Are predicates used in CDT safe?

[artem-ogre]

Discussed in #99

^{Originally posted by msokalski August 10, 2022}
It seams to me that predicates library can be easily fooled by flipping sign and/or swapping x,y coordinates of a point to form 4 exactly concyclic set of points. Results are not very optimistic ... any thoughts? Do you reproduce same behaviour?

// example where adaptive differs from exact predicate
// adaptive seams to be wrong (I think rising xxx_errbound could fix this)

ax	-1.4048103911580639e-11	const double
ay	5.3534412196824579e-12	const double
bx	1.4048103911580639e-11	const double
by	-5.3534412196824579e-12	const double
cx	-1.4048103911580639e-11	const double
cy	-5.3534412196824579e-12	const double
dx	-5.3534412196824579e-12	const double
dy	-1.4048103911580639e-11	const double

predicates::adaptive::incircle<double> returned	1.5785423408550432e-75	double
predicates::exact::incircle<double> returned	0.0000000000000000	double
crude_xa::incircle<double> returned 		0.0000000000000000	double

// ---------------------------------------------------
// example where adaptive & exact predicates seams to be both wrong!

ax	-9.1428964336885931e-10	const double
ay	-5.2844166345548637e-19	const double
bx	-5.2844166345548637e-19	const double
by	-9.1428964336885931e-10	const double
cx	5.2844166345548637e-19	const double
cy	-9.1428964336885931e-10	const double
dx	-5.2844166345548637e-19	const double
dy	9.1428964336885931e-10	const double

predicates::adaptive::incircle<double> returned	1.2392294127517818e-119	double
predicates::exact::incircle<double> returned	1.2392294127517818e-119	double
crude_xa::incircle<double> returned 		0.0000000000000000	double

EDIT:
adding a more representative set of 4 points (both adaptive and exact predicates seams to be wrong here):

	double ax = -1.00;
	double ay = -0.01;
	double bx = -0.01;
	double by = -1.00;
	double cx =  0.01;
	double cy = -1.00;
	double dx = -0.01;
	double dy =  1.00;

I've narrowed the problem to the Dekker's product function. Forcing use of std::fma() solves the problem, probably at the cost of performance drop on platforms not having fma in hardware.