The Cosmic Linear Anisotropy Solving System (CLASS) incorporates improved approximation schemes to enhance both speed and precision in cosmological parameter estimation using Cosmic Microwave Background and Large Scale Structure data. The paper describes three key approximations used by CLASS for basic ΛCDM models: 1. **Tight Coupling Approximation (TCA)**: This approximation simplifies the equations governing the baryon-photon fluid during the tightly coupled regime before recombination. It can be set to first order, second order, or a compromise between the two. The paper derives the full second-order TCA formulas in the synchronous gauge and compares them with those from other codes, showing good agreement. 2. **Ultra-Relativistic Fluid Approximation (UFA)**: This approximation treats massless neutrinos and ultra-relativistic relics as a single species, assuming they are fully decoupled. It introduces a sub-Hubble fluid approximation, where the Boltzmann hierarchy is truncated at a low multipole value deep inside the Hubble radius. The UFA is designed to save computational time during radiation domination and at the beginning of matter domination. 3. **Radiation Streaming Approximation (RSA)**: This approximation accounts for reionization by relativistic relics, such as massless neutrinos. It is implemented to consistently include reionization effects in the cosmological simulations. The paper also discusses the implementation of these approximations in CLASS, their impact on the accuracy and speed of the code, and comparisons with other codes like CAMB. The results show that the improved approximations significantly enhance the efficiency and precision of CLASS, making it a valuable tool for cosmological data analysis.The Cosmic Linear Anisotropy Solving System (CLASS) incorporates improved approximation schemes to enhance both speed and precision in cosmological parameter estimation using Cosmic Microwave Background and Large Scale Structure data. The paper describes three key approximations used by CLASS for basic ΛCDM models: 1. **Tight Coupling Approximation (TCA)**: This approximation simplifies the equations governing the baryon-photon fluid during the tightly coupled regime before recombination. It can be set to first order, second order, or a compromise between the two. The paper derives the full second-order TCA formulas in the synchronous gauge and compares them with those from other codes, showing good agreement. 2. **Ultra-Relativistic Fluid Approximation (UFA)**: This approximation treats massless neutrinos and ultra-relativistic relics as a single species, assuming they are fully decoupled. It introduces a sub-Hubble fluid approximation, where the Boltzmann hierarchy is truncated at a low multipole value deep inside the Hubble radius. The UFA is designed to save computational time during radiation domination and at the beginning of matter domination. 3. **Radiation Streaming Approximation (RSA)**: This approximation accounts for reionization by relativistic relics, such as massless neutrinos. It is implemented to consistently include reionization effects in the cosmological simulations. The paper also discusses the implementation of these approximations in CLASS, their impact on the accuracy and speed of the code, and comparisons with other codes like CAMB. The results show that the improved approximations significantly enhance the efficiency and precision of CLASS, making it a valuable tool for cosmological data analysis.