Logistic Sigmoid Transformation

Easy

The sigmoid function, also known as the logistic function, is one of the most iconic mathematical transformations in machine learning and statistics. It serves as a fundamental activation function that elegantly maps any real-valued number to a probability-like value between 0 and 1. The sigmoid function is mathematically defined as: $$\sigma(z) = \frac{1}{1 + e^{-z}}$$ Where: z is the input value (can be any real number from -∞ to +∞) e is Euler's number (approximately 2.71828) σ(z) is the output, always bounded in the range (0, 1) Key Properties of the Sigmoid Function: S-Shaped Curve: The sigmoid produces a characteristic "S" or sigmoidal shape when plotted, transitioning smoothly from 0 to 1. Symmetric Center Point: At z = 0, the sigmoid returns exactly 0.5, representing the decision boundary. Asymptotic Behavior: As z approaches +∞, σ(z) approaches 1. As z approaches -∞, σ(z) approaches 0. Monotonically Increasing: The function is always increasing—larger inputs always produce larger outputs. Differentiable: The sigmoid has a smooth derivative: σ'(z) = σ(z) × (1 - σ(z)), which is crucial for gradient-based learning. Historical Significance: The sigmoid function has been central to the development of neural networks since the 1980s. While modern deep learning often prefers ReLU and its variants, sigmoid remains essential in output layers for binary classification, gates in LSTM/GRU networks, and probabilistic modeling. Your Task: Write a Python function that computes the sigmoid activation for a given input value. The function should return the result rounded to four decimal places to ensure consistent precision.

Example 1

Input

z = 0

Output

0.5

Explanation

When z = 0, the sigmoid calculation is σ(0) = 1 / (1 + e⁰) = 1 / (1 + 1) = 1 / 2 = 0.5. This represents the exact midpoint of the sigmoid curve, the point of maximum uncertainty or the decision boundary in binary classification.

Example 2

Input

z = 1

Output

0.7311

Explanation

For z = 1, we calculate σ(1) = 1 / (1 + e⁻¹) = 1 / (1 + 0.3679) ≈ 1 / 1.3679 ≈ 0.7311. Notice how a small positive input pushes the output above 0.5 but not all the way to 1.

Example 3

Input

z = -1

Output

0.2689

Explanation

For z = -1, we calculate σ(-1) = 1 / (1 + e¹) = 1 / (1 + 2.7183) ≈ 1 / 3.7183 ≈ 0.2689. This demonstrates the symmetry: σ(-1) = 1 - σ(1), showing how negative inputs produce outputs below the 0.5 midpoint.

Constraints

-1000 ≤ z ≤ 1000 (input range)
Output must be rounded to exactly 4 decimal places
The function must handle edge cases near the asymptotic boundaries
For very large positive z, the output approaches but never equals 1
For very large negative z, the output approaches but never equals 0

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Logistic Sigmoid Transformation

Easy

The sigmoid function, also known as the logistic function, is one of the most iconic mathematical transformations in machine learning and statistics. It serves as a fundamental activation function that elegantly maps any real-valued number to a probability-like value between 0 and 1. The sigmoid function is mathematically defined as: $$\sigma(z) = \frac{1}{1 + e^{-z}}$$ Where: z is the input value (can be any real number from -∞ to +∞) e is Euler's number (approximately 2.71828) σ(z) is the output, always bounded in the range (0, 1) Key Properties of the Sigmoid Function: S-Shaped Curve: The sigmoid produces a characteristic "S" or sigmoidal shape when plotted, transitioning smoothly from 0 to 1. Symmetric Center Point: At z = 0, the sigmoid returns exactly 0.5, representing the decision boundary. Asymptotic Behavior: As z approaches +∞, σ(z) approaches 1. As z approaches -∞, σ(z) approaches 0. Monotonically Increasing: The function is always increasing—larger inputs always produce larger outputs. Differentiable: The sigmoid has a smooth derivative: σ'(z) = σ(z) × (1 - σ(z)), which is crucial for gradient-based learning. Historical Significance: The sigmoid function has been central to the development of neural networks since the 1980s. While modern deep learning often prefers ReLU and its variants, sigmoid remains essential in output layers for binary classification, gates in LSTM/GRU networks, and probabilistic modeling. Your Task: Write a Python function that computes the sigmoid activation for a given input value. The function should return the result rounded to four decimal places to ensure consistent precision.

Example 1

Input

z = 0

Output

0.5

Explanation

Example 2

Input

z = 1

Output

0.7311

Explanation

For z = 1, we calculate σ(1) = 1 / (1 + e⁻¹) = 1 / (1 + 0.3679) ≈ 1 / 1.3679 ≈ 0.7311. Notice how a small positive input pushes the output above 0.5 but not all the way to 1.

Example 3

Input

z = -1

Output

0.2689

Explanation

Constraints

-1000 ≤ z ≤ 1000 (input range)
Output must be rounded to exactly 4 decimal places
The function must handle edge cases near the asymptotic boundaries
For very large positive z, the output approaches but never equals 1
For very large negative z, the output approaches but never equals 0