Softsign Curvature Magnitude Tool

Description

For x!=0, return magnitude=2/(1+abs(x))³ with defined=true.

Softsign uses x/(1+abs(x)); its finite-input range is between -1 and 1. It has a first derivative at zero but different one-sided second derivatives there. Changing an additive denominator scale is distinct from multiplying the function’s input by a scale.

For x!=0, return magnitude=2/(1+abs(x))³ with defined=true.

When to use this tool

Use Softsign Curvature Magnitude when checking the stated softsign inverse, saturation or derivative 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 defined, magnitude. Inspect defined before interpreting the associated values.

For x!=0, return magnitude=2/(1+abs(x))³ with defined=true. At zero return magnitude=0 and defined=false.^[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
defined	Whether the corresponding reporting denominator/value is defined; inspect before interpreting zero placeholders.	Required. Boolean.
magnitude	Absolute second derivative for x!=0; zero placeholder when defined=false at the join.	Required. Finite number.

json

{
  "input": {
    "x": 1
  },
  "output": {
    "defined": true,
    "magnitude": 0.25
  }
}

Limitations and assumptions

Zero magnitude at the join is a placeholder because the two-sided second derivative is undefined, not a statement of zero curvature.
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

Calculation and input contract: activation/softsign-curvature-magnitude — tool.io

Similar or alternative tools

Softsign Output Distance To Saturation
Compute y=x/(1+abs(x)); return positiveGap=1-y and negativeGap=1+y.
Activation: Softsign Scalar Derivative
Return 1/(1+abs(x))² as the first scalar derivative of x/(1+abs(x)), including derivative one at zero.