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 7th decimal is "not bad")

Also, the constant 1 ⁄ √ 2π does it have a name? All I was able to find was that "this expression ensures that the total area under the curve Φ(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 aproximate 16 decimals?

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?

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

Φ(x) = from -∞ to x ∫ ( (1 ⁄ √ ( 2π)) * ( e ^{-½ x²} ) )

This is approximated with what looks like an asymptotic series like here however the expression is slightly different:

Φ(x) ≈ (1 ⁄ ( x √ 2π)) * [ 1 - 1 ⁄x² + 3⁄x⁴ - 15⁄x⁶ + 105⁄x⁸]

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 7th decimal is "not bad")

Also, the constant 1 ⁄ √ 2π does it have a name? All I was able to find was that "this expression ensures that the total area under the curve Φ(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 aproximate 16 decimals?

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 7th decimal is "not bad")

Also, the constant 1 ⁄ √ 2π does it have a name? All I was able to find was that "this expression ensures that the total area under the curve Φ(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 aproximate 16 decimals?