Exponentiation

# 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:
$A_o = \left(1 - \left(\frac{B_i}{B_i+A_i}\right)^{\frac{W_i}{W_o}}\right).B_o$
$$$\begin{gathered} A_i = \left(\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}}-1\right).B_i \end{gathered}$$$
Since solidity does not have fixed point algebra or more complex functions like fractional power we use the following binomial approximation:
$$$\begin{gathered} \left(1+x\right)^\alpha=1+\alpha x+\frac{(\alpha)(\alpha-1)}{2!}x^2+ \frac{(\alpha)(\alpha-1)(\alpha-2)}{3!}x^3+ \cdots = \sum_{k=0}^{\infty}{\alpha \choose k}x^k \end{gathered}$$$
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
$\alpha$
:
$$$\begin{gathered} A_i = \left(1 - \left(\frac{B_o}{B_o-A_o}\right)^{int\left(\frac{W_o}{W_i}\right)}\left(\frac{B_o}{B_o-A_o}\right)^{\frac{W_o}{W_i}\%1}\right).B_i \end{gathered}$$$