正态累积分布函数四位数的实用可逆逼近--对Winitzki近似的修正
Practical Explicitly Invertible Approximation to 4 Decimals of Normal Cumulative Distribution Function Modifying Winitzki's Approximation of erf
摘要：
我们给出了正态累积分布函数的一个新的显式可逆近似：$Φ(x) \simeq 1/2 + 1/2 \sqrt{1-{e}^{-x^2\frac{17+{x}^{2}}{26.694+2x^2}}}$，$\forall x \ge 0$，具有绝对误差$<4.00\cdot 10^{-5}$，相对误差的绝对值$<4.53\cdot 10^{-5}$，它基本上是为实际应用而设计的，比以前公布的公式简单得多，尽管精度较低，但仍达到小数点后4位的精度，并且具有基本上可与直接从Winitzki的erf $(x)$的近似导出的正态累积分布函数$Φ(x)$的近似相比的复杂性，相对于此减小了约36%的绝对误差和约28%的相对误差，克服了4位小数精度的阈值。
英文摘要：
We give a new explicitly invertible approximation of the normal cumulative distribution function: $Φ(x) \simeq 1/2 + 1/2 \sqrt{1-{e}^{-x^2\frac{17+{x}^{2}}{26.694+2x^2}}}$, $\forall x \ge 0$, with absolute error $<4.00\cdot 10^{-5}$, absolute value of the relative error $<4.53\cdot 10^{-5}$, which, beeing designed essentially for practical use, is much simpler than a previously published formula and, though less precise, still reaches 4 decimals of precision, and has a complexity essentially comparable with that of the approximation of the normal cumulative distribution function $Φ(x)$ immediatly derived from Winitzki's approximation of erf$(x)$, reducing about 36% the absolute error and about 28% the relative error with respect to that, overcoming the threshold of 4 decimals of precision.
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