ATANH in Excel: Invert a Bounded Hyperbolic Ratio Safely
Use Excel ATANH with strict endpoint checks, signed examples, a scaled inverse response and rounding cautions.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: ATANH returns the inverse hyperbolic tangent. =ATANH(0.8) is expected to give approximately 1.098612289. The real input must lie strictly between −1 and 1.
Availability and version history
- Introduced: First introduction version is not established by the official history reviewed.
- Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
- Checked: October 14, 2026; official ATANH reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.
Syntax and strict endpoints
=ATANH(number)
The required real argument must satisfy −1<number<1. Microsoft explicitly excludes −1 and 1. An endpoint is not a valid finite inverse, even if a rounded display makes a nearby permitted value appear equal to one.
The operation differs from ATAN, which inverts an ordinary tangent ratio. ATANH’s strict input domain follows the bounded real output of TANH.
Example 1: compare signed fractions
Enter −0.8, −0.4, 0, 0.4, 0.8 in A2:A6 and fill =ATANH(A2) down in B2:B6. Expected values are approximately −1.098612289, −0.423648930, 0, 0.423648930, 1.098612289.
The result can exceed one in magnitude even though the input cannot. Domain and output range are separate ideas. Keep ratio labels on the inputs rather than implying these numbers are degree angles.
Example 2: use an independent logarithm control
For input 0.8:
=0.5*LN((1+0.8)/(1-0.8))
=ATANH(0.8)
Both expected results are approximately 1.098612289. The quotient is 9 and half LN(9) equals LN(3). Applying TANH to the unrounded result reconstructs approximately 0.8.
The identity is useful for a small check. It does not establish that every expanded expression has the same native accuracy near a domain endpoint.
Advanced example: undo a scaled bounded response
Define a fictional forward model y=20*TANH(x/3). For target y values −16, −8, 0, 8, 16 in D2:D6, invert it with:
=3*ATANH(D2/20)
Expected x values are approximately −3.295836866, −1.270946791, 0, 1.270946791, 3.295836866. Substituting each unrounded x into =20*TANH(x/3) reconstructs the corresponding target.
The multiplier 20 and input scale 3 have different roles. Targets of ±20 are the model’s limiting responses and have no finite real inverse. Targets beyond those bounds are incompatible with the model.
Advanced example: make the allowed target rule visible
For ordinary scalar numeric inputs, use:
=IF(ABS(D2)<20,3*ATANH(D2/20),"Review target")
Controls 0, 16, 20, −20, 21 are intended to yield 0, 3.295836866, Review target, Review target, Review target. A strict comparison preserves the endpoint exclusion. The formula is a worksheet policy, not a guarantee about evaluation order in all array contexts.
Rounding and information loss
A large forward TANH result may round to one in stored or displayed arithmetic. Applying ATANH to an endpoint cannot recover the original finite input. Retain unrounded values and inspect whether a source has saturated before trying to invert it.
The direct reference does not certify every native error display or coercion. Separate an invalid domain, lost precision and an incorrectly specified scale when diagnosing an unexpected answer.
References and verification
Worked inputs are fictional. Decimal answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Extreme magnitudes, complete coercion matrices and cross-version numerical accuracy remain untested.