Inverse Error Function erf⁻¹ Calculator

Compute the inverse error function erf⁻¹(y) and inverse complementary erfc⁻¹(y) with Newton iteration. See the verification erf value and the convergence of each step.

Input

Enter y to find the x satisfying erf(x)=y or erfc(x)=y using Newton iteration.

Input value y

Range is -1 ＜ y ＜ 1

Result

Value of erf⁻¹(0.5)

0.4769362762

Check via inverse erf⁻¹

0.5

Residual (absolute error)

Iterations

Convergence

Shows each iterate and how its log-scaled error to the true value shrinks.

Step 00.4769362933

Step 10.4769362762

Step 20.4769362762

How it works

The error function erf(x) = (2/√π)∫₀ˣ e⁻ᵗ² dt, and its inverse erf⁻¹(y) is the x satisfying erf(x)=y.
A rational polynomial approximation gives the initial guess, then Newton iteration refines it; the derivative is (2/√π)e⁻ˣ².
The inverse complementary function uses erfc⁻¹(y)=erf⁻¹(1-y).
erf⁻¹ is defined only for -1 ＜ y ＜ 1, and erfc⁻¹ only for 0 ＜ y ＜ 2.
erf is evaluated with a Taylor series for small |x| and a continued fraction for large |x|.

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