probabilityAt

Solves the quantile equation for the cumulative probability that produces the supplied value.

Feasibility guarantees the quantile function is strictly increasing, so a bracket always exists. The bracket is found by stepping geometrically toward whichever endpoint the value lies past, and the root is then refined by Newton iteration confined to that bracket. Because the derivative of the quantile function is the reciprocal of the density, the Newton step is the residual multiplied by the density. A step that would leave the bracket is replaced by bisection, so the iteration cannot run away.