MATLAB: Identical math operations in C++ and matlab differ by ~1% (10^11 in this case!)

cnumerical precisionroundoff error

Hello,
I have a single (VERY LONG) line of code written in both C++ and matlab that evaluates a mathematical expression. The expression is just under 31,000 characters long, so it's pretty darn complicated.
When I evaluate this expression in matlab vs. C++ I get very different numbers:
MATLAB: 31856235311612.7
C++: 31745938896594.422
However, as you'll probably notice, they're not different enough to think I made a mistake in the transfer. I basically just copied and pasted the expression from C++, so no issues there.
I have two similar expressions of lower order that I also ported; one on the order of 10^9 (differs by about 1%), and one of order 10^7 which does not suffer the same magnitude of error:
MATLAB=3087973.34551207
C++=3087973.3455120716
The question is, which is "correct", and from where are these differences arising? I'm happy to provide code, but the first entry code is a 31,000 character expression!
FYI, in C++ all variables are stored as doubles.

Best Answer

A much simpler example:
MATLAB uses FDLIBM for the calculation of ACOS. I've added this library to a C-mex file also and compared the results of:
a = [ 0.82924747834055368, ...
      0.55849526207902467, ...
     -0.020776474703722032];
b = [-0.82889260818746668, ...
     -0.55895647212964961, ...
      0.022465670623305872];
acos(dot(a, b))
I found differences between the values calculated by C and Matlab up to 32768 EPS, depending on the C-compiler and the input values.
Let's look on the sensitivities:
b_var = b + [0, eps, 0];
acos(dot(a, b)) - acos(dot(a, b_var))
>> 1.2434e-013
This means that tiny differences in the input are amplified by the factor 560. This seems to be small. But together with some cancellation effects, such differences can explode: [EDITED]
r = 1 / (3.13980602793225 - acos(dot(a, b)));
>>  r = 2.2518e+014
r_var = 1 / (3.13980602793225 - acos(dot(a, b_var)));
>>  r_var = 7.7648e+012
Now the analysis of the sensitivity would reveal, that the algorithm is such instable, that it cannot be used to determine a single siginificant digit.
Conclusion: Calculating values using a 31'000 characters term without an analysis of the sensitivity is not meaningful. For a high sensitivity the results have a large larger random part.
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