Description
Return (1+x/hypot(x,sqrt(smoothing)))/2 as the scalar input derivative of squareplus.
Squareplus is the algebraic rectifier (x+sqrt(x²+b))/2. A positive smoothing term b gives a strictly positive output and a smooth inverse, while allowing b=0 admits a clipped rectifier. Here smoothing/bias denotes the term inside the square root, rather than an additive output bias.
Return (1+x/hypot(x,sqrt(smoothing)))/2 as the scalar input derivative of squareplus.
When to use this tool
Use Activation: Squareplus Scalar Derivative when checking the stated squareplus inverse, smoothing or derivative convention.
How to use the tool and how it works
- Enter x, smoothing. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Require smoothing>0. Return (1+x/hypot(x,sqrt(smoothing)))/2 as the scalar input derivative of squareplus.1
Squareplus uses a nonnegative parameter inside its square root. Strictly positive smoothing and the zero-smoothing rectifier case differ; the input domain and feasibility flags above specify which case this helper covers.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| smoothing | Smoothing | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Object containing the operation-specific returned members listed below. | Required. Object. |
| result.derivative | Scalar derivative with respect to the variable and at the fixed coordinates specified above; not a tensor/model gradient. | Calculated result member: number. |
{
"input": {
"x": 0,
"smoothing": 4
},
"output": {
"result": {
"derivative": 0.5
}
}
}Limitations and assumptions
- The smoothing parameter is the quantity inside the square root, not its square root; finite-precision tails can round to zero or one.
- Outputs are nested inside result. 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/squareplus-scalar-derivative — tool.io
-
Squareplus: A Softplus-Like Algebraic Rectifier — Jonathan T. Barron
Similar or alternative tools
- Activation: Squareplus Scalar Inverse
For squareplus(x)=(x+sqrt(x²+smoothing))/2, require y>0 and smoothing>0.
- Squareplus Bias Partial Derivative
Return 1/(4*hypot(x,sqrt(bias))) as its partial derivative with respect to bias.