Bounded Smooth Activation: The Softsign Function

In the realm of neural network architectures, activation functions play a pivotal role in introducing non-linearity and enabling networks to learn complex patterns. Among these, the Softsign activation function stands out as an elegant alternative to traditional hyperbolic tangent (tanh) functions.

The Softsign function is defined mathematically as:

$$\text{softsign}(x) = \frac{x}{1 + |x|}$$

This function possesses several remarkable properties that make it valuable for neural network design:

Key Characteristics:

• Bounded Output: The function outputs values strictly within the open interval (-1, 1), never actually reaching these boundary values
• Zero-Centered: The function passes through the origin (softsign(0) = 0), providing symmetric activation around zero
• Smooth Gradient: Unlike hard threshold functions, Softsign provides continuous and differentiable gradients throughout its domain
• Gentle Saturation: As inputs grow in magnitude, the output asymptotically approaches ±1 more gradually than tanh, which can help mitigate vanishing gradient problems

Mathematical Properties: The Softsign function exhibits polynomial decay rather than the exponential decay seen in tanh. This means:

• For positive values: as x → ∞, softsign(x) → 1
• For negative values: as x → -∞, softsign(x) → -1
• The rate of approach to saturation is slower than tanh, providing better gradient flow in deep networks

Your Task: Implement a Python function that computes the Softsign activation value for any given real number input. Your implementation should correctly handle positive, negative, and zero inputs, returning the result rounded to 4 decimal places.