The equality test is equivalent to deciding if something is 0.
You can probably ask the app to calculate sqrt(2) * sqrt(2). If that doesn't show a 2 as answer, you've lost.
Same goes for sin(arcsin(1/2)). You want the exact answer.
Equality is also needed if you want to know whether "x<y".
Proof: 1) suppose you can decide whether "x<y" for all x,y. 2) Then, for all x,y you can decide whether x==y because that's just "!(x<y) && !(y<x)"
Edit: the paper claims that you need exact equality tests to avoid printing unnecessary zeros at the end of decimal expansions, among other things.
Interesting that this work was done by Hans-J. Boehm, better known for the widely used Boehm garbage collector.