I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a normal distribution function; the original documentation mentions the same/similar formula quoted [here](http://mathworld.wolfram.com/NormalDistributionFunction.html)


$$\phi(x) = {1\over \sqrt{2\pi}}\int\_{-\infty}^x e^{-{1\over 2} x^2} \ dx$$


This is approximated with what looks like an asymptotic series like [here](http://mathworld.wolfram.com/AsymptoticSeries.html)
however the expression is slightly different:


$$\phi(x) \simeq {1\over x\sqrt{2\pi}} \Bigl( 1 - {1\over x^2} + {3\over x^4} - {15\over x^6} + {105 \over x^8}\Bigr)$$


The original document quotes a *"Cambridge statistical tables"* book which I don't have; also, the software uses a precalculated table for values of $x$ between $[-8.2, 8.2]$ and uses the approximation function for values of $x$ outside that interval.


I need to find some reference that explains why this particular formula was chosen, and how accurate the approximation really is.
(The documentation claims that an error less then the $7$th decimal is "not bad")


Also, the constant $1/\sqrt{2\pi}$ does it have a name? All I was able to find was that "this expression ensures that the total area under the curve $\Phi(x)$ is equal to one"


Does it matter if the constant has only $10$ decimals when the double type in Java (IEEE 754) has a precision of approximately $16$ decimals?



Approximation of a normal distribution function
This is the (divergent) asymptotic development for $
f(x)=\int\_x^{\infty} e^{-{1\over 2}t^2} \ dt
$ given by


$$f(x) \sim
e^{-{x^2\over 2}}\ (\ {1\over x}\ + \
\Sigma\_{k=1}^{\infty}\ {(-1)^k(2k-1)!\over 2^{k-1}(k-1)!}\ {1\over x^{2k+1}}\ )
$$
or
$$f(x) \sim
e^{-{x^2\over 2}}\ (\ {1\over x}\ - {1\over x^3} + {3\over x^5} - {15\over x^7} + {105\over x^9} - {945\over x^{11}}...)$$


It is easily obtained from an integration by parts. The remainder is given by an explicit integral. From its expression, one can check that $f(x)$ is in fact squeezed between two consecutive sums of the series. As a result, we have the bound, for all $x>0$,


$$0 \geq f(x) -
e^{-{x^2\over 2}}\ (\ {1\over x}\ - {1\over x^3} + {3\over x^5} - {15\over x^7} + {105\over x^9}) \geq - e^{-{x^2\over 2}}\ {945\over x^{11}}$$


Divergent series were standard tools at the beginning of the XXe century. I can't provide a reference, but the book of Hardy "*Divergent series*" may be a starting point for a bibliographic search.


The divergent asymptotics was chosen because it is "good" at infinity. When truncated to some power (say 9), it has the correct behavior when $x$ goes to infinity, that is, it goes to zero, just like $f(x)$. So it gives an interesting approximation of $f$ for all sufficiently big value of $x$. This of course is not the case of a development obtained by truncating a converging series in positive powers of $x$.


