###---- ALL tests here should return  TRUE !
###
### '##P': This lines give not 'T' but relevant ``Print output''
###

Meps <- .Machine $ double.eps

options(rErr.eps = 1e-30)
rErr <- function(approx, true, eps = .Options$rErr.eps)
{
    if(is.null(eps)) { eps <- 1e-30; options(rErr.eps = eps) }
    ifelse(Mod(true) >= eps,
           1 - approx / true, # relative error
           true - approx)     # absolute error (e.g. when true=0)
}

is.infinite(.Machine$double.base ^ .Machine$double.max.exp)# overflow
abs(1- .Machine$double.xmin * 10^(-.Machine$double.min.exp*log10(2)))/Meps < 1e3
##P (1- .Machine$double.xmin * 10^(-.Machine$double.min.exp*log10(2)))/Meps
log10(.Machine$double.xmax) / log10(2) == .Machine$double.max.exp
log10(.Machine$double.xmin) / log10(2) == .Machine$double.min.exp

## Real Trig.:
cos(0) == 1
sin(3*pi/2) == cos(pi)
x <- rnorm(99)
all( sin(-x) == - sin(x))
all( cos(-x) == cos(x))

x <- 1:99/100
all(abs(1 - x / asin(sin(x))) <= Meps)
all(abs(1 - x / atan(tan(x))) <= Meps)

## Complex Trig.:
abs(Im(cos(acos(1i))) -	 1) < 2*Meps
abs(Im(sin(asin(1i))) -	 1) < 2*Meps
##P (1 - Im(sin(asin(Ii))))/Meps
##P (1 - Im(cos(acos(Ii))))/Meps
abs(Im(asin(sin(1i))) -	 1) < 2*Meps
cos(1i) == cos(-1i)# i.e. Im(acos(*)) gives + or - 1i:
abs(abs(Im(acos(cos(1i)))) - 1) < 4*Meps

.Random.seed <- c(0, 629, 6137, 22167) # want reproducible output
Isi <- Im(sin(asin(1i + rnorm(100))))
all(abs(Isi-1) < 100* Meps)
##P table(2*abs(Isi-1)	/ Meps)
Isi <- Im(cos(acos(1i + rnorm(100))))
all(abs(Isi-1) < 100* Meps)
##P table(2*abs(Isi-1)	/ Meps)
Isi <- Im(atan(tan(1i + rnorm(100)))) #-- tan(atan(..)) does NOT work (Math!)
all(abs(Isi-1) < 100* Meps)
##P table(2*abs(Isi-1)	/ Meps)

## gamma():
abs(gamma(1/2)^2 - pi) < 4* Meps
r <- rlnorm(5000)
all(abs(rErr(gamma(r+1), r*gamma(r))) < 500 * Meps)

n <-   10; all(          gamma(1:n) == cumprod(c(1,1:(n-1))))
n <-   20; all(abs(rErr( gamma(1:n), cumprod(c(1,1:(n-1))))) < 100*Meps)
n <-  120; all(abs(rErr( gamma(1:n), cumprod(c(1,1:(n-1))))) < 1000*Meps)
n <- 10000;all(abs(rErr(lgamma(1:n),cumsum(log(c(1,1:(n-1)))))) < 100*Meps)

## fft():
ok <- TRUE
##test EXTENSIVELY:	for(N in 1:100) {
    cat(".")
    for(n in c(1:30, 1000:1050)) {
        x <- rnorm(n)
        er <- Mod(rErr(fft(fft(x), inverse = TRUE)/n, x*(1+0i)))
        n.ok <- all(er < 1e-8) & quantile(er, 0.95, names=FALSE) < 10000*Meps
        if(!n.ok) cat("\nn=",n,": quantile(rErr, c(.95,1)) =",
                      formatC(quantile(er, prob= c(.95,1))),"\n")
        ok <- ok & n.ok
    }
    cat("\n")
##test EXTENSIVELY:	}
ok