AI 中文总结
该研究提出基本超越函数的统一不动点构造，通过巴拿赫压缩原理建立加倍恒等式，将分析转移到残差函数以实现数值稳定，转化为高效算法，计算研究表明其内核在特定环境中有优势且无需查找表和内存流量。
AI 中文摘要
我们提出了一种基本超越函数的统一不动点构造，涵盖实指数函数、复指数函数（正弦和余弦）以及自然对数函数。每个函数都被表征为通过巴拿赫压缩原理建立的加倍恒等式的唯一解。这些基本恒等式对于指数函数是\(e(2x)=e^2(x)\)，对于对数函数是\(\log(x^{2})=2\log x\)。由于直接迭代这些恒等式在数值上不稳定，核心思想是将分析转移到一个残差函数上，在该函数上算子成为具有显式收敛速率的严格压缩。这种表征转化为用于基本函数机器求值的高效算法，其底层框架产生浮点内核，其精度和迭代深度由理论压缩率控制。我们还进行了计算研究，表明在吞吐量受限的向量化环境中，这些内核与标准生产库具有竞争力，在有利配置下超过它们，正弦 - 余弦内核在每个测试迭代深度都更快。这些实现无需查找表或内存流量，这对现代高性能和节能计算具有架构优势。
英文摘要
We present a unified fixed-point construction of the elementary transcendental functions, encompassing the real exponential, the complex exponential (sine and cosine), and the natural logarithm. Each function is characterized as the unique solution of a duplication identity established through the Banach contraction principle. These foundational identities are $e(2x)=e^2(x)$ for the exponentials, and $\log(x^{2})=2\log x$ for the logarithm. Because a direct iteration of these identities is numerically unstable, owing to local expansiveness at the target, the central idea transfers the analysis to a residual function, on which the operator becomes a strict contraction with an explicit convergence rate. Beyond its theoretical economy, which dispenses with differential equations and power series, this characterization translates into efficient algorithms for the machine evaluation of elementary functions: the underlying framework yields floating-point kernels whose accuracy and iteration depth are governed by the theoretical contraction rate. We also present a computational study showing that, in a throughput-bound vectorized regime, these kernels are competitive with standard production libraries, and in favorable configurations exceed them, with the sine--cosine kernel faster at every tested iteration depth. These implementations operate without lookup tables or memory traffic, an architectural advantage for modern high-performance and energy-efficient computing.
Comments23 pages, 5 tables
