In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$).
$$\operatorname{erf}(x)\approx 1 - e^{-x^2}\sum_{i=1}^5 a_i t^i +\epsilon(x),\quad 0\leq x<\infty,$$ where $t = 1/(1 + px)$. The coefficients $a_i$ and $p$ are some decimal numbers ($p = 0.3275911$, for example) and $|\epsilon(x)|\leq 1.5\cdot 10^{-7}$. The others are of similar form.
- How are these formulas derived?
- How to prove that the relative error $\epsilon(x)$ is really as small as they claim?
- What are the exact values of coefficients?
In the bottom of the page with formulas it is said that Abramowitz and Stegun took these formulas from C. Hastings: Approximations for digital computers. Indeed, the formulas are present there (starting at page $167$), also, there are graphs of $\epsilon(x)$, but again, no derivation or exact values.
Thank you.