Different results between 2013a 32bit and 64bit in single precision

I am squaring and summing two numbers in single precision, yet I get different results (one bit) between Matlab 32bit and Matlab 64bit on the same Win64 PC. Moreover, I get different results in Matlab 32bit between multiple implementations of this same equation...

Can anyone offer some insight?

Code to reproduce:

format hex
a = single([-0.1113907,0.64]);

%Method 1 (wrong on 32bit?)
r1 = a(1)^2 + a(2)^2

%Method 2 (wrong on 32bit?)
r2 = a(1)*a(1) + a(2)*a(2)

%Method 3 (correct)
tmp1 = a(1)^2; tmp2 = a(2)^2;
r3 = tmp1 + tmp2

%Method 4 (correct)
r4 = sum(a.^2)

Matlab 32bit results:

r1 =
   3ed8116a
r2 =
   3ed8116a
r3 =
   3ed8116b
r4 =
   3ed8116b

Matlab 64bit results:

r1 =
   3ed8116b
r2 =
   3ed8116b
r3 =
   3ed8116b
r4 =
   3ed8116b

Answer 1

[Walter Roberson]

Different execution paths and different processors (or even processor modes) have different roundoff effects.

[Jonathan Currie]

While that makes sense, I repeated this problem in C using x86 and x86-64 builds on the same PC and got the same solution (bit identical). Therefore differences in instruction sets should also show between these builds? I have both versions of Matlab using only one core (by setting the affinity) - no threading effects.

Could you expand perhaps what is going on a little more by Matlab? The issue is repeatable across multiple computers, operating systems and processor variations.

Answer 2

[Ken Atwell]

32-bit and 64-bit versions of MATLAB may use registers of differing widths as related in this 64-bit migration page, with single being called out as being particularly sensitive.

I forget the details, but it may have something to due with legacy 80-bit registers that are perhaps not used by 64-bit versions of MATLAB. 64-bit floating point is said to be more consistent that 32-bit floating point. Your compiler may or may not produce different answers depending on its implementation and the compilation flags you pass to it.

[Jonathan Currie]

Thanks Ken I'm happy with that explanation. I know the 80-bit registers are about, but I thought with larger width AVX/AVX2 registers in the later Intel Core series these had gone. Prof. Kahan's paper is also a good read on this topic (although quite dated): http://www.cs.berkeley.edu/~wkahan/MxMulEps.pdf

Also, I was using the default floating point flag (precise) in VC++, as described in the following article: http://msdn.microsoft.com/en-us/library/aa289157(v=vs.71).aspx

I would presume Matlab and its numerical libraries would use the same (or equivalent Intel C++ flags)?