Constant Product Automated Market Maker Proof of Concept Smart Contract

This project was created in order to better understand how the math behind a constant product automated market maker (AMM) works. This is a simplified constant function market maker (CFMM) exchange contract and was created to better understand how the constant product formula works in practice.

NB! This repository was not extensively tested for deploying on the Ethereum mainnet.

Compiling and Running Tests

forge build

forge test -vv

Constant Product Function

This swap contract uses the formula:

$${x * y = k}$$

Calculating change in x from change in y:

From:

$${x * y = k}$$

We can deduce:

$${k = (x+dx) * (y+dy)}$$

We then arrive at the following:

$${dy = \frac{(-dx*y)} {(dx + x)}}$$

Lets say that we have the following scenario:

amount of tokens X in the contract = 5
amount of tokens Y in the contract = 10

A user comes along and they want to give us 1 of token X in exchange for token Y.

Following the formula constant product formula (x * y = k), we need to calculate the change of Y given a change in X.

x = 5
y = 10
k = 50
dx = 1 

k = (x+1) * (y+dy)

50 = (5+1) * (10+dy)
50 = 6 * (10 + dy)
50 = 60 + 6dy
-10 = 6dy
-10/6 = dy
-1.666 = dy


amountOut = (-dx * y) / (dx + x)

Constant Product Function given x = 5, y = 10, dx = 1