auto merge of #10196 : huonw/rust/fix-zig, r=alexcrichton

The code was using (in the notation of Doornik 2005) `f(x_{i+1}) -
f(x_{i+2})` rather than `f(x_i) - f(x_{i+1})`. This corrects that, and
removes the F_DIFF tables which caused this problem in the first place.

They `F_DIFF` tables are a micro-optimisation (in theory, they could
easily be a micro-pessimisation): that `if` gets hit about 1% of the
time for Exp/Normal, and the rest of the condition involves RNG calls
and a floating point `exp`, so it is unlikely that saving a single FP
subtraction will be very useful (especially as more tables means more
memory reads and higher cache pressure, as well as taking up space in
the binary (although only ~2k in this case)).

Closes #10084. Notably, unlike that issue suggests, this wasn't a
problem with the Exp tables. It affected Normal too, but since it is
symmetric, there was no bias in the mean (as the bias was equal on the
positive and negative sides and so cancelled out) but it was visible as
a variance slightly lower than it should be.

New plot:

![exp-density](https://f.cloud.github.com/assets/1203825/1445796/42218dfe-422a-11e3-9f98-2cd146b82b46.png)

I've started writing some tests in [huonw/random-tests](https://github.com/huonw/random-tests) (not in the main repo because they can and do fail occasionally, due to randomness, but it is on Travis and Rust-CI so it will hopefully track the language), unsurprisingly, they're [currently failing](https://travis-ci.org/huonw/random-tests/builds/13313987) (note that both exp and norm are failing, the former due to both mean and variance the latter due to just variance), but pass at the 0.01 level reliably with this change.

(Currently the only test is essentially a quantitative version of the plots I've been showing, which is run on the `f64` `Rand` instance (uniform 0 to 1), and the Normal and Exp distributions.)

Author: bors

src/etc/ziggurat_tables.py

@@ -19,7 +19,7 @@
 import random
 
 # The order should match the return value of `tables`
-TABLE_NAMES = ['X', 'F', 'F_DIFF']
+TABLE_NAMES = ['X', 'F']
 
 # The actual length of the table is 1 more, to stop
 # index-out-of-bounds errors. This should match the bitwise operation
@@ -43,13 +43,10 @@ def tables(r, v, f, f_inv):
 
     # cache the f's
     fvec = [0]*(TABLE_LEN+1)
-    fdiff = [0]*(TABLE_LEN+1)
     for i in range(TABLE_LEN+1):
         fvec[i] = f(xvec[i])
-        if i > 0:
-            fdiff[i] = fvec[i] - fvec[i-1]
 
-    return xvec, fvec, fdiff
+    return xvec, fvec
 
 # Distributions
 # N(0, 1)

src/libstd/rand/distributions.rs

@@ -207,7 +207,6 @@ fn ziggurat<R:Rng>(rng: &mut R,
                    symmetric: bool,
                    X: ziggurat_tables::ZigTable,
                    F: ziggurat_tables::ZigTable,
-                   F_DIFF: ziggurat_tables::ZigTable,
                    pdf: &'static fn(f64) -> f64,
                    zero_case: &'static fn(&mut R, f64) -> f64) -> f64 {
     static SCALE: f64 = (1u64 << 53) as f64;
@@ -237,7 +236,7 @@ fn ziggurat<R:Rng>(rng: &mut R,
             return zero_case(rng, u);
         }
         // algebraically equivalent to f1 + DRanU()*(f0 - f1) < 1
-        if F[i + 1] + F_DIFF[i + 1] * rng.gen() < pdf(x) {
+        if F[i + 1] + (F[i] - F[i + 1]) * rng.gen() < pdf(x) {
             return x;
         }
     }
@@ -288,7 +287,7 @@ impl Rand for StandardNormal {
             rng,
             true, // this is symmetric
             &ziggurat_tables::ZIG_NORM_X,
-            &ziggurat_tables::ZIG_NORM_F, &ziggurat_tables::ZIG_NORM_F_DIFF,
+            &ziggurat_tables::ZIG_NORM_F,
             pdf, zero_case))
     }
 }
@@ -366,7 +365,7 @@ impl Rand for Exp1 {
 
         Exp1(ziggurat(rng, false,
                       &ziggurat_tables::ZIG_EXP_X,
-                      &ziggurat_tables::ZIG_EXP_F, &ziggurat_tables::ZIG_EXP_F_DIFF,
+                      &ziggurat_tables::ZIG_EXP_F,
                       pdf, zero_case))
     }
 }

src/libstd/rand/ziggurat_tables.rs

@@ -145,72 +145,6 @@ pub static ZIG_NORM_F: [f64, .. 257] =
      0.887984660763399880, 0.898095921906304051, 0.908726440060562912, 0.919991505048360247,
      0.932060075968990209, 0.945198953453078028, 0.959879091812415930, 0.977101701282731328,
      1.000000000000000000];
-pub static ZIG_NORM_F_DIFF: [f64, .. 257] =
-    [0.000000000000000000, 0.000782818165911943, 0.001348786815607765, 0.001428899847265509,
-     0.001484430705892882, 0.001528472172127356, 0.001565707298030807, 0.001598388670308183,
-     0.001627786418212004, 0.001654692743837703, 0.001679637706201265, 0.001702994844613767,
-     0.001725038123187510, 0.001745974954326004, 0.001765966477270568, 0.001785140598493315,
-     0.001803600702759419, 0.001821431661060659, 0.001838704088536796, 0.001855477433793579,
-     0.001871802266665008, 0.001887722003144375, 0.001903274226858077, 0.001918491715965767,
-     0.001933403251421835, 0.001948034260540625, 0.001962407334827158, 0.001976542650643127,
-     0.001990458313945481, 0.002004170645086643, 0.002017694415851860, 0.002031043048104267,
-     0.002044228781321551, 0.002057262814738517, 0.002070155428613822, 0.002082916088226049,
-     0.002095553533492583, 0.002108075856553551, 0.002120490569226280, 0.002132804661891696,
-     0.002145024655099026, 0.002157156644953973, 0.002169206343177243, 0.002181179112575302,
-     0.002193079998548175, 0.002204913757158977, 0.002216684880213121, 0.002228397617726446,
-     0.002240055998106505, 0.002251663846325885, 0.002263224800326716, 0.002274742325862292,
-     0.002286219729956393, 0.002297660173134250, 0.002309066680560787, 0.002320442152205823,
-     0.002331789372137141, 0.002343111017035562, 0.002354409664009627, 0.002365687797781804,
-     0.002376947817308683, 0.002388192041889739, 0.002399422716815966, 0.002410642018598946,
-     0.002421852059823287, 0.002433054893654529, 0.002444252518034679, 0.002455446879594508,
-     0.002466639877306970, 0.002477833365903986, 0.002489029159078809, 0.002500229032490808,
-     0.002511434726590794, 0.002522647949281448, 0.002533870378427505, 0.002545103664226889,
-     0.002556349431455662, 0.002567609281597438, 0.002578884794865288, 0.002590177532127119,
-     0.002601489036740262, 0.002612820836305291, 0.002624174444343735, 0.002635551361907296,
-     0.002646953079123743, 0.002658381076686089, 0.002669836827288052, 0.002681321797012387,
-     0.002692837446676144, 0.002704385233135737, 0.002715966610556786, 0.002727583031652520,
-     0.002739235948893221, 0.002750926815690169, 0.002762657087557796, 0.002774428223256353,
-     0.002786241685917290, 0.002798098944155558, 0.002810001473169871, 0.002821950755833219,
-     0.002833948283778004, 0.002845995558475284, 0.002858094092312607, 0.002870245409671041,
-     0.002882451048004164, 0.002894712558920987, 0.002907031509275432, 0.002919409482262880,
-     0.002931848078526783, 0.002944348917277934, 0.002956913637427061, 0.002969543898733384,
-     0.002982241382970874, 0.002995007795115689, 0.003007844864553855, 0.003020754346314269,
-     0.003033738022328147, 0.003046797702715820, 0.003059935227105459, 0.003073152465984053,
-     0.003086451322084072, 0.003099833731808721, 0.003113301666695822, 0.003126857134927052,
-     0.003140502182881588, 0.003154238896738770, 0.003168069404132778, 0.003181995875862154,
-     0.003196020527657495, 0.003210145622009941, 0.003224373470066433, 0.003238706433592253,
-     0.003253146927007733, 0.003267697419501892, 0.003282360437226572, 0.003297138565578506,
-     0.003312034451571411, 0.003327050806304299, 0.003342190407532641, 0.003357456102345890,
-     0.003372850809960137, 0.003388377524629727, 0.003404039318688046, 0.003419839345721265,
-     0.003435780843885239, 0.003451867139373843, 0.003468101650046629, 0.003484487889225119,
-     0.003501029469670069, 0.003517730107746697, 0.003534593627793237, 0.003551623966702611,
-     0.003568825178730639, 0.003586201440546166, 0.003603757056536316, 0.003621496464384588,
-     0.003639424240937217, 0.003657545108379068, 0.003675863940735269, 0.003694385770723563,
-     0.003713115796977806, 0.003732059391668707, 0.003751222108547281, 0.003770609691440940,
-     0.003790228083232539, 0.003810083435355216, 0.003830182117840641, 0.003850530729957835,
-     0.003871136111486317, 0.003892005354668437, 0.003913145816891062, 0.003934565134149914,
-     0.003956271235355358, 0.003978272357543333, 0.004000577062061084, 0.004023194251800533,
-     0.004046133189565926, 0.004069403517661885, 0.004093015278800460, 0.004116978938436600,
-     0.004141305408647655, 0.004166006073685835, 0.004191092817346642, 0.004216578052307351,
-     0.004242474751606884, 0.004268796482457593, 0.004295557442594244, 0.004322772499391836,
-     0.004350457232007221, 0.004378627976825644, 0.004407301876525049, 0.004436496933105327,
-     0.004466232065271192, 0.004496527170598785, 0.004527403192966406, 0.004558882195791591,
-     0.004590987441673855, 0.004623743479123199, 0.004657176237135574, 0.004691313128472929,
-     0.004726183162616859, 0.004761817069491636, 0.004798247435199299, 0.004835508851176451,
-     0.004873638078381815, 0.004912674228345848, 0.004952658963181422, 0.004993636716962402,
-     0.005035654941235035, 0.005078764377854039, 0.005123019362831771, 0.005168478165478940,
-     0.005215203367812893, 0.005263262290042703, 0.005312727468930079, 0.005363677197016692,
-     0.005416196132139284, 0.005470375988385734, 0.005526316321746716, 0.005584125426278286,
-     0.005643921359735682, 0.005705833121505521, 0.005770002010457520, 0.005836583196307310,
-     0.005905747545561058, 0.005977683752542928, 0.006052600837980204, 0.006130731092920838,
-     0.006212333565464245, 0.006297698213369562, 0.006387150879090475, 0.006481059288027780,
-     0.006579840329791975, 0.006683968961788356, 0.006793989182803495, 0.006910527673723577,
-     0.007034310911336661, 0.007166186857056056, 0.007307152748134871, 0.007458391141830445,
-     0.007621317291194862, 0.007797642342679434, 0.007989459040836144, 0.008199360125510702,
-     0.008430605346682607, 0.008687362737884952, 0.008975066840784529, 0.009300967772353674,
-     0.009675004947253041, 0.010111261142904171, 0.010630518154258861, 0.011265064987797335,
-     0.012068570920629962, 0.013138877484087819, 0.014680138359337902, 0.017222609470315398,
-     0.022898298717268672];
 pub static ZIG_EXP_R: f64 = 7.697117470131050077;
 pub static ZIG_EXP_X: [f64, .. 257] =
     [8.697117470131052741, 7.697117470131050077, 6.941033629377212577, 6.478378493832569696,
@@ -344,69 +278,3 @@ pub static ZIG_EXP_F: [f64, .. 257] =
      0.775956852040116218, 0.791527636972496285, 0.808421651523009044, 0.826993296643051101,
      0.847785500623990496, 0.871704332381204705, 0.900469929925747703, 0.938143680862176477,
      1.000000000000000000];
-pub static ZIG_EXP_F_DIFF: [f64, .. 257] =
-    [0.000000000000000000, 0.000287067661533533, 0.000513134928485678, 0.000569030497974398,
-     0.000609667963417335, 0.000642831049855169, 0.000671465984262828, 0.000697030342996893,
-     0.000720360862708599, 0.000741986223663093, 0.000762263730113694, 0.000781447246315807,
-     0.000799724254382053, 0.000817237547791934, 0.000834098656693235, 0.000850396538527769,
-     0.000866203416804620, 0.000881578828420777, 0.000896572504999613, 0.000911226471926952,
-     0.000925576608509206, 0.000939653828282008, 0.000953484986066785, 0.000967093584871414,
-     0.000980500333784669, 0.000993723593313716, 0.001006779734568374, 0.001019683431705467,
-     0.001032447902101660, 0.001045085105172934, 0.001057605908173612, 0.001070020225402434,
-     0.001082337135821582, 0.001094564983022843, 0.001106711460658764, 0.001118783685829211,
-     0.001130788262427001, 0.001142731336065933, 0.001154618641914802, 0.001166455546523074,
-     0.001178247084534012, 0.001189997991027938, 0.001201712730115490, 0.001213395520299268,
-     0.001225050357040701, 0.001236681032901414, 0.001248291155571943, 0.001259884164055092,
-     0.001271463343231895, 0.001283031837006378, 0.001294592660197942, 0.001306148709326875,
-     0.001317702772419903, 0.001329257537945404, 0.001340815602974395, 0.001352379480650950,
-     0.001363951607045839, 0.001375534347457789, 0.001387130002219621, 0.001398740812059381,
-     0.001410368963061376, 0.001422016591266340, 0.001433685786946429, 0.001445378598586011,
-     0.001457097036596827, 0.001468843076792140, 0.001480618663643060, 0.001492425713336909,
-     0.001504266116655995, 0.001516141741693663, 0.001528054436422108, 0.001540006031125918,
-     0.001551998340713470, 0.001564033166917514, 0.001576112300394977, 0.001588237522735750,
-     0.001600410608388780, 0.001612633326513305, 0.001624907442762655, 0.001637234721007311,
-     0.001649616925003372, 0.001662055820012304, 0.001674553174376953, 0.001687110761059388,
-     0.001699730359144919, 0.001712413755316500, 0.001725162745304071, 0.001737979135312442,
-     0.001750864743431488, 0.001763821401032123, 0.001776850954151601, 0.001789955264870927,
-     0.001803136212688003, 0.001816395695889220, 0.001829735632922019, 0.001843157963772116,
-     0.001856664651347151, 0.001870257682870316, 0.001883939071285826, 0.001897710856679738,
-     0.001911575107717528, 0.001925533923102574, 0.001939589433056721, 0.001953743800826108,
-     0.001967999224215228, 0.001982357937151347, 0.001996822211282223, 0.002011394357609747,
-     0.002026076728162574, 0.002040871717710169, 0.002055781765521847, 0.002070809357173103,
-     0.002085957026402963, 0.002101227357025226, 0.002116622984897121, 0.002132146599948981,
-     0.002147800948277823, 0.002163588834309782, 0.002179513123034188, 0.002195576742314159,
-     0.002211782685277469, 0.002228134012792427, 0.002244633856033434, 0.002261285419141418,
-     0.002278091981983449, 0.002295056903017983, 0.002312183622271174, 0.002329475664429648,
-     0.002346936642057179, 0.002364570258941101, 0.002382380313575932, 0.002400370702791893,
-     0.002418545425535629, 0.002436908586812392, 0.002455464401797752, 0.002474217200128692,
-     0.002493171430384328, 0.002512331664766249, 0.002531702603989994, 0.002551289082400404,
-     0.002571096073321844, 0.002591128694658967, 0.002611392214760672, 0.002631892058563845,
-     0.002652633814032662, 0.002673623238910738, 0.002694866267805934, 0.002716369019626269,
-     0.002738137805389534, 0.002760179136428037, 0.002782499733014893, 0.002805106533435520,
-     0.002828006703534697, 0.002851207646767162, 0.002874717014785921, 0.002898542718600849,
-     0.002922692940346749, 0.002947176145699226, 0.002972001096982591, 0.002997176867015228,
-     0.003022712853742948, 0.003048618795714386, 0.003074904788455568, 0.003101581301807876,
-     0.003128659198296080, 0.003156149752600867, 0.003184064672214937, 0.003212416119368622,
-     0.003241216734320596, 0.003270479660111680, 0.003300218568896729, 0.003330447689969929,
-     0.003361181839619420, 0.003392436452949343, 0.003424227617828290, 0.003456572111131984,
-     0.003489487437467131, 0.003522991870580083, 0.003557104497672658, 0.003591845266868621,
-     0.003627235038102472, 0.003663295637722386, 0.003700049917134574, 0.003737521815846301,
-     0.003775736429304177, 0.003814720081962375, 0.003854500406067995, 0.003895106426696382,
-     0.003936568653631844, 0.003978919180756157, 0.004022191793678687, 0.004066422086428989,
-     0.004111647588127876, 0.004157907900659452, 0.004205244848493050, 0.004253702641940915,
-     0.004303328055299205, 0.004354170621502118, 0.004406282845128784, 0.004459720435841752,
-     0.004514542564613699, 0.004570812145417769, 0.004628596145424491, 0.004687965927177740,
-     0.004748997626717266, 0.004811772572194672, 0.004876377748206484, 0.004942906311860507,
-     0.005011458167522187, 0.005082140608288488, 0.005155069033533799, 0.005230367753417398,
-     0.005308170893076836, 0.005388623411430704, 0.005471882252147620, 0.005558117647517014,
-     0.005647514599798176, 0.005740274569295156, 0.005836617404105682, 0.005936783553485037,
-     0.006041036615386131, 0.006149666279423593, 0.006262991739818591, 0.006381365669577810,
-     0.006505178868201678, 0.006634865721946159, 0.006770910649812723, 0.006913855752425535,
-     0.007064309938019209, 0.007222959874423007, 0.007390583214465396, 0.007568064673498798,
-     0.007756415714389786, 0.007956798835585532, 0.008170557788458321, 0.008399255510700199,
-     0.008644722212900025, 0.008909116987305010, 0.009195007664428712, 0.009505475652925033,
-     0.009844255532840629, 0.010215923852312625, 0.010626158965710175, 0.011082105722287849,
-     0.011592898788496009, 0.012170432837851575, 0.012830529553771619, 0.013594766864701180,
-     0.014493463190219380, 0.015570784932380066, 0.016894014550512759, 0.018571645120042057,
-     0.020792203980939394, 0.023918831757214210, 0.028765597544542998, 0.037673750936428774,
-     0.061856319137823523];