The error function (erf) is defined by the following equation:
The complementary error function (erfc) is defined by the following equation:
To obtain these numerical solutions, use the following equations:
Taylor series:
Continued fraction expansion:
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.
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:
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
