fix: take quality's determinant ratio in log space

`quality()` computed `det(ata) / det(middle)` in linear space. Both are
products of `k - 1` diagonal entries, so they leave f64's range long
before their ratio does — and the ratio is the only thing the answer
needs.

Measured at the crate defaults: 150 groups correct at 8.45e-53, 200
returned 0, 250 returned NaN where the truth is 9.51e-88. With a small
beta it bit far sooner: at sigma = beta = 1e-3, 60 groups returned NaN
against a true 1.32e-9 — a value nine orders of magnitude inside the
normal range. Neither `quality()` nor `History::predict_quality` caps the
group count, unlike `predict_outcome`, so those are supported calls.

`Lu::ln_abs_determinant` accumulates `ln|diagonal|` instead of
multiplying, and the call site becomes `exp(e_arg + 0.5 * ln_ratio)`.

Verified against the closed form `(beta / sqrt(beta^2 + sigma^2))^(k-1)`
rather than against recorded output, across three parameter sets and
group counts to 300: every case now agrees to 1e-11 or better, including
9.88e-324 at 300 groups, which is subnormal.

Also documents the remaining panic: every rating at zero sigma with a
zero beta makes `middle` singular and `inverse()` panics. Documented
rather than converted — nothing is uncertain there, so there is no
distribution to take the quality of.

Closes #59

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ

tests/quality.rs

@@ -164,3 +164,52 @@ fn quality_matches_the_reference_implementation() {
     let refs: Vec<&[Gaussian]> = five.iter().map(Vec::as_slice).collect();
     assert!((quality(&refs, beta) - 0.040).abs() < 1e-9);
 }
+
+/// `quality()` used to compute `det(ata) / det(middle)` in linear space. Both
+/// are products of `k - 1` diagonal entries, so they leave `f64`'s range long
+/// before their ratio does — and the ratio is the only thing the answer needs.
+///
+/// Measured before the fix: at the crate defaults 150 groups was correct, 200
+/// returned `0`, and 250 returned `NaN` where the truth is `9.51e-88`. With a
+/// small beta it bit sooner — `sigma = beta = 1e-3` returned `NaN` at 60 groups
+/// against a true `1.32e-9`, a value that is entirely ordinary.
+///
+/// For `k` single-member groups with equal means the answer has a closed form,
+/// `(beta / sqrt(beta^2 + sigma^2))^(k-1)`, so this checks against arithmetic
+/// rather than against a recorded output.
+#[test]
+fn quality_matches_its_closed_form_past_the_overflow_point() {
+    for (sigma, beta) in [(25.0 / 3.0, 25.0 / 6.0), (1e-3, 1e-3), (50.0, 25.0 / 6.0)] {
+        let rating = vec![Gaussian::from_ms(25.0, sigma)];
+        for k in [2usize, 50, 60, 150, 200, 250, 300] {
+            let groups: Vec<&[Gaussian]> = (0..k).map(|_| rating.as_slice()).collect();
+            let got = quality(&groups, beta);
+            let expected = (beta / (beta * beta + sigma * sigma).sqrt()).powi(k as i32 - 1);
+
+            assert!(
+                got.is_finite(),
+                "sigma {sigma}, beta {beta}, {k} groups: got {got}"
+            );
+            // Subnormal results have no relative precision left to check.
+            if expected > f64::MIN_POSITIVE {
+                let rel = ((got - expected) / expected).abs();
+                assert!(
+                    rel < 1e-11,
+                    "sigma {sigma}, beta {beta}, {k} groups: got {got:e}, \
+                     closed form {expected:e}, rel {rel:e}"
+                );
+            }
+        }
+    }
+}
+
+/// The overflow was in the intermediates, never in the answer: every value
+/// above is an ordinary float. This pins the specific case that returned `NaN`
+/// where the true answer is nine orders of magnitude inside the normal range.
+#[test]
+fn a_small_beta_does_not_overflow_at_sixty_groups() {
+    let rating = vec![Gaussian::from_ms(25.0, 1e-3)];
+    let groups: Vec<&[Gaussian]> = (0..60).map(|_| rating.as_slice()).collect();
+    let got = quality(&groups, 1e-3);
+    assert!((got - 1.317_089e-9).abs() / 1.317_089e-9 < 1e-6, "{got:e}");
+}