Is floating-point addition and multiplication associative?

c++ floating-point multiplication addition associativity
Karen Tsirunyan picture Karen Tsirunyan · Apr 29, 2012 · Viewed 11.7k times · Source

I had a problem when I was adding three floating point values and comparing them to 1.

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

Why would these values come out different?

Answer

NPE picture NPE · Apr 29, 2012

Floating point addition is not necessarily associative. If you change the order in which you add things up, this can change the result.

The standard paper on the subject is What Every Computer Scientist Should Know about Floating Point Arithmetic. It gives the following example:

Another grey area concerns the interpretation of parentheses. Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers. For example, the expression (x+y)+z has a totally different answer than x+(y+z) when x = 1e30, y = -1e30 and z = 1 (it is 1 in the former case, 0 in the latter).

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