[1.5.73] Formula for Attributes Maximum Value

[LodWig]

Well I should have doubted more...I would have caught the special case of the Block Rating earlier.

(I just started crafting Shields because the Block Rating is shown with two decimal places, and its A coefficient is the highest amongst the two-decimals-shown attributes, so it appeared to be the less blurry window through which one can look at the minimum values. But because of this halving, it is best to observe the Base Armor of the Shield.)

[LodWig]

74% is too much! For example it gives the minimum base armor of a level 2 uncommon hunter gear at 5.5056, the actual rounded value being 5.50. And let's not talk about 75%...

And (to show you how sensitive the equation is) approximating 17/23 by 0.739 is too little: it gives the minimum base armor of a level 6 uncommon hunter gear at 21.37188 while the rounded actual value is 21.38.

Plus you can compute the Orb increase by replacing the 0.115 constant (a.k.a 23/200) from the Q table by 1/600 (a nice number); and if you want 18 Orbs to cover exactly the full range (as is the case for the General attributes) then the range must be 0.03. (I must admit this "reasoning" is a little shallow.)

Also your variance argument is very good.

Anyway I will add more data from higher level gear crafting, in order to amplify any inaccuracy in the coefficient. Let's hope I craft a level 77 legendary pistol with perfect damage!

And please doubt! That is science!

[MarkHark]

Not to put in doubt your calculations, but just for sake of thoroughness, have you considered possible alternatives? 74% is very very close - 17/23 gives 0.739. 75% (or 3/4), looks like one nice round number a lazy programmer might just resort to.

However, I have the feeling you are right, for ONE simple mathematical reason - 0,085 and 0,115 for min and max give us a nice exact +15% and -15% variance on an exact 0,100 average value.

[LodWig]

I expected the same, but experiments showed otherwise.

I screen-recorded some crafts, and by playing slowly the movie I could see the minimum value of an attribute before it started to move up. (Except for weapon damage, unless (this is strange) the damage is going to end up to be "perfect".)

So I can somehow confirm that the 80% rule I read somewhere in the forums is valid for the General attributes, for every item level. Caveat: I did only a small number of experiment. But I do believe in the laziness of programmers (at least of good ones)!

For the other attributes I saw that the minimum seemed to differ from the expected value (80% of the maximum) by a constant ratio. So I tried to approximate the average of the observed ratios by tweaking the coefficients used to compute the maximum value. The best idea was replacing the 0.115 appearing in the Q table by 0.10625, but then it occurred to me that the 80% coefficient could be incorporated to it, and this gave 0.085, a nice number which is exactly 0.115*17/23.

Since then I have looked at the increase in attribute value a CO offers, and I have more evidence that the range of the Base armor attribute is 6/23 of the maximum value, see next post.

Again, I did only a few experiments (maybe two dozens) and for the weapon damage only two gave me minimum values, so... Use with caution!

[LodWig]

To compute the maximum increase in an attribute value a Celestial Orb can offer, divide its maximum value

by	90	for the General attributes
by	69	for the Base Armor attributes
by	828∕29	for the Block Rating of a Shield

In this way exactly 18 Orbs are needed to up any of these attributes from its minimum value to its maximum value.

About Weapon Damage attributes

The application of an Orb to a Weapon Damage attribute seem buggy, for example only 40 Orbs at most seems to be needed to maximise the max damage of a level 70 legendary Pistol, but 47 may be needed to maximise the max damage of a level 70 legendary Axe, assuming you can still apply Orbs after the 26th which would have maximised the min damage... Plus, the result is a weapon with a skewed min to max damage ratio.

For the weapon damage attributes the best I have come up with is the following: the increase is the maximum average damage, divided by 69, divided by 2. What would have made sense is the maximum average damage, divided by 69, and then split in two unequal parts for the increase of the min and the max, keeping the ratio of min to max damage for the weapon. Or to put it another way : increase the min damage by the maximum min damage divided by 138, and the max damage by the maximum max damage divided by 138. In that way, exactly 36 Orbs would be needed to up both min and max damage from their minimum to their maximum values. (One could ask why 36 Orbs for the damage and only 18 for the other attributes, but that is a game designer's choice. The present situation I hope is not by design.)

Postscriptum

Since version 1.4.48, it is possible to use an Orb on a weapon to augment the maximum damage when the minimum damage is already maxed.

[MarkHark]

How did you reach this 17/23 number? I half expected armor and WD to follow the same 80% general rule as the other attributes.

[LodWig]

To compute the minimum value, multiply the maximum value

by	4∕5	for the General attributes
by	17∕23	for the Weapon and Base Armor attributes

Then for the Block Rating attribute of a Shield, divide by 2.

[MarkHark]

Thank you for all the effort put into this!

[LodWig]

Using that pistol with my push toon and comparing sheet damage to computed damage give ample evidence that the range is indeed 4.60—5.75. Thanks again for the correction.

[LodWig]

Ah yes! My confusion is due to the rounding being to exact number for weapon damage, and because my CO showed me no possible increase for this humble weapon, I mixed things up in my head and thought that the actual range was 5.0 — 6.0! My bad, ty for this! I will delete this paragraph from my post.

[MarkHark]

Great job! Need to save this for future reference. Or perhaps Travis could have this set to sticky?

I have encountered only one case of severe discrepancy between the computed value and the actual value of an item attribute: namely, the damage range of the level 1 common pistol a new hunter is equipped with. (Computed: 4.60—5.75, actual: 5.0—6.0.)

This seems to be merely the effect of normal rounding. I would bet the damage range is indeed [4.60-5.75], regardless of the item showing rounded values.

[gzilla]

[Anjaeka]

I appreciate you guys so much for doing these. Math makes my brain nauseous.

[LodWig]

I have accumulated enough data to be convinced that the following simple formula gives an accurate value M for any attribute maximum:

M = L × A × Q

where the coefficients L, A, and Q are tabulated below. (There lie the complexity.)


Level coefficient

The level coefficient L is to be computed from the item level ℓ as follows:

Range of ℓ	Value of L
1—70	( 179 × ℓ - 110 ) / 69
71—75	15 × ℓ - 870
76—77	13 × ℓ - 720

So the coefficient L is 1 for a level 1 item, and grows by 179/69 per level until reaching 180 for a level 70 item; then it grows by 15 per level reaching 255 for a level 75 item; finally it grows by 13 per level reaching 281 for a level 77 item.


Attribute coefficient

The attribute coefficient A is as follows:

General attributes Armor 1 Block Rating 3 / 2 Critical Damage 1 / 300 Deflect Rating / Life on Hit 2 Elemental Damage 1 / 1200 Experience / Extra Gold 1 / 400 Life Regeneration 1 / 0.83 Movement Speed 1 / 3.33 Other 1 / 2

Weapon attributes Pistol 4.0 5.0 Sword 4.0 5.0 Axe 3.2 5.8 Mace 4.3 4.7 Staff 3.6 5.4 Tome 1.0 1.5 Shield 5.0

Base Armor Cape 1.4 Mage Bracer 0.7 Mage Normal Hood 1 Mage Other Gear 1.4 Hunter Bracer 0.75 Hunter Other Gear 1.5 Warrior Bracer 0.8 Warrior Other Gear 1.6 Shield 4.0

(The Weapon attribute for the Shied is the Block Rating. For the other weapons, the attributes are the Min and Max Damage.)


Quality coefficient

The quality coefficient Q is equal to 1 for the General attributes. For the Weapon and Base Armor attributes, it depends on the quality of the item as follows:

Quality	Value of Q
Common	10 × 0.115 = 1.150
Uncommon	12 × 0.115 = 1.380
Rare	13 × 0.115 = 1.495
Epic	14 × 0.115 = 1.610
Legendary	15 × 0.115 = 1.725

Example

The maximum Base Armor for a Level 51 Rare Leather Cap is

(179×51-110)/69 × 1.5 × 1.495 = 293.1175

PS This result builds heavily on the findings of many, including Nhat, Shilien, Jose Sarmento and MarkHark, also on contributions by many to the collection of data.

PPS You can use the spreadsheet "Quality Control" from my Dropbox to evaluate your items.