TANH Function in Apple Numbers: Formula, Examples, and Saturation

=TANH(0) has an expected result of 0; =TANH(1) has an expected result of about 0.761594. TANH calculates hyperbolic tangent, a smooth transformation that mathematically maps any finite real input to a value strictly between −1 and 1.

Use TANH for models that explicitly require hyperbolic tangent, such as a smoothly bounded transformation of a signed numerical score. It is not the ordinary tangent of an angle: TAN and TANH follow different mathematical rules. The figures in this guide are expected results independently calculated from mathematics and cross-checked with Apple’s examples. They are not observations from Numbers running on a device.

Apple’s English Formulas and Functions Help 15.4, checked 2026-10-11 UTC, documents the following exact expression:

=TANH(num)

• num — required: a number value, as Apple labels the argument. It is the input to the hyperbolic tangent function. No optional second argument, fallback, or default is documented.
• Argument order: the sole argument goes inside the parentheses; for a cell use =TANH(B2). Enter an initial equals sign in a Numbers formula cell. Formula names are not case-sensitive according to Apple’s overview.
• Regional notation: these examples assume an English-style decimal point (0.5, -0.5). In decimal-comma regions enter numbers using the locale-compatible numeric syntax. Apple’s semicolon rule applies when separating multiple arguments, not to this one-argument function. In another function such as IF, commas in English may become semicolons when the locale uses decimal commas.
• Platform/version qualification: 15.4 is the documentation edition, not a verified first-supported Numbers release. This single-function reference does not establish parity across every Mac, iPhone, iPad, iCloud, or earlier Numbers edition; none of those applications was run during preparation.

Official claims: Apple TANH — syntax, numeric parameter, and examples; Apple functions overview — formula entry and locale separators.

1. Create a blank Numbers document and insert a regular table named InputData (or rename a blank table to InputData). Keep row 1 as headers.
2. Enter Label in A1, Input in B1, and Result in C1. Type the numeric B-column values below; do not copy the descriptions into B.
3. Format column B as Number and column C as Number with at least six decimal places. Cell formatting changes the display, not the underlying stored precision.
4. Type =TANH(B2) in C2. Fill the formula down to C10. Check that copied formulas use B3, B4, and so on, rather than B2 repeatedly.
5. Compare the expected six-decimal figures with your cells. The table below was calculated independently using Python math.tanh and is not a record of Numbers execution. If a displayed value disagrees, confirm whether it is a real numeric input and that the correct function was selected.

Row: 2

A: Label
negative five

B: Numeric input
-5

C: Formula
=TANH(B2)

C: Expected six-decimal output
-0.999909

Row: 3

A: Label
negative two

B: Numeric input
-2

C: Formula
=TANH(B3)

C: Expected six-decimal output
-0.964028

Row: 4

A: Label
negative one

B: Numeric input
-1

C: Formula
=TANH(B4)

C: Expected six-decimal output
-0.761594

Row: 5

A: Label
negative half

B: Numeric input
-0.5

C: Formula
=TANH(B5)

C: Expected six-decimal output
-0.462117

Row: 6

A: Label
zero

B: Numeric input
0

C: Formula
=TANH(B6)

C: Expected six-decimal output
0.000000

Row: 7

A: Label
positive half

B: Numeric input
0.5

C: Formula
=TANH(B7)

C: Expected six-decimal output
0.462117

Row: 8

A: Label
one

B: Numeric input
1

C: Formula
=TANH(B8)

C: Expected six-decimal output
0.761594

Row: 9

A: Label
two

B: Numeric input
2

C: Formula
=TANH(B9)

C: Expected six-decimal output
0.964028

Row: 10

A: Label
five

B: Numeric input
5

C: Formula
=TANH(B10)

C: Expected six-decimal output
0.999909

Refer to another table or sheet

If C2 belongs to a different Numbers table from the inputs, an explicit table reference can help: =TANH(InputData::B2). Apple’s documentation also describes sheet-qualified references in the form Sheet Name::Table Name::B2 when needed. Use the formula editor and click the intended cell instead of manually guessing tokenization for names with spaces. A same-table B2 reference generally needs no table qualifier. Apple’s cell-reference guide provides the syntax and copy/paste rules.

The output approaches ±1 without mathematically reaching it

For real finite x, −1 < tanh(x) < 1. The function is odd, so TANH(-x)=-TANH(x). For an input of 5, the expected result is about 0.999909; for −5, about −0.999909. For 20, Python’s independent mathematical check gives about 1.000000 when displayed to six decimals. That rounded display does not establish an exact mathematical result of 1 or an exact Numbers cell value; implementation roundoff may affect large inputs.

Smoothly bound a numeric score

Suppose B2 contains a centred score. =TANH(B2) produces a signed output constrained near −1 and 1. To rescale the theoretical result into (0,1), use =(TANH(B2)+1)/2. At a score of 0 the expected result is 0.500000; at 1 it is approximately 0.880797; at −1 it is about 0.119203. Choose this transformation only if such nonlinear compression makes sense for the metric. Its output is not a calibrated probability without an additional justified model.

Check the relationship to SINH and COSH

The mathematical identity tanh(x)=sinh(x)/cosh(x) means =SINH(B2)/COSH(B2) should agree approximately with =TANH(B2) at moderate inputs. With B2=1, both have an expected value around 0.761594. Dividing separately computed hyperbolic functions can overflow for extreme inputs even though the mathematical ratio remains bounded; prefer TANH rather than the expanded expression for spreadsheet work.

Reverse with ATANH cautiously

Over its open domain (−1,1), the inverse hyperbolic tangent is ATANH. Thus =ATANH(TANH(0.5)) is mathematically 0.5. When the TANH output is extremely close to ±1, floating-point rounding may make this reverse operation lose precision or fail. It does not justify assuming that Numbers returns a safely invertible value for arbitrarily large finite inputs.

Decide whether the input is a number before diagnosing a math error

The Apple function reference specifies a numeric argument. Imported CSV values may look numeric while being stored as text; a custom-format display can also obscure precision. Test with a freshly typed number such as 1 first, then inspect the imported input. This guide does not claim any undocumented implicit conversion of text, empty cells, logical values, or dates into a number.

Numerical and version limitations: The mathematical output range is (−1,1) for finite real x, but observed machine rounding, large-input outputs, error strings, implicit conversions, and the point at which the implementation displays or stores 1 or −1 are not verified on Apple Numbers. Specific Apple Numbers error text, implicit coercion, exact large-value limits, behavior of non-finite inputs, and platform-specific differences have not been tested. Small last-digit differences are possible with floating-point calculation and display rounding; do not require exact string equality for approximate examples.

Excel and Google Sheets also document a function called TANH. In ordinary real-number mathematics the function definition is the same, so the example figures are suitable comparison expectations. This does not establish identical application behavior for imported text, rounding, errors, extreme floating-point values, cross-table syntax, or release support. Check these when transferring a spreadsheet.

• Apple Numbers: uses tables as formula-reference units, so another table can be referenced as InputData::B2.
• Excel / Google Sheets: frequently use sheet-centric references such as Sheet1!B2; those are not a Numbers table-qualified reference.
• Google Sheets: consult Google’s official function list for its function signature. For Excel, Microsoft’s official TANH reference documents a one-number argument, but confirm the intended Excel edition separately.
• Regional settings: test the file’s actual decimal and argument separators in the target spreadsheet. The one-argument TANH formula itself has no separator to translate.

• Apple: TANH function — official syntax, argument type and Apple examples. Formulas and Functions Help 15.4, consulted 2026-10-11 UTC.
• Apple: Functions overview — case conventions, formulas, and locale argument separators; consulted 2026-10-11 UTC.
• Apple: Refer to cells in formulas — same-table and cross-table reference forms; consulted 2026-10-11 UTC.
• Apple: Functions Help welcome — general function browser/platform context, not a first-release guarantee; consulted 2026-10-11 UTC.
• Google Sheets function list — cross-application signature existence; consulted 2026-10-11 UTC.
• Microsoft Excel TANH documentation — documented Excel function name and signature, consulted 2026-10-11 UTC. Exact release/locale parity was not tested.

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

For a software-specific comparison: LibreOffice Calc TANH, Apache OpenOffice Calc TANH, Gnumeric TANH. 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.