Consider a hypothetical automated market maker as a protocol-level price oracle for the trading pair GAS/ETH whose reserve of gas and ether after the n-th trade are g_n and f_n \times g_n, respectively. Moreover, let g_{n+1} = g_n + M/2 - w_n, that is, g += excess. It can be proved that, the limit of f_n as the initial reserve g_0 goes to infinity is given by:
- the Almgren-Chriss additive formula in the case of constant sum market maker,
- and your proposed exponential formula in the case of constant product market maker.
This observation immediately implies that both of these update rules (and any other one based on another constant function market maker) are path independet. Ironically, this is exactly why we have arrived at these formulas in the first place when attempting to solve a simple instance of path dependence attacks.