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Squareplus Bias Partial Derivative Tool

Return the partial derivative of (x+sqrt(x²+bias))/2 with respect to positive bias.

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Description

Return 1/(4*hypot(x,sqrt(bias))) as its partial derivative with respect to bias.

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/(4*hypot(x,sqrt(bias))) as its partial derivative with respect to bias.

When to use this tool

Use Squareplus Bias Partial Derivative when checking the stated squareplus inverse, smoothing or derivative convention.

How to use the tool and how it works

  1. Enter x, bias. Use the units and conventions below.
  2. Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
  3. Read derivative. Use the output convention below.

Require bias>0 in (x+sqrt(x²+bias))/2. Return 1/(4*hypot(x,sqrt(bias))) as its partial derivative with respect to bias.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 fieldMeaning and unitsAccepted input
xFinite scalar input coordinate; use the mathematical scale specified above.Required. Finite number.
biasBiasRequired. Finite number; strictly above 0.
Output fieldInterpretationReturned shape
derivativeScalar derivative with respect to the variable and at the fixed coordinates specified above; not a tensor/model gradient.Required. Finite number.
json
{
  "input": {
    "x": 0,
    "bias": 4
  },
  "output": {
    "derivative": 0.125
  }
}

Limitations and assumptions

  • Input x is held fixed; bias denotes the square-root smoothing term rather than an additive output offset.
  • 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

  1. Calculation and input contract: activation/squareplus-bias-partial-derivative — tool.io

  2. Squareplus: A Softplus-Like Algebraic Rectifier — Jonathan T. Barron

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