Accurate measurements of stellar positions and velocities are crucial for studying galactic and stellar dynamics. We aim to create a Cartesian catalog from Gaia DR3 to serve as a high-precision database for further research using stellar coordinates and velocities. To avoid the negative parallax values, we select 31129169 sources in Gaia DR3 with radial velocity, where the fractional parallax error is less than 20% (0<{sigma}_{varpi}_/{varpi}<0.2). To select the most accurate and efficient method of propagating mean and covariance, we use the Monte Carlo results with 107 samples (MC7) as the benchmark, and compare the precision of linear, second-order, and Monte Carlo error propagation methods. By assessing the accuracy of propagated mean and covariance, we observe that second-order error propagation exhibits mean deviations of at most 0.5% compared to MC7, with variance deviations of up to 10%. Overall, this outperforms linear transformation. Though the Monte Carlo method with 104 samples (MC4) is an order of magnitude slower than second-order error propagation, its covariance propagation accuracy reaches 1% when {sigma}_{varpi}/{varpi} is below 15%. Consequently, we employ second-order error propagation to convert the mean astrometry and radial velocity into Cartesian coordinates and velocities in both equatorial and galactic systems for 30 million Gaia sources, and apply MC4 for covariance propagation. The Cartesian catalog and source code are provided for future applications in high-precision stellar and galactic dynamics.