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Math

Hyperbolic Tangent Calculator.

Computes the hyperbolic tangent of a given real number, useful in math, physics, and machine learning.

Results update live as you type.
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Your inputs

How it works

  1. 1

    Enter the value of x.

  2. 2

    Compute e^(2x) using the exponential function.

  3. 3

    Subtract 1 from e^(2x) to get the numerator.

  4. 4

    Add 1 to e^(2x) to get the denominator, then divide numerator by denominator.

(exp(2*x)-1)/(exp(2*x)+1)

Frequently asked questions

What is the range of tanh(x)?

The hyperbolic tangent always outputs values between -1 and 1, approaching -1 as x goes to negative infinity and 1 as x goes to positive infinity.

How is tanh(x) related to sinh and cosh?

tanh(x) = sinh(x) / cosh(x), where sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2.

Why use tanh in neural networks?

Because its output is zero-centered and bounded, which helps with gradient stability during training.

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Results

Formula checked

tanh(x)

0,762

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How it works

Computes the hyperbolic tangent of a given real number, useful in math, physics, and machine learning.

  1. Enter the value of x.
  2. Compute e^(2x) using the exponential function.
  3. Subtract 1 from e^(2x) to get the numerator.
  4. Add 1 to e^(2x) to get the denominator, then divide numerator by denominator.

Formulas

The math behind this calculator, written out so you can verify the result.

Definition

tanh(x) = (e^(2x) - 1) / (e^(2x) + 1)

This is the standard formula for the hyperbolic tangent, expressed using the exponential function.

Example:

Input: x = 1

Calculation: (e^2 - 1) / (e^2 + 1)

Result: ≈ 0.761594

Real-world use cases

Where this calculation shows up in everyday life.

Activation function

Used in neural networks to introduce non-linearity while keeping outputs bounded.

Example: Squashing neuron outputs to [-1, 1].

Hyperbolic geometry

Appears in formulas for the Gudermannian function and in special relativity.

Example: Rapidity in physics: φ = atanh(v/c).

Tips and common mistakes

Tips

  • For large |x|, tanh(x) approaches ±1; use the formula directly to avoid overflow.
  • If your calculator lacks tanh, use the exponential form.
  • Remember tanh is odd: tanh(-x) = -tanh(x).

Common Mistakes to Avoid

  • Confusing tanh with arctan (inverse tangent).
  • Using degrees instead of radians when x is an angle.
  • Forgetting that tanh(0) = 0, not 1.

Assumptions and limitations

  • Use the stated inputs and units.
  • Results are estimates for planning and education.
  • Check measurements and source data before making an important decision.