Marc B. Reynolds Marc-B-Reynolds

Given two 2D point sets of 'n' elements both on unit disc, taking a point from each: p0=(x0,y0) and p1=(x1,y1)

map to point 'S' on 3-sphere (4D unit sphere):

	(* Mathematica source
	same method as the previous gist for double. This validates that the constants giving here are optimal:
	https://stackoverflow.com/questions/26692859/best-machine-optimized-polynomial-minimax-approximation-to-arctangent-on-1-1
	*)

	(* round(Pi,48,RN) *)
	s = 221069929750889

	(* prime factors of 's' and nearby integers *)
	f0 = FactorInteger[s];

	(*
	Pi correctly rounded to 106 digits via sollya:
	> display=dyadic!;
	> round(Pi,106,RN);
	63719069007931157819013617823235b-104
	*)

	s = 63719069007931157819013617823235

	(* prime factors of 's' and nearby integers *)

	// SEE: https://lemire.me/blog/2019/03/19/the-fastest-conventional-random-number-generator-that-can-pass-big-crush/

	__uint128_t g_lehmer64_state;

	uint64_t lehmer64() {
	uint64_t r = (g_lehmer64_state >> 64);
	g_lehmer64_state *= 0xda942042e4dd58b5;
	return r;
	}

	// Public Domain under http://unlicense.org, see link for details.

	#include <stdint.h>
	#include <stdlib.h>
	#include <stdio.h>
	#include <math.h>
	#include <string.h>

	#include "xmmintrin.h"

	// 1 dep-chain (assuming state words in reg) to result.
	// DON'T USE ME UNLESS YOUR PARINOID AND
	// USING SOMETHING MORE EXPENSIVE (actually inferior
	// to some other formulations...period is only 2^32-1)

	// Zero is illegal for both state words.
	typedef struct { uint32_t s[2]; } rng_state_t;

	// smallcrush: passes
	// crush: 1 systematic failure = 72 LinearComp, r = 29

	<!DOCTYPE html><html><head><meta charset='utf-8'><title>Plot</title><script src='https://cdn.plot.ly/plotly-latest.min.js'></script><script type='text/javascript' src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'></script></head><body>
	<style>body{margin:auto;padding:15px;text-align:justify;line-height:140%; -webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale;font-size:14px;font-smoothing:antialiased;color:#222;font-family:Roboto, sans-serif;}</style><style>h1{border-bottom: 2px solid #666; color: #333; padding: 20px 0 8px }</style><style>h2{border-bottom: 1px solid #888; color: #555; padding: 20px 0 4px }</style></style><style>div.funclist{font-size: 12px}</style><h1>Example report using arctangent</h1>
	This is overkill 'reporting' as demo. Actual intended use is to quickly visualize a small set of approximations over the domain. The script builds the HTML and can automatically open the results in a brower. A fair number of plot traces are so close they ove