erfc

erfc (num [, accuracy ])

The erfc function returns the complementary error function of num (erfc(x) = 1 - erf(x)). Optionally, accuracy can be used to specify the desired fractional accuracy.

In complex expressions the complementary error function is

$$\displaystyle \operatorname{erfc}(z)=1-\operatorname{erf}(z)=1-\frac{2 z}{\sqrt{\pi}} \space{_1} F_{1}\left(\frac{1}{2} ; \frac{3}{2} ;-z^{2}\right),$$

where

$$\displaystyle { }_{1} F_{1}\left(\frac{1}{2} ; \frac{3}{2} ;-z^{2}\right)$$

is the confluent hypergeometric function of the first kind HyperG1F1. In this case the accuracy parameter is ignored.

Details

The accuracy parameter specifies the fractional accuracy that you desire. That is, if you set accuracy to 1e-7, that means that you wish that the absolute value of (f_actual-f_returned)/f_actual be less than 10^-7.

For backwards compatibility, in the absence of accuracy an alternate calculation method is used that achieves fractional accuracy better than 2x10^-7.

If accuracy is present, erfc can achieve fractional accuracy better than 2x10^-16 for num up to 1. From num = 1 to 10 fractional accuracy is better than 2x10^-15.

Higher accuracy takes somewhat longer to calculate. With accuracy set to 1e-16 erfc takes about 50% more time than with accuracy set to 1e-7.

See Also

erf, erfcw, erfcx, inverseErfc, dawson