Crafting

Crafting is a Crafting Method available at Crafting Benches to perform Alchemy.
It allows players to craft higher quality Character and Weapon Enhancement Materials, Weapon Ascension Materials, Talent Level-Up Materials, Potions, and Character Ascension Materials. Gadgets can also be crafted after obtaining their Instructions.
==List of Crafting Recipes==
Ascension Materials are crafted from three units of the same material of one tier lower.
===Character and Weapon Enhancement Materials===
===Weapon Ascension Materials===
===Character Talent Materials===
===Potions===
===Essential Oils===
===Character Ascension Materials===
===Gadgets===
===Bait===
===Food===
===Quest and Event Items===
==Character Bonuses==
Certain characters have Talents that provides a bonus while performing crafting.
===Efficiency===
The bonuses from talents when crafting Character Talent Materials, Weapon Ascension Materials, and Character and Weapon Enhancement Materials are typically one of the following:
* 25% chance to refund one count of one material used
* 10% chance to double the product
Over time, the 10% chance to double the product bonus will yield slightly more products than the 25% chance to refund one material:
In terms of the number of additional products, these bonuses are effectively:
* 25% chance to refund one count of one material used:On average, for every three materials used, products can be obtained additionally if the refunded materials are also used up
* 10% chance to double the product:On average, for every three materials used, 0.10 products can be obtained additionally
In terms of the materials used per products, these bonuses are effectively:
* 25% chance to refund one count of one material used:On average, 0.25 fewer materials required per product
* 10% chance to double the product:On average, fewer materials required per product
====Derivations====
;25% chance refund
Suppose one starts with x_0 materials. For every 3 materials the probability of getting one back is p, so the expected number of materials refunded is x_0\left(\tfrac{p}{3}\right). One can also use these refunded materials to craft even more and get more refunded materials back. The expected number of materials refunded from these is x_0(\tfrac{p}{3})^2. One can continue this process, so that theoretically, the number of materials gained is:
:x_0[\tfrac{p}{3} + (\tfrac{p}{3})^2 + (\tfrac{p}{3})^3 + \ldots] = \tfrac{x_0}{(3/p)-1}
when summing the . This means that \tfrac{1}{(3/p)-1} more materials are gained from these passive talents. In turn, one will be able to craft around \tfrac{1}{3/p-1} portion more products (or \tfrac{1}{(3/p)-1} more products per 3 materials used).
So the total amount of materials that can be used for crafting is x_0 + \tfrac{x_0}{(3/p)-1} = \tfrac{x_0}{1-(p/3)} meaning one can craft \tfrac{x_0}{3-p} products. Thus with the passive talent, the amount of materials used per product is 3-p whereas otherwise it would be 3. Hence 3-(3-p)=p fewer materials per product are used.
For p=1/4=25\% this means 0.25 fewer materials per product are used and \tfrac{1}{11} \approx 9\% more products can be crafted (or .09 more products per 3 materials used).
;10% chance double
For characters which have a 10% chance of yielding double the product, 10% more products will be gained.
So if one starts with x_0 materials, they can gain \tfrac{11x_0}{30} products on average. With this passive, the amount of materials used per product is \tfrac{30}{11} so 3-\tfrac{30}{11} \approx 0.27 fewer materials per product are used on average.
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