Free · Algebra

Logistic Sigmoid and Its Inverse

Calculate the logistic sigmoid of a real input or recover the input from a probability strictly between zero and one, with a correctly sampled sigmoid chart.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

At x=0, the logistic sigmoid equals 0.5. Entering p=0.75 gives x=ln3, about1.098612289.

σ(x)=1/(1+e^(−x)); x=ln(p/(1−p)) for 0<p<1.

Worked example

Enter these known values and leave the other values blank.

Input x: 0

Sigmoid output p: 0.5

Assumptions and limitations

Both original numerical controls remain editable, including the output-to-input inverse. The quantities are unitless. The initial x=0 produces σ=0.5.
The inverse requires a value strictly between 0 and 1. Neither endpoint has a finite inverse: p approaches 0 as x approaches−∞ and p approaches 1 as x approaches+∞. This tool does not accept infinity as a number.
The captured chart shifted and rescaled a fixed offset curve around the entered output; away from zero its samples did not generally equal σ at their displayed x coordinates. This source-owned chart correction evaluates the actual logistic function at the same captured x sample locations. The immutable original callback is retained as evidence.
The chart retains nine samples at x=0, using offsets −0.4 through 0.4. At a nonzero input it retains eleven samples spaced by |x|/10, centered on x. It uses bounded browser plotting arithmetic and does not determine the numeric answer. A chart near an endpoint may be hidden or flatten at browser plotting resolution even when the numeric inverse remains available.
The controls retain 64-digit calculation precision and use scientific notation for nonzero values too small for ordinary fixed-decimal display. Extremely saturated forward results can round to exactly 0 or 1 and are then withheld by the retained open-interval domain. This is finite numerical behavior, not a claim that the mathematical sigmoid attains an endpoint.
Use decimal/scientific notation or supported exact finite expressions. Raw inputs must pass the existing bounded proof; nonzero magnitudes below 10^-10000 are outside the input working range. Clear x before solving from p when you want the inverse to determine it.

Common questions

What is the sigmoid of zero?

It is 0.5, because e⁰=1 and 1/(1+1)=1/2.

How do I invert the sigmoid?

For an output p strictly between 0 and 1, calculate ln(p/(1−p)). For p=0.75, the input is ln3.

Why are zero and one invalid inverse inputs?

They are limiting values reached only as x tends toward negative or positive infinity. There is no finite real inverse at either endpoint.

References

Bind a source-owned sigmoid chart correction to the entire original callback/source: keep its bounded sample count and x locations but evaluate the actual sigmoid at every displayed x instead of shifting/scaling a fixed curve. Preserve both numeric controls, defaults and forward/inverse formulas. Use exact open-interval domain predicates and the existing source/schema-pinned finite-input/scientific display policy.

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