Native Metal SSB redraw precision at the lowest scan frequencies, 2026-10-25

Question

Is the native interactive SSB redraw (ssb-redraw, method ssb, 1–4×) within twice the CUDA reference’s own float32 error, measured against a float64 evaluation of the same definition, on every measured acquisition, fitted optics set and output sampling, at the same drag speed?

Workload and boundary

• Five measured 512 × 512 × 192 × 192 acquisitions at 300 kV and 30 mrad, with 8,800–9,300 BF pixels: three at a 0.373 Å scan step (Auto 3×) and two at 0.525 Å (Auto 4×). Eight fitted optics sets (C10 … C56, sample tilt, depth spread), each reconstructed at 1×, 2× and Auto: 24 images.

• References: the CUDA SSB (float32 corrections, error-free float32 sums met in float64, complex128 inverse FFT) and an independent float64 evaluation of the definition (float64 FFT of the counts, exact chi, complex128 sums and inverse FFT). e_C = rms(CUDA − float64), per image, 3.2e-6 to 8.7e-6 rad.

• Runner: metal-ssb-fit-benchmark --fixture-parity CASE --oracle-dir DIR --methods ssb (rms and maximum of the wrapped phase difference); timing with metal-ssb-redraw-benchmark as in the 2026-10-24 redraw report (synthetic counts on the 0.373 Å geometry, every BF pixel, all 14 terms, tilt, depth; 20 warm calls with a changed C10). Apple M5 Max, macOS 26.4. Before = the same runners built from f829cab4 with the same release flags; before and after alternated run by run under the shared GPU lock. Raw outputs are in the private evidence archive.

Finding

v0.2 was 2.28 and 2.45 e_C from the float64 definition at 1× on the two 0.525 Å acquisitions (1.36–1.51 e_C at 1× on two others). The whole excess sat at the first scan frequencies (frequency index ±1): 8.1e-6 rad of error power there against CUDA’s 2.5e-6, every other frequency band equal to CUDA’s.

There the two sidebands nearly coincide. With δ− = chi(k − q) − chi(k), δ+ = chi(k + q) − chi(k), μ = (δ+ − δ−)/2 and ν = −(δ− + δ+)/2, the sideband difference is gamma = exp(iμ) ((a − b) cos ν + i (a + b) sin ν), so the correction’s direction is set by μ and by the sign of the second difference chi(k + q) + chi(k − q) − 2 chi(k). At index 1 of a 0.525 Å scan (q = 0.0037 / Å) that second difference is a few 1e-5 rad when C10 and C12 partly cancel along the scan direction, while chi reaches tens of radians. v0.2 formed δ± from separately rounded float32 chi values (a row-collapsed absolute polynomial minus chi(k) from the probe kernel), errors of several 1e-6 rad, so the sign flipped for a sizable fraction of BF pixels, each flip adding 2 |G(q, k)|. The 0.373 Å scans have twice the q² at index 1 and cancel less. Trigonometry was not the cause: on the M5 Max metal::fast::sincos and metal::precise::sincos return identical values (within 1.3e-7 of double for |x| < 100), and swapping them changed no output bit. Accumulation order and the float32 inverse FFT contribute equally to both implementations at every other frequency.

Change (v0.3)

• redraw_taylor (once per redraw, one thread per BF pixel) shifts the absolute 7 × 7 chi polynomial to each pixel: δ±(q) = Σ d_ij(k) (±q_row)^i (±q_col)^j, stored coefficient-major; it also writes the pixel’s probe geometry, replacing the probe dispatch. The threadgroup collapses the row power at ±q_row (plus and minus from one read) and Horner runs in ±q_col. Rounding is now proportional to δ± (about 1e-8 rad at index 1) instead of to chi, and chi(k) is no longer subtracted.

• Pairs with both sidebands possible form gamma from μ and ν as above, so the rounding of sin and cos is not amplified by 1/|gamma| where the sidebands nearly cancel. Single-sideband pairs, accumulation, mirror, origin and FFT are unchanged.

Results

Worst over the optics sets of each acquisition; native − float64 / e_C and native − CUDA / e_C (rms). The rule is ≤ 2.

Acquisition	Step	1× v0.2	1× v0.3	2× v0.2	2× v0.3	Auto v0.2	Auto v0.3
1 (three optics sets)	0.373 Å	1.20 / 1.08	0.95 / 1.11	1.13 / 1.00	1.07 / 1.01	1.13 / 1.00	1.08 / 1.01
2	0.373 Å	1.36 / 1.61	0.96 / 1.13	1.14 / 1.26	0.98 / 1.06	1.14 / 1.26	0.99 / 1.06
3	0.373 Å	1.51 / 1.67	0.81 / 1.01	1.18 / 1.30	0.90 / 1.03	1.17 / 1.29	0.90 / 1.02
4	0.525 Å	2.28 / 2.20	0.74 / 1.07	1.45 / 1.40	0.93 / 0.97	1.34 / 1.25	0.95 / 0.93
5 (two optics sets)	0.525 Å	2.45 / 2.53	0.62 / 1.11	1.57 / 1.62	0.82 / 1.00	1.40 / 1.44	0.85 / 0.97

All 24 v0.3 images: native − float64 at most 1.08 e_C, native − CUDA at most 1.13 e_C, maximum absolute differences at most 0.57 × 10 e_C. At index ±1 the error power fell from 8.1e-6 to 1.5e-7 rad (acquisition 4) and from 9.5e-6 to 4.5e-7 (acquisition 5), 7–17 times below CUDA’s; the remaining error is the high-frequency float32 floor, equal to CUDA’s. Synthetic check (testRedrawResolvesNearlyCancellingSidebands, C10 and C12 balanced along the scan rows, |gamma| about 2e-5 at index 1): v0.2 4.5e-3 rad rms against the double-precision definition, v0.3 3.7e-8.

Redraw time per frame, median of the per-run medians / median p90, v0.2 → v0.3, runs alternating on the final builds (other agents’ GPU work ran between runs):

Scan	Order	Output	Rounds	v0.2	v0.3	Paired difference
512²	fifth	3×	12	30.5 / 31.3	30.8 / 31.4	+0.1 ms
512²	low	3×	8	40.3 (loaded; minimum 29.4)	39.9 (minimum 29.8)	−0.1 ms
256²	fifth	3×	8	10.1 / 10.3	10.3 / 10.7	+0.3 ms
512²	fifth	1×	6	23.9	23.7	+0.1 ms

At 512² the difference is within the run-to-run noise; at 256², where one wave of threadgroups exposes the per-batch collapse from device memory (constant memory before), v0.3 costs about 3 %. The 512² fifth-order 3× drag stays within 33 ms per frame.

Conclusion

The native SSB redraw now meets the 2× rule everywhere: below CUDA’s own error on 20 of 24 images and within 8 % of it on the rest, at the same drag speed at 512² (about 3 % slower at 256²). v0.3 replaces v0.2 with no fallback. The in-app headed drag timing is still to be recorded.