我已经实现了记忆所有计算权力的算法，然后在需要时使用它们。比如x^13等于(x^2)^2^2 * x^2 * x其中x^2^2是从表中取出来的而不是再计算一次。这基本上是@Pramod answer的实现(但在c#中)。 需要的乘法数是Ceil(Log n)

public static int Power(int base, int exp)
{
    int tab[] = new int[exp + 1];
    tab[0] = 1;
    tab[1] = base;
    return Power(base, exp, tab);
}

public static int Power(int base, int exp, int tab[])
    {
         if(exp == 0) return 1;
         if(exp == 1) return base;
         int i = 1;
         while(i < exp/2)
         {  
            if(tab[2 * i] <= 0)
                tab[2 * i] = tab[i] * tab[i];
            i = i << 1;
          }
    if(exp <=  i)
        return tab[i];
     else return tab[i] * Power(base, exp - i, tab);
}

如果要取2的a次方。最快的方法是按幂位移位。

2 ** 3 == 1 << 3 == 8
2 ** 30 == 1 << 30 == 1073741824 (A Gigabyte)

int pow(int const x, unsigned const e) noexcept
{
  return !e ? 1 : 1 == e ? x : (e % 2 ? x : 1) * pow(x * x, e / 2);
  //return !e ? 1 : 1 == e ? x : (((x ^ 1) & -(e % 2)) ^ 1) * pow(x * x, e / 2);
}

是的，它是递归的，但是一个好的优化编译器会优化递归。

请注意，平方求幂并不是最优的方法。这可能是一种适用于所有指数值的通用方法，但对于特定的指数值，可能有更好的序列，需要更少的乘法。

例如，如果你想计算x^15，用平方求幂的方法会给你:

x^15 = (x^7)*(x^7)*x 
x^7 = (x^3)*(x^3)*x 
x^3 = x*x*x

这一共有6次乘法。

事实证明，这可以通过“仅仅”5次加法链幂运算来完成。

n*n = n^2
n^2*n = n^3
n^3*n^3 = n^6
n^6*n^6 = n^12
n^12*n^3 = n^15

没有有效的算法来找到这个最优的乘法序列。从维基百科:

The problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations). For example, in the shortest addition chain for a¹⁵ above, the subproblem for a⁶ must be computed as (a³)² since a³ is re-used (as opposed to, say, a⁶ = a²(a²)², which also requires three multiplies).

如果您在编译时知道指数(并且它是一个整数)，您可以使用模板展开循环。这可以更有效，但我想在这里演示基本原则:

#include <iostream>

template<unsigned long N>
unsigned long inline exp_unroll(unsigned base) {
    return base * exp_unroll<N-1>(base);
}

我们使用模板特化来终止递归:

template<>
unsigned long inline exp_unroll<1>(unsigned base) {
    return base;
}

指数需要在运行时已知，

int main(int argc, char * argv[]) {
    std::cout << argv[1] <<"**5= " << exp_unroll<5>(atoi(argv[1])) << ;std::endl;
}

下面是一个计算x ** y的O(1)算法，灵感来自这条评论。它适用于32位有符号int。

对于较小的y值，它使用平方求幂。对于较大的y值，只有少数x值的结果不会溢出。这个实现使用一个查找表来读取结果而不进行计算。

对于溢出，C标准允许任何行为，包括崩溃。但是，我决定对LUT索引进行边界检查，以防止内存访问违反，这可能是令人惊讶和不受欢迎的。

伪代码:

If `x` is between -2 and 2, use special-case formulas.
Otherwise, if `y` is between 0 and 8, use special-case formulas.
Otherwise:
    Set x = abs(x); remember if x was negative
    If x <= 10 and y <= 19:
        Load precomputed result from a lookup table
    Otherwise:
        Set result to 0 (overflow)
    If x was negative and y is odd, negate the result

C代码:

#define POW9(x) x * x * x * x * x * x * x * x * x
#define POW10(x) POW9(x) * x
#define POW11(x) POW10(x) * x
#define POW12(x) POW11(x) * x
#define POW13(x) POW12(x) * x
#define POW14(x) POW13(x) * x
#define POW15(x) POW14(x) * x
#define POW16(x) POW15(x) * x
#define POW17(x) POW16(x) * x
#define POW18(x) POW17(x) * x
#define POW19(x) POW18(x) * x

int mypow(int x, unsigned y)
{
    static int table[8][11] = {
        {POW9(3), POW10(3), POW11(3), POW12(3), POW13(3), POW14(3), POW15(3), POW16(3), POW17(3), POW18(3), POW19(3)},
        {POW9(4), POW10(4), POW11(4), POW12(4), POW13(4), POW14(4), POW15(4), 0, 0, 0, 0},
        {POW9(5), POW10(5), POW11(5), POW12(5), POW13(5), 0, 0, 0, 0, 0, 0},
        {POW9(6), POW10(6), POW11(6), 0, 0, 0, 0, 0, 0, 0, 0},
        {POW9(7), POW10(7), POW11(7), 0, 0, 0, 0, 0, 0, 0, 0},
        {POW9(8), POW10(8), 0, 0, 0, 0, 0, 0, 0, 0, 0},
        {POW9(9), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
        {POW9(10), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
    };

    int is_neg;
    int r;

    switch (x)
    {
    case 0:
        return y == 0 ? 1 : 0;
    case 1:
        return 1;
    case -1:
        return y % 2 == 0 ? 1 : -1;
    case 2:
        return 1 << y;
    case -2:
        return (y % 2 == 0 ? 1 : -1) << y;
    default:
        switch (y)
        {
        case 0:
            return 1;
        case 1:
            return x;
        case 2:
            return x * x;
        case 3:
            return x * x * x;
        case 4:
            r = x * x;
            return r * r;
        case 5:
            r = x * x;
            return r * r * x;
        case 6:
            r = x * x;
            return r * r * r;
        case 7:
            r = x * x;
            return r * r * r * x;
        case 8:
            r = x * x;
            r = r * r;
            return r * r;
        default:
            is_neg = x < 0;
            if (is_neg)
                x = -x;
            if (x <= 10 && y <= 19)
                r = table[x - 3][y - 9];
            else
                r = 0;
            if (is_neg && y % 2 == 1)
                r = -r;
            return r;
        }
    }
}