User:Trang Oul/Utils

Calculating skill params formulae and values per level

This tutorial's goal is to obtain two results:

• a formula for a given skill's property (damage, range, duration and so on)
• values for that property at different levels (1-60, as on skill pages).

1. Obtaining data

First we need to find the formulae. They are stored in data files, in skills.txt and in skillcalc.txt. They are ordinary TSV files, and while there are specialized tools for modding Diablo 2, plain old MS Excel suffices, with features like freezing headers, filtering or searching.

This part is based on Phrozen Keep. File: skills.txt

Columns

Formulae

Formulae are stored in the following columns (this list may not be exhaustive):

• calcX - general skill formulae - see calcX desc for description
• prgcalcX - Assassin finishers; also Shock Web, Blade Fury
• auralencalc - length of aura/curse/buff/debuff, in frames
• aurarangecalc - range of aura/curse/buff/debuff; also used by other skills
• aurastatcalcX - stats of aura/curse/buff/debuff - see aurastatX for stat name
• passivecalcX - passive skills - see passivestatX for stat name
• petmax - max number of pets
• DmgSymPerCalc - synergy to physical damage
• EDmgSymPerCalc - synergy to elemental damage
• ELenSymPerCalc - synergy to the duration of elemental damage
• cltcalcX - client-side calculations - not used for our purpose

Params

• ParamX - numeric param used in formula - see ParamX Description for description
• other columns (including, but not limited toELen..., EMin..., EMax...) - referenced by formulae below.

Multipliers

• hitshift - used to multiply the damage by a power of 2 (2^(hitshift-8); 8 = ×1). See Phrozen Keep guide for details.
• SrcDamage - used as weapon damage multiplier (for skills such as Multiple Shot), in 1/128 (128 = ×1, 96 = ×3/4 and so on).

Formula details

• hardcoded numbers
• basic math operators and functions: arithmetic, min, max
• conditional operator: condition ? value_if_true : value_if_false
• functions and params as defined in skillcalc.txt:

• lnXY, dmXY - linear and diminishing (rational) functions. We will be the most interested in these formulae.

All of them use skill level and additional params.

Currently all those functions are the same per category.

• parX - parameter (column ParamX)
• lvl - soft skill level (i.e. with bonuses)
• blvl - base skill level (i.e. only hard points); used mostly for synergies
• ulvl - character level; used by Druid summons only
• references to other column or columns - used mostly for progression based on level ranges

Examples:

• edns - "Elemental Damage Min (256ths)"

Used by Prayer to calculate life regen.

As per Phrozen Keep:

EMin: and EMax: Controls the basic minimum and maximum elemental damage dealt by this skill at sLvl 1.

EMinLev1: to EMinLev5: and EMaxLevDam1: to EMaxLevDam5: Controls the additional elemental damage added to the skills minimum and maximum damage for every additional sLvl.

LevDam1 is used for sLvls 2-8, LevDam2 for 9-16, LevDam3 for 17-22, LevDam4 for 23-28 and LevDam5 for sLvls above 28.

So, Prayer starts with 2 (col EMin) life points at level 1, increasing by 1 per level at levels 2-8 (EMinLev1), again by 1 at levels 9-16 (EMinLev2) and so on, ending by +3 per each level above 28.

• edmn - "Elemental Damage"

Used for Energy Shield's' Damage % Absorption (min(edmn,95))

It starts at 20% (EMin), increased by 5 percentage points per skill (EMinLev1), then by 2 p.p. (EMinLev2), then by 1 p.p. (EMinLev3, EMinLev4, EMinLev5), capped at 95%.

Notice that although these functions refer to "Elemental Damage" columns, for those skills these columns just hold the values, used for other properties than damage.

• edmx - "Elemental Damage Max"

Used by Dire Wolf's max cold damage (in D2R); cols EMax....

• skill('Skill Name'.fun) - reference to another skill - used for synergies:

  Examples:

• skill('Golem Mastery'.ln12) - ln12 formula calculated using Golem Mastery params.
• skill('Summon Fenris'.par2) - Param2 of Summon Fenris (i.e. Summon Dire Wolf).
• skill('Summon Fenris'.lvl) - (soft) skill level of Summon Fenris.

• stat('stat_name'.fun) - used mostly to make pets inherit character stats

  Examples:

• stat('item_pierce_ELE_immunity'.accr) - Sunder
• stat('passive_ELE_pierce'.accr) - Enemy Resistance
• stat('passive_ELE_mastery'.accr) - Skill Damage

2. Calculation

Examples:

Skill Name	Column description if present	Formula	Params (besides lvl)	Simplified Formula	Values (sample)
Jab	calc1 : Damage %	ln34	Param3=-15, Param4=3	3*lvl - 18	Level Value 1 -15 2 -12 3 -9
Battle Cry	auralencalc	ln12	Param1=300, Param2=75 Since this is duration, we convert it to seconds by dividing by 25 and allowing fractional values.	12*lvl/5 + 48/5 or 2.4*lvl + 9.6	Level Value 1 12.0 2 14.4 3 16.8
Clay Golem	aurastatcalc1 : item_slow	dm34	Param3=0, Param4=75	165*lvl/(2*lvl + 12)	Level Value 1 11 2 20 3 27
Blaze	auralencalc	dm12	Param3=50, Param4=500 Since this is duration, we convert it to seconds by dividing by 25 and allowing fractional values.	(109*lvl + 60)/(5*lvl + 30)	Level Value 1 4.8 2 6.92 3 8.6
Chain Lightning	calc1 : # of target jump hits	ln34/par5	Param3=26, Param4=1, Param5=5	lvl/5 + 5	Level Value 1 5 2 5 5 6 25 10
Strafe	aurastatcalc1 : Minimum # of Missiles created	min(par3 + lvl - 1, par4)	Param3=0, Param4=75	Min(10, lvl + 3)	Level Value 1 4 2 5 7 10
Raise Skeleton	petmax	(lvl < 4) ?lvl:(2+lvl/3)	(none)	Piecewise((lvl, lvl < 4), (lvl/3 + 2, True))	Level Value 1 1 2 2 3 3 4 3
Prayer	aurastatcalc1 : hitpoints	edns	EMin...: 2, 1, 1, 2, 2, 3	Piecewise((lvl + 1, (lvl <= 8) | (lvl <= 16)), (2*lvl - 15, (lvl <= 22) | (lvl <= 28)), (3*lvl - 43, True))	Level Value 1 2 2 3 3 4 60 137
Energy Shield	aurastatcalc1 : Damage % Absorption	min(edmn,95)	EMin...: 20, 5, 2, 1, 1, 1	Min(95, Piecewise((5*lvl + 15, lvl <= 8), (2*lvl + 39, lvl <= 16), (lvl + 55, True)))	Level Value 1 20 2 25 3 30 10 59 40 95

The following Python script can be used to simplify the formula and calculate values per level.

#coding: utf-8
import math
from fractions import Fraction

import sympy

FRAMES_PER_SECOND:int = 25


lvl, par1, par2, par3, par4, par5, par6, par7, par8, base, lev1, lev2, lev3, lev4, lev5 = sympy.symbols('lvl, par1, par2, par3, par4, par5, par6, par7, par8, base, lev1, lev2, lev3, lev4, lev5', integer=True)



# the formulae are the same per category, but for ease of use they are defined with names from skillcalc.txt
ln12: sympy.core.expr.Expr = par1 + (lvl - 1) * par2
ln34: sympy.core.expr.Expr = par3 + (lvl - 1) * par4
ln56: sympy.core.expr.Expr = par5 + (lvl - 1) * par6
ln78: sympy.core.expr.Expr = par7 + (lvl - 1) * par8

# dm12: sympy.core.expr.Expr = sympy.floor(((110 * lvl) * (par2 - par1)) / (100 * (lvl + 6))) + par1 #floor because of the integer division
# dm34: sympy.core.expr.Expr = sympy.floor(((110 * lvl) * (par4 - par3)) / (100 * (lvl + 6))) + par3
# dm56: sympy.core.expr.Expr = sympy.floor(((110 * lvl) * (par6 - par5)) / (100 * (lvl + 6))) + par5
# dm78: sympy.core.expr.Expr = sympy.floor(((110 * lvl) * (par8 - par7)) / (100 * (lvl + 6))) + par7

# A formula different than in skillcalc.txt, but the results match those in the game (credit: Onderduiker).
dm12 = sympy.Min(par1 + sympy.floor((par2 - par1) * sympy.floor((110 * lvl) / (lvl + 6)) / 100), par2)
dm34 = sympy.Min(par3 + sympy.floor((par4 - par3) * sympy.floor((110 * lvl) / (lvl + 6)) / 100), par4)
dm56 = sympy.Min(par5 + sympy.floor((par6 - par5) * sympy.floor((110 * lvl) / (lvl + 6)) / 100), par6)
dm78 = sympy.Min(par7 + sympy.floor((par8 - par7) * sympy.floor((110 * lvl) / (lvl + 6)) / 100), par8)


levelwise3: sympy.core.expr.Expr = sympy.Piecewise(
		(base,                                                                                         lvl == 1),
		(base + lev1 * (lvl-1),                                                                        lvl <= 8),
		(base + lev1 * 7       + lev2 * (lvl-8),                                                       lvl <= 16),
		(base + lev1 * 7       + lev2 * 8       + lev3 * (lvl-16),                                     True)
	)

levelwise5: sympy.core.expr.Expr= sympy.Piecewise(
		(base,                                                                                         lvl == 1),
		(base + lev1 * (lvl-1),                                                                        lvl <= 8),
		(base + lev1 * 7       + lev2 * (lvl-8),                                                       lvl <= 16),
		(base + lev1 * 7       + lev2 * 8       + lev3 * (lvl-16),                                     lvl <= 22),
		(base + lev1 * 7       + lev2 * 8       + lev3 * 6        + lev4 * (lvl-22),                   lvl <= 28),
		(base + lev1 * 7       + lev2 * 8       + lev3 * 6        + lev4 * 6        + lev5 * (lvl-28), True),
	)

def make_formula(formula:str) -> sympy.core.expr.Expr:
	FORMULA_PARTS: dict[str, sympy.core.expr.Expr] = {
		'lvl' : lvl,
		'par1' : par1,
		'par2' : par2,
		'par3' : par3,
		'par4' : par4,
		'par5' : par5,
		'par6' : par6,
		'par7' : par7,
		'par8' : par8,
		'base' : base,
		'lev1' : lev1,
		'lev2' : lev2,
		'lev3' : lev3,
		'lev4' : lev4,
		'lev5' : lev5,
		'ln12' : ln12,
		'ln34' : ln34,
		'ln56' : ln56,
		'ln78' : ln78,
		'dm12' : dm12,
		'dm34' : dm34,
		'dm56' : dm56,
		'dm78' : dm78,
		'levelwise3': levelwise3,
		'levelwise5': levelwise5,
	}
	return sympy.sympify(formula, FORMULA_PARTS)

def levelwise3_params(values:str) -> dict[sympy.core.symbol.Symbol, int]:
	"""
	Builds parameters for levelwise formula from the values directly copied from data file.
	4 values are expected: base, lev1, lev2, lev3, separated with a tab.
	"""
	values = tuple(int(val) for val in values.split('\t'))
	return {base: values[0], lev1: values[1], lev2: values[2], lev3: values[3]}

def levelwise5_params(values:str) -> dict[sympy.core.symbol.Symbol, int]:
	"""
	Builds parameters for levelwise formula from the values directly copied from data file.
	6 values are expected: base, lev1, lev2, lev3, lev5, lev5, separated with a tab.
	"""
	values = tuple(int(val) for val in values.split('\t'))
	return {base: values[0], lev1: values[1], lev2: values[2], lev3: values[3], lev4: values[4], lev5: values[5]}

class Stat:
	def __init__(self, desc:str, formula, params:dict|None=None, hitshift:int=8, seconds:bool=False) -> None:
		if params is None:
			params = {}
		self.desc = desc
		self.params = params
		self.formula = self.make_formula(formula, hitshift, seconds)
		if isinstance(self.formula, sympy.core.expr.Expr):
			print(self.desc)
			self.print_expr(seconds)
	
	@classmethod
	def make_formula(cls, formula, hitshift:int=8, seconds:bool=False):
		if isinstance(formula, (str, int, float, Fraction)):
			formula = make_formula(formula)
		
		factor = sympy.Pow(2, hitshift-8)
		formula = formula * factor
		
		if seconds:
			if isinstance(formula, sympy.core.expr.Expr):
				formula = formula / FRAMES_PER_SECOND
		return formula
	
	
	@classmethod
	def round(cls, val) -> int|float:
		ival: int   = int(val)
		fval: float = float(val)
		return ival if ival == fval else fval
	
	def eval(self, slvl:int) -> int|float:
		return self.round((self.formula.subs({lvl: slvl} | self.params)))
	
	def print_expr(self, seconds:bool=False) -> None:
		expr: sympy.core.expr.Expr = self.formula.subs(self.params)
		expr = expr.simplify()
		print(expr)
		if seconds:
			print(expr.evalf())


def build_table(stats, maxlvl:int=60, splitlvl:int=10):
	'''
	Builds a wiki table of given stats for given level range.
	'''
	
	return '\n\n'.join(
		f'''{{{{Levels {lvl_group+1}-{lvl_group+splitlvl}}}}}
{'\n'.join(f'|-\n!align=left| {stat.desc}\n| {" || ".join(str(stat.eval(lvl_group+lvl_unit)) for lvl_unit in range(1, splitlvl+1))}' for stat in stats)}
|}}''' for lvl_group in range(0, maxlvl, splitlvl)
	)


"""
USAGE
Make a list of stats to be included in the table of values (levels 1-60).
Formulae can be defined in two ways:
1. SymPy: use SymPy expressions - either defined here (fun) or other expressions, using appropriate SymPy functions (for exampe sympy.Min instead of min).
1. Formula as string directly from the data file. Allowed formula parts: lvl, parameters (parX), functions (lnXY and dmXY) and some math functions (min, max). Remove synergies to calculate values without them.
See the examples below. Use only one formula per stat; the repeated stats below are to show how to define formulae.

For level-based values, use "levelwise" formulae, either for 3 or 5 columns.
To convert time values from frames to seconds, set the "seconds" flag.

Simplified formulae are also printed.
For time values formulae with integer and decimal coefficients are printed.
"""


stats: tuple[Stat] = (
	Stat(desc='Inner Sight Duration (Seconds)',  formula=ln34,                                                             params={par3: 300, par4: 150}, seconds=True),
	Stat(desc='Inner Sight Duration (Seconds)',  formula='ln34',                                                                 params={par3: 300, par4: 150}, seconds=True),
	Stat(desc='Slow Missiles [[Missile]] Speed %', formula=                      100-dm12,                                 params={par1: 25,  par2: 75}),
	Stat(desc='Slow Missiles [[Missile]] Speed %', formula=                     '100-dm12',                                      params={par1: 25,  par2: 75}),
	Stat(desc='Impale Durability loss chance %', formula=                           par6-dm34,                             params={par3: 0,   par4: 30,  par6: 50}),
	Stat(desc='Impale Durability loss chance %', formula=                          'par6-dm34',                                  params={par3: 0,   par4: 30,  par6: 50}),
	Stat(desc='Thunderstorm perdelay', formula=                                (100-dm56) * par4/100 + par3,               params={par3: 25,  par4: 100, par5: 0, par6: 100}, seconds=True),
	Stat(desc='Thunderstorm perdelay', formula=                               '(100-dm56) * par4/100 + par3',                    params={par3: 25,  par4: 100, par5: 0, par6: 100}, seconds=True),
	Stat(desc='Battle Cry duration (Seconds)', formula= ln12,                                                              params={par1: 300, par2: 60}, seconds=True),
	Stat(desc='Battle Cry duration (Seconds)', formula='ln12',                                                                   params={par1: 300, par2: 60}, seconds=True),
	Stat(desc='Chain Lightning hits', formula=                              ln34               / par5,                     params={par3: 26,  par4: 1, par5:5}),
	Stat(desc='Chain Lightning hits', formula=                             'ln34/par5',                                          params={par3: 26,  par4: 1, par5:5}),
	Stat(desc='Strafe min hits', formula=                  sympy.Min(par3 + lvl - 1, par4),                                      params={par3: 4,  par4: 10}),
	Stat(desc='Strafe min hits', formula=                       'min(par3 + lvl - 1, par4)',                                     params={par3: 4,  par4: 10}),
	Stat(desc='Prayer healing', formula=levelwise5,                                                                        params={base: 2, lev1: 1, lev2: 1, lev3: 2, lev4: 2, lev5: 3}),
	Stat(desc='Prayer healing', formula='levelwise5',                                                                            params={base: 2, lev1: 1, lev2: 1, lev3: 2, lev4: 2, lev5: 3}),
	Stat(desc='Energy shield absorption',
		formula=sympy.Min(levelwise5, 95),
		params={base: 20, lev1: 5, lev2: 2, lev3: 1, lev4: 1, lev5: 1}
	),
	Stat(desc='Energy shield absorption',
		formula='min(levelwise5, 95)',
		params={base: 20, lev1: 5, lev2: 2, lev3: 1, lev4: 1, lev5: 1}
	),
	Stat(desc='Spirit Wolf coldlength', formula=levelwise3,     params={base: 25,  lev1: 25, lev2: 25, lev3: 25}, seconds=True),
	Stat(desc='Spirit Wolf coldlength', formula='levelwise3',         params={base: 25,  lev1: 25, lev2: 25, lev3: 25}, seconds=True),
	Stat(desc='Chilling Armor cold length', formula=levelwise3, params={base: 100, lev1: 0,  lev2: 25, lev3: 25}, seconds=True),
	Stat(desc='Chilling Armor cold length', formula='levelwise3',     params={base: 100, lev1: 0,  lev2: 25, lev3: 25}, seconds=True),
	Stat(desc='Clay Golem slow', formula=dm34,       params={par3: 0, par4: 75}),
	Stat(desc='Clay Golem slow', formula='dm34',           params={par3: 0, par4: 75}),
	Stat(desc='Blaze length', formula=dm12,       params={par1: 50, par2: 500}, seconds=True),
	Stat(desc='Blaze length', formula='dm12',           params={par1: 50, par2: 500}, seconds=True),
	
	# advanced example: poison damage (Poison Javelin)
	# duration first (will be used in subsequent formulae)
	Stat(desc='Seconds',                 formula=levelwise3, params=levelwise3_params('75	5	5	5'), seconds=True),
	# total poison damage = poison bit rate (from data file, in 1/256 per frame) * duration (from data file, in frames) / 256 (to converts bits into full points); we round it down to the full point
	Stat(desc='Minimum Poison Damage',   formula=sympy.floor(levelwise5.subs(levelwise5_params('196	64	128	208	440	640')) * levelwise3.subs(levelwise3_params('75	5	5	5')) / 256)),
	# poison DPS = (poison bit rate * duration / 256) / duration * 25 (to calculate per second, not per frame)
	# = poison bit rate * 25 / 256; we round it down to the full point
	Stat(desc='Minimum Poison Damage/s', formula=sympy.floor(levelwise5.subs(levelwise5_params('196	64	128	208	440	640')) * sympy.Rational(25, 256))),
	#similar calculations for maximum damage
)
print(build_table(stats))

stats = (
	#build your own table
)
#print(build_table(stats))