In this paper, we introduce a new approximation of the cumulative distribution function of the standard normal distribution based on Tocher's approximation. Also, we assess the quality of the new approximation using two criteria namely the maximum absolute error and the mean absolute error. The approximation is expressed in closed form and it produces a maximum absolute error of 4.43*10^(-10) while the mean absolute error is 9.62*10^(-11). In addition, we propose an approximation of the inverse cumulative function of the standard normal distribution based on Polya approximation and compare the accuracy of our findings with some of the existing approximations. The results show that our approximations surpass other existing ones based on the aforementioned accuracy measures.
翻译：在本文中,我们采用了基于 Tocher 近似值的标准正常分布的累积分布函数的新近似值。 另外,我们使用两个标准,即最大绝对误差和平均绝对误差来评估新近似值的质量。 近似值以封闭形式表示,产生最大绝对误差4.33*10 ⁇ (-10),而平均绝对误差为9.62*10 ⁇ (-11),此外,我们建议以 Polya 近近似值为基础,对标准正常分布的反累积函数近近近似值,并将我们的调查结果与现有的一些近近似值的准确性进行比较。 结果表明,根据上述精确度衡量,我们的近似值超过了其他现有误差。
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