GAMMALN.PRECISE

What it does

GAMMALN.PRECISE(x)

GAMMALN.PRECISE(x) returns the natural logarithm of the gamma function, ln Γ(x), for x > 0.

The mathematics is set out in full on the GAMMALN page — the integral definition, the recurrence ln Γ(x+1) = ln x + ln Γ(x), strict convexity on (0, ∞), the two zeros at x = 1 and x = 2, the single minimum near x ≈ 1.4616, and Stirling's asymptotic series — and is not repeated here. Microsoft's description of this surface is word-for-word the description of the legacy one: "Returns the natural logarithm of the gamma function, Γ(x)."

The interesting content of this page is not the mathematics. It is the pair relationship, and what the word "PRECISE" in the name does and does not promise.

Arguments

Try it

Quirks

The pair, stated precisely

The .PRECISE suffix belongs to the Excel 2010 renaming of the statistical function set, and in the general case the Handbook's rule holds: a modern dotted name is not guaranteed to be the same computation as its legacy counterpart, and proving identity requires evidence. For this particular pair, unusually, some of that evidence exists — upstream's, not the Handbook's — and it is worth laying out with each claim's warrant attached.

1. Documented supersession. Microsoft's GammaLn page carries the replacement banner: the function "has been replaced with one or more new functions that may provide improved accuracy and whose names better reflect their usage", and it names GammaLn_Precise. Note the modal verb — may provide improved accuracy. The banner is boilerplate carried by every function in that 2010 renaming, and it promises nothing measurable.
2. Observed identity, upstream, on one build. EV-MATH-0014 records an Excel-versus-Excel check in which both surfaces resolved to the same implementation and published the same results at every probed point. The record is explicit that this is an identity check and not a pass rate, and that the source publishes no numerator or denominator for it.
3. The direction of inheritance runs toward the legacy name, not away from it. EV-MATH-0013 records that both of the family's held-out tests were captured through this spelling, so a figure measured on GAMMALN.PRECISE transfers to GAMMALN by that argument and not the reverse. That is the opposite of the usual assumption, in which the older, better-studied surface is the measured one.

Result and edge cases

Returns Number. The edge-case inventory is the one on the GAMMALN page: exact zeros at x = 1 and x = 2, the −ln x behaviour as x → 0⁺, Stirling growth at large x, and #NUM! for x ≤ 0.

Errors

Microsoft's documentation for this surface is worded differently from the legacy one, and the difference is worth noting rather than smoothing over:

The modern page does not name the error values; the legacy page does. The two pages describe the same two conditions, but only one of them tells you what lands in the cell. A reader writing an IFERROR/IFNA guard needs the code, and the documentation for the recommended surface is the one that omits it. Recorded as a documentation gap.

How to implement

Numerical notes

The algorithmic content — the three-region split (recurrence for small x, minimax rational in the middle, Stirling above a seam), the unbounded relative error in the two narrow bands around the zeros, and the identified structure with its threshold and seam — is set out on the GAMMALN page.

Two points belong specifically here.

"Precise" is a name, not a measurement. Nothing in Microsoft's documentation quantifies the improvement the 2010 renaming promised, and the banner's own wording is "may provide improved accuracy". Where a .PRECISE surface has been checked against its legacy counterpart in this domain, the result so far has been identity rather than improvement. A reader should not choose this spelling expecting better answers; they should choose it because it is the current name.

Rust code

60 gammaln_precise "GAMMALN.PRECISE" (1, 1) [V "x"] |s, a, sv| {
    out(s, sd::gammaln_precise_kernel(num_with_subnormal(s, a, 0)?))
}

pub fn gammaln_precise_kernel(x: f64) -> Result<f64, WorksheetErrorCode> {
    gammaln_kernel(x)
}

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

Relationships

• GAMMALN — the legacy spelling; the full pair analysis is above, and the numerical treatment lives on that page.
• GAMMA — the value rather than its logarithm, in the same module and not on the same numeric path: upstream's landed log-gamma kernel is wired only to the two GAMMALN surfaces. LN(GAMMA(x)) and GAMMALN.PRECISE(x) are different computations with different domains, and the first overflows where the second does not.
• GAMMA.DIST, BETA.DIST, POISSON.DIST, COMBIN, MULTINOMIAL — consumers of log-gamma, directly or through their normalising constants.
• ERF.PRECISE / ERFC.PRECISE — module siblings that carry the same .PRECISE suffix from the same renaming. Whether those pairs are identical computations is a separate question with its own record, and nothing about this pair transfers to them.

Named variants

No named variants are recorded yet.