&& sqrtFloor.pow(2).equals(x); // slow exact check
        return sqrtFloorIsExact ? sqrtFloor : sqrtFloor.add(BigInteger.ONE);
      case HALF_DOWN:
      case HALF_UP:
      case HALF_EVEN:
        BigInteger halfSquare = sqrtFloor.pow(2).add(sqrtFloor);
        /*
         * We wish to test whether or not x <= (sqrtFloor + 0.5)^2 = halfSquare + 0.25. Since both x

         *            sqrtFloor^2 <= x < (sqrtFloor + 1)^2
         * halfSquare - sqrtFloor <= x < halfSquare + sqrtFloor + 1
         * so |x - halfSquare| <= sqrtFloor.  Therefore, it's safe to treat x - halfSquare as a
         * signed long, so lessThanBranchFree is safe for use.
         */
        return sqrtFloor + lessThanBranchFree(halfSquare, x);
    }
    throw new AssertionError();