homepage

On some older Intel 32-bit hardware, under Linux, floating-point 
operations don't always give correctly rounded results. Here's an 
example involving addition, on SuSE Linux 10.2/Xeon.
Python 2.6a3+ (trunk:63521, May 21 2008, 15:40:39) 
[GCC 4.1.2 20061115 (prerelease) (SUSE Linux)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> 1e16 + 2.999
10000000000000002.0
>>> 1e16 + 2.9999
10000000000000004.0
The second result should really be 1e16+2., not 1e16+4. This appears to 
be related to this GCC issue:
http://gcc.gnu.org/bugzilla/show_bug.cgi?id=323
Various fixes are possible. One possible fix is to add the -ffloat-
store flag to the gcc options. Another is to use the information in 
fpu_control.h, if available, to set the precision control. Yet another 
is to sprinkle some 'volatile' modifiers throughout floatobject.c.
It's not clear to me that this *should* be fixed, but I think the 
problem should at least be documented. Hence this bug report.

Python (the language) makes no guarantee itself on what the precise
semantics of floating-point operations is. This is documented in
http://docs.python.org/ref/types.html
"These represent machine-level double precision floating point numbers.
You are at the mercy of the underlying machine architecture (and C or
Java implementation) for the accepted range and handling of overflow."
If you want to, one could add ", precision" in the sentence; I think it
is fine as it stands.

Okay; so this is definitely not a Python bug---it's a well-known
and well-documented problem with IA32 floating-point. And I accept
that it's really not Python's responsibility to document this, either.
Nevertheless, it was a surprise to me when my (supposedly IEEE 754
compliant) Pentium 4 box produced this. I probably shouldn't have
been surprised. I'm aware of issues with 80-bit extended precision when 
programming in C, but naively expected that Python would be largely 
immune from these, since it's always going to force intermediate results 
from (80-bit) floating-point registers into (64-bit) memory slots.
There's an excellent recent article by David Monniaux, "The pitfalls of
verifying floating-point computations.", that's available online at
http://hal.archives-ouvertes.fr/hal-00128124
that explains exactly what's going on here (it's a case of double-
rounding, as described in section 3.1.2 of that paper).
Do you think a documentation patch that added this reference, along with 
the oft-quoted "What Every Computer Scientist Should Know About 
Floating-Point Arithmetic" by David Goldberg, to Appendix B of the 
tutorial would be acceptable?
One other thing that's worth mentioning: on Pentium 4 and later, the
gcc flags "-mfpmath=sse -msse2" appear to fix the problem, by forcing 
gcc to use the SSE floating-point unit instead of the x87-derived one.
In any case, I guess this report should be closed as 'invalid', but I 
hope that at least others who encounter this problem manage to find this 
bug report.

Closing as invalid.