Intersection of 2 lines not really connected in TikZ

Here I keep the Jake's answer to my beginning question, which might be useful for someone. For this to work, the rectangle shall be "empty" and not in background layer.

documentclass[tikz]{standalone}

begin{document}
begin{tikzpicture}

draw (0,0) -- (2,2);
draw [draw=white,double=black,double distance=pgflinewidth,ultra thick] (0,2) -- (2,0);
draw (0.25,0.5) rectangle (1.75,1.5);

end{tikzpicture}
end{document}

Although an awesome answer from Mark Wibrow, I like things to be more smooth, so I implemented controls to his code (it ends up reducing a bit of the calculations as well):

documentclass[tikz, border=5mm]{standalone}
usetikzlibrary{calc}
usetikzlibrary{intersections}

%%% Adapted from Mark Wibrow
tikzset{%
    connect/.style args={(#1) to (#2) over (#3) by #4}{%
        insert path={%
            let p1=($(#1)-(#3)$), n1={veclen(x1,y1)}, n2={abs(#4)}  in%
                (#1) --%
                ($(#1)!n1-n2!(#3)$) .. controls +(0:n2/2) and +(180:n2/2) ..%
                +(n2,#4)%
                .. controls +(0:n2/2) and +(180:n2/2) .. ($(#1)!n1+n2!(#3)$) -- (#2)%
        }%
    }%
}

begin{document}
begin{tikzpicture}[line join=round]
  coordinate (a) at (-0.5,0.5);
  coordinate (b) at (8,-0.5);
  draw[ultra thick, name path=sine, domain=-1:8, smooth, samples=50] plot (x,{sin(x r)});
  path[name path=line] (a) -- (b);
  path[name intersections={of=sine and line}];
  draw[blue, connect={(a) to ($(a)!.5!(intersection-2)$) over (intersection-1) by 12pt},
        connect={($(a)!.5!(intersection-2)$) to ($(intersection-2)!.5!(intersection-3)$) over (intersection-2) by -8pt},
        connect={($(intersection-2)!.5!(intersection-3)$) to (b) over (intersection-3) by 4pt}
        ];
end{tikzpicture}
end{document}

Here's a solution using the spath3 package (development version at github). This package introduces the ability to split a path where it intersects with another path (or itself), to insert gaps, and to splice in other paths (such as arcs). So we do the following:

1. Save the two paths.
2. Split the over path where it crosses the under path.
3. Insert a decent sized gap at that point.
4. Splice in an arc into that gap.
5. Now split the under path where it crosses under this new version of the over path.
6. Insert a small gap at that point.

Since the gaps are genuine, there's no problem with the background path. If there's a possibility that the arc might be upside down then the development version contains a version of the splicing code that ensures that the arc is always the right way up.

Here's the result:

documentclass{article}
%url{https://tex.stackexchange.com/q/111660/86}
usepackage{tikz}
usetikzlibrary{
  spath3,
  intersections,
  backgrounds
}

begin{document}

begin{tikzpicture}

begin{pgfonlayer}{background}
draw[fill=yellow] (0.25,0.5) rectangle (1.75,1.5);
end{pgfonlayer}
path [spath/save=a] (0,0) -- (2,2);% line that will be repeated
path [spath/save=b] (0,2) -- (2,0);

path[spath/save=arc] (0,0) arc[radius=1cm, start angle=180, delta angle=-180];

tikzset{
  spath/split at intersections with={b}{a},
  spath/insert gaps after components={b}{8pt},
  spath/join components with={b}{arc},
  spath/split at intersections with={a}{b},
  spath/insert gaps after components={a}{4pt},
}

draw[spath/use=a];
draw[spath/use=b];

end{tikzpicture}
end{document}