The need for recognition of numerical (decimal, floating-point) constants in terms of elementary functions emerges in many areas of experimental mathematics, numerical analysis, computer algebra systems, model building, approximation and data compression. However, existing solutions are plagued by lack of any criteria distinguishing between random formula, matching literally decimal expansion (i.e. approximation) and probable "exact" (or at least probable) expression match in the sense of Occam's razor. In particular, convincing STOP criteria for search were never developed. In article, such a criteria, working in statistical sense, are provided. Recognition process can be viewed as (1) enumeration of all formulas in order of increasing Kolmogorov complexity (2) random process with appropriate statistical distribution (3) compression of a decimal string. All three approaches are remarkably consistent, and provide essentially the same limit for practical depth of search. Tested unique formulas count must not exceed 1/sigma, where sigma is relative numerical error of the target constant. Beyond that, further search is pointless, because, in the view of approach (1), number of equivalent expressions within error bounds grows exponentially; in view of (2), probability of random match approaches 1; in view of (3) compression ratio much smaller than 1.

中文翻译：

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数值常数识别标准

在实验数学、数值分析、计算机代数系统、模型构建、逼近和数据压缩的许多领域中，都需要根据初等函数来识别数值（十进制、浮点）常数。然而，现有的解决方案受到缺乏区分随机公式、匹配字面十进制扩展（即近似）和奥卡姆剃刀意义上的可能“精确”（或至少可能）表达式匹配的任何标准的困扰。特别是，从未开发出令人信服的搜索 STOP 标准。在文章中，提供了这样一个在统计意义上起作用的标准。识别过程可以看成（1）按照柯尔莫哥洛夫复杂度递增的顺序对所有公式进行枚举（2）具有适当统计分布的随机过程（3）十进制字符串的压缩。所有三种方法都非常一致，并且为实际搜索深度提供了基本相同的限制。测试的唯一公式计数不得超过 1/sigma，其中 sigma 是目标常数的相对数值误差。除此之外，进一步搜索是没有意义的，因为在方法（1）看来，误差范围内的等价表达式的数量呈指数增长；鉴于（2），随机匹配的概率接近1；鉴于（3）压缩比远小于1。测试的唯一公式计数不得超过 1/sigma，其中 sigma 是目标常数的相对数值误差。除此之外，进一步搜索是没有意义的，因为在方法（1）看来，误差范围内的等价表达式的数量呈指数增长；鉴于（2），随机匹配的概率接近1；鉴于（3）压缩比远小于1。测试的唯一公式计数不得超过 1/sigma，其中 sigma 是目标常数的相对数值误差。除此之外，进一步搜索是没有意义的，因为在方法（1）看来，误差范围内的等价表达式的数量呈指数增长；鉴于（2），随机匹配的概率接近1；鉴于（3）压缩比远小于1。