How MPFR Library Calculation Accuracy Works

I confess that I am having trouble understanding the MPFR library, I am trying to calculate the square root of a large number but I am not sure how to define the precision of the mpft_t root variable or what is the best way of rounding.

my code is as follows:

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


int main(){

    mpfr_t bigNumber, bigNumber2, root2;

    unsigned long int size = 1000000;

    mpfr_init2(bigNumber, size);
    mpfr_init2(bigNumber2, size);
    mpfr_init2(root2, size);

    mpfr_ui_pow_ui(bigNumber, 8, 20000, MPFR_RNDZ);

    mpfr_sqrt(root2, bigNumber, MPFR_RNDA);

    mpfr_pow_ui(bigNumber2, root2, 2, MPFR_RNDA);

    return 0;
}

But regardless of the precision set for mpz_t root, the result is often not satisfactory.

The result of mpfr_pow_ui (bigNumber2, root, 2, MPFR_RNDA) is most often different than mpfr_t bigNumber, and I don't mean the floating value, but the entire part.

How to find out the precision needed to perform a certain calculation?

what is the best form of rounding for this calculation?

why does this inaccuracy happen?

The code works fine (on GCC 9.3.0 and MPFR 4.1.0) and gives back the original number.

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


int main()
{
    mpfr_t bigNumber, bigNumber2, root2;
    unsigned long int size = 100000;
    mpfr_init2(bigNumber, size);
    mpfr_init2(bigNumber2, size);
    mpfr_init2(root2, size);
    mpfr_ui_pow_ui(bigNumber, 8, 20000, MPFR_RNDD);
    mpfr_out_str (stdout, 10, 0, bigNumber, MPFR_RNDD);
    putchar ('\n');
    putchar ('\n');
    putchar ('\n');
    mpfr_sqrt(root2, bigNumber, MPFR_RNDD);
    mpfr_out_str (stdout, 10, 0, root2, MPFR_RNDD);
    putchar ('\n');
    putchar ('\n');
    putchar ('\n');
    mpfr_pow_ui(bigNumber2, root2, 2, MPFR_RNDD);
    mpfr_out_str (stdout, 10, 0, bigNumber2, MPFR_RNDD);
    putchar ('\n');
    putchar ('\n');
    putchar ('\n');
    putchar ('\n');
    return 0;
}