Below 10, GAMMA is a stored product down to a seed on (1,2)

FN-GAMMA-001: Below 10, GAMMA is a stored product down to a seed on (1,2)

Detail for matching Excel's bits · measured · recorded 2026-11-04

With RN(v) meaning v rounded to 64 bits on the x87 unit and then stored to binary64, write x = i + y with GAMMA(1+y) on (1,2). For i from 2 to 9, GAMMA(x) is RN(t*GAMMA(1+y)), where t is the product (x-1)(x-2)...(1+y) taken from the top with each product RN. Each stage rounds, so the answers are not correctly rounded: at the fraction below, GAMMA(3.649...) is 1 unit high and GAMMA(5.649...) 1 unit low. The seed on (1,2) is a chain of stored products that has not been identified, and at x = 10 the route changes (FN-GAMMA-002).

Correct value: mpmath gamma at 800 bits from the binary64 input, correctly rounded.

Correct value: mpmath gamma at 800 bits from the binary64 input, correctly rounded.

Basis: 2,000 of 2,000 fractions at each i from 2 to 9, from Excel's own GAMMA(1+y) (build 20527). The same formula fails at i = 10. Replayed for this note in exact rational arithmetic on 300 of those fractions: 300 of 300 at each i from 2 to 9, 61 of 300 at i = 10. Recorded GAMMA rows match 6,406 of 12,603 with stand-ins for the seed and the Stirling series, the worst by 49 units.

Open: The seed GAMMA(1+y) on (1,2): about five stored smooth quantities, seen as lines in its noise. Cody's rational matches 33% of it. Many closed families were tested and refuted.

Open: x < 1 and negative arguments.