damage term is to maximize the product on the opposite side of the equation:

    [total synergy modifier] * [total mastery and gear modifier]

    or,

    [1 + (X% / 100)] * [1 + (Y% / 100)]

    Many people automatically assume that maximizing this modifier simply means placing points in the skill that gives the biggest bonus per point, e.g. placing points in Lightning Mastery (12% per point) rather than one of the synergies to Chain Lightning (4% per point). Unfortunately, of course, it’s more complicated than this.

    Let’s say for a moment that you are trying to maximize your Chain Lightning damage and have the following skill allocation:

    1 Charged Bolt
    1 Nova
    1 Lightning
    20 Chain Lightning
    19 Lightning Mastery

    Let’s also say that you’re wearing gear with +10 to all skills total, bringing your Lightning Mastery up to slvl 29 (+386%). The 3 points invested in synergies (1 Charged Bolt, 1 Nova, and 1 Lightning; +12% total) combined with the slvl 29 Lightning Mastery result in the following damage modifier:

    [1 + (12 / 100)] * [1 + (386 / 100)] = 5.4432

    Now, if you were to level up and place your next skill point in Lightning Mastery under the assumption that the extra 12% from Lightning Mastery would net you more damage than the extra 4% from a synergy, you would get the following damage modifier:

    [1 + (12 / 100)] * [1 + (398 / 100)] = 5.5776

    If, however, you had put that extra skill point in a synergy to Chain Lightning, you would have gotten the following damage modifier:

    [1 + (16 / 100)] * [1 + (386 / 100)] = 5.6376

    5.6376 is obviously more than 5.5776, but how can 4% net you more damage than 12%? It’s all about proportionality. The mastery bonus is so large at this point (386%) that adding another 12% ends up being relatively small proportionally to the total mastery bonus, whereas the synergy bonus is so small (12%) that another 4% ends up being relatively large proportionally to the total synergy bonus. The same can be shown for the case where the incremental synergy bonus is larger than the incremental mastery bonus, e.g. Fire Ball synergies (14%) and Fire Mastery (7%).

    The key, then, is to find the point at which the proportional incremental increases for the two respective modifiers are equal. Once you reach this point, maximum damage is maintained by alternating points between synergy and mastery one after the other. As I see it, there are three basic ways to go about finding this numerical cusp (if you just want to see the results, skip down to the Calculus section below):


    Trial and Error – Go to a skill planner and play around with the skill allocation (simultaneously add a point to the mastery and remove a point from the synergy for each trial, or vice versa, until you get the maximum damage possible for the number of skill points available). You should see the damage climb to a maximum and then fall back down again. At the maximum damage point, note the difference between total points invested in synergies and displayed slvl mastery. This difference will be your rule for maximum damage.


    Spreadsheet – Setup a spreadsheet with synergy investment (1 to whatever the maximum synergy investment is for that particular skill) at the top of each column and mastery slvl (1 to however high you think you can get your mastery) at the beginning of each row. Each cell formula must then contain the product of the synergy and mastery modifiers based on the numbering of the rows and columns.

    If x% is the incremental synergy bonus, the synergy portion of the modifier will look like the following:

    [1 + ( [column , $1] * x% / 100)]

    The mastery modifiers look a bit different because you need to account for the fact that they start off higher than the incremental bonus and take off incrementally from there.

    Lightning Mastery: [1 + (0.5 + ([$A , row] - 1) * 12 / 100)]

    Fire Mastery: [1 + (0.3 + ([$A , row] - 1) * 7 / 100)]

    If you setup your spreadsheets like I have mine, with synergies along Row 1 and mastery down Column A, your first formula cell (B2) should look like the following:

    Fire Ball: =(1+(0.3+($A2-1)*0.07))*(1+(B$1*0.14))

    Chain Lightning: =(1+(0.5+($A2-1)*0.12))*(1+(B$1*0.04))

    Etc. (Remember that any gear modifiers should be added to the 0.3 or 0.5 term towards the left; gear modifiers are supposed to be lumped in with the mastery bonus. Also, don't get confused by the dollar signs. In Excel all they do is keep that portion of the cell reference constant when you fill down or right, so it's more of a convenience than anything else.)

    If setup correctly, the spreadsheet will show an apparent maximum damage modifier line diagonally from somewhere in the upper-left towards the lower-right. Note the difference between synergy investment and slvl mastery along this line to get the maximum damage rule (again, if you just want to see the results, skip down to the Calculus section below).


    Calculus – Those who have taken calculus know that the local maximum value of a function can be determined by taking the derivative of the function and setting it equal to zero. For this type of analysis we can work off of the formulas entered into the spreadsheet, but we first need to make the transition from two variables (total points invested in synergies and displayed slvl mastery) to one variable (the difference between these two variables). One way this can be accomplished is by setting the midpoint between total points invested in synergies and displayed slvl mastery equal to some constant, C, and setting the difference between that midpoint and either variable equal to D such that:

    [total points invested in synergies] = [C + D]

    and,

    [displayed slvl mastery] = [C - D]

    We then take the derivative of the product of the two damage modifiers using the product rule so that the C terms cancel out, set the derivative equal to zero, and solve for D. The difference rule for the particular spell, i.e. total points invested in synergies minus displayed slvl mastery, is then simply 2D:

    [total points invested in synergies] - [displayed slvl mastery] = [C + D] - [C - D] = C + D - C + D = 2D

    Below is a summary of the synergy-mastery interaction skills analyzed using the calculus method:

    Blessed Hammer synergies with Concentration:
    [1 + (0.6 + [C – D – 1] * 15 / 100) / 2] * [1 + ([C + D] * 14 / 100)]
    = [1.225 + 0.075C – 0.075D] [1 + 0.14C + 0.14D]
    d/dD = [1.225 + 0.075C – 0.075D] [0.14] + [-0.075] [1 + 0.14C + 0.14D] = 0
    0.1715 + 0.0105C – 0.0105D – 0.075 – 0.0105C – 0.0105D = 0
    0.0965 = 0.021D
    D = 4.595
    2D = 9.19 (9 or 10)

    Chain Lightning synergies with Lightning Mastery:
    [1 + (0.5 + [C – D – 1] * 12 / 100)] * [1 + ([C + D] * 4 / 100)]
    = [1.38 + 0.12C – 0.12D] [1 + 0.04C + 0.04D]
    d/dD = [1.38 + 0.12C – 0.12D] [0.04] + [-0.12] [1 + 0.04C + 0.04D] = 0
    0.0552 + 0.0048C – 0.0048D – 0.12 – 0.0048C – 0.0048D = 0
    -0.0648 = 0.0096D
    D = -6.750
    2D = -13.50 (-13 or -14)

    Charged Bolt synergy with Lightning Mastery:
    [1 + (0.5 + [C – D – 1] * 12 / 100)] * [1 + ([C + D] * 6 / 100)]
    = [1.38 + 0.12C – 0.12D] [1 + 0.06C + 0.06D]
    d/dD = [1.38 + 0.12C – 0.12D] [0.06] + [-0.12] [1 + 0.06C + 0.06D] = 0
    0.0828 + 0.0072C – 0.0072D – 0.12 – 0.0072C – 0.0072D = 0
    -0.0372 = 0.0144D
    D = -2.583
    2D = -5.17 (-5 or -6)

    Enchant synergy with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 9 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.09C + 0.09D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.09] + [-0.07] [1 + 0.09C + 0.09D] = 0
    0.1107 + 0.0063C – 0.0063D – 0.07 – 0.0063C – 0.0063D = 0
    0.0407 = 0.0126D
    D = 3.230
    2D = 6.46 (6 or 7)

    Fire Ball synergies with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 14 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.14C + 0.14D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.14] + [-0.07] [1 + 0.14C + 0.14D] = 0
    0.1722 + 0.0098C – 0.0098D – 0.07 – 0.0098C – 0.0098D = 0
    0.1022 = 0.0196D
    D = 5.214
    2D = 10.43 (10 or 11)

    Fire Bolt synergies with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 16 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.16C + 0.16D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.16] + [-0.07] [1 + 0.16C + 0.16D] = 0
    0.1968 + 0.0112C – 0.0112D – 0.07 – 0.0112C – 0.0112D = 0
    0.1268 = 0.0224D
    D = 5.661
    2D = 11.32 (11 or 12)

    Hydra synergies with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 3 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.03C + 0.03D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.03] + [-0.07] [1 + 0.03C + 0.03D] = 0
    0.0369 + 0.0021C – 0.0021D – 0.07 – 0.0021C – 0.0021D = 0
    -0.0331 = 0.0042D
    D = -7.881
    2D = -15.76 (-15 or -16)

    Inferno synergy with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 13 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.13C + 0.13D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.13] + [-0.07] [1 + 0.13C + 0.13D] = 0
    0.1599 + 0.0091C – 0.0091D – 0.07 – 0.0091C – 0.0091D = 0
    0.0899 = 0.0182D
    D = 4.940
    2D = 9.** (9 or 10)

    Lightning synergies with Lightning Mastery:
    [1 + (0.5 + [C – D – 1] * 12 / 100)] * [1 + ([C + D] * 8 / 100)]
    = [1.38 + 0.12C – 0.12D] [1 + 0.08C + 0.08D]
    d/dD = [1.38 + 0.12C – 0.12D] [0.08] + [-0.12] [1 + 0.08C + 0.08D] = 0
    0.1104 + 0.0096C – 0.0096D – 0.12 – 0.0096C – 0.0096D = 0
    -0.0096 = 0.0192D
    D = -0.500
    2D = -1.00 (exactly -1)

    Meteor synergies with Fire Mastery:
    [1 + (0.3 + [C – D – 1] * 7 / 100)] * [1 + ([C + D] * 5 / 100)]
    = [1.23 + 0.07C – 0.07D] [1 + 0.05C + 0.05D]
    d/dD = [1.23 + 0.07C – 0.07D] [0.05] + [-0.07] [1 + 0.05C + 0.05D] = 0
    0.0615 + 0.0035C – 0.0035D – 0.07 – 0.0035C – 0.0035D = 0
    -0.0085 = 0.007D
    D = -1.214
    2D = -2.43 (-2 or -3)

    (Remember again that the above equations assume no gear modifiers. If you want to figure in any damage enhancements from gear you will need to add them in to the mastery side of the initial modifier equation.)
     
  2. Delreich

    Delreich Diabloii.Net Member

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    Very nice. Doing the calculus bit symbolically should give you the simpler formula:
    Code:
    2D = MB/ML - SB/SL
    
    MB = Mastery base and any eq bonuses
    ML = Mastery per level
    SB = Synergy base
    SL = Synergy per level
    Example: Fire Ball
    ML = 0.07
    MB = 1.23 (+30% at lvl1, so 1.30 minus one ML)
    SB = 1 (this is always the case, right?)
    SL = 0.14

    1.23/0.07 - 1/0.14 = 10.43


    This was posted here a few days ago by someone not me, so credit should go to whoever that was.

    [edit]
    Found it. Rynas was the 'whoever'.
     
  3. NoobXDon

    NoobXDon Diabloii.Net Member

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    " (I seem to remember reading, possibly in the Paladin Mage Guide, that there’s a certain point with Fist of the Heavens at which additional points in synergies actually net you more damage than additional points in the skill itself, but as far as I know that’s not the case with any of the Sorceress’ skills.) "

    are you sure about this?

    great guide. sticky melianor?
     
  4. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    Delreich: I assumed someone would be able to work that out more eloquently than I could. I'll check out that link and see what I can do to clean it up.

    NoobXDon:
    Like I said, I don't think any Sorceress skills behave like this only because I've never seen it mentioned, but I could be wrong.
     
  5. melianor

    melianor D3 Wizard Moderator

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    *honk*stickied* very nicely put. will also go into the library after a while. I have explained and copy pasted similiar formulas for ages and i have gotten sick of it ;)

    Thanks for that post PhatTrumpet, please iron out any errors if people find any and then make a final post. From the basic formulas concerning masteries/synergies i could not see anything wrong.

    greets, melianor
     
  6. MageChick

    MageChick Diabloii.Net Member

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    Great Guide

    PhatTrumpet,

    I think you have a winner again. I've seen you explain this to a few people before. This should definatly help lots of people. (Now if we can just get the average open bnet player to come research this forum instead of just spamming every channel out there for advice. :rolleyes:)

    I do have a couple of suggestions. Unfortunatly, I'm afraid that you will still loose many people with your explanations and even your end results. I think it would help to pull the end results out to a seperate section in addition to listing them in the calc area. If you pulled them out to a seperate section, and included a breif explanation of what that means again - I think you could "idiot proof" it.

    Here's what I mean:
    I. Introduction
    II. “Basic†Basics
    III. Interaction Basics
    IV. Optimal Skill Placement Calculation Methods
    a. Trial and error/Skill Calculator
    b. Spreadsheet
    c. Calculus
    V. Final Results

    Below is a summary of the synergy-mastery interaction skills results. In order to maximise your skill damage, your total hard points invested into synergies should be the listed number of points above your slvl mastery (below if the value is negative).

    Blessed Hammer synergies with Concentration: 9 to 10
    Chain Lightning synergies with Lightning Mastery: -13 to -14
    Charged Bolt synergy with Lightning Mastery: -5 to -6
    Lightning synergies with Lightning Mastery: exactly -1
    Enchant synergy with Fire Mastery: 6 to 7
    Fire Ball synergies with Fire Mastery: 10 to 11
    Fire Bolt synergies with Fire Mastery: 11 to 12
    Hydra synergies with Fire Mastery: -15 to -16
    Inferno synergy with Fire Mastery: 9 to 10
    Meteor synergies with Fire Mastery: -2 to -3

    *Keep in mind that some gear (e.g. Eschuta’s Temper, Ormus’ Robes, Tal Rasha's Orb, and Rainbow Facet jewels) adds a percent damage, similar to skill mastery. You should add 1 to 2 levels to your displayed slvl mastery to compensate.
     
  7. melianor

    melianor D3 Wizard Moderator

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    "idiotproof" would be very welcome.

    There is basic textual explanation without formulas, that will be enough for many, but there is more to be said for those of interest.

    Taking Cold Mastery or other -xx% negative resistances into the formula would also be welcome, since quite a few ask about that too. Adding it to the formula is easy by the way. Not much to change. Also adding the influence of monster resistances in general would be great. Damn where do i dig out that formula. Its quite a few times in these forums, but the i posted these often is quite long ago. *goes digging through own posts*

    EDIT: While browsing my posts and not finding anything i realized from viewing some threads that you PhatTrumpet surely have the complete formula including resistances and -xx% resistances gear included up your sleeve. I think it would be interesting to see the changes in damage, once the resistances have been applied, just aswell how big the help of cold mastery and -xx% resistances gear can be.
     
  8. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    I didn't realize this was going to turn into a Spell Damage Calculation Compendium. :rolleyes:

    Maybe I'm reading you wrong, but I don't think factoring -res gear and monster resistances into the damage optimization formula would be possible. I would certainly be willing to include a small section on how -res affects your damage output, but as far as including that in a general rule for optimal damage that might become a bit too complicated.
     
  9. melianor

    melianor D3 Wizard Moderator

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    I didn't mean for you to put it into the optimization formula, but just to add it to the normal damage calculation formula, as an extra point, possibly separate from the formula with the damage calculation. Since the thread title wants to explain synergy/mastery i think this should be in it, or i wish it would. Many people totally disregard the effects that -xx% gear has, well maybe not after all the Infinity builds :p, but still.

    All in all its your guide though, so you should do with it what you think best. When this is final it will go into the library.

    EDIT: Finally found it. This is what i meant to add.

    damage = base damage * (1+synergy%) * (1+mastery%) * (1-(monsterres%-negative resistances sources%)

    just like your formula above, just with monsterres and -xx% resistance sources included.
     
  10. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    OK. Tossing that in shouldn't be much work at all.

    I'll incorporate everyone's comments and post the updated version as soon as it's ready.
     
  11. MageChick

    MageChick Diabloii.Net Member

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    Actually, can you just re-write the players guide that comes with the game. include every stickied thread on dii.net, and all the good tips and tricks too. Be sure to include all the information in the Arreat Summit and update it for 1.12 while your at it. Can you have that on my desk by Friday. :grin:

    Sorry - don't mean to take over your guide.
     
  12. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    Here's revision #1. I had to divide it between two posts because it apprently jumped up just over 20,000 characters.



    Explanation of Synergies, Masteries, Decreased Enemy Resistance, and the Interaction Between Them

    Table of Contents

    Code:
    I.   Introduction
    II.  “Basic†Basics
       A. Synergies
       B. Masteries
       C. Decreased Enemy Resistance
    III. Interaction Basics
    IV.  Determination of Optimal Skill Placement
       A. Trial and Error Method
       B. Spreadsheet Method
       C. Calculus Method
       D. Summary of Results

    I. Introduction


    The effects of elemental masteries have been fairly well understood by the Diablo 2 community for quite some time now (except maybe Cold Mastery, but that’s another story). Each damage spell in the Sorceress’ arsenal would have a given base damage based on its current skill level (slvl), which was then enhanced by the percentage number listed in the skill description of that skill’s respective elemental mastery (Lightning Mastery, Fire Mastery). (Note: this same concept also applies to the Paladin’s Blessed Hammer, which is enhanced by the aura Concentration at half effectiveness.)

    With the advent of patch 1.10, however, came the introduction of synergies. Investing in lower-level, lesser-used skills now boosted the damage of many of the higher-level, commonly-used skills, and vice versa. No longer could a Sorceress simply dump 40 skill points into Nova and Lightning Mastery or Meteor and Fire Mastery, automatically becoming a bovine-slaughtering powerhouse. Sorceresses now needed to make an important decision with respect to their skill development plan: either specialize in one element, dealing impressive amounts of damage in one element while being relatively ineffective against monsters immune to that element; or specialize in two or three different elements, sacrificing huge damage for greater versatility. Items were also introduced in patch 1.10 that enhanced elemental damages beyond the effects of +skills gear (e.g. Eschuta’s Temper, Ormus’ Robes) and reduced enemy resistances beyond the effects of Conviction aura and Lower Resist curse (e.g. Rainbow Facet jewels).

    But how do all these damage modifiers work together? Are they multiplicative like Tiger Strike and Dragon Tail, or are they additive like sources of off-weapon enhanced damage? How should one go about placing skill points to maximize the damage of a particular spell throughout their Sorceress’ development? This guide attempts to answer these questions.


    II. “Basic†Basics

    A. Synergies

    You’ve probably heard it a million times by now: “synergies aren’t affected by +skills… only hard points count as synergies.†What people are trying to say is that if and only if you actually place an available skill point into a synergy skill do you gain exactly one point worth of synergy bonus. By placing one point in a synergy and getting +19 to all skills from gear you do not gain 20 times the synergy bonus; you only gain the one point of synergy bonus you actually invested. Similarly, if you’ve already maxed out a synergy (20 skill points invested) and you decide to put on a piece of gear that gives +1 to all skills, you do not gain any additional synergy bonus; you only benefit from the 20 points already invested. (Your spell damage will increase as a result of the additional +1 to all skills, but your synergy bonus will remain unchanged.) Synergy bonuses are only affected by hard points you actually invest into the synergy. If you want to gain the maximum benefit of a particular synergy, you must max out that synergy. Getting a particular synergy to slvl 20 through the use of +skills gear will not net the same result.

    B. Masteries

    Masteries, however, are just the opposite. You can invest one point into Fire or Lightning Mastery, equip gear that gives +19 to all skills, and reap the benefits of a level 20 Fire or Lightning Mastery. This is an important distinction that needs to be made and understood. This guide refers to total points invested in synergies and displayed slvl mastery because the former is completely independent of +skills gear while the latter is dependent on +skills gear.

    C. Decreased Enemy Resistance

    This is another mod that became very popular starting in patch 1.10, but because the effects of it are highly dependant on enemy resists, it will not be discussed in the Optimum Skill Placement section.

    The negative enemy resist mod applies directly to your target's current resistance to that particular element when the character carrying the mod actually damages said target. Putting 'Obedience' (-25% to enemy fire resistance) on your Fire Sorceress' Merc will not increase your own damage output, nor will it increase the fire damage output of anyone else around you; you/they would have to actually wield the polearm to benefit from the mod. A net positive enemy resistance results in a reduction of the damage you inflict whereas a net negative enemy resistance results in an increase in the damage you inflict. (More on the calculations of this and the other two effects below.)

    (Note that negative enemy resist from gear cannot break immunites; only Conviction and Lower Resist can break immunities, at 20% effectiveness. Once an immunity is broken by Conviction or Lower Resist, however, negative enemy resist from gear will work as usual. Also, resistances are capped at -100%, meaning once a target's resistance is lowered to -100% it cannot be lowered any further.)


    III. Interaction Basics

    All synergy bonuses add together to form a total synergy bonus. This resulting total is applied to a particular skill’s base damage before any other modifiers. If we let V represent the base damage of the skill in question and X% represent the total synergy bonus, the synergy-modified damage, U, would be calculated as follows:

    U = V * [1 + (X% / 100)]

    The above calculated damage is now what the mastery bonus is applied to. The catch here is that any enhanced elemental damage you may have from gear also gets applied in this step. Any and all elemental damage enhancements from gear (items like Eschuta’s Temper, Ormus’ Robes, Rainbow Facet jewels, etc.) are added directly to the displayed mastery bonus. This point warrants a bit of elaboration. Because elemental damage enhancements from gear get lumped together with the elemental mastery, the bigger the mastery is the less effective those gear enhancements become (another proportionality issue). Something like Eschuta’s on a Lightning Sorceress will have relatively little effect on the final damage, whereas something like Death’s Fathom on a Cold Sorceress will boost her damage by monstrous amounts.

    Once all the mastery and gear bonuses are summed together, this resulting total is applied to the above calculated damage as follows, with T representing the fully-modified damage (or final damage) and Y% representing the sum of all gear and mastery bonuses:

    T = U * [1 + (Y% / 100)]

    After calculating the fully-modified spell damage, one can then determine how much damage a particular enemy will recieve based on the enemy's resistance. Let S be the actual inflicted damage, R% be the resistance of the target, and NR% be the total negative resistance from gear, curses, and auras:

    S = T * [1 - (R% - NR%) / 100]

    (Thanks to melianor for suggesting this.)

    Now, if we combine the three above formulas into one formula, we get the following:

    S = U * [1 + (X% / 100)] * [1 + (Y% / 100)] * [1 - (R% - NR%) / 100]

    or,

    [inflicted damage] = [base spell damage] * [total synergy bonus] * [total mastery and gear bonus] * [enemy resistance modifier]

    There are two key points to note here. The first point is that the only operation in the formula is multiplication, meaning that synergies and masteries are multiplicative. As synergies and masteries both increase as you level up, your damage increases exponentially rather than linearly. The second point is that multiplication is a commutative operation, meaning that even though synergies are technically applied to the base spell damage before the sum of masteries and gear bonuses, the order in which either modifier is applied doesn’t matter as far as the final calculated damage is concerned. (You can verify this for yourself by plugging numbers into the final formula.)
     
  13. PhatTrumpet

    PhatTrumpet Diabloii.Net Member

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    IV. Determination of Optimal Skill Placement

    This section addresses how to go about determining the appropriate allocation of skill points between synergies and mastery to attain maximum spell damage at every level of your Sorceress’ development. This discussion assumes that you’ve already invested 20 points in the spell you’re trying to maximize since that’s actually the first and most fundamental step to attaining maximum damage. (According to LordDrift's Mage Paladin Guide v2.1, there’s actually a point with Fist of the Heavens at which additional points in the Holy Shock synergy actually net you more damage than additional points in the skill itself, but as far as I know this is not the case with any of the Sorceress’ skills.)

    Referring back to the final formula that was determined above, we see that we have four terms to contend with:

    [final damage]
    [base damage]
    [total synergy modifier]
    [total mastery and gear modifier]


    The final damage term is obviously what we’re trying to maximize here. Furthermore, because we’ve assumed that the skill itself is already maxed out, the base damage term is actually a constant and can be eliminated from the equation. Thus, the way to maximize the final damage term is to maximize the product on the opposite side of the equation:

    [total synergy modifier] * [total mastery and gear modifier]

    or,

    [1 + (X% / 100)] * [1 + (Y% / 100)]

    Many people automatically assume that maximizing this modifier simply means placing points in the skill that gives the biggest bonus per point, e.g. placing points in Lightning Mastery (12% per point) rather than one of the synergies to Chain Lightning (4% per point). Unfortunately, of course, it’s more complicated than this.

    Let’s say for a moment that you are trying to maximize your Chain Lightning damage with the following skill allocation:

    1 Charged Bolt
    1 Nova
    1 Lightning
    20 Chain Lightning
    19 Lightning Mastery

    Let’s also say that you’re wearing gear with +10 to all skills total, bringing your Lightning Mastery up to slvl 29 (+386%). The 3 points invested in synergies (1 Charged Bolt, 1 Nova, and 1 Lightning; +12% total) combined with the slvl 29 Lightning Mastery result in the following damage modifier:

    [1 + (12 / 100)] * [1 + (386 / 100)] = 5.4432

    Now, if you were to level up and place your next skill point in Lightning Mastery under the assumption that the extra 12% from Lightning Mastery would net you more damage than the extra 4% from a synergy, you would get the following damage modifier:

    [1 + (12 / 100)] * [1 + (398 / 100)] = 5.5776

    If, however, you had put that extra skill point in a synergy to Chain Lightning, you would have gotten the following damage modifier:

    [1 + (16 / 100)] * [1 + (386 / 100)] = 5.6376

    5.6376 is obviously more than 5.5776, but how can 4% net you more damage than 12%? It’s all about proportionality. The mastery bonus is so large at this point (386%) that directly adding another 12% ends up being relatively small proportionally to the total mastery bonus, whereas the synergy bonus is so small (12%) that another 4% ends up being relatively large proportionally to the total synergy bonus. The same can be shown for the case where the incremental synergy bonus is larger than the incremental mastery bonus, e.g. Fire Ball synergies (14% per point) and Fire Mastery (7% per point).

    The key, then, is to find the point at which the proportional incremental increases for the two respective modifiers are equal. Once you reach this point, maximum damage is maintained by alternating points between synergy and mastery one after the other. As I see it, there are three basic ways to go about finding this numerical cusp (if you just want to see the results, skip down to the Summary of Results below):


    A. Trial and Error Method

    Go to a skill planner and play around with the skill allocation, simultaneously adding a point to the mastery and removing a point from the synergy for each trial, or vice versa, until you get the maximum damage possible for the number of skill points available. If you continue doing this you should see the damage climb to a maximum and then fall back down again. At the maximum damage point, note the difference between total points invested in synergies and displayed slvl mastery. This difference will be your rule for maximum damage.


    B. Spreadsheet Method

    Setup a spreadsheet with synergy investment (1 to whatever the maximum synergy investment is for that particular skill) at the top of each column and mastery slvl (1 to however high you think you can get your mastery) at the beginning of each row. Each cell formula must then contain the product of the synergy and mastery modifiers based on the numbering of the rows and columns.

    If x% is the incremental synergy bonus, the synergy portion of the modifier will look like the following:

    [1 + ( [column , $1] * x% / 100)]

    The mastery modifiers look a bit different because you need to account for the fact that they start off higher than the incremental bonus and take off incrementally from there.

    Lightning Mastery: [1 + (0.5 + ([$A , row] - 1) * 12 / 100)]

    Fire Mastery: [1 + (0.3 + ([$A , row] - 1) * 7 / 100)]

    If you setup your spreadsheets like I have mine, with synergies along Row 1 and mastery down Column A, your first formula cell (B2) should look like the following:

    Fire Ball: =(1+(0.3+($A2-1)*0.07))*(1+(B$1*0.14))

    Chain Lightning: =(1+(0.5+($A2-1)*0.12))*(1+(B$1*0.04))

    Etc. (Remember that any gear modifiers should be added to the 0.3 or 0.5 term towards the left; gear bonuses are lumped in with the mastery bonus. Also, don't get confused by the dollar signs. In Excel they simply keep that portion of the cell reference constant when you fill down or right, so it's more of a convenience than anything else.)

    If setup correctly, the spreadsheet will show an apparent maximum damage modifier line diagonally from somewhere in the upper-left towards the lower-right. Note the difference between synergy investment and slvl mastery along this line to get the maximum damage rule (again, if you just want to see the results, skip down to the Summary of Results below).


    C. Calculus Method

    Those who have taken calculus know that the local maximum value of a function can be determined by taking the derivative of the function and setting it equal to zero. For this type of analysis we can work off of the formulas entered into the spreadsheet, but we first need to make the transition from two variables (total points invested in synergies and displayed slvl mastery) to one variable (the difference between these two variables). One way this can be accomplished is by setting the midpoint between total points invested in synergies and displayed slvl mastery equal to some constant, C, and setting the difference between that midpoint and either endpoint equal to D such that:

    [total points invested in synergies] = [C + D]

    and,

    [displayed slvl mastery] = [C - D]

    We then take the derivative of the product of the two damage modifiers using the product rule so that the C terms cancel out, set the derivative equal to zero, and solve for D. The difference rule for the particular spell, i.e. total points invested in synergies minus displayed slvl mastery, is then simply 2D:

    [total points invested in synergies] - [displayed slvl mastery] = [C + D] - [C - D] = C + D - C + D = 2D

    We will first go through the analysis symbollically using a notation first posted by Rynas:

    MB = base mastery bonus plus any gear bonuses
    ML = additional mastery bonus per level
    SB = base synergy bonus
    SL = additional synergy bonus per level

    MB represents the theoretical modifier for an slvl 0 mastery, which is equal to the slvl 1 mastery modifier minus one incremental mastery bonus. MB equals 1.23 for Fire Mastery and 1.38 for Lightning Mastery.

    ML represents the additional mastery bonus for each skill level. ML equals 0.07 for Fire Mastery and 0.12 for Lightning Mastery.

    SB represents the modifier for zero points invested into synergies, which is always equal to 1.

    SL represents the incremental synergy bonus for each additional synergy investment. This number is listed as a percentage in the skill of interest's description (e.g. 0.14 for Fire Ball, 0.04 for Chain Lightning).

    Using this notation, the product of the two modifiers becomes:

    [MB + (C - D) * ML] * [SB + (C + D) * SL]

    or,

    [MB + ML(C) - ML(D)] * [SB + SL(C) + SL(D)]

    Taking the derivative of this with respect to D and setting it equal to zero:

    d/dD = [MB + ML(C) - ML(D)] * [SL] + [-ML] * [SB + SL(C) + SL(D)] = 0

    or,

    MB(SL) + ML(SL)(C) - ML(SL)(D) - ML(SB) - ML(SL)(C) - ML(SL)(D) = 0

    Eliminating the ML(SL)(C) terms and solving for 2D:

    MB(SL) - ML(SB) = 2(ML)(SL)(D)

    or,

    2D = MB/ML - SB/SL

    (Big thanks to Rynas for pointing out this convenient result.)

    As you can see, after all that work, we're left with a very simple equation that can be applied to any skill with a synergy-mastery interaction, where:

    [total points invested in synergies] - [displayed slvl mastery plus any gear bonuses] = [MB / ML] - [1 / SL]

    Below is a summary of the applications of this formula (skills with an asterisk have been detailed):

    Blessed Hammer synergies with Concentration:*
    MB = 1 + [0.60 - 0.15] / 2 = 1.225 (divided by 2 because it works at half effectiveness)
    ML = 0.075 (again, half effectiveness)
    SB = 1
    SL = 0.14
    [MB / ML] - [1 / SL] = [1.225 / 0.075] - [1 / 0.14] = 9.19 (9 or 10

    Chain Lightning synergies with Lightning Mastery:*
    MB = 1 + [0.50 - 0.12] = 1.38
    ML = 0.12
    SB = 1
    SL = 0.04
    [MB / ML] - [1 / SL] = [1.38 / 0.12] - [1 / 0.04] = -13.50 (-13 or -14)

    Charged Bolt synergy with Lightning Mastery:
    [MB / ML] - [1 / SL] = [1.38 / 0.12] - [1 / 0.06] = -5.17 (-5 or -6)

    Enchant synergy with Fire Mastery:
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.09] = 6.46 (6 or 7)

    Fire Ball synergies with Fire Mastery:*
    MB = 1 + [0.30 - 0.07] = 1.23
    ML = 0.07
    SB = 1
    SL = 0.14
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.14] = 10.43 (10 or 11)

    Fire Bolt synergies with Fire Mastery:
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.16] = 11.32 (11 or 12)

    Hydra synergies with Fire Mastery:
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.03] = -15.76 (-15 or -16)

    Inferno synergy with Fire Mastery:
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.13] = 9.** (9 or 10)

    Lightning synergies with Lightning Mastery:
    [MB / ML] - [1 / SL] = [1.38 / 0.12] - [1 / 0.08] = -1.00 (exactly -1)

    Meteor synergies with Fire Mastery:
    [MB / ML] - [1 / SL] = [1.23 / 0.07] - [1 / 0.05] = -2.43 (-2 or -3)


    D. Summary of Results

    Blessed Hammer: total number of points invested in synergies should be 9 or 10 more than your displayed slvl Concentration.

    Chain Lightning: displayed slvl Lightning Mastery should be 13 or 14 more than the total number of points invested in synergies.

    Charged Bolt: displayed slvl Lightning Mastery should be 5 or 6 more than the total number of points invested in the synergy.

    Enchant: total number of points invested in synergies should be 6 or 7 more than your displayed slvl Fire Mastery.

    Fire Ball: total number of points invested in synergies should be 10 or 11 more than your displayed slvl Fire Mastery.

    Fire Bolt: total number of points invested in synergies should be 11 or 12 more than your displayed slvl Fire Mastery.

    Hydra: displayed slvl Fire Mastery should be 15 or 16 more than the total number of points invested in synergies.

    Inferno: total number of points invested in the synergy should be 9 or 10 more than the displayed slvl Fire Mastery.

    Lightning: displayed slvl Lightning Mastery should be 1 more than the total number of points invested in synergies.

    Meteor: displayed slvl Fire Mastery should be 2 or 3 more than the total number of points invested in synergies.

    (Remember again that the above calculations and results assume no gear modifiers. If you want to figure in any damage enhancements from gear you will need to add them in to the mastery modifier and re-calculate.)
     
  14. melianor

    melianor D3 Wizard Moderator

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    Negative resistances: To make it idiot proof could you mention that Cold mastery is calculated together with those negative resistance modifiers? Now it sounds like this is a different thing, which it is not for the damage calculation.

    Otherwise, great. Please make a total repost of this when its finished. I will sticky the final version and add it to the library.
     
  15. Lyndis

    Lyndis Diabloii.Net Member

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    Does anyone know whether it is better to max out cold mastery or a synergy if you could only max one of those out?
     
  16. FrostBurn

    FrostBurn Diabloii.Net Member

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    Depends on who you are running.

    Against 0% CR monsters, maxing a synergy is better.

    Against 50% CR monsters, taking CM to slvl 27 is better.

    You're welcome.
     
  17. Lyndis

    Lyndis Diabloii.Net Member

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    I'll be against meph. So I'll want level 32?
     
  18. FrostBurn

    FrostBurn Diabloii.Net Member

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    Yes.

    more characters.
     
  19. MageChick

    MageChick Diabloii.Net Member

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    You really think that CM slvl 32 will help more than putting those 5 points into a synergy? This is an honest question. I just don't see it - even for a dedicated meph runner.

    I'd still stop your CM at slvl 17 initially, then get at least 1 synergy maxed. Then you can increas CM up to about 27. But I'd stop here until all the synergies are maxed. Now once you have all your synergies maxed, by all means - top your CM off at slvl 32

    (you can probably do all this by the time you're in the mid 80's if you have good gear anyway, so maybe its a moot point.)
     
  20. Lyndis

    Lyndis Diabloii.Net Member

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    I'll probably only hit level 76 or 77 at most, so I guess I'll leave cold mastery at less than optimal levels...
     

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