https://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
BCH formula
$$ \exp(\hat A)\hat B\exp(-\hat A)=\hat B+[\hat A,\hat B]+\frac 1 {2!}[\hat A,[\hat A,\hat B]]+\frac 1 {3!}[\hat A.[\hat A,[\hat A,\hat B]]]+\cdots $$
$$ \begin{aligned} \exp(\hat A)f(\hat C)\exp(-\hat A) & = \sum_{l = 0}^{\infty }\frac{f^{(l)}(0)}{l!} \exp (\hat{A })\hat{C^l}\exp (-\hat{A} ) \\ &=\sum_{l = 0}^{\infty }\frac{f^{(l)}(0)}{l!}[ \exp (\hat{A })\hat{C}\exp (-\hat{A} ) ]^l\\ &=f\left (\exp(\hat A)\hat C\exp(-\hat A)\right). \end{aligned} $$