Why isn't `int pow(int base, int exponent)` in the standard C++ libraries?

Question

As of C++11, special cases were added to the suite of power functions (and others). C++11 [c.math] /11 states, after listing all the float/double/long double overloads (my emphasis, and paraphrased):

Moreover, there shall be additional overloads sufficient to ensure that, if any argument corresponding to a double parameter has type double or an integer type, then all arguments corresponding to double parameters are effectively cast to double.

So, basically, integer parameters will be upgraded to doubles to perform the operation.

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Prior to C++11 (which was when your question was asked), no integer overloads existed.

Since I was neither closely associated with the creators of C nor C++ in the days of their creation (though I am rather old), nor part of the ANSI/ISO committees that created the standards, this is necessarily opinion on my part. I'd like to think it's informed opinion but, as my wife will tell you (frequently and without much encouragement needed), I've been wrong before :-)

Supposition, for what it's worth, follows.

I suspect that the reason the original pre-ANSI C didn't have this feature is because it was totally unnecessary. First, there was already a perfectly good way of doing integer powers (with doubles and then simply converting back to an integer, checking for integer overflow and underflow before converting).

Second, another thing you have to remember is that the original intent of C was as a systems programming language, and it's questionable whether floating point is desirable in that arena at all.

Since one of its initial use cases was to code up UNIX, the floating point would have been next to useless. BCPL, on which C was based, also had no use for powers (it didn't have floating point at all, from memory).

As an aside, an integral power operator would probably have been a binary operator rather than a library call. You don't add two integers with x = add (y, z) but with x = y + z - part of the language proper rather than the library.

Third, since the implementation of integral power is relatively trivial, it's almost certain that the developers of the language would better use their time providing more useful stuff (see below comments on opportunity cost).

That's also relevant for the original C++. Since the original implementation was effectively just a translator which produced C code, it carried over many of the attributes of C. Its original intent was C-with-classes, not C-with-classes-plus-a-little-bit-of-extra-math-stuff.

As to why it was never added to the standards before C++11, you have to remember that the standards-setting bodies have specific guidelines to follow. For example, ANSI C was specifically tasked to codify existing practice, not to create a new language. Otherwise, they could have gone crazy and given us Ada :-)

Later iterations of that standard also have specific guidelines and can be found in the rationale documents (rationale as to why the committee made certain decisions, not rationale for the language itself).

For example the C99 rationale document specifically carries forward two of the C89 guiding principles which limit what can be added:

• Keep the language small and simple.
  
• Provide only one way to do an operation.

Guidelines (not necessarily those specific ones) are laid down for the individual working groups and hence limit the C++ committees (and all other ISO groups) as well.

In addition, the standards-setting bodies realise that there is an opportunity cost (an economic term meaning what you have to forego for a decision made) to every decision they make. For example, the opportunity cost of buying that $10,000 uber-gaming machine is cordial relations (or probably all relations) with your other half for about six months.

Eric Gunnerson explains this well with his -100 points explanation as to why things aren't always added to Microsoft products- basically a feature starts 100 points in the hole so it has to add quite a bit of value to be even considered.

In other words, would you rather have a integral power operator (which, honestly, any half-decent coder could whip up in ten minutes) or multi-threading added to the standard? For myself, I'd prefer to have the latter and not have to muck about with the differing implementations under UNIX and Windows.

I would like to also see thousands and thousands of collection the standard library (hashes, btrees, red-black trees, dictionary, arbitrary maps and so forth) as well but, as the rationale states:

A standard is a treaty between implementer and programmer.

And the number of implementers on the standards bodies far outweigh the number of programmers (or at least those programmers that don't understand opportunity cost). If all that stuff was added, the next standard C++ would be C++215x and would probably be fully implemented by compiler developers three hundred years after that.

Anyway, that's my (rather voluminous) thoughts on the matter. If only votes were handed out based on quantity rather than quality, I'd soon blow everyone else out of the water. Thanks for listening :-)

For any fixed-width integral type, nearly all of the possible input pairs overflow the type, anyway. What's the use of standardizing a function that doesn't give a useful result for vast majority of its possible inputs?

You pretty much need to have an big integer type in order to make the function useful, and most big integer libraries provide the function.

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Edit: In a comment on the question, static_rtti writes "Most inputs cause it to overflow? The same is true for exp and double pow, I don't see anyone complaining." This is incorrect.

Let's leave aside exp, because that's beside the point (though it would actually make my case stronger), and focus on double pow(double x, double y). For what portion of (x,y) pairs does this function do something useful (i.e., not simply overflow or underflow)?

I'm actually going to focus only on a small portion of the input pairs for which pow makes sense, because that will be sufficient to prove my point: if x is positive and |y| <= 1, then pow does not overflow or underflow. This comprises nearly one-quarter of all floating-point pairs (exactly half of non-NaN floating-point numbers are positive, and just less than half of non-NaN floating-point numbers have magnitude less than 1). Obviously, there are a lot of other input pairs for which pow produces useful results, but we've ascertained that it's at least one-quarter of all inputs.

Now let's look at a fixed-width (i.e. non-bignum) integer power function. For what portion inputs does it not simply overflow? To maximize the number of meaningful input pairs, the base should be signed and the exponent unsigned. Suppose that the base and exponent are both n bits wide. We can easily get a bound on the portion of inputs that are meaningful:

• If the exponent 0 or 1, then any base is meaningful.
• If the exponent is 2 or greater, then no base larger than 2^(n/2) produces a meaningful result.

Thus, of the 2^(2n) input pairs, less than 2^(n+1) + 2^(3n/2) produce meaningful results. If we look at what is likely the most common usage, 32-bit integers, this means that something on the order of 1/1000th of one percent of input pairs do not simply overflow.

Because there's no way to represent all integer powers in an int anyways:

>>> print 2**-4
0.0625

That's actually an interesting question. One argument I haven't found in the discussion is the simple lack of obvious return values for the arguments. Let's count the ways the hypthetical int pow_int(int, int) function could fail.

1. Overflow
2. Result undefined pow_int(0,0)
3. Result can't be represented pow_int(2,-1)

The function has at least 2 failure modes. Integers can't represent these values, the behaviour of the function in these cases would need to be defined by the standard - and programmers would need to be aware of how exactly the function handles these cases.

Overall leaving the function out seems like the only sensible option. The programmer can use the floating point version with all the error reporting available instead.

Short answer:

A specialisation of pow(x, n) to where n is a natural number is often useful for time performance. But the standard library's generic pow() still works pretty (surprisingly!) well for this purpose and it is absolutely critical to include as little as possible in the standard C library so it can be made as portable and as easy to implement as possible. On the other hand, that doesn't stop it at all from being in the C++ standard library or the STL, which I'm pretty sure nobody is planning on using in some kind of embedded platform.

Now, for the long answer.

pow(x, n) can be made much faster in many cases by specialising n to a natural number. I have had to use my own implementation of this function for almost every program I write (but I write a lot of mathematical programs in C). The specialised operation can be done in O(log(n)) time, but when n is small, a simpler linear version can be faster. Here are implementations of both:

    // Computes x^n, where n is a natural number.
    double pown(double x, unsigned n)
    {
        double y = 1;
        // n = 2*d + r. x^n = (x^2)^d * x^r.
        unsigned d = n >> 1;
        unsigned r = n & 1;
        double x_2_d = d == 0? 1 : pown(x*x, d);
        double x_r = r == 0? 1 : x;
        return x_2_d*x_r;
    }
    // The linear implementation.
    double pown_l(double x, unsigned n)
    {
        double y = 1;
        for (unsigned i = 0; i < n; i++)
            y *= x;
        return y;
    }

(I left x and the return value as doubles because the result of pow(double x, unsigned n) will fit in a double about as often as pow(double, double) will.)

(Yes, pown is recursive, but breaking the stack is absolutely impossible since the maximum stack size will roughly equal log_2(n) and n is an integer. If n is a 64-bit integer, that gives you a maximum stack size of about 64. No hardware has such extreme memory limitations, except for some dodgy PICs with hardware stacks that only go 3 to 8 function calls deep.)

As for performance, you'll be surprised by what a garden variety pow(double, double) is capable of. I tested a hundred million iterations on my 5-year-old IBM Thinkpad with x equal to the iteration number and n equal to 10. In this scenario, pown_l won. glibc pow() took 12.0 user seconds, pown took 7.4 user seconds, and pown_l took only 6.5 user seconds. So that's not too surprising. We were more or less expecting this.

Then, I let x be constant (I set it to 2.5), and I looped n from 0 to 19 a hundred million times. This time, quite unexpectedly, glibc pow won, and by a landslide! It took only 2.0 user seconds. My pown took 9.6 seconds, and pown_l took 12.2 seconds. What happened here? I did another test to find out.

I did the same thing as above only with x equal to a million. This time, pown won at 9.6s. pown_l took 12.2s and glibc pow took 16.3s. Now, it's clear! glibc pow performs better than the three when x is low, but worst when x is high. When x is high, pown_l performs best when n is low, and pown performs best when x is high.

So here are three different algorithms, each capable of performing better than the others under the right circumstances. So, ultimately, which to use most likely depends on how you're planning on using pow, but using the right version is worth it, and having all of the versions is nice. In fact, you could even automate the choice of algorithm with a function like this:

double pown_auto(double x, unsigned n, double x_expected, unsigned n_expected) {
    if (x_expected < x_threshold)
        return pow(x, n);
    if (n_expected < n_threshold)
        return pown_l(x, n);
    return pown(x, n);
}

As long as x_expected and n_expected are constants decided at compile time, along with possibly some other caveats, an optimising compiler worth its salt will automatically remove the entire pown_auto function call and replace it with the appropriate choice of the three algorithms. (Now, if you are actually going to attempt to use this, you'll probably have to toy with it a little, because I didn't exactly try compiling what I'd written above. ;))

On the other hand, glibc pow does work and glibc is big enough already. The C standard is supposed to be portable, including to various embedded devices (in fact embedded developers everywhere generally agree that glibc is already too big for them), and it can't be portable if for every simple math function it needs to include every alternative algorithm that might be of use. So, that's why it isn't in the C standard.

footnote: In the time performance testing, I gave my functions relatively generous optimisation flags (-s -O2) that are likely to be comparable to, if not worse than, what was likely used to compile glibc on my system (archlinux), so the results are probably fair. For a more rigorous test, I'd have to compile glibc myself and I reeeally don't feel like doing that. I used to use Gentoo, so I remember how long it takes, even when the task is automated. The results are conclusive (or rather inconclusive) enough for me. You're of course welcome to do this yourself.

Bonus round: A specialisation of pow(x, n) to all integers is instrumental if an exact integer output is required, which does happen. Consider allocating memory for an N-dimensional array with p^N elements. Getting p^N off even by one will result in a possibly randomly occurring segfault.

One reason for C++ to not have additional overloads is to be compatible with C.

C++98 has functions like double pow(double, int), but these have been removed in C++11 with the argument that C99 didn't include them.

http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2011/n3286.html#550

Getting a slightly more accurate result also means getting a slightly different result.