# Inverse of the cumulative distribution function¶

ndtri(y)

Returns the argument x for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y.

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