Inverse of the cumulative distribution function¶
-
ndtri(y)¶ Returns the argument
xfor which the area under the Gaussian probability density function (integrated from minus infinity tox) is equal toy.Parameters: y (float) – area under the curve.
Description¶
For small arguments \(0 < y < \exp{(-2)}\), the program computes \(z = \sqrt{-2.0 * \log{y}}\); then the approximation is
\[x = z - \log{(z)}/z - (1/z) P(1/z) / Q(1/z).\]
There are two rational functions P/Q, one for \(0 < y < \exp{(-32)}\) and the other for \(y\) up to \(\exp{(-2)}\). For larger arguments, \(w = y - 0.5\), and
\[x/\sqrt{2\pi} = w + w^3 R(w^2)/S(w^2).\]
Accuracy¶
| Relative error | ||||
|---|---|---|---|---|
| arithmetic | domain | # trials | peak | rms |
| DEC | 0.125, 1 | 5500 | 9.5e-17 | 2.1e-17 |
| DEC | 6e-39, 0.135 | 3500 | 5.7e-17 | 1.3e-17 |
| IEEE | 0.125, 1 | 20000 | 7.2e-16 | 1.3e-16 |
| IEEE | 3e-308, 0.135 | 50000 | 4.6e-16 | 9.8e-17 |
Error messages¶
| message | condition | value returned |
|---|---|---|
| ndtri domain | x <= 0 | -MAXNUM |
| ndtri domain | x >= 1 | MAXNUM |