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heyoka.py is a Python library for the integration of ordinary differential equations (ODEs) via Taylor's method. Notable features include: * support for both double-precision and extended-precision floating-point types (80-bit and 128-bit), * the ability to maintain machine precision accuracy over tens of billions of timesteps, * high-precision zero-cost dense output, * accurate and reliable event detection, * batch mode integration to harness the power of modern SIMD instruction sets, * interoperability with SymPy, * a high-performance implementation of Taylor's method based on automatic differentiation techniques and aggressive just-in-time compilation via LLVM.