ATANH in Google Sheets: Recover Inputs from a Bounded Hyperbolic Response
Calculate Sheets inverse hyperbolic tangent with strict domain checks, a scaled model and explicit endpoint cautions.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: ATANH returns inverse hyperbolic tangent. =ATANH(0.6) is expected to give approximately 0.693147181. Its real argument must be strictly between −1 and 1.
Availability and version history
- Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
- Availability: documented in the rolling Google Sheets service. Examples use ordinary web-spreadsheet numeric cells; other clients and imported files are not separately certified.
- Checked: October 14, 2026; official ATANH reference. Documentation and independent calculations, without native Sheets execution.
Syntax and excluded endpoints
=ATANH(value)
Google explicitly excludes both −1 and 1 from the input domain. This differs from inverse circular sine and cosine, whose real domains include their endpoints. Use the function-specific rule rather than relying on similar names.
The input is a ratio or a defined numeric model value, not automatically an angle measured in degrees. The guide addresses this real-valued operation and does not repeat the help’s broad complex-number sentence as a claim about all Sheets capabilities.
Example 1: compare both signs
Enter −0.6, −0.2, 0, 0.2, 0.6 in A2:A6 and fill =ATANH(A2) down. Expected outputs are approximately −0.693147181, −0.202732554, 0, 0.202732554, 0.693147181.
The sign is preserved. Output magnitudes are not restricted to the input’s strict unit interval; an input closer to one can have a much larger inverse.
Example 2: use a logarithm control
=0.5*LN((1+0.6)/(1-0.6))
=ATANH(0.6)
Both expected results are approximately 0.693147181. The logarithm argument is four, so half LN(4) is LN(2). Applying TANH to the unrounded inverse reconstructs approximately 0.6.
This is a transparent moderate-input identity. It does not establish that an expanded ratio and logarithm have identical native accuracy close to the excluded endpoints.
Advanced example: invert a bounded scaled curve
For the fictional model y=10*TANH(x/2), put target y values −6, −2, 0, 2, 6 in D2:D6. Use:
=2*ATANH(D2/10)
Expected x values are approximately −1.386294361, −0.405465108, 0, 0.405465108, 1.386294361. Forward substitution into the model returns the target sequence within numerical tolerance.
The output scale ten determines the limiting response; the input scale two determines the inverse multiplier. Swapping those scales changes the model. Keep both beside the data.
Advanced example: use a strict target gate
For numeric scalar controls, =IF(ABS(D2)<10,2*ATANH(D2/10),"Outside finite inverse") is intended to accept y=0 and y=6, but label 10, −10 and 11 as Outside finite inverse.
A target at a bound is an asymptotic limit of this real model, not a finite solution. The explicit gate describes an intended worksheet policy, without asserting universal evaluation order for complex arrays.
Rounding and troubleshooting
If a forward TANH output has rounded to an endpoint, the stored result may no longer preserve a usable finite inverse. Do not replace one with an arbitrary fraction just to obtain an answer. Preserve the unrounded source and investigate saturation or source precision.
The official definition pins the domain, while a complete native error display, extreme-value accuracy and import-coercion matrix remain outside these independent examples.
References and verification
Inputs are fictional. Approximate numerical answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Complete coercion, extreme-magnitude and cross-client behavior remain untested.