Are floating point operations in C associative?

问题:

Addition mathematically holds the associative property:

(a + b) + c = a + (b + c)

In the general case, this property does not hold for floating-point numbers because they represent values in a finite precision.

Is a compiler allowed to make the above substitution when generating machine code from a C program as part of an optimization? Where does it exactly say in the C standard?

回答1:

The compiler is not allowed to perform "optimizations", which would result in a different value computed, than the one computed according to abstract machine semantics.

5.1.2.3 Program execution

[#1] The semantic descriptions in this International Standard describe the behavior of an abstract machine in which issues of optimization are irrelevant.

[#3] In the abstract machine, all expressions are evaluated as specified by the semantics.

[#13] EXAMPLE 5 Rearrangement for floating-point expressions is often restricted because of limitations in precision as well as range. The implementation cannot generally apply the mathematical associative rules for addition or multiplication, nor the distributive rule, because of roundoff error, even in the absence of overflow and underflow.

In your example:

(a + b) + c

or even without the parentheses:

a + b + c

we have

   +
  / \
  +  c
 / \
 a  b

and the compiler is required to generate code as if a is summed with b and the result is summed with c.

回答2:

Floating point multiplication in C is not associative.

In C, Floating point multiplication is not associative.

Some evidence is with this C code:

Pick three random float values.
Check if a*(b*c) is ever not equal to (a*b)*c

#include<stdio.h>
#include<time.h>
#include<stdlib.h>
using namespace std;
int main() {
    int counter = 0;
    srand(time(NULL));
    while(counter++ < 10){
        float a = rand() / 100000;
        float b = rand() / 100000;
        float c = rand() / 100000;

        if (a*(b*c) != (a*b)*c){
            printf("Not equal\n");
        }
    }
    printf("DONE");
    return 0;
}

The program prints:

Not equal
Not equal
Not equal
Not equal
DONE
RUN FINISHED; exit value 0; real time: 10ms; user: 0ms; system: 0ms

Conclusion:

For my test, three randomly selected floating point multiplication values are associative about 70% of the time.

回答3:

You can make floating point operations associative with the gcc options:

-funsafe-math-optimizations -O2

Example: double test (double a, double b, double c) {
return (a + b + c) * (a + (b + c)); }

This is reduced to: double temp = a + (b + c); return temp * temp;

Similarly, (a + b + c) - (a + (b + c)) is reduced to zero, ignoring the possibility of INF and NAN.

If I compile with -fassociative-math -O2 instead, I get the weird message: "warning: -fassociative-math disabled; other options take precedence".

The -funsafe-math-optimizations can improve speed if you don't care about the order of the operands anyway, but it may cause loss of precision if the order of operands is important, and you may lose NAN and INF results.