Cauchy integral also for b< e^(1/e)?

[Kouznetsov]

r b< e^(1/e), the algorithm of the evaluation on the imaginary axis through the integral Cauchi equation is not yet ready.
While, you may recover the function from the asymptotic behavior.

There is a plenty of formulas for tetration on this forum. Why not to use them?

Because they did not allow to plot the map of tetration.
Because they do not give 14 decimal digits.
Because they are slow in the evaluation.

better, use the Cauchi; you may evaluate a hundred coefficients within a second.

How?

Use formulas for the derivatives through the Cauchi integrals from Wiki.
Chose the circular contour of integration.
Distribute some thoudand points uniformly at this contour.
Spend halfsecond of CPU time to evaluate function in these points.
Spend another halfsecond to approximate the Cauchi integrals with the finite sums
and get the coefficients in the Taylor expansion.
Compare the truncated Taylor expansion to the values by some of algorithms you have mentioned.