math: trigonometry still wrong for huge arguments #37839
Description
What version of Go are you using (go version)?
$ go version go version devel +ae3f98c51b Sat Mar 7 00:35:52 2020 +0000 linux/amd64
Does this issue reproduce with the latest release?
Dunno. Probably.
What operating system and processor architecture are you using (go env)?
go env Output
$ go env GO111MODULE="" GOARCH="amd64" GOBIN="" GOCACHE="/tmp/freedesktopCache/go-build" GOENV="/home/nsajko/.config/go/env" GOEXE="" GOFLAGS="" GOHOSTARCH="amd64" GOHOSTOS="linux" GOINSECURE="" GONOPROXY="" GONOSUMDB="" GOOS="linux" GOPATH="/home/nsajko" GOPRIVATE="" GOPROXY="https://proxy.golang.org,direct" GOROOT="/home/nsajko/goNew" GOSUMDB="sum.golang.org" GOTMPDIR="" GOTOOLDIR="/home/nsajko/goNew/pkg/tool/linux_amd64" GCCGO="gccgo" AR="ar" CC="gcc" CXX="g++" CGO_ENABLED="1" GOMOD="" CGO_CFLAGS="-g -O2" CGO_CPPFLAGS="" CGO_CXXFLAGS="-g -O2" CGO_FFLAGS="-g -O2" CGO_LDFLAGS="-g -O2" PKG_CONFIG="pkg-config" GOGCCFLAGS="-fPIC -m64 -pthread -fmessage-length=0 -fdebug-prefix-map=/tmp/go-build573035198=/tmp/go-build -gno-record-gcc-switches"
What did you do?
Calculated sines, cosines, etc. of all kind of values, including huge ones. I had a different goal, but in the process I discovered this bug, because I compared Go's math functions' outputs with FriCAS (it can calculate with, e.g., 15000 digits of precision, which provides an effectively accurate result).
What did you expect to see?
Normally, I would just brush aside the meaningless results given huge arguments, but it seems Go actually supports that, having looked at some commit histories, Github issues, and src/math/huge_test.go. In particular, see 98521a5 @bmkessler and #6794.
Thus I would expect math.Sin and company to be accurate up to the last couple of bits of the mantissa (e.g., ULP error < 1000).
What did you see instead?
It seems the Payne-Hanek range reduction is not working properly as some inputs produce meaningless results given arguments above 10^31.
Some data for math.Sin, see attached file for more rangeReductionFailures.txt. Each line records a point where math.Sin is wrong in about 37 bits of the mantissa or more:
EDIT: added excerpt from the file for inputs around 2.6e+307
Input: 2.197000000000063281250000000000e+13 Have: 2.559435063127277198979925287858e-01 Want: 2.562412708565855457010229656589e-01 Diff: 2.977645438578258030304368730867e-04 ULP diff: 5364049155049
Input: 2.197000000000063671875000000000e+13 Have: 2.597176836301987035149352323060e-01 Want: 2.600151377550460329857173746859e-01 Diff: 2.974541248473294707821423799032e-04 ULP diff: 5358457143289
Input: 2.197000000000064062500000000000e+13 Have: 2.634878979753585426237805222627e-01 Want: 2.637850371424114159779605870426e-01 Diff: 2.971391670528733541800647799391e-04 ULP diff: 5352783368066
Input: 2.197000000000064453125000000000e+13 Have: 2.672540918193749837072914488090e-01 Want: 2.675509114946553190783617992565e-01 Diff: 2.968196752803353710703504475532e-04 ULP diff: 5347027915955
Input: 2.197000000000064843750000000000e+13 Have: 2.710162076947636444756994933414e-01 Want: 2.713127033491684048094327863510e-01 Diff: 2.964956544047603337332930095727e-04 ULP diff: 5341190874777
Input: 2.197000000000065234375000000000e+13 Have: 2.747741881962650345450072109088e-01 Want: 2.750703553056352279604368504806e-01 Diff: 2.961671093701934154296395718120e-04 ULP diff: 5335272333596
Input: 2.197000000000065625000000000000e+13 Have: 2.785279759817201883365100911760e-01 Want: 2.788238100269102015538180694421e-01 Diff: 2.958340451900132173079782660352e-04 ULP diff: 5329272382725
Input: 2.197000000000066015625000000000e+13 Have: 2.822775137729460204205622630980e-01 Want: 2.825730102398922305440009949962e-01 Diff: 2.954964669462101234387318982044e-04 ULP diff: 5323191113713
Input: 2.197000000000066406250000000000e+13 Have: 2.860227443566089600146540306014e-01 Want: 2.863178987363989014269805011281e-01 Diff: 2.951543797899414123264705267502e-04 ULP diff: 5317028619355
Input: 2.197000000000066796875000000000e+13 Have: 2.897636105850983079257332519774e-01 Want: 2.900584183740392285599796196038e-01 Diff: 2.948077889409206342463676264742e-04 ULP diff: 5310784993681
Input: 2.197000000000067187500000000000e+13 Have: 2.935000553773978726468385502812e-01 Want: 2.937945120770856788361413691746e-01 Diff: 2.944566996878061893028188933386e-04 ULP diff: 5304460331963
Input: 2.197000000000067578125000000000e+13 Have: 2.972320217199572733868251361855e-01 Want: 2.975261228373449751138934971095e-01 Diff: 2.941011173877017270683609240223e-04 ULP diff: 5298054730706
Input: 2.197000000000067968750000000000e+13 Have: 3.009594526675616332767049243557e-01 Want: 3.012531937150280669790447518608e-01 Diff: 2.937410474664337023398275050567e-04 ULP diff: 5291568287653
Input: 2.197000000000068359375000000000e+13 Have: 3.046822913442007729756255685061e-01 Want: 3.049756678396189357727052993141e-01 Diff: 2.933764954181627970797308080364e-04 ULP diff: 5285001101778
Input: 2.197000000000068750000000000000e+13 Have: 3.084004809439368499646150212357e-01 Want: 3.086934884107423449073337451409e-01 Diff: 2.930074668054949427187239052728e-04 ULP diff: 5278353273288
Input: 2.197000000000069140625000000000e+13 Have: 3.121139647317711651730576249975e-01 Want: 3.124065986990305909820619945094e-01 Diff: 2.926339672594258090043695119675e-04 ULP diff: 5271624903622
Input: 2.197000000000069531250000000000e+13 Have: 3.158226860445099704044480404264e-01 Want: 3.161149420469889781415417928656e-01 Diff: 2.922560024790077370937524392502e-04 ULP diff: 5264816095445
Input: 2.197000000000069921875000000000e+13 Have: 3.195265882916290545168180869950e-01 Want: 3.198184618698606818121277228784e-01 Diff: 2.918735782316272953096358833136e-04 ULP diff: 5257926952653
Input: 2.197000000000070312500000000000e+13 Have: 3.232256149561371638689877272554e-01 Want: 3.235171016564896695477671073604e-01 Diff: 2.914867003525056787793801049702e-04 ULP diff: 5250957580364
Input: 2.197000000000070703125000000000e+13 Have: 3.269197095954384790772451196972e-01 Want: 3.272108049701833998312849871581e-01 Diff: 2.910953747449207540398674609605e-04 ULP diff: 5243908084922
Input: 2.197000000000071093750000000000e+13 Have: 3.306088158421937039932458901603e-01 Want: 3.308995154495738555411321613065e-01 Diff: 2.906996073801515478862711461261e-04 ULP diff: 5236778573896
Input: 2.197000000000071484375000000000e+13 Have: 3.342928774051804885480976281542e-01 Want: 3.345831768094775227062598332850e-01 Diff: 2.902994042970341581622051307932e-04 ULP diff: 5229569156072
Input: 2.197000000000071875000000000000e+13 Have: 3.379718380701520197284537516680e-01 Want: 3.382617328417540925045159383444e-01 Diff: 2.898947716020727760621866764268e-04 ULP diff: 5222279941455
Input: 2.197000000000072265625000000000e+13 Have: 3.416456417006948353964901343716e-01 Want: 3.419351274161644971272266957385e-01 Diff: 2.894857154696617307365613669390e-04 ULP diff: 5214911041273
Input: 2.197000000000072656250000000000e+13 Have: 3.453142322390856389091595701757e-01 Want: 3.456033044812270027534850669326e-01 Diff: 2.890722421413638443254967569374e-04 ULP diff: 5207462567964
Input: 2.197000000000073046875000000000e+13 Have: 3.489775537071463928917580687994e-01 Want: 3.492662080650726919017756699759e-01 Diff: 2.886543579262990100176011765143e-04 ULP diff: 5199934635183
Input: 2.197000000000073437500000000000e+13 Have: 3.526355502070986358553739137278e-01 Want: 3.529237822762995024916676811699e-01 Diff: 2.882320692008666362937674421119e-04 ULP diff: 5192327357797
Input: 2.197000000000073828125000000000e+13 Have: 3.562881659224163555244047074666e-01 Want: 3.565759713048250456601806490653e-01 Diff: 2.878053824086901357759415986948e-04 ULP diff: 5184640851884
Input: 2.197000000000074218750000000000e+13 Have: 3.599353451186776964298985603818e-01 Want: 3.602227194227382023328232207859e-01 Diff: 2.873743040605059029246604040964e-04 ULP diff: 5176875234731
Input: 2.197000000000074609375000000000e+13 Have: 3.635770321444153907464169606101e-01 Want: 3.638639709851493875270023181656e-01 Diff: 2.869388407339967805853575555375e-04 ULP diff: 5169030624831
Input: 2.197000000000075000000000000000e+13 Have: 3.672131714319659123724193250382e-01 Want: 3.674996704310397599435589199857e-01 Diff: 2.864989990738475711395949474536e-04 ULP diff: 5161107141884
Input: 2.197000000000075390625000000000e+13 Have: 3.708437074983175207876229251269e-01 Want: 3.711297622841088772460693689936e-01 Diff: 2.860547857913564584464438667055e-04 ULP diff: 5153104906790
Input: 2.197000000000075781250000000000e+13 Have: 3.744685849459565285535234124836e-01 Want: 3.747541911536214076505757475388e-01 Diff: 2.856062076648790970523350551957e-04 ULP diff: 5145024041657
Input: 2.197000000000076171875000000000e+13 Have: 3.780877484637129581912518005993e-01 Want: 3.783729017352519541361743904417e-01 Diff: 2.851532715389959449225898424629e-04 ULP diff: 5136864669786
Input: 2.197000000000076562500000000000e+13 Have: 3.817011428276043116802895838191e-01 Want: 3.819858388119293790552433165431e-01 Diff: 2.846959843250673749537327239523e-04 ULP diff: 5128626915681
Input: 2.197000000000076953125000000000e+13 Have: 3.853087129016783407564616936725e-01 Want: 3.855929472546790193199228724552e-01 Diff: 2.842343530006785634611787827453e-04 ULP diff: 5120310905039
Input: 2.197000000000077343750000000000e+13 Have: 3.889104036388542073865437487257e-01 Want: 3.891941720234640134101766761887e-01 Diff: 2.837683846098060236329274630407e-04 ULP diff: 5111916764753
Input: 2.197000000000077734375000000000e+13 Have: 3.925061600817626450421471417940e-01 Want: 3.927894581680250740696180855593e-01 Diff: 2.832980862624290274709437653655e-04 ULP diff: 5103444622905
Input: 2.197000000000078125000000000000e+13 Have: 3.960959273635842325944622643874e-01 Want: 3.963787508287190952671608101809e-01 Diff: 2.828234651348626726985457935371e-04 ULP diff: 5094894608772
Input: 2.197000000000078515625000000000e+13 Have: 3.996796507088869465640357248049e-01 Want: 3.999619952373561493352838169812e-01 Diff: 2.823445284692027712480921763927e-04 ULP diff: 5086266852816
Input: 2.197000000000078906250000000000e+13 Have: 4.032572754344617149691032409464e-01 Want: 4.035391367180350963295154542720e-01 Diff: 2.818612835733813604122133256169e-04 ULP diff: 5077561486685
Input: 2.197000000000079296875000000000e+13 Have: 4.068287469501569164620491392270e-01 Want: 4.071101206879781386760441819206e-01 Diff: 2.813737378212222139950426935684e-04 ULP diff: 5068778643214
Input: 2.197000000000079687500000000000e+13 Have: 4.103940107597113806647826095286e-01 Want: 4.106748926583633219067337449815e-01 Diff: 2.808818986519412419511354528368e-04 ULP diff: 5059918456416
Input: 2.197000000000080078125000000000e+13 Have: 4.139530124615858341918794849335e-01 Want: 4.142333982351563137491723409767e-01 Diff: 2.803857735704795572928560432047e-04 ULP diff: 5050981061488
Input: 2.197000000000080468750000000000e+13 Have: 4.175056977497930699172457025270e-01 Want: 4.177855831199401848152774618939e-01 Diff: 2.798853701471148980317593668588e-04 ULP diff: 5041966594804
Input: 2.197000000000080859375000000000e+13 Have: 4.210520124147266174396975202399e-01 Want: 4.213313931107439680445736485126e-01 Diff: 2.793806960173506048761282727355e-04 ULP diff: 5032875193913
Input: 2.197000000000081250000000000000e+13 Have: 4.245919023439879147474584897282e-01 Want: 4.248707741028697748575382320269e-01 Diff: 2.788717588818601100797422986943e-04 ULP diff: 5023706997538
Input: 2.197000000000081640625000000000e+13 Have: 4.281253135232118145481194915192e-01 Want: 4.284036720897182459744101379329e-01 Diff: 2.783585665064314262906464136904e-04 ULP diff: 5014462145575
Input: 2.197000000000082031250000000000e+13 Have: 4.316521920368910913978766075161e-01 Want: 4.319300331636128920109740647604e-01 Diff: 2.778411267218006130974572442938e-04 ULP diff: 5005140779090
Input: 2.197000000000082421875000000000e+13 Have: 4.351724840691988394070222057053e-01 Want: 4.354498035166222691394466437487e-01 Diff: 2.773194474234297324244380433811e-04 ULP diff: 4995743040315
Input: 2.197000000000082812500000000000e+13 Have: 4.386861359048098707447138622229e-01 Want: 4.389629294413812665709428983973e-01 Diff: 2.767935365713958262290361744817e-04 ULP diff: 4986269072646
Input: 2.197000000000083203125000000000e+13 Have: 4.421930939297200602311477268813e-01 Want: 4.424693573319106731922545350244e-01 Diff: 2.762634021906129611068081430858e-04 ULP diff: 4976719020647
Input: 2.197000000000083593750000000000e+13 Have: 4.456933046320648017513121885713e-01 Want: 4.459690336844349678457888330740e-01 Diff: 2.757290523701660944766445027199e-04 ULP diff: 4967093030038
Input: 2.197000000000083984375000000000e+13 Have: 4.491867146029351332003898278344e-01 Want: 4.494619050981988328530292164942e-01 Diff: 2.751904952636996526393886597361e-04 ULP diff: 4957391247702
Input: 2.197000000000084375000000000000e+13 Have: 4.526732705371930287618909005687e-01 Want: 4.529479182762818356700051936059e-01 Diff: 2.746477390888069081142930372152e-04 ULP diff: 4947613821674
Input: 2.197000000000084765625000000000e+13 Have: 4.561529192342845817620400339365e-01 Want: 4.564270200264119448085864405584e-01 Diff: 2.741007921273630465464066219283e-04 ULP diff: 4937760901147
Input: 2.197000000000085156250000000000e+13 Have: 4.596256075990518552565333720850e-01 Want: 4.598991572617768808228788657289e-01 Diff: 2.735496627250255663454936438939e-04 ULP diff: 4927832636463
Input: 2.197000000000085546875000000000e+13 Have: 4.630912826425429562604563216155e-01 Want: 4.633642770018343015614448177075e-01 Diff: 2.729943592913453009884960920317e-04 ULP diff: 4917829179115
Input: 2.197000000000085937500000000000e+13 Have: 4.665498914828206666882692843501e-01 Want: 4.668223263731203220849863555486e-01 Diff: 2.724348902996553967170711985091e-04 ULP diff: 4907750681745
Input: 2.197000000000086328125000000000e+13 Have: 4.700013813457693534481052211049e-01 Want: 4.702732526100561472048866562545e-01 Diff: 2.718712642867937567814351496054e-04 ULP diff: 4897597298139
Input: 2.197000000000086718750000000000e+13 Have: 4.734456995659002132015302777290e-01 Want: 4.737170030557530941983657157834e-01 Diff: 2.713034898528809968354380544042e-04 ULP diff: 4887369183223
Input: 2.197000000000087109375000000000e+13 Have: 4.768827935871549072999187046662e-01 Want: 4.771535251628164497894601936423e-01 Diff: 2.707315756615424895414889761014e-04 ULP diff: 4877066493067
Input: 2.197000000000087500000000000000e+13 Have: 4.803126109637074758751396075240e-01 Want: 4.805827664941468846393490821356e-01 Diff: 2.701555304394087642094746115617e-04 ULP diff: 4866689384876
Input: 2.197000000000087890625000000000e+13 Have: 4.837350993607645865957067599084e-01 Want: 4.840046747237408131248059817153e-01 Diff: 2.695753629762265290992218069732e-04 ULP diff: 4856238016992
Input: 2.856100000000115000000000000000e+14 Have: 3.675982770314665804001208471163e-01 Want: 3.513038755892304854100416378060e-01 Diff: -1.629440144223609499007920931035e-02 ULP diff: 293533841053919
Input: 2.856100000000115625000000000000e+14 Have: 4.249667541022760230973176476255e-01 Want: 4.090962128301270372965348087746e-01 Diff: -1.587054127214898580078283885086e-02 ULP diff: 285898255037673
Input: 2.856100000000116250000000000000e+14 Have: 4.806757450926888997067010222963e-01 Want: 4.652910381146620477643693902792e-01 Diff: -1.538470697802685194233163201716e-02 ULP diff: 277146242453784
Input: 2.856100000000116875000000000000e+14 Have: 5.345077075850710457416425924748e-01 Want: 5.196689118526798267794220009819e-01 Diff: -1.483879573239121896222059149295e-02 ULP diff: 133655989862048
Input: 2.856100000000117500000000000000e+14 Have: 5.862524289179044112430005952774e-01 Want: 5.720174896109305739955175340583e-01 Diff: -1.423493930697383724748306121910e-02 ULP diff: 128216934717058
Input: 2.856100000000118125000000000000e+14 Have: 6.357078470617316678925590167637e-01 Want: 6.221323513135383898031705030007e-01 Diff: -1.357549574819327808938851376297e-02 ULP diff: 122277195185866
Input: 2.856100000000118750000000000000e+14 Have: 6.826808396670568379605015252309e-01 Want: 6.698177994980224481125219426758e-01 Diff: -1.286304016903438984797958255513e-02 ULP diff: 115859965824230
Input: 2.856100000000119375000000000000e+14 Have: 7.326595164261566761609856257564e-01 Want: 7.148876235097005871210740224342e-01 Diff: -1.777189291645608903991160332225e-02 ULP diff: 160074980632440
Input: 2.856100000000120000000000000000e+14 Have: 7.737385279645031532780308225483e-01 Want: 7.571658266503101808453379817365e-01 Diff: -1.657270131419297243269284081180e-02 ULP diff: 149273622926244
Input: 2.856100000000120625000000000000e+14 Have: 8.117961071093062752623836786370e-01 Want: 7.964873134413491539973506405659e-01 Diff: -1.530879366795712126503303807112e-02 ULP diff: 137889354917007
Input: 2.856100000000121250000000000000e+14 Have: 8.466836398284449671791662694886e-01 Want: 8.326985343183861409954715782078e-01 Diff: -1.398510551005882618369469128083e-02 ULP diff: 125966631927676
Input: 2.856100000000121875000000000000e+14 Have: 8.782648910532515618143634128501e-01 Want: 8.656580852388235935990223879344e-01 Diff: -1.260680581442796821534102491569e-02 ULP diff: 113552011936380
Input: 2.856100000000122500000000000000e+14 Have: 9.064165366735379425477958648116e-01 Want: 8.952372598616580301467138269800e-01 Diff: -1.117927681187991240108203783166e-02 ULP diff: 100693973768508
Input: 2.856100000000123125000000000000e+14 Have: 9.310286451156077891511131383595e-01 Want: 9.213205521429876920791457450832e-01 Diff: -9.708092972620097071967393276282e-03 ULP diff: 87442727787940
Input: 2.856100000000123750000000000000e+14 Have: 9.520051066226988822904786502477e-01 Want: 9.438061073846414350541067506128e-01 Diff: -8.198999238057447236371899634833e-03 ULP diff: 73850019826653
Input: 2.856100000000124375000000000000e+14 Have: 9.692640085615263911833494603343e-01 Want: 9.626061199745959973128606179671e-01 Diff: -6.657888586930393870488842367195e-03 ULP diff: 59968929118348
Input: 2.856100000000125000000000000000e+14 Have: 9.827379552893630298981975101924e-01 Want: 9.776471762660116082699346407026e-01 Diff: -5.090779023351421628262869489845e-03 ULP diff: 45853661025182
Input: 2.856100000000125625000000000000e+14 Have: 9.923743313325819048742459926871e-01 Want: 9.888705412559519736248603294371e-01 Diff: -3.503790076629931249385663249996e-03 ULP diff: 31559335366990
Input: 2.856100000000126250000000000000e+14 Have: 9.981355068489545789844896717113e-01 Want: 9.962323879443134577726937095576e-01 Diff: -1.903118904641121211795962153701e-03 ULP diff: 17141771179567
Input: 2.856100000000126875000000000000e+14 Have: 9.999989845713798120030446625606e-01 Want: 9.997039684773230128200793842552e-01 Diff: -2.950160940567991829652783053461e-04 ULP diff: 2657268742525
Input: 2.856100000000127500000000000000e+14 Have: 9.979574876592303134259509533877e-01 Want: 9.992717264072917826567277188587e-01 Diff: 1.314238748061469230776765471091e-03 ULP diff: 11837610272091
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