CAUCHY Function (Gnumeric)

• Introduced: Gnumeric 1.1.7 release notes record its addition.
• Supported in: Gnumeric 1.12.62; no retirement is established.
• Scope: Desktop Gnumeric with its Statistics 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.

CAUCHY evaluates a zero-location Cauchy density when cumulative is FALSE, and the upper-tail probability when it is TRUE. The shorter function has no location argument and no log-probability option. Scale is a positive width parameter, not a standard deviation. Versioned function help.

=CAUCHY(x,scale,cumulative)

• x: observation or cutoff.
• scale: positive width parameter. The location is fixed at zero.
• cumulative: required. FALSE requests density; TRUE requests the upper tail in this implementation.

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.

At x=1 and scale 1, =CAUCHY(1,1,FALSE()) gives density 1/(2π)≈0.159154943092. =CAUCHY(1,1,TRUE()) instead gives upper probability 0.25. The corresponding lower probability is 0.75; the returned 0.25 must not be treated as that lower-tail probability.

At x=−1 the upper probability is 0.75, and at zero it is 0.5. The decreasing sequence as x increases is the survival-function direction. The ordinary Cauchy model is continuous, so equality at the cutoff does not affect these probabilities.

For scale a>0, density is 1/(PI()*a*(1+(x/a)²)); the upper probability is 0.5−ATAN(x/a)/PI(). The central interval from −a to a contains half the probability. Neither a finite mean nor a finite variance exists for the ordinary Cauchy model.

The executed equality check =CAUCHY(1,1,TRUE())-R.PCAUCHY(1,0,1,FALSE()) returns zero. This makes the direction explicit. When a nonzero location is required, the R-style function provides it directly; do not insert an extra location argument into CAUCHY.

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

The help’s general cumulative description does not specify the actual upper-tail selection clearly. Current implementation passes FALSE as the lower-tail argument to its probability helper. Zero or negative scale gives #NUM! in the tests. Numeric flag 2 also selects the upper branch despite the help’s stricter wording; use TRUE/FALSE for clarity. Nonnumeric text gives #VALUE!.

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 dcauchy calculation
• Current pcauchy calculation
• Nonfinite worksheet-value handling

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, NORMDIST.