Re: Floating point problems (accuracy).

[James E Wilson <wilson at specifixinc dot com>]

On Tue, 2005-01-18 at 14:31, Chris Lattner wrote:
> Aren't F1, F2, and F3 identical?

Yes.

The problem case would be something like
  float Y, Z;
  float X = (float)Y + (float)Z;
and
  float X = (float)((double)Y + (double)Z)
giving different results, because the second one is liable to double
rounding error.

Double rounding error can occur when the result of the add is almost
exactly half way between two representable float values.  Using decimal
for convenience, suppose the unrounded result of the add is something
like
	00...005,4999999999,999
where the first comma represents the rounding point for float, and the
second comma represents the rounding point for double.  If we round this
directly to float, we get 00...005.  If we round this to double, we get
00...005,500..00.  If we then round this to float, we get 00...006. 
Thus double rounding can introduce a single bit error in the least
significant bit of the result.

If you are using x87 long double arithmetic, and converting the result
to float/double, then you will get double rounding errors.  Double
rounding errors will be rare, and probably will affect few programs, but
it does mean that the results are still not exactly the same as the
results you will get on sparc/mips/power/x86_64/etc.  If the only
problem you have is double rounding, then you are certainly much better
off than the current x87 FP gcc support though.

The only way to avoid double rounding is to have separate float, double,
and long double arithmetic instructions, which unfortunately the x87 FP
unit doesn't have.
-- 
Jim Wilson, GNU Tools Support, http://www.SpecifixInc.com