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Apr 10 2009 11:59pm
Countdown to Hellfire



- a technical approach to the Übers


Table of Contents
Introduction
Chance to Get Hit
Beefing up Resistances
Life Tap
Character Skills and Equipment
Smite Damage
Crushing Blow
Open Wounds
Countdown to Hellfire



Introduction

Six weapons: Black, Grief, Last Wish, Astreon's Iron Ward, Heaven's Light, Stormlash. One objective: crush the Übers.

Last Wish is often regarded as the ultimate weapon for an über smiter. But it isn't the better weapon of choice, no matter how improbable it may sound. Players tend to overlook the importance of their remaining equipment and thus end up concluding that the modifiers on LW alone make it outshine any other weapon. We will assemble a proper setup for each of the above weapons, ensuring high performance while not neglecting safety. Based on our choice of equipment, it is possible to give fairly accurate estimates of kill speed.

Our Smite Paladin will do battle in Über Tristram alone, hence the Übers have, on average, 655 000 HP each (650 000-660 000).


Chance to Get Hit

Our Defense is only checked when the to-Hit equation is considered (physically delivered attacks),
  • Chance to Hit = 2 * Level_att / (Level_att + Level_def) * AR / (AR + Defense)
capped below at 5 % and above at 95 %. Level_att, Level_def: attacker and defender levels; AR: Attack Rating (attacker); Defense (defender).

The to-Hit equation has a level component: 2 * Level_att / (Level_att + Level_def) and an Attack-Defense component: AR / (AR + Defense). Note that the important figures are the ratios (attacker to defender levels and AR to Defense), not the individual values. For instance, it makes no sense to ask whether X Defense at level Y is low, high, or enough, without reference to the attacker.

The Übers are level 110 with an AR of ~ 10-15 k (Lilith: AR ~ 30 k). Suppose our Paladin is level 80-90 with a Defense of ~ 15-30 k (1-point Defiance without Exile, or max Defiance with an average Exile).

Code
                         Attacker: Pandemonium Über

Level_att      Level_def      AR      Defense       CtH      Final CtH *)

   110             80        10 k      15 k        46.3 %      11.6 %
                                       30 k        28.9 %       7.2 %
                             15 k      15 k        57.9 %      14.5 %
                                       30 k        38.6 %       9.6 %
                             30 k      15 k        77.2 %      19.3 %
                                       30 k        57.9 %      14.5 %

   110             85        10 k      15 k        45.1 %      11.3 %
                                       30 k        28.2 %       7.1 %
                             15 k      15 k        56.4 %      14.1 %
                                       30 k        37.6 %       9.4 %
                             30 k      15 k        75.2 %      18.8 %
                                       30 k        56.4 %      14.1 %

   110             90        10 k      15 k        44.0 %      11.0 %
                                       30 k        27.5 %       6.9 %
                             15 k      15 k        55.0 %      13.8 %
                                       30 k        36.7 %       9.2 %
                             30 k      15 k        73.3 %      18.3 %
                                       30 k        55.0 %      13.8 %


*) Final Chance to Hit = CtH/4, after 75 % (max) block

Max block reduces the difference in CtH between 15 k and 30 k Defense from a rather significant ~ 17-20 % to just ~ 4-5 %. Besides, by far the most potent attacks in the game are of elemental nature, against which Defense does not offer any protection. Therefore, spend only 1 point in Defiance; there are better options for leftover skill points.


Beefing up Resistances

Fade : http://classic.battle.net/diablo2exp/skills/assassin-shadow.shtml#fade
  • Last Wish
    6 % CtC Level 11 Fade [240 s] when struck
    [All Resistances +56, 75 % curse length reduction, +11 % DR]
  • Treachery
    5 % CtC Level 15 Fade [288 s] when struck
    [All Resistances +60, 79 % curse length reduction, +15 % DR]
The numbers speak for themselves. Considering the low cost of Treachery ('ShaelThulLem'), it makes no sense to use Fade as an argument for Last Wish ('JahMalJahSurJahBer'). Trigger Fade before engaging in combat, as the chance is too risky to rely on during the fight.

Salvation : http://classic.battle.net/diablo2exp/skills/paladin-defense.shtml#salvation

It takes +300 % elemental resistance to maintain 75+ % elemental resistance in Hell (-100 %) against Über Mephisto's Level 20 Conviction (-125 %). Salvation offers ~ +80-100 % elemental resistance for 1-5 points and skill adders. The remaining ~ +200-220 % can be covered by equipment, resistance charms, and/or Fade (+60 %).

Some players suggest dominating Über Mephisto's aura by skilling a higher level (> 20) Conviction, thus rendering his aura ineffective. Personally, I consider it a waste to devote ~ 22 skill points, including 4 additional prerequisite skills, just for that single purpose. Reducing enemy Defense and elemental resistances does not benefit a physical damage skill with built-in auto-hit. Not even the slightest.

Salvation basically provides the same degree of safety for a few skill points.

Thawing Potion, Antidote
Temporary [30 s, stackable duration] bonus: +50 % CR, +50 % PR, including +10 % to MCR, +10 % to MPR. Gulp a few before taking on Über Duriel, Über Izual, and Lilith. Repeat in Über Tristram.


Life Tap

http://classic.battle.net/diablo2exp/skills/necromancer-curses.shtml#lifetap

Charges
Wands with Life Tap charges are shopable from Act III Normal or later. Depending on character level, it is possible to shop up to 105 charges of Level 6 Life Tap [28.0 s]. Below are ideal odds of the initial LT wearing off before we re-apply the curse from Dracul's Grasp (5 % CtC Level 10 Life Tap on striking). Assuming a 7-frame Smite (3.57 attacks per second), the minimum speed we will tolerate.

Code
Ideal odds of NOT overriding the initial Life Tap in n hits (7 FPA), wearing Dracul's Grasp

Level      duration      n        0.95^n
(LT)
1           16.0 s       57       5.38 %
2           18.4 s       65       3.57 %
3           20.8 s       74       2.25 %
4           23.2 s       82       1.49 %
5           25.6 s       91       0.94 %
6           28.0 s      100       0.60 %

The n hits assume a perfect Smite assault (ignoring blocking with HS [2 frames], hit recovery [5-6 frames], or whatever might occasionally hinder us in attacking). Regardless, either we proc LT on striking within the initial Tap's 16-28-second window and otherwise (as we will see in the final section), only the ashes of Über Mephisto or the corpse of one of his brothers remain at that time.

X % Chance to Cast Level Y Life Tap on striking
  • Dracul's Grasp
    5 % CtC Level 10 Life Tap [37.6 s, 8 2/3 yds] on striking
  • Last Wish
    10 % CtC Level 18 Life Tap [56.8 s, 14 yds] on striking
  • Exile
    15 % CtC Level 5 Life Tap [25.6 s, 5 1/3 yds] on striking
This time assuming we prefer to not use a charged wand.

Code
Odds of cursing Life Tap on target AT LEAST ONCE in n hits

      (Dracul's)      (Dracul's + LW)   (Dracul's + Exile)    (Dracul's + LW + Exile)
n     1 - 0.95^n        1 - 0.855^n        1 - 0.8075^n            1 - 0.72675^n

1        5.0 %            14.5 %               19.2 %                  27.3 %
2        9.7 %            26.8 %               34.7 %                  47.1 %
3       14.2 %            37.4 %               47.3 %                  61.6 %
4       18.5 %            46.5 %               57.4 %                  72.1 %
5       22.6 %            54.3 %               65.6 %                  79.7 %
6       26.4 %            60.9 %               72.2 %                  85.2 %
7       30.1 %            66.5 %               77.6 %                  89.2 %
8       33.6 %            71.4 %               81.9 %                  92.2 %
9       36.9 %            75.5 %               85.4 %                  94.3 %
10      40.1 %            79.1 %               88.2 %                  95.8 %

Note: Dracul's, LW, and Exile proc LT of different levels. Thus, when combined, the chance of not applying LT is ~ 72.7 % (0.95 * 0.9 * 0.85), not 70 % (1 - {0.05 + 0.10 + 0.15}).

One might argue that Last Wish and Exile provide decent odds of applying Life Tap within 5-10 hits (2-4 seconds), and usually they do proc LT within this time. It is my experience that Dracul's suffice with max resistances and MLR. Last Wish isn't worth the trouble for the purpose of Life Tap. 15 % from Exile is great, but on the other hand, the shield lacks both resistances and blocking compared to HoZ and Sanctuary. We would have to make up for the loss of resistances. And a socket in HoZ could provide additional safety or improve our Smite speed (+15 % IAS) in certain cases.


Character Skills and Equipment

Base Stats
  • Strength: 100 (enough for equipment, using this number for calculation purposes)
  • Dexterity: D (enough for max block, 136+ with equipment when using a Phase Blade)
  • Vitality: remaining stat points
Skill Points
  • Mandatory
    20 Smite
    3 prerequisites
    20 Holy Shield
    3 prerequisites
    20 Fanaticism
    1+ Salvation (+ is optional)
  • Optional (recommended)
    10 Resist Fire
    10 Resist Lightning
    3 prerequisites
    1 Vigor
    1 Sacrifice
    1 Zeal
Assuming this distribution, at level 90 with all skill quests completed, we have 8 points left. Options ? Sanctuary, Meditation, and Redemption, or all in Resist Fire (max at level 92). There is no reason whatsoever to spend any points in Resist Cold, as Raven Frost and Thawing Potions already handle any problems we might have against cold-based attacks.

Equipment
  • Guillaume's Face (Winged Helm)
    35 % Chance of CB, +30 % FHR, +15 to Strength
  • Highlord's Wrath (Amulet)
    +1 to All Skills, +20 % IAS, +35 % LR
  • Duress ('ShaelUmThul')
    15 % Chance of CB, 33 % Chance of OW, +10-20 % off-weapon ED, +15 % FR + LR + PR, +45 % CR, +40 % FHR, +150-200 % Enhanced Defense
    Fortitude ('ElSolDolLo')
    +300 % off-weapon ED, All Resistances +25-30, +1-1.5 to Life per Character Level, +5 % to MLR, +200 % Enhanced Defense
  • Herald of Zakarum (Gilded Shield)
    +2 to Paladin Skill Levels, +2 to Combat Skills (Paladin Only), +30 % ICB, All Resistances +50, +20 to Vitality, +20 to Strength
    Smite Damage: 20 to 28 {46 to 46, upgrade: Zakarum Shield}
  • Dwarf Star (Ring), Raven Frost (Ring)
    (15 % Fire Absorb, 12-15 MDR, +40 to Life), CBF, 20-(40) % Cold Absorb, +15-20-(30-40) to Dexterity
    Dwarf + Raven or 2 x Raven
  • Dracul's Grasp (Vampirebone Gloves)
    5 % CtC Level 10 Life Tap on striking, 25 % Chance of OW, +10-15 to Strength
  • Thundergod's Vigor (War Belt)
    +10 % to MLR, +20 Lightning Absorb, +20 to Vitality, +20 to Strength
  • Goblin Toe (Light Plated Boots)
    25 % Chance of CB
    Gore Rider (War Boots)
    15 % Chance of CB, 10 % Chance of OW
    Sandstorm Trek (Scarabshell Boots)
    +20 % FHR, +10-15 to Vitality, +10-15 to Strength
    Waterwalk (Sharkskin Boots)
    +5 % to MFR, +45-65 to Life, +15 to Dexterity
Assuming +15 to Strength on Dracul's in the calculations. Using Sandstorm Trek (+15 to Strength) with Last Wish; the obvious alternative is Waterwalk. As you read along, it should become apparent exactly why I chose the above equipment.

Weapon modifiers and use of sockets are covered in the following section.
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Apr 11 2009 12:01am
Smite Damage

Modifiers that do NOT apply to Smite
  • Weapon's base damage
  • +% Enhanced Damage on the weapon
  • + to Minimum/Maximum Damage *)
  • +% Damage to Demons/Undead
  • % Critical/Deadly Strike
  • % Life/Mana Stolen per Hit
  • Magic, Fire, Lightning, Cold, and Poison Damage
  • Cold and poison length
Ref: Basin Wiki, Paladin Combat Skills, http://wiki.theamazonbasin.com/index.php/Paladin_Combat_Skills#Smite

*) including Adds Y-Z Damage (read: +Y to Minimum Damage, +Z to Maximum Damage) whether on the weapon or off-weapon (War Traveler).

How is Smite Damage calculated ?
  1. Take the shield's Smite Damage
    Add bonus Smite Damage from Holy Shield (HS)
    Add X from Damage +X (Grief, Astreon's Iron Ward, The Redeemer, Stone Crusher)
  2. Add the sum of Enhanced Damage bonuses from the following sources
    +% ED from Strength (in a 1:1 ratio, regardless of weapon used)
    +% ED from skills (Smite, Might, Concentration, Fanaticism)
    +% ED which is off-weapon (e.g. Fortitude)
Smite Damage Formula
  • Physical Damage = [ ( {Shield_min + HS_min , Shield_max + HS_max} + X ) * ( 1 + Strength/100 + (skill +% ED)/100 + (off-weapon +% ED)/100 ) ]
{,}: minimum and maximum values, [ ]: final values are truncated

Smite Speed Modifiers
  • Weapon Speed Modifier (WSM), the base weapon speed
  • Increased Attack Speed (IAS) from equipment
  • Effective IAS (EIAS) from Fanaticism
Ref: Smite FAQ, http://forums.battle.net/thread.html?topicId=10971909421&sid=3000

Which base weapon for Black, Grief, and Last Wish ?
  • Black (club, hamm, mace)
    The fastest of the allowed weapon types with 3-socket potential are Flail, Knout, and Scourge (-10 WSM). We pick the Flail below, as it is usually the easiest to obtain.
  • Grief (swor, axe)
    Phase Blade (-30 WSM) is the fastest base weapon available (and it is even indestructible by default).
  • Last Wish (swor, hamm, axe)
    Phase Blade (-30 WSM), by the same reasoning.
To calculate the Average Damage Taken (the average physical damage inflicted to the Übers after Damage Resist), we summarise the weapon modifiers applying to Smite in the following table,

Code
Weapon            Modifier                                       WSM      7 FPA  1)      6 FPA  2)      Smite ED        HS_min-max

Black             40 % Chance of Crushing Blow                            + 60 %         +129 % /       +375 %          +77-80
(Flail)           +15 % IAS                                      -10                     + 34 %

Grief             Damage +340-400                                         + 24 %         + 65 % /       +375 %          +77-80
(Phase Blade)     +30-40 % IAS                                   -30                     +  7 %

Last Wish         60-70 % Chance of Crushing Blow                         + 24 %         + 65 % /       +375 %          +77-80
(Phase Blade)     Level 17 Might Aura when equipped              -30                     +  7 %
                  [+200 to skill +% ED]

Astreon's IW      33 % Chance of Crushing Blow                            + 60 %         +129 % /       +405-435 %      +(85-93)-(88-96)
(Caduceus)        Damage +40-85                                  -10                     + 34 %
                  +2-4 to Combat Skills (Paladin Only)
                  +10 % IAS

Heaven's Light    33 % Chance of Crushing Blow                            + 86 %         +180 % /       +405-420 %      +(85-89)-(88-92)
(Mighty Scepter)  +2-3 to Paladin Skill Levels                     0                     + 52 %
                  Socketed (1-2)
                  +20 % IAS

Stormlash  3)     33 % Chance of Crushing Blow                            + 60 %         +129 % /       +375 %          +77-80
(Scourge)         20 % CtC Level 18 Tornado on striking          -10                     + 34 %
                  15 % CtC Level 10 Static Field on striking
                  +30 % IAS


1) Minimum IAS for 7-frame Smite (7 FPA) without Fanaticism (while running Salvation against Über Mephisto, we want at least 7 FPA)
2) Minimum IAS for 6-frame Smite (6 FPA) without Fanaticism / with Fanaticism: Level 23 (25-26 for Heaven's Light; still +36 % EIAS)
   Smite IAS breakpoints based on TitanSeal's Attack Speed Calculator
   25 frames = 1 second
3) Level 18 Tornado (unsynergised): 233-253 (ø243) physical damage before Damage Resist (DR)
   Über Mephisto: 20 % DR, 110 % LR (immune to lightning). Static Field will have no effect
   Pandemonium Diablo, Über Baal: 50 % DR, 75 % LR. Static Field is reduced to 25 % effect

Note: even without Fanaticism and even with the lowest possible IAS on Grief (+30 %), our Smite speed reaches the maximum 6 FPA with a Grief PB, Highlord's Wrath (+20 %), and a +15 % IAS Jewel (Helm or HoZ).

Thus, while running Salvation against Über Mephisto, we lose only the skill +% ED from Fanaticism when using Grief.

The other weapons suffer a minor loss of speed, dropping from 6 FPA (4.16 attacks per second) to 7 FPA (3.57 attacks per second), given sufficient IAS. Even a Heaven's Light can easily reach 7 FPA (+86 % IAS) thanks to the 2 sockets. Once Mephisto is no more, we switch to Fanaticism and maintain a 6-frame Smite with either weapon.

Crushing Blow (CB) + Average Damage Taken (ADT)
CB and OW odds, ADT, and Smite speed values are given in the table below. Indicated weapon stats used. Assuming +15 % off-weapon ED on Duress (Grief).

Code
                                               Full equipment

          Chance of Crushing Blow (CB %), Chance of Open Wounds (OW %), and Average Damage Taken (ADT)

                                      Herald of Zakarum (Gilded Shield)

Weapon              Stats      CB %  4)      OW %      Strength      Physical Damage      DR      ADT      FPA  4)

Black                          100           25          170             916-1020         20 %    774       7
(Flail)                                                                 1327-1478         50 %    701       6

Grief               +370        75           58          170            3082-3154         20 %   2494       6
(Phase Blade)                                                           5062-5181         50 %   2560       6

Last Wish           65+ %      100           25          185            1125-1252         20 %    950       7
(Phase Blade)                                                           1536-1710         50 %    811       6

Astreon's IW        +62,3      100           35          170            1692-1801         20 %   1397       7
(Caduceus)                                                              2417-2573         50 %   1247       6

Heaven's Light      +3         100           35          170            1079-1188         20 %    906       7
(Mighty Scepter)                                                        1596-1758         50 %    838       6

Stormlash                      100           35          170             916-1020         20 %    774       7
(Scourge)                                                               1327-1478         50 %    701       6


                                      Herald of Zakarum (Zakarum Shield)

Weapon              Stats      CB %  4)      OW %      Strength      Physical Damage      DR      ADT      FPA  4)

Black                          100           25          170            1162-1190         20 %    940       7
(Flail)                                                                 1683-1724         50 %    851       6

Grief               +370        75           58          170            3253-3273         20 %   2610       6
(Phase Blade)                                                           5344-5376         50 %   2680       6

Last Wish           65+ %      100           25          185            1426-1461         20 %   1154       7
(Phase Blade)                                                           1948-1995         50 %    985       6

Astreon's IW        +62,3      100           35          170            1950-1980         20 %   1572       7
(Caduceus)                                                              2785-2828         50 %   1403       6

Heaven's Light      +3         100           35          170            1336-1366         20 %   1080       7
(Mighty Scepter)                                                        1977-2021         50 %    999       6

Stormlash                      100           35          170            1162-1190         20 %    940       7
(Scourge)                                                               1683-1724         50 %    851       6


4) Required use of sockets for 100 % Chance of Crushing Blow and a 7-frame Smite without Fanaticism,

   Guillaume's Face: +15 % IAS
   Astreon's IW: 'Ber'
   Heaven's Light: 'ShaelBer'
   Stormlash: 'Ber'
   Herald of Zakarum: +15 % IAS (required for Black, Astreon's IW, and Heaven's Light)

   Using Fortitude with either weapon except Grief (Guillaume's Face + Duress + Goblin Toe: 75 % CB)

Mind the low weapon range when using Heaven's Light (rangeadder 0): http://wiki.theamazonbasin.com/index.php/Melee_range , as Smite will miss if the target is out of range.


Crushing Blow

http://classic.battle.net/diablo2exp/items/magic.shtml#sma (scroll down)

A Crushing Blow (CB) is a fixed percentage modification of a unit's current Hit Points (Static Field is basically the lightning version of Crushing Blow). The more HP a unit has left, the greater the loss of HP from the next CB. Since the Übers initially have ~ 655k HP each, we are interested in knowing just how effective CB is.

In order to examine the effect of CB, we discard Physical Damage, Open Wounds, and monster HP regeneration (note that OW eliminates regeneration; see below). Under these conditions,
  • 1st Crushing Blow
    HP_max is reduced by a factor of 1/8 * (1 - DR%/100), i.e. 1/10th for Über Mephisto (DR% = 20) and 1/16th for his two brothers (DR% = 50)
  • 2nd Crushing Blow
    (HP_max - HP lost from the 1st CB) is reduced by 1/10th or 1/16th
  • n'th Crushing Blow
    (HP_max - HP lost from the previous n-1 CBs) is reduced by 1/10th or 1/16th
Thus, Über Mephisto has 9/10th of HP_max after the 1st CB, 9/10th of (HP_max - HP lost from the 1st CB) after the 2nd CB etc. Formally,

Code
Remaining Hit Points after n CBs *)

     HP_1 = 9/10 * HP_max
     HP_2 = 9/10 * HP_1 = (9/10)^2 * HP_max
     HP_3 = 9/10 * HP_2 = (9/10)^3 * HP_max
     .
     .
     .
     HP_n = 9/10 * HP_n-1 = (9/10)^n * HP_max


Cumulative loss of HP from n CBs *)

  HP_lost = HP_max - HP_n

          = HP_max * {1 - (9/10)^n}

*) For Pandemonium Diablo and Über Baal, replace 9/10 by 15/16

The cumulative loss of HP increases asymptotically (for Über Mephisto: 10 %, 19 %, 27 %, ... of HP_max), tending towards HP_max, while the remaining HP decrease exponentially (90 %, 81 %, 73 %, ... of HP_max). In a mere 10 CB hits, Über Mephisto's initial HP are cut to ~ 33 % (his brothers' HP would be nearly halved). A perfect 7 FPA Smite assault (3.57 attacks per second) with 100 % Chance of CB could ideally achieve this within 3 seconds.

The ideal effect of CB is graphed below (HP_max = 655k), along with a graph representing the HP left after n hits with no CB and 2.5k ADT (Grief 0 CB). Note that while CB quickly decimates the initial HP, it struggles to finish off the target. Smite Damage becomes increasingly important towards the end.


A few questions remain: How do we estimate the kill speed when both CB and ADT are involved and 0 < Chance of CB < 100 % ? Even with our Helm, Body Armor, and Boots providing CB, for Grief to be competitive, we cannot reach 100 % Chance of CB with Grief. And how about monster HP regeneration ?

Let's deal with the latter first ...


Open Wounds

http://classic.battle.net/diablo2exp/items/magic.shtml#sma

Open Wounds 'Damage' (negative life regeneration) is largely inferior to Physical Damage and CBs. Even at clvl 90, the OW 'damage' inflicted over 8 seconds is comparable to the ADT with Astreon's in the Smite Damage section. Over a time span of 8 seconds (200 frames; ideally 28 hits at 7 FPA or 33 hits at 6 FPA), Physical Damage and CB together can easily apply several hundred times the damage equivalent of OW.

The reason for OW is the fact that the modifier Prevent Monster Heal (PMH) does not work against the Übers and we want to eliminate HP regeneration. In MonStats.txt, the Übers have twice the regeneration rate of most monsters (according to the DiabloWiki, standard regeneration is 1.2 % HP/s). Based on 655k HP, this amounts to ~ 16k HP/s (~ 630 HP/frame). Massive regeneration which would decrease our kill speed.
  • Open Wounds may be re-applied within the 8 seconds (200 frames)
  • Open Wounds 'Damage' doesn't stack; if re-applied, the timer resets
Check the DiabloWiki for the remaining details. For a more technical approach, read Open Wounds Explained (v1.11b), http://phrozenkeep.planetdiablo.gamespy.com/forum/viewtopic.php?t=44308

Odds of applying Open Wounds
For the estimated kill speed to be fairly accurate, it is imperative that we use at least Dracul's Grasp (25 % Chance of OW) with either weapon. Since we cannot know in advance if a hit applies OW or not, we consider the odds of inflicting OW at least once in n hits. Setting p = 1 - OW%/100, the odds of n consecutive failed OW checks are p^n and so, the chance of at least one OW strike is 1 - p^n, graphed below.


Within a couple of seconds (7-8 hits @ 6-7 FPA), the odds are approximately 90 % with Dracul's Grasp alone and 95-100 % with 35 % Chance of OW or higher.

Based on this, we consider it reasonably safe to ignore regeneration.
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Apr 11 2009 12:02am
Countdown to Hellfire

No matter which of the 6 weapons is wielded, we will absolutely crush the Übers in a matter of seconds. In this sense, neither weapon is "crappy". Rather, some are slightly more effective than others, due to a higher ADT. But it is probably surprising to some of you that Grief wins on an average basis, despite 'just' 75 % Chance of CB. How is that possible ?

Black, Last Wish, Astreon's Iron Ward, Heaven's Light
Crushing Blow is applied before normal damage. Since we have 100 % Chance of CB with either weapon, we proceed as follows (compare the Crushing Blow section),

Code
Remaining Hit Points after n hits *)

     HP_1 = 9/10 * HP_max - ADT
     HP_2 = 9/10 * HP_1 - ADT = (9/10)^2 * HP_max - {1 + 9/10} * ADT
     HP_3 = 9/10 * HP_2 - ADT = (9/10)^3 * HP_max - {1 + 9/10 + (9/10)^2} * ADT
     .
     .
     .
     HP_n = 9/10 * HP_n-1 - ADT = (9/10)^n * HP_max - {1 + 9/10 + (9/10)^2 + ... + (9/10)^(n-1)} * ADT
         
          = (9/10)^n * HP_max - {((9/10)^n - 1) / (9/10 - 1)} * ADT

          = (9/10)^n * HP_max - 10 * {1 - (9/10)^n} * ADT

*) For Pandemonium Diablo and Über Baal, replace 9/10 by 15/16

HP_n is graphed below (HP_max = 655k), using values of ADT from the Smite Damage section. The difference between either weapon is hardly visible until the end.



ÜMeph, Herald of Zakarum (Zakarum Shield) : http://i39.tinypic.com/wsmzba.jpg

PDÜB, Herald of Zakarum (Gilded Shield) : http://i43.tinypic.com/2q8murk.jpg

PDÜB, Herald of Zakarum (Zakarum Shield) : http://i39.tinypic.com/qpfdy1.jpg


Grief, Stormlash
  • Grief: 75 % Chance of CB
  • Stormlash: 20 % CtC Level 18 Tornado on striking, 15 % CtC Level 10 Static Field on striking
Tornado does 243 average physical damage which is cut to 194 (ÜMeph: 20 % DR) or 121 (PDÜB: 50 % DR). Über Mephisto is immune to lightning (110 %). Pandemonium Diablo and Über Baal have 75 % Lightning Resist, meaning SF removes 0.25 * (1 - 0.75) = 6.25 % of their current HP. Furthermore, SF cannot lower HP < 1/2 * HP_max (Hell). How to estimate the effect of these randomly applied modifiers ? With Grief (CB % = 75), we expect to deliver a CB on every 3 of 4 hits on average. With Stormlash, SF on ~ 1 of 7 hits (PDÜB) and Tornado on 1 of 5 hits.

The extremes are graphed below,
  • Grief
    ADT1: CB on the last 3 of 4 hits
    ADT4: CB on the first 3 of 4 hits
  • Stormlash
    T1: Tornado on the first of 5 hits
    T5: Tornado on the last of 5 hits
    SF1T1: SF on the first of 7 hits + Tornado on the first of 5 hits
    SF7T5: SF on the last of 7 hits + Tornado on the last of 5 hits
Using CB % = 100 for Stormlash and ADT values from the Smite Damage section.

For more details on % CtC modifiers, read Interaction of % Chance to Cast Modifiers, http://wiki.theamazonbasin.com/index.php/Diablo2_Technical_Information

We assume that CB and SF are additive (HP lost due to either modifier is based on the same current HP), not multiplicative. Formally, if CB removes X % of HP and SF removes Y % of HP, then
  • Additive: 1 - (X + Y)/100
    (X + Y) % of current HP removed
  • Multiplicative: (1 - X/100) * (1 - Y/100) = 1 - {(X + Y) - X*Y/100}/100
    (X + Y - X*Y/100) % of current HP removed
In our case, X = Y = 6.25, so the difference is hardly noticeable: X*Y/100 = 0.390625. Even on the first hit (HP_max), this amounts to a difference in HP of just 2.56 k, which subsequently decreases with every hit (due to CB). Ultimately, the same number of hits are required to kill PDÜB whether CB and SF are additive or multiplicative.



ÜMeph, Herald of Zakarum (Zakarum Shield) : http://i39.tinypic.com/15rxo9u.jpg

PDÜB, Herald of Zakarum (Gilded Shield) : http://i42.tinypic.com/wteuxk.jpg

PDÜB, Herald of Zakarum (Zakarum Shield) : http://i41.tinypic.com/18eixc.jpg


1 minute and counting ...
Based on the above and in descending order,

Code
                                              Kill speed estimates  5)

                                  Über Mephisto                  Pandemonium Diablo / Über Baal

                            hits      FPA      time  6)            hits      FPA      time  6)

1. Grief                    38-39      6        9.4 s              52-54      6       12.7 s
   (Phase Blade)

2. Astreon's Iron Ward      36-37      7       10.4 s              53-55      6       13.0 s
   (Caduceus)

3. Heaven's Light           40-41      7       11.5 s              58-61      6       14.4 s
   (Mighty Scepter)

4. Stormlash                40-42      7       11.5 s              58-62      6       14.4 s
   (Scourge)

5. Last Wish                39-41      7       11.2 s              59-62      6       14.6 s
   (Phase Blade)

6. Black                    41-43      7       11.8 s              61-64      6       15.1 s
   (Flail)


5) Depending on the time of application of Open Wounds, the ideal kill speed may be lower than indicated.
   The values cannot and should not be regarded as exact; they are ideal estimates on an average basis.
6) A perfect (uninterrupted) Smite assault on the Über Three would in theory last for about 35-45 seconds
   (not counting the time required to move around). Practice may differ.

And our reward ?



Hellfire Torch (Large Charm)
Required Level: 75
25 % Chance to cast Connection Interrupted on striking
+3 to Flaming Skills
+10-20 to Accusation Attributes
Maturity Resistances +10-20
+8 to Lag Radius
Level 30 Desink (10 Charges)

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Apr 11 2009 12:12am
TL; DR

Just kidding, I did read the majority of it (skimming over the heavy math sections), well written as usual, though it's not a guide I'd recommend to the average person looking for an uber smiter guide ^_^

What was very interesting was the rate of uber regeneration, 16k per second is astonishing,
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Apr 11 2009 12:14am
All of your guides should be stickied under a separate category. Good work!

This post was edited by inkanddagger on Apr 11 2009 12:15am
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Apr 11 2009 12:27am
Way too much attention towards such an easy goal imo. Res/Cb/Tap/Juvs > Ubers

This post was edited by young2093 on Apr 11 2009 12:27am
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Apr 11 2009 01:08am
Quote (Faelwen @ Sat, 11 Apr 2009, 08:02)
Level 30 Desink (10 Charges)


10/10

amazing guide! iso sticky. ;)
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Apr 11 2009 01:21am
lol'd at pala image^^
good guide for those people who search such information
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Apr 11 2009 02:04am
nice guide, i love the use of charts and graphs

10/10
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Apr 11 2009 02:31am
good as maths
<3 it all

and love the torch stats
25% chance to cast connection interupted =)
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