Benchmark RHA vs IEEE 754: Geometric and Arithmetic Precision in Rational Harmonic Arithmetic

This technical benchmark compares Rational Harmonic Arithmetic (RHA) with classical IEEE 754 double-precision floating-point arithmetic. Conducted on a 64-bit Google Colab environment, the benchmark evaluates both geometric and arithmetic tasks: the orthographic projection of a harmonic 3D heptagon using rational trigonometric interpolation, and an iterative accumulation of a non-binary rational value. RHA achieves an arithmetic relative error of 1.12 × 10⁻¹², matching NASA’s requirements for long-term simulations. Geometric errors, originally between 100% and 400% in early implementations, are now reduced to under 8.5% thanks to piecewise rational interpolation of sin₇ and cos₇. This improvement positions RHA as a viable framework for precision-critical applications in physics, quantum modeling, and 3D rendering. The benchmark highlights the trade-offs of current software-based RHA (notably memory usage and execution time) and calls for the development of a dedicated Rational Harmonic Arithmetic Processor (RHAP), capable of executing exact rational operations in hardware. Comparisons with LAPACK and supercomputing architectures show that, while those tools dominate in speed, they remain limited by cumulative floating-point errors. The complete Python code is provided for full reproducibility.