y1=(x1-1)^2, y2=-7/8-x2^2
d^2=(y2-y1)^2 + (x2-x1)^2 = [-7/8-x2^2-(x1-1)^2]^2-(x2-x1)^2
to find minimum distance = minimum(d^2)
min{[-7/8-x2^2-(x1-1)^2]^2-(x2-x1)^2}
yields {x1,x2} = {3/4,1/4}
(you can get this min result in
www.wolframalpha.com)
then,
y1=(3/4-1)^2=1/16
y2=-7/8-(1/4)^2=-15/16
d=sqrt{[(1/16)-(-15/16)]^2+[(1/4)-(3/4)]^2}
d=sqrt(5)/2 = 1.11803398875...