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

### --> d-p-q-r-tests.R  for distribution things

opt.conformance <- 0
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
if(opt.conformance)#fails at least on SGI/IRIX 6.5
abs(1- .Machine$double.xmax * 10^(-.Machine$double.max.exp*log10(2)))/Meps < 1e3

log10(.Machine$double.xmax) / log10(2) == .Machine$double.max.exp
log10(.Machine$double.xmin) / log10(2) == .Machine$double.min.exp

## More IEEE  Infinity/NaN checks
i1 <- pi / 0
i1 == (i2 <- 1:1 / 0:0)
is.infinite( i1) & is.infinite( i2) &   i1 > 12   &   i2 > 12
is.infinite(-i1) & is.infinite(-i2) & (-i1) < -12 & (-i2) < -12

is.nan(n1 <- 0 / 0)
is.nan( - n1)

i1 ==  i1 + i1
i1 ==  i1 * i1
is.nan(i1 - i1)
is.nan(i1 / i1)

1/0 == Inf & 0 ^ -1 == Inf
1/Inf == 0 & Inf ^ -1 == 0

!is.na(Inf) & !is.nan(Inf) &   is.infinite(Inf) & !is.finite(Inf)
!is.na(-Inf)& !is.nan(-Inf)&   is.infinite(-Inf)& !is.finite(-Inf)
 is.na(NA)  & !is.nan(NA)  &  !is.infinite(NA)  & !is.finite(NA)
 is.na(NaN) &  is.nan(NaN) &  !is.infinite(NaN) & !is.finite(NaN)

all(!is.nan(c(1,NA)))
all(c(F,T,F) == is.nan(c   (1,NaN,NA)))
all(c(F,T,F) == is.nan(list(1,NaN,NA)))


##  log() and "pow()" -- POSIX is not specific enough..
log(0) == -Inf
is.nan(log(-1))# TRUE and warning

rp <- c(1:2,Inf); rn <- rev(- rp)
r <- c(rn, 0, rp)
all(r^0 == 1)
all((rn ^ -3) == -((-rn) ^ -3))
#
all(c(1.1,2,Inf) ^ Inf == Inf)
all(c(1.1,2,Inf) ^ -Inf == 0)
.9 ^ Inf == 0
.9 ^ -Inf == Inf
## Wasn't ok in 0.64:
all(1^c(-Inf,Inf) == 1)
all(is.nan(rn ^ .5))# in some C's : (-Inf) ^ .5 gives Inf, instead of NaN


## 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

## var():
for(n in 2:10)
    print(all.equal(n*(n-1)*var(diag(n)),
		    matrix(c(rep(c(n-1,rep(-1,n)),n-1), n-1), nr=n, nc=n),
		    tol = 20*Meps))# use tol=0	to see rel.error