Spell Success

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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% success 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