Description
Evaluate s=sigmoid(x/temperature) using sign-stable exponential branches; return -x*s*(1-s)/temperature², or zero when s rounds to an endpoint.
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.
Evaluate s=sigmoid(x/temperature) using sign-stable exponential branches; return -x*s*(1-s)/temperature², or zero when s rounds to an endpoint.
When to use this tool
Use Sigmoid Temperature Sensitivity when checking the stated sigmoid/logit scalar derivative or stable product convention.
How to use the tool and how it works
- Enter x, temperature. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read derivative. Use the output convention below.
Require temperature>0. Evaluate s=sigmoid(x/temperature) using sign-stable exponential branches; return -x*s*(1-s)/temperature², or zero when s rounds to an endpoint.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| temperature | Positive dimensionless logit/input temperature multiplier under the stated convention. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| derivative | Partial derivative of sigmoid(x/temperature) with respect to temperature, holding x fixed. | Required. Finite number. |
{
"input": {
"x": 1.0986122886681096,
"temperature": 1
},
"output": {
"derivative": -0.2059898041252706
}
}Limitations and assumptions
- This is temperature sensitivity at fixed x, not an input derivative; floating-point saturation is distinct from the mathematical tail.
- 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 Scalar Third Derivative
With s=sigmoid(x), return s*(1-s)*(1-6*s+6*s²), the third scalar input derivative.
- Sigmoid Logit Jacobian Sign
Return sigmoidDerivative=p*(1-p), inverse-logit derivative=1/(p*(1-p)) and sign=1.