erfc_real returns the value of the complementary error function, .

For full information please refer to the NAG Library document for s15ad


The argument of the function.


The value of the complementary error function, .


erfc_real calculates an approximate value for the complement of the error function

Let be the root of the equation (then ). For the value of is based on the following rational Chebyshev expansion for :

where denotes a rational function of degree in the numerator and in the denominator.

For the value of is based on a rational Chebyshev expansion for : for the value is based on the expansion

and for it is based on the expansion

For each expansion, the specific values of and are selected to be minimal such that the maximum relative error in the expansion is of the order , where is the maximum number of decimal digits that can be accurately represented for the particular implementation (see machine.decimal_digits).

For there is a danger of setting underflow in . For , erfc_real returns ; for it returns .


NIST Digital Library of Mathematical Functions

Cody, W J, 1969, Rational Chebyshev approximations for the error function, Math.Comp. (23), 631–637