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math: trigonometry still wrong for huge arguments #37839

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nsajko opened this issue Mar 13, 2020 · 3 comments
Closed

math: trigonometry still wrong for huge arguments #37839

nsajko opened this issue Mar 13, 2020 · 3 comments
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@nsajko
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nsajko commented Mar 13, 2020 •
edited

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
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@robpike
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robpike commented Mar 13, 2020

I believe this is a dup of #31566, for which https://go-review.googlesource.com/c/go/+/172838 is underway.

@nsajko
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nsajko commented Mar 13, 2020 •
edited

I am not an expert, but #31566 says package math is not using Payne-Hanek enough, while my issue is about Payne-Hanek (or the implementation) being wrong. https://go-review.googlesource.com/c/go/+/172838 likewise just expands the use of Payne-Hanek (by lowering the threshhold), while the fix for my issue would presumably belong to either func trigReduce or var mPi4. Thus these issues are actually complementary, or even opposite. EDIT: since this bug happens with the entire possible range of exponents, var mPi4 is probably correct and func trigReduce is probably wrong.

BTW, I uploaded the entire file with the diffs now, so the input values in it now go up to +Inf, approximately. And, BTW, the file does not list all the points for which this bug happens. And to repeat my self for clarity, it features results for math.Sin in every line.

@nsajko
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nsajko commented Mar 13, 2020

Oops: turns out converting from the high precision floating point type to binary64 and back again fixes the Fricas results. In other words I did sin(HUGEFLOATVAL) instead of sin(HUGEFLOATVAL::DoubleFLoat::Float) in Fricas, where HUGEFLOATVAL was obtained from Go's fmt.Printf("%35.30e", ...), and this greatly affects results for huge arguments.

@nsajko nsajko closed this as completed Mar 13, 2020
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