Building upon our previous work on rational number representations over elliptic curves via equivalence relations, we extend the framework to encompass real number computations through precision-controlled rational approximations. We present four fundamental contributions: (1) a systematic two-layer architecture clearly separating external real number interfaces from internal rational computation engines, (2) advanced bootstrapping techniques for noise reduction and computational capacity extension, (3) a comprehensive noise management framework with concrete bounds and mitigation strategies, and (4) a complete elliptic curve-based fully homomorphic encryption (EC-FHE) scheme that supports arbitrary-precision real arithmetic while maintaining cryptographic security. Our construction enables privacy-preserving computations on real-valued data with formal guarantees on both computational correctness and cryptographic security, opening new applications in secure scientific computing, privacy-preserving machine learning, and confidential financial modeling.