Question

I’m trying to determine the asymptotic run-time of one of my algorithms, which uses exponents, but I’m not sure of how exponents are calculated programmatically.

I’m specifically looking for the pow() algorithm used for double-precision, floating point numbers.

Answer 1

I’ve had a chance to look at fdlibm’s implementation. The comments describe the algorithm used:

 *                    n  * Method:  Let x =  2   * (1+f)  *      1. Compute and return log2(x) in two pieces:  *              log2(x) = w1 + w2,  *         where w1 has 53-24 = 29 bit trailing zeros.  *      2. Perform y*log2(x) = n+y' by simulating muti-precision  *         arithmetic, where |y'|<=0.5.  *      3. Return x**y = 2**n*exp(y'*log2) 

followed by a listing of all the special cases handled (0, 1, inf, nan).

The most intense sections of the code, after all the special-case handling, involve the log2 and 2** calculations. And there are no loops in either of those. So, the complexity of floating-point primitives notwithstanding, it looks like a asymptotically constant-time algorithm.

Floating-point experts (of which I’m not one) are welcome to comment. 🙂