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Dokchiter's Paper: attachment:dokchiter.png 
Dokchitser Project for Sage Days 11
From Jen:
Here's the version (closest to Dokchitser's original pari code) that still uses continued fraction approximation:
http://sage.math.washington.edu/home/jen/sage3.0.5x86_64Linux/l4.py
(needs gamma_series.py to run:
http://sage.math.washington.edu/home/jen/sage3.0.5x86_64Linux/gamma_series.py)
The version with Pade approximation (l5.py) has a negligible speedup but only really works for low precision. I'm not sure if Pade gives us a means of computing bounds (I think Mike Rubinstein said that continued fractions won't). Also, l4.py doesn't work for imaginary inputs yet  some coercion with SymbolicRing that I didn't try.
Dokchiter's Paper: attachment:dokchiter.png