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What is the number of floating point operations needed to perform exponentiation (power of)?
Assuming multiplication of two floats use one FLOP, the number of operations for $x^n$ will be $n-1$. However, is there a faster way to do this? How does it work if $n$ isn't an integer?
Assuming multiplication between two numbers use one FLOP, the number of operations for $x^n$ will be $n-1$. However, is there a faster way to do this ...
There most certainly is a faster way to do this for non-negative integer powers. For example, $x^{14}=x^{8}x^{4}x^{2}$. It takes one multiplication to compute $x^2$, one more to compute $x^4$, one more to compute $x^8$, and two more to multiple those three numbers. This suggests a simple cost and a simple algorithm.
This yields a concise algorithm for any non-negative integer power. This algorithm is the most efficient, up to $x^{14}$. This algorithm suggests six multiplications are needed to compute $x^{15}$ since $x^{15}=x^8x^4x^2x$. However, 15 is 120 in base 3 and 30 in base 5, both of which imply that only five multiplications are needed to compute $x^{15}$: $x^{15}=(x^3)^4x^3$ from the base three representation, and $x^{15}=(x^5)^3$ from the base five representation. The minimum number of multiplications needed to compute $x^n$ where $n$ is a non-negative integer is in fact an NP-complete problem. But it's a whole lot less than $n-1$ multiplications.
... and how does it work if $n$ isn't an integer?
There are some tricks one can use if $n$ is a rational. But if $x$ is real and $n$ is a non-negative real, one must resort to approximation techniques. (For example, approximation techniques are used twice-fold in calculating $\exp(n\ln(x))$.)
Using n-1 multiplications would be rather daft. For example, if n = 1024, you just square x ten times. Worst case is 2 * log_2 (n). You can look up Donald Knuth, Art of Computer Programming, for some details how to do it faster. There are some situations, like n = 1023, where you would square x ten times giving x^1024, then divide by x.
You can use the formula $$ x^y = \exp (y \ln x). $$
If you want to use only multiplications, when $n$ is a natural number you can use repeated squaring, that uses $O(\log n)$ multiplications. For other $n$, multiplication alone doesn’t suffice.
People told you what happens when $n$ is an integer.
Regarding when it isn't, there may not even exist a way to do floating-point exponentiation at all.
It's called the Table-Maker's Dilemma, which says the amount of memory you'd need is unbounded:
No general way exists to predict how many extra digits will have to be carried to compute a transcendental expression and round it correctly to some preassigned number of digits.
Even the fact (if true) that a finite number of extra digits will ultimately suffice may be a deep theorem.
pow(x,y) for some inputs. It tried very hard (arguably tried too hard) to chase down the elusive half an ULP accuracy.
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– David Hammen
Mar 3 at 19:14
If you are serious about the problem, you may not try to find a solution with the lowest number of multiplications, but with the lowest execution time.
Consider a model where you can start a multiplication in every cycle, but each multiplication takes a fixed number of cycles, say 3 cycles. A method calculating x^n with k multiplications might take 3k cycles (if each multiplication depends on a result that was calculated just before), while a a method using more multiplications might run quicker.
For example: To calculate x^15, you might calculate in that order x^2 = x*x, x^3 = (x^2)*x, x^6 = (x^3)^2, x^7 = x^6 * x, x^14 = (x^7)^2, x^15 = x^14 * x. Six multiplications, but each dependent on the previous one.
Or you calculate x^2, x^4 = (x^2)^2, x^3 = x^2 * x, x^5 = (x^4)*x, x^15 = x^5 * x^3, so you have only four multiplications depending on previous results.
