(12C Platinum) Normal Distribution

[Dieter]

(12-02-2018 09:11 AM)Dieter Wrote:  Fine, even with less digits. But it can still be tweaked a bit:

a1 = 0,433563
a2 = 0,084144052
a3 = -9,6366 E-5
b1 = 2,4629583
b2 = 1,1328872
b3 = 0,2059062

This way the error seems to stay a tiny bit below 1,12E-6.

There now is a new approximation with focus on the absolute error. But in a somewhat different way: With the following coefficients the error is less than 6,5 units in the 7th significant digit.

a1 = 0,4367117
a2 = 0,0851264
a3 = -9,34575 E-5
b1 = 2,46927
b2 = 1,1397766
b3 = 0,2085059

The valid domain is even a bit wider than before: the above error threshold is maintained up to x=5,1993..., which is the point where the CDF reaches 1 E–7. Which is easy to memorize: any result down to 1 E–7 has an error less than 7 units in its 7th significant digit. ;-)

This rational approximation can be complemented with the above CF expansion. Replace the constant 0,66 with 0,6444 and switch between both methods at x=5. This way the error stays below 7 units in the 7th significant digit for all x≥0.

Dieter