[x/gamm][StableSwap] Figure out LP share amount out derivation for stableswap

[ValarDragon]

No description provided.

[catShaark]

Can you provide more description for this issue ?

[AlpinYukseloglu]

Note: since our stableswap CFMM's constant does not scale linearly with pool assets like our balancer one's does, we cannot use the k value as the total number of pool shares. I have still derived a general formula for how k scales below, but we will have to figure out a way to either scale down our CFMM to make k linear or find some other way to calculate LP shares.

EDIT: It turns out for all-asset joins LP share calculation is completely independent from k value. We can just use the same process that we do for Balancer (start with some arbitrary value for all new pools, then if you increase liquidity by 10% you get 10% of the total share supply minted to you).

Leaving the calculations below in case we do need to eventually linearize k to be able to support single-asset joins.

General formula

The general formula is: c^(4+j) * k, where c is the factor we scale the total liquidity by (e.g. increasing pool liquidity by 10% would make c = 1.1), and j is the number of assets in the pool that aren't being traded against.

A more intuitive representation might be c^(2 + numAssets) * k

Examples:

• Two asset pool, increasing liquidity by 15% (c = 1.15, j = 0): k increases to k*(1.15^4)
• Six-asset pool, decreasing liquidity by 4% (c = 0.96, j = 4): k decreases to k*(0.96^8)

Derivation

Two-asset CFMM:

Multi-asset CFMM (derived example from 4-asset pool with assets x, y, m, and n):