sexp(strip) is winding around the fixed points

[Kouznetsov]

.. would be really interesting to see slog(G) (with non-winding cut) with image range up to 20i on the imaginary axis. Whether it intersects the strip there or not.

I have implemented the funciton fsexp, that approximates sexp, and fslog, that approximates slog. I plot \( f= |\mathrm{fslog}(\mathrm{fsexp}(z))-z| \).
The level \( f=10^{-2} \) is shown with dark yellow.
The level \( f=10^{-4} \) is shown with thick yellow.
The level \( f=10^{-6} \) is shown with thin yellow.
The level \( f=10^{-8} \) is shown with black.
The level \( f=10^{-10} \) is shown with red.
The level \( f=10^{-12} \) is shown with pink.
The level \( f=10^{-14} \) is shown with green.
   
The plot below indicates the range of values of slog. The \( \mathrm{slog}(G) \) is well inside the range of values; it intersects the strip \( S=\{z\in \mathbb{C} : -1<\Re(z)\le 0 \} \) only once. In particular, the slog never has value 20i; but it may have value -4+20i.
The levels indicate the precision of the fast numerical implementaitons. While the imaginary part does not exceed 5, the errors are regular, and, perhaps, are due to the approximation fslog. Currently fslog is made of 2 elementary functions (expansion in vicinity of the fixed point and expansion in vicinity of unity); fsexp uses 3 functions (expansion at \( i \infty \), expansion at 3i and expansion at zero). I estimate, in the worst case (distance of order of 0.1 from the fixed point), fslog returns at least 10 significant figures; in vicinity of the real axis, each of functions fslog and fsexp return values what deviate from slog and sexp in 15th decimal digit.
While the imaginary part exceeds 8, the deviation of \( f= |\mathrm{fslog}(\mathrm{fsexp}(z))-z| \) from zero is mainly due to the smallness of derivatives of sexp; it is difficult to distinguish values of the sexp from the fixed point.