[HP49/50g] ERF, ERF inverse, Q, Q inverse in SysRPL
01-01-2024, 07:56 AM (This post was last modified: 01-02-2024 05:04 AM by LdBeth.)
Post: #1
 LdBeth Junior Member Posts: 4 Joined: Dec 2023
[HP49/50g] ERF, ERF inverse, Q, Q inverse in SysRPL
There are seven functions of the type % --> % provided, they are ERF
function, Q function, and their inverse implemented using numerical
approximation methods. Internally they are all computed using extended
float.

ERF, IERF

Implementation of error function and its inverse using method from
Sergei Winitzki's paper.

ERFIV

Implementation of inverse error function using method from Mike Giles'
paper.

QNORM, IQNORM

Implementation of Q function and its inverse using ERF and IERF.
Note the UPTN function actually gives a more acurrate result for
Q function, but the inverse Q function is much faster than
computing the inverse of Q using numerical solver, while
the accuracy of both methods are approximately the same.

QBEST

A slightly better Q function approximation from [Dao Ngoc Phong et al].
If you ask me the result is still not as good as UPTN.

IQSML

This function from Pingyi Fan's paper gives decent approximation to
inverse Q function when the input is small enough, and can compute the
inverse when IQNORM gives infinity error.

Reference

[Sergei Winitzki] A handy approximation for the error
function and its inverse, February 6, 2008.

[Dao Ngoc Phong, Nguyen Xuan Hoai, R.I. (Bob) McKay] Evolving
the Best Known Approximation to the Q Function.

[Mike Giles] Approximating the erfinv function.

[Pingyi Fan] New inequalities of Mill's ratio and Its Application to
The Inverse Q-function Approximation