Squareplus Output Bias Feasibility Tool

Description

Solve candidate bias=4*output*(output-x); feasible means bias>=0 for output>0, or x<=0 for output=0.

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.

Solve candidate bias=4*output*(output-x); feasible means bias>=0 for output>0, or x<=0 for output=0.

When to use this tool

Use Squareplus Output Bias Feasibility when checking the stated squareplus inverse, smoothing or derivative convention.

How to use the tool and how it works

Enter x, output. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read bias, feasible. Inspect feasible before interpreting the associated values.

Require output>=0. Solve candidate bias=4*output*(output-x); feasible means bias>=0 for output>0, or x<=0 for output=0.^[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.
output	Output	Required. Finite number; minimum 0.

Output field	Interpretation	Returned shape
bias	Candidate smoothing parameter 4output(output-x), not additive output bias; can be negative when infeasible.	Required. Finite number.
feasible	Feasible	Required. Boolean.

json

{
  "input": {
    "x": 1,
    "output": 2
  },
  "output": {
    "bias": 8,
    "feasible": true
  }
}

Limitations and assumptions

This feasibility check permits zero bias, unlike strictly smoothed squareplus tools; a negative candidate bias is returned with feasible=false.
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-output-bias-feasibility — tool.io
Squareplus: A Softplus-Like Algebraic Rectifier — Jonathan T. Barron

Similar or alternative tools

Squareplus Bias Partial Derivative
Return 1/(4*hypot(x,sqrt(bias))) as its partial derivative with respect to bias.
Activation: Squareplus Scalar Derivative
Return (1+x/hypot(x,sqrt(smoothing)))/2 as the scalar input derivative of squareplus.