u/Key_Bad_323

Hi everyone,

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I wanted to share an approximation for the classic Gaussian function $e\^{-x\^2}$ that I've been working on. It completely avoids traditional exponential evaluations, replacing them with basic algebra and a hyperbolic secant.

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\### The Idea

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I started by taking the derivative of the well-known Vedjer (Winitzki) approximation for $\\text{erf}(x)$:

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$$\\text{erf}(x) \\approx \\tanh\\left( \\frac{2}{\\sqrt{\\pi}} x + 0.147 x\^3 \\right)$$

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Directly differentiating this baseline configuration and isolating $e\^{-x\^2}$ yields a maximum absolute error of 1.88% (peaking around $x \\approx 0.63$). The issue is that the parameter $k = 0.147$ was mathematically tuned for the integral, not the slope.

\### The Optimization

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To fix this, I performed a non-linear parameter optimization specifically to fit the Gaussian curve.

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By tuning the parameter to $k \\approx 0.10307$, the final proposed formula becomes:

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$$e\^{-x\^2} \\approx \\left( 1 + 0.27403 x\^2 \\right) \\text{sech}\^2\\left( 1.12838 x + 0.10307 x\^3 \\right)$$

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\### The Results

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This single parameter shift causes the maximum absolute error to plummet to a staggering 0.00082 (0.082%) across the entire real line—marking a 20-fold precision increase compared to the baseline derivative!

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Here is a quick look at the pointwise residual behavior:

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\* x = 0.0: Exact = 1.00000 | Approx = 1.00000 | Error = 0.00000

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\* x = 0.5: Exact = 0.77880 | Approx = 0.77953 | Error = 0.00073

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\* x = 1.0: Exact = 0.36788 | Approx = 0.36735 | Error = 0.00053

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\* x = 2.0: Exact = 0.01832 | Approx = 0.01844 | Error = 0.00012

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I have made the full 2-page Research Note in LaTeX:

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https://drive.google.com/file/d/18fcq9Zoz9BsOlhAVgMcctRnW1bnvn89A/view?usp=drivesdk

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Would love to hear your thoughts on this!