Complex AGM

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

Hello PARI developers,

I noticed the complex AGM function in PARI do not always return the so-called 'optimal' AGM.
However this is documented:

??agm
...
In the case of complex or negative numbers, the principal square root is always chosen.
...
So I am not sure whether this should be fixed.
However, do we have a proof of that PARI agm never diverges to zero ?

Example of bad case are:
agm(1,-.00001)
which gives -0.0403257332 - 0.00327376459*I.
but the optimal AGM is
0.114955789 + 0.0279973724*I

Cheers,
Bill.