Softplus Calculator
Compute the Softplus activation function, a smooth approximation of ReLU.
About Softplus Calculator
Softplus is a smooth approximation of ReLU: Softplus(x) = ln(1 + e^x), which behaves almost identically to max(0, x) for inputs far from zero, but replaces ReLU's sharp corner with a smooth, continuously differentiable curve. Its derivative is exactly the sigmoid function, softplus'(x) = σ(x), which is a convenient closed form. Because softplus is always positive and never exactly zero, it's commonly used to parameterize quantities that must stay strictly positive (like a standard deviation in a probabilistic model), and it also appears as a building block inside the Mish activation function.
Formula
Softplus(x) = ln(1 + e^x)
- x
- Input value
- e
- Euler's number (≈2.718)
Worked example
With the default values below, Softplus Calculator returns Softplus(x) of 1.313262. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Input (x)accepts -100–100
- 1
Results
- Softplus(x)
- 1.313262
- Softplus'(x)
- 0.731059
Step by step
- 1Softplus(x) = ln(1 + e^x)= 1.313262
- 2Derivative: softplus'(x) = σ(x)= 0.731059
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How is Softplus related to ReLU?
Softplus is a smooth, differentiable approximation of ReLU — for large positive or negative x, Softplus(x) closely matches max(0,x), but near x=0 it transitions smoothly instead of having ReLU's sharp corner, and it never outputs exactly 0 (it asymptotically approaches 0 for very negative x).
Why is softplus's derivative exactly the sigmoid function?
This falls directly out of calculus: differentiating ln(1+e^x) gives e^x/(1+e^x), which is algebraically identical to 1/(1+e^(−x)) — the sigmoid function — making softplus and sigmoid a mathematically paired activation/derivative combination.
Where is softplus used in practice?
It's commonly used to constrain a neural network's output to be strictly positive (e.g. predicting a variance or scale parameter in a probabilistic model, since variance can't be negative), and it's also a core building block inside the Mish activation function.