HP calcs are really not that accurate..

[ttw]

One must use the "Interval Newton Method" which isn't Newton's method converted to interval arithmetic (as I described above).

https://ww2.ii.uj.edu.pl/~zgliczyn/cap07/krawczyk.pdf
http://www2.math.uni-wuppertal.de/~xsc/xsc/node12.html

Starting with the interval (0,1) one computes x0=(0+1)/2; assuming the 0 and 1 are exact and that d is the unit of rounding, one uses F=x^3-2x+2 and dF=3x^2-2 starting at (0+1)/2 or 1/2 with the Newton formula:

(1/2-(3/8-1+2)/(3/4-2) giving (1/2-(11/8)/(-5/4)) => (1/2+(11/8)*(4/5)) => (1/2+11/10)=>(5/8+d,5/8-d). The last interval is intersected with (0,1) and the process continued. This assumes that I did my arithmetic correctly. The formulas are in the links.