ATANH Function in Apple Numbers: Formula, Examples, and Domain Rules

=ATANH(0) returns 0; =ATANH(0.5) is expected to return approximately 0.549306. Undo a hyperbolic tangent on a real input strictly between −1 and +1. It is useful for recovering a hyperbolic parameter from a bounded transformed value.

Important: Real-valued domain: −1 < num < 1. -1, 1, and values outside this open interval do not have a finite real ATANH result. Do not assume an exact app error string without a native Numbers run. These are expected mathematical results, not screenshots or observed outputs from an Apple Numbers installation.

Apple’s current English Formulas and Functions Help 15.4 documents the function below. This guide does not establish the first Numbers release supporting it, or certify exact feature parity on all devices and historical versions. The documentation was accessed 2026-10-11 UTC.

=ATANH(num)

One required numeric argument, num, strictly greater than −1 and strictly less than +1. Neither endpoint is valid. There is no optional parameter or default.

Enter an equals sign first. Apple prints uppercase function names but allows lowercase names when typed. No second argument, separator, or optional setting is defined for this function. Decimal notation in the examples uses a period, as in 0.5. Apple’s general documentation says that for formulas with multiple arguments, a decimal-comma locale uses a semicolon separator; this particular one-argument formula has no argument separator. Change example constants to the decimal convention accepted by your configured locale, and confirm the stored B-column values are numeric. ATANH official syntax; Apple function entry rules.

Real-valued domain: −1 < num < 1. -1, 1, and values outside this open interval do not have a finite real ATANH result. Do not assume an exact app error string without a native Numbers run.

1. Open an editable Numbers sheet and add a regular table (or reuse a blank table). Name it InputData to make references unambiguous.
2. Keep the first row as a header, with Label, Input, and Result in columns A, B and C.
3. Enter the exact sample numbers from the table below into B2 downward. Format B and C as Number and show at least six decimal places in C before comparing.
4. Enter =ATANH(B2) into C2. Confirm it references the input number (not its text label).
5. Copy the C2 formula down through the last valid test row. Use the invalid input only as a controlled domain troubleshooting exercise, not as a successful-output test.
6. Compare each cell with the expected numbers below. Last-digit differences can arise from finite precision; do not compare rounded display text as exact mathematical equality.

Row: 2

A: Label
negative high magnitude

B: Input
-0.95

C: Formula
=ATANH(B2)

C: Expected output (6 decimals)
-1.831781

Row: 3

A: Label
negative half

B: Input
-0.5

C: Formula
=ATANH(B3)

C: Expected output (6 decimals)
-0.549306

Row: 4

A: Label
negative tenth

B: Input
-0.1

C: Formula
=ATANH(B4)

C: Expected output (6 decimals)
-0.100335

Row: 5

A: Label
zero

B: Input
0

C: Formula
=ATANH(B5)

C: Expected output (6 decimals)
0.000000

Row: 6

A: Label
positive tenth

B: Input
0.1

C: Formula
=ATANH(B6)

C: Expected output (6 decimals)
0.100335

Row: 7

A: Label
positive half

B: Input
0.5

C: Formula
=ATANH(B7)

C: Expected output (6 decimals)
0.549306

Row: 8

A: Label
positive high magnitude

B: Input
0.95

C: Formula
=ATANH(B8)

C: Expected output (6 decimals)
1.831781

Row: 9

A: Label
invalid upper boundary

B: Input
1

C: Formula
Do not apply: outside real domain

C: Expected output (6 decimals)
No finite real result; actual Numbers error not observed

The figures above were calculated independently with Python’s math.atanh function for the same real inputs. They are predictions, not observations in Numbers. The body values can be entered as constants without any external connection or data import.

Reading a value from another Numbers table

If InputData is a different table on the same sheet, use =ATANH(InputData::B2) rather than a bare B2 where ambiguity could arise. If it is on a different sheet, Apple documents the pattern SheetName::TableName::Cell, for example =ATANH(Sheet 2::InputData::B2). Use the formula editor’s reference selection to avoid mistakes with spaces and headers. Apple reference syntax.

Invert a bounded score without treating it as a circular angle

Suppose a modeling step stores TANH(t)=0.5. =ATANH(0.5) recovers an expected real parameter near 0.549306. The output is not a compass bearing or angle in degrees. The bound is fundamental: the input must lie strictly between −1 and +1.

Evaluate near-boundary sensitivity

ATANH(0.95) is expected to be about 1.831780; ATANH(0.99) about 2.646652. Small input changes near either endpoint can produce large output changes. Compare these values with a number in the middle, such as ATANH(0.1) ≈ 0.100335. This can amplify measurement and rounding noise; do not silently treat stored 1 or −1 as safe inputs.

Guard the strict domain explicitly

=IF(ABS(B2)>=1,"Outside ATANH domain",ATANH(B2)) displays a readable warning for numerical out-of-range values. The guard is a new illustrative formula, not an observation from a Numbers run. It does not establish what Numbers does when B2 contains text or an error; validate input types separately.

Check the inverse and the logarithmic definition

For valid x, =TANH(ATANH(x)) should approximately reproduce x. For real finite t, =ATANH(TANH(t)) approximately reproduces t while sufficient floating-point precision remains. Algebraically, atanh(x)=0.5*ln((1+x)/(1-x)); a Numbers worksheet expression is =LN((1+B2)/(1-B2))/2 for −1 < B2 < 1. The longer form is a mathematical cross-check, not more stable at the boundaries.

Numbers’ exact displayed error wording, numerical underflow/overflow thresholds, and any implicit conversion of nonnumeric cells have not been tested here. Do not infer those implementation details from Python or from another spreadsheet product. The real-domain restrictions in this guide come from Apple’s official function documentation.

Excel documents ATANH(number) with the exclusive interval −1 to 1; Google Sheets documents ATANH(value) with the same open interval. Do not assume the software products share identical error messages or imported text coercion.

• Numbers: ATANH (Apple Formulas and Functions Help 15.4). One argument called num.
• Excel: ATANH (Microsoft’s function reference). Numeric argument is called number.
• Google Sheets: ATANH (Google Docs Editors Help). Numeric argument is called value.

As a portability check, a numeric constant and one argument will avoid table-reference differences. Numbers uses InputData::B2 to disambiguate cross-table references; Excel or Sheets worksheet/cell references use different conventions. No cross-product application executions have been performed for this article. Excel, Sheets and Numbers have their own locale and calculation settings; compare reasonable numeric tolerances, not formatting alone.

ATAN; TANH. These references explain the related mathematical operations in Apple’s documentation.

For syntax details, see Apple’s functions overview, cell-reference formats and argument value types. All linked Apple function pages above were accessible in the English 15.4 documentation during the documentation check. No native Numbers screenshots are included; the guide does not use mock application images.

Why is ATANH(1) not zero or infinity in an ordinary cell? Its real mathematical limit grows without bound; the function does not have a finite value at 1. The precise app error text requires native testing.

How is it different from ATAN? ATAN is inverse ordinary tangent and accepts any real input; ATANH is inverse hyperbolic tangent and requires −1<x<1.

Can I use degrees? The result is not an ordinary geometric angle. Converting with DEGREES generally mislabels its mathematical meaning.

Official documentation: ATANH; Apple Functions Overview; Apple Cell References; Apple Argument Types.

• ACOSH — continue with a related Apple Numbers calculation.
• ASINH — continue with a related Apple Numbers calculation.
• COSH — continue with a related Apple Numbers calculation.
• SINH — continue with a related Apple Numbers calculation.
• TANH — continue with a related Apple Numbers calculation.

For a software-specific comparison: LibreOffice Calc ATANH, Apache OpenOffice Calc ATANH, Gnumeric ATANH. A shared name does not establish identical argument, error or version behaviour.

The linked official function documentation was reviewed on October 12, 2026. Apple’s Formulas and Functions Help 15.4 identifies the documentation edition, not a minimum Numbers release or a guarantee of historical platform parity. Selected worked examples were checked using independent mathematical or text-processing models; results are expected values, not observations from the spreadsheet application. Native Apple Numbers execution was not performed. Undocumented coercion, exact error messages and platform-specific behaviour remain unverified.

View reproducible example inputs and independent check results. This record is not a native spreadsheet test log.