#### Fast Power

/**
* @file
* @brief Faster computation for \f$a^b\f$
*
* Program that computes \f$a^b\f$ in \f$O(logN)\f$ time.
* It is based on formula that:
* 1. if \f$b\f$ is even:
*  \f$a^b = a^\frac{b}{2} \cdot a^\frac{b}{2} = {a^\frac{b}{2}}^2\f$
* 2. if \f$b\f$ is odd: \f$a^b = a^\frac{b-1}{2} * \cdot a^\frac{b-1}{2} \cdot a = {a^\frac{b-1}{2}}^2 \cdot a\f$
*
* We can compute \f$a^b\f$ recursively using above algorithm.
*/

#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdlib>
#include <ctime>
#include <iostream>

/**
* algorithm implementation for \f$a^b\f$
*/
template <typename T>
double fast_power_recursive(T a, T b) {
// negative power. a^b = 1 / (a^-b)
if (b < 0)
return 1.0 / fast_power_recursive(a, -b);

if (b == 0)
return 1;
T bottom = fast_power_recursive(a, b >> 1);
// Since it is integer division b/2 = (b-1)/2 where b is odd.
// Therefore, case2 is easily solved by integer division.

double result;
if ((b & 1) == 0)  // case1
result = bottom * bottom;
else  // case2
result = bottom * bottom * a;
return result;
}

/**
Same algorithm with little different formula.
It still calculates in \f$O(\log N)\f$
*/
template <typename T>
double fast_power_linear(T a, T b) {
// negative power. a^b = 1 / (a^-b)
if (b < 0)
return 1.0 / fast_power_linear(a, -b);

double result = 1;
while (b) {
if (b & 1)
result = result * a;
a = a * a;
b = b >> 1;
}
return result;
}

/**
* Main function
*/
int main() {
std::srand(std::time(nullptr));
std::ios_base::sync_with_stdio(false);

std::cout << "Testing..." << std::endl;
for (int i = 0; i < 20; i++) {
int a = std::rand() % 20 - 10;
int b = std::rand() % 20 - 10;
std::cout << std::endl << "Calculating " << a << "^" << b << std::endl;
assert(fast_power_recursive(a, b) == std::pow(a, b));
assert(fast_power_linear(a, b) == std::pow(a, b));

std::cout << "------ " << a << "^" << b << " = "
<< fast_power_recursive(a, b) << std::endl;
}

int64_t a, b;
std::cin >> a >> b;

std::cout << a << "^" << b << " = " << fast_power_recursive(a, b)
<< std::endl;

std::cout << a << "^" << b << " = " << fast_power_linear(a, b) << std::endl;

return 0;
}