This here basically says it all:

>>> import cmath;[cmath.asinh(i*1e-17).real for i in range(0,20)]
[4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16, 4.4408920985006257e-16,
4.4408920985006257e-16, 4.4408920985006257e-16]

The boost.math toolkit at [2] is an implementation that does better in
the above (real-only) aspect.
[2] http://freespace.virgin.net/boost.regex/toolkit/html/index.html

Tim Peters remarks in [1] that basically all of cmath is unsound.
http://mail.python.org/pipermail/python-bugs-list/2001-February/004126.html

I just wanted to make sure that this issue remains on the radar.