Tanh Calculator — Hyperbolic Tangent
Compute tanh(x) = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ) for any real x, and see sinh(x), cosh(x), the derivative sech²(x) = 1 − tanh²(x), and the characteristic S-curve.
Hyperbolic tangent — always strictly between −1 and 1 for any real x
- 1
sinh(x) = (eˣ − e⁻ˣ) ÷ 2
(e^1 − e^−1) ÷ 2 = 1,175201 - 2
cosh(x) = (eˣ + e⁻ˣ) ÷ 2
(e^1 + e^−1) ÷ 2 = 1,543081 - 3
tanh(x) = sinh(x) ÷ cosh(x)
1,175201 ÷ 1,543081 = 0,761594Always strictly between −1 and 1 for any real x.
Як працює цей калькулятор?
tanh(x) = (eˣ−e⁻ˣ)/(eˣ+e⁻ˣ). Always strictly between −1 and 1, zero at x=0, odd function, derivative sech²(x) = 1−tanh²(x) peaks at 1 when x=0. Related to logistic sigmoid by tanh(x) = 2σ(2x)−1. Widely used as a neural-network activation function for its zero-centred output and analytic derivative.
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).
Поширені запитання
The logistic sigmoid σ(x) = 1/(1+e⁻ˣ) maps to (0, 1), while tanh maps to (−1, 1). They are related by tanh(x) = 2σ(2x) − 1. Tanh is zero-centred, which generally helps gradient descent converge faster in neural network training.
d/dx tanh(x) = sech²(x) = 1 − tanh²(x). It is always positive (tanh is strictly increasing), reaches its maximum of 1 at x = 0, and approaches 0 as |x| → ∞ — reflecting the flattening of the S-curve near its horizontal asymptotes at ±1.
tanh is an odd function: tanh(−x) = −tanh(x). Its graph has 180°-rotational symmetry about the origin. This follows from sinh being odd and cosh being even, so sinh/cosh is odd.
TG we-Calculate Editorial Team. (2026). Tanh Calculator — Hyperbolic Tangent [Online calculator]. TG we-Calculate. https://we-calculate.com/uk/calculator/tanh-calculator
TG we-Calculate Editorial Team. "Tanh Calculator — Hyperbolic Tangent." TG we-Calculate. 2026. https://we-calculate.com/uk/calculator/tanh-calculator.
TG we-Calculate Editorial Team, "Tanh Calculator — Hyperbolic Tangent," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/uk/calculator/tanh-calculator
@misc{wecalculate_tanh_calculator, title = {Tanh Calculator — Hyperbolic Tangent}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/uk/calculator/tanh-calculator}}, year = {2026}, note = {TG we-Calculate} }
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