(For three summands see PG's post in the thread where you originally brought up this question.) However, I am getting strange results depending on whether I use -O2 optimization or not, even though the code itself is serial. True in the case of IEEE-754: ... -Blas Pérez Pinilla Apr 20 '18 at 12:48 $\begingroup$ @BorbonJuggler Please clarify whether the question is restricted to floating-point arithmetic as defined by the IEEE-754 standard. Technically, it’s a bit more complex, but not as much as you might think, because IEEE 754 floats are designed to maintain their order when their bit patterns are interpreted as integers. However, this method does require the programming language to support conversion between floating-point values and integer bit patterns. Share. I've found it really hard to implement that simple test in Prolog. Floating-point addition and multiplication are both commutative (\(a+b = b+a\) and \(a\times b = b\times a\)), but they are not necessarily associative (the sum \((a+b)+c\) may differ from \(a+(b+c)\)) or distributive (\((a+b)\times c\) may not be the same as \(ac+bc\)). However, I am getting strange results depending on whether I use -O2 optimization or not, even though the code itself is serial. I would be surprised to see such a case (with two summands) in SAS 9.4 on Windows. Fun fact 2: there exist floating-point implementations different than IEEE 754 where addition is not necessarily commutative. Hello @LineMoon,. Commutativity of Floating Point Arithmetic That FP arithmetic can be tricky in multi-threaded environment is well known. Single precision: 32 bits. Cite. Three different “kinds” of floating point numbers based on the exp field: Computing with Floating Point Numbers As mentioned earlier, computers cannot represent real numbers precisely since there are only a finite number of bits for storing a real number. is implementation-dependent. 1 comment Open Floating point addition is not commutative - Unexpected behavior - leads to … However, in 2012 I encountered strange results with SAS 9.2 TS2M3 on Windows Server 2008, which came close to this (different results, in a sense, even without modifying the … Improve this question. Follow edited Jan 22 '16 at 15:48. exp field is 8 bits. That FP arithmetic can be tricky in multi-threaded environment is well known. The more interesting question "are the floating point operations $+$ and $\cdot$ commutative?" frac field is 52 bits. I've started to compare how different programming languages treat the following floating point comparison case: a=1 b=1/(1-epsilon) (a==b) == (b==a) That's a test to check if the equality operator ('==') is commutative. It later mentions "the importance of preserving parentheses". Double precision: 64 bits. Floating Point Precision; Floating Point Numbers. In particular, in the enclosed sample code, four complex arrays, z1, z2, z3, and z4, are initialized with random numbers so that their value ranges are pairwise at different scales. Therefore, any number that has infinite number of digits such as 1/3, the square root of 2 and PI cannot be represented completely. exp field is 11 bits. The only direct reference to the arithmetic axioms is a single sentence: "Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers." ... $\begingroup$ Floating point addition and multiplication is commutative but not associative in … example, floating-point addition and multiplication are commutative but not associative, i.e. But it is not commutative! abstract-algebra examples-counterexamples binary-operations associativity. • If a number, that is not exactly represent able as a floating-point number, is entered into the This is my first post to this list. : (a + b) + c ≠ (a + c) + b • A real arithmetic operation on two floating-point numbers does not necessarily result in another floating-point number. frac field is 23 bits. frac field encodes \(M\) (but is not equal to \(M\)) Precision options. Hi guys! For example, according to Towards sensible floating-point arithmetic multiplication is not commutative for Cray-1. 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