tk-yoshimura / errorfunctionapproximation

Error function approximation

ErrorFunctionApproximation

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Definition

The error function (erf) is defined by the following equation:

The complementary error function (erfc) is defined by the following equation:

Convergence Equation

To obtain these numerical solutions, use the following equations:

Taylor series:

Continued fraction expansion:

Taylor series

First, Taylor series convergence becomes slow as the absolute value of z increases.

The absolute value of each term is as follows:

The behavior for each z:

The ratio to the next term is less than 1:

The number of terms required to obtain arbitrary precision is as follows:
In addition, in order to obtain the expected value, 128-bit expansion was performed to absorb cancellation.

Continued fraction expansion

Next, in Continued fraction expansion, when the absolute value of z is small, the convergence becomes slow.

Replace continued fractions:

The recursive part can be written below:

The fixed points at n = 1 are:

These approximate the true values:

Solve the continued fraction F1 (z) backward with X4 (z) as the initial value.
Let N be the recursion times (Fn+4(z) to Fn(z)), and N until convergence is as follows:

Double Precision (IEEE 754) Approx

The calculation error is at most 2 epsilons.
C# code

erf result

erfc result

inverse erf result

inverse erfc result

The following is used to approximate the error of machine epsilon in the entire domain.

near zero:

greater than 1/2:

pade erf precision16
pade erf precision32
pade erfc precision16
pade erfc precision32

Inverse Function

InverseErfc NearZero

pade inverfc precision16
pade inverfc precision32

InverseErf NearZero

series inverf e20
series inverf e40
series inverf e80