# Saddlepoint approximation of the distribution function of the 
# GAL distribution
# 
# Author: Francois Pelletier
#
# LGPL 3.0
###############################################################################

#' Saddlepoint approximation of the distribution function of the 
#' GAL distribution
#' @param x vector of quantiles
#' @param param Parameter vector
#' @param eval.time Time of the process
#' @param type Choose between "mu" or "kappa" parametrization
#' @param log Logical for log-parameters
#' @return Saddlepoint approximation of the distribution function
#' @export psaddleapproxGAL
#' @author Francois Pelletier
psaddleapproxGAL <- function(x,param,eval.time=1,type="mu",log=FALSE)
{
	s <- saddlepointGAL(x,param,eval.time,type,log)
	u <- s * sqrt(diffcgfGAL(s,2,param,eval.time,type,log))
	w <- sign(s)*sqrt(2*(s*x-cgfGAL(s,param,type,log)))
	
	(x==mGAL(param,1,type,log)) * (1/2 + diffcgfGAL(0,3,param,eval.time,type,log)/
				(6*sqrt(2*pi)*diffcgfGAL(0,2,param,eval.time,type,log)^(3/2))) + 
			(x!=mGAL(param,1,type,log)) * pnorm(w)+dnorm(w)*(1/w-1/u)
			
}