Read the equation
The sigmoid maps every finite real input to (0,1), gives 0.5 at zero, and increases monotonically. Its derivative is σ(x)(1−σ(x)), with maximum slope one quarter at zero. Numerical implementations can avoid overflow for large negative inputs by using a sign-dependent equivalent expression.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| x | Real-valued input |
| σ | Logistic activation |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
## 12.2 Sigmoid
```
σ(x) = 1 / (1 + e^{-x})
```
**Range**: (0, 1)
**Explanation**: Classic S-curve activation for probability-like outputs.
## 12.3 Clamp
```
clamp₀₁(x) = max(0, min(1, x))
```
**Range**: [0, 1]
**Explanation**: Hard boundary function ensuring outputs stay in valid range.
## 12.4 Smooth Norm
```
smooth_norm(a, b) = √(a² + b²)Exact original notation
σ(x) = 1 / (1 + e^{-x})