Sigmoid Calculator

Compute the sigmoid activation function output for a given input value.

About Sigmoid Calculator

The sigmoid function squashes any real-valued input into the range (0, 1): σ(x) = 1/(1+e^(−x)), making it a natural choice for representing probabilities and for binary classification output layers. Its derivative, σ'(x) = σ(x)·(1−σ(x)), is conveniently expressed in terms of the function's own output, which made it historically popular for hand-derived backpropagation. However, the derivative is at most 0.25 (at x=0) and shrinks toward 0 for large |x|, causing the 'vanishing gradient' problem in deep networks — this is why ReLU and its variants have largely replaced sigmoid in hidden layers, though sigmoid remains standard for binary output layers and gates (e.g. LSTM gates).

Formulas

Sigmoid function

sigma(x) = 1 / (1 + e^(-x))

x: Input value
e: Euler's number (≈2.718)

Sigmoid derivative

sigma'(x) = sigma(x) × (1 - sigma(x))

sigma(x): Sigmoid output at x

Worked example

With the default values below, Sigmoid Calculator returns σ(x) of 0.731059. Change any input above and every figure here is recalculated in your browser.

Inputs used

Input (x)accepts -100–100: 1

Results

σ(x): 0.731059
σ'(x): 0.196612

Step by step

1σ(x) = 1 / (1 + e^(−x))= 0.731059
2Derivative: σ(x) × (1 − σ(x))= 0.196612

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

Why does sigmoid cause vanishing gradients?

The sigmoid derivative σ(x)×(1−σ(x)) has a maximum value of only 0.25 (at x=0) and approaches 0 as |x| grows large in either direction, so in deep networks, gradients get multiplied by these small values at every layer during backpropagation, shrinking exponentially and making early layers learn very slowly.

When should I still use sigmoid?

Sigmoid remains the standard choice for binary classification output layers (interpreting the output as a probability) and for gate mechanisms inside LSTM and GRU cells, where its bounded (0,1) output naturally represents a 'how much to let through' gating signal.

What is the output range and midpoint of sigmoid?

Sigmoid outputs values strictly between 0 and 1 (never reaching either endpoint), with σ(0) = 0.5 exactly — this midpoint is why 0.5 is the conventional decision threshold for binary classifiers using a sigmoid output.