####	d|ensity
####	p|robability (cumulative)
####	q|uantile
####	r|andom number generation
####
####	Functions for  ``d/p/q/r'' === "regression" tests  (no *.Rout.save)

options(warn = 2)
##      ======== No warnings, unless explicitly asserted via
assertWarning <- tools::assertWarning

as.nan <- function(x) { x[is.na(x) & !is.nan(x)] <- NaN ; x }
###-- these are identical in ./arith-true.R ["fixme": use source(..)]
opt.conformance <- 0
onWindows <- .Platform$OS.type == "windows"
b64 <- .Machine$sizeof.pointer >= 8 # 64 (or more) bits
str(.Machine[grep("^sizeof", names(.Machine))]) ## also differentiate long-double..
options(rErr.eps = 1e-30)
rErr <- function(approx, true, eps = getOption("rErr.eps", 1e-30))
{
    ifelse(Mod(true) >= eps,
	   1 - approx / true, # relative error
	   true - approx)     # absolute error (e.g. when true=0)
}
## Numerical equality: Here want "rel.error" almost always:
All.eq <- function(x,y) {
    all.equal.numeric(x,y, tolerance = 64*.Machine$double.eps,
                      scale = max(0, mean(abs(x), na.rm=TRUE)))
}
if(!interactive())
    set.seed(123)

.ptime <- proc.time()

## The prefixes of ALL the PDQ & R functions
PDQRinteg <- c("binom", "geom", "hyper", "nbinom", "pois","signrank","wilcox")
PDQR <- c(PDQRinteg, "beta", "cauchy", "chisq", "exp", "f", "gamma",
	  "lnorm", "logis", "norm", "t","unif","weibull")
PQonly <- c("tukey")


### (Extreme) tail tests added more recently:
All.eq(1, -1e-17/ pexp(qexp(-1e-17, log=TRUE),log=TRUE))
abs(pgamma(30,100, lower=FALSE, log=TRUE) + 7.3384686328784e-24) < 1e-36
All.eq(1, pcauchy(-1e20)           /  3.18309886183791e-21)
All.eq(1, pcauchy(+1e15, log=TRUE) / -3.18309886183791e-16)## PR#6756
x <- 10^(ex <- c(1,2,5*(1:5),50,100,200,300,Inf))
for(a in x[ex > 10]) ## improve pt() : cbind(x,t= pt(-x, df=1), C=pcauchy(-x))
    stopifnot(all.equal(pt(-a, df=1), pcauchy(-a), tolerance = 1e-15))
## for PR#7902:
ex <- -c(rev(1/x), ex)
All.eq(-x, qcauchy(pcauchy(-x)))
All.eq(+x, qcauchy(pcauchy(+x, log=TRUE), log=TRUE))
All.eq(1/x, pcauchy(qcauchy(1/x)))
All.eq(ex,  pcauchy(qcauchy(ex, log=TRUE), log=TRUE))
II <- c(-Inf,Inf)
stopifnot(pcauchy(II) == 0:1, qcauchy(0:1) == II,
          pcauchy(II, log=TRUE) == c(-Inf,0),
          qcauchy(c(-Inf,0), log=TRUE) == II)
## PR#15521 :
p <- 1 - 1/4096
stopifnot(all.equal(qcauchy(p), 1303.7970381453319163, tolerance = 1e-14))

pr <- 1e-23 ## PR#6757
stopifnot(all.equal(pr^ 12, pbinom(11, 12, prob= pr,lower=FALSE),
                    tolerance = 1e-12, scale= 1e-270))
## pbinom(.) gave 0 in R 1.9.0
pp <- 1e-17 ## PR#6792
stopifnot(all.equal(2*pp, pgeom(1, pp), scale= 1e-20))
## pgeom(.) gave 0 in R 1.9.0

x <- 10^(100:295)
sapply(c(1e-250, 1e-25, 0.9, 1.1, 101, 1e10, 1e100),
       function(shape)
       All.eq(-x, pgamma(x, shape=shape, lower=FALSE, log=TRUE)))
x <- 2^(-1022:-900)
## where all completely off in R 2.0.1
all.equal(pgamma(x, 10, log = TRUE) - 10*log(x),
          rep(-15.104412573076, length(x)), tolerance = 1e-12)# 3.984e-14 (i386)
all.equal(pgamma(x, 0.1, log = TRUE) - 0.1*log(x),
          rep(0.0498724412598364, length(x)), tolerance = 1e-13)# 7e-16 (i386)

All.eq(dpois(  10*1:2, 3e-308, log=TRUE),
       c(-7096.08037610806, -14204.2875435307))
All.eq(dpois(1e20, 1e-290, log=TRUE), -7.12801378828154e+22)
## all gave -Inf in R 2.0.1


x <- c(outer(1:12, 10^c(-3:2, 6:9, 10*(2:30))))
for(nu in c(.75, 1.2, 4.5, 999, 1e50)) {
    lfx <- dt(x, df=nu, log=TRUE)
    stopifnot(is.finite(lfx), All.eq(exp(lfx), dt(x, df=nu)))
}## dt(1e160, 1.2, log=TRUE) was -Inf  up to R 2.15.2

## pf() with large df1 or df2
## (was said to be PR#7099, but that is about non-central pchisq)
nu <- 2^seq(25, 34, 0.5)
target <- pchisq(1, 1) # 0.682...
y <- pf(1, 1, nu)
stopifnot(All.eq(pf(1, 1, Inf), target),
          diff(c(y, target)) > 0, # i.e. pf(1, 1, *) is monotone increasing
          abs(y[1] - (target - 7.21129e-9)) < 1e-11) # computed value
## non-monotone in R <= 2.1.0

stopifnot(pgamma(Inf, 1.1) == 1)
## didn't not terminate in R 2.1.x (only)

## qgamma(q, *) should give {0,Inf} for q={0,1}
sh <- c(1.1, 0.5, 0.2, 0.15, 1e-2, 1e-10)
stopifnot(Inf == qgamma(1, sh))
stopifnot(0   == qgamma(0, sh))
## the first gave Inf, NaN, and 99.425 in R 2.1.1 and earlier

## In extreme left tail {PR#11030}
p <- 10:123*1e-12
qg <- qgamma(p, shape=19)
qg2<- qgamma(1:100 * 1e-9, shape=11)
stopifnot(diff(qg, diff=2) < -6e-6,
          diff(qg2,diff=2) < -6e-6,
	  abs(1 - pgamma(qg, 19)/ p) < 1e-13,
          All.eq(qg  [1], 2.35047385139143),
          All.eq(qg2[30], 1.11512318734547))
## was non-continuous in R 2.6.2 and earlier

f2 <- c(0.5, 1:4)
stopifnot(df(0, 1, f2) == Inf,
          df(0, 2, f2) == 1,
          df(0, 3, f2) == 0)
## only the last one was ok in R 2.2.1 and earlier

x0 <- -2 * 10^-c(22,10,7,5) # ==> d*() warns about non-integer:
assertWarning(fx0 <- dbinom(x0, size = 3, prob = 0.1))
stopifnot(fx0 == 0,  pbinom(x0, size = 3, prob = 0.1) == 0)

## very small negatives were rounded to 0 in R 2.2.1 and earlier

## dbeta(*, ncp):
db.x <- c(0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805,
	  50000, 73205, 103680, 142805, 192080, 253125, 327680)
a <- rlnorm(100)
stopifnot(All.eq(a, dbeta(0, 1, a, ncp=0)),
	  dbeta(0, 0.9, 2.2, ncp = c(0, a)) == Inf,
	  All.eq(65536 * dbeta(0:16/16, 5,1), db.x),
	  All.eq(exp(16 * log(2) + dbeta(0:16/16, 5,1, log=TRUE)), db.x)
          )
## the first gave 0, the 2nd NaN in R <= 2.3.0; others use 'TRUE' values
stopifnot(all.equal(dbeta(0.8, 0.5, 5, ncp=1000),# was way too small in R <= 2.6.2
		    3.001852308909e-35),
	  all.equal(1, integrate(dbeta, 0,1, 0.8, 0.5, ncp=1000)$value,
		    tolerance = 1e-4),
          all.equal(1, integrate(dbeta, 0,1, 0.5, 200, ncp=720)$value),
          all.equal(1, integrate(dbeta, 0,1, 125, 200, ncp=2000)$value)
          )

## df(*, ncp):
x <- seq(0, 10, length=101)
h <- 1e-7
dx.h <- (pf(x+h, 7, 5, ncp= 2.5) - pf(x-h, 7, 5, ncp= 2.5)) / (2*h)
stopifnot(all.equal(dx.h, df(x, 7, 5, ncp= 2.5), tolerance = 1e-6),# (1.50 | 1.65)e-8
          All.eq(df(0, 2, 4, ncp=x), df(1e-300, 2, 4, ncp=x))
          )

## qt(p ~ 0, df=1) - PR#9804
p <- 10^(-10:-20)
qtp <- qt(p, df = 1)
## relative error:
table(rerr <- abs(1 - p / pt(qtp, df=1))) # 64-bit F34 : 1 x 0, 10 x 2.22e-16
stopifnot(rerr < 1e-14)

## Similarly for df = 2 --- both for p ~ 0  *and*  p ~ 1/2
## P ~ 0
stopifnot(all.equal(qt(-740, df=2, log=TRUE), -exp(370)/sqrt(2)))
## P ~ 1 (=> p ~ 0.5):
p.5 <- 0.5 + 2^(-5*(5:8))
stopifnot(all.equal(qt(p.5, df = 2),
		    c(8.429369702179e-08, 2.634178031931e-09,
		      8.231806349784e-11, 2.572439484308e-12)))
## qt(<large>, log = TRUE)  is now more finite and monotone (again!):
stopifnot(all.equal(qt(-1000, df = 4, log=TRUE),
                    -4.930611e108, tolerance = 1e-6))
qtp <- qt(-(20:850), df=1.2, log=TRUE, lower=FALSE)
##almost: stopifnot(all(abs(5/6 - diff(log(qtp))) < 1e-11))
stopifnot(abs(5/6 - quantile(diff(log(qtp)), pr=c(0,0.995))) < 1e-11)

## close to df=1 (where Taylor steps are important!):
stopifnot(all.equal(-20, pt(qt(-20, df=1.02, log=TRUE),
                          df=1.02, log=TRUE), tolerance = 1e-12),
          diff(lq <- log(qt(-2^-(10:600), df=1.1, log=TRUE))) > 0.6)
lq1 <- log(qt(-2^-(20:600), df=1, log=TRUE))
lq2 <- log(qt(-2^-(20:600), df=2, log=TRUE))
stopifnot(mean(abs(diff(lq1) - log(2)      )) < 1e-8,
	  mean(abs(diff(lq2) - log(sqrt(2)))) < 4e-8)
## Case, where log.p=TRUE was fine, but log.p=FALSE (default) gave NaN:
lp <- 40:406
stopifnot(all.equal(lp, -pt(qt(exp(-lp), 1.2), 1.2, log=TRUE), tolerance = 4e-16))
## Other log.p cases, gave NaN (all but 1.1) in R <= 4.2.1, PR#18360 [NB: *still* inaccurate: tol=0.2]
q <- exp(seq(200, 500, by=1/2))
for(df in c(1.001, 1 + (1:10)/100)) {
    pq  <- pt(q,  df = df, log = TRUE)
    qpq <- qt(pq, df = df, log = TRUE)
    cat("df = ", df, ": all.equal(., tol=0): "); print(all.equal(q,qpq, tol=0)) # ~0.17!
    ## plot(lp, 1-qpq/q, main=paste0("relErr(qt(., df=",df,"))"), type="l")
    stopifnot(all.equal(q,qpq, tol = 0.2)) # Lnx 64b: 1.001 shows 0.179
    if(any(ina <- is.na(qpq))) { cat("NaN in q-range: [", range(q[ina]),"]\n") }
}


## pbeta(*, log=TRUE) {toms708} -- now improved tail behavior
x <- c(.01, .10, .25, .40, .55, .71, .98)
pbval <- c(-0.04605755624088, -0.3182809860569, -0.7503593555585,
           -1.241555830932, -1.851527837938, -2.76044482378, -8.149862739881)
stopifnot(all.equal(pbeta(x, 0.8, 2, lower=FALSE, log=TRUE), pbval),
          all.equal(pbeta(1-x, 2, 0.8, log=TRUE), pbval))
qq <- 2^(0:1022)
df.set <- c(0.1, 0.2, 0.5, 1, 1.2, 2.2, 5, 10, 20, 50, 100, 500)
for(nu in df.set) {
    pqq <- pt(-qq, df = nu, log=TRUE)
    stopifnot(is.finite(pqq))
}
## PR#14230 -- more extreme beta cases {should no longer rely on denormalized}
x <- (256:512)/1024
P <- pbeta(x, 3, 2200, lower.tail=FALSE, log.p=TRUE)
stopifnot(is.finite(P), P < -600,
	  -.001 < (D3P <- diff(P, diff = 3)), D3P < 0, diff(D3P) < 0)
## all but the first 43 where -Inf in R <= 2.9.1
stopifnot(All.eq(pt(2^-30, df=10),
                 0.50000000036238542))
## = .5+ integrate(dt, 0,2^-30, df=10, rel.tol=1e-20)

## rbinom(*, size) gave NaN for large size up to R <= 2.6.1
M <- .Machine$integer.max
set.seed(7) # as M is large, now "basically" rbinom(n, *) := qbinom(runif(n), *) :
(tt <- table(rbinom(100,    M, pr = 1e-9 )) ) # had values in {0,2} only
(t2 <- table(rbinom(100, 10*M, pr = 1e-10)) )
stopifnot(0:6 %in% names(tt), sum(tt) == 100, sum(t2) == 100) ## no NaN there
## related qbinom() tests:
k <- 0:32
for(n in c((M+1)/2, M, 2*M, 10*M)) {
    for(pr in c(1e-8, 1e-9, 1e-10)) {
        nDup <- !duplicated( pb <- pbinom(k, n, pr) )
        qb <- qbinom(pb[nDup], n, pr)
        pn1 <- pb[nDup] < if(b64) 1 else 1 - 3*.Machine$double.eps
        stopifnot(k[nDup][pn1] == qb[pn1]) ##^^^^^ fudge needed (Linux 32-b)
    }
}
## qbinom()  gave  NaN in R 4.0.2


## qf() with large df1, df2  and/or  small p:
x <- 0.01; f1 <- 1e60; f2 <- 1e90
stopifnot(qf(1/4, Inf, Inf) == 1,
	  all.equal(1, 1e-18/ pf(qf(1e-18, 12,50), 12,50), tolerance = 1e-10),
	  abs(x - qf(pf(x, f1,f2, log.p=TRUE), f1,f2, log.p=TRUE)) < 1e-4)

## qbeta(*, log.p) for "border" case:
stopifnot(is.finite(q0 <- qbeta(-1e10, 50,40, log.p=TRUE)),
          1 ==            qbeta(-1e10,  2, 3, log.p=TRUE, lower=FALSE))
## infinite loop or NaN in R <= 2.7.0

## phyper(x, 0,0,0), notably for huge x
stopifnot(all(phyper(c(0:3, 1e67), 0,0,0) == 1))
## practically infinite loop and NaN in R <= 2.7.1 (PR#11813)

## plnorm(<= 0, . , log.p=TRUE)
stopifnot(plnorm(-1:0, lower.tail=FALSE, log.p=TRUE) == 0,
	  plnorm(-1:0, lower.tail=TRUE,	 log.p=TRUE) == -Inf)
## was wrongly == 'log.p=FALSE' up to R <= 2.7.1 (PR#11867)


## pchisq(df=0) was wrong in 2.7.1; then, upto 2.10.1, P*(0,0) gave 1
stopifnot(pchisq(c(-1,0,1), df=0) == c(0,0,1),
          pchisq(c(-1,0,1), df=0, lower.tail=FALSE) == c(1,1,0),
	  ## for ncp >= 80, gave values >= 1 in 2.10.0
	  pchisq(500:700, 1.01, ncp = 80) <= 1)

## dnbinom for extreme  size and/or mu :
mu <- 20
d <- dnbinom(17, mu=mu, size = 1e11*2^(1:10)) - dpois(17, lambda=mu)
stopifnot(d < 0, diff(d) > 0, d[1] < 1e-10)
## was wrong up to 2.7.1
## The fix to the above, for x = 0, had a new cancellation problem
mu <- 1e12 * 2^(0:20)
stopifnot(all.equal(1/(1+mu), dnbinom(0, size = 1, mu = mu), tolerance = 1e-13))
## was wrong in 2.7.2 (only)
mu <- sort(outer(1:7, 10^c(0:10,50*(1:6))))
NB <- dnbinom(5, size=1e305, mu=mu, log=TRUE)
P  <- dpois  (5,                mu, log=TRUE)
stopifnot(abs(rErr(NB,P)) < 9*.Machine$double.eps)# seen 2.5*
## wrong in 3.1.0 and earlier


## Non-central F for large x
x <- 1e16 * 1.1 ^ (0:20)
dP <- diff(pf(x, df1=1, df2=1, ncp=20, lower.tail=FALSE, log=TRUE))
stopifnot(-0.047 < dP, dP < -0.0455)
## pf(*, log) jumped to -Inf prematurely in 2.8.0 and earlier


## Non-central Chi^2 density for large x
stopifnot(0 == dchisq(c(Inf, 1e80, 1e50, 1e40), df=10, ncp=1))
## did hang in 2.8.0 and earlier (PR#13309).


## qbinom() .. particularly for large sizes, small prob:
p.s <- c(.01, .001, .1, .25)
pr <- (2:20)*1e-7
sizes <- 1000*(5000 + c(0,6,16)) + 279
k.s <- 0:15; q.xct <- rep(k.s, each=length(pr))
for(sz in sizes) {
    for(p in p.s) {
	qb <- qbinom(p=p, size = sz, prob=pr)
	pb <- qpois (p=p, lambda = sz * pr)
	stopifnot(All.eq(qb, pb))
    }
    pp.x <- outer(pr, k.s, function(pr, q) pbinom(q, size = sz, prob=pr))
    qq.x <- apply(pp.x, 2, function(p)     qbinom(p, size = sz, prob=pr))
    stopifnot(qq.x == q.xct)
}
## do_search() in qbinom() contained a thinko up to 2.9.0 (PR#13711)


## pbeta(x, a,b, log=TRUE)  for small x and a  is ~ log-linear
x <- 2^-(200:10)
for(a in c(1e-8, 1e-12, 16e-16, 4e-16))
    for(b in c(0.6, 1, 2, 10)) {
        dp <- diff(pbeta(x, a, b, log=TRUE)) # constant approximately
        stopifnot(sd(dp) / mean(dp) < 0.0007)
    }
## had  accidental cancellation '1 - w'

## qgamma(p, a) for small a and (hence) small p
## pgamma(x, a) for very very small a
a <- 2^-seq(10,1000, .25)
q.1c <- qgamma(1e-100,a,lower.tail=FALSE)
q.3c <- qgamma(1e-300,a,lower.tail=FALSE)
p.1c <- pgamma(q.1c[q.1c > 0], a[q.1c > 0], lower.tail=FALSE)
p.3c <- pgamma(q.3c[q.3c > 0], a[q.3c > 0], lower.tail=FALSE)
x <- 1+1e-7*c(-1,1); pg <- pgamma(x, shape = 2^-64, lower.tail=FALSE)
stopifnot(qgamma(.99, .00001) == 0,
          abs(pg[2] - 1.18928249197237758088243e-20) < 1e-33,
	  abs(diff(pg) + diff(x)*dgamma(1, 2^-64)) < 1e-13 * mean(pg),
	  abs(1 - p.1c/1e-100) < 10e-13,# max = 2.243e-13 / 2.442 e-13
	  abs(1 - p.3c/1e-300) < 28e-13)# max = 7.057e-13
## qgamma() was wrong here, orders of magnitude up to R 2.10.0
## pgamma() had inaccuracies, e.g.,
## pgamma(x, shape = 2^-64, lower.tail=FALSE)  was discontinuous at x=1

stopifnot(all(qpois((0:8)/8, lambda=0) == 0))
## gave Inf as p==1 was checked *before* lambda==0

## extreme tail of non-central chisquare
stopifnot(all.equal(pchisq(200, 4, ncp=.001, log.p=TRUE), -3.851e-42))
## jumped to zero too early up to R 2.10.1 (PR#14216)
## left "extreme tail"
lp <- pchisq(2^-(0:200), 100, 1, log=TRUE)
stopifnot(is.finite(lp), lp < -184,
	  all.equal(lp[201], -7115.10693158))
dlp <- diff(lp)
dd <- abs(dlp[-(1:30)] - -34.65735902799)
stopifnot(-34.66 < dlp, dlp < -34.41, dd < 1e-8)# 2.2e-10 64bit Lnx
## underflowed to -Inf much too early in R <= 3.1.0
for(e in c(0, 2e-16))# continuity at 80 (= branch point)
stopifnot(all.equal(pchisq(1:2, 1.01, ncp = 80*(1-e), log=TRUE),
		    c(-34.57369629, -31.31514671)))

## logit() == qlogit() on the right extreme:
x <- c(10:80, 80 + 5*(1:24), 200 + 20*(1:25))
stopifnot(All.eq(x, qlogis(plogis(x, log.p=TRUE),
			   log.p=TRUE)))
## qlogis() gave Inf much too early for R <= 2.12.1
## Part 2:
x <- c(x, seq(700, 800, by=10))
stopifnot(All.eq(x, qlogis(plogis(x, lower=FALSE, log.p=TRUE),
			   lower=FALSE, log.p=TRUE)))
# plogis() underflowed to -Inf too early for R <= 2.15.0

## log upper tail pbeta():
x <- (25:50)/128
pbx <- pbeta(x, 1/2, 2200, lower.tail=FALSE, log.p=TRUE)
d2p <- diff(dp <- diff(pbx))
b <- 2200*2^(0:50)
y <- log(-pbeta(.28, 1/2, b, lower.tail=FALSE, log.p=TRUE))
stopifnot(-1094 < pbx, pbx < -481.66,
            -29 < dp,   dp < -20,
           -.36 < d2p, d2p < -.2,
          all.equal(log(b), y+1.113, tolerance = .00002)
          )
## pbx had two -Inf; y was all Inf  for R <= 2.15.3;  PR#15162

## dnorm(x) for "large" |x|
stopifnot(abs(1 - dnorm(35+3^-9)/ 3.933395747534971e-267) < 1e-15)
## has been losing up to 8 bit precision for R <= 3.0.x

## pbeta(x, <small a>,<small b>, .., log):
ldp <- diff(log(diff(pbeta(0.5, 2^-(90+ 1:25), 2^-60, log.p=TRUE))))
stopifnot(abs(ldp - log(1/2)) < 1e-9)
## pbeta(*, log) lost all precision here, for R <= 3.0.x (PR#15641)
##
## "stair function" effect (from denormalized numbers)
a <- 43779; b <- 0.06728
x. <- .9833 + (0:100)*1e-6
px <- pbeta(x., a,b, log=TRUE) # plot(x., px) # -> "stair"
d2. <- diff(dpx <- diff(px))
stopifnot(all.equal(px[1], -746.0986886924, tol=1e-12),
          0.0445741 < dpx, dpx < 0.0445783,
          -4.2e-8 < d2., d2. < -4.18e-8)
## were way off in R <= 3.1.0

c0 <- system.time(p0 <- pbeta(  .9999, 1e30, 1.001, log=TRUE))
cB <- max(.001, c0[[1]])# base time
c1 <- system.time(p1 <- pbeta(1- 1e-9, 1e30, 1.001, log=TRUE))
c2 <- system.time(p2 <- pbeta(1-1e-12, 1e30, 1.001, log=TRUE))
stopifnot(all.equal(p0, -1.000050003333e26, tol=1e-10),
          all.equal(p1, -1e21, tol = 1e-6),
          all.equal(p2, -9.9997788e17),
          c(c1[[1]], c2[[1]]) < 1000*cB)
## (almost?) infinite loop in R <= 3.1.0


## pbinom(), dbinom(), dhyper(),.. : R allows "almost integer" n
for (FUN in c(function(n) dbinom(1,n,0.5), function(n) pbinom(1,n,0.5),
              function(n) dpois(n, n), function(n) dhyper(n+1, n+5,n+5, n)))
    try( lapply(sample(10000, size=1000), function(M) {
    ## invisible(lapply(sample(10000, size=1000), function(M) {
        n <- (M/100)*10^(2:20); if(anyNA(P <- FUN(n)))
            stop("NA for M=",M, "; 10ex=",paste((2:20)[is.na(P)], collapse=", "))}))
## check was too tight for large n in R <= 3.1.0 (PR#15734)

## [dpqr]beta(*, a,b) where a and/or b are Inf
stopifnot(pbeta(.1, Inf, 40) == 0,
          pbeta(.5, 40, Inf) == 1,
          pbeta(.4, Inf,Inf) == 0,
          pbeta(.5, Inf,Inf) == 1,
          ## gave infinite loop (or NaN) in R <= 3.1.0
          qbeta(.9, Inf, 100) == 1, # Inf.loop
          qbeta(.1, Inf, Inf) == 1/2)# NaN + Warning
## range check (in "close" cases):
assertWarning(qN <- qbeta(2^-(10^(1:3)), 2,3, log.p=TRUE))
assertWarning(qn <- qbeta(c(-.1, -1e-300, 1.25), 2,3))
stopifnot(is.nan(qN), is.nan(qn))

## lognormal boundary case sdlog = 0:
p <- (0:8)/8; x <- 2^(-10:10)
stopifnot(all.equal(qlnorm(p, meanlog=1:2, sdlog=0),
		    qlnorm(p, meanlog=1:2, sdlog=1e-200)),
	  dlnorm(x, sdlog=0) == ifelse(x == 1, Inf, 0))

## qbeta(*, a,b) when  a,b << 1 : can easily fail
q1 <- qbeta(2^-28, 0.125, 2^-26) # gave 1000 Newton it + warning
stopifnot(all.equal(2^-28, pbeta(q1, 0.125, 2^-26), tol= 2^-50))
a <- 1/8; b <- 2^-(4:200); alpha <- b/4
qq <- qbeta(alpha, a,b)# gave warnings intermediately
pp <- pbeta(qq, a,b)
stopifnot(pp > 0, diff(pp) < 0, ## pbeta(qbeta(alpha,*),*) == alpha:
          abs(1 - pp/alpha) < 4e-15)# seeing 2.2e-16

## orig. qbeta() using *many* Newton steps; case where we "know the truth"
a <- 25; b <- 6; x <- 2^-c(3:15, 100, 200, 250, 300+100*(0:7))
pb <- c(## via Rmpfr's roundMpfr(pbetaI(x, a,b, log.p=TRUE, precBits = 2048), 64) :
    -40.7588797271766572448, -57.7574063441183625303, -74.9287878018119846216,
    -92.1806244636893542185, -109.471318248524419364, -126.781111923947395655,
    -144.100375042814531426, -161.424352961544612370, -178.750683324909148353,
    -196.078188674895169383, -213.406281209657976525, -230.734667259724367416,
    -248.063200048177428608, -1721.00081201679567511, -3453.86876341665894863,
    -4320.30273911659058550, -5186.73671481652222237, -6919.60466621638549567,
    -8652.47261761624876897, -10385.3405690161120427, -12118.2085204159753165,
    -13851.0764718158385902, -15583.9444232157018631, -17316.8123746155651368)
stopifnot(all.equal(pb, pbeta(x,a,b, log.p=TRUE), tol=8e-16))# seeing {1.5|1.6|2.0}e-16
qp <- qbeta(pb, a,b, log.p=TRUE)
## x == qbeta(pbeta(x, *), *) :
stopifnot(qp > 0, all.equal(x, qp, tol= 1e-15))# seeing {2.4|3.3}e-16

## qbeta(), PR#15755
a1 <- 0.0672788; b1 <- 226390
curve(pbeta(x, a1,b1), from=1e-50, n=4001, log="x",
      main = sprintf("(a1=%g, b1=%g) -- needs log-x scale",a1,b1))
## zoom into part where it jumps from const = 0 to const = 5.562685e-309 to regular growth:
curve(qbeta(x, a1,b1), from=1e-21, to=6e-21, type="o", cex=1/4) -> qb1
rle(qb1$y)
p <- 0.6948886
qp <- qbeta(p, a1,b1)
stopifnot(qp < 2e-8, # was '1' (with a warning) in R <= 3.1.0
          All.eq(p, pbeta(qp, a1,b1)))
## less extreme example, same phenomenon:
a <- 43779; b <- 0.06728
stopifnot(All.eq(0.695, pbeta(qbeta(0.695, b,a), b,a)))
x <- -exp(seq(0, 14, by=2^-9))
qx <- qbeta(x, a,b, log.p=TRUE)# used to be slow
pqx <- pbeta(qx, a,b, log=TRUE)
stopifnot(diff(qx) < 0,
          all.equal(x, pqx, tol= 2e-15)) # seeing {3.51|3.54}e-16
## note that qx[x > -exp(2)] is too close to 1 to get full accuracy:
i2 <- x > -exp(2); all.equal(x[i2], pqx[i2], tol= 0)#-> 5.849e-12
## was Inf, and much slower, for R <= 3.1.0
x3 <- -(15450:15700)/2^11
pq3 <- pbeta(qbeta(x3, a,b, log.p=TRUE), a,b, log=TRUE)
stopifnot(mean(abs(pq3-x3)) < 4e-12,# 1.46e-12
          max (abs(pq3-x3)) < 8e-12)# 2.95e-12
##
.a <- .2; .b <- .03; lp <- -(10^-(1:323))
qq <- qbeta(lp, .a,.b, log=TRUE) # warnings in R <= 3.1.0
assertWarning(qN <- qbeta(.5, 2,3, log.p=TRUE))
assertWarning(qn <- qbeta(c(-.1, 1.25), 2,3))
stopifnot(1-qq < 1e-15, is.nan(qN), is.nan(qn))# typically qq == 1  exactly
## failed in intermediate versions
##
a <- 2^-8; b <- 2^(200:500)
pq <- pbeta(qbeta(1/8, a, b), a, b)
stopifnot(abs(pq - 1/8) < 1/8)
## whereas  qbeta() would underflow to 0 "too early", for R <= 3.1.0
#
## very extreme tails on both sides
x <- c(1e-300, 1e-12, 1e-5, 0.1, 0.21, 0.3)
stopifnot(0 == qbeta(x, 2^-12, 2^-10))## gave warnings
a <- 10^-(8:323)
length(uq <- unique(qb <- qbeta(0.95, a, 20)))# typically just 1 !
head(uq) # 5.562685e-309
## had warnings and wrong value +1; also NaN
ct2 <- system.time(q2 <- qbeta(0.95, a,a))[1]
stopifnot(is.finite(qb), qb < 1e-300, q2 == 1)
if(ct2 > 0.020) { cat("system.time:\n"); print(ct2) }
## had warnings and was much slower for R <= 3.1.0

## qt(p, df= Inf, ncp)  <==> qnorm(p, m=ncp)
p <- (0:32)/32
stopifnot(all.equal(qt(p, df=Inf, ncp=5), qnorm(p, m=5)))
## qt(*, df=Inf, .)  gave NaN in  R <= 3.2.1

## rhyper(*, <large>);  PR#16489
ct3 <- system.time(N <- rhyper(100, 8000, 1e9-8000, 1e6))[1]
table(N)
stopifnot(exprs = {
    identical(c(table(N)),
              `names<-`(as.integer(c(1, 1, 2, 2, 8, 21, 9, 18, 15, 10, 1, 7, 1, 2, 2)),
                        as.character((0:15)[-2])))
    abs(mean(N) - 8) < 1.5
})
if(ct3 > 0.02) { cat("system.time:\n"); print(ct3) }
## N were all 0 and took very long for R <= 3.2.1
set.seed(17)
stopifnot(rhyper(1, 3024, 27466, 251) == 25,
          rhyper(1,  329,  3059, 225) == 22)
## failed for a day after a "thinko" in the above bug fix.

## *chisq(*, df=0, ncp=0) == Point mass at 0
stopifnot(rchisq(32,        df=0, ncp=0) == 0,
          dchisq((0:16)/16, df=0, ncp=0) == c(Inf, rep(0, 16)))
## gave all NaN's  for R <= 3.2.2

## pchisq(*, df=0, ncp > 0, log.p=TRUE) :
th <- 10*c(1:9,10^c(1:3,7))
pp <- pchisq(0, df = 0, ncp=th, log.p=TRUE)
stopifnot(all.equal(pp, -th/2, tol=1e-15))
## underflowed at about th ~= 60  in R <= 3.2.2

## pnbinom (-> C's bratio())
op <- options(warn = 1)# -- NaN's giving warnings
L <- 1e308; p <- suppressWarnings(pnbinom(L, L, mu = 5)) # NaN or 1 (for 64 / 32 bit)
stopifnot(is.nan(p) || p == 1)
## gave infinite loop on some 64b platforms in R <= 3.2.3

## [dpqr]nbinom(*, mu, size=Inf) -- PR#16727
## (extended wrt changes for PR#15628 )
L <- 1e308; mu <- 5
x <- c(0:3, 1e10, 1e100, L, Inf)
MxM <- .Machine$double.xmax
xL <- c(2^(1000+ c(1:23, 23.5, 23.9, 24-1e-13)), MxM, Inf)
(dP <- dpois(xL, mu)) # all 0
(pP <- ppois(xL, mu)) # all 1
stopifnot(dP == 0, pP == 1, identical(pP, pgamma(mu, xL + 1, 1., lower.tail=FALSE)))
## cbind(xL, dP, pP)

(dLmI <- dnbinom(xL, mu = 1, size = Inf))  # all ==  0
## FIXME ?!:  MxM/2 seems +- ok ??
(dLmM <- dnbinom(xL, mu = 1, size = MxM))  # all NaN but the last
(dLpI <- dnbinom(xL, prob=1/2, size = Inf))#  ditto
(dLpM <- dnbinom(xL, prob=1/2, size = MxM))#  ditto

d <- dnbinom(x,  mu = mu, size = Inf) # gave NaN (for 0 and L), now all 0
p <- pnbinom(x,  mu = mu, size = Inf) # gave all NaN, now uses ppois(x, mu)
pp <- (0:16)/16
q <- qnbinom(pp, mu = mu, size = Inf) # gave all NaN
set.seed(1); NI <- rnbinom(32, mu = mu, size = Inf)# gave all NaN
set.seed(1); N2 <- rnbinom(32, mu = mu, size = L  )
stopifnot(exprs = {
    all.equal(d, c(0.006737947, 0.033689735, 0.0842243375, 0.140373896, 0,0,0,0), tol = 9e-9)# 7.6e-10
    all.equal(p, c(0.006737947, 0.040427682, 0.1246520195, 0.265025915, 1,1,1,1), tol = 9e-9)# 7.3e-10
    all.equal(d, dpois(x, mu))# current implementation: even identical()
    all.equal(p, ppois(x, mu))
    q == c(0, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 9, Inf)
    q == qpois(pp, mu)
    identical(NI, N2)
})
options(op)
## size = Inf -- mostly gave NaN  in R <= 3.2.3

## qpois(p, *) for invalid 'p' should give NaN -- PR#16972
stopifnot(is.nan(suppressWarnings(c(qpois(c(-2,3, NaN), 3), qpois(1, 3, log.p=TRUE),
                                    qpois(.5, 0, log.p=TRUE), qpois(c(-1,pi), 0)))))
## those in the 2nd line gave 0 in R <= 3.3.1
## Similar but different for qgeom():
stopifnot(qgeom((0:8)/8, prob=1) == 0, ## p=1 gave Inf in R <= 3.3.1
          is.nan(suppressWarnings(qgeom(c(-1/4, 1.1), prob=1))))

## all our RNG  r<dist>() functions:
##' catch all: value and warnings or error <-- demo(error.catching) :
tryCatch.W.E <- function(expr) {
    W <- NULL
    w.handler <- function(w){ # warning handler
	W <<- w
	invokeRestart("muffleWarning")
    }
    list(value = withCallingHandlers(tryCatch(expr, error = function(e) e),
				     warning = w.handler),
	 warning = W)
}
.stat.ns <- asNamespace("stats")
Ns <- 4
for(dist in PDQR) {
    fn <- paste0("r",dist)
    cat(sprintf("%-9s(%d, ..): ", fn, Ns))
    F <- get(fn, envir = .stat.ns)
    nArg <- length(fms <- formals(F))
    if(dist %in% c("nbinom", "gamma")) ## cannot specify *both* 'prob' & 'mu' / 'rate' & 'scale'
        nArg <- nArg - 1
    nA1 <- nArg - 1 # those beside the first (= 'n' mostly)
    expected <- rep(if(dist %in% PDQRinteg) NA_integer_ else NaN, Ns)
    for(ia in seq_len(nA1)) {
        aa <- rep(list(1), nA1)
        aa[[ia]] <- NA
        cat(ia,"")
        R <- tryCatch.W.E( do.call(F, c(Ns, aa)) )
        if(!inherits(R$warning, "simpleWarning")) stop(" .. did *NOT* give a warning! ")
	if(!(identical(R$value, expected))) { ## allow NA/NaN mismatch in these cases for now:
	  if(!(dist %in% c("beta","f","t") && all(is.na(R$value))))
	    stop(" .. not giving expected NA/NaN's ")
        }
    }
    cat(" [Ok]\n")
}


## qbeta() in very asymmetric cases
sh2 <- 2^seq(9,16, by=1/16)
p <- 1e-10
qbet <- qbeta(p, 1.5, shape2=sh2, lower.tail=FALSE)
plot(sh2, pbeta(qbet, 1.5, sh2, lower.tail=FALSE)/p -1 -> rE, log="x", main="rel.Error")
dqb <- diff(qbet); d2qb <- diff(dqb); d3qb <- diff(d2qb)
stopifnot(all.equal(qbet[[1]], 0.047206901483498, tol=1e-12),
	  print(max(abs(rE))) < 1e-12, # Lx 64b: 2.4e-13
	  0 > dqb, dqb > -0.002,
	  0 < d2qb, d2qb < 0.00427,
	  -3.2e-8 > d3qb, d3qb > -3.1e-6,
	  diff(d3qb) > 1e-9)
## had discontinuity (from wrong jump out of Newton) in R <= 3.3.2


## rt() [PR#17306];  rf() and rbeta() [PR#17375] with non-scalar 'ncp'
nc <- c(NA, 1); iN <- is.na(rep_len(nc, 3))
## each gives warning  "NAs produced":
assertWarning(T <- rt   (3, 4,   ncp = nc))
assertWarning(F <- rf   (3, 4,5, ncp = nc))
assertWarning(B <- rbeta(3, 4,5, ncp = nc))
stopifnot(identical(iN, is.na(T)), identical(iN, is.na(F)), identical(iN, is.na(B)))
## was not handled correctly, notably with NA's in ncp, in R <= 3.4.(2|3)


## check old version of walker_Probsample is being used for old sample kind
suppressWarnings(RNGversion("3.5.0"))
set.seed(12345)
p <- c(2, rep(1, 200))
x <- sample(length(p), 100000, prob = p, replace = TRUE)
stopifnot(sum(x == 1) == 994)

## check for failure of new walker_Probsample
RNGversion("3.6.0")
set.seed(12345)
epsilon <- 1e-10
p201 <- proportions( rep( c(1, epsilon), c(201, 999-201)))
x <- sample(length(p201), 100000, prob = p201, replace = TRUE)
stopifnot(sum(x <= 201) == 100000)

arch <- Sys.info()[["machine"]]
## had if(!(onWindows && arch == "x86"))
## PR#17577 - dgamma(x, shape)  for shape < 1 (=> +Inf at x=0) and very small x
stopifnot(exprs = {
    all.equal(dgamma(2^-1027, shape = .99 , log=TRUE), 7.1127667376, tol=1e-10)
    all.equal(dgamma(2^-1031, shape = 1e-2, log=TRUE), 702.8889158,  tol=1e-10)
    all.equal(dgamma(2^-1048, shape = 1e-7, log=TRUE), 710.30007699, tol=1e-10)
    all.equal(dgamma(2^-1048, shape = 1e-7, scale = 1e-315, log=TRUE),
              709.96858768, tol=1e-10)
})
## all gave Inf in R <= 3.6.1
## } else cat("PR#17577 bug fix not checked, as it may not work on this platform\n")


## if(!(onWindows && arch == "x86")) {
  ## This gave a practically infinite loop (on 64-bit Lnx, Windows; not in 32-bit)
  suppressWarnings(# typically warns, but not on Apple clang 14.0.3
    p <- pchisq(1.00000012e200, df=1e200, ncp=100)
  )
  stopifnot(p == 1)
## }


## Extreme tails  for  qnorm(*, log.p=TRUE)   :
qs <- 2^seq(0, 155, by=1/8)
lp  <- pnorm( qs, log.p=TRUE, lower.tail=FALSE)
lp. <- pnorm(-qs, log.p=TRUE)
stopifnot(all.equal(lp, lp., tol= 4e-16)) # actually exactly via code-identity
## Both these gave NaNs (and warned about it):
qpU <- qnorm(lp, log.p=TRUE, lower.tail=FALSE)
qp. <- qnorm(lp, log.p=TRUE)
## Show the (mostly) small differences :
all.equal( qs, qpU, tol=0)
all.equal(-qs, qp., tol=0)
all.equal(-qp.,qpU, tol=0) # typically TRUE (<==> exact equality)
## however,
range(qpU/qs - 1) # -5.68e-6  5.41e-6  in R <= 4.2.1
stopifnot(exprs = {
    all.equal( qs,  qpU, tol=1e-15)
    all.equal(-qs,  qp., tol=1e-15)
    all.equal(-qp., qpU, tol=1e-15)# diff of 4.71e-16 in 4.1.0 w/icc (Eric Weese)
    max(abs(qpU/qs - 1)) < 1e-15 # see  4.44e-16  {was 5.68e-6 in R <= 4.2.1; much larger in R <= 4.0.x)
})
## both failed very badly in  R <= 4.0.x

## pnorm(x, log.p=TRUE) now works for larger x, |x| <= 1.896..e154 {before: |x| < 1.34e154}
x <- seq(134.5, 189, by=.5)
px <- pnorm(-x * 1e152, log.p=TRUE)# all these underflowed previously
stopifnot(exprs = {
    all.equal(-1.79769313486073e+308, pnorm(-1.896150381621e154, log.p=TRUE), tol=1e-14)
    is.finite(px)
    abs(1 - diff(diff(px)) / -2.5e303) < 3e-11 * (1 + (.Machine$sizeof.longdouble < 12))
})
## all these where -Inf  in R <= 4.0.x


## qnbinom(*, size=<large>, mu=<small>) -- PR#18095:
qi  <- 0:16
qiL <- c(0:99, 10L*(10:20), 50L*(5:20))
verbose <- interactive()
for(N in c(10^c(-300, -100, -20, -9:-7, -2, 2, 7:9), 9.1e15, 10^c(20,30,50, 100, 300))) {
    if(verbose) cat("N =", N,": ")
    pbi  <- pnbinom(qi,  size=N, mu=1)
    pbiU <- pnbinom(qiL, size=N, mu=1, lower.tail=FALSE, log.p=TRUE)
    stopifnot(exprs = { ## whne pbinom(.) gave 1, quntile is Inf
        pbi == 1 | qi  == qnbinom(pbi,  size=N, mu=1)
        qiL == qnbinom(pbiU, size=N, mu=1, lower.tail=FALSE, log.p=TRUE)
    })
    if(verbose) cat("[Ok]\n")
}

## qnbinom(*) gave all 0 in R <= 4.1.0
##
##
## Fix qnbinom(*, scale = <<small>>), partly reported to R-devel, Aug.7, 2020
## by Constantin Ahlmann-Eltze: size=1e-10 already took 30 sec on his computer:
st <- system.time(qn <- qnbinom(0.5, mu = 3, size = 2^-(10:59)))[[1]]
stopifnot(exprs = {
    st < .5 # = 500 ms; observed 0 sec, i.e., '< 1 ms'
    qn == 0
    print(qnbinom(.9999, mu=3,   size=1e-4)) == 7942 # was 8044 (MM on R-devel)
    print(qnbinom(.9999, mu=1e5, size=1e10)) == 101178 # was off by 1
    { ## Even larger:                 ^^^^ large size
        size <- 1e13; mu <- size/4; k <- 2500002265485 + (-100:100)
        pnb <- pnbinom( k , mu=mu, size=size) # around 0.9
        qnb <- qnbinom(pnb, mu=mu, size=size)
        qnb == k ## in R <= 4.0.2, qnb == k+23  wrongly
    }
    { ## small size, large k : needs "p fuzz factor 16 instead of 64 :
        k <- 75000 + -1000:1000; size <- 1e-9; mu <- 30
        pnb <- pnbinom( k , mu=mu, size=size) # = 0.999999987678 (for k=75'000)
        qnb <- qnbinom(pnb, mu=mu, size=size)
        qnb == k ## in R <= 4.0.2, qnb == k-1  wrongly
    }
})
## For log.p=TRUE  and or lower.tail=FALSE
##
## all three  qpois(), qbinom() & qnbinom() were *far* from good in R <= 4.1.0


## PR#18072 : dnbinom() underflow
stopifnot(dnbinom(1:40, size=2^58, prob = 1) == 0)
## gave mostly 1 in R <= 4.1.0
x <- unique(sort(c(1:12, 15, outer(c(1,2,5), 10^(1:11)))))
sz <- 2^70 ; prb <- .9999999
summary(dn <- dnbinom(x, size=sz, prob = prb, log=TRUE))
dL <- 118059167912526.5
summary(dl.dn1 <- diff(log(dn[-1] + dL)))
stopifnot(dn + dL > 0,
          0.09 < dl.dn1, dl.dn1 < 0.93)
## accuracy loss of 6 and more digits in R <= 4.1.0
##---- reverse case, very *small* size ---------------
dS <- dnbinom(1:90, size=1e-15, mu=200, log=TRUE)
d4S <- diff(d3S <- diff(ddS <- diff(dS)))
stopifnot(-39.1 < dS,  dS < -34.53
    ,     -0.7 < ddS, ddS < -0.01116
    , 0.000126 < d3S, d3S < 0.287683
    ,    -0.17 < d4S, d4S < -2.8859e-6
    , all.equal(c(-48.40172, -49.155492, -49.905797, -50.653012, -51.397452),
                dnbinom(16:20, size=1e-15, prob=1/2, log=TRUE))
)
## failed in R 4.1.1 (and R-devel) only


## PR#15628 ebd0() for dpois() [for now] \\  dpois(x, *) for  2\pi x >= DMAX
stopifnot(exprs = {
 print(abs(dpois(  40, 7.5)/ 6.81705586862060455e-17 -1)) < 6e-16 # 2.66e-15 in R <= 4.1.1
 print(abs(dpois(1400,1000)/ 1.46677334419656833e-33 -1)) < 8e-16 # 1.71e-14 in R <= 4.1.1
})
## very large x:
x <- trunc(2^(1000+ head(seq(1,24, by=1/64), -1)))
L <- tail(x,1)
dpxx <-  dpois(x,x) ## had ended in many 0's
ldpxx <- dpois(x,x, log=TRUE) # ... -Inf
(d <- mean(dlp <- diff(ldpxx)))# -0.005415
stopifnot(exprs = {
    dpxx > 0
    is.finite(ldpxx)
    print(abs(print(dpois(L,L))/ (1/sqrt(2*pi)/sqrt(L)) -1)) < 1e-15 # see 1.11e-16
    abs(range(dlp) - d) < 1e-12 # seen 4.4e-14, was NaN in R <= 4.1.1
    all.equal(ldpxx, log(dpxx), tol = 1e-15)
})
## dpois(x,x) underflowed to zero in R <= 4.1.1 for such large x.



cat("Time elapsed: ", proc.time() - .ptime,"\n")
