Error Functions and Inverses

Functions to compute the error function, complementary error function, and their inverses.

Error functions appear frequently in probability, statistics, and partial differential equations, particularly in the study of normal distributions and diffusion processes.

Error Function:

The error function is defined by the integral: $$erf(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt$$

The error function is an odd function (erf(-z) = -erf(z)). Implementation uses rational approximations optimised for absolute error, particularly for |z| <= 0.5.

Complementary Error Function:

The complementary error function is defined as: $$erfc(z) = 1 - erf(z) = \frac{2}{\sqrt{\pi}} \int_z^\infty e^{-t^2} dt$$

Key reflection formulas:

• erfc(-z) = 2 - erfc(z) (preferred when -z < -0.5)

• erfc(-z) = 1 + erf(z) (preferred when -0.5 <= -z < 0)

For large z, uses exponential scaling to maintain numerical stability.

Inverse Functions:

• erf_inv(p): Returns x such that p = erf(x), where -1 <= p <= 1

• erfc_inv(p): Returns x such that p = erfc(x), where 0 <= p <= 2

Inverse functions use rational approximations with different formulas for different ranges of p, achieving accuracy to less than ~2 epsilon for standard precision types.

Usage

erf(x)

erfc(x)

erf_inv(p)

erfc_inv(p)

Arguments

x

Input numeric value

p

Probability value for the inverse functions

Value

A single numeric value with the computed error function, complementary error function, or their inverses.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Error function
erf(0.5)
#> [1] 0.5204999
# Complementary error function
erfc(0.5)
#> [1] 0.4795001
# Inverse error function
erf_inv(0.5)
#> [1] 0.4769363
# Inverse complementary error function
erfc_inv(0.5)
#> [1] 0.4769363