How is pow() calculated in C?

Question

Our professor said that you can\'t calculate a^b if a<0 using pow() because pow() uses natural logarithms to calculate it (a^b

Answer 1

If you're curious how the pow function might be implemented in practice, you can look at the source code. There is a kind of "knack" to searching through unfamiliar (and large) codebases to find the section you are looking for, and it's good to get some practice.

One implementation of the C library is glibc, which has mirrors on GitHub. I didn't find an official mirror, but an unofficial mirror is at https://github.com/lattera/glibc

We first look at the math/w_pow.c file which has a promising name. It contains a function __pow which calls __ieee754_pow, which we can find in sysdeps/ieee754/dbl-64/e_pow.c (remember that not all systems are IEEE-754, so it makes sense that the IEEE-754 math code is in its own directory).

It starts with a few special cases:

if (y == 1.0) return x;
if (y == 2.0) return x*x;
if (y == -1.0) return 1.0/x;
if (y == 0) return 1.0;

A little farther down you find a branch with a comment

/* if x<0 */

Which leads us to

return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */

So you can see, for negative x and integer y, the glibc version of pow will compute pow(-x,y) and then make the result negative if y is odd.

This is not the only way to do things, but my guess is that this is common to many implementations. You can see that pow is full of special cases. This is common in library math functions, which are supposed to work correctly with unfriendly inputs like denormals and infinity.

The pow function is especially hard to read because it is heavily-optimized code which does bit-twiddling on floating-point numbers.

The C Standard

The C standard (n1548 §7.12.7.4) has this to say about pow:

A domain error occurs if x is finite and negative and y is finite and not an integer value.

So, according to the C standard, negative x should work.

There is also the matter of appendix F, which gives much tighter constraints on how pow works on IEEE-754 / IEC-60559 systems.