Tanh Calculator — Hyperbolic Tangent

Формула

How this is calculated

The hyperbolic tangent is defined as tanh(x) = sinh(x) / cosh(x) = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ). Because the numerator is always strictly smaller in magnitude than the denominator (both exponentials are positive, and the denominator is their sum), tanh(x) is bounded strictly between −1 and 1 for every real x. It passes through zero at x = 0, approaches +1 as x → +∞, and approaches −1 as x → −∞, tracing the smooth S-shaped curve shown in the plot.

This smooth saturation behaviour makes tanh a popular activation function in neural networks: it is zero-centred (unlike the logistic sigmoid, which maps to (0, 1) and can slow gradient descent), differentiable everywhere, and its derivative sech²(x) = 1 − tanh²(x) can be computed directly from the output — convenient for backpropagation. However, tanh still suffers from the vanishing-gradient problem for very large |x|, where the curve flattens and the derivative approaches zero.

The fundamental hyperbolic identity cosh²(x) − sinh²(x) = 1 implies that 1 − tanh²(x) = sech²(x), the derivative shown in the stat grid. For large |x|, tanh saturates rapidly: tanh(3) ≈ 0.9951, tanh(5) ≈ 0.9999. The function is strictly increasing and odd: tanh(−x) = −tanh(x).