I read the following paragraph which claims to be the Taylor expansion of standard normal CDF for positive $x$.

$1 - \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} (a_1k + a_2k^2 + a_3k^3 + a_4k^4 + a_5k^5)$,

where $k:=\frac{1}{1+0.2316419x}, a_1=0.319381530, a_2=-0.356563782, a_3=1.781477937, a_4=-1.821255978, a_5=1.330274429.$

I do not get how this is derived even though I am numerically convinced of its truth. Could anyone explain this to me, please?


This is no Taylor expansion. It is a rational approximation listed as formula 26.2.17 in Abramowitz / Stegun (see http://people.math.sfu.ca/~cbm/aands/page_932.htm) and attributed to Hastings (1955), Approximations for Digital Computers.

  • $\begingroup$ Thanks. Where to find how this is derived, please? $\endgroup$ – LaTeXFan Nov 9 '16 at 10:17
  • $\begingroup$ @LaTeXFan. Probably curve fit after having chosen the form of the approximation. $\endgroup$ – Claude Leibovici Nov 9 '16 at 10:21
  • $\begingroup$ May be you have access to the Hastings reference in a public library, I do not know a derivation. $\endgroup$ – gammatester Nov 9 '16 at 10:22

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