####--------------- bd0() & ebd0() ------------------------------------------------ ## ## was 2nd part of ./stirlerr-tst.R require(DPQ) for(pkg in c("Rmpfr", "DPQmpfr")) if(!requireNamespace(pkg)) { cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n") q("no") } n0 <- numeric() stopifnot(identical(n0, dpois_raw(1, n0))) # gave an error before 2025-05-14 require("Rmpfr") options(warnPartialMatchArgs = FALSE) source(system.file(package="DPQ", "test-tools.R", mustWork=TRUE)) ## => showProc.time(), ... list_() , loadList() , readRDS_() , save2RDS() ##_ options(conflicts.policy = list(depends.ok=TRUE, error=FALSE, warn=FALSE)) require(sfsmisc) # masking 'list_' *and* gmp's factorize(), is.whole() ##_ options(conflicts.policy = NULL) pks <- c("sfsmisc", "DPQ", "Rmpfr", "DPQmpfr") sapply(lapply(setNames(,pks), packageVersion), format) showProc.time() (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home())) ## ebd0 constants: the column sums of "bd0_scale": log(n / 1024) for all these n ## ---- according to the *comments* in the C code -- so here I test that at least the *sums* are correct bd0.n <- c(2048,2032,2016,2001,1986,1971,1956,1942,1928,1913,1900,1886,1872,1859, 1846,1833,1820,1808,1796,1783,1771,1759,1748,1736,1725,1713,1702,1691, 1680,1670,1659,1649,1638,1628,1618,1608,1598,1589,1579,1570,1560,1551, 1542,1533,1524,1515,1507,1498,1489,1481,1473,1464,1456,1448,1440,1432, 1425,1417,1409,1402,1394,1387,1380,1372,1365,1358,1351,1344,1337,1331, 1324,1317,1311,1304,1298,1291,1285,1279,1273,1266,1260,1254,1248,1242, 1237,1231,1225,1219,1214,1208,1202,1197,1192,1186,1181,1176,1170,1165, 1160,1155,1150,1145,1140,1135,1130,1125,1120,1116,1111,1106,1101,1097, 1092,1088,1083,1079,1074,1070,1066,1061,1057,1053,1049,1044,1040,1036, 1032,1028,1024) stopifnot( all.equal(bd0.n, 1024 * exp(colSums(DPQ:::logf_mat))) ) ## on lynne (64-bit, Fedora 32, 2021) they are even *identical* identical(bd0.n, 1024 * exp(colSums(DPQ:::logf_mat))) # amazingly to me ## Also, the numbers themselves decrease monotonely, ## their differences are close to, but *not* monotone: diff(bd0.n) # -16 -16 -15 -15 -15 -15 -14 -14 -15 -13 -14 ... # ^^^^^^^^^^^^^^ (etc) do.pdf <- TRUE do.pdf <- !dev.interactive(orNone = TRUE) do.pdf if(do.pdf) { pdf.options(width = 9, height = 6.5)# for all {9/6.5 = 1.38 ; 4/3 < 1.38 < sqrt(2) [A4] pdf("diff-bd0_tab.pdf") } plot(diff(bd0.n), type="b") c2 <- adjustcolor(2, 1/2) par(new=TRUE) plot(diff(bd0.n, differences = 2), type="b", col=c2, axes=FALSE, ann=FALSE) axis(4, at=-1:2, col=c2, col.axis=c2) showProc.time() ## use functionality originally in ~/R/MM/NUMERICS/dpq-functions/15628-dpois_raw_accuracy.R ## now -- require(Rmpfr) ## transition till DPQmpfr exports this *and* that version is on CRAN, to ease maintainer("DPQ"): ## vvvvvvvvvvvvvvvvvvvvvvvvvvv DPQmpfr 0.3-3 has, but does *not* export dpoisEr() if(file.exists(ff <- "~/R/Pkgs/DPQmpfr/R/dpoisEr.R")) withAutoprint({ #------------- source(ff) str(dpoisEr) # ==> prBits = 1536 __large__ default: found that 256 was too small for lambda = 1e100 (??) ## but should prBits not depend on lambda ##-- ----- *or* move to vignette ../vignettes/log1pmx-etc.Rnw <<<<<<<<<<<<<<< ##-------- small lambda --- is dpois_simpl0() good ? range(dpE40 <- dpoisEr(40.25, x=0:200)) # integer only: dpois(x, ..) is 0 for non-int !!! ## -2.442882e-16 3.645529e-16 was -4.401959e-16 3.645529e-16 str(attributes(dpE40)) p.dpoisEr(dpE40) # showing errors to be small, as w/ range(..) above ## dpois_simpl0() uses "old" direct formula on original scale: factorial(x) stopifnot(factorial(170) < Inf, factorial(171) == Inf) xS <- 0:170 # the full range of "sensible" x values for dpois_simpl0 range(dpE40simpl <- dpoisEr(40.25, x=xS, dpoisFUN = dpois_simpl0)) str(attributes(dpE40simpl)) ## -1.299950e-13 1.118291e-13 p.dpoisEr(dpE40simpl) ## --> suprising: errors are *very* small up to x <= 49, then in in the order of 1e-13 ## zoom in [y- range only] p.dpoisEr(dpE40simpl, ylim = c(-1,1)*4e-16) ## zoom into small x --- integer x only: range(dpE40simpl2 <- dpoisEr(40.25, x=0:49, dpoisFUN = dpois_simpl0)) ## -2.889661e-16 2.076597e-16 p.dpoisEr(dpE40simpl2) # --- almost all in [-eps, +eps] ## zoom into small x and use non-integer x: range(dpE40simpl2d <- dpoisEr(40.25, x=seq(0, 49, by=1/8), dpoisFUN = dpois_simpl0)) ## [1] -3.861877e-14 3.080750e-14 == Oops ! blown up to p.dpoisEr(dpE40simpl2d) }) ### MM: moved much of this to Rmpfr vignette: ### " ~/R/D/R-forge/Rmpfr/pkg/vignettes/gamma-inaccuracy.Rnw " ##~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ## and the R code actually to the R script, also pkg {Rmpfr} " ~/R/Pkgs/Rmpfr/vignettes/gamma-inaccuracy_R/plot-factErr-def.R " ##=======> gamma(x) itself suffers from the fact that exp(y) has a *large* relative error, ## -------- when |y| ~ 100 or so, more specifically, the ## relative error of exp(y) = |y| * {rel.err(y)} , since ## exp(((1+ eps)*y) = exp(y) * exp(eps*y) >= exp(y) (1 + eps*y) and indeed, ## the inaccuracy of y (i.e. eps) is blown up by a factor |y| which is not small here! ## close to over-/underflow ------- ### Large lambda == np == M ------- if(do.pdf) { dev.off(); pdf("bd0-ebd0.pdf") } ##-- TODO ----- *or* move to vignette ---> ../vi??????????? LL <- 1e20 dput(x1 <- 1e20 - 2e11) # 9.99999998e+19 (P1 <- dpois (x1, LL)) # was 3.989455e-11; now 5.520993e-98 (P1m <- Rmpfr::dpois(mpfr(x1, 128), LL)) # 5.52099285934214335003128935..e-98 ## However -- the ebd0() version (P1e <- dpois_raw(x1, LL, version="ebd0_v1"))## was 3.989455e-11, but now good! asNumeric(relErr(P1m, P1)) # 3.218894e-14 asNumeric(relErr(P1m, P1e)) # 3.218894e-14 -- the same, as R's dpois() now *does* ebd0 stopifnot(exprs = { all.equal(P1 , 5.520992859342e-98, tol=1e-12) all.equal(P1e, P1, tol=1e-12) all.equal(P1m, P1, tol=1e-12) }) options(digits = 9) ## indeed: regular bd0() works "ok" ... and ebd0() now does, too (bd.1 <- bd0(x1, LL, verbose=2)) ## bd0(1e+20, 1e+20): T.series w/ 2 terms -> bd0=200 ## [1] 200 (bd.1M <- bd0(x1, mpfr(LL, 128), verbose=2)) # we checked, 128 is sufficient ## bd0(1e+20, 1e+20): T.series w/ 3 terms -> bd0=200 ## ---> 199.9999919413334091607468236761591740489 asNumeric(bd.1 / bd.1M - 1)# -1.82e-17 -- suggests bd0() is really accurate here stopifnot(abs(bd.1 / bd.1M - 1) < 3e-16, all.equal(199.999991941333, bd.1, tolerance=1e-14)) (ebd1 <- sum((ebd.1 <- ebd0(x1, LL, verbose=TRUE))))# fixed since June 6, 2021 asNumeric(relErr(bd.1M, ebd1)) # 1.603e-16 ... ebd0() slightly *less* accurate than bd0() !! showProc.time() ### Large x -- small np == M ------------------------------------ mpfrPrec <- 1024 mpfrPrec <- 256 yy <- bd0 (1e307, 10^(-5:1), verbose=TRUE) yhl <- ebd0 (1e307, 10^(-5:1), verbose=TRUE) yhlC<- ebd0C(1e307, 10^(-5:1)) ## newly returns data.frame stopifnot(yy == Inf, yhl$yh == Inf, yhlC == yhl) yM <- bd0(mpfr(1e307, mpfrPrec), 10^(-5:1)) roundMpfr(range(yM), 12) ## 7.0363e+309 7.1739e+309 -- *are* larger than DBL_MAX ### Now *BOTH* x and lambda are large : --------------------------------------- ##' lseq()--like sequence of length ~= n2ex around "round(x)" to use as `np` in bd0(x, np) np_bd0 <- function(x, rnd.N = 256, ex2M = 2, n2ex = 256) { stopifnot(is.numeric(x), x >= 0, length(x) == 1L) l2x <- round(rnd.N*log2(x))/rnd.N ## 2 ^ seq() i.e., lseq()--like: 2^(l2x + seq(-ex2M, ex2M, by=1/(n2ex/2/ex2M))) } ## (FIXME?? Small loss for ebd0, see below) <<< ??? ## is bd0(, *) really accurate --- (currently, it seems even precBits = 128 is fine) ## it uses it's own convergent series approximation for |x-np| < .. ???? ##' Compute relative errors (wrt MPFR) for "many" bd0() versions ##' @title ##' @param x, np main arguments for computing bd0*(x, np) = D_0(x, np) == D_0(x, M) ##' @param mpfrPrec positive integer, typically >= 64, `precBits` bit precision for mpfr-number arithmetic ##' @param delta ##' @param tol_logcf ##' @param ... ##' @param chkVerb logical, TRUE by default, indicating if some checking + verbosity should happen ##' @param keepMpfr logical, FALSE by default, indication if full precision mpfr result should be storted ##' @return a \code{list()}, .. ##' @author Martin Maechler bd0ver <- function(x, np = np_bd0(x), mpfrPrec, delta = 0.1, tol_logcf = 1e-14, ..., chkVerb=TRUE, keepMpfr=FALSE) { ### passed to log1pmx() stopifnot(length(mpfrPrec <- as.integer(mpfrPrec)) == 1, !is.na(mpfrPrec), mpfrPrec >= 64, x >= 0, np >= 0) yy <- cbind(bd0 = bd0 (x, np, delta = delta), bd0.l = bd0_l1pm (x, np, tol_logcf=tol_logcf, ...), bd0.p1l1 = bd0_p1l1 (x, np, tol_logcf=tol_logcf, ...), bd0.p1.d = bd0_p1l1d (x, np, tol_logcf=tol_logcf, ...), bd0.p1.d1= bd0_p1l1d1(x, np, tol_logcf=tol_logcf, ...)) yhl <- ebd0 (x, np) yhlC <- ebd0C(x, np) if(chkVerb) { yhl. <- ebd0 (x, np, verbose=TRUE) yhlC. <- ebd0C(x, np, verbose=TRUE) stopifnot(identical(yhl., yhl), identical(yhlC., yhlC)) } epsC <- .Machine$double.eps aeq0 <- all.equal(yhl, yhlC, tol = 0) aeq4 <- all.equal(yhl, yhlC, tol = 4*epsC) if(!isTRUE(aeq4)) warning("the C and R versions of ebd0() differ:", aeq4) stopifnot(is.whole(yhl [["yh"]]), is.whole(yhlC[["yh"]])) yM <- bd0(mpfr(x, mpfrPrec), mpfr(np,mpfrPrec), verbose=chkVerb)# more accurate ! (?? always ??) relE <- relErrV(target = yM, # the mpfr one cbind(ebd0 = yhl [["yh"]] + yhl [["yl"]], ebd0C= yhlC[["yh"]] + yhlC[["yl"]], yy)) relE <- structure(asNumeric(relE), dim=dim(relE), dimnames=dimnames(relE)) ## return: list(x=x, np=np, delta = delta, bd0=yy, ebd0=yhl, ebd0C=yhlC, bd0M=if(keepMpfr) yM, # <- expensive aeq0=aeq0, aeq4=aeq4, relE = relE) } x. <- 1e307 cbind(log10.M=300:308, ebd0(x., 10^(300:308))) # yl are all 0 ; and bd0(x.,x.) = 0 bd0v.7 <- bd0ver(x., mpfrPrec = 128, chkVerb = FALSE) bd0v.10 <- bd0ver(x., mpfrPrec = 1024) stopifnot( all.equal(bd0v.7, bd0v.10, tol=0), bd0v.7$aeq0, # even tol=0 equality ! bd0v.7$aeq4 ) ## ==> 256 bit gives the *same* (asNumeric() - double-prec accuracy) as 1024 bits ! ## so, at least here rm(bd0v.10) showProc.time() ##' Plot the result of bd0ver() -- i.e., relative errors for "many" bd0() versions ##' @title ##' @param bd0v result of bd0ver() ##' @param dFac ##' @param log a string, "", "x", "y", or "xy" ##' @param type plot `type`, typically "b" or "l" ##' @param add logical ##' @param col.add ##' @param smooth ##' @param f.lowsm lowess() smoothing parameter when adding smooths' ##' @return ##' @author Martin Maechler p.relE <- function(bd0v, dFac = if(max(np) >= 8e307) 1e10 else 1, log = "x", type="b", add = FALSE, col.add = adjustcolor(k+2, 2/3), smooth = TRUE, f.lowsm = 1/16) { stopifnot(length(x <- bd0v$x) == 1 # for now , is.numeric(x), is.numeric(np <- bd0v$np), length(np) > 1 , is.numeric(dFac), dFac > 0, length(dFac) == 1 , is.matrix(relE <- bd0v$relE) , (k <- ncol(relE)) >= 1 , sum(iOk <- local({ y <- bd0v$bd0 ay <- apply(y, 1L, \(ro) any(is.finite(ro) & ro != 0)); ay })) > 1 ) np. <- np[iOk]/dFac if(add) { ## add only lines for relE[,"bd0"] relE <- relE[iOk, "bd0"] ##--- only (with varying delta, typically) abs <- par("ylog") if(abs) relE <- abs(relE) lines(np., relE, col = col.add, lwd=3) } else { ## full txtRE <- "relative Errors" if(abs <- grepl("y", log)) { txtRE <- paste0("|", txtRE, "|") relE <- abs(relE) yli <- pmax(2^-54, range(relE[iOk,], finite = TRUE)) } else yli <- range(relE[iOk,], finite = TRUE) ## */dFac : otherwise triggering axis() error ## log - axis(), 'at' creation, _LARGE_ range: invalid {xy}axp or par; nint=5 ## axp[0:1]=(1e+299,1e+308), usr[0:1]=(7.28752e+298,inf); i=9, ni=1 pc <- 1:k matplot(np., relE[iOk,], type=type, log=log, pch=pc, col=1+pc, main = paste(txtRE, "WRT bd0()"), xlim = range(np)/dFac, # show full range ylim = yli, xlab = paste0("np[iOk]", if(dFac != 1) sprintf("/ dFac, dFac=%g",dFac)), ## could use sfsmisc::pretty10exp(1e10, drop.1=TRUE) xaxt="n"); eaxis(1, sub10=3) mtext(sprintf("bd0(x, np), x = %g", x)) if(k >= 2) legend("topright", colnames(relE), pch=pc, lty=1:2, col=1+pc, bty="n") if(abs) { abline(h = 2^(-54:-51), lty = 3, lwd = c(1,1,2,1), col="gray") axis(4, at= 2^(-54:-51), las = 1, col.axis="gray", tick = FALSE, cex = 3/4, hadj = +1/2, expression(epsilon[C]/4, epsilon[C]/2, epsilon[C], 2*epsilon[C])) } } # not add if(abs && smooth) { # add smoothed relE if(add) { smRE <- lowess(np., relE[iOk], f = f.lowsm)$y lines(np., smRE, col = adjustcolor(col.add, 2/3), lwd = 2.5, lty = "dashed") lines(np., smRE, col = adjustcolor("gray44", 1/2), lwd = 4)#lty = 1 } else { smRE <- apply(relE[iOk,], 2L, function(y) lowess(np., y, f = f.lowsm)$y) matlines(np., smRE, col = adjustcolor(1+pc, 2/3), lwd = 2.5, lty = "dashed") matlines(np., smRE, col = adjustcolor("gray44", 1/2), lwd = 4, lty = 1) } } colD <- if(add) "skyblue" else 2 jO <- if(add) 5 else 1 delta <- bd0v$delta inR <- abs(x - np) <= delta * (x + np) ## ----- rngTayl <- range(np[inR])/dFac; f.rng <- format(rngTayl, digits = 4) message(sum(inR), " np[]/dFac values inside delta-range = [", f.rng[1],", ", f.rng[2],"]") abline(v = rngTayl, col = adjustcolor(colD, 2/3), lwd = 2, lty = 3) u4 <- par("usr")[3:4] %*% c(jO,64-jO)/64; yUp <- if(abs) 10^u4 else u4 arrows(rngTayl[1], yUp, rngTayl[2], yUp, code = 3, lwd = 2, col = adjustcolor(colD, 2/3)) text(rngTayl[1], yUp, substitute(delta == D, list(D = delta)), col=colD, adj = c(5/4, 3/4)) if(!add) { rug(np[!iOk]/dFac, col=2) axis(1, at=x/dFac, quote(x), col=2, col.axis=2, lwd=2, line=-1) } } if(do.pdf) { dev.off(); pdf("p.relE_bd0ver.pdf") } system.time(bd0v.7.d.40 <- bd0ver(x., mpfrPrec = 128, delta = 0.40, chkVerb = FALSE) ) ## larger delta --> some "long taking" Taylor sums p.relE(bd0v.7) p.relE(bd0v.7, log = "xy") p.relE(bd0v.7.d.40, add=TRUE, col.add = adjustcolor("steelblue", .8)) ## ==> NOTE: a whole small (extreme) range where bd0() is *better* than ebd0() !!! ## correct #{digits} with(bd0v.7, cbind(log2.lam = log2(np), np, round(-log10(abs(relE)), 1))) ## around 2^[1018, 1021] with(bd0v.7, stopifnot(yhl[["yl"]] == 0)) # which is not really good and should maybe change ! ## Fixed now : both have 4 x Inf and then are equal {but do Note relE difference above!} stopifnot(all.equal(bd0v.7$ebd0[["yh"]], bd0v.7$bd0[,"bd0"], tolerance = 4 * .Machine$double.eps)) showProc.time() ## bd0() and ebd0() are _Inf_ for first three lambda's .. but they *must* be as truly > DBL_MAX ## At the 4th value, Llam = 2^993, bd0() no longer overflows; _FIXME_ ebd0() shld *not* overflow: yM = 1.7598e+308 summary(Llam <- 2^c(990:1023, 1024 - 1e-12)) bd0M <- bd0ver(x., Llam, mpfrPrec = 256, keepMpfr=TRUE) with(bd0M, data.frame(log2.L = log2(np), bd0 = bd0, ebd0, bd0M. = format(bd0M, digits=8))) matplot(log2(Llam), with(bd0M, cbind(bd0 = bd0/x., yh=ebd0[["yh"]]/x., asNumeric(bd0M/x.))), type="o", ylab = "*bd0*(x., L) / x.", pch=1:3, main= paste("ebd0(x., Lam) and bd0(*) for x=",format(x.)," *and* larg Lam")) abline(h=0, lty=3, col=adjustcolor("gray20", 1/2)) axis(1, at=log2(x.), labels="log2(x.)", line=-1, tck=-1/16, col=2, col.axis=2) legend("topright", c("bd0()", "ebd0()", "MPFR bd0()"), bty="n", lty=1:3, pch=1:3, col=1:3) dMax <- .Machine$double.xmax abline(h = dMax / x., col=4, lty=3) ; ux <- par("usr")[1:2] text(c(ux %*% c(3,1))/4, dMax/x., pos=3, sprintf("bd0(.) > DBL_MAX = %.5g", dMax), col=4) showProc.time() bd0.2 <- bd0ver(x., mpfrPrec = 512, delta = 0.25, keepMpfr=TRUE) p.relE(bd0.2)#, dFac=1) #=========== p.relE(bd0.2, log = "xy") ### --- 2025-05 (inspired by ~/R/MM/NUMERICS/dpq-functions/dbinom_Lrg-bug.R ): ## dbinom_raw(x=1.2e+308, n=1.72e+308, p=0.2, q=0.8, give_log=1): --> str(bd0_fns <- sfsmisc::list_(bd0, ebd0, bd0_l1pm, bd0_p1l1d, bd0_p1l1d1)) stopifnot(sapply(bd0_fns, is.function)) # fails e.g. when have 'bd0' matrix ## x ~= M = np = 3.44e307 ## tail(x34.7 <- 1e307*seq(1, 18, by=1/32)) # the last is Inf ==> have fixed all bd0() functions ## for speed, interactive tail(x34.7 <- 1e307*seq(1, 18, by=1/ 4)) lBM <- lapply(bd0_fns, \(BD0) BD0(x34.7, 3.44e+307)) ## gave if( ) error ! ## because of the following error [because it called log1pmx(NaN, ..) giving NaN]; now ok: Inf bd0_l1pm(Inf, 3.44e+307) stopifnot(identical(Inf, bd0_l1pm(Inf, 3.44e+307)), identical(-Inf, log1pmx(Inf))) # now log1pmx(Inf) |--> -Inf correctly mBM <- cbind(do.call(cbind, lBM[-2]), ebd0 = with(lBM$ebd0, yl+yh)) cbind(x=x34.7, mBM) # after fixes: too early overflow to Inf *only* happens for ebd0() ## and *bd0*(Inf, ) \--> Inf now stopifnot(tail(mBM,1) == Inf) if(!doExtras) # gets too expensive quit("no") ## FIXME: do not quit: use less precision, etc options(digits = 6, width = 130) # width: for tables ## zoom in more: L.3 <- 2^c(seq(1019, 1021, by=1/128)) if(FALSE) ## too slow system.time(bd0.3 <- bd0ver(x., L.3, mpfrPrec = 1024, chkVerb=FALSE))# 10.4 sec (was only 7.3s !?) system.time(bd0.3 <- bd0ver(x., L.3, mpfrPrec = 256, chkVerb=FALSE)) # 2.25 s p.relE(bd0.3) # up to 1e-11 rel.error !! p.relE(bd0.3, log="xy") system.time(bd0.3.d25 <- bd0ver(x., L.3, delta = .25, mpfrPrec = 256, chkVerb=FALSE)) # 2.14 s p.relE(bd0.3.d25, add = TRUE) ## different x : system.time(bd0.2.2e307 <- bd0ver(2e307, mpfrPrec = 256, delta = 0.15, chkVerb=FALSE)) # 1.44 s p.relE(bd0.2.2e307, log="xy") system.time(bd0.2.2e307.d.50 <- bd0ver(2e307, mpfrPrec = 256, delta = 0.50, chkVerb=FALSE)) # 1.44 s p.relE(bd0.2.2e307.d.50, add=TRUE) ## less large x .. still same problem: ebd0() is worse than bd0() system.time(bd0.2.2e305 <- bd0ver(2e305, np=np_bd0(2e305, ex2M = 7),# <- enlarge x-range mpfrPrec = 256, delta = 0.40, chkVerb=FALSE)) # 1.9 s p.relE(bd0.2.2e305, log="xy") # interesting: quite different behavior on the left & right ! ## less large x .. still same problem: ebd0() is worse than bd0() x <- 1e250 bd0.2.1e250 <- bd0ver(x, mpfrPrec = 256, delta = 0.3, chkVerb=FALSE) p.relE(bd0.2.1e250, log="xy") ## the pd0_l*() and pd0_p*() log1pmx() and logcf() using versions are quite *stable* ## *and* differ on the left and right of np = x x <- 1e120 bd0.2.1e120 <- bd0ver(x, mpfrPrec = 256, delta = 0.25, chkVerb=FALSE) ## p.relE(bd0.2.1e120) # still rel.E -8e-13 p.relE(bd0.2.1e120, log = "xy", f.lowsm = 1/8) x <- 1e20 bd0.2.1e20 <- bd0ver(x, mpfrPrec = 256, delta = 0.15, chkVerb=FALSE) ## p.relE(bd0.2.1e20) # rel.E -8e-12 p.relE(bd0.2.1e20, log = "xy") # rel.E -8e-12 bd0.2.1e20.d40 <- bd0ver(x, mpfrPrec = 256, delta = .40, chkVerb=FALSE) p.relE(bd0.2.1e20.d40, add=TRUE) ## again: quite different behavior left and right of np = x apply(bd0.2.1e20$relE, 2, quantile) ## ebd0 ebd0C bd0 ## 0% -1.92048e-12 -1.92048e-12 -8.57215e-16 ## 25% -4.95045e-16 -4.95045e-16 -1.25294e-16 ## 50% -9.57122e-17 -9.57122e-17 -1.52898e-17 ## 75% 2.86728e-16 2.86728e-16 1.12103e-16 ## 100% 8.79272e-13 8.79272e-13 7.24720e-16 x <- 1e14 bd0.2.1e14 <- bd0ver(x, mpfrPrec = 256, delta = 0.25, chkVerb=FALSE) ## p.relE(bd0.2.1e14) # rel.E -2.7e-12 p.relE(bd0.2.1e14, log = "xy") apply(bd0.2.1e14$relE, 2, quantile) ## ebd0 ebd0C bd0 ## 0% -2.68404e-13 -2.68404e-13 -7.64861e-16 ## 25% -1.13517e-16 -1.13517e-16 -1.11198e-16 ## 50% 3.98896e-17 3.98896e-17 1.82322e-17 ## 75% 2.28878e-16 2.28878e-16 1.43407e-16 ## 100% 1.05107e-12 1.05107e-12 9.37795e-16 # = not soo large ===> see that ebd0() is *better* than bd0() -- outside Taylor range! x <- 1e9 bd0.2.1e9 <- bd0ver(x, mpfrPrec = 256, chkVerb=FALSE) bd0.2.1e9.d25 <- bd0ver(x, mpfrPrec = 256, delta = 0.25, chkVerb=FALSE) ## p.relE(bd0.2.1e9, log = "xy") # rel.E -2.7e-12 p.relE(bd0.2.1e9.d25, log = "xy") # bd0() still better _inside_ delta-range; but *worse* outside bd0.2.1e9.d50 <- bd0ver(x, mpfrPrec = 256, delta = 0.50, chkVerb=FALSE) bd0.1e9.RE <- cbind(bd0.2.1e9$relE, bd0.d.25 = bd0.2.1e9.d25$relE[,"bd0"], bd0.d.50 = bd0.2.1e9.d50$relE[,"bd0"]) apply(bd0.1e9.RE, 2, quantile) |> print(digits = 3) ## ebd0 ebd0C bd0 bd0.l bd0.p1l bd0.p1.d1 bd0.d.25 bd0.d.50 ## 0% -5.65e-13 -5.65e-13 -5.61e-15 -8.33e-16 -5.37e-16 -7.59e-16 -7.14e-16 -6.34e-16 ## 25% -1.54e-16 -1.54e-16 -1.61e-16 -1.12e-16 -7.88e-17 -1.26e-16 -1.41e-16 -1.38e-16 ## 50% -1.63e-17 -1.63e-17 -4.60e-18 -3.02e-17 3.53e-17 2.80e-18 -2.03e-17 -2.38e-17 ## 75% 1.22e-16 1.22e-16 1.74e-16 8.25e-17 1.63e-16 1.63e-16 1.21e-16 1.09e-16 ## 100% 2.33e-13 2.33e-13 5.04e-15 5.98e-16 1.13e-15 1.13e-15 8.91e-16 6.08e-16 apply(abs(bd0.1e9.RE), 2, quantile) |> print(digits = 3) ## ebd0 ebd0C bd0 bd0.l bd0.p1l bd0.p1.d1 bd0.d.25 bd0.d.50 ## 0% 6.14e-19 6.14e-19 6.14e-19 2.94e-19 2.94e-19 1.24e-18 6.14e-19 6.14e-19 ## 25% 5.68e-17 5.68e-17 7.08e-17 5.20e-17 5.65e-17 7.52e-17 5.90e-17 5.73e-17 ## 50% 1.34e-16 1.34e-16 1.70e-16 1.04e-16 1.33e-16 1.49e-16 1.30e-16 1.30e-16 ## 75% 4.14e-16 4.14e-16 3.63e-16 1.75e-16 2.39e-16 2.61e-16 2.41e-16 2.23e-16 ## 100% 5.65e-13 5.65e-13 5.61e-15 8.33e-16 1.13e-15 1.13e-15 8.91e-16 6.34e-16 ## ===> bd0( delta = 0.50) is "best" here ## = x <- 1e6 bd0.2.1e6 <- bd0ver(x, mpfrPrec = 256, chkVerb=FALSE) bd0.2.1e6.30 <- bd0ver(x, mpfrPrec = 256, delta = .30, chkVerb=FALSE) ## p.relE(bd0.2.1e6) # rel.E +- 4e-12 p.relE(bd0.2.1e6, log="xy", f.lowsm = 1/8) p.relE(bd0.2.1e6.30, log="xy", f.lowsm = 1/8) apply(abs(bd0.2.1e6.30$relE), 2, quantile) ## is not ideal summary, as "left" and "right" differ so much x <- 1e3 bd0.2.1e3 <- bd0ver(x, mpfrPrec = 256, chkVerb=FALSE) ## p.relE(bd0.2.1e3) # rel.E +- 4e-12 p.relE(bd0.2.1e3, log="xy") # rel.E +- 4e-12 bd0.2.1e3.d30 <- bd0ver(x, mpfrPrec = 256, delta = 0.30, chkVerb=FALSE) p.relE(bd0.2.1e3.d30, add = TRUE, f.lowsm = 1/8) ## --> ebd0() is superior in the very "outskirts" ## --> delta = 0.30 is clearly still too small ## - {number of correct digits}: ## matplot(L.x, log10(abs(bd0.2.1e3$relE[,-1])), type="l") ## abline(v = range(np[abs(x - np) <= 0.1 * (x + np)]), col=adjustcolor(2, 2/3), lwd=2, lty=3) ## now extend to the full "lambda" / np range [slowish !] str(np.x <- seq(.5, 4*x, length.out=1001)) bd0..1e3 <- bd0ver(x, np.x, mpfrPrec = 256, chkVerb=FALSE) ## this is slow: ## p.relE(bd0..1e3, log="") # rel.E +- 4e-12 bd0..1e3.d.40 <- bd0ver(x, np.x, delta = 0.40, mpfrPrec = 256, chkVerb=FALSE) if(do.pdf) { dev.off(); pdf("p.relE_bd0ver_x=1000.pdf") } p.relE(bd0..1e3, log="y") # rel.E +- 4e-12 p.relE(bd0..1e3.d.40, add = TRUE, f.lowsm = 1/8) # rel.E +- 4e-12 mtext(shortRversion(), adj=1, cex=2/3); mtext('p.relE(bd0..1e3, log="y")', adj=0, cex=2/3) apply(bd0..1e3$relE, 2, quantile) ## even here, bd0(x, np) is more accurate around np ~= x ## but in the flanks, ebd0 is better : matplot(np.x, log10(abs(bd0..1e3$relE[,-1])), type="l") if(do.pdf) dev.off() ### all the above till `` quit("no") '' is *only* run if(doExtras)