Difference between revisions of "Spell Success"

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Spell success rate is determined by a complex combination of:
 
Spell success rate is determined by a complex combination of:
  
−
* Spell-related skills
+
# Spell level
−
* [[Intelligence]]
+
# Spell-related skills, [[Intelligence]]
−
* Spell level
+
# Penalties from [[body armour]] and [[shields]] (mitigated by [[Armour (skill)|Armour skill]] / [[Shields (skill)|Shields skill]])
−
* Armour and Shield penalty, mitigated by [[Armour (skill)|Armour skill]] and [[Shields (skill)|Shields skill]]
+
# Modifiers like [[Wizardry]] and [[Good_mutations#Wild_Magic|Wild Magic]]
−
* Mutation Penalty ([[Good_mutations#Wild_Magic|Wild Magic]])
 
−
* Piety with [[Vehumet]] for [[Conjurations]] spells
 
−
* Any equipped wizardry items
 
−
 
 
−
== Spell success calculations ==
 
−
 
 
−
It is debatable whether the following formulae are of real use during a game. However, inflection points such when the extra armour EV penalty disappears are fairly easy to discern. Calculating this stuff on the fly is only for savants! However, one can refer example section of this page for Lv 9 magics, which requires tremendous amount of experience to be stabilized.
 
−
 
 
−
=== Base chance ===
 
  
−
The source code talks in terms of chance of ''failure'', not success. Therefore, we want the spellFailure chance to be as low as possible - in fact, to reach a Perfect success rate, your chance to fail has to be '''negative'''. More on that below.
+
==Base Chance==
 +
The source code talks in terms of chance of ''failure'', not success. Therefore, we want the spellFailure chance to be as low as possible - in fact, to reach a 0% failure rate, your chance to fail has to be '''negative'''. More on that below.
  
−
To begin with, there's a 60% chance of failure. From this, two things are subtracted - twice the caster's Intelligence, and a value calculated from the user's relevant skills. This value is similar to (but simpler than) the [[spell power]]. Note that these two factors are the only ones that '''directly''' improve spell success, while other factors described below only serve to ''mitigate'' spellcasting penalties. Therefore, focus on these areas if you want to maximize your spellcasting chances, particularly spell skills.
+
The base failure before ''any'' modifiers is as follows:<ref>{{source ref|0.30.0|spl-cast.cc|406}}</ref>
−
 
 
−
Here's an overview of the basic formula, before stepdown and miscellaneous penalties and enhancers.
 
  
 
   spellFailure = 60
 
   spellFailure = 60
−
               - [6 * spell skills]
+
               - [6 * spell_skills]
 
               - [2 * Intelligence]
 
               - [2 * Intelligence]
 
               + Spell difficulty
 
               + Spell difficulty
−
               + Armour/shield penalty
+
               + Encumbrance penalties
 +
 
 +
* 60 is an arbitrary base chance of failure.
 +
* Increasing intelligence and spell skills (skills respective to the spell ''and'' [[Spellcasting]]) will decrease failure.
 +
* Higher level spells are more difficult, wearing armour/shields make it more difficult.
  
 
=== Spell skills ===
 
=== Spell skills ===
−
 
+
"Spell_skills" takes into account Spellcasting and an average of a spell's schools (e.g. a Conjurations / Fire spell will take the average of the [[Conjurations]] skill and [[Fire Magic]] skill). This is calculated with the following equation:<ref>{{source ref|0.30.0|spl-cast.cc|413}}</ref>
−
Now they use very familiar equation with spellcasting equation. The main difference is, spell skill boost effect of potion of brilliance is not counted here.
+
<ref>{{source ref|0.30.0|spl-cast.cc|377}}</ref>
−
 
+
   spell_skills = [Spellcasting / 2] + [Average(Skill_level) * 2]
−
   S_0 = [Spellcasting / 2] + [Average(SpellSkills) * 2]
 
−
 
 
−
  spell skills = 50 * log_2 (1 + S_0 / 50)
 
−
 
 
−
This value (6 * spell skills) is naturally capped at 370, assuming level 27 on relevant spell skills and spellcasting.
 
  
 
=== Spell difficulty ===
 
=== Spell difficulty ===
−
 
+
Higher level spells are more difficult. Each level of spell adds a number to your spell failure rate:<ref>{{source ref|0.30.0|spl-cast.cc|422}}</ref>
−
A number is added, making spellcasting more difficult, dependent on the spell's level.
 
  
 
   spellDifficulty =  3 (level 1)
 
   spellDifficulty =  3 (level 1)
Line 56: Line 43:
 
                     200 (level 7)  (+50)
 
                     200 (level 7)  (+50)
 
                     260 (level 8)  (+60)
 
                     260 (level 8)  (+60)
−
                     330 (level 9)  (+70)
+
                     340 (level 9)  (+80)
 +
 
 +
=== Encumbrance penalties ===
 +
 
 +
Both Armour and Shield penalties for spells are derived from their [[EV]] penalties. First, calculate the ev penalty, then calculate the spell penalty. Armour and Shields do not influence each other.
  
−
=== Armour and shield penalties (Under Construction) ===
+
'''Armour'''<ref>{{source ref|0.30.0|player.cc|5723}}</ref><ref>{{source ref|0.30.0|player.cc|2131}}</ref>
  
−
   Armour/Shield Penalty = 19 * (armour penalty) + 19 * (shield penalty)
+
   ev penalty = 1/225 * [[encumbrance]]^2 * (90 - 2 * [[Armour (skill)|armour_skill]]) / ([[str]] + 3)
 +
  spell penalty = 19 * ev penalty
  
−
  armour penalty = 0.4 * (Encumbrance Rating)^2
+
'''Shield'''<ref>{{source ref|0.30.0|player.cc|5739}}</ref><ref>{{source ref|0.30.0|player.cc|2134}}</ref>
−
                  * (45 - Armour skill Lv) / 45
 
−
                  / (Str + 3)
 
  
−
   shield penalty = max(0, (EV Penalty) - (Shield Skill Lv) / (Player species factor) )
+
   ev penalty = 2/5 * [[Shields#EV_Penalty|encumbrance]]^2 / (5 + str) * (27 - [[Shields_(skill)|shield_skill]]) / 27
 +
  spell penalty = 19 * ev penalty
  
−
Player species factor is 5 for normal species, 3 for large species, and 7 for small species, respectively. Spriggans have 9, though they only can use bucklers.
+
== Step down ==
 +
At this point, the spell failure is put through a complicated post-processing curve:<ref>{{source ref|0.30.0|spl-cast.cc|463}}</ref>
  
−
=== Step down ===
+
(x^3 + 426x^2 + 82670x + 6983254) / 262144 IF spellFailure < 43
  
−
At this point, the spell failure is put through a step down curve. If it's over 45 (just into the 'Fair' range), it's unaffected, otherwise it needs to be progressively lower to improve the overall result. This is a stepwise curve, and complex to describe. In the table is the listed native chance to reach certain points (i.e., spell success bands), although note that there are a couple of further steps performed after this.
+
This replicates a step-down curve:
 +
[[File:Spell_success_stepdown.png|thumb|center|800px|x-axis = spellFailure, y-axis = Raw Failure Rate]]
  
−
{| class="prettytable" border=1
+
This curve has the following points:
 +
{| class="prettytable" style="text-align: center" border=1
 
|-
 
|-
−
!Description!!Failure rate!!Needed fail chance
+
!SpellFailure!!Raw fail chance
 
|-
 
|-
−
|Fair||45%||45
+
|43 || 43%
 
|-
 
|-
−
|Good||30%||10
+
|25 || 36.6%
 
|-
 
|-
−
|Very Good||20%||−24
+
|0 || 27.7%
 
|-
 
|-
−
|Great||10%||−80
+
|<nowiki>-</nowiki>50 || 15.5%
 
|-
 
|-
−
|Excellent||4%||−140
+
|<nowiki>-</nowiki>72 || 10%
 
|-
 
|-
−
|Perfect||0%||−180
+
|<nowiki>-</nowiki>100 || 8.5%
 +
|-
 +
|<nowiki>-</nowiki>173 || 0%
 
|-
 
|-
 
|}
 
|}
  
 +
==Modifiers==
 +
The following modifiers are added after the STEPPED DOWN failure rates above. However, it is still considered the ''raw spell failure chance'', which is adjusted by a sigmoid function later.<ref>{{source ref|0.30.0|spl-cast.cc|466}}</ref>
  
−
From −60 to −180 it's linear, with each difference of 20 giving 2% in the final spell success chance. So penalties affect bad wizards more than good ones. Penalty mitigation (for armour) applies before this, so the stepping function doesn't change that.
+
'''Positive (+spell failure):'''
−
 
+
*[[Wild Magic (mutation)|Wild Magic]]: +4% per level
−
=== Mutations ===
+
*Disrupted Magic (-Wiz) [[Ru|sacrifice]]: +4% per level
−
 
+
*[[Orb]] of [[energy]]: +10%
−
From now on, we work with STEPPED DOWN failure rates.
+
**[[Crystal ball of Wucad Mu]]: +10%
−
 
+
*[[Vertigo]]: +7%
−
  Placid magic: Failure Rate - 2%
 
−
  Wild magic:   Failure Rate + 4%
 
−
  Anti-Wizadry: Failure Rate + 4%
 
−
 
 
−
The unrandart [[Hat of the High Council]] increases failure by 7% rather than 4%.
 
−
 
 
−
=== Other Penalties ===
 
−
 
 
−
The [[Sap Magic]] status has a 50% chance to increase your spell failure rates by 12% each time you cast a spell. This maxes out at a penalty of 36%.
 
−
 
 
−
The [[Vertigo]] status increases failure by 7%.
 
−
 
 
−
=== Wizardry, Vehumet, and other factors ===
 
  
−
Finally (phew!) spellcasting success boosts from items and other sources are applied. Note that these are actually calculated as a reduction in the fail chance, so in the table below, a low number is better.
+
'''Negative (-spell failure):'''
 +
*[[Subdued Magic]]: -2% per level
 +
*[[Wizardry]]: multiplied by ×<code>6 / (7 + sources)</code>%, assuming you have at least one source. (×75% for Wiz x1, ×66.6% for Wiz x2, and so on.)
 +
*[[Vehumet]]: multiplied by ×66.6%.<ref>{{source ref|0.30.0|spl-cast.cc|335}}</ref> (Note: this is not considered a source of wizardry, but it multiplicatively stacks with wizardry.)
  
−
* Vehumet knocks 1/3 off the fail chance for Conjurations for disciples in good standing (piety over 70).
+
The wizardry and Vehumet multipliers are applied after all others.<ref>{{source ref|0.30.0|spl-cast.cc|475}}</ref>
  
−
* Rings of wizardry and staves of wizardry give some assistance, given by their "Wizardry" bonus in that column, but they suffer from decreasing cumulative effects. Other spell enhancers, such as rings of fire/ice or staves of various kinds, do not affect spell success chances, only their power. Wizardry effect is determined using this formula:
+
After all modifiers have applied, you will get a cumulative ''raw spell failure chance''. This is converted into the final success via the function below.
  
−
  Reduced Failure rate = Failure rate * 6 / (7 + # of Wizardry Items)
+
==Final Step==
−
 
+
In short, the game doesn't compare your raw failure chance with a number from 0 to 99. Instead, it compares the raw failure to the sum of three numbers divided by three.<ref>{{source ref|0.30.0|spl-cast.cc|2533}}</ref>
−
* Potion of brilliance: Failure rate / 2  (Directly to max level of reduction)
 
−
 
 
−
Note that there is an hard cap of 50% for the effect of all enhancers. For Vehumet worshippers, one wizardry bonus brings a character to the 50% limit, eliminating the need for multiple Wizardry effects..
 
−
 
 
−
===The final step===
 
−
 
 
−
The number we have obtained isn't the final chance of failure. The game doesn't compare that number with a random number between 0 and 99, but with the sum of three numbers divided by three.
 
 
  (1d101 + 1d101 + 1d100 - 3)/3 < fail chance
 
  (1d101 + 1d101 + 1d100 - 3)/3 < fail chance
  
Line 134: Line 115:
 
[[Image:Spell_success_transformation.png]]
 
[[Image:Spell_success_transformation.png]]
  
−
This sigmoid function makes it more difficult to decrease your failure rate when it is high or low, but it will go down very quickly when it is in the middle. Up to raw failure rate of 33%, one can use this formula:
+
This sigmoid function makes it more difficult to decrease your failure rate when it is high or low, but it will go down very quickly when it is in the middle. For raw failures below 33%, one can use this formula:
  
 
   N                = (raw_failure_rate) * 3
 
   N                = (raw_failure_rate) * 3
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This can be tabulated for usefulness.
 
This can be tabulated for usefulness.
  
−
{| class="prettytable" border=1
+
{| class="prettytable" style="text-align: center" border=1
 
|-
 
|-
−
!Raw Fail chance!!Actual fail chance
+
!SpellFailure Req'd*!!Raw fail chance!!Actual fail chance
 
|-
 
|-
−
|32%||14.9%
+
|13||32%||14.9%
 
|-
 
|-
−
|30%||12.3%
+
|7||30%||12.3%
 
|-
 
|-
−
|28%||10.0%
+
|1||28%||10.0%
 
|-
 
|-
−
|26%||8.1%
+
|<nowiki>-</nowiki>6||26%||8.1%
 
|-
 
|-
−
|24%||6.4%
+
|<nowiki>-</nowiki>12||24%||6.4%
 
|-
 
|-
−
|22%||4.9%
+
|<nowiki>-</nowiki>20||22%||4.9%
 
|-
 
|-
−
|20%||3.7%
+
|<nowiki>-</nowiki>28||20%||3.7%
 
|-
 
|-
−
|18%||2.7%
+
|<nowiki>-</nowiki>37||18%||2.7%
 
|-
 
|-
−
|16%||1.9%
+
|<nowiki>-</nowiki>47||16%||1.9%
 
|-
 
|-
−
|14%||1.3%
+
|<nowiki>-</nowiki>58||14%||1.3%
 
|-
 
|-
−
|12%||0.8%
+
|<nowiki>-</nowiki>71||12%||0.8%
 
|-
 
|-
 
|}
 
|}
 +
<nowiki>*</nowiki>SpellFailure required to reach a given raw fail chance, ''assuming you don't have any modifiers like [[wizardry]]''. E.g. if you have wizardry x1, having 1 SpellFailure = (28% * 0.75) raw failure = 21% raw failure.
 +
 +
== Strategy ==
 +
It is debatable whether these formulae are of any practical use during a game.
 +
 +
It may be nice to know a few things from a strategic point of view.
 +
*High level spells take a massive amount of XP to train, and it can be useful to know how many levels you'd need.
 +
*You can quantize your level of armour [[encumbrance]]. A caster could say "a set of [[fire dragon scales]] gives a spell penalty equal to 6 levels of each spell school". Then, you could determine if wearing FDS is worth it, both short-term ("can I cast a spell in this ''right now''?") and long-term ("can I cast a spell in this ''when I reach Zot''?").
 +
*You can calculate the relative value of a skill. For example, you can see when a level of <Spell School> would be more valuable than a level of Armour skill, or when a point of [[strength]] is worth more than a point of [[intelligence]].
 +
 +
Of course, this isn't necessary. You can derive most (if not all) of this information by just checking your spell failure in game. If you wear a pierce of armour, and determine that your spell failure is unacceptable, then don't wear that armour - no need to calculate it! After you trained your skill a bit, you can wear a desired armour again, and see if it's any better.
 +
 +
==Analysis==
 +
 +
===Skill Req'd to Cast L9 Spells===
 +
Say you wanted to [[Fire Storm]] reliably. For sake of example, we're trying to reach a 10% failure rate. You can look at the table above: to get a 10% actual failure rate, you'd need 28% raw failure. If you have no wizardry, 28% raw failure = 1 SpellFailure.
 +
 +
Using the [[#Base Chance|SpellFailure]] calculations at the top of the page, you can calculate what's needed to get to 1 SpellFailure. For this example, assume you have 30 intelligence, no wizardry, and 0 encumbrance from armour/shields.
 +
 +
60 (base failure)
 +
+ 340 (level 9 spell)
 +
- 6 * [0.5 * spellcasting + 2 * avg_skill] (spell skills)
 +
- 30 * 2 (30 intelligence)
 +
+ 0 (encumbrance)<br><br>= 340 - (3 * spellcasting) - (12 * avg_skill) = SpellFailure
 +
 +
We want a SpellFailure = 1, so <code>340 - (3 * spellcasting) - (12 * avg_skill) = 1</code>. So if you had an Spellcasting skill of 14, you'd need 24.75 in avg_skill - that is, Fire Magic and Conjurations. If you had a Spellcasting skill of 22.7, you'd need 22.7 in Fire Magic and Conjurations.
  
−
==Example: Stabilizing [[Fire Storm]] within 90% success rate==
+
Wizardry reduces your raw failure rate by x75% for 1 stack, x66% for 2 stacks, etc.. So if you had wizardry x1, you'd need a (28 / .75) = 37.3% raw failure before wizardry (which, in turn, equals 29.21 SpellFailure). The same calculations can be done for other amounts of wizardry.
  
−
For pure-magic build, stabilizing Fire Storm before the Vault is strictly required. In this example, required skill level for reaching is calculated. For the failure rate of 10%, raw failure rate of 28% is required. Let's set Int=32, a decent value for spellcasters but not sufficient for end-game annihilator. Assuming no armour/shield penalty. This argument can be applied with any of Lv 9 spells, if one exclude Vehumet effect.
+
So, assuming you have 30 int and 0 encumbrance, the following skills would reach the desired 10% actual failure rate:
 +
*'''No Wizardry''': {14 Spellcasting, 24.75 avg_skill}, {22.7 Spellcasting, 22.7 avg_skill}
 +
*'''Wizardry x1''': {14 Spellcasting, 22.49 avg_skill}, {20.7 Spellcasting, 20.7 avg_skill}
 +
*'''Wizardry x2''' / '''Vehumet''': {14 Spellcasting, 21.5 avg_skill}, {20 Spellcasting, 20 avg_skill}
 +
*'''Vehumet + Wizardry x1''': {14 Spellcasting, 20.2 avg_skill}, {18.9 Spellcasting, 18.9 avg_skill}
  
−
* Without any aid: Raw rate of 2% is stepped down to 28%, so 2% is the point that should be achieved. This is equivalent to 54.0 point of spell skills, requiring '''(school skill*2 + spellcasting*0.5) = 55.7'''. Even if the spellcasting is maxed out to 27, the average school skill value of 21.1 is required to cast Fire storm within 10% failure.  
+
These numbers apply to any level 9 spell in the game, not just Fire Storm... again, ''assuming you have 30 int / 0 encumbrance''. If you're wearing [[swamp dragon scales]] and a [[kite shield]], for instance, it'll take more skill to cast. Change the values of spell level, intelligence, encumbrance as you see fit.
  
−
* With an aid of a wizardry ring: A wizardry option knocks 1/4 of failure rate before sigmoid function, so the required raw fail rate is 38%. Raw rate of 32% is stepped down to 38%, and '''(school skill*2 + spellcasting*0.5) = 48.6'''. This value is quite easy to be achieved now, and average of 19.4 Lv including spellcasting is sufficient for 10% failure rate.
+
===Measuring Encumbrance===
 +
The effect of encumbrance on your actual % to fail is non-linear. However, it can be measured by "levels of a spell school needed to compensate".
  
−
* Vehumet worshipper, or two wizardry rings: These enhancements knock off 1/3 of failure rate before sigmoid. Now the raw fail rate required before stepdown is 38%, and '''(school skill*2 + spellcasting*0.5) = 47.3'''. Required average skill Lvs, including spellcasting, is 18.9. Worshipping Vehumet really offers tremendous benefits for high-level spells.
+
'''Armour'''
  
−
* Vehumet with a wizardry ring: Maxed out enhancer effect. Raw fail rate of 56% can be used without stepdown, and '''(school skill*2 + spellcasting*0.5) = 43.3'''. Required average skill Lvs, including spellcasting, is 17.3. If one regulates spellcasting up to 15, then 17.9 Lv of each school level is sufficient!
+
Body armour increases SpellFailure by <code>19/225 * encumbrance^2 * (90 - 2 * Armour) / (str + 3)</code>. Having 1 level of every spell school (except Spellcasting) reduces SpellFailure by 12. Therefore, armour encumbrance can be measured in terms of skill levels, using the formula [SpellFailure Increase] / 12.
  
−
* [[Potion of brilliance]]: Maxed out enhancer effect with increased intelligence. Raw fail rate of 66% can be used without stepdown. '''(school skill*2 + spellcasting*0.5) = 41.1''' is required, and this reduction is huge if the caster is not aided by any of enhancers!
+
As an example: say you have 6 strength, are wearing [[troll leather armour]] (encumbrance = 4), and have 0 Armour skill. Your SpellFailure would be <code>19/225 * 4^2 * (90 - 0) / (6 + 3)</code> = 13.51 SpellFailure, or 1.13 levels in each skill. That really isn't much in the grand scheme of things. By the time you get troll leather armour, you can negate a ~1 level penalty.
 +
 
 +
For reference:
 +
*10 strength, [[leather armour]], 0 Armour skill: -0.78 levels in each skill
 +
*10 strength, [[ring mail]] or [[swamp dragon scales]], 5 Armour skill:  -2.12 levels in each skill
 +
*10 strength, [[fire dragon scales]], 5 Armour skill: -5.24 levels in each skill
 +
 
 +
Remember that things change based on strength and skill. Also remember that you can train Armour skill to lower the penalty (see [[#Armour skill vs Spell skill|Armour skill vs Spell skill]] for more).
 +
 
 +
'''Shields'''
 +
 
 +
Shield increases SpellFailure by <code>38/5 * encumbrance^2 / (str + 5) * (27 - Shields)/27</code>. Using the same method as body armour, as seen above, you can get a number for shield encumbrance, too. Relative to armour, strength matters a little less, but Shields skill matters more.
 +
 
 +
If you have 6 strength, a [[buckler]] (encumbrance = 5), and have 0 Shields skill, SpellFailure equals <code>38/5 * 5^2 / (6 + 5) * 27/27</code> = 17.27, or 1.44 levels in each skill. A [[kite shield]] would be 4x that, a [[tower shield]] would be 9x that.
 +
 
 +
===Armour skill vs Spell skill===
 +
*Armour skill reduces SpellFailure by <code>-Armour * 38/225 * encumbrance^2 / (str + 3)</code>.
 +
*Having 1 of every spell school (except Spellcasting) reduces SpellFailure by 12. Divide by number of spell schools if needed.
 +
*Therefore, each level in Armour is worth <code>19/1350 * encumbrance^2 / (str + 3)</code>, or roughly <code>1/71 * encumbrance^2 / (str + 3)</code>, levels in each spell school.
 +
 
 +
For example, if you have 11 strength and are wearing [[fire dragon scales]], the equation <code>1/71 * 11^2 / 14</code> = 0.122, or 8.21 Armour : 1 in each spell school. If you have a dual-school spell, like [[Fireball]], 4.1 Armour skill = 1 Earth Magic OR 1 Conjurations, while 8.21 Armour = 1 Earth AND Conjurations.
 +
 
 +
Of course, this '''ONLY''' takes into account spell failure. Training Armour skill increases your defenses; training spell schools increases your power.
 +
 
 +
===Strength vs Intelligence===
 +
*For a given change in strength 'x', strength reduces both armour and shield encumbrance by:
 +
**Armour: <code>(1/<str+3> - 1/<str+x+3>) * 19/225 * encumbrance^2 * (90 - 2 * Armour)</code>
 +
**Shield: <code>(1/<str+5> - 1/<str+x+5>) * 38/5 * encumbrance^2 * (27 - Shields)/27</code>
 +
*Intelligence reduces SpellFailure by 2 per point.
 +
 
 +
Say you have 11 strength, fire dragon scales, and 0 Armour skill. On level up, you have a choice of 2 str or 2 int. The impact of +2 str is equal to <code>(1/14 - 1/16) * 19/225 * 11^2 * (90 - 0)</code> = 8.21 SpellFailure, greater than 2*2 = 4 SpellFailure from intelligence.
 +
 
 +
If this 11 strength character is also wearing a [[kite shield]], with 9 Shields skill, +2 strength would also give <code>(1/16 - 1/18) * 38/5 * 10^2 * 18/27</code> = 3.52 SpellFailure.
 +
 
 +
Of course, intelligence also increases [[spell power]], while strength reduces [[EV]] penalties.
  
 
==See also==
 
==See also==
 +
*[[Spell power]]
 +
*[[Miscast effect]]
 +
 +
==History==
 +
*Prior to [[0.31]], the raw failure rate multiplier from wizardry/Vehumet was capped at 50%. Therefore, you could only have a maximum of 5 stacks of wizardry, or 1 stack of wizardry with Vehumet's bonus. Also, some [[transmutations]] increased your spell failure: [[Spider Form]] (+10%) and [[Blade Hands (spell)|Blade Hands]] (+20%).
 +
*Prior to [[0.22]], spell_schools went through a stepdown, <code>50 * log2(1 + spell_schools/50)</code>, which penalized the caster if <code>(Spellcasting/2) + (2* AvgSpellSchool) > 50</code>. However, spellFailure for level 9 spells was 330 instead of 340.
 +
*Prior to [[0.20]], the [[#Step down|Step down]] function was different, relying on breakpoints rather than a polynomial. Also, more breakpoints existed elsewhere in the spell failure calculation.
  
−
* [[Spell power]]
+
==References==
−
* [[Miscast effect]]
+
<references/>
  
 
[[Category:Magic]]
 
[[Category:Magic]]

Latest revision as of 06:01, 27 March 2024

Version 0.31: This article is up to date for the latest stable release of Dungeon Crawl Stone Soup.

Spell success is the rate at which casting a spell can be expected to succeed. Spell success rate is determined by a complex combination of:

  1. Spell level
  2. Spell-related skills, Intelligence
  3. Penalties from body armour and shields (mitigated by Armour skill / Shields skill)
  4. Modifiers like Wizardry and Wild Magic

Base Chance

The source code talks in terms of chance of failure, not success. Therefore, we want the spellFailure chance to be as low as possible - in fact, to reach a 0% failure rate, your chance to fail has to be negative. More on that below.

The base failure before any modifiers is as follows:[1] 

 spellFailure = 60
              - [6 * spell_skills]
              - [2 * Intelligence]
              + Spell difficulty
              + Encumbrance penalties
  • 60 is an arbitrary base chance of failure.
  • Increasing intelligence and spell skills (skills respective to the spell and Spellcasting) will decrease failure.
  • Higher level spells are more difficult, wearing armour/shields make it more difficult.

Spell skills

"Spell_skills" takes into account Spellcasting and an average of a spell's schools (e.g. a Conjurations / Fire spell will take the average of the Conjurations skill and Fire Magic skill). This is calculated with the following equation:[2] [3] 

 spell_skills = [Spellcasting / 2] + [Average(Skill_level) * 2]

Spell difficulty

Higher level spells are more difficult. Each level of spell adds a number to your spell failure rate:[4] 

 spellDifficulty =   3 (level 1)
                    15 (level 2)   (+12)
                    35 (level 3)   (+20)
                    70 (level 4)   (+35)
                   100 (level 5)   (+30)
                   150 (level 6)   (+50)
                   200 (level 7)   (+50)
                   260 (level 8)   (+60)
                   340 (level 9)   (+80)

Encumbrance penalties

Both Armour and Shield penalties for spells are derived from their EV penalties. First, calculate the ev penalty, then calculate the spell penalty. Armour and Shields do not influence each other.

Armour[5][6] 

 ev penalty = 1/225 * encumbrance^2 * (90 - 2 * armour_skill) / (str + 3)
 spell penalty = 19 * ev penalty

Shield[7][8] 

 ev penalty = 2/5 * encumbrance^2 / (5 + str) * (27 - shield_skill) / 27
 spell penalty = 19 * ev penalty

Step down

At this point, the spell failure is put through a complicated post-processing curve:[9] 

(x^3 + 426x^2 + 82670x + 6983254) / 262144 IF spellFailure < 43

This replicates a step-down curve:

x-axis = spellFailure, y-axis = Raw Failure Rate

This curve has the following points:

SpellFailure Raw fail chance
43 43%
25 36.6%
0 27.7%
-50 15.5%
-72 10%
-100 8.5%
-173 0%

Modifiers

The following modifiers are added after the STEPPED DOWN failure rates above. However, it is still considered the raw spell failure chance, which is adjusted by a sigmoid function later.[10]

Positive (+spell failure):

Negative (-spell failure):

  • Subdued Magic: -2% per level
  • Wizardry: multiplied by ×6 / (7 + sources)%, assuming you have at least one source. (×75% for Wiz x1, ×66.6% for Wiz x2, and so on.)
  • Vehumet: multiplied by ×66.6%.[11] (Note: this is not considered a source of wizardry, but it multiplicatively stacks with wizardry.)

The wizardry and Vehumet multipliers are applied after all others.[12]

After all modifiers have applied, you will get a cumulative raw spell failure chance. This is converted into the final success via the function below.

Final Step

In short, the game doesn't compare your raw failure chance with a number from 0 to 99. Instead, it compares the raw failure to the sum of three numbers divided by three.[13] 

(1d101 + 1d101 + 1d100 - 3)/3 < fail chance

This is equivalent to applying this transformation to the chance of failure:

Spell success transformation.png

This sigmoid function makes it more difficult to decrease your failure rate when it is high or low, but it will go down very quickly when it is in the middle. For raw failures below 33%, one can use this formula: 

 N                 = (raw_failure_rate) * 3
 Real Failure Rate = N * (N+1) * (N+2) / 6 / 101 / 101 (in unit of %)

This can be tabulated for usefulness.

SpellFailure Req'd* Raw fail chance Actual fail chance
13 32% 14.9%
7 30% 12.3%
1 28% 10.0%
-6 26% 8.1%
-12 24% 6.4%
-20 22% 4.9%
-28 20% 3.7%
-37 18% 2.7%
-47 16% 1.9%
-58 14% 1.3%
-71 12% 0.8%

*SpellFailure required to reach a given raw fail chance, assuming you don't have any modifiers like wizardry. E.g. if you have wizardry x1, having 1 SpellFailure = (28% * 0.75) raw failure = 21% raw failure.

Strategy

It is debatable whether these formulae are of any practical use during a game.

It may be nice to know a few things from a strategic point of view.

  • High level spells take a massive amount of XP to train, and it can be useful to know how many levels you'd need.
  • You can quantize your level of armour encumbrance. A caster could say "a set of fire dragon scales gives a spell penalty equal to 6 levels of each spell school". Then, you could determine if wearing FDS is worth it, both short-term ("can I cast a spell in this right now?") and long-term ("can I cast a spell in this when I reach Zot?").
  • You can calculate the relative value of a skill. For example, you can see when a level of <Spell School> would be more valuable than a level of Armour skill, or when a point of strength is worth more than a point of intelligence.

Of course, this isn't necessary. You can derive most (if not all) of this information by just checking your spell failure in game. If you wear a pierce of armour, and determine that your spell failure is unacceptable, then don't wear that armour - no need to calculate it! After you trained your skill a bit, you can wear a desired armour again, and see if it's any better.

Analysis

Skill Req'd to Cast L9 Spells

Say you wanted to Fire Storm reliably. For sake of example, we're trying to reach a 10% failure rate. You can look at the table above: to get a 10% actual failure rate, you'd need 28% raw failure. If you have no wizardry, 28% raw failure = 1 SpellFailure.

Using the SpellFailure calculations at the top of the page, you can calculate what's needed to get to 1 SpellFailure. For this example, assume you have 30 intelligence, no wizardry, and 0 encumbrance from armour/shields. 

60 (base failure)
+ 340 (level 9 spell)
- 6 * [0.5 * spellcasting + 2 * avg_skill] (spell skills)
- 30 * 2 (30 intelligence)
+ 0 (encumbrance)

= 340 - (3 * spellcasting) - (12 * avg_skill) = SpellFailure

We want a SpellFailure = 1, so 340 - (3 * spellcasting) - (12 * avg_skill) = 1. So if you had an Spellcasting skill of 14, you'd need 24.75 in avg_skill - that is, Fire Magic and Conjurations. If you had a Spellcasting skill of 22.7, you'd need 22.7 in Fire Magic and Conjurations.

Wizardry reduces your raw failure rate by x75% for 1 stack, x66% for 2 stacks, etc.. So if you had wizardry x1, you'd need a (28 / .75) = 37.3% raw failure before wizardry (which, in turn, equals 29.21 SpellFailure). The same calculations can be done for other amounts of wizardry.

So, assuming you have 30 int and 0 encumbrance, the following skills would reach the desired 10% actual failure rate:

  • No Wizardry: {14 Spellcasting, 24.75 avg_skill}, {22.7 Spellcasting, 22.7 avg_skill}
  • Wizardry x1: {14 Spellcasting, 22.49 avg_skill}, {20.7 Spellcasting, 20.7 avg_skill}
  • Wizardry x2 / Vehumet: {14 Spellcasting, 21.5 avg_skill}, {20 Spellcasting, 20 avg_skill}
  • Vehumet + Wizardry x1: {14 Spellcasting, 20.2 avg_skill}, {18.9 Spellcasting, 18.9 avg_skill}

These numbers apply to any level 9 spell in the game, not just Fire Storm... again, assuming you have 30 int / 0 encumbrance. If you're wearing swamp dragon scales and a kite shield, for instance, it'll take more skill to cast. Change the values of spell level, intelligence, encumbrance as you see fit.

Measuring Encumbrance

The effect of encumbrance on your actual % to fail is non-linear. However, it can be measured by "levels of a spell school needed to compensate".

Armour

Body armour increases SpellFailure by 19/225 * encumbrance^2 * (90 - 2 * Armour) / (str + 3). Having 1 level of every spell school (except Spellcasting) reduces SpellFailure by 12. Therefore, armour encumbrance can be measured in terms of skill levels, using the formula [SpellFailure Increase] / 12.

As an example: say you have 6 strength, are wearing troll leather armour (encumbrance = 4), and have 0 Armour skill. Your SpellFailure would be 19/225 * 4^2 * (90 - 0) / (6 + 3) = 13.51 SpellFailure, or 1.13 levels in each skill. That really isn't much in the grand scheme of things. By the time you get troll leather armour, you can negate a ~1 level penalty.

For reference:

Remember that things change based on strength and skill. Also remember that you can train Armour skill to lower the penalty (see Armour skill vs Spell skill for more).

Shields

Shield increases SpellFailure by 38/5 * encumbrance^2 / (str + 5) * (27 - Shields)/27. Using the same method as body armour, as seen above, you can get a number for shield encumbrance, too. Relative to armour, strength matters a little less, but Shields skill matters more.

If you have 6 strength, a buckler (encumbrance = 5), and have 0 Shields skill, SpellFailure equals 38/5 * 5^2 / (6 + 5) * 27/27 = 17.27, or 1.44 levels in each skill. A kite shield would be 4x that, a tower shield would be 9x that.

Armour skill vs Spell skill

  • Armour skill reduces SpellFailure by -Armour * 38/225 * encumbrance^2 / (str + 3).
  • Having 1 of every spell school (except Spellcasting) reduces SpellFailure by 12. Divide by number of spell schools if needed.
  • Therefore, each level in Armour is worth 19/1350 * encumbrance^2 / (str + 3), or roughly 1/71 * encumbrance^2 / (str + 3), levels in each spell school.

For example, if you have 11 strength and are wearing fire dragon scales, the equation 1/71 * 11^2 / 14 = 0.122, or 8.21 Armour : 1 in each spell school. If you have a dual-school spell, like Fireball, 4.1 Armour skill = 1 Earth Magic OR 1 Conjurations, while 8.21 Armour = 1 Earth AND Conjurations.

Of course, this ONLY takes into account spell failure. Training Armour skill increases your defenses; training spell schools increases your power.

Strength vs Intelligence

  • For a given change in strength 'x', strength reduces both armour and shield encumbrance by:
    • Armour: (1/<str+3> - 1/<str+x+3>) * 19/225 * encumbrance^2 * (90 - 2 * Armour)
    • Shield: (1/<str+5> - 1/<str+x+5>) * 38/5 * encumbrance^2 * (27 - Shields)/27
  • Intelligence reduces SpellFailure by 2 per point.

Say you have 11 strength, fire dragon scales, and 0 Armour skill. On level up, you have a choice of 2 str or 2 int. The impact of +2 str is equal to (1/14 - 1/16) * 19/225 * 11^2 * (90 - 0) = 8.21 SpellFailure, greater than 2*2 = 4 SpellFailure from intelligence.

If this 11 strength character is also wearing a kite shield, with 9 Shields skill, +2 strength would also give (1/16 - 1/18) * 38/5 * 10^2 * 18/27 = 3.52 SpellFailure.

Of course, intelligence also increases spell power, while strength reduces EV penalties.

See also

History

  • Prior to 0.31, the raw failure rate multiplier from wizardry/Vehumet was capped at 50%. Therefore, you could only have a maximum of 5 stacks of wizardry, or 1 stack of wizardry with Vehumet's bonus. Also, some transmutations increased your spell failure: Spider Form (+10%) and Blade Hands (+20%).
  • Prior to 0.22, spell_schools went through a stepdown, 50 * log2(1 + spell_schools/50), which penalized the caster if (Spellcasting/2) + (2* AvgSpellSchool) > 50. However, spellFailure for level 9 spells was 330 instead of 340.
  • Prior to 0.20, the Step down function was different, relying on breakpoints rather than a polynomial. Also, more breakpoints existed elsewhere in the spell failure calculation.

References

  1. ↑ spl-cast.cc:406 (0.30.0)
  2. ↑ spl-cast.cc:413 (0.30.0)
  3. ↑ spl-cast.cc:377 (0.30.0)
  4. ↑ spl-cast.cc:422 (0.30.0)
  5. ↑ player.cc:5723 (0.30.0)
  6. ↑ player.cc:2131 (0.30.0)
  7. ↑ player.cc:5739 (0.30.0)
  8. ↑ player.cc:2134 (0.30.0)
  9. ↑ spl-cast.cc:463 (0.30.0)
  10. ↑ spl-cast.cc:466 (0.30.0)
  11. ↑ spl-cast.cc:335 (0.30.0)
  12. ↑ spl-cast.cc:475 (0.30.0)
  13. ↑ spl-cast.cc:2533 (0.30.0)
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