Error Function Erfc Approximation
Evaluate real erfc directly to preserve small positive tails without subtracting a saturated erf from one on |x|≤26; Numerical Recipes approximation with roughly 1.5e−7 absolute accuracy, not arbitrary precision.
Description
The complementary error function represents a small positive tail for large positive arguments.
The complementary error function represents a small positive tail for large positive arguments. Computing it directly avoids subtracting an erf value already rounded to one.1
Evaluate real erfc directly to preserve small positive tails without subtracting a saturated erf from one on |x|≤26; Numerical Recipes approximation with roughly 1.5e−7 absolute accuracy, not arbitrary precision.
When to use Error Function Erfc Approximation
Use this tool to evaluate real erfc directly to preserve small positive tails without subtracting a saturated erf from one on |x|≤26; Numerical Recipes approximation with roughly 1.5e−7 absolute accuracy, not arbitrary precision.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Error Function Erfc Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter x using the field table and calculation rules, then run the tool.
- Read complementaryErrorFunctionValue. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| x | X | number | Required |
Use real x in [−26,26]. Return 1 at zero; use the near-zero erf Taylor complement for |x|<0.5 and the direct exponential approximation otherwise, applying the negative-argument relation.
erfc(x)=1−erf(x); erfc(−x)=2−erfc(x) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| complementaryErrorFunctionValue | Complementary error function value | number |
{
"input": {
"x": 1
},
"output": {
"complementaryErrorFunctionValue": 0.15729921223999332
}
}Limitations and assumptions
Outside the Taylor regime the implementation describes roughly 1.5×10⁻⁷ absolute accuracy. It does not provide arbitrary precision or a per-result certificate.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Error Function Erf Approximation: Evaluate real erf using direct near-zero Taylor series, odd symmetry and complementary approximation on |x|≤26; roughly 1.5e−7 absolute accuracy outside Taylor regime, not arbitrary precision. For a complementary task, use Upper Incomplete Gamma Approximation: Evaluate upper tail Γ(a,x) and Q separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.
References
-
Error Function Erfc Approximation — tool.io
Similar or alternative tools
- Error Function Erf Approximation
Evaluate real erf using direct near-zero Taylor series, odd symmetry and complementary approximation on |x|≤26; roughly 1.5e−7 absolute accuracy outside Taylor regime, not arbitrary precision.
- Upper Incomplete Gamma Approximation
Evaluate upper tail Γ(a,x) and Q separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.