Why Doesn't Floating-Point Addition Always Follow the Associative Law?

Floating-Point Arithmetic: Exploring Associativity in Floating-Point Operations

In the realm of programming, floating-point numbers are essential for working with continuous values. However, floating-point arithmetic can sometimes behave differently from the mathematics we learned in school, especially when it comes to associativity.

Consider the example of adding three floating-point values and comparing them to 1:

Curiously, the two expressions produce different results. This discrepancy stems from the fact that floating-point addition is not always associative. In other words, changing the order in which we add values can alter the final result.

The phenomenon of non-associativity arises due to the limitations of floating-point representation. Floating-point numbers use a binary format with a finite number of bits, which limits their accuracy. As a result, computations with floating-point numbers can introduce rounding errors that accumulate over multiple operations.

In the example above, the addition of 0.7, 0.2, and 0.1 results in slightly different intermediate values depending on the order of operations. These slight differences propagate through the subsequent addition, ultimately leading to different final results.

To understand this concept further, let's consider an example from David Goldberg's seminal paper "What Every Computer Scientist Should Know about Floating Point Arithmetic":

In this case, the parentheses deeply affect the result. Adding 1e30 and -1e30 in parentheses results in a near-zero value due to rounding, which then gets added to 1. On the other hand, adding 1e30 and the result of -1e30 1 produces a non-zero value.

Therefore, when working with floating-point arithmetic, it is crucial to be aware of potential non-associativity and to be cautious when manipulating expressions involving repeated additions or other associative operations.

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