A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle  $ABEF$?

[asy]
pair A,B,C,D,I,F,G,H;
A=(-1,1);
B=(1,1);
H=(-2.41,-0.41);
C=(2.41,-0.41);
G=(-2.41,-2.41);
D=(2.41,-2.41);
F=(-1,-3.82);
I=(1,-3.82);
draw(A--B--C--D--I--F--G--H--cycle,linewidth(0.7));
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,E);
label("$E$",I,S);
label("$F$",F,S);
label("$G$",G,W);
label("$H$",H,W);
[/asy]
Let $O$ be the intersection of the diagonals of $ABEF$. Since the octagon is regular, $\triangle AOB$ has area $1/8$. Since $O$ is the midpoint of $\overline{AE}$, $\triangle OAB$ and $\triangle
BOE$ have the same area. Thus  $\triangle ABE$ has area $1/4$, so $ABEF$ has area $\boxed{\frac{1}{2}}$. [asy]
pair A,B,C,D,I,F,G,H;
A=(-1,1);
B=(1,1);
H=(-2.41,-0.41);
C=(2.41,-0.41);
G=(-2.41,-2.41);
D=(2.41,-2.41);
F=(-1,-3.82);
I=(1,-3.82);
draw(A--B--C--D--I--F--G--H--cycle,linewidth(0.7));
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,E);
label("$E$",I,S);
label("$F$",F,S);
label("$G$",G,W);
label("$H$",H,W);
draw(A--F);
draw(B--I);
draw(A--I,dashed);
draw(B--F,dashed);
label("$o$",(0,-1.42),E);
[/asy]