THE PRACTIAL APPROXIMATIONS TO NORMAL DISTRIBUTION FUNCTION AND ITS INVERSE

The present author proposes a compensation algorithm based on the principle of replacement of integral region and gives some practical approximations to the normal distribution and its inverse as follows: Y_m(u)=\left1-\frac4\pi \sum_i=1^m \theta_i e^-b_i u^2\left(1+c_m u^4\right)\right^\frac12 \sim 2 \Phi(u)-1, m=1,2,, \cdots with maximum error less than 9. 3×10^-5 for m=1 and 4×10^-6 for m=2, u=\leftu_0^2+d_1 u_0^4+d_2 u_0^0\right^\frac12, \quad u_0^2=-\frac\pi2 \ln (4 p(1-p)) with maximum error less than 1×10^-4. These approximations are better than Hastings’ and Yamuti’s formulas.