Pokémon Emerald's capture formula (pret/pokeemerald)

Cmd_handleballthrow reads the target species' catch rate (45 for Bulbasaur), except with a Safari Ball, which uses the Safari Zone's catch factor times 1275 / 100; the factor starts at the species' catch rate times 100 / 1275. pokeemerald-capture/catch-rate

The ball multiplier is in tenths: 20 for an Ultra Ball, 15 for a Great or Safari Ball, 10 for a Poké Ball; a Net Ball 30 against a Water or Bug type, a Dive Ball 35 underwater, a Nest Ball 40 less the target's level below level 40 (at least 10), a Repeat Ball 30 for a species already caught, a Timer Ball 10 plus the turn count up to 40, and 10 for the Luxury and Premier Balls and for a Net, Dive, Nest, or Repeat Ball outside its condition. pokeemerald-capture/ball-modifier

{
 "kind": "ball-modifier",
 "unit": "tenths",
 "ultra": 20,
 "great": 15,
 "poke": 10,
 "safari": 15,
 "net": [
  30,
  10
 ],
 "dive": [
  35,
  10
 ],
 "nest": "max(40 - level, 10) below level 40, else 10",
 "repeat": [
  30,
  10
 ],
 "timer": "min(10 + turns, 40)",
 "luxury": 10,
 "premier": 10
}

Emerald computes $a=\lfloor \lfloor Cm/10\rfloor (3M-2H)/3M\rfloor$ from catch rate $C$, ball multiplier $m$, max HP $M$, and current HP $H$, then doubles it for sleep or freeze, or multiplies it by 15/10 (rounded down) for poison, bad poison, burn, or paralysis. pokeemerald-capture/odds

{
 "kind": "odds",
 "maxFactor": 3,
 "hpFactor": 2,
 "ballUnit": 10,
 "status": {
  "sleepOrFreeze": 2,
  "poisonBurnParalysis": 1.5
 }
}

Cited by: Capture

An $a$ over 254 catches at once. Otherwise Emerald turns it into a threshold $b=\lfloor 1048560/\mathrm{Sqrt}(\mathrm{Sqrt}(\lfloor 16711680/a\rfloor ))\rfloor$, with Sqrt the GBA BIOS call, and draws up to four 16-bit Random() values; each one below $b$ is a shake. Four shakes catch; a throw that fails shows the shakes it passed, 0 to 3. A Master Ball's shakes are set to four after the draw. pokeemerald-capture/shake-checks

{
 "kind": "shake-checks",
 "catchAbove": 254,
 "numerator": 1048560,
 "inner": 16711680,
 "rolls": 4,
 "rollRange": [
  0,
  65535
 ]
}

Cited by: Capture

Computed, counting every Random() value as equally likely and taking the BIOS square root as rounding down: a throw catches with chance ${(b/65536)}^4$ when $a\le 254$. Since 16,711,680 is 255 × 65,536 and 1,048,560 is 16 × 65,535, $b$ is close to $65535{(a/255)}^{1/4}$ and four passes close to $a/255$, higher for small $a$ because both roots round down. A Bulbasaur (catch rate 45) with max HP 100 in a Poké Ball has $a=15$ and $b=32767$ at full HP, about 6.25%, and $a=44$, $b=43690$ at 1 HP, about 19.8%. pokeemerald-capture/chance

{
 "kind": "example",
 "catchRate": 45,
 "maxHP": 100,
 "ball": "poke",
 "status": "none",
 "fullHP": 0.0625,
 "oneHP": 0.1975,
 "thresholds": {
  "full": 32767,
  "one": 43690
 }
}

Ruby and Sapphire run the same formula and shake checks as Emerald, except that their status test leaves out bad poison (STATUS1_TOXIC_POISON, a separate status bit from poison), so a badly poisoned Pokémon gets no bonus. Emerald's test includes it. pokeemerald-capture/ruby-sapphire-toxic

About: Pokémon Ruby and Sapphire Capture

Cited by: Capture

FireRed and LeafGreen run Emerald's formula, status test (bad poison included), and shake checks unchanged; before them, a ball thrown in a battle flagged BATTLE_TYPE_GHOST is dodged. pokeemerald-capture/firered-leafgreen-same

About: Pokémon FireRed and LeafGreen Capture

Cited by: Capture

Computed, for a Bulbasaur (catch rate 45) with max HP 100 and no status in a Poké Ball: at full HP it is caught about 6.0% of the time in Red and Blue, 6.25% in Crystal, and 6.25% in Emerald; at 1 HP, 18.0%, 17.6%, and 19.8%. Here the three formulas land close together, though they roll twice in Red and Blue, once in Crystal, and up to four times in Emerald. pokeemerald-capture/across-generations

{
 "kind": "comparison",
 "catchRate": 45,
 "maxHP": 100,
 "ball": "poke",
 "fullHP": {
  "red-blue": 0.0604,
  "crystal": 0.0625,
  "emerald": 0.0625
 },
 "oneHP": {
  "red-blue": 0.1797,
  "crystal": 0.1758,
  "emerald": 0.1975
 }
}

Cited by: Capture