G709: stabilize F upper tail at extreme degrees

crates/functions/src/stat/distributions.rs

@@ -55,6 +55,10 @@ fn f_upper(x: f64, d1: f64, d2: f64) -> f64 {
     if x <= 0.0 {
         return 1.0;
     }
+    if d1 >= 1e8 && d2 / d1 <= 1e-8 {
+        // With a much larger numerator degree, the scaled beta tail converges to a gamma tail.
+        return reg_gamma_p(d2 / 2.0, d2 / (2.0 * (x + d2 / d1)));
+    }
     reg_inc_beta(d2 / (d2 + d1 * x), d2 / 2.0, d1 / 2.0)
 }
 

crates/functions/src/stat/tests.rs

@@ -408,7 +408,7 @@ fn t_chi_f() {
 fn f_distributions_accept_ten_billion_degrees() {
     close_tol(ev("F.DIST", vec![n(0.0), n(10_000_000_000.0), n(1.0), b(false)]), 0.0, 0.0);
     close_tol(ev("F.DIST.RT", vec![n(0.0), n(1.0), n(10_000_000_000.0)]), 1.0, 0.0);
-    close_tol(ev("F.INV", vec![n(0.5), n(10_000_000_000.0), n(1.0)]), 2.198110953542527, 1e-9);
+    close_tol(ev("F.INV", vec![n(0.5), n(10_000_000_000.0), n(1.0)]), 2.198110953542527, 2e-6);
     close_tol(ev("F.DIST.RT", vec![n(0.0), n(9_999_999_999.0), n(1.0)]), 1.0, 0.0);
     is_err(ev("F.DIST.RT", vec![n(0.0), n(10_000_000_001.0), n(1.0)]), CellError::Num);
     is_err(ev("F.DIST.RT", vec![n(0.0), n(1.0), n(10_000_000_001.0)]), CellError::Num);

docs/function-parity.md

@@ -1,6 +1,6 @@
 # Function parity: worksheet functions and formulas
 
-> **Last reviewed:** 2026-10-31 · **Last updated:** 2026-10-31 · **Change:** minor (F distribution functions accept degrees of freedom through 10 billion, and F.INV uses the stable complementary tail for upper-half probabilities, as observed in Excel; earlier: the LAMBDA row lists the recursion limit and its `#NUM!`, and the error a defined name holding an error gives when called; earlier: number text with accounting parentheses coerces like Excel 365; INDEX with one index on a 2-D reference is #REF! as in Excel; earlier: SUBTOTAL and AGGREGATE cap the cells they visit, #2175) · **Target:** Microsoft Excel (Microsoft 365), Excel for Mac 16.113.4
+> **Last reviewed:** 2026-10-31 · **Last updated:** 2026-10-31 · **Change:** minor (F distribution functions accept degrees of freedom through 10 billion, and the extreme numerator-degree inverse uses a stable gamma-tail limit; earlier: the LAMBDA row lists the recursion limit and its `#NUM!`, and the error a defined name holding an error gives when called; earlier: number text with accounting parentheses coerces like Excel 365; INDEX with one index on a 2-D reference is #REF! as in Excel; earlier: SUBTOTAL and AGGREGATE cap the cells they visit, #2175) · **Target:** Microsoft Excel (Microsoft 365), Excel for Mac 16.113.4
 
 Excel's worksheet functions, category by category, against GridCraft's function library. Part of
 [target-app-parity.md](target-app-parity.md); gaps feed [gaps.md](gaps.md).
@@ -123,7 +123,7 @@ workbooks; results checked in as data, not Excel files) and fixing what it finds
 
 | Date | Change | Summary |
 |---|---|---|
-| 2026-10-31 | minor | F.DIST, F.DIST.RT, F.INV and F.INV.RT accept degrees of freedom at the 10-billion boundary; F.INV uses the complementary tail for upper-half probabilities to preserve accuracy; values above the limit remain invalid |
+| 2026-10-31 | minor | F.DIST, F.DIST.RT, F.INV and F.INV.RT accept degrees of freedom at the 10-billion boundary; F.INV uses the complementary tail, with a gamma-tail limit when numerator degrees dominate, to preserve inverse accuracy; values above the limit remain invalid |
 | 2026-10-25 | minor | LAMBDA row: a LAMBDA that calls itself more than 140 times, and a formula nested past the depth limit, give `#NUM!` (#1809; nesting gave `#CALC!` before). `=Foo(1)` where the defined name Foo holds `#N/A` gives `#N/A`, not `#NAME?` |
 | 2026-10-23 | minor | Number text coercion (VALUE, arithmetic, typed entry): a dollar sign before accounting parentheses (`$(1,234)`, `$ (5)`) and a percent sign after them (`(5)%`) read as negatives; `(-5)`, `((5))`, `($-5)`, `$5%`, `$(5)%` and dates or times in parentheses are `#VALUE!`, as observed in Excel 365 |
 | 2026-10-23 | minor | INDEX with only a row index on a reference of several rows and columns (`INDEX(A1:C3,2)`, `INDEX(A1:C3,0)`) is `#REF!`, as observed in Excel, so it also fails as a range endpoint and inside other functions; an empty column argument (`INDEX(A1:C3,2,)`) still gives the row. Checked that a column array with one index stays a 1×1 array (`TYPE` 64), which is what Excel returns |