log_ndtr

→Tensor

log_ndtr(x: Tensor)

source

Numerically stable logarithm of the standard normal CDF.

Computes $\log \Phi(x)$ without losing precision in the deep left tail, where the direct composition log(ndtr(x)) underflows. Essential for log-likelihood evaluation of probit / Tobit / censored models that need to handle $x \ll 0$ cleanly.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$\log \Phi(x)$ element-wise, same shape and dtype as x; values lie in $(-\infty, 0]$ .

Notes

The implementation switches strategies on a branch boundary at x = -1:

\log \Phi(x) = \begin{cases} \log \Phi(x), & x \ge -1 \\[2pt] \log\!\left(\tfrac{1}{2}\, \mathrm{erfc}(-x/\sqrt{2})\right), & x < -1. \end{cases}

For $x \ge -1$ we have $\Phi(x) \gtrsim 0.16$ so the direct log(ndtr(x)) is safe. Below $x = -1$ the identity $\Phi(x) = \tfrac{1}{2}\,\mathrm{erfc}(-x/\sqrt{2})$ is used, sidestepping the catastrophic cancellation in $1 + \mathrm{erf}(x/\sqrt{2})$ .

Examples

>>> import lucid
>>> from lucid.special import log_ndtr
>>> log_ndtr(lucid.tensor([-10.0, -1.0, 0.0, 1.0]))
Tensor([-52.6651, -1.8410, -0.6931, -0.1727])