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PDIF
Physical Difference or "pDIF" is a series of functions that consider the ratio of player attack vs. monster defense and determine how much melee damage is dealt every time your character swings its weapon or gets swung at. There are at least three different kinds of pDIF equations. The formulas below mostly refer to melee pDIF, with a nod here and there to ranged pDIF. They completely ignore pet/monster/avatar pDIF, which is thought to follow different equations or have different caps.
Prerandomizer pDIF Caps by weapon type  

Type  noncrit  crit 
H2H  3.5  4.5 
1H  3.25  4.25 
2H  3.75  4.75 
Marksmanship  3.5  3.5*1.25 
Archery and Thrown  3.25  3.25*1.25 
Contents
Attack/Defense Ratio
 Ratio = Attacker's Attack/ Target's Defense
 1H melee weapons have an effective ratio cap of 3.625
 H2H melee weapons have an effective ratio cap of 3.875
 2H melee weapons have an effective ratio cap of 4.125
 Archery and Throwing have an unknown ratio cap (perhaps approximately 3.2375)
 Marksmanship also has an unknown ratio cap (perhaps approximately 3.475)
 Example: 1H Player Earthworm Jim has 1000 Attack and is fighting a monster that has 500 Defense. His Ratio is 1000/500, or 2.00.
 Additional Factors during this stage
 Attack Bonuses (ex: Tachi: Gekko)
 Ignore Defense Traits (ex: Quietus)
 Estimating Ratio via /check command
 If your Ratio is less than 1, the monster will check "High Defense."
 If it is higher than 1.25, the monster will check "Low Defense."
Level Correction Function (cRatio)
Level correction function imposes a penalty to attack/defense ratio if the target's level is higher than attacker's level (or item level). After the Attack/Defense Ratio has been corrected, it is referred to as "cRatio" or "corrected Ratio." This correction does not apply in all zones (not in Adoulin zones, Legion, or zones in Escha/Reisenjima). Monsters are also affected by level correction when they attack players, but their formulas give a bonus to ratio when they are over the player's level. It is unclear whether this mechanic applies in the aforementioned zones, but it probably does not.
 Melee attacks have 0.05/level ratio penalty
 cRatio = Ratio  (target's level  attacker's level) × 0.05
 Ranged attacks have 0.025/level ratio penalty
 cRatio = Ratio  (target's level  attacker's level) × 0.025
These equations are one of the reasons that Rangers are preferred for fighting targets many levels above you.
 Example: 1H Player Earthworm Jim has 2.00 Ratio, but the enemy he's fighting is level 124 and he's item level 119. The 5 level difference loses him .25 (=5×.05) Ratio, dropping him to 1.75 cRatio.
After Ratio and Level Correction (Melee)
Beyond this point, there are several competing models that explain how cRatio translates into damage. The below description of melee pDIF is from Motenten's model, extended due to the 2/2016 pDIF changes. For a review of model differences, please see the talk page. Ranged damage is better understood (or at least our understanding of it is challenged less frequently), so it will be handled separately at the bottom.
Motenten has expressed that this model is complete in an algorithmic manner, but explicitly incomplete.
1. wRatio
 1. Calculate wRatio using your cRatio
 If you critical hit: wRatio = (cRatio + 1)
 If you do not critical hit: wRatio = (cRatio + 0)
 Example 1: 1H Player Earthworm Jim has 2.0 cRatio. If he crits, he will have 3.0 wRatio. If he doesn't crit, he will have 2.0 wRatio.
 Example 2: 1H Player Earthworm Jim has 3.0 cRatio. If he crits, he will have 4.0 wRatio. If he doesn't crit, he will have 3.0 wRatio.
 Optional: You can represent this mathematically write this using a logical expression, which will be 1 when true and 0 when false. Rand represents a random number from 0 to 1.
 wRatio = cRatio + ( Rand < CritRate% )
 ex: If your critical hit rate is 25% (0.25) and Rand (A random number between 01) is 0.1, the equation becomes wRatio = cRatio + 1
2. qRatio
 1. Determine the Upper and Lesser Limit
 Upper Limit (See UL Equations)
 Lesser Limit (See LL Equations)
 Min/Max is only there for testing convenience; do not bother with them if they are confusing
 It is currently unknown if the cap per weapon type is applied to the UL and the LL. Doing so would only be noticeable when the UL is above the cap but the LL is not, resulting in a lower average result on the following step of qRatio.

 2. Randomly select a value between LL and UL
 If the value is less than zero, set to zero
 If the value is greater than the cap shown for your weapon type in the list below, set to the cap value.
 1H qRatio caps at 3.25 for noncrits and 4.25 for crits
 H2H qRatio caps at 3.5 for noncrits and 4.5 for crits
 2H qRatio caps at 3.75 for noncrits and 4.75 for crits
3. Multiply the final value by a random number between 1 and 1.05
 For example, if the final qRatio is 3, you'll end up with 3, 3.15, or a value between 3 and 3.15
 (3 ・ 1) = 3
 (3 ・ 1.05) = 3.15
After Ratio and Level Correction (Ranged )
Ranged damage uses cRatio, which is calculated as described above. Then it picks a random value between the below damage limits. It is notable that there is no secondary random multiplier for ranged damage, but qRatio is capped differently for different weapon types:
 Archery and Throwing qRatio is capped at 3.25
 Marksmanship qRatio is capped at 3.50

Ranged Critical Hit Damage
Unlike melee hits (which have additive damage from critical hits), Ranged critical hits are implemented as a +25% multiplier that bypasses the cap.
 Critical ranged pDIF = Normal Ranged pDIF × 1.25
Distance Correction
Also unlike melee hits, ranged attacks have Distance Correction and utilize True Shot, which uniquely affects damage multipliers for Ranged Attacks and also is not capped.
Ex: Master RNG with 4% True Shot gear 7% from traits and 8% from gifts would have 19% True Shot. The multiplier for archery would then become 3.25 * 1.19 = 3.8675 and the multiplier for marksmanship would become 3.5 * 1.19 = 4.165 if fired from proper distance ranges.
References
 Motenten's pDIF graphs and testing (Motenten, BG Forums)
 Pchan's pDIF testing (Pchan, BG Forums)
 Most of the information for Masamune's pDIF model was obtained from a longterm largescale effort by Masamune of BG (or Masamunai on some other forums). It is currently represented in an excel sheet, which makes it difficult to cite as the wiki cannot host it and it contains personally identifying information. Send him a PM for information.
 pDIF formulas 22016 adjustment (Fwahm and Byrth, BG)