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

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

This is approximated with what looks like an asymptotic series like here 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?