Suspect behavior of remez/plot (from gforge #8866)
Imported issue: Initially reported by @chevilla in https://gforge.inria.fr/tracker/?group_id=1015&aid=8866
remez behaves unexpectedly well on the following example:
b=round(pi,53,RD);p=remez(1,[|1,...,6|],[-1/8,b],1/sin(x));
Note that we exhibit a bug/problem/whatever in plot on the same example:
plot(p/sin(x)-1,[-1/8, 3.25]);
The curve seems regular when, obviously, it is not, since it has a pole in x=pi.
We can see the pole, by zooming extremely close to pi:
u=round(pi,80,RD); v=round(pi,80,RU); plot(p( (v-u)*x+u )/sin( (v-u)*x+u ), [0,1]);
Finally, note that, to the scale of double numbers, we see almost nothing:
u=round(pi,53,RD); v=round(pi,53,RU); plot(p( (v-u)*x+u )/sin( (v-u)*x+u ), [0,1]);
From the floating-point point of view, p/sin(x)-1 is indeed continuous!
I would be curious to know what it gives from the point of view of the evaluation. I expect p to be very ill-conditionned around b, but I am not sure to trust myself any more on this subject.