Why doesn't GCC optimize a*a*a*a*a*a to (a*a*a)*(a*a*a)?

Question

I am doing some numerical optimization on a scientific application. One thing I noticed is that GCC will optimize the call pow(a,2) by compiling it into a

Answer 1

Fortran (designed for scientific computing) has a built-in power operator, and as far as I know Fortran compilers will commonly optimize raising to integer powers in a similar fashion to what you describe. C/C++ unfortunately don't have a power operator, only the library function pow(). This doesn't prevent smart compilers from treating pow specially and computing it in a faster way for special cases, but it seems they do it less commonly ...

Some years ago I was trying to make it more convenient to calculate integer powers in an optimal way, and came up with the following. It's C++, not C though, and still depends on the compiler being somewhat smart about how to optimize/inline things. Anyway, hope you might find it useful in practice:

template<unsigned N> struct power_impl;

template<unsigned N> struct power_impl {
    template<typename T>
    static T calc(const T &x) {
        if (N%2 == 0)
            return power_impl<N/2>::calc(x*x);
        else if (N%3 == 0)
            return power_impl<N/3>::calc(x*x*x);
        return power_impl<N-1>::calc(x)*x;
    }
};

template<> struct power_impl<0> {
    template<typename T>
    static T calc(const T &) { return 1; }
};

template<unsigned N, typename T>
inline T power(const T &x) {
    return power_impl<N>::calc(x);
}

_{Clarification for the curious: this does not find the optimal way to compute powers, but since finding the optimal solution is an NP-complete problem and this is only worth doing for small powers anyway (as opposed to using pow), there's no reason to fuss with the detail.}

Then just use it as power<6>(a).

This makes it easy to type powers (no need to spell out 6 as with parens), and lets you have this kind of optimization without -ffast-math in case you have something precision dependent such as compensated summation (an example where the order of operations is essential).

You can probably also forget that this is C++ and just use it in the C program (if it compiles with a C++ compiler).

Hope this can be useful.

EDIT:

This is what I get from my compiler:

For a*a*a*a*a*a,

    movapd  %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm1, %xmm0

For (a*a*a)*(a*a*a),

    movapd  %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm1, %xmm0
    mulsd   %xmm0, %xmm0

For power<6>(a),

    mulsd   %xmm0, %xmm0
    movapd  %xmm0, %xmm1
    mulsd   %xmm0, %xmm1
    mulsd   %xmm0, %xmm1

Answer 2

There are already a few good answers to this question, but for the sake of completeness I wanted to point out that the applicable section of the C standard is 5.1.2.2.3/15 (which is the same as section 1.9/9 in the C++11 standard). This section states that operators can only be regrouped if they are really associative or commutative.

Answer 3

Because Floating Point Math is not Associative. The way you group the operands in floating point multiplication has an effect on the numerical accuracy of the answer.

As a result, most compilers are very conservative about reordering floating point calculations unless they can be sure that the answer will stay the same, or unless you tell them you don't care about numerical accuracy. For example: the -fassociative-math option of gcc which allows gcc to reassociate floating point operations, or even the -ffast-math option which allows even more aggressive tradeoffs of accuracy against speed.

Answer 4

Lambdageek correctly points out that because associativity does not hold for floating-point numbers, the "optimization" of a*a*a*a*a*a to (a*a*a)*(a*a*a) may change the value. This is why it is disallowed by C99 (unless specifically allowed by the user, via compiler flag or pragma). Generally, the assumption is that the programmer wrote what she did for a reason, and the compiler should respect that. If you want (a*a*a)*(a*a*a), write that.

That can be a pain to write, though; why can't the compiler just do [what you consider to be] the right thing when you use pow(a,6)? Because it would be the wrong thing to do. On a platform with a good math library, pow(a,6) is significantly more accurate than either a*a*a*a*a*a or (a*a*a)*(a*a*a). Just to provide some data, I ran a small experiment on my Mac Pro, measuring the worst error in evaluating a^6 for all single-precision floating numbers between [1,2):

worst relative error using    powf(a, 6.f): 5.96e-08
worst relative error using (a*a*a)*(a*a*a): 2.94e-07
worst relative error using     a*a*a*a*a*a: 2.58e-07

Using pow instead of a multiplication tree reduces the error bound by a factor of 4. Compilers should not (and generally do not) make "optimizations" that increase error unless licensed to do so by the user (e.g. via -ffast-math).

Note that GCC provides __builtin_powi(x,n) as an alternative to pow( ), which should generate an inline multiplication tree. Use that if you want to trade off accuracy for performance, but do not want to enable fast-math.

Answer 5

As Lambdageek pointed out float multiplication is not associative and you can get less accuracy, but also when get better accuracy you can argue against optimisation, because you want a deterministic application. For example in game simulation client/server, where every client has to simulate the same world you want floating point calculations to be deterministic.

Answer 6

GCC does actually optimize a*a*a*a*a*a to (a*a*a)*(a*a*a) when a is an integer. I tried with this command:

$ echo 'int f(int x) { return x*x*x*x*x*x; }' | gcc -o - -O2 -S -masm=intel -x c -

There are a lot of gcc flags but nothing fancy. They mean: Read from stdin; use O2 optimization level; output assembly language listing instead of a binary; the listing should use Intel assembly language syntax; the input is in C language (usually language is inferred from input file extension, but there is no file extension when reading from stdin); and write to stdout.

Here's the important part of the output. I've annotated it with some comments indicating what's going on in the assembly language:

; x is in edi to begin with.  eax will be used as a temporary register.
mov  eax, edi  ; temp = x
imul eax, edi  ; temp = x * temp
imul eax, edi  ; temp = x * temp
imul eax, eax  ; temp = temp * temp

I'm using system GCC on Linux Mint 16 Petra, an Ubuntu derivative. Here's the gcc version:

$ gcc --version
gcc (Ubuntu/Linaro 4.8.1-10ubuntu9) 4.8.1

As other posters have noted, this option is not possible in floating point, because floating point arithmetic is not associative.