Correct the scale of the continued fraction's convergence test

[andreasnoack]

Stacked on #550.

En_cf_nogamma's convergence test compared abs(Aprev*B - A*Bprev) against 10*eps(real(B))*abs(B*Bprev). Three things are wrong with that:

• eps(real(B)) is eps of the value z + ν, not of the type, so the tolerance grows with |z| — 7.1e-15 at |z| = 40.
• Dividing through by |B*Bprev| shows the test to be |Aprev/Bprev - A/B| < ϵ, i.e. absolute, on a convergent of size |e^z E_ν(z)| ≈ 1/|z|. With the first point that gives an effective relative tolerance of about 10*eps(one(T))*|z|².
• B*Bprev is the largest of the three products (|B| ≈ |z||A|), so it reaches floatmax while the numerator's mixed A·B products are still finite, and finite < Inf then succeeds unconditionally.

The last one is #545. Running the original loop on the Float32 seed points the real(z) < 0 branch picks for the reported cases, every one exits on an overflowed threshold:

seed z	iters	rescales	rel. err
−3.5 + 0.00121i	34	1	0.874
−4.0 + 0.00138i	221	21	0.581
−6.0 + 0.00207i	196	18	0.174
−8.0 + 0.00276i	78	5	0.0554
−10.0 + 0.00345i	20	0	0.0179

The fix uses eps(T) and abs(A*Bprev), moves the rescaling above the test, tests both |A| and |B|, and lowers the threshold to sqrt(floatmax(T))/4. Each of those is load-bearing: the bound has to hold when the test consumes the products rather than one iteration earlier (for Float16, floatmax^(1/4) is 16, so one iteration's growth can overflow them); A overtakes B by up to 112x near the cut, so checking A alone understates the maximum by |z|; and the margin covers the overshoot between checks, there being two recurrence steps per loop iteration.

Effect

Against Complex{BigFloat} references at 200 bits, grid ±60 × ±30:

ν	worst before	worst after	mean before	mean after
1	2.01e-12	4.25e-15	7.96e-14	3.68e-16
2	1.55e-12	2.62e-15	1.14e-13	3.48e-16
2.5	1.93e-12	2.63e-15	1.21e-13	3.79e-16
1+i	1.17e-12	4.05e-15	1.03e-13	3.39e-16

Both half-planes improve by comparable factors — the right half-plane never touches the real(z) < 0 walk, so the fix stands on its own there.

Narrow types, which is what #545 reports:

	before	after
#545's eight rows (Float32)	worst 0.851	worst 8.79e-7
Float32 grid, ν = 2.5, 844 points	83 above 1e-5, worst 0.864	0 above 1e-5, worst 1.23e-6
Float16 grid, ν = 2, 458 points	90 above 1e-2, worst 0.651	0 above 1e-2, median 1.8 ulps

Cost: unchanged per iteration (3.50 → 3.88 ns real, 16.7 → 17.0 ns complex, for En_cf at ν = 2.5); the ~15% more iterations are the ones the old test skipped.

Tests

A new expint continued fraction testset covers nine points with abs2(z) > 9, for ν = 2, 2.5, 3, 5 and 10, against MPFR's incomplete gamma at rtol = 50*eps(Float64). All nine fail on the base branch, by 74 to 1313 ulps, and pass here, within 8.

Full suite passes: expint 6373 pass, 1 pre-existing @test_broken.

Not fixed here

#453 improves about 5x (worst 2.75e-13 at z = 31.8 → 5.54e-14 at z = 2.89) but stays open: its residual is the cancellation in the series about the origin below |z| = 3, which is #551's mechanism. #546 is untouched. Details and measurements in #453 (comment).

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.68%. Comparing base (ef23f33) to head (b10bc46).

Additional details and impacted files

@@                    Coverage Diff                     @@
##           fix/expint-same-precision     #553   +/-   ##
==========================================================
  Coverage                      94.68%   94.68%           
==========================================================
  Files                             14       14           
  Lines                           3010     3010           
==========================================================
  Hits                            2850     2850           
  Misses                           160      160           

Flag	Coverage Δ
unittests	94.68% <100.00%> (ø)

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[stevengj]

LGTM.