# golden precision oracle table
# generated by scripts/gen_golden_precision.py
# each line: <input_raw>\t<expected_raw>
# input_raw is the storage integer of x at the tier scale.
# expected_raw is the half-to-even rounding of f(x) at the tier scale,
# computed by mpmath at max(700, 2*SCALE + 64)-digit precision.
40704400255883940956300986862591354850418	63800000200536003569086284943562290531
3276090458537197077496664154161186842419	18099973642348756408027281490335625335
65124906237406121032497167884134443167129	80700003864563799060696157236509909069
30030409934658380226825632681516966299433	54800009064468575347554833985596076693
50268102712448889859911453943591307176991	70900001912869431078432193512230458109
4452092153129399381486353443448508349046	21099981405511710501276944056807658915
25603596640950494884972093722740222816064	50599996680781012562080264860324642041
73273605807924954761287519114301857364413	85600003392479462425302855922653819278
99800095300305122960078344349213185252626	99899997647800334149972804854563190584
27562494718543962999992608205582302666558	52499994970041628566838499714437717776
22502022454518684102040429094667194494	1500067413635756708088643852696975538
90250002882291325204966395886617274050404	95000001516995422206429097859998160632
98406399693593471098034093224061824634479	99199999845561225231613277068778724689
1322505220247567474635847793270176934241	11500022696706156825791977631085063555
2402507727428603191833345687797513274116	15500024927168998562227473773459318453
20249996738002268701469663257104302697648	44999996375557930372962780482105715308
18490006414680945576942976969939241054924	43000007458930685140085255495956716467
43164895787231139565183564820752207139157	65699996793935340012692180080239883874
12744901026096292456696228870050081515710	35700001437109624375147764920542090085
2402507106736507597019388694368998032411	15500022924939522801606927686056126534
14212892344474047339805444552283231068373	37699989846781188940693888682742069443
10956095471644985859342694547921659346018	33099993159583863967392074917192086672
8236891313122874613712954770304347093502	28699984866063735633178424387526958473
32490004801582155482693005493303262286185	57000004211914015860084417111463854904
2992896280863189330055255205472225967084	17299989251046340838172217112724380995
57912103094821964090936575920562127768874	76100002033391539253695832349157514822
12460896276383087275182246716734266258472	35299994725754687108422154573591200911
51984091949846685085115012759384772950881	72099994417369191003777751193644776004
7728403885911697119309322605250097012792	27800006989048936423127635670385067871
11155607037155346815862863015607743574704	33400010534662031474283965803491619884
16080102196951328327889963330420969623862	40100002739340714822960653943554233788
15132092389070111020514382626085997163704	38899990217312537288913177094175598230
518394359089254062306132987703337715526	7199960826902144071873831806589656724
1368904681776827502603396597672992173084	11700020007576172955385600328776490797
4409998332936612965579552746712171543626	20999996030801084333760074967057537372
9120405804804602336616870315192148664035	30200009610602117052791965760138577378
37576901012977445408702454645823147665012	61300000826245871600228426820237804014
66585603604041206812863711265034952484395	81600002208358552723277412625337447586
70056891909936481401089295825351827901717	83699995167225954506594711994268927521
57456409227755328609747014507632360813588	75800006086909602971676693284620332122
1742394868940599559926355874653087448956	13199980564154628935884381555120296187
93702397007983952064649912552945038929416	96799998454537152985586672279951290841
5198408571510418457215861508353432314476	22800018797164221761815926937287982983
4708907620817516154763589068607744997360	21700017559480259910814762886774093042
19184399376536354935313860595119073454727	43799999288283504417705661821574328231
57456408857308958203412218444676282831847	75800005842551858080006645714574611693
28902237519796991274982903753988040134	1700065808132055553781495647289162622
688461109204318769464457159652893148745772467672382601474621398617700402112	8297355658306559354090528086635677940892073576035716729
137058599780058872013212814605242557095626599423120370407039898694567584076	3702142619889985707966752820029483239577443480931051051
876600233084764573885177174087418800810425361023415788584693179352745791587	9362693165349191202155335340046850280862020018077651887
572107922303984983705876325551152449740195675256605989736986328706022367311	7563781609115806456355753661989526639721688246797325367
913111600392871023850256377012193132746809550669213634615302356707874704172	9555687313808834671386835917052461971352237481042518672
249460124581608204656391846556830027896376580135446813081912599805226584829	4994598328010053349142822560310400930656131004098578378
791118950231647386638569745332699788551762600930995538657740673548185409800	8894486776827808554783988108202479597129317402128003649
212083021066990020013395383001894977809732027844358425660312787946845283913	4605247236218594377469352744032381838272592672001604966
110955979843849590859224248327775016710473502005617845045462016616513566328	3331005551539198363622254131368840982704807564337905631
184018984459167950088896945672972785154032503796982817255255783446835442799	4289743400940992765208779991726546204019660199478541851
891542941660872091989401863275021718768455823766254816321912569560337164002	9442155165325721822314179880455846210749663768245640712
337308121154233237978912648085075243099782216880083267398910690943534295594	5807823354357751452521474080866673324314526911025108218
221569084354140252455639892116790515936137115860824689588025426840220535968	4707112536939607230656735616207565387470921374945865545
410846830862807576262868664610545668152446077800190694389702349274794289969	6409733464527269844212109735585029027098873001779261615
521801287404238747456167694783353433272257325326763066827658366578387452033	7223581434470291534156157707907719205426324325965569211
578795512515589776250503883221255193171244393888867551497694300158255090372	7607861148283331913996842529429166518319117938845576705
531099998563718825525591357593347854293405758424233627158086124094240312133	7287660794546620674415842027844247346708502192325056274
666388235696774608433664751711268586046558152689738580679401280096911752377	8163260596702610525045206680133684023403514412399008372
466655029057917813475307215629839474344850558524151596963541994760142938664	6831215331534483355626859034634477105550130297819332904
468238913820542216173311676312175210129286838501579829401962361766395927049	6842798505147891728130704008503318843929880942562668845
639463608578349621424419211144290094340764771056380415034014126864272457781	7996646850889125148852320368762273688584339869546250445
274248438062802119082866185940467166995900040600982501648957701659213832788	5236873476252812039889609953461244455440814773876547498
808981058857864175970792909222326254402584841549259462766045595673961585929	8994337434507692384671852381649195823863060015433177239
116159889506151495896863774448346380943461251312899766651670331556003227918	3408223723674129580934192498862570721516959957784706745
218923781015067283985266435941545846778370723467961844359541809864605899757	4678929161838927477494122749944742414611011223233843573
456016675379800171439875965128978602345434356775846441527069643879658593313	6752900675856266223570244106046370434100423310918306987
364465197230838115617345808781257619518183594539156179330213764358893646739	6037095305118498090922596349551372706086636426274812938
467897075913102736349329365484768074653018507585229008943361249039529251325	6840300255932503554923509250876577583161984239444597426
48620608460635327225058250456759704170195614053201010774790296653704877076	2205008128344095860490955769499312957027537114593432399
704101980896247959535983604268914211517037966242620001142861718351941955664	8391078481913084571099045891101843250697270412764568488
900359801471659161551968992328158407159108144296810822992049517446231612658	9488729111275435507377982439807226682291844319143641661
63966331433479630521931690188156795887962941941004626979734686494491780962	2529156607121821361609800973204664367870558574211437646
643578806011583721304592918467270514178863499660345802146319867723148700733	8022336355523767998081212273660211709853175315633447414
994751552047252501031179895485936683092290508469516108583921751933962715186	9973723236822107151883381646475429768491997796031026639
158860209498486480072706838113422844425257843830572968743236203994541640496	3985727154466126775301765882076867008568906306758124862
842741264694018077291659726386378845390309568071615785835285971525581274541	9180094033799534437597122885687446473524512824464428979
910252018066554815468162205846040456124083075693145381191593333596450594026	9540712856315060567352481980173724336339172983865657136
588722529662205232388958667625198927737704180438470078671869504924954975407	7672825618129251035643643695559414596923241476157366381
524633195312789494533206125427272047350495612148397715957371068845752370490	7243156738003047622115436490139269870198847351502901272
377403937774626946618200236394034703242503477007351052495820884295713295620	6143321070680149195047477854830704644398573840427233056
893936496317337566255680235442667568937385800302069671136170044326243846539	9454821501844112372833652814102761832794229095985315361
790827313876910627393083968511744066453315433315905824952312936229403310349	8892847203662675252414840032701916575072103454891706022
541890014651181851128202095085079685358457917523524102648803223860682621236	7361317916318937594483759850654769716776928901970078620
703761945826671203246476756035884144808853555132355738872118343091451015192	8389052066989876557983197287951112374132039938381597845
950216654373409859742163305581240702373560356478093610068088988910630598121	9747905694934732285549263855674510985594751134705313007
982824211627014448721432915058078036737163920388506664642703367835132527010	9913749097223584913220913103989754948223064294752785164
235052660078186440612098456181802533054453511837055825407341240635945964788	4848222974226602550763633349979631967739749544221537849
22176153610801677456948148999315757216742580283058147623204119235717686855	1489165995139617652454964166387153018544800434011247591
438561660268129039319403475189986155982083290558469345790387235127259753928	6622398812123361083672918003207812465512963731879259279
243934998857943378282830838467788524318008355971580431488094052982811404818	4938977615437666450437185462414832903930491673204138748
45962245924387187883740161186493042713370626112533962410824737290914148999	2143880731859568438985880837544754331578566978042866736
573649357685325766854434718293348150100106058412939718581823637917114733220	7573964336365241737607125055681312031782418783639458369
151280400578699570420133923660170288448174661137069419001681039303796488326	3889478121531210906987797314609734321599146304616117893
189136490155191686093112486705702412257096903358101700809288919089851338731	4348982526467445880592248557966220088768003302625988533
818439291664661918463245944216372110916749728230820937066187210451432129001	9046763463607644906486230416504178660428435467303086129
653945794221154760376030684791848518121845721111149166773012327716881212799	8086691500367469566352817749817049647884759553366888029
245744578552531364458141698192834467096003557247776035460766986072480025765	4957263141618884895748481642392789586103268014139232358
740665877676814721060945707545132685420007324012708743243260439615551656060	8606194732149713105636020410961001151528412534501786431
646176891893871335879336365744811936621024656999218742811430654560911244841	8038512871756014896126554827489057594904582067817994893
897367037341653627583536064748302129349432093580236286536356415187126665704	9472945884684730043892476607851231972318409501559275891
449273345225532517441658715624952601676850177760322916547364422511772199933	6702785579335896654456671304479757678160044381306883556
466651842600329779332545315414623144262556475424335689608146769501418388180	6831192008722414613523819477716951399035229223222429867
853279735981498214868305427290315464080940332961115977219985265468210390679	9237314198301897183704490075390139055881351423523954639
958771179219485443688615133542051586113539292214495444442059093463835626588	9791686163370869821541479532082249281531824461169025732
356127300574896088564462026382096152924566565702074095384546733952500564815	5967640241962446654843506934082804634257908582931884620
231583252561166767661004120628788075836533863641811928477082777151717431319	4812309763109257367966368262485420266772681041664073495
294832179689061794058747453707052928283276601377448134157569807295678633126	5429845114633213863848878247518172864585301047139925870
430883389783252170979459725677286161825644636164445144127545281787407195310	6564170852310687464251992771507258362281969450420716168
679951664494175152441339285405599854787795909443840612054931128736427089521	8245918168974120691709550410526417712860904161648748380
584028736837957377531336707859800711136432825699050324390460617966519671748	7642177286859795020750741015213887563606201125096717885
974433402768184268759634561777929815572046194905052685278544049392508753361	9871339335511591401527973631132174285478654899141200340
401506823015673456605399866422529653911724643571427174069045788405722077062	6336456604567520089988286739769223574735702117336652459
739959115660591901342573151367494483262108658659796598985469351762769406568	8602087628364360607634381824757769857917762672877657795
669552107980149613725780822830938449151537142365062627967851184826438255566	8182616378519462363600753979761860649577632458311852006
705041050809495015462503846436150896535809518536280549473787792388245452241	8396672262328064781513247905229825115750222536128292832
430786791594361107524494326417622301154509169803500739979723218449517774872	6563435012204821911353716061323259476425402864862815314
440209240809721708854875176930197371247291843688038392037684487899565914119	6634826605192645943341152713781729229011691598509827234
549802191630980463192843957295275038756626143197838783394175442919557619790	7414864743412252312214189485492652519879947880488398154
259953974669460657821649537934708138307168072216793540292593780177565396189	5098568178120801179384597891762280562372161724577872538
237276228910845705123386980370273123120109029742322769964687651277960668066	4871100788434229320256690912721135474787352802344549712
923210928295327048114071531781871337699676123952075957416972119718316398446	9608386588264062182597592228660488282068581199012184478
127760194551316398124973509732573998152935561324177768751479953942314494305	3574355809811278401024901761436235137257538680512839607
668136173100597472659493745180556955381425813642539326838094735809233382465	8173959708125539033421279892926488738461765044428698271
361629347670443649127921452795221275811097727169327849490013337570192074676	6013562568648002760447227602702387026956688653197829629
44646930123512959076961362310378159735079588928418475479528527025489926545	2112982018937050942479452695994544286847269635269443689
8973135221131075576140473763970805663397619490380870967889850113844987709	947266341697575338068217090865732134094179097109651723
168650129673532981016685414524446529380994023757495936181322125498268916557	4106703418479754072350005953425809113446958645721017493
92778687758094112670913776980405859545511836401678847809619431163781587867	3045959417951823478806613686940179902615056340802049134
750552963601914429229168254570757217894769971870557566371876082033390607245	8663445986453164407061582127652930834799223703585163924
537991969936113442736324342894082715894850018475743082759884922853838681108	7334793589025620581415912689510871945193530666386501942
877706820226797020253590455531105087880455211537306654715360571175607802781	9368600857261435386727496330334758823616674006765291248
796636554338586490209106788151380635186295165837917757460932810104735781288	8925449872911653354702722315395540204240234951126393836
415570844797527261362937706949908247096886099235169478274657123648842238975	6446478455696003187899380814530677269858315961422955646
953157706186349118061871757125065043894394980482686842583031385964917020451	9762979597368567266045641306019936459091042197279502634
86288959735055492349247875312922028040373	92891850953167841561966164649507840247
14024331758476599758740711223343325635298	37449074432456404514241517920430302821
53728391751304096880850627385167517663589	73299653308391641314998164490343964853
18981470281638571606296433590365552208381	43567729205959967734491451801903598551
87186866616249300397019451226207680474833	93373907820252066601906742308222755689
45883454140927615556081664677395428813913	67737326593930185573424474509497976748
38924097801621577568806015547004505930550	62389179992705127370061494541308482223
28605330208707130294029972251623862552842	53483951058899089429031659325180719151
31805022861331708636987885830878351181081	56395942106974069625925718296853980827
67230914303111651465405689935177641330577	81994459753761199756099511161865305850
86785751000376677060982211126406913077282	93158870216623321317511460658608654657
88072327824763304098094280449104797411239	93846858138545748599353766270763927188
686689841785017134872250544283013833230	8286675097920861135421022842448512613
58379305838516053467273742965712261080573	76406351724523566252974024637320954987
88122815825539911676715219171160583732639	93873753427430348848281472247905460240
72758001729882298109608904822290960784271	85298301114314286271030969518219808834
67470948971536341416470607121557604639043	82140701830174510273885678207627196807
50326759453869034978753160727194342302385	70941355677678612416037263078734400338
43357754654635516913877092989261431959069	65846605572827759026118869667166857252
15930710603434308752063928220839505495930	39913294280771048317560701062933377118
21267530320767338589976519520009205957108	46116732669137932532297379597534802841
8799741592459045725950172942454990515095	29664358399363782170906695240026544009
34203200946558261443830930026611268793463	58483502756382728006509098720630676195
44323227130066083970830541116587272117435	66575691607422362526594936988852297819
39838023196262472260055374432298683127920	63117369397228899609351102375432758766
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# ─── hard-input sections (see scripts/gen_hard_inputs.py) ───
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