What is the range of tanh(x)?

tanh(x) always produces a value strictly between −1 and 1. As x → +∞, tanh(x) → 1; as x → −∞, tanh(x) → −1. It equals 0 when x = 0.

Why is arctanh undefined outside (−1, 1)?

The formula arctanh(x) = 0.5 × ln((1+x)/(1−x)) requires the argument of the logarithm to be positive. When |x| ≥ 1 the logarithm is undefined (or infinite), so arctanh has no real-number result.

How does tanh differ from the regular tangent (tan)?

tan is a trigonometric function defined via the unit circle with period π and vertical asymptotes. tanh is a hyperbolic function defined via the unit hyperbola; it has no asymptotes in the usual sense and is bounded between −1 and 1 for all real inputs.

Why is tanh used as a neural network activation function?

tanh is zero-centered (outputs range from −1 to 1) which helps gradients flow better than the 0–1 sigmoid during training. It is differentiable everywhere and its derivative, 1 − tanh²(x), is easy to compute.

Can I enter degrees instead of radians?

Hyperbolic functions take a dimensionless real number, not an angle. The degree/radian toggle converts your input angle to radians first (x_rad = x_deg × π/180) so you can work in whichever unit feels natural.

Hyperbolic Tangent Calculator

Enter any real number to instantly compute its hyperbolic tangent (tanh) and inverse hyperbolic tangent (arctanh).

Calculate tanh(x)

Input as: Number Degrees Radians

Value of x

Batch Input

Enter a value on the left and click Calculate.

Batch Results

x	tanh(x)	arctanh(x)

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Summary

The hyperbolic tangent function tanh(x) maps any real number to the open interval (−1, 1) using the formula (e^x − e^−x) / (e^x + e^−x). It appears in neural networks as an activation function, in physics for describing relativistic velocities, and in engineering for modeling wave propagation. This calculator also computes the inverse function arctanh(x) — valid only when |x| < 1 — which reverses tanh to recover the original input.

How it works

Enter any real number in the input field.
The calculator evaluates tanh(x) = (e^x − e^−x) / (e^x + e^−x) instantly.
If your input is in the range (−1, 1), the inverse arctanh(x) = 0.5 × ln((1+x)/(1−x)) is also shown.
Results are displayed to 10 significant figures with a radians/degrees angle toggle.
Use the batch section to compute tanh for a comma-separated list of values at once.

Use cases

Verifying tanh activation function values when debugging neural networks.
Computing relativistic velocity additions in special relativity (tanh of rapidity).
Solving hyperbolic geometry problems in mathematics courses.
Checking arctanh results when fitting sigmoid curves to data.
Teaching hyperbolic functions interactively with instant feedback.
Engineering signal analysis involving hyperbolic wave models.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu