Description
Return exp(-abs(x))/(1+exp(-abs(x)))², the stable product sigmoid(x)*(1-sigmoid(x)).
The logistic sigmoid is 1/(1+exp(-x)) and its inverse logit is log(p/(1-p)) on 0<p<1. Temperature and input sensitivities use different derivatives. The logarithm of a derivative is distinct from the derivative of a logarithm, especially when numerical tails underflow.
Return exp(-abs(x))/(1+exp(-abs(x)))², the stable product sigmoid(x)*(1-sigmoid(x)).
When to use this tool
Use Sigmoid Complement Product when checking the stated sigmoid/logit scalar derivative or stable product convention.
How to use the tool and how it works
- Enter x. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read product. Use the output convention below.
Return exp(-abs(x))/(1+exp(-abs(x)))², the stable product sigmoid(x)*(1-sigmoid(x)).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| product | Product | Required. Finite number. |
{
"input": {
"x": 1.0986122886681096
},
"output": {
"product": 0.18749999999999997
}
}Limitations and assumptions
- The product equals the unit-scale sigmoid input derivative; it can still underflow to zero for sufficiently large finite magnitude.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the complete explicit function and the intended parameter, branch and numerical conventions for symbolic, tensor or model-level differentiation; assess training behavior independently.
References
Similar or alternative tools
- Sigmoid Log Derivative
Return -abs(x)-2*log1p(exp(-abs(x))), the natural logarithm of sigmoid's first derivative.
- Sigmoid Logit Jacobian Sign
Return sigmoidDerivative=p*(1-p), inverse-logit derivative=1/(p*(1-p)) and sign=1.