Hl! I have problems with this exercises

Let $\displaystyle \varepsilon$ the Stereographic projection of $\displaystyle \mathbb{S}^1- \{i \}$, that is

$\displaystyle \varepsilon : \mathbb{S}^1 - \{i \} \longrightarrow{\mathbb{R}}$

where

$\displaystyle \varepsilon(z)=\displaystyle\frac{RE(z)}{1-Im(z)}$

Find a formula for $\displaystyle \varepsilon^{-1} :\mathbb{R}\rightarrow{\mathbb{S}^1}- \{i \} $

I tried to take the line that passes through i and a point on the real axis to see where the circle intersects but I can not find the formula

Thanks