Hyperbolic Tangent Activation Function (Easy) — Practice with Code Visualizer

The hyperbolic tangent function (commonly written as tanh) is one of the most fundamental activation functions used in neural networks and deep learning architectures. It serves as a smooth, differentiable non-linearity that maps any real-valued input to an output in the range (-1, 1).

The mathematical definition of the hyperbolic tangent function is:

$$\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$

This can also be expressed in terms of the sigmoid function:

$$\tanh(x) = 2 \cdot \sigma(2x) - 1$$

where σ(x) is the sigmoid function.

Key Properties:

Zero-centered output: Unlike the sigmoid function which outputs values in (0, 1), tanh outputs values centered around zero in (-1, 1). This property can help with faster convergence during training.
Smooth gradient: The derivative of tanh is (1 - \tanh^2(x)), which provides smooth gradients for backpropagation.
Saturation behavior: For large positive or negative inputs, the function saturates to +1 or -1 respectively, which can lead to vanishing gradients in deep networks.
Symmetric around origin: tanh(-x) = -tanh(x), making it an odd function.

Your Task: Write a Python function that computes the hyperbolic tangent activation value for a given real-valued input. Your implementation should:

Use the exponential-based mathematical formula to compute the result
Return the result rounded to 4 decimal places
Handle edge cases appropriately (very large or very small inputs)

For x = 1.0, we compute the exponentials:

• e¹ = 2.7183 • e⁻¹ = 0.3679

Applying the tanh formula:

tanh(1) = (e¹ - e⁻¹) / (e¹ + e⁻¹) = (2.7183 - 0.3679) / (2.7183 + 0.3679) = 2.3504 / 3.0862 = 0.7616

The function maps the input 1.0 to approximately 0.7616, demonstrating the compression of values toward the range boundaries.

For x = 0.0, both exponentials equal 1:

• e⁰ = 1.0 • e⁻⁰ = 1.0

Applying the tanh formula:

tanh(0) = (e⁰ - e⁻⁰) / (e⁰ + e⁻⁰) = (1.0 - 1.0) / (1.0 + 1.0) = 0 / 2 = 0.0

This demonstrates that the hyperbolic tangent function passes through the origin, a key property that makes it zero-centered.

For x = -1.0, we compute:

• e⁻¹ = 0.3679 • e¹ = 2.7183

Applying the tanh formula:

tanh(-1) = (e⁻¹ - e¹) / (e⁻¹ + e¹) = (0.3679 - 2.7183) / (0.3679 + 2.7183) = -2.3504 / 3.0862 = -0.7616

Notice that tanh(-1) = -tanh(1), demonstrating the odd function property. This symmetry is valuable in neural networks as it treats negative and positive inputs symmetrically.

The mathematical definition of the hyperbolic tangent function is:

$$\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$

This can also be expressed in terms of the sigmoid function:

$$\tanh(x) = 2 \cdot \sigma(2x) - 1$$

where σ(x) is the sigmoid function.

Key Properties:

Zero-centered output: Unlike the sigmoid function which outputs values in (0, 1), tanh outputs values centered around zero in (-1, 1). This property can help with faster convergence during training.
Smooth gradient: The derivative of tanh is (1 - \tanh^2(x)), which provides smooth gradients for backpropagation.
Saturation behavior: For large positive or negative inputs, the function saturates to +1 or -1 respectively, which can lead to vanishing gradients in deep networks.
Symmetric around origin: tanh(-x) = -tanh(x), making it an odd function.

Your Task: Write a Python function that computes the hyperbolic tangent activation value for a given real-valued input. Your implementation should:

Use the exponential-based mathematical formula to compute the result
Return the result rounded to 4 decimal places
Handle edge cases appropriately (very large or very small inputs)

For x = 1.0, we compute the exponentials:

• e¹ = 2.7183 • e⁻¹ = 0.3679

Applying the tanh formula:

tanh(1) = (e¹ - e⁻¹) / (e¹ + e⁻¹) = (2.7183 - 0.3679) / (2.7183 + 0.3679) = 2.3504 / 3.0862 = 0.7616

The function maps the input 1.0 to approximately 0.7616, demonstrating the compression of values toward the range boundaries.

For x = 0.0, both exponentials equal 1:

• e⁰ = 1.0 • e⁻⁰ = 1.0

Applying the tanh formula:

tanh(0) = (e⁰ - e⁻⁰) / (e⁰ + e⁻⁰) = (1.0 - 1.0) / (1.0 + 1.0) = 0 / 2 = 0.0

This demonstrates that the hyperbolic tangent function passes through the origin, a key property that makes it zero-centered.

For x = -1.0, we compute:

• e⁻¹ = 0.3679 • e¹ = 2.7183

Applying the tanh formula:

tanh(-1) = (e⁻¹ - e¹) / (e⁻¹ + e¹) = (0.3679 - 2.7183) / (0.3679 + 2.7183) = -2.3504 / 3.0862 = -0.7616

Notice that tanh(-1) = -tanh(1), demonstrating the odd function property. This symmetry is valuable in neural networks as it treats negative and positive inputs symmetrically.

Hyperbolic Tangent Activation Function

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Hyperbolic Tangent Activation Function

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