R.PCAUCHY Function (Gnumeric)

• Introduced: First release not established; present in the retained 1.12.57 source.
• Supported in: Gnumeric 1.12.62; no retirement is established.
• Scope: Desktop Gnumeric with its Statistical Functions From R plugin. English comma-separated formulas are shown.
• Checked: October 14, 2026. Tables and the named advanced check use 1.12.57 ssconvert. Current 1.12.62 source was inspected; that release was not executed.

R.PCAUCHY returns a Cauchy lower-tail probability by default. lower_tail=FALSE requests the upper tail, and log_p=TRUE requests natural log probability. Location is the distribution’s center and median; scale is its positive width parameter. Versioned function help.

=R.PCAUCHY(x,location,scale[,lower_tail[,log_p]])

• x, location, scale: required numeric inputs. Scale must be positive.
• lower_tail: optional, TRUE by default.
• log_p: optional, FALSE by default; supply lower_tail first.

Brackets mark optional arguments; do not type them. Current worksheet implementation.

These self-contained formulas were recalculated natively and checked against independent mathematics. Results are rounded to twelve decimal places, or thirteen significant digits for small nonzero values. This is presentation precision, not a universal accuracy guarantee.

For location 0 and scale 1, the lower probabilities at x=−1,0,1 are 0.25,0.5,0.75. At x=1, the upper probability is 0.25. The executed lower-plus-upper check returns 1.

For location 10 and scale 2, the cutoff 12 is one scale unit above the center and again has lower probability 0.75. The central interval from location−scale to location+scale contains half the distribution’s probability.

The lower probability is 0.5+ATAN((x−location)/scale)/PI(). The upper probability has the minus sign. Far into a positive tail, requesting the upper tail directly avoids subtracting a nearly-one lower result from 1.

At x=10,000,000,000 with location 0 and scale 1, the native upper output is approximately 3.183098861838E−11. The equivalent expression atan(scale/(x−location))/π explains that small positive tail. Cauchy tails decrease much more slowly than normal tails; location and scale do not create a finite mean or variance.

The saved advanced-check result for =R.PCAUCHY(1,0,1)+R.PCAUCHY(1,0,1,FALSE()) was 1.

Zero or negative scale gives #NUM!. Numeric flag 2 acts as TRUE in the tested calls, but TRUE/FALSE is clearer. Nonnumeric text gives #VALUE!. Gnumeric’s shorter CAUCHY function uses an upper tail when its cumulative flag is TRUE; do not assume its default orientation matches this function.

F5 is genuinely empty, rather than a cell containing an empty string. #NUM! can represent an unrepresentable infinite mathematical endpoint as well as an invalid numeric domain; the distinctions above matter.

• Current worksheet callback
• Tested-version worksheet callback
• Current pcauchy calculation
• Nonfinite worksheet-value handling
• Statistical Functions From R plugin declaration

The relevant worksheet callback is unchanged between the inspected versions. That does not establish later-version numerical execution. Both stale and uncached SUM controls recalculated correctly. Native XML identities were checked; secondary ODF exports retain these functions with the ORG.GNUMERIC namespace. GUI entry, other locales and other spreadsheet products were not tested.

ATAN, PI, LN.