Exponentiation

The main formulas used in Balancer protocol make use of a form of exponentiation where both the base and exponent are fixed-point (non-integer) values. Take for example the swap functions, where the weights in both the exponent and the base are fractions:

$$A_o = \left(1 - \left(\frac{B_i}{B_i+A_i}\right)^{\frac{W_i}{W_o}}\right).B_o$$

$$\begin{equation} \begin{gathered} A_i = \left(\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}}-1\right).B_i \end{gathered} \end{equation}$$

Since solidity does not have fixed point algebra or more complex functions like fractional power we use the following binomial approximation:

$$\begin{equation} \begin{gathered} \left(1+x\right)^\alpha=1+\alpha x+\frac{(\alpha)(\alpha-1)}{2!}x^2+ \frac{(\alpha)(\alpha-1)(\alpha-2)}{3!}x^3+ \cdots = \sum_{k=0}^{\infty}{\alpha \choose k}x^k \end{gathered} \end{equation}$$

which converges for ${|x| < 1}$.

When $\alpha>1$ we split the calculation into two parts for increased accuracy, the first is the exponential with the integer part of $\alpha$ (which we can calculate exactly) and the second is the exponential with the fractional part of $\alpha$:

$$\begin{equation} \begin{gathered} A_i = \left(1 - \left(\frac{B_o}{B_o-A_o}\right)^{int\left(\frac{W_o}{W_i}\right)}\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}\%1}\right).B_i \end{gathered} \end{equation}$$