Recursive exponentiation - c

Question

so I have to write a recursive algorithm for exponentiation and I have to use this to make the algorithm faster: and then I'd have to figure out how many time multiplication is happening. I wrote it, but I am not sure if I am right - also I need some help with figuring out the multiplication part.
#include <stdio.h>
#include <math.h>
double intpower(double x, int n)
{
double result;
if(n>1&&n%2!=0) {result=x*intpower(x,(n-1)/2)*intpower(x,(n-1)/2);}
if(n>1&&n%2==0) {result=intpower(x,n/2)*intpower(x,n/2);}
if(n==1) return x;
else return result;
}
int main()
{
int n;
double x,result;
printf("x\n");
scanf("%lf", &x);
printf("n\n");
scanf("%d", &n);
printf("result = %.2f\n", intpower(x,n));
return 0;
}

Answer 1

The inductive definitions are saying
If k is even, then x^k = [ x^(k/2) ] ^ 2
If k is odd, then x^k = x * [ x^(floor(k)/2) ] ^ 2
With these it's a bit easier to see how to arrange the recursion:
#include <stdio.h>
double int_pwr(double x, unsigned k)
{
if (k == 0) return 1;
if (k == 1) return x; // This line can be omitted.
double y = int_pwr(x, k/2);
return (k & 1) ? x * y * y : y * y;
}
int main(void)
{
double x;
unsigned k;
scanf("%lf%u", &x, &k);
printf("x^k=%lg\n", int_pwr(x, k));
return 0;
}
I've changed types to be a bit more logical and saved an exponential (in k) amount of work that the OP's solution does by making two recursive calls at each level.
As to the number of multiplications, it's pretty easy to see that if k's highest order bit is 2^p (i.e. at position p), then you'll need p multiplications for the repeated squarings. Another way of saying this is p = floor(log_2(k)). For example if k=4=2^2, you'll square the square to get the answer: 2 multiplications. Additionally you'll need q-1 more, where q is the number of 1's in k's binary rep. This is the number of times the check for "odd" will be true. I.e. if k = 5 (which has 2 bits that are 1's), you'll square the square and then multiply the result by x one more time. To summarize, the number of multiplications is p + q - 1 with p and q as defined above.

Answer 2

To figure out how many times multiplication is happening, you could count them in intpower().
static int count = 0;
double intpower(double x, int n) {
double result;
if(n>1&&n%2!=0) {result=x*intpower(x,(n-1)/2)*intpower(x,(n-1)/2); count += 2;}
if(n>1&&n%2==0) {result=intpower(x,n/2)*intpower(x,n/2); count++;}
if(n==1) return x;
else return result;
}
int main() {
int n;
double x,result;
printf("x\n");
scanf("%lf", &x);
printf("n\n");
scanf("%d", &n);
mcount = 0;
printf("result = %.2f\n", intpower(x,n));
printf("multiplcations = %d\n", mcount);
return 0;
}

Answer 3

Try this
double intpower(double x, int n)
{
if(n == 0) return 1;
if(n == 1) return x;
if(n%2!=0)
{
return x*intpower(x,(n-1));
}
else
{
x = intpower(x,n/2);
return x*x;
}
}
or you can reduce your function to one line
double intpower(double x, int n)
{
return n == 0 ? 1 : n%2 != 0 ? x*intpower( x, (n-1) ) : (x = intpower(x, n/2), x*x);
}