快速求幂算法

#include <stdio.h>
#include <math.h>
//递归算法
int recursion(int a,int b)
{
    int tem = 1;
    if(b==0)return 1;
    else if(b==1)return a;
    tem = recursion(a,b>>1);
    tem = tem*tem;
    if(b&1) tem = tem * a;
    return tem;
}
//循环算法
int loop(int a,int b)
{
    int tem=1,ret=a;
    while(b>0)
    {
        if(b&1) tem = tem * ret;
        ret = ret*ret;
        b>>=1;
    }
    return tem;
}
int main()
{
    int a,b;
    while(1)
    {
    printf("For a^b input a and b:");
    scanf("%d%d",&a,&b);
    printf("The recursion method:\n");
    printf("%d\n",recursion(a,b));
    printf("The loop method:\n");
    printf("%d\n",loop(a,b));
    }
    return 0;
}

原理：

直接乘要做998次乘法。但事实上可以这样做，先求出2^k次幂：

3 ^ 2 = 3 * 3
3 ^ 4 = (3 ^ 2) * (3 ^ 2)
3 ^ 8 = (3 ^ 4) * (3 ^ 4)
3 ^ 16 = (3 ^ 8) * (3 ^ 8)
3 ^ 32 = (3 ^ 16) * (3 ^ 16)
3 ^ 64 = (3 ^ 32) * (3 ^ 32)
3 ^ 128 = (3 ^ 64) * (3 ^ 64)
3 ^ 256 = (3 ^ 128) * (3 ^ 128)
3 ^ 512 = (3 ^ 256) * (3 ^ 256)

再相乘：

3 ^ 999
= 3 ^ (512 + 256 + 128 + 64 + 32 + 4 + 2 + 1)
= (3 ^ 512) * (3 ^ 256) * (3 ^ 128) * (3 ^ 64) * (3 ^ 32) * (3 ^ 4) * (3 ^ 2) * 3