不仅仅是泰勒级数It’s not just Taylor series
针对近期 X 上关于近似公式 exp(−x²) ≈ (1 + cos(sin(x) + x))/2 的热烈讨论，有人认为其精准度仅是因为泰勒级数在 x^6 项之前都保持一致。作者明确反驳了这一观点，指出这种数学近似现象背后的机制远比简单的泰勒展开复杂。这提醒人们不要用过度简化的数学工具去解释所有巧合的近似关系。
John
There is still active discussion on X about the approximation
exp(−x²) ≈ (1 + cos(sin(x) + x))/2
and some are saying this can just be explained by Taylor series: the series for the two sides differ for the first time at the x6 term, so that’s why you get a good approximation. As I wrote yesterday, that’s only part of it.
If it were just about Taylor series you could use
exp(−x²) ≈ 1 − x² + x4/2
which also has error O(x6). But this approximation is only good for fairly small x, say in [−0.5, 0.5], whereas the approximation at the top of the post is good over [−4, 4]. When x = 4, the error in the cosine approximation is 0.002579 but the error in the Taylor approximation is 113, five orders of magnitude larger.
If the accuracy of the cosine approximation were due to Taylor series, then we’d expect the function
exp(−x²) − (1 + cos(sin(x) + x))/2
to be small not just over the interval [−4, 4] but over a disk of radius 4 in the complex plane. But it’s not. When x = 4i the function is on the order of 1013.
Both the cosine approximation and the Taylor approximation are good for small disks, say of radius 0.5. They’re both bad for much larger disks, and in fact the cosine approximation is worse.
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