Mathematically, Dfyn V2 uses the same formulas as Dfyn V1, but they have been enhanced and augmented in comparison to V1.

Liquidity Math:

To facilitate transitioning between price ranges, streamline liquidity management, and avoid rounding errors, Dfyn V2 introduces the following concepts:

$$ L = \sqrt{xy} $$

The liquidity of a pool is determined by the product of two numbers, which represent the amount of token reserves in the pool. The measure of liquidity, represented by the variable $L$, is the square root of the product of these two numbers ($x$ and $y$). This means that $L$ is the geometric mean of $x$ and $y$, and when $L$ is multiplied by itself, the result is equal to the product of $x$ and $y$ $(k)$. This formula can be written as $L =$ $\sqrt{xy}$ , where x and y are the token reserves and k is their product.

Price Math:

$$ \sqrt{P} = \sqrt{\frac{y}{x}} $$

${y/x}$ is the price of token 0 in terms of 1. Since token prices in a pool are reciprocals of each other, we can use only one of them in calculations (and by convention DfynV2 uses $y/x$).

The price of token 1 in terms of token 0 is simply. $\frac{1}{y/x}=\frac{x}{y}$ Similarly, $\frac{1}{\sqrt{P}} = \frac{1}{\sqrt{y/x}} = \sqrt{\frac{x}{y}}$

Why using $\sqrt{P}$ instead of P? There are two reasons:

1. Square root calculation is not precise and causes rounding errors. Thus, it’s easier to store the square root without calculating it in the contracts (we will not store $x$ and $y$ in the contracts).
2. $\sqrt{P}$ has an interesting connection to $L: L$ is also the relation between the change in output amount and the change in $\sqrt{P}.$

$$ L=ΔPΔy

$$

$$ Proof: \\ \sqrt{xy}=\frac{y_1−y_0}{\sqrt{P_1}-\sqrt{P_0}} \\ \sqrt{xy}({\sqrt{P_1}-\sqrt{P_0}})= {y_1−y_0} \\ \sqrt{xy}(\sqrt{\frac{y_1}{x_1}}-\sqrt{\frac{y_0}{x_0}})={y_1−y_0} \\ Since,\sqrt{x_0y_0}=\sqrt{x_1y_1}=\sqrt{xy}=L \\ \sqrt{\frac{x_1y_1y_1}{x_1}}-\sqrt{\frac{x_0y_0y_0}{x_0}} = {y_1−y_0} \\ \sqrt{y_1^2}-\sqrt{y_0^2} = {y_1−y_0} \\ {y_1−y_0}={y_1−y_0}

$$

Again, we don’t need to calculate actual prices–we can calculate output amount right away. Also, since we’re not going to track and store $x$ **and $y$ our calculation will be based only on $L$ and $\sqrt{P}$. From the formula above we can calculate $Δy$.

$$ Δy = Δ\sqrt{P}L $$

As we discussed above, prices in a pool are reciprocals of each other. Thus $Δx$ is:

$$ Δx = Δ\frac{1}{\sqrt{P}}L $$