Test case

To avoid any ambiguity, here is my test case for complex-valued coefficients:

$$\begin{eqnarray} e^A &=& 2+Z \\ e^B &=& 3+iZ \\ R \quad = \quad e^U &=& e^{\overline{B}(1/Z)+A(Z)} \quad = \quad-2i/Z +(6-i)+3Z\end{eqnarray}$$ (6) (7) (8)

with an output of

$$\begin{eqnarray} e^A &=& 1+Z/2 \\ e^B &=& 1+iZ/3 \end{eqnarray}$$ (9) (10)

For this case I found the errors are less than 1% when n=8. Generally, the larger the dynamic range of |R| on the unit circle the greater the need for a higher n.