R Under development (unstable) (2024-08-21 r87038 ucrt) -- "Unsuffered Consequences"
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Platform: x86_64-w64-mingw32/x64

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> require("DPQ")
Loading required package: DPQ
> require("Rmpfr")
Loading required package: Rmpfr
Loading required package: gmp

Attaching package: 'gmp'

The following objects are masked from 'package:base':

    %*%, apply, crossprod, matrix, tcrossprod

C code of R package 'Rmpfr': GMP using 64 bits per limb


Attaching package: 'Rmpfr'

The following object is masked from 'package:gmp':

    outer

The following object is masked from 'package:DPQ':

    log1mexp

The following objects are masked from 'package:stats':

    dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm

The following objects are masked from 'package:base':

    cbind, pmax, pmin, rbind

> 
> source(system.file(package="DPQ", "extraR", "relErr-plots.R", mustWork=TRUE))
Loading required package: sfsmisc

Attaching package: 'sfsmisc'

The following objects are masked from 'package:gmp':

    factorize, is.whole

> ##-> drawEps.h() , mtextVersion(), ...
> (doExtras <- DPQ:::doExtras())
[1] FALSE
> (noLdbl <- (.Machine$sizeof.longdouble <= 8)) ## TRUE when --disable-long-double
[1] FALSE
> options(width = 100, nwarnings = 1e5)
> 
> k1 <- 44300 + (1:800)
> p1 <- (63/64)
> x <- p1; stopifnot(  (1-x)-1 == -x , 1-x == 2^-6 )
> px <- p1 ^ k1
> isNorm <- px >= 2^-1022 # is "normal", i.e., not subnormal
> k1 <- k1[isNorm]
> px <- px[isNorm]
> dput(px,, "digits17")
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1.7857054158744765e-307, 1.7578037687514377e-307, 1.7303380848646965e-307, 
1.7033015522886858e-307, 1.6766874655341749e-307, 1.6504892238852036e-307, 
1.6247003297619972e-307, 1.599314387109466e-307, 1.5743250998108806e-307, 
1.5497262701263355e-307, 1.5255117971556116e-307, 1.5016756753250552e-307, 
1.4782119928981012e-307, 1.4551149305090683e-307, 1.4323787597198642e-307, 
1.4099978415992413e-307, 1.3879666253242531e-307, 1.3662796468035616e-307, 
1.3449315273222559e-307, 1.3239169722078458e-307, 1.3032307695170981e-307, 
1.2828677887433936e-307, 1.2628229795442779e-307, 1.2430913704888986e-307, 
1.2236680678250097e-307, 1.2045482542652438e-307, 1.1857271877923494e-307, 
1.167200200483094e-307, 1.1489626973505456e-307, 1.1310101552044434e-307, 
1.113338121529374e-307, 1.0959422133804773e-307, 1.0788181162964075e-307, 
1.0619615832292761e-307, 1.0453684334913186e-307, 1.0290345517180169e-307, 
1.0129558868474229e-307, 9.971284511154318e-308, 9.8154831906675317e-308, 
9.6621162658133508e-308, 9.5111456991600184e-308, 9.3625340476106433e-308, 
9.216244453116726e-308, 9.0722406335367785e-308, 8.9304868736377661e-308, 
8.7909480162371756e-308, 8.6535894534834692e-308, 8.5183771182727905e-308, 
8.3852774757997778e-308, 8.254257515240406e-308, 8.1252847415647756e-308, 
7.9983271674778251e-308, 7.8733533054859845e-308, 7.7503321600877656e-308, 
7.6292332200863951e-308, 7.5100264510225443e-308, 7.3926822877253171e-308, 
7.2771716269796092e-308, 7.163465820308053e-308, 7.0515366668657394e-308, 
6.9413564064459622e-308, 6.8328977125952446e-308, 6.7261336858359432e-308, 
6.6210378469947573e-308, 6.5175841306354638e-308, 6.4157468785942845e-308, 
6.315500833616249e-308, 6.2168211330909951e-308, 6.1196833028864483e-308, 
6.0240632512788482e-308, 5.9299372629776157e-308, 5.8372819932435903e-308, 
5.7460744620991591e-308, 5.6562920486288598e-308, 5.5679124853690338e-308, 
5.4809138527851425e-308, 5.3952745738353754e-308, 5.3109734086191969e-308, 
5.2279894491095222e-308, 5.1463021139671865e-308, 5.0658911434364491e-308, 
4.9867365943202544e-308, 4.9088188350340006e-308, 4.8321185407365938e-308, 
4.7566166885375845e-308, 4.6822945527791846e-308, 4.6091337003920103e-308, 
4.5371159863233849e-308, 4.4662235490370825e-308, 4.3964388060833778e-308, 
4.3277444497383252e-308, 4.2601234427111635e-308, 4.1935590139188015e-308, 
4.1280346543263203e-308, 4.0635341128524717e-308, 4.000041392339152e-308, 
3.9375407455838524e-308, 3.8760166714341046e-308, 3.8154539109429469e-308, 
3.7558374435844633e-308, 3.697152483528456e-308, 3.6393844759733243e-308, 
3.5825190935362406e-308, 3.5265422326997369e-308, 3.4714400103138038e-308, 
3.4171987601526508e-308, 3.3638050295252652e-308, 3.3112455759389332e-308, 
3.2595073638148874e-308, 3.2085775612552798e-308, 3.1584435368606661e-308, 
3.1090928565972181e-308, 3.0605132807128865e-308, 3.0126927607017478e-308, 
2.9656194363157828e-308, 2.9192816326233486e-308, 2.8736678571136091e-308, 
2.8287667968462087e-308, 2.7845673156454869e-308, 2.741058451338526e-308, 
2.6982294130363616e-308, 2.6560695784576682e-308, 2.6145684912942676e-308, 
2.5737158586177946e-308, 2.5335015483268913e-308, 2.4939155866342835e-308, 
2.4549481555931231e-308, 2.4165895906619802e-308, 2.378830378307887e-308, 
2.3416611536468261e-308, 2.3050726981210945e-308, 2.2690559372129525e-308, 
2.2336019381940003e-308)
> dput(px,, "hex")
c(0x1.648865267f463p-1007, 0x1.5ef64391e5492p-1007, 0x1.597a6a839db3fp-1007, 
0x1.541480d98f3d2p-1007, 0x1.4ec42ed629003p-1007, 0x1.49891e1ad05c3p-1007, 
0x1.4462f9a2651acp-1007, 0x1.3f516dbbdb865p-1007, 0x1.3a542804ec184p-1007, 
0x1.356ad764d867ep-1007, 0x1.30952c0745064p-1007, 0x1.2bd2d75727f22p-1007, 
0x1.27238bf9cb526p-1007, 0x1.2286fdc9e4251p-1007, 0x1.1dfce1d2bc948p-1007, 
0x1.1984ee4b71a23p-1007, 0x1.151eda9243dbap-1007, 0x1.10ca5f27facc3p-1007, 
0x1.0c8735ab5ae1p-1007, 0x1.085518d4ad758p-1007, 0x1.0433c4715abfbp-1007, 
0x1.0022f55f9554bp-1007, 0x1.f844d3142dfebp-1008, 0x1.f063bfc7dd46bp-1008, 
0x1.e8a230c8bdd1ap-1008, 0x1.e0ffa8059ada5p-1008, 0x1.d97ba965846efp-1008, 
0x1.d215babfee5d3p-1008, 0x1.cacd63d4eea3cp-1008, 0x1.c3a22e459ae93p-1008, 
0x1.bc93a58c847d8p-1008, 0x1.b5a156f6526b9p-1008, 0x1.aecad19a7921ep-1008, 
0x1.a80fa6540f3d6p-1008, 0x1.a16f67babf006p-1008, 0x1.9ae9aa1bd4046p-1008, 
0x1.947e037364b45p-1008, 0x1.8e2c0b6597218p-1008, 0x1.87f35b3800c5p-1008, 
0x1.81d38dcb20c1ep-1008, 0x1.7bcc3f93f43eep-1008, 0x1.75dd0e95a46dep-1008, 
0x1.70059a5b4ddc3p-1008, 0x1.6a4583f1e0a4cp-1008, 0x1.649c6de219222p-1008, 
0x1.5f09fc2a90bdap-1008, 0x1.598dd439e67abp-1008, 0x1.54279ce8fee0cp-1008, 
0x1.4ed6fe755ae54p-1008, 0x1.499ba27b8579ap-1008, 0x1.447533f19763cp-1008, 
0x1.3f635f21d1063p-1008, 0x1.3a65d1a549c21p-1008, 0x1.357c3a5eb49b1p-1008, 
0x1.30a6497539c8ap-1008, 0x1.2be3b04f64e18p-1008, 0x1.2734218e274ep-1008, 
0x1.22975107eeb0cp-1008, 0x1.1e0cf3c3cef6p-1008, 0x1.1994bff4bfba2p-1008, 
0x1.152e6cf4ecbb4p-1008, 0x1.10d9b34119085p-1008, 0x1.0c964c7414a43p-1008, 
0x1.0863f3424451ap-1008, 0x1.044263753b406p-1008, 0x1.003159e766535p-1008, 
0x1.f86128ff91741p-1009, 0x1.f07fa45b932e4p-1009, 0x1.e8bda5ca24e19p-1009, 
0x1.e11aaf32fc4ep-1009, 0x1.d9964476305cdp-1009, 0x1.d22feb64579b6p-1009, 
0x1.cae72bb6c63cfp-1009, 0x1.c3bb8f07eb24p-1009, 0x1.bcaca0cbcb777p-1009, 
0x1.b5b9ee489c499p-1009, 0x1.aee3068f79d86p-1009, 0x1.a8277a753bf1p-1009, 
0x1.a186dc8b67014p-1009, 0x1.9b00c11939654p-1009, 0x1.9494be14d47fap-1009, 
0x1.8e426b1c812dap-1009, 0x1.880961700f28fp-1009, 0x1.81e93bea4eec5p-1009, 
0x1.7be196faa5b0ap-1009, 0x1.75f2109ebb19ep-1009, 0x1.701a485c402d7p-1009, 
0x1.6a59df3acf2ccp-1009, 0x1.64b077bde3f01p-1009, 0x1.5f1db5deec605p-1009, 
0x1.59a13f0770aecp-1009, 0x1.543aba0b52ec1p-1009, 0x1.4ee9cf2325a06p-1009, 
0x1.49ae27e69909ep-1009, 0x1.44876f46fea5bp-1009, 0x1.3f755189e2ab2p-1009, 
0x1.3a777c43bb207p-1009, 0x1.358d9e52ac33fp-1009, 0x1.30b767d961832p-1009, 
0x1.2bf48a39fbfd1p-1009, 0x1.2744b811140d2p-1009, 0x1.22a7a530cfbcfp-1009, 
0x1.1e1d069c0c7dfp-1009, 0x1.19a492819c4cp-1009, 0x1.153e003795dadp-1009, 
0x1.10e90836b7836p-1009, 0x1.0ca56415dca55p-1009, 0x1.0872ce858532cp-1009, 
0x1.0451034b6f1dfp-1009, 0x1.003fbf3e41618p-1009, 0x1.f87d808290b7fp-1010, 
0x1.f09b8a8086751p-1010, 0x1.e8d91c56845b4p-1010, 0x1.e135b7e52a49dp-1010, 
0x1.d9b0e10595a0ap-1010, 0x1.d24a1d817f4a2p-1010, 0x1.cb00f50b794dp-1010, 
0x1.c3d4f1374b67cp-1010, 0x1.bcc59d726e3a3p-1010, 0x1.b5d286fca4814p-1010, 
0x1.aefb3ce0b1ef4p-1010, 0x1.a83f4fed2f278p-1010, 0x1.a19e52ad7a6aep-1010, 
0x1.9b17d962c4813p-1010, 0x1.94ab79fd396f3p-1010, 0x1.8e58cc1544897p-1010, 
0x1.881f68e4ef775p-1010, 0x1.81feeb415bb97p-1010, 0x1.7bf6ef94564a9p-1010, 
0x1.760713d604f16p-1010, 0x1.702ef786acddap-1010, 0x1.6a6e3ba8922a2p-1010, 
0x1.64c482b9efe18p-1010, 0x1.5f3170af0821fp-1010, 0x1.59b4aaec4c017p-1010, 
0x1.544dd8409ad16p-1010, 0x1.4efca0df98662p-1010, 0x1.49c0ae5c1a049p-1010, 
0x1.4499aba2a99c7p-1010, 0x1.3f8744f41ef6p-1010, 0x1.3a8927e04e7a3p-1010, 
0x1.359f0340cd404p-1010, 0x1.30c88733ca0b4p-1010, 0x1.2c056516fae31p-1010, 
0x1.27554f829ef79p-1010, 0x1.22b7fa44947bbp-1010, 0x1.1e2d1a5b8229cp-1010, 
0x1.19b465f214211p-1010, 0x1.154d945a4bd09p-1010, 0x1.10f85e08e2a15p-1010, 
0x1.0cb47c90bf16dp-1010, 0x1.0881aa9e7c1a7p-1010, 0x1.045fa3f4022ap-1010, 
0x1.004e256432216p-1010, 0x1.f899d99d42b1bp-1011, 0x1.f0b77236cda6fp-1011, 
0x1.e8f4946df2705p-1011, 0x1.e150c21c3aa69p-1011, 0x1.d9cb7f13c9bbfp-1011, 
0x1.d26451177a95p-1011, 0x1.cb1abfd31caabp-1011, 0x1.c3ee54d3d038p-1011, 
0x1.bcde9b8080f72p-1011, 0x1.b5eb21127ef34p-1011, 0x1.af13748e34f78p-1011, 
0x1.a85726bbfc23ap-1011, 0x1.a1b5ca210c331p-1011, 0x1.9b2ef2f888024p-1011, 
0x1.94c2372ca5e24p-1011, 0x1.8e6f2e4ff34abp-1011, 0x1.88357196b37d8p-1011, 
0x1.82149bd058af9p-1011, 0x1.7c0c4961174cdp-1011, 0x1.761c183b92efap-1011, 
0x1.7043a7daa4a3ep-1011, 0x1.6a82993b3a115p-1011, 0x1.64d88ed64d291p-1011, 
0x1.5f452c9af3f46p-1011, 0x1.59c817e888249p-1011, 0x1.5460f788e604p-1011, 
0x1.4f0f73aac26bfp-1011, 0x1.49d335dc17624p-1011, 0x1.44abe904a704cp-1011, 
0x1.3f9939609468ap-1011, 0x1.3a9ad47b1217p-1011, 0x1.35b0692925ceap-1011, 
0x1.30d9a78481377p-1011, 0x1.2c1640e66f329p-1011, 0x1.2765e7e2d575cp-1011, 
0x1.22c850434a1ffp-1011, 0x1.1e3d2f023cf77p-1011, 0x1.19c43a4634039p-1011, 
0x1.155d295d1b338p-1011, 0x1.1107b4b7a6c6bp-1011, 0x1.0cc395e4c82bap-1011, 
0x1.0890878d350afp-1011, 0x1.046e456f0036cp-1011, 0x1.005c8c594435ep-1011, 
0x1.f8b6344fbe4a2p-1012, 0x1.f0d35b7e7f50fp-1012, 0x1.e9100e108553bp-1012, 
0x1.e16bcdd8433e6p-1012, 0x1.d9e61ea0e2316p-1012, 0x1.d27e86265ea8ap-1012, 
0x1.cb348c0dc52ep-1012, 0x1.c407b9dd8e194p-1012, 0x1.bcf79af617e0ep-1012, 
0x1.b603bc8a3f816p-1012, 0x1.af2bad9816835p-1012, 0x1.a86efee1b6295p-1012, 
0x1.a1cd42e62f50ap-1012, 0x1.9b460dda96936p-1012, 0x1.94d8f5a32c391p-1012, 
0x1.8e8591cc9f883p-1012, 0x1.884b7b856d0a1p-1012, 0x1.822a4d975755ep-1012, 
0x1.7c21a460f9f89p-1012, 0x1.76311dcf7610bp-1012, 0x1.7058595838387p-1012, 
0x1.6a96f7f2d7579p-1012, 0x1.64ec9c130bfa3p-1012, 0x1.5f58e9a2bfca4p-1012, 
0x1.59db85fc34cb2p-1012, 0x1.547417e443f7fp-1012, 0x1.4f224784b2e81p-1012, 
0x1.49e5be66a01c7p-1012, 0x1.44be276d059cp-1012, 0x1.3fab2ecf51859p-1012, 
0x1.3aac8214143f7p-1012, 0x1.35c1d00bc3ee7p-1012, 0x1.30eac8cb94decp-1012, 
0x1.2c271da8668b4p-1012, 0x1.27768131c4f11p-1012, 0x1.22d8a72cfddd5p-1012, 
0x1.1e4d449049e5ep-1012, 0x1.19d40f7e08be4p-1012, 0x1.156cbf40109b5p-1012, 
0x1.11170c431058ep-1012, 0x1.0cd2b01204178p-1012, 0x1.089f6551bc072p-1012, 
0x1.047ce7bc7517p-1012, 0x1.006af41d8342ap-1012, 0x1.f8d2909a1a6b3p-1013, 
0x1.f0ef4657b2018p-1013, 0x1.e92b893e53398p-1013, 0x1.e186db1959ecap-1013, 
0x1.da00bfacf484fp-1013, 0x1.d298bcae40b2dp-1013, 0x1.cb4e59bb87b01p-1013, 
0x1.c421205499915p-1013, 0x1.bd109bd3472bp-1013, 0x1.b61c5963fa0e6p-1013, 
0x1.af43e7fe6a262p-1013, 0x1.a886d85e707d8p-1013, 0x1.a1e4bcfcf6bb9p-1013, 
0x1.9b5d2a0902e0ap-1013, 0x1.94efb560ded52p-1013, 0x1.8e9bf68b5b59dp-1013, 
0x1.886186b12dec6p-1013, 0x1.824000966934bp-1013, 0x1.7c3700940f8fep-1013, 
0x1.76462491bf51ap-1013, 0x1.706d0bff78546p-1013, 0x1.6aab57cf7a73p-1013, 
0x1.6500aa703c894p-1013, 0x1.5f6ca7c67b971p-1013, 0x1.59eef52761a8cp-1013, 
0x1.54873952c4221p-1013, 0x1.4f351c6d79119p-1013, 0x1.49f847fbc32d5p-1013, 
0x1.44d066dbd4209p-1013, 0x1.3fbd254064d01p-1013, 0x1.3abe30ab633cdp-1013, 
0x1.35d337e8b5afep-1013, 0x1.30fbeb0912d92p-1013, 0x1.2c37fb5cee8dcp-1013, 
0x1.27871b6f7ad38p-1013, 0x1.22e8ff01bce83p-1013, 0x1.1e5d5b05b5f49p-1013, 
0x1.19e3e5999f1ccp-1013, 0x1.157c560338a05p-1013, 0x1.112664ab2bbddp-1013, 
0x1.0ce1cb187f0edp-1013, 0x1.08ae43ec1d12ap-1013, 0x1.048b8adc6c9e5p-1013, 
0x1.00795cb0faebdp-1013, 0x1.f8eeee7c6e005p-1014, 0x1.f10b32c27c485p-1014, 
0x1.e94705f772573p-1014, 0x1.e1a1e9df948ddp-1014, 0x1.da1b6238163bap-1014, 
0x1.d2b2f4af35e2bp-1014, 0x1.cb6828dc790b2p-1014, 0x1.c43a88390726fp-1014, 
0x1.bd299e18230a6p-1014, 0x1.b634f79fc27e3p-1014, 0x1.af5c23c143743p-1014, 
0x1.a89eb3323e666p-1014, 0x1.a1fc3865756cdp-1014, 0x1.9b744783df972p-1014, 
0x1.95067665d018cp-1014, 0x1.8eb25c8c38d86p-1014, 0x1.8877931a07f4fp-1014, 
0x1.8255b4cd9fd52p-1014, 0x1.7c4c5dfa6955dp-1014, 0x1.765b2c827fb08p-1014, 
0x1.7081bfd075b1bp-1014, 0x1.6abfb8d133dafp-1014, 0x1.6514b9edef0b8p-1014, 
0x1.5f806706374f5p-1014, 0x1.5a02656a1e722p-1014, 0x1.549a5bd475f85p-1014, 
0x1.4f47f26524207p-1014, 0x1.4a0ad29b8f8ffp-1014, 0x1.44e2a7512151bp-1014, 
0x1.3fcf1cb3dccc6p-1014, 0x1.3acfe0410d593p-1014, 0x1.35e4a0c00923dp-1014, 
0x1.310d0e3d08ff4p-1014, 0x1.2c48da0414db4p-1014, 0x1.2797b69c0487dp-1014, 
0x1.22f957c19475bp-1014, 0x1.1e6d72628e23ep-1014, 0x1.19f3bc9903eb5p-1014, 
0x1.158beda69fdbap-1014, 0x1.1135bdf0055c3p-1014, 0x1.0cf0e6f84546cp-1014, 
0x1.08bd235c6431bp-1014, 0x1.049a2ecef2a0ep-1014, 0x1.0087c613b6d66p-1014, 
0x1.f90b4df6cff61p-1015, 0x1.f12720bef4b63p-1015, 0x1.e962843bf8e36p-1015, 
0x1.e1bcfa2b08ffdp-1015, 0x1.da3606425cdbdp-1015, 0x1.d2cd2e2953686p-1015, 
0x1.cb81f970ae1acp-1015, 0x1.c453f18aeb625p-1015, 0x1.bd42a1c4bfb4dp-1015, 
0x1.b64d973dacb5fp-1015, 0x1.af7460e0b6032p-1015, 0x1.a8b68f5d332b1p-1015, 
0x1.a213b51fbe5e6p-1015, 0x1.9b8b664b3f64fp-1015, 0x1.951d38b212675p-1015, 
0x1.8ec8c3cf4a1dcp-1015, 0x1.888da0c00cf54p-1015, 0x1.826b6a3d0cc17p-1015, 
0x1.7c61bc94188e7p-1015, 0x1.767035a1c82c3p-1015, 0x1.709674cb410b8p-1015, 
0x1.6ad41af814075p-1015, 0x1.6528ca8c33b73p-1015, 0x1.5f94276202e85p-1015, 
0x1.5a15d6c47adcbp-1015, 0x1.54ad7f6968f14p-1015, 0x1.4f5ac96bc34d8p-1015, 
0x1.4a1d5e4614404p-1015, 0x1.44f4e8ccfbef4p-1015, 0x1.3fe11529c7ff9p-1015, 
0x1.3ae190d520df9p-1015, 0x1.35f60a91cc5c1p-1015, 0x1.311e3267852aap-1015, 
0x1.2c59b99de715fp-1015, 0x1.27a852b76f79ap-1015, 0x1.2309b16c91bbbp-1015, 
0x1.1e7d8aa6df74cp-1015, 0x1.1a03947c43f6fp-1015, 0x1.159b862a52e71p-1015, 
0x1.11451811a99b8p-1015, 0x1.0d0003b162f51p-1015, 0x1.08cc03a29d693p-1015, 
0x1.04a8d39412f39p-1015, 0x1.00963045c2a7cp-1015, 0x1.f927af09573a5p-1016, 
0x1.f143104d31dd6p-1016, 0x1.e97e040bfd15fp-1016, 0x1.e1d80bfbcd219p-1016, 
0x1.da50abcbdded1p-1016, 0x1.d2e7691cae756p-1016, 0x1.cb9bcb783bbb8p-1016, 
0x1.c46d5c4a5acc9p-1016, 0x1.bd5ba6d931616p-1016, 0x1.b666383dcc9bep-1016, 
0x1.af8c9f5cd5697p-1016, 0x1.a8ce6cdf6213dp-1016, 0x1.a22b332be48b8p-1016, 
0x1.9ba2865f34f95p-1016, 0x1.9533fc45b8256p-1016, 0x1.8edf2c54a144dp-1016, 
0x1.88a3afa34ebfcp-1016, 0x1.828120e4c184cp-1016, 0x1.7c771c612e7ebp-1016, 
0x1.76853fefa9c4bp-1016, 0x1.70ab2aefeb1dap-1016, 0x1.6ae87e442b713p-1016, 
0x1.653cdc4b1ac36p-1016, 0x1.5fa7e8d9ee585p-1016, 0x1.5a294936869efp-1016, 
0x1.54c0a411ac848p-1016, 0x1.4f6da18165d26p-1016, 0x1.4a2feafb603b2p-1016, 
0x1.45072b4f72ba3p-1016, 0x1.3ff30ea234ef5p-1016, 0x1.3af34267ac1b9p-1016, 
0x1.3607755e0d6b2p-1016, 0x1.312f578895357p-1016, 0x1.2c6a9a2a72e0ap-1016, 
0x1.27b8efc1c9152p-1016, 0x1.231a0c02c1f0cp-1016, 0x1.1e8da3d2b6e9p-1016, 
0x1.1a136d436c0d6p-1016, 0x1.15ab1f8e5e5d2p-1016, 0x1.1154731024e3bp-1016, 
0x1.0d0f2143e4502p-1016, 0x1.08dae4bed4beep-1016, 0x1.04b7792bd96bep-1016, 
0x1.00a49b472a064p-1016, 0x1.f94411b41abc4p-1017, 0x1.f15f016d4a515p-1017, 
0x1.e999856795281p-1017, 0x1.e1f31f51f6d37p-1017, 0x1.da6b52d4aef82p-1017, 
0x1.d301a5895c3c4p-1017, 0x1.cbb59ef336cb5p-1017, 0x1.c486c87769f02p-1017, 
0x1.bd74ad558c486p-1017, 0x1.b67edaa036174p-1017, 0x1.afa4df35b53eep-1017, 
0x1.a8e64bb8de69ep-1017, 0x1.a242b289faf04p-1017, 0x1.9bb9a7bfd3048p-1017, 
0x1.954ac120d3b86p-1017, 0x1.8ef5961c50698p-1017, 0x1.88b9bfc3df27ep-1017, 
0x1.8296d8c4cfab4p-1017, 0x1.7c8c7d61bc6c9p-1017, 0x1.769a4b6c357aep-1017, 
0x1.70bfe23e84a4fp-1017, 0x1.6afce2b58a926p-1017, 0x1.6550ef2ab4682p-1017, 
0x1.5fbbab6e09967p-1017, 0x1.5a3cbcc051702p-1017, 0x1.54d3c9cd502a6p-1017, 
0x1.4f807aa61ae9bp-1017, 0x1.4a4278bb827e1p-1017, 0x1.45196ed894741p-1017, 
0x1.4005091d32224p-1017, 0x1.3b04f4f8bd59cp-1017, 0x1.3618e124da645p-1017, 
0x1.31407da046facp-1017, 0x1.2c7b7ba9c5deep-1017, 0x1.27c98dbb1ec76p-1017, 
0x1.232a6784324c4p-1017, 0x1.1e9dbde621831p-1017, 0x1.1a2346ee88fdp-1017, 
0x1.15bab9d2ced91p-1017, 0x1.1163ceeb839dbp-1017, 0x1.0d1e3fafd58f3p-1017, 
0x1.08e9c6b11638fp-1017, 0x1.04c61f9651e01p-1017, 0x1.00b30717f8989p-1017, 
0x1.f96075f7316c6p-1018, 0x1.f17af41f54a6bp-1018, 0x1.e9b5084ed7541p-1018, 
0x1.e20e342d9bf6cp-1018, 0x1.da85fb5ce586fp-1018, 0x1.d31be36f71f0dp-1018, 
0x1.cbcf73e1b4291p-1018, 0x1.c4a036122d586p-1018, 0x1.bd8db539e4a3p-1018, 
0x1.b6977e64fd108p-1018, 0x1.afbd206b691c3p-1018, 0x1.a8fe2be9bb77cp-1018, 
0x1.a25a333a1489ep-1018, 0x1.9bd0ca6d2c37cp-1018, 0x1.956187437786ep-1018, 
0x1.8f0c012669a8cp-1018, 0x1.88cfd121d0022p-1018, 0x1.82ac91dd48c22p-1018, 
0x1.7ca1df95d39f1p-1018, 0x1.76af58177c509p-1018, 0x1.70d49ab71e5f5p-1018, 
0x1.6b11484c41e5dp-1018, 0x1.6565032b10de4p-1018, 0x1.5fcf6f1e649acp-1018, 
0x1.5a503161eb086p-1018, 0x1.54e6f09c635c3p-1018, 0x1.4f9354d9f1cecp-1018, 
0x1.4a5507868a079p-1018, 0x1.452bb3686fdf7p-1018, 0x1.4017049ace1ffp-1018, 
0x1.3b16a88862e77p-1018, 0x1.362a4de6415bdp-1018, 0x1.3151a4aea8566p-1018, 
0x1.2c8c5e1bedb51p-1018, 0x1.27da2ca37dfe3p-1018, 0x1.233ac3f0f0064p-1018, 
0x1.1eadd8e12c462p-1018, 0x1.1a33217da7951p-1018, 0x1.15ca54f7b0f6bp-1018, 
0x1.11732ba3d232ep-1018, 0x1.0d2d5ef542ea1p-1018, 0x1.08f8a9796dde7p-1018, 
0x1.04d4c6d38826fp-1018, 0x1.00c173b83a065p-1018, 0x1.f97cdbd2b23c7p-1019, 
0x1.f196e86367738p-1019, 0x1.e9d08cc1d9d5bp-1019, 0x1.e2294a8ed26e6p-1019, 
0x1.daa0a5649724ap-1019, 0x1.d33622cf04c81p-1019, 0x1.cbe94a43c8b4fp-1019, 
0x1.c4b9a51ab9922p-1019, 0x1.bda6be864eabdp-1019, 0x1.b6b0238c35712p-1019, 
0x1.afd562fe049b6p-1019, 0x1.a9160d720c88fp-1019, 0x1.a271b53c4456dp-1019, 
0x1.9be7ee6753457p-1019, 0x1.95784eadb5f86p-1019, 0x1.8f226d72ff208p-1019, 
0x1.88e5e3bd3324p-1019, 0x1.82c24c2e3e577p-1019, 0x1.7cb742fd855e1p-1019, 
0x1.76c465f18f489p-1019, 0x1.70e95459c90b7p-1019, 0x1.6b25af0861e74p-1019, 
0x1.6579184c405fap-1019, 0x1.5fe333eb0f5e3p-1019, 0x1.5a63a71b6320bp-1019, 
0x1.54fa187ef5943p-1019, 0x1.4fa6301cf9bdep-1019, 0x1.4a67975c85d6ep-1019, 
0x1.453df8ff13bf9p-1019, 0x1.4029011b17709p-1019, 0x1.3b285d16ab12dp-1019, 
0x1.363bbba250668p-1019, 0x1.3162ccb3c724ep-1019, 0x1.2c9d4180f8085p-1019, 
0x1.27eacc7af4283p-1019, 0x1.234b214908579p-1019, 0x1.1ebdf4c3e4363p-1019, 
0x1.1a42fcf0d4a55p-1019, 0x1.15d9f0fd1152cp-1019, 0x1.118289391d0d7p-1019, 
0x1.0d3c7f1438994p-1019, 0x1.09078d17e7b6ep-1019, 0x1.04e36ee38818p-1019, 
0x1.00cfe127f9f7ap-1019, 0x1.f9994346b41f8p-1020, 0x1.f1b2de39994fp-1020, 
0x1.e9ec12c0b2e9dp-1020, 0x1.e2446275b01e2p-1020, 0x1.dabb50ebd95dbp-1020, 
0x1.d35063a829f83p-1020, 0x1.cc03221989505p-1020, 0x1.c4d31591232b1p-1020, 
0x1.bdbfc93ade9e6p-1020, 0x1.b6c8ca15f323fp-1020, 0x1.afeda6ed9b576p-1020, 
0x1.a92df051e4eap-1020, 0x1.a28938909d565p-1020, 0x1.9bff13ae5ae1p-1020, 
0x1.958f175fa1758p-1020, 0x1.8f38db0222efap-1020, 0x1.88fbf7961a63ep-1020, 
0x1.82d807b7c1fa5p-1020, 0x1.7ccca798e2f27p-1020, 0x1.76d974fa7f66ap-1020, 
0x1.70fe0f2695691p-1020, 0x1.6b3a16e9fb136p-1020, 0x1.658d2e8e53271p-1020, 
0x1.5ff6f9d419da8p-1020, 0x1.5a771decc9731p-1020, 0x1.550d4175164d4p-1020, 
0x1.4fb90c6f41f41p-1020, 0x1.4a7a283d84ec4p-1020, 0x1.45503f9c8ed89p-1020, 
0x1.403afe9e1c9d3p-1020, 0x1.3b3a12a3a42abp-1020, 0x1.364d2a59159a1p-1020, 
0x1.3173f5afb143ap-1020, 0x1.2cae25d8f27e9p-1020, 0x1.27fb6d418eb4ap-1020, 
0x1.235b7f8c8879dp-1020, 0x1.1ece118e5657ep-1020, 0x1.1a52d9481cfe8p-1020, 
0x1.15e98de2fc8a8p-1020, 0x1.1191e7ab70986p-1020, 0x1.0d4ba00cc2d6p-1020, 
0x1.0916718c8fcaap-1020, 0x1.04f217c65d8b8p-1020, 0x1.00de4f6744155p-1020, 
0x1.f9b5ac534e09fp-1021, 0x1.f1ced5a200d1cp-1021, 0x1.ea079a4b78ce8p-1021, 
0x1.e25f7be24aeb4p-1021, 0x1.dad5fdf2c1bf9p-1021, 0x1.d36aa5faf6b8ap-1021, 
0x1.cc1cfb630addbp-1021, 0x1.c4ec87757eb24p-1021, 0x1.bdd8d557a8b77p-1021, 
0x1.b6e172024a14ap-1021, 0x1.b005ec3a40ec4p-1021, 0x1.a945d48957e89p-1021, 
0x1.a2a0bd373288fp-1021, 0x1.9c163a4255bedp-1021, 0x1.95a5e1594c67dp-1021, 
0x1.8f4f49d3e7363p-1021, 0x1.89120cac97996p-1021, 0x1.82edc479e53afp-1021, 
0x1.7ce20d67fda61p-1021, 0x1.76ee85325daf7p-1021, 0x1.7112cb1d9438bp-1021, 
0x1.6b4e7ff11de7dp-1021, 0x1.65a145f159703p-1021, 0x1.600ac0d9940a7p-1021, 
0x1.5a8a95d62dba5p-1021, 0x1.55206b7ed5036p-1021, 0x1.4fcbe9d0d9af5p-1021, 
0x1.4a8cba2996489p-1021, 0x1.45628740efef7p-1021, 0x1.404cfd23ec2fbp-1021, 
0x1.3b4bc92f5c7efp-1021, 0x1.365e9a0a9f0dp-1021, 0x1.31851fa27490cp-1021, 
0x1.2cbf0b23eabe8p-1021, 0x1.280c0ef75b139p-1021, 0x1.236bdebb7da74p-1021, 
0x1.1ede2f408fb0ap-1021, 0x1.1a62b6838d71ep-1021, 0x1.15f92ba97f3c1p-1021, 
0x1.11a146fad93f2p-1021, 0x1.0d5ac1deedda2p-1021, 0x1.092556d77222cp-1021, 
0x1.0500c17c145a3p-1021, 0x1.00ecbe762408dp-1021, 0x1.f9d216f896f15p-1022, 
0x1.f1eace9cb4959p-1022, 0x1.ea23236241c33p-1022, 0x1.e27a96d4b8bc2p-1022, 
0x1.daf0ac7965d93p-1022, 0x1.d384e9c78041dp-1022, 0x1.cc36d6206240dp-1022, 
0x1.c505fac7e0b7cp-1022, 0x1.bdf1e2dcc134ep-1022, 0x1.b6fa1b514e301p-1022, 
0x1.b01e32e408f75p-1022, 0x1.a95dba1878d37p-1022, 0x1.a2b8433016f03p-1022, 
0x1.9c2d622356946p-1022, 0x1.95bcac9ac93a1p-1022, 0x1.8f65b9e85e153p-1022, 
0x1.89282300bc9cep-1022, 0x1.83038274b9aa6p-1022, 0x1.7cf7746ae6c3cp-1022, 
0x1.770396993b28bp-1022, 0x1.7127883ed63c1p-1022, 0x1.6b62ea1ddae32p-1022, 
0x1.65b55e7563779p-1022, 0x1.601e88fb8de9bp-1022, 0x1.5a9e0ed79fb21p-1022, 
0x1.5533969c41334p-1022, 0x1.4fdec841d02e7p-1022, 0x1.4a9f4d20c8edcp-1022, 
0x1.4574cfec45cap-1022, 0x1.405efcac94b2ep-1022, 0x1.3b5d80b9e2601p-1022, 
0x1.36700ab6fad69p-1022, 0x1.31964a8c1eeb3p-1022, 0x1.2ccff161ee6f9p-1022, 
0x1.281cb19c66b5dp-1022, 0x1.237c3ed5f51afp-1022, 0x1.1eee4dda9d468p-1022, 
0x1.1a7294a332d17p-1022, 0x1.1608ca50a6062p-1022, 0x1.11b0a727636e1p-1022, 
0x1.0d69e48ac5e05p-1022, 0x1.09343cf89ac8dp-1022, 0x1.050f6c04b85dbp-1022, 
0x1.00fb2e54a57c4p-1022)
> 
> 
> x <- p2 <- 1023/1024
> stopifnot(  (1-x)-1 == -x , 1-x == 2^-10 )
> 
> k2 <- 700000 + (1:1000)
> dput(p2^ k2, control = "digits17") ## last 2-3 digits printed *differ*
c(9.4086285016650735e-298, 9.3994403878939149e-298, 9.3902612468901133e-298, 
9.3810910698911961e-298, 9.3719298481432567e-298, 9.3627775729009291e-298, 
9.353634235427393e-298, 9.3444998269943586e-298, 9.3353743388820597e-298, 
9.3262577623792452e-298, 9.3171500887831706e-298, 9.3080513093995932e-298, 
9.2989614155427581e-298, 9.2898803985353915e-298, 9.2808082497086976e-298, 
9.2717449604023414e-298, 9.2626905219644486e-298, 9.2536449257515922e-298, 
9.2446081631287874e-298, 9.2355802254694829e-298, 9.2265611041555478e-298, 
9.2175507905772709e-298, 9.2085492761333477e-298, 9.1995565522308734e-298, 
9.1905726102853361e-298, 9.1815974417206032e-298, 9.1726310379689232e-298, 
9.1636733904709069e-298, 9.1547244906755259e-298, 9.1457843300401003e-298, 
9.1368529000302959e-298, 9.1279301921201101e-298, 9.1190161977918671e-298, 
9.1101109085362109e-298, 9.1012143158520938e-298, 9.092326411246769e-298, 
9.0834471862357862e-298, 9.0745766323429774e-298, 9.0657147411004556e-298, 
9.0568615040485994e-298, 9.0480169127360514e-298, 9.0391809587197075e-298, 
9.0303536335647077e-298, 9.0215349288444301e-298, 9.0127248361404811e-298, 
9.0039233470426868e-298, 8.9951304531490915e-298, 8.9863461460659372e-298, 
8.9775704174076702e-298, 8.9688032587969195e-298, 8.9600446618645013e-298, 
8.9512946182493991e-298, 8.9425531195987653e-298, 8.9338201575679075e-298, 
8.9250957238202816e-298, 8.9163798100274887e-298, 8.9076724078692581e-298, 
8.8989735090334488e-298, 8.8902831052160342e-298, 8.8816011881210956e-298, 
8.872927749460822e-298, 8.8642627809554895e-298, 8.8556062743334619e-298, 
8.8469582213311833e-298, 8.838318613693165e-298, 8.8296874431719801e-298, 
8.8210647015282565e-298, 8.812450380530671e-298, 8.8038444719559346e-298, 
8.7952469675887898e-298, 8.7866578592220038e-298, 8.7780771386563562e-298, 
8.7695047977006378e-298, 8.7609408281716332e-298, 8.7523852218941227e-298, 
8.7438379707008654e-298, 8.7352990664326042e-298, 8.7267685009380401e-298, 
8.7182462660738429e-298, 8.7097323537046305e-298, 8.7012267557029654e-298, 
8.69272946394935e-298, 8.6842404703322107e-298, 8.6757597667479021e-298, 
8.6672873451006876e-298, 8.6588231973027382e-298, 8.650367315274122e-298, 
8.6419196909427993e-298, 8.633480316244614e-298, 8.62504918312328e-298, 
8.6166262835303865e-298, 8.6082116094253773e-298, 8.5998051527755477e-298, 
8.5914069055560394e-298, 8.5830168597498333e-298, 8.574635007347733e-298, 
8.5662613403483699e-298, 8.5578958507581856e-298, 8.5495385305914311e-298, 
8.5411893718701491e-298, 8.5328483666241828e-298, 8.5245155068911504e-298, 
8.516190784716452e-298, 8.5078741921532524e-298, 8.4995657212624794e-298, 
8.4912653641128088e-298, 8.4829731127806659e-298, 8.474688959350217e-298, 
8.4664128959133508e-298, 8.4581449145696868e-298, 8.4498850074265517e-298, 
8.4416331665989859e-298, 8.4333893842097303e-298, 8.4251536523892124e-298, 
8.4169259632755516e-298, 8.4087063090145402e-298, 8.4004946817596422e-298, 
8.3922910736719863e-298, 8.3840954769203535e-298, 8.3759078836811748e-298, 
8.3677282861385166e-298, 8.3595566764840845e-298, 8.3513930469172063e-298, 
8.3432373896448256e-298, 8.3350896968815008e-298, 8.3269499608493892e-298, 
8.3188181737782481e-298, 8.3106943279054179e-298, 8.3025784154758223e-298, 
8.2944704287419596e-298, 8.2863703599638908e-298, 8.2782782014092382e-298, 
8.2701939453531748e-298, 8.2621175840784162e-298, 8.2540491098752135e-298, 
8.2459885150413519e-298, 8.2379357918821316e-298, 8.2298909327103717e-298, 
8.2218539298463962e-298, 8.2138247756180309e-298, 8.2058034623605914e-298, 
8.1977899824168795e-298, 8.189784328137177e-298, 8.1817864918792296e-298, 
8.1737964660082544e-298, 8.1658142428969175e-298, 8.1578398149253391e-298, 
8.1498731744810765e-298, 8.141914313959121e-298, 8.1339632257618958e-298, 
8.1260199022992375e-298, 8.118084335988398e-298, 8.1101565192540355e-298, 
8.1022364445282014e-298, 8.094324104250342e-298, 8.0864194908672844e-298, 
8.0785225968332337e-298, 8.0706334146097643e-298, 8.0627519366658095e-298, 
8.05487815547766e-298, 8.0470120635289504e-298, 8.0391536533106609e-298, 
8.0313029173211e-298, 8.0234598480659031e-298, 8.0156244380580257e-298, 
8.0077966798177348e-298, 7.9999765658726005e-298, 7.9921640887574908e-298, 
7.9843592410145632e-298, 7.9765620151932596e-298, 7.9687724038502979e-298, 
7.9609903995496633e-298, 7.9532159948626032e-298, 7.945449182367619e-298, 
7.9376899546504643e-298, 7.929938304304126e-298, 7.9221942239288279e-298, 
7.9144577061320222e-298, 7.9067287435283793e-298, 7.8990073287397757e-298, 
7.8912934543953047e-298, 7.883587113131247e-298, 7.8758882975910794e-298, 
7.8681970004254632e-298, 7.8605132142922351e-298, 7.8528369318564026e-298, 
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4.3652437548028982e-298, 4.3609808214485987e-298, 4.3567220511151528e-298, 
4.3524674397371102e-298, 4.3482169832529923e-298, 4.3439706776052841e-298, 
4.3397285187404348e-298, 4.335490502608853e-298, 4.3312566251648985e-298, 
4.3270268823668859e-298, 4.3228012701770747e-298, 4.3185797845616671e-298, 
4.3143624214908065e-298, 4.3101491769385695e-298, 4.305940046882965e-298, 
4.3017350273059313e-298, 4.2975341141933272e-298, 4.2933373035349353e-298, 
4.2891445913244523e-298, 4.2849559735594867e-298, 4.2807714462415578e-298, 
4.2765910053760873e-298, 4.2724146469724e-298, 4.2682423670437163e-298, 
4.2640741616071496e-298, 4.2599100266837056e-298, 4.2557499582982725e-298, 
4.2515939524796218e-298, 4.2474420052604032e-298, 4.2432941126771408e-298, 
4.2391502707702298e-298, 4.2350104755839304e-298, 4.2308747231663683e-298, 
4.2267430095695263e-298, 4.2226153308492431e-298, 4.2184916830652111e-298, 
4.2143720622809676e-298, 4.2102564645638959e-298, 4.2061448859852202e-298, 
4.2020373226200003e-298, 4.1979337705471291e-298, 4.1938342258493294e-298, 
4.1897386846131488e-298, 4.1856471429289558e-298, 4.1815595968909398e-298, 
4.177476042597101e-298, 4.1733964761492517e-298, 4.1693208936530127e-298, 
4.1652492912178048e-298, 4.1611816649568498e-298, 4.1571180109871654e-298, 
4.1530583254295608e-298, 4.1490026044086333e-298, 4.1449508440527652e-298, 
4.14090304049412e-298, 4.1368591898686376e-298, 4.1328192883160316e-298, 
4.1287833319797852e-298, 4.1247513170071485e-298, 4.1207232395491343e-298, 
4.1166990957605115e-298, 4.112678881799808e-298, 4.1086625938293004e-298, 
4.1046502280150142e-298, 4.1006417805267181e-298, 4.0966372475379222e-298, 
4.0926366252258743e-298, 4.0886399097715521e-298, 4.084647097359665e-298, 
4.0806581841786502e-298, 4.0766731664206635e-298, 4.0726920402815802e-298, 
4.0687148019609931e-298, 4.0647414476622028e-298, 4.0607719735922208e-298, 
4.0568063759617592e-298, 4.0528446509852337e-298, 4.0488867948807562e-298, 
4.0449328038701303e-298, 4.0409826741788508e-298, 4.0370364020360981e-298, 
4.0330939836747346e-298, 4.0291554153313024e-298, 4.025220693246018e-298, 
4.0212898136627704e-298, 4.017362772829115e-298, 4.0134395669962744e-298, 
4.0095201924191291e-298, 4.0056046453562201e-298, 4.0016929220697393e-298, 
3.9977850188255309e-298, 3.9938809318930839e-298, 3.989980657545532e-298, 
3.9860841920596476e-298, 3.9821915317158396e-298, 3.9783026727981478e-298, 
3.9744176115942433e-298, 3.9705363443954215e-298, 3.9666588674965972e-298, 
3.9627851771963077e-298, 3.9589152697967021e-298, 3.955049141603541e-298, 
3.9511867889261941e-298, 3.9473282080776336e-298, 3.9434733953744328e-298, 
3.9396223471367618e-298, 3.9357750596883866e-298, 3.9319315293566597e-298, 
3.9280917524725222e-298, 3.9242557253704979e-298, 3.9204234443886911e-298, 
3.91659490586878e-298, 3.9127701061560174e-298, 3.9089490415992244e-298, 
3.9051317085507878e-298, 3.9013181033666568e-298, 3.8975082224063371e-298, 
3.8937020620328936e-298, 3.8898996186129396e-298, 3.8861008885166378e-298, 
3.8823058681176957e-298, 3.8785145537933625e-298, 3.8747269419244232e-298, 
3.8709430288952007e-298, 3.8671628110935447e-298, 3.8633862849108366e-298, 
3.8596134467419779e-298, 3.8558442929853943e-298, 3.8520788200430255e-298, 
3.8483170243203276e-298, 3.8445589022262643e-298, 3.8408044501733089e-298, 
3.8370536645774366e-298, 3.8333065418581226e-298, 3.8295630784383397e-298, 
3.825823270744552e-298, 3.8220871152067155e-298, 3.8183546082582717e-298, 
3.8146257463361444e-298, 3.8109005258807378e-298, 3.8071789433359326e-298, 
3.8034609951490811e-298, 3.7997466777710058e-298, 3.7960359876559952e-298, 
3.7923289212617996e-298, 3.7886254750496299e-298, 3.7849256454841519e-298, 
3.781229429033484e-298, 3.7775368221691932e-298, 3.7738478213662936e-298, 
3.7701624231032407e-298, 3.7664806238619287e-298, 3.7628024201276885e-298, 
3.7591278083892829e-298, 3.7554567851389026e-298, 3.7517893468721654e-298, 
3.7481254900881104e-298, 3.7444652112891964e-298, 3.7408085069812966e-298, 
3.7371553736736979e-298, 3.7335058078790946e-298, 3.7298598061135876e-298, 
3.7262173648966799e-298, 3.7225784807512728e-298, 3.7189431502036641e-298, 
3.7153113697835435e-298, 3.7116831360239892e-298, 3.7080584454614661e-298, 
3.7044372946358197e-298, 3.700819680090277e-298, 3.6972055983714388e-298, 
3.6935950460292795e-298, 3.6899880196171413e-298, 3.686384515691734e-298, 
3.6827845308131288e-298, 3.6791880615447568e-298, 3.6755951044534041e-298, 
3.6720056561092114e-298, 3.6684197130856676e-298, 3.6648372719596073e-298, 
3.6612583293112089e-298, 3.657682881723991e-298, 3.6541109257848077e-298, 
3.6505424580838456e-298, 3.6469774752146233e-298, 3.6434159737739842e-298, 
3.6398579503620953e-298, 3.6363034015824449e-298, 3.6327523240418369e-298, 
3.6292047143503897e-298, 3.6256605691215319e-298, 3.6221198849719992e-298, 
3.6185826585218316e-298, 3.6150488863943686e-298, 3.6115185652162493e-298, 
3.6079916916174052e-298, 3.6044682622310601e-298, 3.6009482736937249e-298, 
3.597431722645196e-298, 3.5939186057285503e-298, 3.5904089195901437e-298, 
3.5869026608796062e-298, 3.5833998262498412e-298, 3.579900412357019e-298, 
3.5764044158605765e-298, 3.5729118334232128e-298, 3.5694226617108854e-298, 
3.5659368973928083e-298, 3.5624545371414482e-298, 3.5589755776325211e-298, 
3.555500015544989e-298, 3.5520278475610586e-298, 3.5485590703661745e-298, 
3.5450936806490204e-298)
> dput(p2^ k2, control = "hex")
c(0x1.3b0b694f990ecp-987, 0x1.3abca67545287p-987, 0x1.3a6df74ba7d73p-987, 
0x1.3a1f5bcdd4ed3p-987, 0x1.39d0d3f6e178p-987, 0x1.39825fc1e3bfap-987, 
0x1.3933ff29f346bp-987, 0x1.38e5b22a28c9ep-987, 0x1.389778bd9e3fbp-987, 
0x1.384952df6ed82p-987, 0x1.37fb408ab6fc6p-987, 0x1.37ad41ba944eap-987, 
0x1.375f566a25a99p-987, 0x1.37117e948b202p-987, 0x1.36c3ba34e5fd6p-987, 
0x1.3676094658c3ep-987, 0x1.36286bc4072dbp-987, 0x1.35dae1a9162bep-987, 
0x1.358d6af0abe65p-987, 0x1.35400795efbb6p-987, 0x1.34f2b7940a3f7p-987, 
0x1.34a57ae6253cep-987, 0x1.345851876bb39p-987, 0x1.340b3b7309d8ap-987, 
0x1.33be38a42d163p-987, 0x1.33714916040aep-987, 0x1.33246cc3be89ep-987, 
0x1.32d7a3a88d9a4p-987, 0x1.328aedbfa376ep-987, 0x1.323e4b04338ep-987, 
0x1.31f1bb7172812p-987, 0x1.31a53f0296248p-987, 0x1.3158d5b2d57efp-987, 
0x1.310c7f7d68c99p-987, 0x1.30c03c5d896f6p-987, 0x1.30740c4e720dp-987, 
0x1.3027ef4b5e708p-987, 0x1.2fdbe54f8b98ep-987, 0x1.2f8fee5637b6p-987, 
0x1.2f440a5aa2281p-987, 0x1.2ef839580b7f8p-987, 0x1.2eac7b49b57cap-987, 
0x1.2e60d02ae30f4p-987, 0x1.2e1537f6d8568p-987, 0x1.2dc9b2a8daa07p-987, 
0x1.2d7e403c3069cp-987, 0x1.2d32e0ac215dbp-987, 0x1.2ce793f3f6555p-987, 
0x1.2c9c5a0ef957cp-987, 0x1.2c5132f875996p-987, 0x1.2c061eabb77cp-987, 
0x1.2bbb1d240c8e2p-987, 0x1.2b702e5cc38bp-987, 0x1.2b2552512c5a2p-987, 
0x1.2ada88fc980fp-987, 0x1.2a8fd25a58e9p-987, 0x1.2a452e65c252cp-987, 
0x1.29fa9d1a28e23p-987, 0x1.29b01e72e258p-987, 0x1.2965b26b459f6p-987, 
0x1.291b58feaacep-987, 0x1.28d112286b235p-987, 0x1.2886dde3e1088p-987, 
0x1.283cbc2c68104p-987, 0x1.27f2acfd5cf64p-987, 0x1.27a8b0521d9fp-987, 
0x1.275ec62609179p-987, 0x1.2714ee747f955p-987, 0x1.26cb2938e2757p-987, 
0x1.2681766e943cdp-987, 0x1.2637d610f897cp-987, 0x1.25ee481b74599p-987, 
0x1.25a4cc896d7c8p-987, 0x1.255b63564b212p-987, 0x1.25120c7d758e6p-987, 
0x1.24c8c7fa5630fp-987, 0x1.247f95c8579b7p-987, 0x1.243675e2e5858p-987, 
0x1.23ed68456ccc2p-987, 0x1.23a46ceb5b70fp-987, 0x1.235b83d0209a1p-987, 
0x1.2312acef2c91fp-987, 0x1.22c9e843f0c6cp-987, 0x1.228135c9dfca9p-987, 
0x1.2238957c6d52ap-987, 0x1.21f007570e375p-987, 0x1.21a78b553873cp-987, 
0x1.215f21726325ap-987, 0x1.2116c9aa068cep-987, 0x1.20ce83f79c0b3p-987, 
0x1.208650569e243p-987, 0x1.203e2ec2887cbp-987, 0x1.1ff61f36d7da9p-987, 
0x1.1fae21af0a249p-987, 0x1.1f6636269e621p-987, 0x1.1f1e5c9914ba7p-987, 
0x1.1ed69501ee754p-987, 0x1.1e8edf5cadf9ap-987, 0x1.1e473ba4d6ce3p-987, 
0x1.1dffa9d5ed987p-987, 0x1.1db829eb781d1p-987, 0x1.1d70bbe0fd3fp-987, 
0x1.1d295fb204ffbp-987, 0x1.1ce2155a187e7p-987, 0x1.1c9adcd4c1f86p-987, 
0x1.1c53b61d8cc7ep-987, 0x1.1c0ca1300564ap-987, 0x1.1bc59e07b9635p-987, 
0x1.1b7eaca03774fp-987, 0x1.1b37ccf50f672p-987, 0x1.1af0ff01d2234p-987, 
0x1.1aaa42c211aebp-987, 0x1.1a639831612a5p-987, 0x1.1a1cff4b54d2p-987, 
0x1.19d6780b81fcdp-987, 0x1.1990026d7f1c5p-987, 0x1.19499e6ce3bc8p-987, 
0x1.19034c0548839p-987, 0x1.18bd0b3247317p-987, 0x1.1876dbef7a9fbp-987, 
0x1.1830be387ec1p-987, 0x1.17eab208f0a15p-987, 0x1.17a4b75c6e653p-987, 
0x1.175ece2e97499p-987, 0x1.1718f67b0ba3cp-987, 0x1.16d3303d6ce0dp-987, 
0x1.168d7b715d85ap-987, 0x1.1647d812812e4p-987, 0x1.1602461c7c8dfp-987, 
0x1.15bcc58af56edp-987, 0x1.1577565992b17p-987, 0x1.1531f883fc4ccp-987, 
0x1.14ecac05db4dbp-987, 0x1.14a770dad9d6ep-987, 0x1.146246fea3206p-987, 
0x1.141d2e6ce377ap-987, 0x1.13d82721483ecp-987, 0x1.139331177fecbp-987, 
0x1.134e4c4b3a0cbp-987, 0x1.130978b8273e3p-987, 0x1.12c4b659f9346p-987, 
0x1.1280052c62b61p-987, 0x1.123b652b179d7p-987, 0x1.11f6d651ccd78p-987, 
0x1.11b2589c38645p-987, 0x1.116dec0611563p-987, 0x1.1129908b0fd1ep-987, 
0x1.10e54626ed0dfp-987, 0x1.10a10cd56352ap-987, 0x1.105ce4922df9dp-987, 
0x1.1018cd59096e5p-987, 0x1.0fd4c725b32bfp-987, 0x1.0f90d1f3e9bf3p-987, 
0x1.0f4cedbf6cc4cp-987, 0x1.0f091a83fce99p-987, 0x1.0ec5583d5bea5p-987, 
0x1.0e81a6e74c935p-987, 0x1.0e3e067d92c03p-987, 0x1.0dfa76fbf35b8p-987, 
0x1.0db6f85e345ebp-987, 0x1.0d738aa01cd19p-987, 0x1.0d302dbd74ca6p-987, 
0x1.0cece1b2056d3p-987, 0x1.0ca9a67998ebdp-987, 0x1.0c667c0ffa859p-987, 
0x1.0c236270f686fp-987, 0x1.0be059985a495p-987, 0x1.0b9d6181f432cp-987, 
0x1.0b5a7a2993b5bp-987, 0x1.0b17a38b0950cp-987, 0x1.0ad4dda2268e7p-987, 
0x1.0a92286abe04dp-987, 0x1.0a4f83e0a3555p-987, 0x1.0a0cefffab2c7p-987, 
0x1.09ca6cc3ab41bp-987, 0x1.0987fa287a56ep-987, 0x1.09459829f0384p-987, 
0x1.090346c3e5bc3p-987, 0x1.08c105f234c2dp-987, 0x1.087ed5b0b8359p-987, 
0x1.083cb5fb4c079p-987, 0x1.07faa6cdcd349p-987, 0x1.07b8a82419c14p-987, 
0x1.0776b9fa10badp-987, 0x1.0734dc4b9236ap-987, 0x1.06f30f147f521p-987, 
0x1.06b15250ba324p-987, 0x1.066fa5fc2603bp-987, 0x1.062e0a12a6fa3p-987, 
0x1.05ec7e9022507p-987, 0x1.05ab03707e47ep-987, 0x1.056998afa2284p-987, 
0x1.05283e49763fcp-987, 0x1.04e6f439e3e23p-987, 0x1.04a5ba7cd5693p-987, 
0x1.0464910e3633ep-987, 0x1.042377e9f2a65p-987, 0x1.03e26f0bf829ap-987, 
0x1.03a17670352bap-987, 0x1.03608e12991e5p-987, 0x1.031fb5ef1478p-987, 
0x1.02deee0198b2fp-987, 0x1.029e3646184ccp-987, 0x1.025d8eb886c6bp-987, 
0x1.021cf754d8a4fp-987, 0x1.01dc7017036edp-987, 0x1.019bf8fafdadfp-987, 
0x1.015b91fcbeee8p-987, 0x1.011b3b183fbedp-987, 0x1.00daf44979aeep-987, 
0x1.009abd8c67507p-987, 0x1.005a96dd0436ap-987, 0x1.001a80374cf59p-987, 
0x1.ffb4f32e7e44ap-988, 0x1.ff3505f1b2a51p-988, 0x1.feb538b036387p-988, 
0x1.fe358b620a2aep-988, 0x1.fdb5fdff31a85p-988, 0x1.fd36907fb1dbep-988, 
0x1.fcb742db91ef7p-988, 0x1.fc38150adb0afp-988, 0x1.fbb9070598543p-988, 
0x1.fb3a18c3d6ee2p-988, 0x1.fabb4a3da5f86p-988, 0x1.fa3c9b6b168eep-988, 
0x1.f9be0c443bc94p-988, 0x1.f93f9cc12aba5p-988, 0x1.f8c14cd9fa6fap-988, 
0x1.f8431c86c3f1p-988, 0x1.f7c50bbfa24p-988, 0x1.f7471a7cb2577p-988, 
0x1.f6c948b6132aep-988, 0x1.f64b9663e5a61p-988, 0x1.f5ce037e4cacbp-988, 
0x1.f5508ffd6d198p-988, 0x1.f4d33bd96dbe4p-988, 0x1.f456070a7762dp-988, 
0x1.f3d8f188b4c4fp-988, 0x1.f35bfb4c5297cp-988, 0x1.f2df244d7f832p-988, 
0x1.f2626c846c234p-988, 0x1.f1e5d3e94b083p-988, 0x1.f1695a7450b57p-988, 
0x1.f0ed001db3a14p-988, 0x1.f070c4ddac345p-988, 0x1.eff4a8ac74c95p-988, 
0x1.ef78ab8249ac2p-988, 0x1.eefccd576919bp-988, 0x1.ee810e24133f6p-988, 
0x1.ee056de08a3a9p-988, 0x1.ed89ec8512181p-988, 0x1.ed0e8a09f0d38p-988, 
0x1.ec9346676e575p-988, 0x1.ec182195d47bbp-988, 0x1.eb9d1b8d6f06ap-988, 
0x1.eb2234468baadp-988, 0x1.eaa76bb97a07fp-988, 0x1.ea2cc1de8ba97p-988, 
0x1.e9b236ae14068p-988, 0x1.e937ca2068818p-988, 0x1.e8bd7c2de0676p-988, 
0x1.e8434cced4ef4p-988, 0x1.e7c93bfba13a1p-988, 0x1.e74f49aca251cp-988, 
0x1.e6d575da37292p-988, 0x1.e65bc07cc09b6p-988, 0x1.e5e2298ca16b3p-988, 
0x1.e568b1023e42ep-988, 0x1.e4ef56d5fdb35p-988, 0x1.e4761b004833ep-988, 
0x1.e3fcfd798821dp-988, 0x1.e383fe3a29bfcp-988, 0x1.e30b1d3a9b355p-988, 
0x1.e2925a734c8e9p-988, 0x1.e219b5dcafbb6p-988, 0x1.e1a12f6f388f7p-988, 
0x1.e128c7235cc15p-988, 0x1.e0b07cf193ea2p-988, 0x1.e03850d257853p-988, 
0x1.dfc042be22ef4p-988, 0x1.df4852ad73669p-988, 0x1.ded08098c809bp-988, 
0x1.de58cc78a1d7bp-988, 0x1.dde1364583af4p-988, 0x1.dd69bdf7f24e5p-988, 
0x1.dcf263887451cp-988, 0x1.dc7b26ef9234ap-988, 0x1.dc040825d6502p-988, 
0x1.db8d0723ccda8p-988, 0x1.db1623e203e75p-988, 0x1.da9f5e590b665p-988, 
0x1.da28b68175238p-988, 0x1.d9b22c53d4c63p-988, 0x1.d93bbfc8bfd1p-988, 
0x1.d8c570d8cda11p-988, 0x1.d84f3f7c976dap-988, 0x1.d7d92bacb847cp-988, 
0x1.d7633561cd19bp-988, 0x1.d6ed5c9474a67p-988, 0x1.d677a13d4f894p-988, 
0x1.d602035500356p-988, 0x1.d58c82d42af55p-988, 0x1.d5171fb375eaap-988, 
0x1.d4a1d9eb890d2p-988, 0x1.d42cb1750e2aep-988, 0x1.d3b7a648b0e75p-988, 
0x1.d342b85f1ebb1p-988, 0x1.d2cde7b106f36p-988, 0x1.d25934371ab1bp-988, 
0x1.d1e49dea0cebp-988, 0x1.d17024c29267cp-988, 0x1.d0fbc8b961c33p-988, 
0x1.d08789c7336acp-988, 0x1.d01367e4c19dep-988, 0x1.cf9f630ac86d7p-988, 
0x1.cf2b7b3205bb6p-988, 0x1.ceb7b0533939fp-988, 0x1.ce440267246bap-988, 
0x1.cdd071668aa28p-988, 0x1.cd5cfd4a30ffep-988, 0x1.cce9a60ade73ap-988, 
0x1.cc766ba15bbcp-988, 0x1.cc034e0673651p-988, 0x1.cb904d32f1c83p-988, 
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0x1.1b0fdd3ef4de1p-988, 0x1.1ac91947a520dp-988, 0x1.1a82670153379p-988, 
0x1.1a3bc66792e2cp-988, 0x1.19f53775f8fep-988, 0x1.19aeba281b7fcp-988, 
0x1.19684e799178ep-988, 0x1.1921f465f3148p-988, 0x1.18dbabe8d997cp-988, 
0x1.189574fddf616p-988, 0x1.184f4fa09fe98p-988, 0x1.18093bccb7c19p-988, 
0x1.17c3397dc493ap-988, 0x1.177d48af65227p-988, 0x1.1737695d39493p-988, 
0x1.16f19b82e1faep-988, 0x1.16abdf1c01426p-988, 0x1.166634243a421p-988, 
0x1.16209a9731338p-988, 0x1.15db12708b673p-988, 0x1.15959babef445p-988, 
0x1.1550364504488p-988, 0x1.150ae23773077p-988, 0x1.14c59f7ee52abp-988, 
0x1.14806e1705716p-988, 0x1.143b4dfb7fbp-988, 0x1.13f63f2800d02p-988, 
0x1.13b1419836cfep-988, 0x1.136c5547d0c23p-988, 0x1.13277a327ecep-988, 
0x1.12e2b053f22e5p-988, 0x1.129df7a7dd31cp-988, 0x1.12595029f33a7p-988, 
0x1.1214b9d5e8bdbp-988, 0x1.11d034a773438p-988, 0x1.118bc09a4966ap-988, 
0x1.11475daa22d45p-988, 0x1.11030bd2b84bap-988, 0x1.10becb0fc39d8p-988, 
0x1.107a9b5cffacap-988, 0x1.10367cb6286cbp-988, 0x1.0ff26f16fae2ap-988, 
0x1.0fae727b3523ep-988, 0x1.0f6a86de96569p-988, 0x1.0f26ac3cdeb1p-988, 
0x1.0ee2e291cf795p-988, 0x1.0e9f29d92b057p-988, 0x1.0e5b820eb4babp-988, 
0x1.0e17eb2e310d8p-988, 0x1.0dd4653365814p-988, 0x1.0d90f01a18a7ep-988, 
0x1.0d4d8bde1221cp-988, 0x1.0d0a387b1a9d3p-988, 0x1.0cc6f5ecfbd69p-988, 
0x1.0c83c42f80979p-988, 0x1.0c40a33e74b77p-988, 0x1.0bfd9315a51a4p-988, 
0x1.0bba93b0dfb1p-988, 0x1.0b77a50bf3791p-988, 0x1.0b34c722b07c3p-988, 
0x1.0af1f9f0e7d01p-988, 0x1.0aaf3d726b962p-988, 0x1.0a6c91a30efb3p-988, 
0x1.0a29f67ea6377p-988, 0x1.09e76c01068dfp-988, 0x1.09a4f226064c4p-988, 
0x1.096288e97ccabp-988, 0x1.09203047426b8p-988, 0x1.08dde83b309aep-988, 
0x1.089bb0c121cecp-988, 0x1.085989d4f1865p-988, 0x1.081773727c49fp-988, 
0x1.07d56d959faadp-988, 0x1.0793783a3a42fp-988, 0x1.0751935c2bb46p-988, 
0x1.070fbef754a97p-988, 0x1.06cdfb0796d44p-988, 0x1.068c4788d4ee9p-988, 
0x1.064aa476f2b95p-988, 0x1.060911cdd4fcap-988, 0x1.05c78f8961876p-988, 
0x1.05861da57f2fp-988, 0x1.0544bc1e15cf4p-988, 0x1.05036aef0e49cp-988, 
0x1.04c22a1452863p-988, 0x1.0480f989cd719p-988, 0x1.043fd94b6afe3p-988, 
0x1.03fec95518237p-988, 0x1.03bdc9a2c2dd7p-988, 0x1.037cda305a2cbp-988, 
0x1.033bfaf9ce163p-988, 0x1.02fb2bfb0fa2ap-988, 0x1.02ba6d3010decp-988, 
0x1.0279be94c4da8p-988, 0x1.023920251fa95p-988, 0x1.01f891dd16616p-988, 
0x1.01b813b89f1bdp-988, 0x1.0177a5b3b0f4p-988, 0x1.013747ca4407cp-988, 
0x1.00f6f9f85176cp-988, 0x1.00b6bc39d3626p-988, 0x1.00768e8ac4ed9p-988, 
0x1.003670e7223c5p-988, 0x1.ffecc695d0e78p-989, 0x1.ff6ccb642b735p-989, 
0x1.feecf03152687p-989, 0x1.fe6d34f54613dp-989, 0x1.fded99a808c25p-989, 
0x1.fd6e1e419ec02p-989, 0x1.fceec2ba0e587p-989, 0x1.fc6f87095fd4ep-989, 
0x1.fbf06b279d7cep-989, 0x1.fb716f0cd3958p-989, 0x1.faf292b11060ap-989, 
0x1.fa73d60c641c9p-989, 0x1.f9f53916e1038p-989, 0x1.f976bbc89b4b4p-989, 
0x1.f8f85e19a9247p-989, 0x1.f87a200222ba2p-989, 0x1.f7fc017a22317p-989, 
0x1.f77e0279c3a8fp-989, 0x1.f70022f92538p-989, 0x1.f68262f066eebp-989, 
0x1.f604c257aad4fp-989, 0x1.f587412714ea4p-989, 0x1.f509df56cb25p-989, 
0x1.f48c9cdef5724p-989, 0x1.f40f79b7bdb4ep-989, 0x1.f39275d94fc57p-989, 
0x1.f315913bd9718p-989, 0x1.f298cbd78a7b2p-989, 0x1.f21c25a494988p-989, 
0x1.f19f9e9b2b736p-989, 0x1.f12336b384a88p-989, 0x1.f0a6ede5d7c76p-989, 
0x1.f02ac42a5e516p-989, 0x1.efaeb97953b9dp-989, 0x1.ef32cdcaf564ep-989, 
0x1.eeb7011782a79p-989, 0x1.ee3b53573cc6ep-989, 0x1.edbfc48266f7bp-989, 
0x1.ed445491465dfp-989, 0x1.ecc9037c220c6p-989, 0x1.ec4dd13b4303dp-989, 
0x1.ebd2bdc6f4331p-989, 0x1.eb57c91782761p-989, 0x1.eadcf3253c957p-989, 
0x1.ea623be873464p-989, 0x1.e9e7a35979297p-989, 0x1.e96d2970a2cb3p-989, 
0x1.e8f2ce2646a27p-989, 0x1.e8789172bd10dp-989, 0x1.e7fe734e60619p-989, 
0x1.e78473b18cc97p-989, 0x1.e70a9294a0664p-989, 0x1.e690cfeffb3e2p-989, 
0x1.e6172bbbff3f5p-989, 0x1.e59da5f1103f8p-989, 0x1.e5243e8793fb7p-989, 
0x1.e4aaf577f2168p-989, 0x1.e431caba9419fp-989, 0x1.e3b8be47e574fp-989, 
0x1.e33fd018537b9p-989, 0x1.e2c700244d66bp-989, 0x1.e24e4e6444535p-989, 
0x1.e1d5bad0ab424p-989, 0x1.e15d4561f7177p-989, 0x1.e0e4ee109e99bp-989, 
0x1.e06cb4d51a72p-989, 0x1.dff499a7e52b7p-989, 0x1.df7c9c817b322p-989, 
0x1.df04bd5a5ad35p-989, 0x1.de8cfc2b043cap-989, 0x1.de1558ebf97b9p-989, 
0x1.dd9dd395be7d3p-989, 0x1.dd266c20d90d9p-989, 0x1.dcaf2285d0d75p-989, 
0x1.dc37f6bd2f631p-989, 0x1.dbc0e8bf80174p-989, 0x1.db49f88550373p-989, 
0x1.dad326072ee33p-989)
> 
> x <- p3 <- (1- 2^-14)
> stopifnot(  (1-x)-1 == -x , 1-x == 2^-14 )
> log(1e-300)/ log(p3) # 11317321
[1] 11317321
> 
> k3 <- 11321012 + (1:1000)
> dput(p3^ k3, control = "digits17") ## last 2-3 digits printed *differ*
c(7.9823091750159774e-301, 7.9818219735282437e-301, 7.981334801776929e-301, 
7.9808476597602195e-301, 7.9803605474762984e-301, 7.9798734649233519e-301, 
7.9793864120995648e-301, 7.9788993890031234e-301, 7.9784123956322125e-301, 
7.9779254319850184e-301, 7.9774384980597273e-301, 7.9769515938545225e-301, 
7.9764647193675935e-301, 7.9759778745971235e-301, 7.975491059541302e-301, 
7.9750042741983121e-301, 7.9745175185663418e-301, 7.9740307926435775e-301, 
7.9735440964282056e-301, 7.973057429918414e-301, 7.9725707931123891e-301, 
7.9720841860083173e-301, 7.9715976086043865e-301, 7.9711110608987832e-301, 
7.9706245428896954e-301, 7.9701380545753094e-301, 7.9696515959538148e-301, 
7.9691651670233981e-301, 7.9686787677822456e-301, 7.9681923982285486e-301, 
7.9677060583604918e-301, 7.9672197481762666e-301, 7.9667334676740576e-301, 
7.9662472168520562e-301, 7.9657609957084505e-301, 7.9652748042414266e-301, 
7.9647886424491761e-301, 7.9643025103298852e-301, 7.9638164078817452e-301, 
7.9633303351029443e-301, 7.9628442919916703e-301, 7.9623582785461147e-301, 
7.9618722947644655e-301, 7.9613863406449122e-301, 7.960900416185643e-301, 
7.9604145213848508e-301, 7.9599286562407236e-301, 7.9594428207514512e-301, 
7.9589570149152231e-301, 7.9584712387302307e-301, 7.9579854921946637e-301, 
7.9574997753067135e-301, 7.9570140880645679e-301, 7.9565284304664184e-301, 
7.956042802510458e-301, 7.9555572041948746e-301, 7.9550716355178612e-301, 
7.9545860964776076e-301, 7.954100587072305e-301, 7.9536151073001448e-301, 
7.9531296571593184e-301, 7.9526442366480169e-301, 7.9521588457644318e-301, 
7.9516734845067561e-301, 7.9511881528731811e-301, 7.9507028508618964e-301, 
7.9502175784710968e-301, 7.9497323356989734e-301, 7.9492471225437194e-301, 
7.9487619390035244e-301, 7.9482767850765831e-301, 7.9477916607610883e-301, 
7.9473065660552316e-301, 7.9468215009572058e-301, 7.9463364654652039e-301, 
7.9458514595774189e-301, 7.9453664832920439e-301, 7.9448815366072734e-301, 
7.9443966195212988e-301, 7.9439117320323131e-301, 7.9434268741385125e-301, 
7.9429420458380901e-301, 7.9424572471292373e-301, 7.9419724780101502e-301, 
7.9414877384790202e-301, 7.9410030285340454e-301, 7.940518348173417e-301, 
7.9400336973953296e-301, 7.9395490761979796e-301, 7.93906448457956e-301, 
7.9385799225382637e-301, 7.9380953900722887e-301, 7.9376108871798281e-301, 
7.937126413859078e-301, 7.9366419701082315e-301, 7.9361575559254866e-301, 
7.9356731713090362e-301, 7.935188816257075e-301, 7.9347044907678026e-301, 
7.9342201948394103e-301, 7.9337359284700962e-301, 7.9332516916580564e-301, 
7.9327674844014857e-301, 7.9322833066985803e-301, 7.9317991585475366e-301, 
7.9313150399465508e-301, 7.9308309508938193e-301, 7.93034689138754e-301, 
7.9298628614259076e-301, 7.9293788610071183e-301, 7.9288948901293719e-301, 
7.9284109487908628e-301, 7.9279270369897892e-301, 7.9274431547243472e-301, 
7.9269593019927366e-301, 7.926475478793152e-301, 7.9259916851237913e-301, 
7.9255079209828541e-301, 7.9250241863685367e-301, 7.9245404812790356e-301, 
7.9240568057125518e-301, 7.9235731596672818e-301, 7.9230895431414235e-301, 
7.9226059561331748e-301, 7.9221223986407354e-301, 7.9216388706623016e-301, 
7.9211553721960746e-301, 7.9206719032402525e-301, 7.9201884637930331e-301, 
7.9197050538526162e-301, 7.9192216734172012e-301, 7.9187383224849846e-301, 
7.9182550010541692e-301, 7.917771709122953e-301, 7.9172884466895341e-301, 
7.9168052137521152e-301, 7.9163220103088927e-301, 7.915838836358068e-301, 
7.9153556918978405e-301, 7.9148725769264117e-301, 7.9143894914419794e-301, 
7.9139064354427466e-301, 7.9134234089269112e-301, 7.9129404118926745e-301, 
7.9124574443382379e-301, 7.9119745062618009e-301, 7.9114915976615664e-301, 
7.9110087185357324e-301, 7.9105258688825019e-301, 7.9100430487000745e-301, 
7.9095602579866531e-301, 7.9090774967404373e-301, 7.90859476495963e-301, 
7.9081120626424343e-301, 7.907629389787048e-301, 7.9071467463916741e-301, 
7.9066641324545171e-301, 7.9061815479737768e-301, 7.9056989929476544e-301, 
7.9052164673743544e-301, 7.9047339712520782e-301, 7.9042515045790287e-301, 
7.9037690673534071e-301, 7.9032866595734181e-301, 7.9028042812372612e-301, 
7.9023219323431428e-301, 7.9018396128892657e-301, 7.9013573228738296e-301, 
7.9008750622950408e-301, 7.900392831151102e-301, 7.8999106294402164e-301, 
7.8994284571605885e-301, 7.8989463143104196e-301, 7.8984642008879142e-301, 
7.8979821168912786e-301, 7.8975000623187141e-301, 7.8970180371684269e-301, 
7.8965360414386184e-301, 7.8960540751274964e-301, 7.8955721382332621e-301, 
7.895090230754122e-301, 7.8946083526882804e-301, 7.8941265040339405e-301, 
7.89364468478931e-301, 7.893162894952592e-301, 7.8926811345219926e-301, 
7.8921994034957148e-301, 7.8917177018719666e-301, 7.8912360296489524e-301, 
7.890754386824877e-301, 7.8902727733979464e-301, 7.8897911893663671e-301, 
7.8893096347283436e-301, 7.8888281094820838e-301, 7.8883466136257923e-301, 
7.8878651471576753e-301, 7.8873837100759391e-301, 7.8869023023787917e-301, 
7.8864209240644375e-301, 7.8859395751310828e-301, 7.8854582555769373e-301, 
7.8849769654002055e-301, 7.8844957045990953e-301, 7.8840144731718113e-301, 
7.8835332711165647e-301, 7.8830520984315602e-301, 7.8825709551150055e-301, 
7.8820898411651087e-301, 7.881608756580076e-301, 7.881127701358117e-301, 
7.8806466754974379e-301, 7.8801656789962482e-301, 7.8796847118527542e-301, 
7.8792037740651656e-301, 7.8787228656316901e-301, 7.8782419865505358e-301, 
7.8777611368199121e-301, 7.8772803164380253e-301, 7.8767995254030867e-301, 
7.8763187637133041e-301, 7.8758380313668855e-301, 7.8753573283620421e-301, 
7.8748766546969801e-301, 7.8743960103699124e-301, 7.8739153953790453e-301, 
7.8734348097225883e-301, 7.8729542533987527e-301, 7.872473726405748e-301, 
7.8719932287417821e-301, 7.8715127604050663e-301, 7.8710323213938118e-301, 
7.8705519117062265e-301, 7.8700715313405215e-301, 7.8695911802949082e-301, 
7.8691108585675945e-301, 7.8686305661567932e-301, 7.8681503030607155e-301, 
7.8676700692775693e-301, 7.8671898648055677e-301, 7.86670968964292e-301, 
7.8662295437878392e-301, 7.8657494272385365e-301, 7.8652693399932215e-301, 
7.8647892820501071e-301, 7.8643092534074027e-301, 7.8638292540633231e-301, 
7.8633492840160776e-301, 7.8628693432638793e-301, 7.862389431804941e-301, 
7.8619095496374722e-301, 7.8614296967596859e-301, 7.8609498731697966e-301, 
7.8604700788660138e-301, 7.859990313846552e-301, 7.8595105781096226e-301, 
7.8590308716534401e-301, 7.8585511944762139e-301, 7.8580715465761604e-301, 
7.8575919279514924e-301, 7.8571123386004211e-301, 7.856632778521161e-301, 
7.8561532477119251e-301, 7.8556737461709262e-301, 7.8551942738963805e-301, 
7.8547148308864993e-301, 7.854235417139497e-301, 7.8537560326535866e-301, 
7.8532766774269843e-301, 7.8527973514579029e-301, 7.852318054744557e-301, 
7.8518387872851611e-301, 7.8513595490779297e-301, 7.8508803401210775e-301, 
7.8504011604128173e-301, 7.8499220099513668e-301, 7.8494428887349391e-301, 
7.8489637967617487e-301, 7.8484847340300134e-301, 7.8480057005379461e-301, 
7.8475266962837629e-301, 7.8470477212656785e-301, 7.8465687754819107e-301, 
7.8460898589306723e-301, 7.8456109716101813e-301, 7.8451321135186521e-301, 
7.844653284654301e-301, 7.8441744850153458e-301, 7.8436957146000011e-301, 
7.843216973406483e-301, 7.8427382614330078e-301, 7.8422595786777933e-301, 
7.8417809251390556e-301, 7.8413023008150111e-301, 7.8408237057038775e-301, 
7.8403451398038711e-301, 7.8398666031132097e-301, 7.8393880956301095e-301, 
7.8389096173527883e-301, 7.838431168279464e-301, 7.8379527484083529e-301, 
7.8374743577376744e-301, 7.8369959962656447e-301, 7.8365176639904817e-301, 
7.8360393609104032e-301, 7.8355610870236288e-301, 7.8350828423283763e-301, 
7.8346046268228635e-301, 7.8341264405053083e-301, 7.8336482833739302e-301, 
7.8331701554269472e-301, 7.8326920566625769e-301, 7.8322139870790407e-301, 
7.8317359466745563e-301, 7.8312579354473417e-301, 7.8307799533956179e-301, 
7.8303020005176028e-301, 7.829824076811516e-301, 7.8293461822755787e-301, 
7.8288683169080069e-301, 7.8283904807070253e-301, 7.8279126736708483e-301, 
7.8274348957977004e-301, 7.8269571470857979e-301, 7.8264794275333635e-301, 
7.8260017371386168e-301, 7.8255240758997773e-301, 7.8250464438150662e-301, 
7.8245688408827045e-301, 7.8240912671009136e-301, 7.8236137224679112e-301, 
7.8231362069819202e-301, 7.8226587206411634e-301, 7.8221812634438587e-301, 
7.8217038353882272e-301, 7.8212264364724934e-301, 7.8207490666948768e-301, 
7.8202717260535987e-301, 7.8197944145468818e-301, 7.8193171321729473e-301, 
7.8188398789300163e-301, 7.8183626548163118e-301, 7.8178854598300565e-301, 
7.8174082939694716e-301, 7.8169311572327783e-301, 7.816454049618201e-301, 
7.8159769711239626e-301, 7.8154999217482826e-301, 7.8150229014893872e-301, 
7.8145459103454975e-301, 7.8140689483148363e-301, 7.8135920153956282e-301, 
7.8131151115860942e-301, 7.8126382368844606e-301, 7.8121613912889469e-301, 
7.8116845747977792e-301, 7.8112077874091819e-301, 7.8107310291213747e-301, 
7.8102542999325869e-301, 7.809777599841038e-301, 7.8093009288449544e-301, 
7.8088242869425586e-301, 7.8083476741320753e-301, 7.8078710904117305e-301, 
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7.5266427362892563e-301, 7.526183346473809e-301, 7.5257239846972918e-301, 
7.5252646509579905e-301, 7.5248053452541975e-301, 7.5243460675841986e-301, 
7.5238868179462847e-301, 7.5234275963387432e-301, 7.522968402759865e-301, 
7.5225092372079391e-301, 7.5220500996812547e-301, 7.5215909901780994e-301, 
7.5211319086967655e-301, 7.5206728552355422e-301, 7.5202138297927169e-301, 
7.5197548323665822e-301, 7.5192958629554272e-301, 7.518836921557541e-301, 
7.5183780081712161e-301, 7.5179191227947416e-301, 7.5174602654264066e-301, 
7.5170014360645036e-301, 7.5165426347073218e-301, 7.516083861353152e-301, 
7.5156251160002865e-301, 7.5151663986470146e-301, 7.5147077092916288e-301, 
7.5142490479324197e-301, 7.5137904145676765e-301, 7.5133318091956934e-301, 
7.5128732318147612e-301, 7.5124146824231706e-301, 7.5119561610192142e-301, 
7.5114976676011843e-301, 7.5110392021673701e-301, 7.5105807647160657e-301, 
7.5101223552455637e-301)
> dput(p3^ k3, control = "hex")
c(0x1.11b31a015e2efp-997, 0x1.11aed334f6297p-997, 0x1.11aa8c79a9559p-997, 
0x1.11a645cf776efp-997, 0x1.11a1ff3660311p-997, 0x1.119db8ae63579p-997, 
0x1.11997237809ep-997, 0x1.11952bd1b7cp-997, 0x1.1190e57d08792p-997, 
0x1.118c9f397285p-997, 0x1.11885906f59f4p-997, 0x1.118412e591836p-997, 
0x1.117fccd545ed2p-997, 0x1.117b86d61298p-997, 0x1.117740e7f73fcp-997, 
0x1.1172fb0af39fep-997, 0x1.116eb53f07741p-997, 0x1.116a6f843277fp-997, 
0x1.116629da74672p-997, 0x1.1161e441ccfd5p-997, 0x1.115d9eba3bf62p-997, 
0x1.11595943c10d3p-997, 0x1.115513de5bfe3p-997, 0x1.1150ce8a0c84cp-997, 
0x1.114c8946d25c9p-997, 0x1.11484414ad414p-997, 0x1.1143fef39cee9p-997, 
0x1.113fb9e3a1202p-997, 0x1.113b74e4b9919p-997, 0x1.11372ff6e5febp-997, 
0x1.1132eb1a26231p-997, 0x1.112ea64e79ba8p-997, 0x1.112a6193e0809p-997, 
0x1.11261cea5a311p-997, 0x1.1121d851e687bp-997, 0x1.111d93ca85401p-997, 
0x1.11194f543616p-997, 0x1.11150aeef8c52p-997, 0x1.1110c69acd094p-997, 
0x1.110c8257b29e1p-997, 0x1.11083e25a93f4p-997, 0x1.1103fa04b0a8ap-997, 
0x1.10ffb5f4c895ep-997, 0x1.10fb71f5f0c2cp-997, 0x1.10f72e0828eafp-997, 
0x1.10f2ea2b70ca5p-997, 0x1.10eea65fc81c9p-997, 0x1.10ea62a52e9d7p-997, 
0x1.10e61efba408bp-997, 0x1.10e1db63281a2p-997, 0x1.10dd97dbba8d8p-997, 
0x1.10d954655b1eap-997, 0x1.10d5110009893p-997, 0x1.10d0cdabc589p-997, 
0x1.10cc8a688ed9fp-997, 0x1.10c847366537bp-997, 0x1.10c40415485e2p-997, 
0x1.10bfc1053809p-997, 0x1.10bb7e0633f42p-997, 0x1.10b73b183bdb5p-997, 
0x1.10b2f83b4f7a6p-997, 0x1.10aeb56f6e8d2p-997, 0x1.10aa72b498cf6p-997, 
0x1.10a6300acdfdp-997, 0x1.10a1ed720dd1dp-997, 0x1.109daaea58099p-997, 
0x1.10996873ac603p-997, 0x1.1095260e0a918p-997, 0x1.1090e3b972596p-997, 
0x1.108ca175e3739p-997, 0x1.10885f435d9cp-997, 0x1.10841d21e08e9p-997, 
0x1.107fdb116c071p-997, 0x1.107b9911ffc16p-997, 0x1.107757239b796p-997, 
0x1.107315463eeafp-997, 0x1.106ed379e9d1fp-997, 0x1.106a91be9bea5p-997, 
0x1.1066501454efep-997, 0x1.10620e7b149e8p-997, 0x1.105dccf2dab23p-997, 
0x1.10598b7ba6e6dp-997, 0x1.10554a1578f83p-997, 0x1.105108c050a25p-997, 
0x1.104cc77c2da1p-997, 0x1.104886490fb05p-997, 0x1.10444526f68c1p-997, 
0x1.10400415e1f03p-997, 0x1.103bc315d198bp-997, 0x1.10378226c5417p-997, 
0x1.10334148bca65p-997, 0x1.102f007bb7836p-997, 0x1.102abfbfb5948p-997, 
0x1.10267f14b695bp-997, 0x1.10223e7aba42dp-997, 0x1.101dfdf1c057fp-997, 
0x1.1019bd79c890fp-997, 0x1.10157d12d2a9cp-997, 0x1.10113cbcde5e8p-997, 
0x1.100cfc77eb6bp-997, 0x1.1008bc43f98b5p-997, 0x1.10047c21087b7p-997, 
0x1.10003c0f17f75p-997, 0x1.0ffbfc0e27bafp-997, 0x1.0ff7bc1e37825p-997, 
0x1.0ff37c3f47097p-997, 0x1.0fef3c71560c5p-997, 0x1.0feafcb46447p-997, 
0x1.0fe6bd0871757p-997, 0x1.0fe27d6d7d53ap-997, 0x1.0fde3de3879dbp-997, 
0x1.0fd9fe6a900f9p-997, 0x1.0fd5bf0296655p-997, 0x1.0fd17fab9a5afp-997, 
0x1.0fcd40659bac9p-997, 0x1.0fc901309a162p-997, 0x1.0fc4c20c9553bp-997, 
0x1.0fc082f98d216p-997, 0x1.0fbc43f7813b3p-997, 0x1.0fb80506715d2p-997, 
0x1.0fb3c6265d436p-997, 0x1.0faf875744a9fp-997, 0x1.0fab4899274cep-997, 
0x1.0fa709ec04e84p-997, 0x1.0fa2cb4fdd383p-997, 0x1.0f9e8cc4aff8bp-997, 
0x1.0f9a4e4a7ce5fp-997, 0x1.0f960fe143bcp-997, 0x1.0f91d1890436fp-997, 
0x1.0f8d9341be12ep-997, 0x1.0f89550b710bfp-997, 0x1.0f8516e61cde2p-997, 
0x1.0f80d8d1c145bp-997, 0x1.0f7c9ace5dfebp-997, 0x1.0f785cdbf2c53p-997, 
0x1.0f741efa7f557p-997, 0x1.0f6fe12a036b7p-997, 0x1.0f6ba36a7ec36p-997, 
0x1.0f6765bbf1196p-997, 0x1.0f63281e5a29ap-997, 0x1.0f5eea91b9b03p-997, 
0x1.0f5aad160f695p-997, 0x1.0f566fab5b111p-997, 0x1.0f5232519c63ap-997, 
0x1.0f4df508d31d3p-997, 0x1.0f49b7d0fef9ep-997, 0x1.0f457aaa1fb5fp-997, 
0x1.0f413d94350d7p-997, 0x1.0f3d008f3ebcap-997, 0x1.0f38c39b3c7fap-997, 
0x1.0f3486b82e12bp-997, 0x1.0f3049e61331fp-997, 0x1.0f2c0d24eb99ap-997, 
0x1.0f27d074b706p-997, 0x1.0f2393d575332p-997, 0x1.0f1f574725dd4p-997, 
0x1.0f1b1ac9c8c0bp-997, 0x1.0f16de5d5d999p-997, 0x1.0f12a201e4241p-997, 
0x1.0f0e65b75c1c8p-997, 0x1.0f0a297dc53f1p-997, 0x1.0f05ed551f48p-997, 
0x1.0f01b13d69f38p-997, 0x1.0efd7536a4fdep-997, 0x1.0ef93940d0234p-997, 
0x1.0ef4fd5beb2p-997, 0x1.0ef0c187f5b06p-997, 0x1.0eec85c4ef908p-997, 
0x1.0ee84a12d87ccp-997, 0x1.0ee40e71b0316p-997, 0x1.0edfd2e1766aap-997, 
0x1.0edb97622ae4dp-997, 0x1.0ed75bf3cd5c2p-997, 0x1.0ed320965d8cep-997, 
0x1.0ecee549db337p-997, 0x1.0ecaaa0e460cp-997, 0x1.0ec66ee39dd2fp-997, 
0x1.0ec233c9e2447p-997, 0x1.0ebdf8c1131cfp-997, 0x1.0eb9bdc93018ap-997, 
0x1.0eb582e238f3ep-997, 0x1.0eb1480c2d6bp-997, 0x1.0ead0d470d3a4p-997, 
0x1.0ea8d292d81e1p-997, 0x1.0ea497ef8dd2bp-997, 0x1.0ea05d5d2e148p-997, 
0x1.0e9c22dbb89fcp-997, 0x1.0e97e86b2d30ep-997, 0x1.0e93ae0b8b843p-997, 
0x1.0e8f73bcd356p-997, 0x1.0e8b397f0462bp-997, 0x1.0e86ff521e66ap-997, 
0x1.0e82c536211e2p-997, 0x1.0e7e8b2b0c45ap-997, 0x1.0e7a5130df997p-997, 
0x1.0e7617479ad5fp-997, 0x1.0e71dd6f3db78p-997, 0x1.0e6da3a7c7fa9p-997, 
0x1.0e6969f1395b7p-997, 0x1.0e65304b91968p-997, 0x1.0e60f6b6d0684p-997, 
0x1.0e5cbd32f58dp-997, 0x1.0e5883c000c13p-997, 0x1.0e544a5df1c12p-997, 
0x1.0e50110cc8496p-997, 0x1.0e4bd7cc84164p-997, 0x1.0e479e9d24e43p-997, 
0x1.0e43657eaa6fap-997, 0x1.0e3f2c711474fp-997, 0x1.0e3af37462b0ap-997, 
0x1.0e36ba8894df1p-997, 0x1.0e3281adaabccp-997, 0x1.0e2e48e3a4061p-997, 
0x1.0e2a102a80778p-997, 0x1.0e25d7823fcd8p-997, 0x1.0e219eeae1c48p-997, 
0x1.0e1d66646619p-997, 0x1.0e192deecc876p-997, 0x1.0e14f58a14cc3p-997, 
0x1.0e10bd363ea3ep-997, 0x1.0e0c84f349caep-997, 0x1.0e084cc135fdcp-997, 
0x1.0e0414a002f8ep-997, 0x1.0dffdc8fb078ep-997, 0x1.0dfba4903e3a2p-997, 
0x1.0df76ca1abf92p-997, 0x1.0df334c3f9727p-997, 0x1.0deefcf726629p-997, 
0x1.0deac53b3285fp-997, 0x1.0de68d901d992p-997, 0x1.0de255f5e758bp-997, 
0x1.0dde1e6c8f811p-997, 0x1.0dd9e6f415cedp-997, 0x1.0dd5af8c79fe8p-997, 
0x1.0dd17835bbcc9p-997, 0x1.0dcd40efdaf5ap-997, 0x1.0dc909bad7364p-997, 
0x1.0dc4d296b04aep-997, 0x1.0dc09b8365f02p-997, 0x1.0dbc6480f7e28p-997, 
0x1.0db82d8f65deap-997, 0x1.0db3f6aeafa11p-997, 0x1.0dafbfded4e65p-997, 
0x1.0dab891fd56bp-997, 0x1.0da75271b0ebap-997, 0x1.0da31bd46724ep-997, 
0x1.0d9ee547f7d34p-997, 0x1.0d9aaecc62b36p-997, 0x1.0d967861a781ep-997, 
0x1.0d924207c5fb4p-997, 0x1.0d8e0bbebddc2p-997, 0x1.0d89d5868ee13p-997, 
0x1.0d859f5f38c6fp-997, 0x1.0d816948bb4a1p-997, 0x1.0d7d334316272p-997, 
0x1.0d78fd4e491adp-997, 0x1.0d74c76a53e1ap-997, 0x1.0d70919736385p-997, 
0x1.0d6c5bd4efdb8p-997, 0x1.0d6826238087cp-997, 0x1.0d63f082e7f9cp-997, 
0x1.0d5fbaf325ee2p-997, 0x1.0d5b85743a218p-997, 0x1.0d5750062450ap-997, 
0x1.0d531aa8e4381p-997, 0x1.0d4ee55c79948p-997, 0x1.0d4ab020e4229p-997, 
0x1.0d467af6239fp-997, 0x1.0d4245dc37c67p-997, 0x1.0d3e10d320559p-997, 
0x1.0d39dbdadd091p-997, 0x1.0d35a6f36d9dap-997, 0x1.0d31721cd1cffp-997, 
0x1.0d2d3d57095cap-997, 0x1.0d2908a214008p-997, 0x1.0d24d3fdf1783p-997, 
0x1.0d209f6aa1806p-997, 0x1.0d1c6ae823d5ep-997, 0x1.0d18367678355p-997, 
0x1.0d1402159e5b7p-997, 0x1.0d0fcdc59604fp-997, 0x1.0d0b99865eeeap-997, 
0x1.0d076557f8d52p-997, 0x1.0d03313a63754p-997, 0x1.0cfefd2d9e8bbp-997, 
0x1.0cfac931a9d53p-997, 0x1.0cf69546850e9p-997, 0x1.0cf2616c2ff48p-997, 
0x1.0cee2da2aa43cp-997, 0x1.0ce9f9e9f3b91p-997, 0x1.0ce5c6420c114p-997, 
0x1.0ce192aaf3091p-997, 0x1.0cdd5f24a85d4p-997, 0x1.0cd92baf2bcaap-997, 
0x1.0cd4f84a7d0dfp-997, 0x1.0cd0c4f69be4p-997, 0x1.0ccc91b388099p-997, 
0x1.0cc85e81413b7p-997, 0x1.0cc42b5fc7367p-997, 0x1.0cbff84f19b75p-997, 
0x1.0cbbc54f387afp-997, 0x1.0cb79260233e1p-997, 0x1.0cb35f81d9bd8p-997, 
0x1.0caf2cb45bb61p-997, 0x1.0caaf9f7a8e4ap-997, 0x1.0ca6c74bc106p-997, 
0x1.0ca294b0a3d7p-997, 0x1.0c9e622651147p-997, 0x1.0c9a2facc87b3p-997, 
0x1.0c95fd4409c81p-997, 0x1.0c91caec14b7ep-997, 0x1.0c8d98a4e9079p-997, 
0x1.0c89666e8673fp-997, 0x1.0c853448ecb9dp-997, 0x1.0c8102341b962p-997, 
0x1.0c7cd03012c5bp-997, 0x1.0c789e3cd2056p-997, 0x1.0c746c5a59122p-997, 
0x1.0c703a88a7a8bp-997, 0x1.0c6c08c7bd862p-997, 0x1.0c67d7179a672p-997, 
0x1.0c63a5783e08cp-997, 0x1.0c5f73e9a827cp-997, 0x1.0c5b426bd8812p-997, 
0x1.0c5710feced1cp-997, 0x1.0c52dfa28ad68p-997, 0x1.0c4eae570c4c5p-997, 
0x1.0c4a7d1c52f02p-997, 0x1.0c464bf25e7eep-997, 0x1.0c421ad92eb56p-997, 
0x1.0c3de9d0c350ap-997, 0x1.0c39b8d91c0dap-997, 0x1.0c3587f238a93p-997, 
0x1.0c31571c18e04p-997, 0x1.0c2d2656bc6fep-997, 0x1.0c28f5a22314fp-997, 
0x1.0c24c4fe4c8c6p-997, 0x1.0c20946b38933p-997, 0x1.0c1c63e8e6e65p-997, 
0x1.0c1833775742bp-997, 0x1.0c14031689655p-997, 0x1.0c0fd2c67d0b3p-997, 
0x1.0c0ba28731f14p-997, 0x1.0c077258a7d47p-997, 0x1.0c03423ade71dp-997, 
0x1.0bff122dd5866p-997, 0x1.0bfae2318ccfp-997, 0x1.0bf6b2460408dp-997, 
0x1.0bf2826b3af0cp-997, 0x1.0bee52a13143dp-997, 0x1.0bea22e7e6bf1p-997, 
0x1.0be5f33f5b1f7p-997, 0x1.0be1c3a78e221p-997, 0x1.0bdd94207f83dp-997, 
0x1.0bd964aa2f01dp-997, 0x1.0bd535449c592p-997, 0x1.0bd105efc746ap-997, 
0x1.0bccd6abaf879p-997, 0x1.0bc8a77854d8dp-997, 0x1.0bc47855b6f78p-997, 
0x1.0bc04943d5a0ap-997, 0x1.0bbc1a42b0914p-997, 0x1.0bb7eb5247868p-997, 
0x1.0bb3bc729a3d6p-997, 0x1.0baf8da3a873p-997, 0x1.0bab5ee571e46p-997, 
0x1.0ba73037f64e9p-997, 0x1.0ba3019b356ecp-997, 0x1.0b9ed30f2f01ep-997, 
0x1.0b9aa493e2c53p-997, 0x1.0b9676295075ap-997, 0x1.0b9247cf77d06p-997, 
0x1.0b8e198658928p-997, 0x1.0b89eb4df2792p-997, 0x1.0b85bd2645415p-997, 
0x1.0b818f0f50a84p-997, 0x1.0b7d6109146bp-997, 0x1.0b7933139046ap-997, 
0x1.0b75052ec3f86p-997, 0x1.0b70d75aaf3d5p-997, 0x1.0b6ca99751d2ap-997, 
0x1.0b687be4ab755p-997, 0x1.0b644e42bbe2ap-997, 0x1.0b6020b182d7bp-997, 
0x1.0b5bf3310011bp-997, 0x1.0b57c5c1334dbp-997, 0x1.0b5398621c48ep-997, 
0x1.0b4f6b13bac07p-997, 0x1.0b4b3dd60e718p-997, 0x1.0b4710a917194p-997, 
0x1.0b42e38cd474fp-997, 0x1.0b3eb68146419p-997, 0x1.0b3a89866c3c8p-997, 
0x1.0b365c9c4622dp-997, 0x1.0b322fc2d3b1bp-997, 0x1.0b2e02fa14a66p-997, 
0x1.0b29d64208be1p-997, 0x1.0b25a99aafb5fp-997, 0x1.0b217d04094b3p-997, 
0x1.0b1d507e153b1p-997, 0x1.0b192408d342bp-997, 0x1.0b14f7a4431f7p-997, 
0x1.0b10cb50648e6p-997, 0x1.0b0c9f0d374cdp-997, 0x1.0b0872dabb17fp-997, 
0x1.0b0446b8efadp-997, 0x1.0b001aa7d4c94p-997, 0x1.0afbeea76a29fp-997, 
0x1.0af7c2b7af8c4p-997, 0x1.0af396d8a4ad9p-997, 0x1.0aef6b0a494afp-997, 
0x1.0aeb3f4c9d21dp-997, 0x1.0ae7139f9fef6p-997, 0x1.0ae2e8035170ep-997, 
0x1.0adebc77b1639p-997, 0x1.0ada90fcbf84dp-997, 0x1.0ad665927b91dp-997, 
0x1.0ad23a38e547ep-997, 0x1.0ace0eeffc645p-997, 0x1.0ac9e3b7c0a46p-997, 
0x1.0ac5b89031c56p-997, 0x1.0ac18d794f849p-997, 0x1.0abd6273199f5p-997, 
0x1.0ab9377d8fd2fp-997, 0x1.0ab50c98b1dcbp-997, 0x1.0ab0e1c47f79fp-997, 
0x1.0aacb700f867fp-997, 0x1.0aa88c4e1c641p-997, 0x1.0aa461abeb2bap-997, 
0x1.0aa0371a647bfp-997, 0x1.0a9c0c9988126p-997, 0x1.0a97e22955ac4p-997, 
0x1.0a93b7c9cd06ep-997, 0x1.0a8f8d7aeddfbp-997, 0x1.0a8b633cb7f3fp-997, 
0x1.0a87390f2b011p-997, 0x1.0a830ef246c47p-997, 0x1.0a7ee4e60afb5p-997, 
0x1.0a7abaea77632p-997, 0x1.0a7690ff8bb94p-997, 0x1.0a72672547bb1p-997, 
0x1.0a6e3d5bab26p-997, 0x1.0a6a13a2b5b75p-997, 0x1.0a65e9fa672c7p-997, 
0x1.0a61c062bf42ep-997, 0x1.0a5d96dbbdb7ep-997, 0x1.0a596d656248ep-997, 
0x1.0a5543ffacb36p-997, 0x1.0a511aaa9cb4bp-997, 0x1.0a4cf166320a3p-997, 
0x1.0a48c8326c717p-997, 0x1.0a449f0f4ba7cp-997, 0x1.0a4075fccf6a9p-997, 
0x1.0a3c4cfaf7775p-997, 0x1.0a382409c38b7p-997, 0x1.0a33fb2933646p-997, 
0x1.0a2fd25946bf9p-997, 0x1.0a2ba999fd5a8p-997, 0x1.0a2780eb56f28p-997, 
0x1.0a23584d53453p-997, 0x1.0a1f2fbff20fep-997, 0x1.0a1b074333101p-997, 
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0x1.02135a783036p-997, 0x1.020f522ac6554p-997, 0x1.020b49ed7daa3p-997, 
0x1.020741c055f43p-997, 0x1.020339a34ef2ep-997, 0x1.01ff31966865ap-997, 
0x1.01fb2999a20cp-997, 0x1.01f721acfba57p-997, 0x1.01f319d074f18p-997, 
0x1.01ef12040dafbp-997, 0x1.01eb0a47c59f8p-997, 0x1.01e7029b9c806p-997, 
0x1.01e2faff9211fp-997, 0x1.01def373a613bp-997, 0x1.01daebf7d8451p-997, 
0x1.01d6e48c2865bp-997, 0x1.01d2dd3096351p-997, 0x1.01ced5e52172bp-997, 
0x1.01cacea9c9de3p-997, 0x1.01c6c77e8f371p-997, 0x1.01c2c063713cdp-997, 
0x1.01beb9586faf1p-997, 0x1.01bab25d8a4d5p-997, 0x1.01b6ab72c0d72p-997, 
0x1.01b2a498130c2p-997, 0x1.01ae9dcd80abdp-997, 0x1.01aa97130975dp-997, 
0x1.01a69068ad29bp-997, 0x1.01a289ce6b86fp-997, 0x1.019e8344444d4p-997, 
0x1.019a7cca373c3p-997, 0x1.0196766044135p-997, 0x1.019270066a924p-997, 
0x1.018e69bcaa78ap-997, 0x1.018a63830385fp-997, 0x1.01865d597579ep-997, 
0x1.0182574000141p-997)
> 
> require(sfsmisc) # for relErrV(), eaxis()
> ## relative error of  x ^ k  computation - comparing with truth = <x_mpfr_{precBits])
> relEP <- function(x, k, precBits, # see below about  precBits needed
+                   mp = mpfr(x, precBits)^k,
+                   FN = function(x,k) x^k) {
+     stopifnot(is.numeric(x), inherits(mp, "mpfr"))
+     asNumeric(relErrV(mp, FN(x,k)))
+ }
> 
> (osV <- abbreviate(gsub("[^[:alnum:]]", '', sub("\\(.*", '', osVersion)), 12))
WindowsServer2022x64 
      "WndwsS202264" 
> if(!dev.interactive(TRUE)) pdf(paste0("pow-tst_", osV, ".pdf"),
+                                width = 9, height=5)
> 
> stopifnot(.Machine$double.xmin == 2^-1022) # will use  2^-1022
> 
> re1. <- relEP(p1, k1, precBits = 2^12) # high precision
> re1  <- relEP(p1, k1, precBits = 256)
> stopifnot(re1. == re1) # ===> 256 bit suffice here
> summary(re1)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-2.105e-16 -1.404e-16 -1.012e-16 -1.022e-16 -6.464e-17  5.573e-18 
> summary(abs(re1))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
2.172e-18 6.464e-17 1.012e-16 1.023e-16 1.404e-16 2.105e-16 
> 
> re1pow <- relEP(p1, k1, precBits = 256, FN = pow)
> re1pdi <- relEP(p1, k1, precBits = 256, FN = pow_di)
> stopifnot(identical(re1pow, re1pdi))# because pow() smartly uses pow_di() here
> ## however
> all.equal(re1, re1pdi, tolerance = 0) # 2201.618 !!
[1] "Mean relative difference: 886.8181"
> summary(re1pdi) # negatively biased !
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-9.179e-14 -9.111e-14 -9.081e-14 -9.079e-14 -9.047e-14 -8.992e-14 
> ##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
> ## -9.179e-14 -9.106e-14 -9.070e-14 -7.740e-14 -9.025e-14  0.000e+00
> plot(k1, re1pdi, type="l"); abline(v = min(k1[p1^k1 < 2^-1022]), lty=3)
Warning message:
In min(k1[p1^k1 < 2^-1022]) :
  no non-missing arguments to min; returning Inf
> 
> re1pow <- relEP(p1, k1, precBits = 256, FN = .pow)
> summary(re1pow)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-2.105e-16 -1.404e-16 -1.012e-16 -1.022e-16 -6.464e-17  5.573e-18 
> ##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max.  (Linux Fedora 38):
> ## -1.070e-16 -3.082e-17  0.000e+00  7.063e-19  3.507e-17  1.075e-16
> 
> ## Explore more about pow_di()'s "flaw"
> k1. <- c(2:64, as.integer(unique(round(lseq(66, 44980, length = 1000)))))
> re1.pdi <- relEP(p1, k1., precBits = 256, FN = pow_di)
> plot(k1., re1.pdi, type="l", log="x", xaxt="n",
+      main = "rel.error {wrt MPFR} of (63/64)^k -- using R_pow_di()")
> abline(h=0, lty=3); eaxis(1, sub10=3)
> 
> plot(k1., re1.pdi, type="o", cex=1/4, log="x", ylim = c(-50, 2)*2^-53, xaxt="n",
+      main = "rel.error of (63/64)^k -- R_pow_di() -- zoomed in")
> abline(h=(-2:2)*2^-53, lty=c(3,3,1), col=adjustcolor(1, 1/2)); eaxis(1, sub10=3)
> 
> all.equal(p1^k1., pow(p1, k1., FALSE), tolerance=0) # TRUE (everywhere?)
[1] TRUE
> 
> ## comparison:  R's  x^y i.e. R_pow() is *much* better -- starting already at ca k >= 50 (for this x=p1):
> plot(k1., abs(re1.pdi), type="l", log="xy", yaxt="n", ylim = c(1e-17, max(abs(re1.pdi))),
+      xlab = quote(k), main = "|rel.error| {wrt MPFR} of"~ (frac(63,64))^k)
Warning message:
In xy.coords(x, y, xlabel, ylabel, log) :
  7 y values <= 0 omitted from logarithmic plot
> lines(k1., abs(relEP(p1, k1., 256)), col = adjustcolor(2, 1/2))
> legend("top", paste("using", c("R_pow_di(x,k)", "x ^ k")), lwd = 1,
+        col = c(palette()[[1]], adjustcolor(2, 1/2)), bty="n")
> eaxis(2); drawEps.h(); abline(h = 8e-18, col=adjustcolor(4, 1/2), lwd=2)
> rug(k1.[k1. <= 50], col="gray30")
> 
> plot(k1, re1, type="l", main = "rel.error {wrt MPFR} of  (63/64)^k")
> mtextVersion()
> abline(v = min(k1[p1^k1 < 2^-1022]), lty=3)
Warning message:
In min(k1[p1^k1 < 2^-1022]) :
  no non-missing arguments to min; returning Inf
> ll <- c(1,1,2,1,1)
> abline(h=(-2:2)*2^-53, lty=3-ll, lwd=ll, col=adjustcolor(ll, 1/2))
> ## looks perfect (in Linux; *not* windows
> 
> ## ditto (pow_di exploration) for an even more critical p
> p1 <- 0.999 # not exactly representable
> print(p1, digits=19)
[1] 0.9989999999999999991
> k <- c(2:50, as.integer(lseq(55, 700000, length=1000)))
> pk <- p1^k
> tail(cbind(k, pk))
             k            pk
[1044,] 667657 7.855476e-291
[1045,] 674004 1.371912e-293
[1046,] 680411 2.256365e-296
[1047,] 686879 3.491300e-299
[1048,] 693408 5.082294e-302
[1049,] 700000 6.946374e-305
> plot(pk ~ k, type = "l", log="xy")
> re.pdi <- relEP(p1, k, precBits = 256, FN = pow_di)
> 
> plot(k, abs(re.pdi), type="l", log="xy", xaxt="n", yaxt="n", ylim = c(1e-17, max(abs(re.pdi))),
+      xlab = quote(k), main = substitute(abs(rel.error) ~~ "of" ~~ P^k, list(P = format(p1))))
> lines(k, abs(relEP(p1, k, 256)), col = adjustcolor(2, 1/2))
> legend("top", paste("using", c("R_pow_di(x,k)", "x ^ k")), lwd = 1,
+        col = c(palette()[[1]], adjustcolor(2, 1/2)), bty="n")
> eaxis(1, sub=3); eaxis(2); drawEps.h(); abline(h = 8e-18, col=adjustcolor(4, 1/2), lwd=2)
> rug(k[k <= 50], col="gray30")
> axis(1, at=2:6)
> mtextVersion()
> 
> ## finaly: pow_di exploration an "easy" p:
> p1 <- 0.001 # not exactly representable
> k <- 2:105
> pk <- p1^k
> tail(cbind(k, pk)) # subnormals ..
         k     pk
 [99,] 100 1e-300
[100,] 101 1e-303
[101,] 102 1e-306
[102,] 103 1e-309
[103,] 104 1e-312
[104,] 105 1e-315
> plot(pk ~ k, type = "l", log="xy")
> re.pdi <- relEP(p1, k, precBits = 256, FN = pow_di)
> 
> plot(k, abs(re.pdi), type="l", log="xy", xaxt="n", yaxt="n", ylim = c(1e-17, max(abs(re.pdi))),
+      xlab = quote(k), main = substitute(abs(rel.error) ~~ "of" ~~ P^k, list(P = format(p1))))
> lines(k, abs(relEP(p1, k, 256)), col = adjustcolor(2, 1/2))
> legend("top", paste("using", c("R_pow_di(x,k)", "x ^ k")), lwd = 1,
+        col = c(palette()[[1]], adjustcolor(2, 1/2)), bty="n")
> eaxis(1, sub=3); eaxis(2); drawEps.h(); abline(h = 8e-18, col=adjustcolor(4, 1/2), lwd=2)
> rug(k, col="gray30")
> axis(1, at=2:6)
> mtextVersion()
> 
> 
> re2  <- relEP(p2, k2, 2^ 9)
> re2. <- relEP(p2, k2, 2^14)
> stopifnot(re2. == re2)
> summary(re2)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
1.268e-15 1.335e-15 1.380e-15 1.378e-15 1.419e-15 1.482e-15 
> summary(abs(re2))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
1.268e-15 1.335e-15 1.380e-15 1.378e-15 1.419e-15 1.482e-15 
> 
> str(rat2 <- c(MASS:::.rat(p2) $ rat))
 num [1:2] 1023 1024
> plot(k2, re2, type="l", xlab = quote(k), main = substitute("rel.Err of " ~ (N/Z)^k,
+                                                            list(N=rat2[1], Z=rat2[2])),
+      ylim = range(0, re2))
> if(any((px <- p2^k2) < 2^-1022)) abline(v = min(k2[px < 2^-1022]), lty=3)
> abline(h=(-2:2)*2^-53, lty=3-ll, lwd=ll, col=adjustcolor(ll, 1/2))
> 
> re3 <- relEP(p3, k3, prec = 256)
> summary(re3)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
1.475e-14 1.480e-14 1.485e-14 1.485e-14 1.491e-14 1.496e-14 
> str(rat3 <- c(MASS:::.rat(p3, max.denominator=1e5) $ rat))
 num [1:2] 16383 16384
> 
> plot(k3, re3, type="l", xlab = quote(k),  ylim = range(0, re3),
+      main = substitute("rel.Err of " ~ (N/Z)^k, list(N=rat3[1], Z=rat3[2])))
> if(any((px <- p3^k3) < 2^-1022)) abline(v = min(k3[px < 2^-1022]), lty=3)
> abline(h=(-2:2)*2^-53, lty=3-ll, lwd=ll, col=adjustcolor(ll, 1/2))
> 
> ##===> all these are perfect on Linux Fedora 36 & 38;
> ## but increasingly (p1 --> p2 --> p3)  worse on Windows
> 
> ## even more extreme
> e2 <- -(10:21)
>  ps <- 1 - 2 ^ e2
> (pM <- 1 - mpfr(2, 256) ^ e2) # 12 'mpfr' numbers of precision  256   bits
12 'mpfr' numbers of precision  256   bits 
 [1]            0.9990234375           0.99951171875          0.999755859375         0.9998779296875
 [5]        0.99993896484375       0.999969482421875      0.9999847412109375     0.99999237060546875
 [9]    0.999996185302734375   0.9999980926513671875  0.99999904632568359375 0.999999523162841796875
> ##  0.9990234375  0.99951171875 ... 0.999999523162841796875
> stopifnot( (1 - ps) -1 == -ps, ps == pM) ## <==> all ps are exactly representable 1 - 2^{-m}
> 
> print(pM, scientific = FALSE, drop0trailing = TRUE)
12 'mpfr' numbers of precision  256   bits 
 [1]            0.9990234375           0.99951171875          0.999755859375         0.9998779296875
 [5]        0.99993896484375       0.999969482421875      0.9999847412109375     0.99999237060546875
 [9]    0.999996185302734375   0.9999980926513671875  0.99999904632568359375 0.999999523162841796875
> 
> 
> ## The range of interesting k -- such that
> ## ps^k  does *not* underflow (in double prec) but *is* ~= 10^{-300}:
> (ks <- as.integer(round( log(1e-300)/ log(ps) )))
 [1]     707009    1414363    2829071    5658488   11317321   22634987   45270320   90540985
 [9]  181082315  362164975  724330295 1448660934
> ## [1]    707009    1414363    2829071    5658488   11317321  22634987  45270320
> ## [8]  90540985  181082315  362164975  724330295 1448660934
> (k0 <- as.integer(signif(ks, 4)))
 [1]     707000    1414000    2829000    5658000   11320000   22630000   45270000   90540000
 [9]  181100000  362200000  724300000 1449000000
> 
> cbind(psN <- setNames(seq_along(ps), paste0("1-2^",e2)))
        [,1]
1-2^-10    1
1-2^-11    2
1-2^-12    3
1-2^-13    4
1-2^-14    5
1-2^-15    6
1-2^-16    7
1-2^-17    8
1-2^-18    9
1-2^-19   10
1-2^-20   11
1-2^-21   12
> reL  <- lapply(psN, function(i) relEP(ps[i], k0[i] + 0:999, precBits = 2^8))
> ## can use names(reL) to reconstruct ps exactly:
> stopifnot(identical(ps, vapply(lapply(names(reL), str2lang), eval, 0.9)))
> if(doExtras) { ## much higher precision -- just to show it's *not* needed
+     print(system.time(
+         reLL <- lapply(psN, function(i) relEP(ps[i], k0[i] + 0:999, precBits = 2^ 14))
+     )) # 2.407 sec
+     stopifnot(all.equal(reL, reLL, tolerance = 0)) # exactly the same
+ }
> 
> absreL <- lapply(reL, abs)
> t(sapply(reL,    summary))
                 Min.       1st Qu.        Median          Mean       3rd Qu.          Max.
1-2^-10  1.280447e-15  1.349201e-15  1.394657e-15  1.391766e-15  1.431874e-15  1.500609e-15
1-2^-11  2.670517e-15  2.743236e-15  2.785959e-15  2.783925e-15  2.826796e-15  2.889592e-15
1-2^-12 -3.078755e-15 -3.026682e-15 -2.978716e-15 -2.978479e-15 -2.932214e-15 -2.874825e-15
1-2^-13 -1.226433e-14 -1.221933e-14 -1.217776e-14 -1.217671e-14 -1.213259e-14 -1.209147e-14
1-2^-14  1.475064e-14  1.480261e-14  1.485386e-14  1.485506e-14  1.490780e-14  1.495633e-14
1-2^-15  1.858175e-14  1.861723e-14  1.865487e-14  1.865398e-14  1.868825e-14  1.872753e-14
1-2^-16  9.337607e-15  9.379792e-15  9.421699e-15  9.420715e-15  9.462557e-15  9.504124e-15
1-2^-17 -1.849254e-13 -1.848816e-13 -1.848382e-13 -1.848413e-13 -1.848018e-13 -1.847590e-13
1-2^-18 -1.280356e-13 -1.279937e-13 -1.279501e-13 -1.279491e-13 -1.279062e-13 -1.278583e-13
1-2^-19 -3.789652e-13 -3.789186e-13 -3.788728e-13 -3.788747e-13 -3.788312e-13 -3.787879e-13
1-2^-20 -6.530429e-13 -6.530030e-13 -6.529615e-13 -6.529630e-13 -6.529222e-13 -6.528822e-13
1-2^-21  4.054088e-13  4.054558e-13  4.055043e-13  4.055048e-13  4.055534e-13  4.056033e-13
> t(sapply(absreL, summary)) # very nice on Linux:
                Min.      1st Qu.       Median         Mean      3rd Qu.         Max.
1-2^-10 1.280447e-15 1.349201e-15 1.394657e-15 1.391766e-15 1.431874e-15 1.500609e-15
1-2^-11 2.670517e-15 2.743236e-15 2.785959e-15 2.783925e-15 2.826796e-15 2.889592e-15
1-2^-12 2.874825e-15 2.932214e-15 2.978716e-15 2.978479e-15 3.026682e-15 3.078755e-15
1-2^-13 1.209147e-14 1.213259e-14 1.217776e-14 1.217671e-14 1.221933e-14 1.226433e-14
1-2^-14 1.475064e-14 1.480261e-14 1.485386e-14 1.485506e-14 1.490780e-14 1.495633e-14
1-2^-15 1.858175e-14 1.861723e-14 1.865487e-14 1.865398e-14 1.868825e-14 1.872753e-14
1-2^-16 9.337607e-15 9.379792e-15 9.421699e-15 9.420715e-15 9.462557e-15 9.504124e-15
1-2^-17 1.847590e-13 1.848018e-13 1.848382e-13 1.848413e-13 1.848816e-13 1.849254e-13
1-2^-18 1.278583e-13 1.279062e-13 1.279501e-13 1.279491e-13 1.279937e-13 1.280356e-13
1-2^-19 3.787879e-13 3.788312e-13 3.788728e-13 3.788747e-13 3.789186e-13 3.789652e-13
1-2^-20 6.528822e-13 6.529222e-13 6.529615e-13 6.529630e-13 6.530030e-13 6.530429e-13
1-2^-21 4.054088e-13 4.054558e-13 4.055043e-13 4.055048e-13 4.055534e-13 4.056033e-13
> ##                 Min.      1st Qu.       Median         Mean      3rd Qu.         Max.
> ## 1-2^-10 4.698362e-20 1.991498e-17 4.117806e-17 4.292841e-17 6.362789e-17 1.085069e-16
> ## 1-2^-11 2.644379e-20 2.143401e-17 4.148793e-17 4.352628e-17 6.504935e-17 1.048650e-16
> ## 1-2^-12 5.185148e-21 2.465148e-17 4.910034e-17 4.719418e-17 6.823981e-17 1.001215e-16
> ## 1-2^-13 1.732619e-19 2.110998e-17 4.188601e-17 4.175295e-17 6.264898e-17 8.759273e-17
> ## 1-2^-14 1.028253e-19 2.692311e-17 5.463435e-17 5.234885e-17 7.736664e-17 1.027465e-16
> ## 1-2^-15 2.173905e-20 1.919186e-17 3.737047e-17 3.683430e-17 5.442273e-17 7.258842e-17
> ## 1-2^-16 7.547357e-20 1.964997e-17 4.034049e-17 4.078038e-17 6.051778e-17 8.354223e-17
> ## 1-2^-17 6.468787e-20 2.113894e-17 4.282366e-17 4.184208e-17 6.196293e-17 8.264685e-17
> ## 1-2^-18 3.053764e-20 2.343907e-17 4.581431e-17 4.515700e-17 6.764085e-17 8.893000e-17
> ## 1-2^-19 3.693339e-20 1.922513e-17 4.318434e-17 4.283787e-17 6.588232e-17 8.900600e-17
> ## 1-2^-20 5.398257e-20 2.190955e-17 4.092451e-17 4.095826e-17 6.083059e-17 8.062014e-17
> ## 1-2^-21 4.708387e-20 2.429069e-17 4.717160e-17 4.758539e-17 7.079720e-17 9.742768e-17
> 
> d.absre <- as.data.frame(absreL, optional=TRUE)
> stopifnot(is.numeric(m.absrel <- as.matrix(d.absre)))
> if(doExtras) { # not particularly interesting ..
+     matplot(0:999, m.absrel, type="l", log="y", ylim = c(1e-17, max(m.absrel)))
+     drawEps.h(lty=2)
+                                         #
+     ## smoothed rel.errors
+     s.absre <- d.absre
+     s.absre[] <- lapply(absreL, function(y) lowess(y, f = 1/20)$y)
+     matlines(0:999, s.absre, lwd=3, col = adjustcolor(1:12, 1/2))
+ }
> 
> ## different "log-scale"  k sets for different ps / ks:
> k2p   <- lapply(setNames(ks, names(psN)), \(k)
+                 as.integer(2^seq(7, floor(log2(k)), by = 1/8)))
> stopifnot( sapply(k2p, is.integer) ) # -> so pow_di() / __powi() .. could be used
> reL2  <- lapply(psN, function(i) relEP(ps[i], k2p[[i]], precBits = 2^8))
> absre2 <- lapply(reL2, abs)
> 
> K <- length(ks) # 12 here
> 
> plot(k2p[[K]], reL2[[K]], type = "l", log = "x", col=K,
+      xlab = quote(k), ylab = "relErr{ p ^ k }",
+      ylim = range((-1:1)*2^-53, unlist(reL2)), xaxt="n"); eaxis(1)
> for(i in rev(seq_along(ks))[-1L])
+     lines(k2p[[i]], reL2[[i]], col=i)
> abline(h = (-2:2)*2^-53, lty=c(3,3,2,3,3), lwd=c(1,1,2,1,1), col=adjustcolor(c(1,1,2,1,1), 1/2))
> 
> plot(k2p[[K]], absre2[[K]], type = "l", log = "xy", col=K,
+      xlab = quote(k), ylab = "|relE|", main = expression(abs(rel.Err( p ^ k ))),
+      ylim = range(2^-53, pmax(8e-18, unlist(absre2))), xaxt="n", yaxt="n"); eaxis(1); eaxis(2)
> (pExpr <- as.call(c(quote(list), lapply(names(k2p), str2lang))))
list(1 - 2^-10, 1 - 2^-11, 1 - 2^-12, 1 - 2^-13, 1 - 2^-14, 1 - 
    2^-15, 1 - 2^-16, 1 - 2^-17, 1 - 2^-18, 1 - 2^-19, 1 - 2^-20, 
    1 - 2^-21)
> mtext(substitute(p == group("{", EE, "}"), list(EE = pExpr)), cex= 2/3)
> for(i in rev(seq_along(ks))[-1L])
+     lines(k2p[[i]], absre2[[i]], col=i)
> drawEps.h(lty=3, lwd=2)
> ## looks perfect (in Linux; *not* windows
> 
> t(sapply(absre2, summary))
                Min.      1st Qu.       Median         Mean      3rd Qu.         Max.
1-2^-10 9.390336e-19 2.346296e-17 5.335712e-17 1.441629e-16 1.342227e-16 9.784937e-16
1-2^-11 4.641186e-19 3.282452e-17 7.167211e-17 2.609528e-16 2.482842e-16 1.976921e-15
1-2^-12 2.868086e-19 2.748112e-17 5.638903e-17 2.564391e-16 2.393033e-16 2.212747e-15
1-2^-13 1.669060e-19 3.489569e-17 7.429264e-17 9.099490e-16 5.979160e-16 8.969010e-15
1-2^-14 4.684314e-19 2.672919e-17 7.947764e-17 1.044787e-15 6.875389e-16 1.098898e-14
1-2^-15 5.037089e-20 3.045571e-17 6.374634e-17 1.232263e-15 7.890786e-16 1.388033e-14
1-2^-16 3.167997e-19 2.161611e-17 4.928212e-17 5.970448e-16 2.882474e-16 7.062778e-15
1-2^-17 9.995104e-19 3.067666e-17 2.011080e-16 1.079877e-14 5.103947e-15 1.370574e-13
1-2^-18 6.595734e-19 3.191051e-17 1.078896e-16 7.108505e-15 2.940018e-15 9.487097e-14
1-2^-19 4.170203e-19 3.588180e-17 1.818854e-16 2.002418e-14 7.360301e-15 2.807288e-13
1-2^-20 7.930429e-22 3.537090e-17 2.548155e-16 3.295712e-14 1.064094e-14 4.840320e-13
1-2^-21 1.651936e-19 3.257837e-17 8.943541e-17 1.958272e-14 5.556943e-15 3.004945e-13
> summary(are2 <- unlist(absre2))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
0.000e+00 3.120e-17 7.450e-17 9.579e-15 1.154e-15 4.840e-13 
> 
> if(.Platform$OS.type == "unix") { # && !noLdbl
+     stopifnot(m.absrel < 2.23e-16, # see max(.) == 1.085e-16 < 2^-53 = 1.11022e-16
+               are2     < 2.23e-16) # see max(.) == 1.032e-16
+ }
> 
> ###---- x ^ y -- with *non-integer* y ---------------
> 
> ## keep x = ps  { above (1-2^-k) }
> 
> yp <- lapply(setNames(ks, names(psN)), \(k)
+              2^seq(7, floor(log2(k)), by = 1/8) - 1/4) # -1/4: sure *not* be int
> 
> reLd  <- lapply(psN, function(i) relEP(ps[i], yp[[i]], precBits = 2^8))
> absred <- lapply(reLd, abs)
> 
> K <- length(ks) # 12 here
> 
> plot(yp[[K]], reLd[[K]], type = "l", log = "x", col=K,
+      xlab = quote(k), ylab = "relErr{ p ^ k }",
+      ylim = range((-1:1)*2^-53, unlist(reLd)), xaxt="n"); eaxis(1)
> for(i in rev(seq_along(ks))[-1L])
+     lines(yp[[i]], reLd[[i]], col=i)
> abline(h = (-2:2)*2^-53, lty=c(3,3,2,3,3), lwd=c(1,1,2,1,1), col=adjustcolor(c(1,1,2,1,1), 1/2))
> 
> palROBG <- colorRampPalette(c("red", "darkorange2", "blue", "seagreen"), space = "Lab")
> palette(adjustcolor(palROBG(12), 3/4))
> ## pmax(.) for 'y': to "show" the '0' also
> plot(yp[[K]], pmax(1e-20, absred[[K]]), type = "l", log = "xy", col=K,
+      xlab = quote(y), ylab = "|relE|", main = quote(list(abs(rel.Err( x ^ y )), y~"*not* integer")),
+      ylim = range(2^-53, pmax(8e-18, unlist(absred))), xaxt="n"); eaxis(1, sub10=3)
> mtext(substitute(x == group("{", EE, "}"), list(EE = pExpr)), cex= 2/3)
> for(i in rev(seq_along(ks))[-1L])
+     lines(yp[[i]], absred[[i]], col=i)
> abline(h = c(1,2,4)*2^-53, lty=3, lwd=2, col=adjustcolor(2, 1/2))
> axis(4, las=2, line=-1, at=c(1,2,4)*2^-53, labels = expression(2^-53, 2^-52, 2^-51),
+      col.axis=adjustcolor(2, 1/2), col=NA, col.ticks=NA)
> legend("bottomright", legend=do.call(expression, as.list(pExpr[-1])),
+        col=palette(), lwd=3, cex=2/3)
> palette("default")# revert
> ## looks perfect (in Linux; *not* Windows)
> 
> t(sapply(absred, summary))
                Min.      1st Qu.       Median         Mean      3rd Qu.         Max.
1-2^-10 2.295971e-20 1.832048e-17 3.855744e-17 3.915690e-17 5.467917e-17 8.732578e-17
1-2^-11 6.862345e-19 1.877996e-17 3.510510e-17 3.725985e-17 5.295532e-17 9.126286e-17
1-2^-12 1.074726e-18 1.791156e-17 3.807128e-17 3.924264e-17 5.746482e-17 9.370526e-17
1-2^-13 3.722164e-19 1.405626e-17 3.343672e-17 3.497505e-17 5.113932e-17 9.576086e-17
1-2^-14 1.509081e-19 1.956440e-17 3.444407e-17 3.851610e-17 5.571175e-17 1.010061e-16
1-2^-15 3.707998e-19 1.395077e-17 2.725726e-17 3.390525e-17 5.110957e-17 8.905101e-17
1-2^-16 1.095571e-19 1.549693e-17 3.246712e-17 3.423402e-17 5.057892e-17 9.432710e-17
1-2^-17 3.397116e-19 1.510450e-17 2.837466e-17 3.191959e-17 4.479704e-17 1.040953e-16
1-2^-18 1.299536e-18 1.566903e-17 3.114549e-17 3.376463e-17 4.856218e-17 1.018159e-16
1-2^-19 4.484504e-19 1.506882e-17 3.370768e-17 3.443235e-17 4.969964e-17 1.033987e-16
1-2^-20 4.724846e-20 1.685262e-17 3.291611e-17 3.499646e-17 5.254457e-17 9.747515e-17
1-2^-21 6.103077e-19 1.718093e-17 3.248586e-17 3.437850e-17 4.887190e-17 9.572765e-17
> c(summary(ared <- unlist(absred)))
        Min.      1st Qu.       Median         Mean      3rd Qu.         Max. 
2.295971e-20 1.662607e-17 3.319100e-17 3.525278e-17 5.113847e-17 1.040953e-16 
> 
> if(.Platform$OS.type == "unix") { # && !noLdbl
+     stopifnot(ared < 2.23e-16) # see max(.) == 1.033987e-16 < 2^-53
+ }
> 
> 
> proc.time()
   user  system elapsed 
   1.92    0.09    2.00