pylib cannot reach the RTL's Power, so complex magnitudes lose a few ulp

Split out 2026-08-16 from [[bug-n-pow-expected-predates-the-complex-type-and-pxx-differs-from-cpython]], whose main defect (a -0 real part from a cancelling argument reduction) is fixed. This is the residual, and it is an ARCHITECTURE question rather than a numerics tweak, which is why it is not folded in.

Measured, after the quadrant-boundary fix

Real parts are now right everywhere. What remains is the MAGNITUDE.

Cause

pycomplex_pow computes z**w = exp(w * ln z):

lnr := PyMathLn(PyCxSqrt(a.FRe*a.FRe + a.FIm*a.FIm));
...
m := PyMathExp(xr);

CPython's _Py_c_pow instead computes len = pow(vabs, b.real) — one correctly-rounded call where pxx takes a logarithm and an exponential, each rounded, on top of pylib's own series implementations of both. For (-8.0)**0.5 that is exp(0.5*ln 8) versus pow(8, 0.5), and the difference is the ~4 ulp above.

Why it is not simply fixed: compiler/builtin/pylib.pas uses builtin, exceptions, pypal, promocore, typinfo — it cannot see lib/rtl/math.pas, where the correctly-rounded Power, Sqrt, Sin and Cos live. That boundary is exactly why pylib carries PyMathLn, PyMathExp, PyCxSqrt and PyCxCosSin as its own series implementations. A NilPy program calling math.cos gets the good one by name match; complex arithmetic inside pylib gets the local copy. builtin has no trig at all, so there is nowhere better to reach today.

The fork, which is why this is filed rather than patched

1. Write a correctly-rounded pow inside pylib. Closes the rows, and makes a THIRD implementation of the same function in the tree. Against the standing rule about counting mechanisms per concept.
2. Let compiler/builtin reach the RTL's numerics, or hoist the correctly-rounded kernels somewhere both can see. Deletes the duplicates rather than adding one, and would also fix PyCxArcTan and the series PyMathLn/PyMathExp at the same time — but it moves a boundary that the self-host bootstrap depends on, so it is a Track A design call.
3. Accept it. A few ulp on the magnitude of a complex power is the low-priority mechanical accuracy class by the standing rule, and no test now asserts those digits.

Recommendation: (3) for now, (2) whenever the builtin/RTL numerics boundary is revisited for another reason — the win then is several duplicate kernels deleted, not this one row.

Note

test_nilpy_pow_matches_cpython.npy deliberately asserts this value to 12 decimals rather than by repr, with the reason in the test, so nothing is silently pinned to the current ulps.

Gate

Whichever option is taken: the four rows above matching CPython, make test-nilpy, self-host fixedpoint.