Floating point behavior

[eool <eoolsson@gmail.com>]

Hi all,

I was reading the C++ FAQ at http://www.parashift.com/c++-faq-lite/ when I
encountered an interesting curiosity in the "newbie questions" section. The
FAQ warns against directly comparing floating point numbers and, in section
29.18, gives an example of code (see below) that produces behavior that
seems unnatural to many (most?) programmers. (Below code copied from
http://www.parashift.com/c++-faq-lite/newbie.html#faq-29.18)

 #include <cmath>
 
 void foo(double x, double y)
 {
   if (cos(x) != cos(y)) {
     std::cout << "Huh?!?\n";  ← you might end up here when x == y!!
   }
 }
 
 int main()
 {
   foo(1.0, 1.0);
   return 0;
 } 

Basically, the reason given in the FAQ is that the result of cos(x)
(intermediate, higher precision value) will be stored in a register and thus
truncated. Then the result of cos(y) is directly compared with the truncated
value located in a register, resulting in this strange (?) behavior. Now, my
question is, why isn't gcc able to tell that it is about to compare two
floating point numbers, one of which has been truncated, and then similarly
truncate the second value before the comparison? (In fact, since the return
type of cos() is double, shouldn't the intermediate value be truncated to
fit a double in any case?)

Second (this question might be even more naive than the previous one), if
you are never, ever, supposed to compare floating point numbers using ==; if
you are always supposed to check the difference of the numbers against an
epsilon value; then how come the compiler can't automatically do this? Is it
because analyzing the source code to track the accumulated floating point
errors (to appropriately adjust epsilon) might be prohibitively
difficult/slow? Or are there actually any situations where you'd want to
compare two floating point numbers directly with each other?

Thanks for taking the time to answer my "newbie questions" :)

/Erik
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