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.
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. :-)