Could someone tell me what the precise definition of tolerant floor is
(or should be) on arbitrary complex arguments?

The way J implements floor (<.) is not completely consistent,

   <.a=: _1e_14 , 100-1e_13
_1 100
   <.a j.0
0 99

and has a shape something like this around any Gaussian integer (look
in fixed width font):

   '....@+m'{~(~.@,i.]) <. 0j0+ j./~i:1e_13j20
.@@@@@@@@@@@@@@++++++
..@@@@@@@@@@@@+++++++
...@@@@@@@@@@++++++++
....@@@@@@@@+++++++++
.....@@@@@@++++++++++
......@@@@+++++++++++
.......@@@+++++++++++
........@@+++++++++++
.........@+++++++++++
..........+++++++++++
.....++++++++++++++++
....+++++++++++++++++
...++++++++++++++++++
..+++++++++++++++++++
.++++++++++++++++++++
+++++++++++++++++++++
+++++++++++++++++++++
+++++++++++++++++++++
+++++++++++++++++++++
+++++++++++++++++++++
+++++++++++++++++++++

But that's all I know about the issue.

Ambrus
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