ACOTH Function (Gnumeric)

ACOTH is Gnumeric’s inverse hyperbolic cotangent. Enter =ACOTH(2) for a mathematically expected result of approximately 0.5493061443. For real finite results, the mathematical input domain is x < −1 or x > 1. Unlike the ordinary inverse cotangent ACOT, this is a hyperbolic function: its input is a ratio/value, not an angle in degrees. The official Gnumeric reference calls it the inverse hyperbolic cotangent and documents exactly one numeric argument. Official ACOTH synopsis.

=ACOTH(x)

Gnumeric’s short manual entry does not specify the exact display error at the boundaries, text coercion, rounding or the numeric limits of its implementation. Do not import another application’s error codes as if they had been tested in Gnumeric. The relevant mathematical identity, for real |x|>1, is ACOTH(x) = ATANH(1/x) = 0.5*LN((x+1)/(x-1)). This identity is an independent mathematical derivation, not a record of native evaluation. Gnumeric ACOTH · ATANH.

Type each numeric value in A2:A6. Put the formula shown into the corresponding B cell (or enter =ACOTH(A2) in B2 and fill down). Format column B to display at least nine decimals.

These outputs were computed using two independent formulations in Python: atanh(1/x) and 0.5*log((x+1)/(x-1)); the model values agreed within floating-point tolerance. This does not verify how an installed Gnumeric build formats or evaluates them.

Why negative values matter: the real-valued inverse hyperbolic cotangent is an odd function, so ACOTH(-2) mathematically equals -ACOTH(2). Using both signs helps expose mistakes in formulas or transcribed inputs.

Suppose a simplified dimensionless signal model is r = COTH(d/L) where d is a distance and L is a positive reference length. Solving mathematically for distance gives d = L*ACOTH(r) for r > 1. This is a deliberately synthetic example, not advice for engineering measurements.

Expected model result for C10: 1.694595721 metres. You can independently check the number using =B10*ATANH(1/A10) or the logarithmic identity above. A value r=1 is a singular limiting boundary: do not treat it as a finite distance.

To reject out-of-domain ratios before trying the inverse, an illustrative guard is:

=IF(ABS(A10)<=1,"Ratio must have magnitude > 1",B10*ACOTH(A10))

The English argument separators in this two-branch IF example may change with locale, and the wrapper has not been executed in Gnumeric. Guarding numeric data types and positive reference length is additional validation for a real workbook.

• |x|<1: ACOTH has no finite real mathematical answer there; do not show a fabricated Gnumeric error token. The official ACOTH snippet does not specify its error code.
• x=1 or x=-1: logarithmic divergence; there is no finite real result. Record the native behavior only after testing the exact version and locale.
• x close to ±1: the magnitude grows quickly and roundoff sensitivity increases. Keep intermediate numeric precision; compare with a tolerance.
• Text, blank and boolean inputs: the official argument type is number; implicit coercion and exact cell error output need native testing. Use explicit numeric values.
• Large |x|: the inverse tends toward zero; displayed zeros can simply reflect formatting rather than an exactly zero result.

Microsoft’s current Excel reference includes ACOTH(number) and specifies a real domain with absolute value greater than one. This supports the existence of a comparable function in Excel, not identical coercion, all error codes, numerical precision or old-release availability in Gnumeric. Microsoft ACOTH. The Gnumeric public catalogue marks ACOTH Complete / No Testsuite—a catalogue classification, not evidence of independent runtime coverage. Gnumeric catalogue.

COTH, ATANH, TANH, LN, IF.

The official Gnumeric reference establishes the function’s purpose and argument structure. The arguments and worked examples were editorially reviewed and the stated expected results were independently calculated. The tagged Gnumeric 1.12.62 numerical implementation was additionally checked for the hyperbolic-cotangent and inverse-hyperbolic-cotangent identities.

Review date: 2026-10-12. No native application run: the worked outputs are documentation-based or independently calculated expectations, not measurements from installed Gnumeric. Ordinary numeric cases and the stated text-construction examples are checked; blank/text coercion, machine extremes and workbook round trips are not certified. The formula editor in your installed version is the authority for localized separators.

Example-check record · Editorial standards.