fix: compute binomial and multinomial coefficients exactly#413
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agene0001 wants to merge 1 commit into
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fix: compute binomial and multinomial coefficients exactly#413agene0001 wants to merge 1 commit into
agene0001 wants to merge 1 commit into
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Both were computed as `floor(0.5 + exp(ln n! - ln k! - ln (n-k)!))`, which
returns the wrong integer well before f64 runs out of precision:
C(50, 25) 126410606437750 exact 126410606437752 (off by 2)
C(60, 30) ...863664 exact ...861424 (off by 2240)
C(67, 33) ...170752 exact ...288370 (off by 116736, 57 ulp)
`C(50, 25)` is exactly representable in f64, so this is not a representation
limit.
Uses the recurrence `C(n, i+1) = C(n, i) * (n - i) / (i + 1)` in `u128`. The
division is exact at every step because `C(n, i+1)` is an integer, so the whole
computation stays in integer arithmetic; `u128 -> f64` is correctly rounded, so
the result is the nearest double even past 2^53. Only coefficients that overflow
`u128` fall back to logs.
`checked_multinomial` gets the same treatment via
`n! / (n1! ... nk!) == prod_i C(s_i, n_i)` with `s_i` the running prefix sums,
sharing the same kernel.
Tested by brute force: every `C(n, k)` for `n <= 170` round-trips exactly
against a `u128` reference, and every two-part multinomial equals the
corresponding binomial.
Cost: the exact path for `C(67, 33)` is 190 ns against ~25 ns before; small-k
cases are unaffected (`C(1000, 3)` is 8 ns).
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binomialandchecked_multinomialcomputedfloor(0.5 + exp(ln n! − ln k! − ln (n−k)!)), which returns the wrong integer well before f64 runs out of precision:C(50, 25)is exactly representable in f64, so this is not a representation limit — the exp/ln round trip just isn't integer-accurate.Uses
C(n, i+1) = C(n, i)·(n−i)/(i+1)inu128; the division is exact at every step (the intermediate is itself a binomial coefficient), so the whole computation stays in integer arithmetic, andu128 → f64is correctly rounded, so results past 2^53 are the nearest double. Coefficients overflowingu128fall back to the log form.checked_multinomialshares the kernel vian!/(n₁!…nₖ!) = Π C(sᵢ, nᵢ)over prefix sums.Tested by brute force: every
C(n, k)for n ≤ 170 round-trips exactly against a u128 reference, every two-part multinomial equals the corresponding binomial, and the fallback agrees withln_binomialto 1e-11.Cost: the worst exact case
C(67, 33)is 190 ns vs ~25 ns before; small-k cases are unaffected (C(1000, 3): 8 ns).🤖 Generated with Claude Code