The error function is described in Abramowitz & Stegun, Chapter 7.
This routine computes the error function \(\operatorname{erf}(x)\), where
This routine computes the complementary error function
This routine computes the logarithm of the complementary error function \(\log(\operatorname{erfc}(x))\).
The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.
This routine computes the Gaussian probability density function \(Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)\).
This routine computes the upper tail of the Gaussian probability function \(Q(x) = (1/\sqrt{2\pi}) \int_x^\infty \exp(-t^2/2) dt\).
The hazard function for the normal distribution, also known as the inverse Mills’ ratio, is defined as,
It decreases rapidly as \(x\) approaches \(-\infty\) and asymptotes to \(h(x) \sim x\) as \(x\) approaches \(+\infty\).
This routine computes the hazard function for the normal distribution.