Is it always faster to use `x*x...` instead of `pow(x, i)` for exponentiation?

Determining the Efficiency of Exponentiation Techniques

Often, it is efficient to multiply a number by itself instead of using a function dedicated to exponentiation, like pow(). However, there may be exceptions to this rule, especially with certain exponents.

Consider the following code to test the performance of xx... and pow(x,i) for different exponents 'i':

#include <cstdlib>
#include <cmath>
#include <boost/date_time/posix_time/posix_time.hpp>

inline boost::posix_time::ptime now()
{
    return boost::posix_time::microsec_clock::local_time();
}

#define TEST(num, expression) \
double test##num(double b, long loops) \
{ \
    double x = 0.0; \
    boost::posix_time::ptime startTime = now(); \
    for (long i=0; i<loops; ++i) \
    { \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
        x += expression; \
    } \
    boost::posix_time::time_duration elapsed = now() - startTime; \
    std::cout << elapsed << " "; \
    return x; \
}

// Test cases for exponentiation using x*x...
TEST(1, b)
TEST(2, b*b)
TEST(3, b*b*b)
TEST(4, b*b*b*b)
TEST(5, b*b*b*b*b)

// Test cases for exponentiation using pow()
template <int exponent>
double testpow(double base, long loops)
{
    double x = 0.0;
    boost::posix_time::ptime startTime = now();
    for (long i = 0; i < loops; ++i)
    {
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
        x += std::pow(base, exponent);
    }
    boost::posix_time::time_duration elapsed = now() - startTime;
    std::cout << elapsed << " ";
    return x;
}

int main()
{
    long loops = 100000000l;
    double x = 0.0;
    std::cout << "1 ";
    x += testpow<1>(rand(), loops);
    x += test1(rand(), loops);

    std::cout << "\n2 ";
    x += testpow<2>(rand(), loops);
    x += test2(rand(), loops);

    std::cout << "\n3 ";
    x += testpow<3>(rand(), loops);
    x += test3(rand(), loops);

    std::cout << "\n4 ";
    x += testpow<4>(rand(), loops);
    x += test4(rand(), loops);

    std::cout << "\n5 ";
    x += testpow<5>(rand(), loops);
    x += test5(rand(), loops);
    std::cout << "\n" << x << "\n";
}

Results

The test results, which measure the time taken for each exponentiation method, demonstrate that xx... is indeed faster.

However, there are certain exceptions

Regarding pow(x, 3), the results may vary depending on the compiler and optimization flags, especially when using floating-point variables like "double".

In certain situations, xxx*... is not always faster than pow(x, 3), as observed in some reported benchmarks. This is due to optimization techniques employed by modern compilers.

Conclusion

While xx... remains generally faster for small exponent values like 1, 2, or 3, it is essential to consider specific implementation details and compiler optimization optimizations when determining the most efficient approach for your code.

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