Item formula

$Recurve Bow = 15 \times \frac{{100}}{{100 + EnemyMR}}$

$Kircheis Shard = 60 \times \frac{{100}}{{100 + EnemyMR}}$

$Unending Despair = (30 + (20 \div 17) \times (Level - 1) + 0.03 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Muramana \bigg|_{RANGED}^{MELEE} = \frac{{((MaxMana \times 0.035) + (0.06 \times AD)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{((MaxMana \times 0.027) + (0.06 \times AD)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Zeke's Convergence = 50 \times \frac{{100}}{{100 + EnemyMR}}$

$Sheen = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Sunfire Aegis = (15 + 0.0175 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Ravenous Hydra \bigg|_{RANGED}^{MELEE} = \frac{{0.4 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.2 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Thornmail = (10 + (0.25 \times BonusArmor)) \times \frac{{100}}{{100 + EnemyMR}}$

$Bramble Vest = 6 \times \frac{{100}}{{100 + EnemyMR}}$

$Tiamat \bigg|_{RANGED}^{MELEE} = \frac{{0.4 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.2 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Trinity Force = (BaseAD \times 2) \times \frac{{100}}{{100 + EnemyArmor}}$

$Heartsteel = (100 + ((MaxHP - BaseHP) \times 0.12)) \times \frac{{100}}{{100 + EnemyArmor}}$

$Runaan Hurricane = 15 \times \frac{{100}}{{100 + EnemyMR}}$

$Statikk Shiv = 90 \times \frac{{100}}{{100 + EnemyMR}}$

$Rapid Firecannon = 60 \times \frac{{100}}{{100 + EnemyMR}}$

$Stormrazor = 100 \times \frac{{100}}{{100 + EnemyMR}}$

$Lich Bane = (0.75 \times BaseAD + (AP \times 0.45)) \times \frac{{100}}{{100 + EnemyMR}}$

$Redemption = 0.1 \times EnemyHP$

$Nashor's Tooth = (15 + (0.2 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Malignance = (60 + (0.05 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Guinsoo's Rageblade = 30 \times \frac{{100}}{{100 + EnemyMR}}$

$Deathfire Grasp = 0.15 \times EnemyHP \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Alternator = (50 + 75 \div 17 \times (Level - 1)) \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Gunblade = (175 + 78 \div 17 \times (Level - 1) + (0.3 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Rocketbelt = (100 + (0.1 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Blade of The Ruined King \bigg|_{RANGED}^{MELEE} = \frac{{EnemyHP - ((0.12 \times EnemyHP \times (1 - 0.12 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1 - AD + AD \times (1 - 0.12 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1) \div 0.12)}}{{EnemyHP - ((0.09 \times EnemyHP \times (1 - 0.09 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1 - AD + AD \times (1 - 0.09 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1) \div 0.09)}}$

$Hullbreaker \bigg|_{RANGED}^{MELEE} = \frac{{(1.4 \times BaseAD + (0.035 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(0.7 \times BaseAD + (0.035 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Terminus = 30 \times \frac{{100}}{{100 + EnemyMR}}$

$Ardent Censer = 15 \times \frac{{100}}{{100 + EnemyMR}}$

$Essence Reaver = (1.4 \times BaseAD + (0.2 \times BonusAD)) \times \frac{{100}}{{100 + EnemyArmor}}$

$Dead Man's Plate = (40 + BaseAD) \times \frac{{100}}{{100 + EnemyArmor}}$

$Titanic Hydra \bigg|_{RANGED}^{MELEE} = \frac{{(0.015 \times MaxHP) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(0.0075 \times MaxHP) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Dream Maker = 90 \times \frac{{100}}{{100 + EnemyMR}}$

$Zaz'Zak's Realmspike = (50 + 0.15 \times AP + (0.03 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Bloodsong = BaseAD \times 1.5 \times \frac{{100}}{{100 + EnemyArmor}}$

$Imperial Mandate = 0.12 \times EnemyHP \times \frac{{100}}{{100 + EnemyMR}}$

$Hellfire Hatchet = (0.12 \times BonusAD + 0.1 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}$

$Night Harvester = (125 + (0.15 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Demonic Embrace \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.016) \times \frac{{100}}{{100 + EnemyMR}}}}{{(EnemyHP \times 0.01) \times \frac{{100}}{{100 + EnemyMR}}}}$

$Stormsurge \bigg|_{RANGED}^{MELEE} = \frac{{(140 + AP \times 0.2) \times \frac{{100}}{{100 + EnemyMR}}}}{{(105 + AP \times 0.15) \times \frac{{100}}{{100 + EnemyMR}}}}$

$Ironspike Whip = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Sundered Sky = CritDamage \div 100 \times AD \times \frac{{100}}{{100 + EnemyArmor}}$

$Goredrinker = 1.75 \times BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Stridebreaker \bigg|_{RANGED}^{MELEE} = \frac{{0.4 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.2 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Divine Sunderer \bigg|_{RANGED}^{MELEE} = \frac{{(BaseAD \times 1.6 + (0.04 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(BaseAD \times 1.6 + (0.02 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Liandry's Torment = EnemyHP \times 0.01 \times \frac{{100}}{{100 + EnemyMR}}$

$Luden's Companion = (45 + (0.04 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Everfrost = (100 + (0.30 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Bami's Cinder = (12 + 0.01 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Iceborn Gauntlet = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Hollow Radiance = (10 + 0.0175 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Rageknife = 20 \times \frac{{100}}{{100 + EnemyMR}}$

$Eclipse \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.08) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(EnemyHP \times 0.04) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Prowlers Claw \bigg|_{RANGED}^{MELEE} = \frac{{(85 + (BonusAD \times 0.55)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(65 + (BonusAD \times 0.35)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Profane Hydra \bigg|_{RANGED}^{MELEE} = \frac{{0.5 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.25 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Voltaic Cyclosword = 100 \times \frac{{100}}{{100 + EnemyArmor}}$

$Syzygy \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.06)\times \frac{{100}}{{100 + EnemyArmor}}}}{{(EnemyHP \times 0.03) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Enmity of the Masses = (60 + (0.05 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Frozen Fist = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Swordnado = 100 \times \frac{{100}}{{100 + EnemyArmor}}$

$Vespertide = (125 + (0.15 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Upgraded Aeropack = (125 + (0.15 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Liandry's Lament = EnemyHP \times 0.01 \times \frac{{100}}{{100 + EnemyMR}}$

$Force of Arms = (45 + (0.04 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Eternal Winter = (100 + (0.30 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Ceaseless Hunger = 1.75 \times BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Stridebreaker \bigg|_{RANGED}^{MELEE} = \frac{{0.4 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.2 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Deicide \bigg|_{RANGED}^{MELEE} = \frac{{(BaseAD \times 1.6 + (0.04 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(BaseAD \times 1.6 + (0.02 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Infinity Force = (BaseAD \times 2) \times \frac{{100}}{{100 + EnemyArmor}}$

$Leviathan = (100 + ((MaxHP - BaseHP) \times 0.12)) \times \frac{{100}}{{100 + EnemyArmor}}$

$The Unspoken Parasite = (80 + 0.07 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Seething Sorrow = 30 \times \frac{{100}}{{100 + EnemyMR}}$

$Hope Adrift = (30 + (20 \div 17) \times (Level - 1) + 0.03 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Heavensfall = CritDamage \div 100 \times AD \times \frac{{100}}{{100 + EnemyArmor}}$

$Eye of the Storm = 100 \times \frac{{100}}{{100 + EnemyMR}}$

$The Baron's Gift = (15 + (0.2 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Recurve Bow_{ARENA} = 15 \times \frac{{100}}{{100 + EnemyMR}}$

$Kircheis Shard_{ARENA} = 60 \times \frac{{100}}{{100 + EnemyMR}}$

$Muramana_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{((MaxMana \times 0.045) + (0.06 \times AD)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{((MaxMana \times 0.035) + (0.06 \times AD)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Sheen_{ARENA} = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Sunfire Aegis_{ARENA} = (15 + 0.0175 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Ravenous Hydra_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(0.4 \times AD + 0.24 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(0.2 \times AD + 0.12 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Thornmail_{ARENA} = (10 + (0.25 \times BonusArmor)) \times \frac{{100}}{{100 + EnemyMR}}$

$Bramble Vest_{ARENA} = 6 \times \frac{{100}}{{100 + EnemyMR}}$

$Tiamat_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{0.4 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}{{0.2 \times AD \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Trinity Force_{ARENA} = (BaseAD \times 2) \times \frac{{100}}{{100 + EnemyArmor}}$

$Heartsteel_{ARENA} = (125 + (MaxHP \times 0.06)) \times \frac{{100}}{{100 + EnemyArmor}}$

$Runaan Hurricane_{ARENA} = 15 \times \frac{{100}}{{100 + EnemyMR}}$

$Rapid Firecannon_{ARENA} = 60 \times \frac{{100}}{{100 + EnemyMR}}$

$Stormrazor_{ARENA} = (90 + 0.25 \times AD) \times \frac{{100}}{{100 + EnemyMR}}$

$Lich Bane_{ARENA} = (BaseAD + (AP \times 0.66)) \times \frac{{100}}{{100 + EnemyMR}}$

$Redemption_{ARENA} = 0.1 \times EnemyHP$

$Nashor's Tooth_{ARENA} = (15 + (0.2 \times AP) + (0.33 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Guinsoo's Rageblade_{ARENA} = 30 \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Alternator_{ARENA} = (50 + 75 \div 17 \times (Level - 1)) \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Gunblade_{ARENA} = (175 + 78 \div 17 \times (Level - 1) + (0.3 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Hextech Rocketbelt_{ARENA} = (100 + (0.1 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Blade of The Ruined King_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{EnemyHP - ((0.105 \times EnemyHP \times (1 - 0.105 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1 - AD + AD \times (1 - 0.105 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1) \div 0.105)}}{{EnemyHP - ((0.08 \times EnemyHP \times (1 - 0.08 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1 - AD + AD \times (1 - 0.08 \times \frac{{100}}{{100 + EnemyArmor}}) \times\times 1) \div 0.08)}}$

$Hullbreaker_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(1.4 \times BaseAD + (0.035 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(0.7 \times BaseAD + (0.035 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Ardent Censer_{ARENA} = 25 \times \frac{{100}}{{100 + EnemyMR}}$

$Essence Reaver_{ARENA} = (1.3 \times BaseAD + (0.6 \times BonusAD)) \times \frac{{100}}{{100 + EnemyArmor}}$

$Dead Man's Plate_{ARENA} = (40 + BaseAD) \times \frac{{100}}{{100 + EnemyArmor}}$

$Titanic Hydra_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(0.015 \times MaxHP) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(0.0075 \times MaxHP) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Imperial Mandate_{ARENA} = 0.12 \times EnemyHP \times \frac{{100}}{{100 + EnemyMR}}$

$Night Harvester_{ARENA} = (125 + (0.15 \times AP) + (0.25 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Demonic Embrace_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.016) \times \frac{{100}}{{100 + EnemyMR}}}}{{(EnemyHP \times 0.01) \times \frac{{100}}{{100 + EnemyMR}}}}$

$Ironspike Whip_{ARENA} = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Goredrinker_{ARENA} = (1.75 \times BaseAD + 1.05 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}$

$Divine Sunderer_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(BaseAD \times 1.6 + (0.04 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(BaseAD \times 1.6 + (0.02 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Liandry's Torment_{ARENA} = EnemyHP \times 0.01 \times \frac{{100}}{{100 + EnemyMR}}$

$Luden's Companion_{ARENA} = (40 + (0.08 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Everfrost_{ARENA} = (100 + (0.30 \times AP) + (0.5 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Bami's Cinder_{ARENA} = (12 + 0.01 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Iceborn Gauntlet_{ARENA} = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Rageknife_{ARENA} = 20 \times \frac{{100}}{{100 + EnemyMR}}$

$Eclipse_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.08) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(EnemyHP \times 0.04) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Prowlers Claw_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(40 + (BonusAD \times 0.2) + (0.12 \times AP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(30 + (BonusAD \times 0.15) + (0.09 \times AP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Syzygy_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(EnemyHP \times 0.06) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(EnemyHP \times 0.03) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Frozen Fist_{ARENA} = BaseAD \times \frac{{100}}{{100 + EnemyArmor}}$

$Vespertide_{ARENA} = (125 + (0.15 \times AP) + (0.25 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Upgraded Aeropack_{ARENA} = (125 + (0.15 \times AP)) \times \frac{{100}}{{100 + EnemyMR}}$

$Liandry's Lament_{ARENA} = (12.5 + (AP \times 0.015) + (0.025 \times BonusAD) + (EnemyHP \times 0.06)) \times \frac{{100}}{{100 + EnemyMR}}$

$Eye of Luden_{ARENA} = (100 + (0.1 \times AP) + (0.16 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Eternal Winter_{ARENA} = (100 + (0.30 \times AP) + (0.5 \times BonusAD)) \times \frac{{100}}{{100 + EnemyMR}}$

$Ceaseless Hunger_{ARENA} = (1.75 \times BaseAD + 1.05 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}$

$Dreamshatter_{ARENA} = (1.75 \times BaseAD + 1.05 \times AP) \times \frac{{100}}{{100 + EnemyArmor}}$

$Deicide_{ARENA} \bigg|_{RANGED}^{MELEE} = \frac{{(BaseAD \times 1.6 + (0.04 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}{{(BaseAD \times 1.6 + (0.02 \times EnemyHP)) \times \frac{{100}}{{100 + EnemyArmor}}}}$

$Infinity Force_{ARENA} = (BaseAD \times 2) \times \frac{{100}}{{100 + EnemyArmor}}$

$Leviathan_{ARENA} = (125 + (MaxHP \times 0.06)) \times \frac{{100}}{{100 + EnemyArmor}}$

$The Unspoken Parasite_{ARENA} = (80 + 0.07 \times BonusHP) \times \frac{{100}}{{100 + EnemyMR}}$

$Seething Sorrow_{ARENA} = 30 \times \frac{{100}}{{100 + EnemyMR}}$

Blade of The Ruined King Complex

$R_B = h-\frac{hy\cdot \left(1-ym\right)^x-a+a\cdot \left(1-ym\right)^x}{y}$

$R_x\:=\:\frac{\ln \left(\frac{a}{hy}\right)}{\ln \left(1-ym\right)}$

$f\left(n\right)=K^n\:\cdot \:f\left(1\right)\:-\:t\:\cdot \:\sum _{i = 0}^{n - 1}\left(K^i\:=\:\frac{1\:-\:K^n}{1\:-\:K}\right), K = \left(\:1-\frac{100\:\cdot \:y}{100\:+\:s}\right)^{\frac{1}{n}}$

$K_B = h-\frac{hy\cdot \left(1-ym\right)^x-t+t\cdot \left(1-ym\right)^x}{my}, \:f\left(1\right) = h$

$K_x\:=\:\frac{\ln \left(\frac{t}{hym + t}\right)}{\ln \left(1-ym\right)}$