`v_w` returned `w` and let `trunc` form `1 - w`. `w` tends to 1 out in
the tail, so that subtraction lost about log10(alpha^2) digits — and the
quantity it was destroying is perfectly representable.
Two separate cancellations, fixed separately.
The non-tie half: `half_line_truncation` now returns `1 - w` computed
symbolically rather than as `1 - v*gap`. With `alpha*gap = 1 - inv^2*b`
the leading ones cancel on paper instead of in floating point. Measured
against the exact truncated variance:
alpha before after
1e6 8.9e-5 rel 0.0 rel (exact)
1e8 returns 0.0 0.0 rel (exact)
At 1e8 the old form gave `sigma_trunc = 0`, and `from_ms(mu, 0.0)` is a
point mass whose `mu()` is inf/inf = NaN. `beta(1e-8).sigma(1e-8)` with
priors 1000 apart went from Err + NaN skills to a finite fit.
The tie half is a different subtraction — `w = v^2 - u`, where both grow
as alpha^2 while their difference stays O(1). The existing escape hatch
could not cover it: it keys on `alpha * width >= HALF_LINE_WINDOW`, how
many window-widths from the mean the window sits, and a NARROW window
fails that however deep it is. Measured at alpha 1e6 with a 1e-6 window
it kept four digits and returned `1 - w = -2.4e-4` where the truth is
+2.8e-13. One step earlier it was quietly wrong instead: `1 - w = 1.0`
exactly, a truncation reported as a no-op, where the truth was 5e-17.
Over a narrow window the density is a truncated exponential in
`s = (x - alpha)/width`, whose mean and variance are closed forms, so
`v = alpha + width*m(t)` and `1 - w = width^2 * V(t)` with no large
subtraction at all. Validated against high-precision quadrature: v exact
to 4e-10, `1 - w` to 4e-10 across the region it is used in.
The crossover is on `alpha / width` rather than on either alone, because
that ratio is what says how many digits the subtraction has left — and
the approximation is most accurate exactly where the subtraction is
worst, since both improve as the window narrows.
Defaults are bit-identical (pi 0.02398318151216503 before and after).
Tests: the three reproductions from the issue, the narrow-window form
against pinned quadrature values, and a continuity sweep across all three
tie branches — a misplaced crossover is the real risk here, and a jump at
a boundary is visible even without pinning absolute values.
Closes#60
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
logaritmiskandClaude Opus 5
6139061740