C loop optimization help for final assignment

So for my final assignment in my Computer Systems class, we need to optimize these forloops to be faster than the original. The basic grade is under 7 seconds and the full grade is under 5 seconds with our linux server. This code that I have right here gets about 5.6 seconds. I am thinking I may need to use pointers with this in some way to get it to go faster but I'm not really sure. Could anyone offer any tips or options that I have? Thank you so much!

QUICKEDIT: The file must remain 50 lines or less and I am ignoring those commented lines the instructor has included.

#include <stdio.h>
#include <stdlib.h>

// You are only allowed to make changes to this code as specified by the comments in it.

// The code you submit must have these two values.
#define N_TIMES     600000
#define ARRAY_SIZE   10000

int main(void)
{
    double  *array = calloc(ARRAY_SIZE, sizeof(double));
    double  sum = 0;
    int     i;

    // You can add variables between this comment ...
    register double sum1 = 0, sum2 = 0, sum3 = 0, sum4 = 0, sum5 = 0, sum6 = 0, sum7 = 0, sum8 = 0, sum9 = 0;
    register int j;
    // ... and this one.

    printf("CS201 - Asgmt 4 - \n");

    for (i = 0; i < N_TIMES; i++)
    {
        // You can change anything between this comment ...
        for (j = 0; j < ARRAY_SIZE; j += 10)
        {
            sum += array[j];
            sum1 += array[j + 1];
            sum2 += array[j + 2];
            sum3 += array[j + 3];
            sum4 += array[j + 4];
            sum5 += array[j + 5];
            sum6 += array[j + 6];
            sum7 += array[j + 7];
            sum8 += array[j + 8];
            sum9 += array[j + 9];
        }
        // ... and this one. But your inner loop must do the same
        // number of additions as this one does.
    }                   

    // You can add some final code between this comment ...
    sum += sum1 + sum2 + sum3 + sum4 + sum5 + sum6 + sum7 + sum8 + sum9;
    // ... and this one.

    return 0;
}

via Chebli Mohamed