Fast Fourier Transform Optimization - How To Pre Calculate Exponential?

Question

Actually, I think that I get the discrete Fourier transform some basics. And now I have some problems with the fast Fourier transform algorithm.

I don't want to share all the functions so as not to complicate the problem. But if you don't understand some parts I can edit the question.

Slow Fourier transform:

void slowft (float *x, COMPLEX *y, int n)
{
    COMPLEX tmp, z1, z2, z3, z4;
    int m, k;

/* Constant factor -2 pi */
    cmplx (0.0, (float)(atan (1.0)/n * -8.0), &tmp);
    
    printf (" constant factor -2 pi %f ", (float)(atan (1.0)/n * -8.0));
    for (m = 0; m<=n; m++)
    {
      
      cmplx (x[0], 0.0, &(y[m]));
      for (k=1; k<=n-1; k++)
      {
/* Exp (tmp*k*m) */
        cmplx ((float)k, 0.0, &z2);
        cmult (tmp, z2, &z3);
        cmplx ((float)m, 0.0, &z2);
        cmult (z2, z3, &z4);
        cexp (z4, &z2);
/* *x[k] */
        cmplx (x[k], 0.0, &z3);
        cmult (z2, z3, &z4);
/* + y[m] */
        csum (y[m], z4, &z2);
        y[m].real = z2.real; y[m].imag = z2.imag;
      }
    }
}

to make clear: cmplx is creating a complete number, cmult is complex multiplication and cexp is taking exponent. that's all.

And some optimizations:

void newslowft (double *x, COMPLEX *y, int n)
{
    COMPLEX tmp, z1, z2, z3, z4, *pre;
    long  m, k, i, p;

    pre = (COMPLEX *)malloc(sizeof(struct cpx)*1024);

/* Constant factor -2 pi */
    cmplx (0.0, atan (1.0)/n * -8.0, &z1);  
    cexp (z1, &tmp);

/* Pre-compute most of the exponential */
    cmplx (1.0, 0.0, &z1);          /* Z1 = 1.0; */
    //n=1024
    for (i=0; i<n; i++)
    {
      cmplx (z1.real, z1.imag, &(pre[i]));
      cmult (z1, tmp, &z3);
      cmplx (z3.real, z3.imag, &z1);
    }

/* Double loop to compute all Y entries */
    for (m = 0; m<n; m++)
    {
      cmplx (x[0], 0.0, &(y[m]));
      for (k=1; k<=n-1; k++)
      {
/* Exp (tmp*k*m) */
        p = (k*m % n);

/* *x[k] */
        cmplx (x[k], 0.0, &z3);
        cmult (z3, pre[p], &z4);
/* + y[m] */
        csum (y[m], z4, &z2);
        y[m].real = z2.real; 
        y[m].imag = z2.imag;
      }
    }
}

The problem: The first step of the optimization:

"precalculating some exponential inside the loop".

So this is actually what I ask. How does this algorithm calculate the all exponential? I think we are calculating the following exponentials: e^0 e^1 e^2.... e^1023 So where are the other exponentials?

I mean, in the first algorithm, inside the for loops we are using m (m=0; m<=1024; m ) and k(k=0; k<1023-1; k ) but, where is the e^1000*900?

As far as I understand, the second algorithm takes the mode according to n. I think this is the key point right? But I didn't get how to work?

Thanks in advance masters.