> ## Documentation Index
> Fetch the complete documentation index at: https://offthepace.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# normal_cdf

> Standard normal CDF Phi(x).

Standard normal CDF Phi(x). DuckDB has no erf/normal CDF builtin, so we use the Zelen & Severo rational approximation (Abramowitz & Stegun 26.2.17): max absolute error \~7.5e-8 over the whole real line far below any calibration tolerance we care about. Phi(x) = 1 - phi(x) \* t \* (b1 + t(b2 + t(b3 + t(b4 + t b5)))),  t = 1/(1+p|x|) with phi(x) = exp(-x^2/2)/sqrt(2\*pi); reflected for x \< 0 via Phi(x)=1-Phi(-x). Pass a simple column / scalar expression (it is inlined several times).

## Signature

```sql theme={null}
{{ normal_cdf(x) }}
```