Either you misinterpreted or either I poorly explained myself, but there is effectively some disagreement.
What I wanted to say writing "NormInv is not so fast" is more precisely : "NormInv doesn't dramatically reduce the time to estimate quantiles for it's barely twice as fast as the latter"
Please, return to my first post and look to the factor 30 between what you said is a "magic" approximation and Quantile.
Of course this speed comes to a price, which is that the approximations are less precise than which you obtain.
So, from the problematic of fast approximations (and the second term is very important which means "we accept to have an error of, let's say, 10-6), the "magic" approximation outperforms NormInv.
You can object that the most important point is accuracy and that NormInv ia about twice faster than Quantile while being more precise, and I agree with that. But if you have, lets say, 106 or more, quantiles to estimate the "magic" method becomes really more interesting than NormInv.
You compute quantiles for probabilities as high as 1-10-20 and, I agree with that too, NormInv is by far better than the "magic" method. But "in practice", that is in usual statistics, you never have to compute such extremely extreme quantiles.
Look, in advanced technology the target reliability of systems is generally between 10-8 and 10-6 , and even: very small failure probabilities are often obtained by using redundancy or connecting such systems in some adhoc way.
What you need is then to estimate probabilities of this magnitude or, reciprocally, to estimate quantiles which (in the normal case) do not exceed +/-6 (which is in fact the idea of this popular, in some circles, "6 sigmas" method).
To set a final point to this debate:
- NormInv outperforms Quantile and it could become an improvement of this latter in future Maple's version
- Despite it's qualities NormInv doesn't fit the objective of much faster approximations of Quantile in the sense I tried to detial above
- It may be a pity that you spent so much time developing, in a direction that was not what I expected, my initial post which the main objective was to put attention on an improvement of the Abramowitz-Stegun's 26.2.23 formula (more generally to point out that this book can contain little errors).
I should have been clearer... but you quickly steered the debate in a particular direction saying "I find it hard to believe that any of these "magic coefficients" methods are generally better than this simple Newton's method, which only took 10 minutes to write, can be understood immediately, and works for any setting of Digits or in evalhf mode".
Looking retrospectively to the whole time spent, it was really much ado about nothing
Thank's for your involvement.
Here is the last test performed.