Is Floating-Point Arithmetic Associative?

The Associativity of Floating-Point Arithmetic

Floating-point arithmetic is widely employed in scientific computing, yet a peculiar question arises: Is it associative for addition and multiplication? This seemingly simple query harbors hidden complexities.

In the realm of floating-point addition, associativity is not always guaranteed. A striking example is the following code:

cout << ((0.7 + 0.2 + 0.1) == 1) << endl;     //output is 0
cout << ((0.7 + 0.1 + 0.2) == 1) << endl;     //output is 1

To our astonishment, these two statements produce different results. Why is this so?

The explanation lies in the limitations of floating-point representation. Floating-point numbers are stored as an approximation of the actual value, introducing a degree of imprecision. When adding multiple floating-point values, the order of operations matters due to accumulated errors and rounding.

As the referenced standard paper, "What Every Computer Scientist Should Know about Floating Point Arithmetic," aptly states:

"Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers."

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