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:
- Spell level
- Spell-related skills, Intelligence
- Penalties from body armour and shields (mitigated by Armour skill / Shields skill)
- Modifiers like Wizardry and Wild Magic
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% success rate, your chance to fail has to be negative. More on that below.
The base failure before any modifiers is as follows:
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.
spellFailure, at this stage, is capped at 1400.
"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: 
spell_skills = [Spellcasting / 2] + [Average(Skill_level) * 2]
spell_skills = Spellcasting * 2.5
With 27 in Spellcasting and 27 in relevant skills, the total value of (6 * spell_skills) is capped at 405.
Higher level spells are more difficult. Each level of spell adds a number to your spell failure rate:
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)
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.
ev penalty = 1/225 * encumbrance^2 * (90 - 2 * armour_skill) / (str + 3) spell penalty = 19 * ev penalty
ev penalty = 2/5 * encumbrance^2 / (5 + str) * (27 - shield_skill) / 27 spell penalty = 19 * ev penalty
At this point, the spell failure is put through a complicated post-processing curve:
(x^3 + 426x^2 + 82670x + 6983254) / 262144 IF spellFailure < 43
This replicates a step-down curve:
This curve has the following points:
|SpellFailure||Raw Fail chance|
This is not the final spell rate; there are more modifiers.
The following modifiers are added after the STEPPED DOWN failure rates. However, they are still raw spell failure chance, and are adjusted by a sigmoid function later. 
Positive (+spell failure):
- Wild Magic: +4% per level
- Disrupted Magic (-Wiz) sacrifice: +4% per level
- Orb of energy: +10%
- Crystal ball of Wucad Mu: +20%
- Vertigo: +7%
- Certain Transformations:
Negative (-spell failure):
- Subdued Magic: -2% per level
- Wizardry: x75% for 1 source, x66.6% for 2, x50% for 3 (applied after all other factors)
Vehumet gives 2 levels of wizardry for destructive spells.
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.
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.
(1d101 + 1d101 + 1d100 - 3)/3 < fail chance
This is equivalent to applying this transformation to the chance of failure:
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||Raw Fail chance||Actual fail chance|
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. For example, a caster could say "a set of fire dragon scales gives a spell penalty equal to 9 levels of a 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. 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.
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".
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, you can divide the SpellFailure increase by 12 in order to get levels of compensation.
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.
- 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 for more).
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 fire dragon scales,
1/71 * 11^2 / 14 = 0.122. That's a ratio of 8.21 Armour skill : 1 in each spell school. If you have a dual-school spell, like Iron Shot, 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:
(1/<str+3> - 1/<str+x+3>) * 19/225 * encumbrance^2 * (90 - 2 * armour)
(1/<str+5> - 1/<str+x+5>) * 38/5 * encumbrance^2 * (27 - Shields)/27
- Intelligence reduces SpellFailure by 2 per point.
For example, 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 you have 11 strength, kite shield, and 9 Shields skill, +2 strength gives
(1/16 - 1/18) * 38/5 * 10^2 * 18/27 = 3.52 SpellFailure.
- 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.