GAMMA.DIST's prefactor carries GAMMALN(a+1) in the exponent, with its error

FN-GAMMA.DIST-001: GAMMA.DIST's prefactor carries GAMMALN(a+1) in the exponent, with its error

Excel differs from the mathematical answer · inferred · recorded 2026-11-04

With RN(v) meaning v rounded to 64 bits on the x87 unit and then stored to binary64, the series arm and the density of GAMMA.DIST at integer shape a from 2 to 6 share r = RN(exp(t - x - GAMMALN(a+1))), with ln x stored, t = RN(a*ln x), and the difference held in the 64-bit registers. The cumulative answer is RN(r*S) with S the stored forward sum, and the density RN(r*a)/x, the division last; with a scale, y = RN(x/beta) comes first. So GAMMALN's own error is in every answer: at shape 2 the constant is GAMMALN(3), one unit above stored ln 2 (FN-GAMMALN-001).

Correct value: mpmath regularized lower incomplete gamma at 800 bits, correctly rounded.

Correct value: mpmath regularized lower incomplete gamma at 800 bits, correctly rounded.

Basis: Noise lines on windows of consecutive doubles (Excel build 20527) put ln x stored and multiplied by a, not a power of x. Storing t - x adds a slow line of amplitude 5, and leaving it unstored takes the noise at small x from 13 units of 2^-53 to 1.5. A scan of one constant factor per shape peaks at -0.875, +0.25, -1.25, +1.625 and -2 of 2^-53 for a = 2 to 6, and the first is the error of GAMMALN(3). This form matches 15,364 of 26,776 rows at a = 2, 15,588 of 26,770 at a = 3, 1,088 of 2,051 at a = 4, 654 of 1,142 at a = 5, 691 of 1,151 at a = 6, and over all shapes 47,647 of 77,975 recorded rows.

Inferred: The form is read from noise and scored in part: one stored stage of the prefactor or the sum is still missing, with a root mean square of about 0.8 of 2^-53 at small x. Non-integer shapes depend on GAMMALN below 3, which is not identified. An earlier reading of a chopped exponential in this function was this constant.

Open: The missing stored stage of the prefactor or the sum.

Open: Shapes outside 2 to 6, and the upper arm of the distribution.