erfcx

→Tensor

erfcx(x: Tensor)

source

Scaled complementary error function.

Computes $\mathrm{erfcx}(x) = e^{x^2}\, \mathrm{erfc}(x)$ , the Mills-ratio-friendly scaling of the complementary error function. Whereas $\mathrm{erfc}(x)$ underflows to zero for moderately large $x$ , erfcx decays only as $1/(x\sqrt\pi)$ and remains finite, which is essential for tail-probability evaluations of Gaussian-related distributions.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$e^{x^2}\,\mathrm{erfc}(x)$ element-wise, same shape and dtype as x.

Notes

Mathematical definition:

\mathrm{erfcx}(x) = e^{x^2}\,\mathrm{erfc}(x) = e^{x^2} \cdot \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt.

The implementation forms the product of exp(x*x) and erfc(x) directly, which is accurate for $|x| \lesssim 5$ . For very large positive x the two factors hit opposite floating-point extremes and accuracy degrades — use a dedicated continued-fraction expansion if extreme-value precision is required.

Examples

>>> import lucid
>>> from lucid.special import erfcx
>>> erfcx(lucid.tensor([0.0, 1.0, 5.0]))
Tensor([1.0000, 0.4276, 0.1107])