erfinv

erfinv denotes the inverse error function. It is defined as the inverse of erf, the standard error function erf(x) = (2/√π) ∫_0^x e^{-t^2} dt. Consequently erfinv is defined for values y in (-1, 1) and yields real x with erf(x) = y. As y approaches ±1, erfinv(y) diverges to ±∞.

Key identities: y = erf(x) implies x = erfinv(y). The derivative of erfinv is (erfinv)'(y) = (√π/2) exp[(erfinv(y))^2]. The error function is linked to the standard normal distribution by erf(z/√2) = 2Φ(z) - 1, i.e., Φ(z) = (1 + erf(z/√2))/2. Therefore the inverse relation gives z = √2 erfinv(2p - 1) for p in (0,1).

Applications: erfinv is used to transform probabilities to Gaussian quantiles, to solve diffusion equations, and in statistics and signal processing. It is common in generating standard normal deviates via inverse transform methods and in expressing tails or hit probabilities for the normal distribution.

Computational notes: erfinv has no simple closed form and is computed numerically. Many languages provide it as part of special functions libraries (for example SciPy's erfinv, MATLAB's erfinv). Implementations typically use Newton iterations with initial approximations or series/continued-fraction expansions to cover different y regions.