[Sad Catapella N1][Numerical correctness] Float histogram binning loses finite near-zero values due to scale overflow/underflow #27

Scope / classification

Normalization Batch 1 at f784c95e3049483f87466150ffc5764375fa631d. Confirmed floating-point arithmetic edge case by source inspection and isolated formula probe, not a package benchmark or test run.

Evidence

• _float_bins, lines 657–679 computes half_range = hi*0.5 - lo*0.5, then (n / half_range) and multiplies it by (value*0.5 - lo*0.5) before clipping. When hi-lo is a tiny but nonzero finite float64 range, the first factor can become inf or half_range can underflow to zero, despite hi != lo.
• _bin_index, lines 850–863 repeats this unstable form for range-check statistics. PlaneHistogram.representatives, lines 159–172 and max_representative_error can also collapse subnormal half widths.
• Existing large-magnitude float test covers [-1e308,1e308], not very small finite ranges.

Minimal counterexample

An array np.array([0., 1e-308], dtype=np.float64) with 100 bins has half_range=5e-309; the ratio 100 / half_range overflows to infinity, so the bin-index computation can yield NaN/infinity and cast to a sentinel integer before clipping. The two distinct finite values can therefore collapse to bin 0 instead of first and last. For np.nextafter(0.,1.), halving the difference becomes zero. Isolated NumPy arithmetic confirmed this behavior.

Impact

Incorrect histogram counts/range checks for valid subnormal or near-zero floating data without an explicit refusal. Rare in normal raw microscopy but plausible in transformed scientific arrays; the library advertises float64 support without an exclusion.

Fix / acceptance

Develop a scale-safe normalized coordinate calculation that avoids both overflow of n/range and underflow of intermediate halves, retaining established bin boundary/rounding conventions for normal values. Do not add an arbitrary epsilon or coerce small numbers to zero. Compare with an exact/high-precision independent oracle for subnormals, 1e-300, mixed signs, extreme opposite signs, constant planes, and robust-mode coincident limits. Ensure min/max, counts, quantiles, warnings, error bounds and block-vs-whole agreement. Benchmark the stable method before replacing hot loops.