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.

Description

The error function is an odd integral of a Gaussian kernel.

The error function is an odd integral of a Gaussian kernel. Near zero, direct series evaluation avoids cancellation in a complementary-function subtraction.^[1]

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.

When to use Error Function Erf Approximation

Use this tool to 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.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Error Function Erf 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 errorFunctionValue. 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

Enter real x in [−26,26]. For |x|<0.5 sum the Taylor series for up to 40 steps; otherwise use the complementary approximation and odd symmetry.

erf(x)=(2/sqrt(π))∫[0..x]exp(−t²)dt

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
errorFunctionValue	Error function value	number

json

{
  "input": {
    "x": 1
  },
  "output": {
    "errorFunctionValue": 0.8427007877600067
  }
}

Limitations and assumptions

The approximation describes roughly 1.5×10⁻⁷ absolute accuracy outside the Taylor regime. No per-result roundoff bound is returned.

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 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. For a complementary task, use Lower Incomplete Gamma Approximation: Evaluate lower integral γ(a,x) and P 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 Erf Approximation — tool.io

Similar or alternative tools

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.
Lower Incomplete Gamma Approximation
Evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.