text: #include "tommath_private.h"
  #ifdef BN_MP_SQRT_C
  /* LibTomMath, multiple-precision integer library -- Tom St Denis */
  /* SPDX-License-Identifier: Unlicense */
  
  #ifndef NO_FLOATING_POINT
  #include <math.h>
  #if (MP_DIGIT_BIT != 28) || (FLT_RADIX != 2) || (DBL_MANT_DIG != 53) || (DBL_MAX_EXP != 1024)
  #define NO_FLOATING_POINT
  #endif
  #endif
  
  /* this function is less generic than mp_n_root, simpler and faster */
  mp_err mp_sqrt(const mp_int *arg, mp_int *ret)
  {
     mp_err err;
     mp_int t1, t2;
  #ifndef NO_FLOATING_POINT
     int i, j, k;
     volatile double d;
     mp_digit dig;
  #endif
  
     /* must be positive */
     if (arg->sign == MP_NEG) {
        return MP_VAL;
     }
  
     /* easy out */
     if (MP_IS_ZERO(arg)) {
        mp_zero(ret);
        return MP_OKAY;
     }
  
  #ifndef NO_FLOATING_POINT
  
     i = (arg->used / 2) - 1;
     j = 2 * i;
     if ((err = mp_init_size(&t1, i+2)) != MP_OKAY) {
        return err;
     }
  
     if ((err = mp_init(&t2)) != MP_OKAY) {
        goto E2;
     }
  
     for (k = 0; k < i; ++k) {
        t1.dp[k] = (mp_digit) 0;
     }
  
     /* Estimate the square root using the hardware floating point unit. */
  
     d = 0.0;
     for (k = arg->used-1; k >= j; --k) {
        d = ldexp(d, MP_DIGIT_BIT) + (double)(arg->dp[k]);
     }
  
     /*
      * At this point, d is the nearest floating point number to the most
      * significant 1 or 2 mp_digits of arg. Extract its square root.
      */
  
     d = sqrt(d);
  
     /* dig is the most significant mp_digit of the square root */
  
     dig = (mp_digit) ldexp(d, -MP_DIGIT_BIT);
  
     /*
      * If the most significant digit is nonzero, find the next digit down
      * by subtracting MP_DIGIT_BIT times thie most significant digit.
      * Subtract one from the result so that our initial estimate is always
      * low.
      */
  
     if (dig) {
        t1.used = i+2;
        d -= ldexp((double) dig, MP_DIGIT_BIT);
        if (d >= 1.0) {
           t1.dp[i+1] = dig;
           t1.dp[i] = ((mp_digit) d) - 1;
        } else {
           t1.dp[i+1] = dig-1;
           t1.dp[i] = MP_DIGIT_MAX;
        }
     } else {
        t1.used = i+1;
        t1.dp[i] = ((mp_digit) d) - 1;
     }
  
  #else
  
     if ((err = mp_init_copy(&t1, arg)) != MP_OKAY) {
        return err;
     }
  
     if ((err = mp_init(&t2)) != MP_OKAY) {
        goto E2;
     }
  
     /* First approx. (not very bad for large arg) */
     mp_rshd(&t1, t1.used/2);
  
  #endif
  
     /* t1 > 0  */
     if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) {
        goto E1;
     }
     if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) {
        goto E1;
     }
     if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) {
        goto E1;
     }
     /* And now t1 > sqrt(arg) */
     do {
        if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) {
           goto E1;
        }
        if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) {
           goto E1;
        }
        if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) {
           goto E1;
        }
        /* t1 >= sqrt(arg) >= t2 at this point */
     } while (mp_cmp_mag(&t1, &t2) == MP_GT);
  
     mp_exch(&t1, ret);
  
  E1:
     mp_clear(&t2);
  E2:
     mp_clear(&t1);
     return err;
  }
  
  #endif
  