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PWR031: Replace pow by multiplication, division and/or square root

Issue​

The pow function (** operator in Fortran) is computationally expensive and in many cases can be replaced by faster mathematical operations.

Actions​

Replace pow with equivalent calculations using multiplications, divisions and/or square roots where possible.

Relevance​

The pow function is commonly used in scientific computations but is generally more expensive than alternative methods. When the exponent value is known at compile time, runtime performance can be significantly improved by substituting pow with a combination of simpler operations like multiplications, divisions, or square roots.

note

Some compilers under some circumstances (e.g. relaxed IEEE 754 semantics) can do this optimization automatically. However, doing it manually will guarantee best performance across all the compilers.

Code example​

C​

The following code invokes pow to calculate x to the power of 1.5:

// example.c
#include <math.h>
#include <stdio.h>

__attribute__((const)) double raise_x_to_the_power_of_1_point_5(double x) {
  return pow(x, 1.5);
}

int main() {
  printf("2 raised to the power of 1.5 is: %0.15f\n",
         raise_x_to_the_power_of_1_point_5(2.0));
  return 0;
}

This can also be accomplished by multiplying x by its square root, which is faster:

// solution.c
...

__attribute__((const)) double raise_x_to_the_power_of_1_point_5(double x) {
  return x * sqrt(x);
}

...

Moreover, we can verify that the computation retains full precision by running the following commands and comparing the results:

$ gcc --version
gcc (GCC) 14.2.1 20240910
$ gcc example.c -lm -o example
$ gcc solution.c -lm -o solution
$ ./example
2 raised to the power of 1.5 is: 2.828427124746190
$ ./solution
2 raised to the power of 1.5 is: 2.828427124746190

Fortran​

The following code uses the ** operator to compute x to the power of 1.5:

! example.f90
program main
  use iso_fortran_env, only : real64
  implicit none
  !
  print '(A, F0.15)', '2 raised to the power of 1.5 is: ', &
      raise_x_to_the_power_of_1_point_5(2.0_real64)
contains
  pure function raise_x_to_the_power_of_1_point_5(x)
    implicit none
    ! function return type
    real(kind=real64) :: raise_x_to_the_power_of_1_point_5
    ! dummy args
    real(kind=real64), intent(in) :: x
    !
    raise_x_to_the_power_of_1_point_5 = x ** 1.5_real64
  end function raise_x_to_the_power_of_1_point_5
end program main

This can be optimized by replacing ** with multiplication and the square root:

! solution.f90
program main
  ...
contains
  pure function raise_x_to_the_power_of_1_point_5(x)
    ...
    raise_x_to_the_power_of_1_point_5 = x * sqrt(x)
  end function raise_x_to_the_power_of_1_point_5
end program main

Moreover, we can verify that the computation retains full precision by running the following commands and comparing the results:

$ gfortran --version
GNU Fortran (GCC) 14.2.1 20240910
$ gfortran example.f90 -o example
$ gfortran solution.f90 -o solution
$ ./example
2 raised to the power of 1.5 is: 2.828427124746190
$ ./solution
2 raised to the power of 1.5 is: 2.828427124746190

References​