Floating point problems (accuracy).

[Chris Lattner]

On Tue, 18 Jan 2005, James E Wilson wrote:

> 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.

Ah, interesting, I did not know that.  I guess if you wanted REALLY 
inefficient code, the way to do this would be to play with the X86 
precision register.

> 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.

I see.

> 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.

Okay, thanks for the clarification.  You're absolutely right, we are 
susceptible to double rounding errors.

> 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.

I believe changing the precision fields would work, but at some point 
you're better off using soft-float...

-Chris

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