Complex ** is a few ulp off CPython

• Type: bug (float accuracy — low prio, mechanical, per the standing rule that float-handling accuracy bugs are Track B). The code is compiler/builtin/pylib.pas, so landing it needs a re-pin.
• Found: 2026-08-16, implementing [[bug-nilpy-no-complex-number-type]]. It is the one line of that ticket's CPython oracle that does not agree, which is why it is not in the checked-in differential test.

Measured

             (-8) ** (1/3)
exact        1                  + 1.7320508075688772935…j
CPython      1.0000000000000002 + 1.7320508075688772j
pxx          1                  + 1.7320508075688767j

Note which way each errs: pxx's real part is exactly right and CPython's is 2 ulp high; pxx's imaginary part is 5 ulp low. So this is not "pxx is wrong and CPython is right" — both are last-ulp approximations of the same exact value, and they disagree. What we owe is CPython's answer, because NilPy's contract is upward compatibility with it, not mathematical optimality.

Everything else in the complex surface matches CPython exactly — all three repr shapes, every arithmetic form, abs, ==, and integer-exponent ** (which takes the repeated-multiplication path CPython also uses, so 2j ** 2 is exactly (-4+0j)).

Cause

pycomplex_pow computes the principal branch exp(w · ln z), needing ln, exp, sqrt, sin, cos and atan2. pylib cannot uses math — the RTL's Abs overload set hides pylib's own, so abs(2.5) stops resolving (measured 2026-08-16, recorded beside PyPowHook). So sqrt/sin/cos/atan are hand-rolled next to the existing PyMathLn/PyMathExp, with quadrant reduction and a 12-term Taylor pair. Good to a few ulp; not correctly rounded.

Fix shape

The sanctioned way around the dependency wall already exists and is already used by this exact operator: the hook pattern. The frontend auto-uses math whenever it sees ** and installs @Power into PyPowHook (pyparser.inc, PyPowHook in pylib.pas). Complex pow needs the same for Sqrt, Sin, Cos and ArcTan2, whose RTL versions are correctly rounded — four more hooks installed at the same site, with the hand-rolled series staying as the fallback for a program that pulls no math unit.

That is mechanical, and it would also let pycomplex_abs use the RTL's Hypot rather than sqrt(re²+im²), which overflows for large components where CPython does not — worth folding into the same pass.

Sibling of [[bug-nilpy-float-power-is-a-ulp-off-the-rtl-already-has-the-fix]]: same wall, same answer already written down on the other side of it.

Gate

The (-8) ** (1/3) row added to test/test_nilpy_complex.npy (which is diffed live against CPython, so it needs no separate .expected), plus a sweep of fractional and complex exponents; gate.sh quick; make stabilize-fast && make pin.