1999-06-10 19:49:32 +02:00
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-- ******************************
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-- * Test suite for the Postgres NUMERIC data type
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-- ******************************
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2000-03-30 09:13:20 +02:00
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-- Must drop tables created by short numeric test.
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DROP TABLE num_data;
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DROP TABLE num_exp_add;
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DROP TABLE num_exp_sub;
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DROP TABLE num_exp_div;
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DROP TABLE num_exp_mul;
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DROP TABLE num_exp_sqrt;
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DROP TABLE num_exp_ln;
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DROP TABLE num_exp_log10;
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DROP TABLE num_exp_power_10_ln;
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DROP TABLE num_result;
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1999-06-10 19:49:32 +02:00
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CREATE TABLE num_data (id int4, val numeric(1000,800));
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CREATE TABLE num_exp_add (id1 int4, id2 int4, expected numeric(1000,800));
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CREATE TABLE num_exp_sub (id1 int4, id2 int4, expected numeric(1000,800));
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CREATE TABLE num_exp_div (id1 int4, id2 int4, expected numeric(1000,800));
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CREATE TABLE num_exp_mul (id1 int4, id2 int4, expected numeric(1000,800));
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CREATE TABLE num_exp_sqrt (id int4, expected numeric(1000,800));
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CREATE TABLE num_exp_ln (id int4, expected numeric(1000,800));
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CREATE TABLE num_exp_log10 (id int4, expected numeric(1000,800));
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CREATE TABLE num_exp_power_10_ln (id int4, expected numeric(1000,800));
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CREATE TABLE num_result (id1 int4, id2 int4, result numeric(1000,800));
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-- ******************************
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-- * The following EXPECTED results are computed by bc(1)
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-- * with a scale of 1000
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-- ******************************
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BEGIN TRANSACTION;
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INSERT INTO num_exp_add VALUES (0,0,'0');
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INSERT INTO num_exp_sub VALUES (0,0,'0');
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INSERT INTO num_exp_mul VALUES (0,0,'0');
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INSERT INTO num_exp_div VALUES (0,0,'NaN');
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INSERT INTO num_exp_add VALUES (0,1,'85243.39540024977626076239847863600785982737155858270959890014613035727868293618673807776733416230953723818527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_sub VALUES (0,1,'-85243.39540024977626076239847863600785982737155858270959890014613035727868293618673807776733416230953723818527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (0,1,'0');
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INSERT INTO num_exp_div VALUES (0,1,'0');
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INSERT INTO num_exp_add VALUES (0,2,'-994877526002806872754342148749241.04353023451866590387054959174736129501310680280823383331007646306243540953499740615246583399296334239109936336446284803020643582102868247857009494139535009572740621288230740389545481395');
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INSERT INTO num_exp_sub VALUES (0,2,'994877526002806872754342148749241.04353023451866590387054959174736129501310680280823383331007646306243540953499740615246583399296334239109936336446284803020643582102868247857009494139535009572740621288230740389545481395');
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INSERT INTO num_exp_mul VALUES (0,2,'0');
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INSERT INTO num_exp_div VALUES (0,2,'0');
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INSERT INTO num_exp_add VALUES (0,3,'-60302029489319384367663884408085757480.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (0,3,'60302029489319384367663884408085757480.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (0,3,'0');
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INSERT INTO num_exp_div VALUES (0,3,'0');
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INSERT INTO num_exp_add VALUES (0,4,'5329378275943663322215245.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_sub VALUES (0,4,'-5329378275943663322215245.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (0,4,'0');
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INSERT INTO num_exp_div VALUES (0,4,'0');
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INSERT INTO num_exp_add VALUES (0,5,'-652755630.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_sub VALUES (0,5,'652755630.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_mul VALUES (0,5,'0');
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INSERT INTO num_exp_div VALUES (0,5,'0');
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INSERT INTO num_exp_add VALUES (0,6,'.0469370721950711508944806393077762204079964905145503836835397203739563036579760026190241480514409364');
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INSERT INTO num_exp_sub VALUES (0,6,'-.0469370721950711508944806393077762204079964905145503836835397203739563036579760026190241480514409364');
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INSERT INTO num_exp_mul VALUES (0,6,'0');
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INSERT INTO num_exp_div VALUES (0,6,'0');
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INSERT INTO num_exp_add VALUES (0,7,'-818934540071845742');
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INSERT INTO num_exp_sub VALUES (0,7,'818934540071845742');
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INSERT INTO num_exp_mul VALUES (0,7,'0');
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INSERT INTO num_exp_div VALUES (0,7,'0');
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INSERT INTO num_exp_add VALUES (0,8,'8496986223.64063724353165506167988570717591150432282238795309964705925283285060558038824227595710139960766584401003765241409149793494330798800');
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INSERT INTO num_exp_sub VALUES (0,8,'-8496986223.64063724353165506167988570717591150432282238795309964705925283285060558038824227595710139960766584401003765241409149793494330798800');
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INSERT INTO num_exp_mul VALUES (0,8,'0');
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INSERT INTO num_exp_div VALUES (0,8,'0');
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INSERT INTO num_exp_add VALUES (0,9,'54863480.34685027005508022756223282084742813020271603840941647350440860843570182437301045468670059279379903480024743452620396345637401505220786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_sub VALUES (0,9,'-54863480.34685027005508022756223282084742813020271603840941647350440860843570182437301045468670059279379903480024743452620396345637401505220786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_mul VALUES (0,9,'0');
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INSERT INTO num_exp_div VALUES (0,9,'0');
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INSERT INTO num_exp_add VALUES (1,0,'85243.39540024977626076239847863600785982737155858270959890014613035727868293618673807776733416230953723818527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_sub VALUES (1,0,'85243.39540024977626076239847863600785982737155858270959890014613035727868293618673807776733416230953723818527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (1,0,'0');
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INSERT INTO num_exp_div VALUES (1,0,'NaN');
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INSERT INTO num_exp_add VALUES (1,1,'170486.79080049955252152479695727201571965474311716541919780029226071455736587237347615553466832461907447637054203186991790701615551214692555785671028648640897898741246882118067609728317430043806625387779037980513762118868084887015059202190301421555269486602797852927777567694581746398790609996101506730430853942556475840126871131898407356048450541232591147357021858041662012293323494543567675306406079659294204054863522259037763051870433216859794083051717080761509518250300466106939998045710070');
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INSERT INTO num_exp_sub VALUES (1,1,'0');
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INSERT INTO num_exp_mul VALUES (1,1,'7266436459.363324713115467666113895787027372854351303425444968800459979742082292257107107767894843498525848597439323325297125474674300428669958003640228730876886174255457103020291514229439701871032118057857763809224712818579091741996335014138185389554630910658876423205103697147288306070059640369158894028731728589073730895396494400175420670713113234800826523252075036892246807434088405522834549449664122407363485486902219500109237667016524913027290777216477989904700729228025571098410870506256758678625928245828210775042611512394316804583459576285681159178280400209217948833631961377519855502763611693070238579591463373484424582723121059964236704135695706864890193388054537703767833595331866551990460050750959493829603581882430597105627056085260296454181999581594565113210481151487049158699087454047624433576922179904629');
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INSERT INTO num_exp_div VALUES (1,1,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
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INSERT INTO num_exp_add VALUES (1,2,'-994877526002806872754342148663997.64812998474240514147207095573950146764154822009863493316394610578375247334825932838513167168342610420582834742950389452212867974756590355021495169819086060202117180229196935525386766373096687306110481009743118940565957556492470398904849289222365256698601073536111216152709126800604695001949246634784573028721762079936564434050796321975774729383704426321489070979168993853338252728216162346796960170352897972568238870481118474064783391570102958474141459619245240874849766946530000977144965');
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INSERT INTO num_exp_sub VALUES (1,2,'994877526002806872754342148834484.43893048429492666626902822775522112238466538551783273345620682034111834572173548391979999630250058057637037929942180153828419189449146140692523818459983958943364062347264545253704196416903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (1,2,'-84806738323879544552397401815149740513.8505875535743013876823142649666132764556588225959336097903898464616542203793600590311980154402068027051522932586050753865288419084437796768749509032177577451738712965496693249429231838833655025794915864261585848007162358912070811805298210095333433397862313304655108809804359760907473898420016370058274978588765092161529583480924554820756527238472641797198545539410039895140087686344382628317530286295498797849942258314364503000942821309916954725689781458590617068629906894951122301020797266469357701283289275708774593896770378558232444454118891917258610753077932026885574920166837998049508644891327208474213193224700658584824407382455480657734911543930195324144216374573825');
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INSERT INTO num_exp_div VALUES (1,2,'-.000000000000000000000000000085682300757901809257711279577127388124986344391495296640171942990079130291883279872719240502687189411421655284515420074848478500192127657883342858267913417679786356766341637336955924836847768457039175660279784295612167899455618405343686908907695358239088351870495830739180518509859269437015797489301844593920484927630172344269378248455657186218762679357609204333669024237648538465053048724383898528808961206696787294681884412485427843796696788390072124570957047672341581447744981862017791206857428430183366004980966398716823512288330174863890117558744630102020144500158878244146399686532935435591262767487823942606452349972401012308378888947381934278131785907155692007064636085000405504866631011593239041758448995933095907216863744502344014999804306234830774259496097549717476344048');
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INSERT INTO num_exp_add VALUES (1,3,'-60302029489319384367663884408085672236.83687099063256754698860828386302509843815398979402006244388708674093244201278399438376682321121138429850885935540924586964982855913223221441591310211730902799041126800414795030815514254713522692405212716783388698431088814919226444677188004928663343696636297536500970117716818423689175692808344185016908913828066250587407384563498516598672584120143890364303296142744031320345312431817858545326010704685255237541162931904446804064783391570102958474141459619245240874849766946530000977144965');
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INSERT INTO num_exp_sub VALUES (1,3,'60302029489319384367663884408085842723.62767149018508907178556555587874475318127115521321786273614780129829831438626014991843514783028586066905089122532715288580534070605779007112619958852628801540288008918482404759132944298520148080184250697297150817299173701934285646867489426483932830299434150464278537812298564822479785688909850915447762856384542090714278516461905872647123125352735037721325154184406043613668806975385533851732090363979459292404685190942209855935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (1,3,'-5140349743195574373979577554212527512597024.162480344833040409158673429491690439298506850052285119390701002577176786023622062742050099464897084793357329597395417632908812044304066963549928478520702505283307379218587635434673128958824348493758429380623577527186462464399974242800361134191519694694139153279582776168995426125926314513926640766117733774558011741611075336271613675760116784769700605008122422944290652448956922432960815546502965310676913079866511016221573557684245901002643719965652152439520727383305120298495304784052489867651462175349450610643411043707261107569691076730261762793560088893354750383257372118118753366377402045596735023445172252225346164608897913115394905485106225627590643805003075069931177395059698550161546962768768895596088478488887530518018212441345360153523733317120037436403475909117998647781920105313938836144009539683');
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INSERT INTO num_exp_mul VALUES (2,0,'0');
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INSERT INTO num_exp_div VALUES (2,0,'NaN');
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INSERT INTO num_exp_sub VALUES (2,2,'0');
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INSERT INTO num_exp_sub VALUES (2,9,'-994877526002806872754342203612721.39038050457374613143278241259478942521582284121765030681448507149813723390800786083916642678676237719134679789066681148658045087323654637787610377226547625566084597844703238942080799221554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_add VALUES (3,0,'-60302029489319384367663884408085757480.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,0,'-60302029489319384367663884408085757480.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,0,'0');
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INSERT INTO num_exp_div VALUES (3,0,'NaN');
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INSERT INTO num_exp_add VALUES (3,1,'-60302029489319384367663884408085672236.83687099063256754698860828386302509843815398979402006244388708674093244201278399438376682321121138429850885935540924586964982855913223221441591310211730902799041126800414795030815514254713522692405212716783388698431088814919226444677188004928663343696636297536500970117716818423689175692808344185016908913828066250587407384563498516598672584120143890364303296142744031320345312431817858545326010704685255237541162931904446804064783391570102958474141459619245240874849766946530000977144965');
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INSERT INTO num_exp_sub VALUES (3,1,'-60302029489319384367663884408085842723.62767149018508907178556555587874475318127115521321786273614780129829831438626014991843514783028586066905089122532715288580534070605779007112619958852628801540288008918482404759132944298520148080184250697297150817299173701934285646867489426483932830299434150464278537812298564822479785688909850915447762856384542090714278516461905872647123125352735037721325154184406043613668806975385533851732090363979459292404685190942209855935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (3,1,'-5140349743195574373979577554212527512597024.162480344833040409158673429491690439298506850052285119390701002577176786023622062742050099464897084793357329597395417632908812044304066963549928478520702505283307379218587635434673128958824348493758429380623577527186462464399974242800361134191519694694139153279582776168995426125926314513926640766117733774558011741611075336271613675760116784769700605008122422944290652448956922432960815546502965310676913079866511016221573557684245901002643719965652152439520727383305120298495304784052489867651462175349450610643411043707261107569691076730261762793560088893354750383257372118118753366377402045596735023445172252225346164608897913115394905485106225627590643805003075069931177395059698550161546962768768895596088478488887530518018212441345360153523733317120037436403475909117998647781920105313938836144009539683');
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INSERT INTO num_exp_div VALUES (3,1,'-707409990019504668223608170643582.082425157530076679823177950190511141917761066423266390864536360056345386873500583953954967225431526056199231768143978526582904071798714789552447782850723926323452633811653766838064983821149041415149067433978085927687765773012158659685363079191901396502099956189371719135315616249471739677995520904113581848295732911534266040260836644379296158092198514963023001686666281725991605685524015227112003429486755206848316731257322742428352116058878710728614841247581716185886403744830796740424927494009978599974431617064012221450054532987372285996679180090592706458366967534834069977644215413076082570497451654516268857039718730203921980307096740864747006176117071983875364434497517026142488015705391255750729200497229031250705777282987863242056223584453312226818451807347197583925624299372040413470456696588043062815');
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INSERT INTO num_exp_add VALUES (3,2,'-60303024366845387174536638750234506721.2758014749274942132576365116182462208228193753118527959000939070820507877345194783035668195137119648748792386548310474079340204536236936213411512867171486174240518914767934028451971067161683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,2,'-60301034611793381560791130065937008239.1887410058901624055165373281235236307966057696953851292799409809571799686645246659986351515277852800926805119259053513475211488115663286642009614039264484259692394657121785950542874788161683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,2,'59993133911282372667149627097418449223835595194300848703012380022306762.154418449236691515146061305380465061074531890529497774836941002526095632166401249277270674802626154774328055399254982998368191676630276960361274433270795772477146870294928855773172789856196219950097157391050424577381777627004101100872747943673762087675405200265837631665464736842180920496158545887039337399558993437594084473932658319914390365451919627956823980800124880375978662052111797881386060353490432427832058851094210488804887183034572364751639107535041308434932952695103493677600969712634416241541391613699710826602011076372592299807609658979777598672141389319098817824624950794758296679318319299142035');
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INSERT INTO num_exp_div VALUES (3,2,'60612.515523995516156897729403721504966784736064970538891936016753206905080265887046037910122269129293912171105589512464185386239562077778499936203155976336284324712221812806801062157592930664021782540155687632208890794166119782594464410498356083266087045927038416810562596141871858142749062925965665039981381277808608946877852933015970874447235220989360704166270479475802673572039541121473138382812420076284458769543418652217394352637294823914346726065145538710933281768776286965107974980550163605068693568717671571780028113969794125200592691656568731359981803586296135840575095063824258761205175762907549288801963550628589530419118771779395037240198270853609924445368393952404606326559485235840170339343865253618184271158932135392539396160392488927771488269959497352568205940636180870805982484030168838833607478593');
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INSERT INTO num_exp_add VALUES (3,3,'-120604058978638768735327768816171514960.4645424808176566187741738397417698516194251450072379251800348880392307563990441443022019710414972449675597505807363987554551692651900222855421126906435970433932913571889719978994845855323367077258946341408053951573026251685351209154467743141259617399607044800077950793001538324616896138171819510046467177021260834130168590102540438924579570947287892808562845032715007493401411940720339239705810106866471452994584812284665666');
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INSERT INTO num_exp_sub VALUES (3,3,'0');
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INSERT INTO num_exp_mul VALUES (3,3,'3636334760530744652235488357607657374520053530993537920755375319352615385278.023608692512217812784472508939511216316773023870624171279878340621219698109986095090336065266376220109007718694455520948311677863167090936408887147442375455695868593092154861636486745490748828207939155392396090682312136290864359484540126174821846208064763823279315343506148025281475729723686566174395516982893064510403581479746673749128344955124070957545815390178764940816628194640888255387443237798761377617383817511745005525149990207764725040109364671749403389999498572538135588695345112358160274671918953118753964073105250116426665508214894805722798842017943220605600452911496071424281587802689830031742105619630787641205011894680546049982654601956546154572720177337696285354350903475239411654436042931409507429892682706228354459580412759920815932840348933425754970917910500027837428631661182510071352138858');
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INSERT INTO num_exp_div VALUES (3,3,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
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INSERT INTO num_exp_add VALUES (3,4,'-60302029489314054989387940744763542234.98295358053252401308872309802346144227050959966671157134780970446370197110016237152333448347415674483796371931316021552756816073493808344537122580089676304958104270609762310229182150728136567294798680824019082599362332377530165818229609055765904048195574142709698758095302560470195171027219786996322461803443213101532716728918363951912367135900414238535625075942525108530051828834829820554490477645701692374399416239080329365045332525699055300921341010989742896430768506909949340276549373661076950964959025967328861569387160956730002517417236732463510495205173523163676450203614971844583064927040066684531931069310935516821795449174271052747559395296525950219449541557191520903507653089998307641491381797101485104546410643');
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INSERT INTO num_exp_sub VALUES (3,4,'-60302029489324713745939828071407972725.48158890028513260568545074171830840934891554534052635383222518357552878529888177277886748756734050012959603126757618322788700853025193884017088688974683399381224865109134889560766307825097103477790782590061456916367930139323346273315068375646692125800496305291080749834712822775973790354498408104142209966769395239768969172107040437333428573572464689550003374384624966403962290572373571842567623422963022155546431883766327294954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (3,4,'-321372325955692885069615337209737469749246561535004445508427591.072860243358366933071485495726715620133686420023451450292996945184959542770492705998350644739298629407567812798540119555932604687814429669592481327761428042980782672136901602006622227365754036664912989085940235439697789102358431343119457114603363936544931303133371137532006899162833369543279729021228901466728220729625107362063321334489394782322741444425117731922691457341543446841167138481424319752111748042440994701571955325673470021626946676976482516292402239416632497972073915818846704053624707839813514171497746804751780741682011937606462260710753056669269928580460921188286249923152921382198282201761171043384698319895970192114563900025573490442674225227682235790590616707857188385274186584856872573669591460447105688151281208238908470285147895678001948902280493477604361481216667716971590499226735103039');
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INSERT INTO num_exp_add VALUES (3,6,'-60302029489319384367663884408085757480.1853341682137571584926062805631087054017160819890685789064777236456590745415460695320768374693076860837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,6,'-60302029489319384367663884408085757480.2792083126038994602815675591786611462177090630181693462735571643935716818574980747701251335721895588837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,6,'-2830400711649493468815157129316992649.40542786074520931471973065281957756940496588853021620372179463538053123396140685749478530925306163968207226329985017644835203709485594362663495728106061878665324856417118064730721101615473194292620972173690618491026470353143141125614124440035267592258385099934706896692953497971326605145704135723011753705907329979207428661473172503098296622281647255008204864404416199384701720347319806375450632245634238172654086373193251877533131784268854289406126119630708578053354762596511353053106459297339360827562281168219966099848212');
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INSERT INTO num_exp_div VALUES (3,6,'-1284742031601444539630782308463065726620.121021225455596762466053504195700643301310745151565435123335541550963124666304408503436412726848834604336377169205828654564329888653766451656774534718709065521243637375270687684572524302099749018591530352756390467862377335526634920857924031482455373589053524922608255779040656019538392173139295812160325688504210040741075388404155144782519528791757450256668977268409265390016721724966592135644698341754332845002439113523127047593325646484654291494607100188094186116001064043796216982681807318598789324900462932294782971663150070521334398542559480877366424630693734132836518604260869235580641521264976411493166969530737254118968281271908306432918913600567757535151861421384835424322504855607676315840963696944683182767935565256136130185809101891760917733694553800748568697830680328155128016670099315391685422333');
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INSERT INTO num_exp_add VALUES (3,7,'-60302029489319384368482818948157603222.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,7,'-60302029489319384366844949868013911738.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,7,'49383414785234649002982046297226894664526726187218771083.0993243619030008310875293647868815940421844461627295157812843657782639833900543200310573708100000958929315945039020410482966753145208427035917753919085618457760620513481628641658765820294863970581642745379331727722585319163262763708386199720411053619449096019862596221607526610103408936214184850115071874430846697061554769773328338028749631552202705583855831155461651414320570061181212214810086436100771547030013079997847086');
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INSERT INTO num_exp_div VALUES (3,7,'73634737013325927185.787791148221519354461791539553527545166847382784629235192342551464898036004011575416717008403527685470842765455409054592207142526523023201841973047779202013398235864494503216973882479116841765663948294836180515686647139678530220909072497288527276378202532400736141014848907023234659020093073127450778982904578906877634654521825977382116752537063128793631412296206704078569268566614023846282524151679028060869175439188773864994186109445961525301841201265289707928211114515861536069733921800160245586536759625418951427346236213019358749196674633237197452976517130405065120577692737021174118093373953642724512531935525024447977867020930500433287279183436509990047372809400167546185096048971157700858970777301410692908939206693154161335335755844997198191427289546263182822280127912118140820265025555165337881999926');
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INSERT INTO num_exp_add VALUES (3,8,'-60302029489319384367663884399588771256.5916339968771732477072012126949734214868901845505193155307646111690097978112797961939995859130827784737422228762767014427842766445950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,8,'-60302029489319384367663884416582743703.8729084839404833710669726270467964301325349604567186096492702768702209585877643481082023851284144664938175277044596973126708926205950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,8,'-512385513828318260570283740065493064477880918352.732624553690077857674083796435724202494963885926573907185100543184828131859183999195040110586155435203949963570735841632689374488877298209082579317039061893012560130258753218955057387206477423088065663401594359617882154814262843273526859406265633827109554791772242178864873774889091687515990672487380368975556580539271333144212685871370972163560839446696514092637412587953506052848750866803569213269271165856310101244342151576488190595936869490659700946174362872797854591188391982770203203644172999264143929484089237665313698600170041324566984832357000400');
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INSERT INTO num_exp_div VALUES (3,8,'-7096872691348467943606706217.907270287823269424282176534343841939501231816905820949045946136373255017076943323578903040918266385724756894003692978391468202345397178445216069294845721607024056189567609414049207292919519881725733381453217071918292453682942046440563446278374996563501512335133749731529362537349288419883140401056747081065947774593869673146309163791076953204291951821124894409171722911526435445719071769008713367057971351892550570642991097981458696464929009464411568672010548002196406312721789582428747564855324072212842315229302959908665089850886951261233852165624100634055045684536311382452553544676139507899503993644452161529145849579200003677255968757773363970434791501820320494192909660871475590637419913907191608957830524390049664686282439567943053924245852983990958276537000732363895444894582579142752920882750130052682');
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INSERT INTO num_exp_add VALUES (3,9,'-60302029489319384367663884408030893999.8854209703537480818248540990234567956069965340942024890856088355839135538265116174644003927269495876835324407641642359213535695803871472434650475144516723617632059130297610134243891145006222068960999879308472500422640481972089756410157246974765071949782242392661524488959954348903412713930092273629207697480131360047867213863018127928853922173643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (3,9,'-60302029489319384367663884408140620960.5791215104639085369493197407183130560124286109130354360944260524553172025725325268378015783145476572840273098165721628341015996848028750420770651761919246816300854441592109844750954710317145008297946462099581451150385769713261452744310496166494545449824802407416426304041583975713483424241727236417259479541129474082301376239522310995725648773643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (3,9,'-3308379209762459471107480259839508279070920437.883503980178028214343751083865562028455061662673132221930429904398963590401793045470444301883103141901787466923883803951815572606105617157736442670792467625964359169270739534412932791178258858918086886061702512427989129732248215348301444245772127142869263635282888226326427510486246184233225114523636171202034558843515894542952126988613018789833835507734620046994907453602573865012044120483116345444810078666601100257620969379968264504287700045822481492526688635364586344704730579892342786173395802035361824932075736340405960099542224953439044947229246847140957298841482874444906129049023002897135347878048572628834749795298712449864571996898774444932083319581439741625832405434317985988163261591679157437224404970927012111196724239860528859217322132733404472897289');
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INSERT INTO num_exp_div VALUES (3,9,'-1099128766678422054524173986658.839339966689456265703816212189145237878729886466041806078542573981227645802109969871638687985985845489422516004202630099080709709893022100481258818112345013009059633421290241583864468453396484606925071369550998772875840640325758308835852391176503689677263605949075815552026731067384737231681068134099746550363063940273625924224721503126912810251607546172009765059506591787282558727077669973711491157840340631805422942099954647016059576777054339588421998882440726473698513560202030309804089250300097589174314677765341104767702983421063649104691583044460507666600260994707192787133590502137391691330098102374713996115782701417107878938473243874299874872852713499024851414757892169376458916467621226859152075901273014182163212783658933754507272478777304254191033562324994395916168496097385872331012258027431094381');
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INSERT INTO num_exp_add VALUES (4,0,'5329378275943663322215245.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_sub VALUES (4,0,'5329378275943663322215245.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,0,'0');
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INSERT INTO num_exp_div VALUES (4,0,'NaN');
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INSERT INTO num_exp_add VALUES (4,1,'5329378275943663322300488.64471790965256505869684245785528331091076155554650629138833809683459634328609777839510066435612911583108717191216693735823717997111970662575497378762952496582183738308720094529950793570383580785385569873278068217936841324404119828637880370718028782103860007754579779716996004352284614661690063919125301052941328989181561787543541920734755989452320799185700078241880935083616978140555713297241612718277766918005268951861880490889884082730841740604517529391011862694381726143520658746305661338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_sub VALUES (4,1,'5329378275943663322130001.85391741010004353389988518583956365616764439012730849109607738227723047091262162286043233973705463946054514004224903034208166782419414876904468730122054597840936856190652484801633363526576955397606531892764306099068756437389060626447578949162759295501062154826802212022414257953494004665588557188694447110384853149054690655645134564686305448219729651828678220200218922790293483596988037990835533058983562863141746692824117439019450865871047657552800448629502344444081260036580660700595591338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,1,'454294299613767152878025320780.534199313974295807138790763501115780294529340799108297697573066187975311338382917022391830256203305238757334106943821060545424417350991354829668286194840925251162479496893943917530660694097932059166013476064988623431110002057735318529554555260199417935495388243829261809007709919225000608711536928171687251088217591210419208480251102484043683131687013687838713055660405381318396419588727500715930145098362997142075433472039319292466570912777345841400769387321465602989947078951135489852486382469990409873227894248208197179481868230244584527040573428134962626267135732247029762468417273891700661832893497067151409134724061246612631376075173287264787886064622106855886785805818642123776489793586531950438285720668411465570116161790343538663297713926678759640594912243360541590368666922379919514826022141331900181');
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INSERT INTO num_exp_div VALUES (4,1,'62519544780217042176.800424689664850775296526267109332647921183817056683200043718160298562843864918741523494444361916531159341418970534833628106062976341639276761669219281771109561175175033739624472497927501467465456946098280878993371659461957361369508794842102784763955539708800574418468150309301129490186416766691183270872711413796386178009615777589066235359283212636467980113350635181915492452697347977967985810294150853782607014649150457138118264698071689065469752702524632313088938504181640435324554007553994564705401249228914199354821595855823113730697333390936834057091883654016371107974899726642500486005445063301647520527084320363513388355471718583708935211830796440056542408492723718088396437530207347815505844074508948817594746824098278470533148171941442049323578854023683167934569551595335539887777638716651319134577441');
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INSERT INTO num_exp_add VALUES (4,2,'-994877520673428596810678826533995.79421257464236160757218576989993781147390382997132644206786872350652200243563770552469933194637146474528320738725486418004701192337175478117026439697031462361180324038544450723753402846519731908503949116978812841497201119103409772457270340059605961197538918709309004130294868847110690336360689446090125918336908930881873778405661757289469281163974774492810850778950071063044769131228124355961427111369335109426492177657001035045332525699055300921341010989742896430768506909949340276549373661076950964959025967328861569387160956730002517417236732463510495205173523163676450203614971844583064927040066684531931069310935516821795449174271052747559395296525950219449541557191520903507653089998307641491381797101485104546410643');
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INSERT INTO num_exp_sub VALUES (4,2,'994877531332185148698005470964486.29284789439497020016891341359478477855230977564514122455228420261834881663435710678023233603955522003691551934167083188036585971868561017596992548582038556784300918537917030055337559943480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,2,'-5302078674303935968062773235453828254014583744527466365136.236414807326868572353809920518232561005161225922028750078608989965741402418802255050636954800114792425419735155504035469350521800895164087027043476055514245942961100610551646034472084954313670284875310691807937254054948742125729353864014122131419164449567115006621212424805182687707372956385102095255735458593389920872596796806885847543910224476727171570873698525606016990229936284811067826588349092841322512643043008589065847223683467371925773023109720951609815041012521485326120380123169545818055967455575736140138663815073081494226676896278654189873597341203197903408668523514375373841493189836809506003729379742035629498519683885268256481104619815130659628225053833297766479068686119691010593208135616363994230674606991733148502293102108193522604968743948323130517040609601859735899914987426089053869350663');
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INSERT INTO num_exp_div VALUES (4,2,'-.000000005356818439105666775800262590702859770599410113087721172791624002387236505438218124867814437523686300450045582100868990117124343222534568799037421944272316277130975314766456260710406160143182498931595199129228915695802952695510723443157825968340043198200740606202264287904755124946591110599335909404657109057432686191440989434662797205973563889238804413861126260401987949920244286377128599413927273444061572120561496904543200956508673923547626768641271397088562966176629018606103663605145666976048261236691866387601532424530473754175270500777679603569715192364542901360534980926452487443629100484491344001509360344122933911316486556042277769848194790964257060927912344609376571637126617813506411190014141992988288983968823792971270853369317867326071952900448455162898476163801382836761898292684175721846');
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INSERT INTO num_exp_add VALUES (4,3,'-60302029489314054989387940744763542234.98295358053252401308872309802346144227050959966671157134780970446370197110016237152333448347415674483796371931316021552756816073493808344537122580089676304958104270609762310229182150728136567294798680824019082599362332377530165818229609055765904048195574142709698758095302560470195171027219786996322461803443213101532716728918363951912367135900414238535625075942525108530051828834829820554490477645701692374399416239080329365045332525699055300921341010989742896430768506909949340276549373661076950964959025967328861569387160956730002517417236732463510495205173523163676450203614971844583064927040066684531931069310935516821795449174271052747559395296525950219449541557191520903507653089998307641491381797101485104546410643');
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INSERT INTO num_exp_mul VALUES (4,3,'-321372325955692885069615337209737469749246561535004445508427591.072860243358366933071485495726715620133686420023451450292996945184959542770492705998350644739298629407567812798540119555932604687814429669592481327761428042980782672136901602006622227365754036664912989085940235439697789102358431343119457114603363936544931303133371137532006899162833369543279729021228901466728220729625107362063321334489394782322741444425117731922691457341543446841167138481424319752111748042440994701571955325673470021626946676976482516292402239416632497972073915818846704053624707839813514171497746804751780741682011937606462260710753056669269928580460921188286249923152921382198282201761171043384698319895970192114563900025573490442674225227682235790590616707857188385274186584856872573669591460447105688151281208238908470285147895678001948902280493477604361481216667716971590499226735103039');
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INSERT INTO num_exp_sub VALUES (4,4,'0');
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INSERT INTO num_exp_mul VALUES (4,4,'28402272808100253242547006276715304015308580784958.804614276533085644370816876160290159450291717634111299841065255625515058118012211808741402904995080624675460593676923639082981788732031193774047612589113654423166826140872334380708795266307037944059108148612979119729408762532396036043629484049508789880964586236575769826806092391573178899640321403656891487586452524427223891405519836671312830183895761747460911777623703557946796784873885800089025388390522992806365773290733075927321101736155663727528284512100509273076328103465333687228713897893434161293693971954442699482857938492961830350598789444266860160794913830991304996676299650460125000959751177037694425217989910261807246272771711816326991282202653917488360776928533800529297474279497910326579608191975246060946079639658615178160271122713225105861574160788280907842327681375920919676063500116492292319');
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INSERT INTO num_exp_div VALUES (4,4,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
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INSERT INTO num_exp_add VALUES (4,5,'5329378275943662669459614.81475694159581596077242547133292502869630735172901157043010370467618244548786897684821457816189831652076071977025794948484549600736179389638319303817478693948215387894509009504287664213474693208847025374388286162907794727810231557001266897729978691844410171412189947386181530441402903608214502713480332746271552746231631136145916685939539173054989927058122097304419584979598595477177513004218594211597809300517607260841648610322863666300637648662611916496850248528515936635845594390453288113296413254893687029540384176335735114863908372780241463999450547422213639667099644505472777149095004849805371205203850993689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_sub VALUES (4,5,'5329378275943663974970875.68387837815679263182430217236192193838209859394480321205431177443564436871085042440731842593128543877087159218415801821547335178795206149841646805067528400474905206604863569827296492883485842974145076391654088154097803033982948898084192422150809385760511991169192044353228731864375715719064118394339415417054629392004621307042759799481522264617060523956256201137680272894311866260366238283858551565663520480629408383844349319586471282301251749494706061523663958609947049544255725056447964564549684815188261035801892684889942971676086592385285071073528462167439314005547455087297279161738865296114495425732286867689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,5,'-3478781676337858247983014311182511.567538638808357215203593479841446379226774481291286361639429856698999485760647422501864626078375852610019829111004807806660731243672830787729048847342063218718651165150612717759770504648306347926061960607388621011846314969634048226452709389995594961695723139571002939804473057725442880410434039783304583526414509590532906062732322732569475349107437896717416548237633532805602064623969799081086996320156575550896200848758685986331692388099427314008504506503745527468550106879602399030419569897808150076298414568875477195447656904373310322813412927463518325927626891046356679526447117311923853482118502868148386882363449163182892615259995945992014431502761210899772725227648729095696228388558331052524469604046072203605897109629560683446827492904111565278516043939137760721315953500281379039771826554155511347152');
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INSERT INTO num_exp_sub VALUES (4,6,'5329378275943663322215245.20238058768123314540388318253964726313120648232235700755866801918195710344138369800874235399515094124581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,6,'250145412892811547138949.592621291590152419206270097656346630226508074074623894951308487425470437268130465956063593951784820669318897182831355375451719125809800516979013437732298382708070979871283132689492336823087794373113039154669229889503700598930220858275174342776478898670277868700384853696009897221747924643343353942154528501454689084608965009561564638167714973711022212547096732831847202912862290958304510651828842182545311077713664465815992616213663619529378061133917572474298028065850515876361609671565914027186063801852554353160801534696062207299890867876199323530337336273950892723090754719547285920090419070001019943385293110663922226230169381423410428577990604776655422105400452217085311617728003688836185608912367677734364834577573255789160419371322775733777518997638403409000055707558465286469808848200141192627396502735');
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INSERT INTO num_exp_sub VALUES (4,7,'5329379094878203394060987.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_mul VALUES (4,7,'-4364411947278810125327066890819882483326918.05664098958260550284395870948992407314161088028674246708928421994893923699743452802989464864039994566042797942433140378990308345483670828497915478397481687305406460330009319949623844175096007381662809083363069100235985794575399268709260901964834244796150883807308976949196661411035264619638771824190014274817662519438658481432363824187693821267613212631153175155634316128036152465184903927860719447693468054624663668062006049759837326188252927823612718163916100588143128358998656306593393889422386501730237442526450419990376323903182669190482615734972147533221144682538647497701130447816148459762464395194383090936159579764712919396391813914821973715879062992249315474841639591907249142779103650773383644785606333916967894');
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INSERT INTO num_exp_add VALUES (5,0,'-652755630.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_sub VALUES (5,0,'-652755630.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_mul VALUES (5,0,'0');
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INSERT INTO num_exp_div VALUES (5,0,'NaN');
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INSERT INTO num_exp_add VALUES (5,1,'-652670387.03916046850422757312745971450663862747133703839829692066597367760104802542475264601221776157515632293978442027199108085723617181683235487266149426304575903892721468296143475297345699313102262188759506518376019936160961709578829069446312051432780603656651983414612264636232727512091101057374054475214114364113300402823059519499217878746766275164739724770556122895799337810694888119810524986616938847385753562624139431982468828696587199570410008890188532132652095915565323400735066310142303225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_sub VALUES (5,1,'-652840873.82996096805674909792441698652235828221445420381749472095823439215841389779822880154688608619423079931032645214190898787339168396375791272937178074945473802633968350414211085025663129356908887576538544498889782055029046596593888271636613472988050090259449836342389832330814473910881711053475561205644968306669776242949930651397625234795216816397330872127577980937461350104018382663378200293023018506679957617487661691020231880567020416430204091941905612894161614165865789507675064355852373225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_mul VALUES (5,1,'-55643106304872.575994253221940844841058071061962511162776681458310912066379595519265546225338405882027547140476045378015935579066580347282075024392379464189067155567624835346798806677988850250198082355055954078446421075165109896091047534711081616362392995575466807084807876544560268050611445006601394735810211678919646667455478469014906335433468365011768049600750224822391684377238242162320161552720449713229523135506671063115436813348612986916614320012995541575293478341408982118538094438068036422562665160411591652618670802973618768526197813319204816293073794413317669922144705633308090832805914096147659820167569140291210526520361556881576175809360614782817717579318298657744021133210954279487777567785280633309576696708168342539425395482429923273623865667723482418178781573723597156804085501875735112311466228778929147929');
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INSERT INTO num_exp_div VALUES (5,1,'-7657.550797567691019915353529993301413746369700087741672762343206271266232635965032053368224472333368713006346867984576168784127503674579531243603836945595880917241997606783133673324236134063757452734295148763280059050480246827193380861494669624151921824660313516974440913733511526807313019192263170823268678149435664224184903925632177789052038092611394447709922076676981043877747276056677801802695466205531230350209787298926245402046182150996849906836743231861317120171583577624262765589605263477198809166390259128339127005924586833372241946051704497188891325715185091060185547236923494393813210904033520844572880475265306843414506359253445517738473745552980984097762509546161690823646176501838559393690565709795724159196133663168004773260451322595899506776323262195323943138344537866088159583331807728944620284996');
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INSERT INTO num_exp_add VALUES (5,2,'-994877526002806872754342801504871.47809095279915423939648794226185974985600242391612965412218049794216637114648812993201775787765690351615479957141288239552036371132381627958673244764559862836085530643408020551049895730005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_sub VALUES (5,2,'994877526002806872754341495993610.60896951623817756834461124123286284017021118170033801249797242818270444792350668237291391010826978126604392715751281366489250793073354867755345743514510156309395711933053460228041067059994425117350974491367099004404995846913641329458537237789584653041949090121498951516476399288513593944575192159570458664608461677113504914551578443229008454218964701550932948083369656042643364608405637360180021322967144409944099438498649645368196191999692949583952927486593144959284443545794934667002661774373364219852712996869245745722896071593910890197478196462961042627387162830776094709087748993678069776845437889735782063');
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INSERT INTO num_exp_mul VALUES (5,2,'649411906691138274293985410502516861224852.2323455192714410716272307781034189160865613770320102043319541634113746032638191509585045862973333645830298922352816245477556264222094036953195419857712804755170632292914187367964994214922001758104594052499795564860466055599417895782179851297585155129541589802249540436678824225950907268084876110445460948679383611117263673106597132046331719468816839434908155684738864149955129235751738204036443603521478609787295079710078973503970964790273461142497259987849074597264522099648376356902360358310245001183020992360260836105404118742418040965190000718736837422434593694808973939805954329718232693154128543253581495885789333274488461716809104532693754070810202831113003978085636579574171344721710232931261731022478029314435363413498991740750878099825781577297965642009156858479681236085226911858782115');
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INSERT INTO num_exp_sub VALUES (5,3,'60302029489319384367663884407433001849.79771052212833997386114856935638647096681695139572314177791340913988441658803134837154906163605506135872443908341816501241365674229987734175441883907154998906319658504271319733469814941611260503645706198407368762270127105340397375230875953495882740039984314121888705481484090911598074635434289709802794549714765847764347865064280637851906308955404165593747173246944693509650424312007333558709071857299501674917023499921977975368196191999692949583952927486593144959284443545794934667002661774373364219852712996869245745722896071593910890197478196462961042627387162830776094709087748993678069776845437889735782063');
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INSERT INTO num_exp_mul VALUES (5,3,'39362489275784146262776411377472433635883331946.794473520543457442955620133347015506556162839462623905489255080102447195050109095701660164272430316804466254467810714209179752718730906325952685817112992943656292503112803950215110778476301809440329937774061163668461957943313261962261081942055908935814323069621279128270849852239727888939033546870208376394878842958202403235309372240005941467570230067124830916866857395233038346727879951123599893174252558078732888910139309038957525961212820831321973219557165558911222848692996406741318948607549825343491479728117062814094258484536263158005174429922237853707635743736923521032098496725445243775790161216159399180889906705265012270270348146530113428221072591696851818281866095288773371414866822270689959827332258348570976075184933893434327278299820594014788148344260948638847457822697682605612771344335201258128');
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INSERT INTO num_exp_div VALUES (5,3,'.000000000000000000000000000010824770508763323320533297369674519056450544793568147911931789010432012750062661590994728968589403602468229106206242395792957238667714358401601098858606386995096923432407249369639633268143022787987190106724545750803196130511146323174462918572423414631798141263222875752767731279138952850500369328934959764805948568471324562210715908420467881411844098258193571194910997918428786213948547748701831331312040839544355427357749520227124858111324859160114175254197992204974033767300989488517391063188153561391320190653403747521648794370679322504188364455328709488846777004202196382575648619395139553279192346251133156445942281048959845827006761160755031086836046398020850814350246219929303018051720203943879538087954853996826539712240458022307680912400297508925714946398031304516583939283');
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INSERT INTO num_exp_sub VALUES (5,5,'0');
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INSERT INTO num_exp_sub VALUES (5,6,'-652755630.48149779047555948642041898982227467525089211162244620449564375525368726526946672639857607193613449752505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_mul VALUES (5,6,'-30638438.151446159804025029882398388155309149089870990062944469684482366692824338098201222171115395923414887930224163525189097571163687285244255335505387733673499447610577050114902372990462064696637481657064525319516004273769831260452832960893174173254560250804003884280384718123289136453955482855362019158401218620018346500189769819687260476334734259702665316562988639223597110627626759216850014150105605927773639897638043177685498804811787888811168524202700283461266793154726325540776914500415140842975457394524215869103737379109516024460317825645645301237375972914247141703084877141866316168268901439172491577729880760950895760711857112463508064820414904611059588717092145484656103798852859978690742216940980929562068');
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INSERT INTO num_exp_add VALUES (5,7,'-818934540724601372.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_div VALUES (5,7,'.000000000797079129642393611556079160915147221153735075943759104977169600937534508973732991117540626046659124172765761873705978811124901421049332579161931652390647472911517923131800238903184679028518657818755558526885018755394697157094867449047655737107085020874974955627907737126958129710597811740696534189608639914753884882702680512272194316887744972931453458445314561564591875764930680945589486999586667912816485821717403892703364322658245615895415781719033810595358092343690359557942948213374234065052300866661453767599465059289920067095083062096458980564265691295895672503728815182981118876144075942348853666085714846210822847053889733510154276933759200630639642310562242207518883342516103725757482864105340008709446643820864294556778969997115586027866760708448174502158738150605938364482719960251612464993');
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INSERT INTO num_exp_mul VALUES (5,8,'-5546455599206321494.0676583421119904300307105296377723816472192007866147764761501865875232824814135783697976183493106885436876081315217834621720906478074798596116645640251460842350553806256223963023430631066024389364515688765194373161385579258482225808660340732705687558150699172147896486727530192499184101617379930846663835628510376484675411350654979679181852179924386290069790336316958202582966248703889464308649631486542724072047294216362186036638115240070658004553260251510288423749333873893917690832829128021808383128393431810674177390352413548658782609064839524756041501835115152819802758773711821322162752064589750295542985780512921839490040396053737870038534216948323935020460307350020911362024271167085905714873548388570602799432705061561572854498075600');
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INSERT INTO num_exp_add VALUES (5,9,'-597892150.08771044822540810796370552966707032464017958269847934730769542644402913723848026909285133109089452632480800168074607090893991283808726990171062867538012237270000932798704781608969096508450960185964292594677356241956277714380500188870696516251767979457838109804726539408115452577436052503866633026489282425086547752714324273565900641436632912781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_mul VALUES (5,9,'-35812445701642379.972368737320206275515144213236752803936806738624588812089615098329765811617509505790110909629109400553415312470540217508070421816878544125783329593128638405659896184248784794258084116406472768709113030915308410565617764394827427154923321461158387012978726512246146545834669665093228316853342805604075936530371665576147966721599968786161939347726656168798065647411457701453987215491345496003650288850096338695703984042549594979897253521041581573388369367579323607093487743440894765114619634001789457486407909224339065748496715380572175183589195611952939575073075140094901024063428239223964510824958346570603142906309198033196987949067156046076497974760641964978711558209708743776024313916111738542765749928287600981397080809041007714387564206594515733287925008053261840295560398311905155157989225181164097547541');
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INSERT INTO num_exp_add VALUES (6,0,'.0469370721950711508944806393077762204079964905145503836835397203739563036579760026190241480514409364');
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INSERT INTO num_exp_sub VALUES (6,0,'.0469370721950711508944806393077762204079964905145503836835397203739563036579760026190241480514409364');
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INSERT INTO num_exp_mul VALUES (6,0,'0');
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INSERT INTO num_exp_div VALUES (6,0,'NaN');
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INSERT INTO num_exp_add VALUES (6,1,'85243.44233732197133191329295927531563604777955507322414928382967007765263923984471408038635831036097817458527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_sub VALUES (6,1,'-85243.34846317758118961150399799670008360696356209219504851646259063690472663252876207514831001425809630178527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
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INSERT INTO num_exp_mul VALUES (6,3,'-2830400711649493468815157129316992649.40542786074520931471973065281957756940496588853021620372179463538053123396140685749478530925306163968207226329985017644835203709485594362663495728106061878665324856417118064730721101615473194292620972173690618491026470353143141125614124440035267592258385099934706896692953497971326605145704135723011753705907329979207428661473172503098296622281647255008204864404416199384701720347319806375450632245634238172654086373193251877533131784268854289406126119630708578053354762596511353053106459297339360827562281168219966099848212');
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INSERT INTO num_exp_mul VALUES (7,4,'-4364411947278810125327066890819882483326918.05664098958260550284395870948992407314161088028674246708928421994893923699743452802989464864039994566042797942433140378990308345483670828497915478397481687305406460330009319949623844175096007381662809083363069100235985794575399268709260901964834244796150883807308976949196661411035264619638771824190014274817662519438658481432363824187693821267613212631153175155634316128036152465184903927860719447693468054624663668062006049759837326188252927823612718163916100588143128358998656306593393889422386501730237442526450419990376323903182669190482615734972147533221144682538647497701130447816148459762464395194383090936159579764712919396391813914821973715879062992249315474841639591907249142779103650773383644785606333916967894');
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INSERT INTO num_exp_mul VALUES (7,6,'-38438389630389612.0042045464692275627184627672063157323631169405883031379129843031477339360597564128205768842448328088');
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INSERT INTO num_exp_div VALUES (7,6,'-17447499423661151023.558342555162228919125358089491573318627107322332520978657843895009110781773496490472817700487707134216424855867015781267287628022535529641238372370292374146871103236048507252055787621394728096799222976387108688980537900309311204203302960751747509648304056939321473462375648710590981564101023812800603438271190184064874290215309040519813024962909469701968804925443161094255632624090623433640078421818321246597728308302979223833487133268472455479442002005374793705431817866798804822885690193667521606781156962792120052947767160957903073698536973292205899421787948529970837601521657406211962967291912148632072929662185840265855612193255596825032457033402506154930851214421895488796227471490998190312007513478459049382774782886773158311656817014322925167278223360446454868236479549745612973293185989975394307678926');
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INSERT INTO num_exp_add VALUES (7,7,'-1637869080143691484');
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INSERT INTO num_exp_sub VALUES (7,7,'0');
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INSERT INTO num_exp_mul VALUES (7,7,'670653780922685519356619170643530564');
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INSERT INTO num_exp_div VALUES (7,7,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
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INSERT INTO num_exp_add VALUES (7,8,'-818934531574859518.35936275646834493832011429282408849567717761204690035294074716714939441961175772404289860039233415598996234758590850206505669201200');
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INSERT INTO num_exp_sub VALUES (7,8,'-818934548568831965.64063724353165506167988570717591150432282238795309964705925283285060558038824227595710139960766584401003765241409149793494330798800');
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INSERT INTO num_exp_mul VALUES (7,8,'-6958475505053954666339703437.48985528725312694198056665033448258303533387675711770743843194274181580881296671866212320171337132096489224277825857521033238709600');
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INSERT INTO num_exp_add VALUES (7,9,'-818934540016982261.65314972994491977243776717915257186979728396159058352649559139156429817562698954531329940720620096519975256547379603654362598494779213610069399116912987384006656023443527501447464682173445385303315267086044455246361273561294141518329233754041352632499787199926225490924591851865949646448441825186059741089695009429827829188117479084665641367');
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INSERT INTO num_exp_mul VALUES (7,9,'-44929599044588573810654775.83678007633232843418115790847152455559258007804727916986432256198687661496804050903769496933400455947645400628259699874770581538122521805603947464462448454681701547899144129061961394870320463199545502030106801911915987309444301341575451240764927967432593181449618816978119423290767783843864768557371257918447461479570164065303599994081990686');
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INSERT INTO num_exp_mul VALUES (8,0,'0');
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INSERT INTO num_exp_div VALUES (8,0,'NaN');
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INSERT INTO num_exp_add VALUES (8,7,'-818934531574859518.35936275646834493832011429282408849567717761204690035294074716714939441961175772404289860039233415598996234758590850206505669201200');
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INSERT INTO num_exp_div VALUES (8,7,'-.000000010375659845651632013446652385870617923988120764298690164486716047614260682259722116360931978511176121353975789418625836899338225571166376573732227571704071000348895791547943896682585450808398324252224265156214259224488248639550967292466343168350213394398101712526534464002532408445204630441167137710565437434313424987517531891145368203998329086865151248833625645567863740298397742783405267970015165358620026813812552194344790169289440822038223606218360105618852154152168496637886434061050281055613760360200323363465925493033734895631921307644481639236601187225135325401868178006133838932915485272554505684060229409404902185944047523033315868230944723282246159741659387362889777495094736963530708159604929268812778894177095572578862150793098548829744006499229853198046828954650334595737117597239208825268');
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INSERT INTO num_exp_sub VALUES (8,8,'0');
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INSERT INTO num_exp_mul VALUES (8,8,'72198774884738777393.8687539247642452953425155400068591498151280875559609979248583367700231031634872342122563819478919600402159024059794279536786611373504966204744811722007869415559012475160471227957857756325962941799428857291371597146319816910515366298862558849452235442246081440000');
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INSERT INTO num_exp_div VALUES (8,8,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
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INSERT INTO num_exp_add VALUES (8,9,'8551849703.98748751358673528924211852802333963452553842636251612056366144128630740476125273064380199240146487881028508694029546139131732304020786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_mul VALUES (9,0,'0');
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INSERT INTO num_exp_div VALUES (9,0,'NaN');
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INSERT INTO num_exp_div VALUES (9,2,'-.000000000000000000000000055145964114074763360265614481666934002579974728749248345352023099030383962250681574081874554842623852433135871821620640200582985140388676650602814646133317791813938390695683843848260103199745295436998313216878337673674660966362155480524935736646623766057029148471463569162153009963312016563281545776175277904913263614668092319707343286073000287493274965714031678784835459999763925833141049057636632430975424499618419962303087175237320046300285962065818926167792812657620724550768858763098967149546312995222223400007044549870620849992226072041407997925405957501929449911416474388622107825120486594723448780503829317691081601820425151593487431389373265285594626753418140874747955925763163132984655078996173911578832035721963554569605730262976354029623260224710106409129114204296314733036');
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INSERT INTO num_exp_add VALUES (9,3,'-60302029489319384367663884408030893999.8854209703537480818248540990234567956069965340942024890856088355839135538265116174644003927269495876835324407641642359213535695803871472434650475144516723617632059130297610134243891145006222068960999879308472500422640481972089756410157246974765071949782242392661524488959954348903412713930092273629207697480131360047867213863018127928853922173643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_sub VALUES (9,3,'60302029489319384367663884408140620960.5791215104639085369493197407183130560124286109130354360944260524553172025725325268378015783145476572840273098165721628341015996848028750420770651761919246816300854441592109844750954710317145008297946462099581451150385769713261452744310496166494545449824802407416426304041583975713483424241727236417259479541129474082301376239522310995725648773643946404281422516357503746700705970360169619852905053433235726497292406142332833');
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INSERT INTO num_exp_mul VALUES (9,3,'-3308379209762459471107480259839508279070920437.883503980178028214343751083865562028455061662673132221930429904398963590401793045470444301883103141901787466923883803951815572606105617157736442670792467625964359169270739534412932791178258858918086886061702512427989129732248215348301444245772127142869263635282888226326427510486246184233225114523636171202034558843515894542952126988613018789833835507734620046994907453602573865012044120483116345444810078666601100257620969379968264504287700045822481492526688635364586344704730579892342786173395802035361824932075736340405960099542224953439044947229246847140957298841482874444906129049023002897135347878048572628834749795298712449864571996898774444932083319581439741625832405434317985988163261591679157437224404970927012111196724239860528859217322132733404472897289');
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INSERT INTO num_exp_div VALUES (9,3,'-.000000000000000000000000000000909811507365065002714756487495210579371808512079908127938523896001746219475805196061435010714649189975968123072269549018826343830061696154665503565341929634172463095299662727352635590451263034658630449260378893723785917860125051787451512267088404686342938118993621396641623525252649748977992770709930435013456855344203854749977414354164157192885125263071636468941596567220391082793700307461350484216679632552883058303710297475827456761138832914743429330069022439380297715971317819244718196187172770061156794130040674050533617155253444764036426045091327368023602807193742585178432544430741520636125146531502042579276206322507516332917325631822606079220413965396706334639331097621824106950192993127113903265025719013680733760540930122186345919977470628988674677630636632053583144327');
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INSERT INTO num_exp_add VALUES (9,4,'5329378275943663377078725.59616792993138452386059664269485161374191901124632386474661634799161523147237015531446709484039091244606359050341194730653343894986479159670583937529516163204904273806158788218327396375034882788180783796976731912141525319602448709213495905899041406302673881364465504945113279286939663215197485367850132991968081639290297033476859158044889351836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_sub VALUES (9,4,'-5329378275943663267351764.90246738982122406873613100099999535333648693442749091773779913112021158272634924594106590925279284284556872145100402039378540884544906379809382171355490931218216320693213791113256760721925653394811317969065642404864072442190731745871963413981746671302248281216916486794296983018838956112081135739969615171358100498945955409711817327376172085836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
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INSERT INTO num_exp_div VALUES (9,4,'.000000000000000010294536718194523982241053267404812827031741197656209184880073175960433631103885281961037127283726462743623757855378209281373475473018922090781553213750339001555832360656399849031527008437303091226051008068950896796359518673740801770866360774945096397034708173365378527676779736929035450380795854046109380272505550244458858231227568118355064007614608452292270378691774826689216790090661497154742954386244856792006376222923780801296832612827123778915598893970651480451509706836620045721191411824060983487064555397842027454385628620582036592315345973096405447742002746762099231557054678593446667904250189208490698468539396733604833688133512716508825505666644390119877423938820483653319376926639295680552194966870285838815705038244628263602997511842285889300557188773128635554621378148419364876651');
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INSERT INTO num_exp_sub VALUES (9,5,'707619110.78141098833556856308817117136192658504561165951731229431651264331543278598450117846625251667849259592530287073315399782168794294250299770032264633712037469256688885911649778714039732161560189579333758422588445749233730591792217152212229008169062714458263709952275557558931748845536759606982982654369800245696528893058665897330942472105350178781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
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INSERT INTO num_exp_div VALUES (9,5,'-.084049034261605466896663277055600903951276881294745183935726262038673990196778002490449355450474227878560465916800470848046625257516764244432096856845087412397406701521972651300484716852035267197801389708234913163750232707469240634303111868882057393120649919262424619226282082184091177505826009374043368623853156698509808569378758387708910629731005691079770517679511879694426434724918004419953301426679939010592502325130576915399009756468717124460489039474155719834555522581553817856854607844133431854471292027873672356863673617090151801474016666978499651970627896504709551656249007718965259502928591648533670568214972768900993459927860068104745163979267716597907297073374689384723943955361288974065531322408839914599555769945298758102515352082822617428033648130099822033393662643586331479103933840387663729387');
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INSERT INTO num_exp_add VALUES (9,6,'54863480.39378734225015137845671346015520435061071252892396685718794832880965812803098645730572474084523997120024743452620396345637401505220786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_sub VALUES (9,6,'54863480.29991319786000907666775218153965190979471954789486608982086888806174552071503445206767644474235809840024743452620396345637401505220786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
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INSERT INTO num_exp_div VALUES (9,6,'1168873084.346566233232746391559830634361431940000227460271861554316197556566224118756340501278103405856646766537018954185964066240457859194626558143313125824412559635129130086906976028635444060218797992547370132082916380788496584864016645155338102476357490305222392452114945853620686975383081427840791892729407194179236897452655907829255937027286698570784397487382242990326347080472574546312522326038419753951437799831430690304084087684303035538181812523230890783372773953961677974396907303758903934808035747944477277528267001070234880092255363221274303820343225415479126819937070570562654065195009839593938440374000473302075568746771126391307584779249330981594640387657042725725493800876630516005713789705652827210295338592985225924959199657729900181287069808881130884115897407246324220524401243575641227725030779990490');
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INSERT INTO num_exp_add VALUES (9,7,'-818934540016982261.65314972994491977243776717915257186979728396159058352649559139156429817562698954531329940720620096519975256547379603654362598494779213610069399116912987384006656023443527501447464682173445385303315267086044455246361273561294141518329233754041352632499787199926225490924591851865949646448441825186059741089695009429827829188117479084665641367');
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INSERT INTO num_exp_mul VALUES (9,7,'-44929599044588573810654775.83678007633232843418115790847152455559258007804727916986432256198687661496804050903769496933400455947645400628259699874770581538122521805603947464462448454681701547899144129061961394870320463199545502030106801911915987309444301341575451240764927967432593181449618816978119423290767783843864768557371257918447461479570164065303599994081990686');
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INSERT INTO num_exp_add VALUES (9,9,'109726960.69370054011016045512446564169485626040543207681883294700881721687140364874602090937340118558759806960049486905240792691274803010441572779861201766174025231986687953112944997105070635653109229393369465827911089507277452877411716963341532491917294735000425600147549018150816296268100707103116349627880517820609981140344341623765041830668717266');
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INSERT INTO num_exp_sub VALUES (9,9,'0');
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INSERT INTO num_exp_mul VALUES (9,9,'3010001475769225.8286280957637941018500905354415197182850820227163907782811814730309044010416886791014702373809932926301368137684091094408663914110947072451332976891128659038142954192986392936981664792370678656287232795203974766040821110221158579481177539669363513848425151485663431478439528936592701070340012569297177488556353760756495238304538439278682066056721729656193616571456456325016960870401748115848423105783116854283646624807603476682295234280408938557209608025246638166902335016025467565869375885610813662767004038102486303756741615124814580306266901273803721191779461890468156043551004644728343579032524687612403663816107770451694666844862368101122025340182510019516924578414085461628689');
|
|
|
|
INSERT INTO num_exp_div VALUES (9,9,'1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000');
|
|
|
|
COMMIT TRANSACTION;
|
|
|
|
BEGIN TRANSACTION;
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (0,'0');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (1,'291.964716019333021494947753821238960905461614737525349376826064492714634914263808902604580614735501799528494357560837535773816469841426747889103714048646989532842972129124080559131220979335403729022278994440514872845756198274805589586120535745968205107562348427941379641465378272611453955517402598409789621997041856848783989993820946766177453801729783316269310186191833995557234577548740940419224137195404391193633808203715191863638616433190672511651125299379882126530500870287424768024674231651229908224729856278167033444719242144302972892419034855417126978468296581589282861879645409909873113678361180607775255758820910366926076380306290306477790931129670172989289536405788838857428768869345763784112862591549008321546447442552533919976570125718481191724503352619626562352280522949665158335559389298720990302071');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (2,'31541679188064906.712574384704440356216787857626740375004266523720148374188511622980520374202725176835435173058936870163875556102907654264048353814040480579464700545975346621546520503928314632418705230212623378642743044255181848913683862360044189531298446109955034944189751302497670367665492719604026161836224535961347218522748523360100432275693829501972749859329753224444694962089604095212784768854310289429208671271394086829270986183171968944659703708706544668326267327938226750760690620258967209626420981505237183055363540806281098871221581265173394406715458619627534396065960117454160969749739483126059760636526242783235685190739315590041294766649891987044641492234243404608847939002062827210734973778130441825067858641461599799772535304379732674727995848518807202053316225824685704785148921785964036119338754973714515974054');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (3,'7765438138915239878.949520541017683429203286303188179443533225547096446554008374834292278237558244698868300666061834105683999048386497322007336816482648302911579331582895326423063492240235074387242190187374869842856897538718280497895072291181675294000739548676781615025944675912072664211455701112700937190832332966000160156597821149428032612782336278939437593991008833233156511435294360065004167893309428565243314846456225604669764879344135321428948841659419438769652686215993544390780212859309497190065178705035652106614050448518931820975038314187040226298661787490226917902356569717171481159691409131778764973037046501816919243659681416263730519167614043077472097520207347950292377914586524327206547377189493301153212000966249655331053184913579513686655963686155890934436604123384536027235444923674128269748280097789270784333442');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (4,'2308544622905.016172868282330339228589083058636874526727829838244942341440716909466939214393597311710652963849541394758298277969240038668406494621950956862959196896847352631445328917063551082418729435554972200530109505384839391233286173517804321019323644218483570886304028175359854335870835404627608254205407525763332087823548640923282031978903399118139052814618531713327991857575390136755426466065839913887477577516426991104516201265995293600539957187007068885368699949673989051443005684755994465547159213587471972139403333249259808344536605314911144950465968669770276463111776581675944967401948957460097365849699783091843609965345747287667911324039374314413430490112443463386381631812537639503425989372084906324702158112088898424705684574998783112519152403201231176840068666882123684602080460378627639651465436618032671756');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (5,'25549.082770905117529972076915050747181125832857399138345044265535151111965091602789684342996759657333588444489085160336703294705499665424408218434077722506748278242942379566431768762487954917389137120540138359870652558814224523699917122023018717544160579704907452934297025088008618627873220397030397424422097405152321366495319708580932627092620533785271831833326130796638935296720064431288560292191928489034307645738331451165431755179025359993690642194334018457793169983249853388987495489562746304107188105521296156525984787815685365255240654972150342496329030279439124533240114879332406941960563154881888172285475336782757262639979527682925214971861707635327995621436598536743180180978457735632181738067997521785965451385630326464388080990200265186437768409003553910194212076755448477164192901658547251079126833187');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (6,'.216649653115510782473161631235601739254284877523828136703593069337209747459679979369185882839688430004369697316986054374456779366220242645866798278985273820408495361607183119980716020227424205519727777568954933592987351750339481522149106749713967143685591960510946511796062486795368200503801097611436787402191532618456991115230272084771674098613479989808680789347124789253499967359190605681912854639520917409710307182238065185749856554472717209097115325999946728168357936779767099041518574001682560265549916593333117469681763348860131760281253987626822958726920016922608371657319505153308390495179319529587670415367205193280809809356733443291197315823747505896510820272670040485083775482983378341120809542502350385555577946098824446199419354197416933858522419312733314383889554606932774046771497129486979593226');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (7,'904950020.759072496304165474991957396337281699986101765045213964054286624338102141970514306010139529492299343393832200631760194440206005974547202512275476562767685193838576516154915404389465528270010938533075930081897392863141132529694804621418663424569202655893682412466871297412964570322984865326770090075582481194532433411398133265643849129084449161396724635797324126396071308557057830046688990212282866035593809633839882468628249964862932050189148498591642162462777480125024786829078066012617362076651920045684345679767223337287825546294839320770903419463644110383560050404456170063805115223954191445548226706113970164823214416171441655706141596091717118495955441099867737827763335880891937222647408575142200256804313345924443344596462585960919126827045197885802122062165934504665811115031150357820196176799560314653');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (8,'92179.098626752893864900181023972781406074846653380680747862421481598042923358730531575438403865501429843141967819802251116774924400485954931201776260931315313253827346015775662310076094882239170765060649024538403329505426563390044695320714825481746233901773893996663258170360232639353378395244461670781152793416950717050461856097473105730100523010642696332151571372764781034028324977128554099993021459338419164426784774496292405945103200724413639660488309795423335142455569853549710795692020963174011003447023610692365550245567840477105794884132665155376243735213346877116105595296043532605899184658904822980397411096930267453332143879534914237169761039374689145860503772331147367757318826885494994339695470190886515765452545019167989882527248872835783707554463866334705735781549392895480816605355996057201589681125');
|
|
|
|
INSERT INTO num_exp_sqrt VALUES (9,'7406.988615277484686670011157489572203134420118818648711986549881046321377798441006745317356200279801348355202517703531020643333388857073977704009782384103170022716610432579974132111487533733493986910583223121269323909760573942980360508642443245341392335557152177332615977623338526935953706604224108508582338123915133189529507760875123300397933931420500010248194253078118618381590347297853307090813639981736227771834732256867579490224181748450683295253634852775448770576585177080941820456051588076218688792321741398867304684922665590162004919486643750098085197190000638539994723704724550600891137853975703823903659121582583388450687255538838161486019214242094423895463814933532217776443473765708693285683261505695170847285063013324823850724236845500162436661946026097459146424122412596018946436589967013641971183281');
|
|
|
|
COMMIT TRANSACTION;
|
|
|
|
BEGIN TRANSACTION;
|
|
|
|
INSERT INTO num_exp_ln VALUES (0,'NaN');
|
|
|
|
INSERT INTO num_exp_ln VALUES (1,'11.353265918833698201334218522735144514838241118349715803442713722607336732214173255618762341321138898556011520430414052782971985419141860417968593746833898952016980791997105866598425597066404919489902082738711038276194174786383758877067916049129476352925010880025206629976454341252818402788928939407784629386362069592202090897264194883276572978998896242281239126931595483958092059051047739223830394259082355969005503976135238921488192773135287876801394308064862257453262299764712613486466254696464150007113953810688169396432889052881763511661127351872408811370081346456019961324265446884877073712053408327408917588393884214304220369626106333713688792094943405258431214313197283237071070354654837081449831786573831004911008790533179001070424813584405346221388686999574752038655226138085374176702005198770598232862');
|
|
|
|
INSERT INTO num_exp_ln VALUES (2,'75.980172429959420723484178622920965327708652620924912610122049843800380131746381968266727388919414524075492921510147435877107720844487333947572033626887969846858337336557672107987074468763307953130616555202495401302128216460637786993535376622372745654109623249396257174895352222213037880060756992073605135503615371392439827458529942230210514752764526895030759481226199720092008002458654297737883219558685499445394647863430593136350562417924068100891680398878483362058595716232013516337079804607378041880078724811071904523716775991447489914128580100888252698281559809224785596795038122963619830942475652745611551345360922016753939774272970008770647516790944335173711498988149783075646985898883858697162003144539047532603946093022417842140993960433780913606807466518632121884254341907122163281927271483110212890483');
|
|
|
|
INSERT INTO num_exp_ln VALUES (3,'86.992429107491709045555322727377654177072455841678650084144967727028762699430180506209786297136121512625728883607972513154010138109866327600596617277403558404624813332464431424791338402731178416819791932126837396086742033973404980654712734845137075562739300866280737071167943367603243180515859476717635339619107593771719314284984269343476343816253634799874584843436046260962736006310389088154751401911743739429257286834178656182340416539923956100441369280015412718483971113838923221170027312390404790743389872757674342133486652087007983701950040432125562287337697971646750563062524010514537132255605131615248097901911480464339325353279118429890601202554448469387179349495284716473293965884844451619766312048304583068386805927433174443889441171878078987788018564357316138422561213329104267180509029624308926098065');
|
|
|
|
INSERT INTO num_exp_ln VALUES (4,'56.935276817066740776567329017240462885579486075188456418197311631774373422196025180114152248099799048545382060930401786002025479108787121595516444894009593031141335985913019897883627990503003577804436730367402618412514152465206336556967419434371593632864308139215157721913158949066717186782560422199668568894551013785702491365073449320535603830475158258853167712460432995074161536886421366716995573365924430692151761737886552457036412140640821310927642146210426044265504978418405684030862182425702683702307323138985481047994648222224089112998195621687911787785594701557252468626097576375468916953563766801336922479861708649876362257086586679701715813254414915314296890025577780265459584203893089574567331742100451277992780400302806430264717887468808962517029442262560742822875484362427192693300423729233467613910');
|
|
|
|
INSERT INTO num_exp_ln VALUES (5,'20.296713391219923821414834924710998522858242536565236229645868008008504475111229451635162536658197320282791428572861452713483981402773630985812066048575864982038046409484905688236579134672910905547858248343712686247795669280482288748331949478864729205285910525962001251260319741279139167559906461672936902355959755164523720443059989357054368460911050707727029320725144824995614445423492687177126412520389766864793826362309254124276325522276592246655562770110024099522184080118637524912964002223613671995639705240767929562023556724031894855094820328152633412077228479168557819219970917880393852962560319397442566813746504969336443969816954424715197797253670026862362130664772772977978222813915593329422557592316429203293264572088112274848838446633519530653849595288125585730314673691986554304725866754516304420665');
|
|
|
|
INSERT INTO num_exp_ln VALUES (6,'-3.058947463851998053084898503420969773173569760507671013593014983772013099601022840164736581595033399273677583253456908293015637115395777673836877852797643436458673662566205707359569792482081945396989472318998080581824382006377064185813936544714612287417301161454496258176319380348780934551188852900784476213986897306897793456700682073399936398243222895442594762628402487110466705108765286617060826203345783502301472192906817785365563881556293576463515218574477264521950513789471494214626744754200844840310516235570475410854073969787604451971790833680742315518808178608136598148628107328076871698598743664423452623124027059698038466681488746505289551548778131621576387262707147068500249466398507704796800459013580425992071957391417767257856002976954566094297724379688683375704613872658653366052459242767328235849');
|
|
|
|
INSERT INTO num_exp_ln VALUES (7,'41.246780548917246608934265057073076900048579756649769602488660179351587788197892095257027979113051775079905924990472069951828742350559917110289416201523653941731339141666097617614477426376799479821365070373247490598890520285155435501242427296281987676879064510605563522117334502131946383957407685328562874307957108543536378261847119286989184256009392692140821396916222386573424618796707564187152459973446833193743614720624765332006827171872712331032607870580880807058576154429597725560836582655488602546786785520452359711161305828045237044625934404295366273012300148250900116489718279757540843657039519736455668388572899273464839528462223812926410544976290646668870192676914370659142463304861500879195867873346447316374869974900582948166687948531910220128160490935170837209017355954301127162240133341813847180541');
|
|
|
|
INSERT INTO num_exp_ln VALUES (8,'22.862977375646110045361670561177818139082238721442691850491173190000619222046296383571431877856442345505931635735363450488731186880557789439424987680284612480261693386095598289519783790826332183796775862215503493910816035128476952347072320869461206895223935484838130924268616681347949695029657753251443811448783435000569829291535036468240771401957519222523032235686030017496209956550934543164421459898155836108824017735809352580723262896259290484291175350770265895317482371895188221452083719817251845416195168686335127805092334984596224320638378502008767433534450949989322562311171685891891122105437154553106840103473941148230953978989145470651955269817951560544095229079088083494695756914405635176899994279484466773598435268700064279990885608144109747858515514066444373797446449729058958270758597627587968112958');
|
|
|
|
INSERT INTO num_exp_ln VALUES (9,'17.820358481980064387183481028572263407130633079314879566896470101569251997264841660326428805413719418277889123643557369421967068805165885825106611310020187894256310674762734896979157570968168599492401269694048046876387337971177513661006711375440365724346137980004810780215236524986274043416621637509807126148966029923572853117418545426960105154053049098579812135003711132897895016476695223444397389521434633067499404903493027304737402519428197015899833229473322655155458942323004249812974150129789653469524573801259946118454333405580647485894435301530550214095993989552176497867244278699359917247910082169086524111229983698975613609318418313798992088206507831757327320958918656453341769110558376097374227592021075267882222057385413453949580066342977546145482215220982989992069525148522710254796105001938615214263');
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|
|
|
COMMIT TRANSACTION;
|
|
|
|
BEGIN TRANSACTION;
|
|
|
|
INSERT INTO num_exp_log10 VALUES (0,'NaN');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (1,'4.930660740129727276654889314296515979425461685461970306647398411855044094312185293195497201658739777714943974003690119189101973212927970410047992001003936259467465542044528955416040460487922970233600641954269411521809500203864460110903973264337093883907933081597350982496469748131390809569321256206859934619579029279954574676601709408712255490686948453752571699579252140062805776361984468580258289509013081691778727372026090522694670379557247829136504595898935235926069699309392675806881162434168418505908116911054206058735257796918687777716036307205415038158583184624809880157060625643069601549803887864772092583549388533013233603450097615537162442973385137488450178790573546382354482351187412256794374383453695483855501587939419102008302408157959291557415763034668013452188944554607063362933134950906875499201');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (2,'32.997769620388965086774969704518222090258389987679691893351902336370051104718852164011301929506188893338106627980171059175447833290713847317665944354651476245003161501753612545484635275306181777040447675475670149066399611203341262105766118892586541910243351018829302798733989560900125591073082441126709911019648451232244139674063434385451279378543163944005973452562993913383659295688375546058256196254319767218634546732685705517341998116744642480938405113447415486950667007645850519659606476727681944251201236366198374488204017630268083077471516734133869728427050843306716313813724061560369884508660845630727190444623729815564381063131729592825825486515070406390371638817503915214206586939112681762984038333298146999891250107667687034785493312416966635780188163871680959873288697497561452228182734430749066579749');
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|
|
INSERT INTO num_exp_log10 VALUES (3,'37.780331928743475574895606142114739140772838801045013007323050327909196792739138159615327729728110344767302636436234256468332011934881494997184865617793179255006442447189720642997935223133982347184994174261506212652322213673745795726283311685835974151422721233207287206894148660531800622455957268888702309499182978182878524951883775154983702898237404558813230370364953160102391101897560104513279410610948028599674950811462114131673380477843456965645417025376374320207504913806546872166094337441573669261285052323206348035827948287081776955945081345131570610652073053464020209215624179904586956137079321655773178387441622685682721151900601340680061607114354850640946256225260430676099781727317540719923791064452012925902993317349390523278687089530234444415688602090547516647302454865526291471706301790881694022223');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (4,'24.726676547286224970759328746582840552419566534667446425423046931401641497155587075591229106937829957279943690528061985864558314570189069764367933957499905044566413640017549478921384160584906257607957223101377816440084188042395098536074479064548620374152344954289432050971466476174493306432228880930006524504974367146536665170956555486181410864034862861231267121149652317599303804477688621597163730470970207231328339082779056152481480926452142005969020950341307977091850953883445808399574256295803245530993204179747743812544604144379381347499056545148243304041538981954204310612049423688645476667184129189153715486929216331980316967699254518020077226689317148303152585009031597809279387172427408557115400021035692880631275593381822805377317270568779655383061987766693697518921188619814204902583361096973421134004');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (5,'8.814750626578650238811431417807018895270298639823442501111235973209197727215795256506525221092818797578008152140054383421240180435087611869193019443372556081555311825248667278358330916098378127100899126895012782320751838528480712942601038190627182482614147263228588284866661508052724762701223357327343090598060805245853527435948381893458352744679795853650453594546267600486696643924152372736774331080527157374379043696696647158270918245668579680394279565181670004245143555617589138267976417280970718829942998800499312890580011246294669585429723974582350357991472101919333996770115834067969654217063942059882195268353998096891812525364797586486311202350700339609637274043915687880562465121559531284337603363356183320193656553931871200575467929714875483123706358278876389849119105053294688326141759401230994901405');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (6,'-1.328484003982869642690619298690906747763234110040562640557173509402512757735587333095924652711056556491908059708986413635120656426593745303715671199761364516107844087845783714418487426723538440387069985879601248897538855843115404484229652166941838283489828419407478748732927617251897244190697443966424660881366993754577233476597163021768156814527570512834684713730559883782625870597080940193303268818336816535968869931456641949301731046034660616615392129109391145214470757259042172416816936479713743188047425796931722546185493217275537303458837771965375448968719169174136287532752370175863826715450565025635651343928205805494319778539652563499901671319955144823432132740582617949774638538594081514904904341299199113721131520557004571803778698005652464301037962272085633628653321081368256925971558076970172779715');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (7,'17.913249188669140643510654105014358282516966474257460687880559542190804665566625978925406311113121982595279826214959603627387555578965653325278444455875162277940655989601428868642914577248262147833499137348602966573601719040813549936948178463592211685237720748377879836890106515699728652218324794927458352954247096536337594789471529493944292143186953509162522579060020018226817623648563806559917579317916242706559131476179714031602207057714677845347616752450567251644277767418397621490301286115159509360375419599968738067461569666699939732107480135216621373057421990702923042287910730395998082514702629760389192370666675364405730936537832803383367187639209534697198515928978064543150195911463663617683085348965065679311986715357338675515370634753254774665197233934933271954463040729779956682570415317734489164385');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (8,'9.929264914121995501917993119394933531225401243275938207624866270551448544301376913376130982251708700134720886862945040266148728213253651323129942781577143957084726727561987639140151337848818195806259935747329665025823709044567138449084349729747202164413995795609659711723455165142329822773177102845804114214340046404641970845707372809306219463962664551623665322610139794354769767829380018857313559373283673392337954610346290037758389035140213224696023751541663171574697035012610534455189013755134090933979479069288110010954211669067225249755249337768792642303351914884187159646984708862430789018895140670365476746734456807215043628059581947593694929159076346249490593187993386780521089745819640214783614157516171005086731241769146397577246387886107367648843380733370112546792442909347322732196805316614555689762');
|
|
|
|
INSERT INTO num_exp_log10 VALUES (9,'7.739283354261751283625223433456284905560931805428759681411970457812279544250432389511382263439324085689734710188041049046660480575958686859942980599595036769090747781359217248301544587434077376812293034848418204834388504169166350770257248896025815531248627658465029806509131631454856186387892627989218208026727504548130018922325585619738185507999433763118148418722504204066578294826264005398891049629199412773138457218976050467479292777172717500219850781664314597312411301296201533610562886229900497272268364496763758868455934979903774531992886483396489868888731578355541611359130188566524240259770918423445785338175040098706500034487703124623745259139247432324145633151895802637182446905097253961951018926565652497920605819785424451050191604602898777804133717341512568151920576684198443843944721398831404081859');
|
|
|
|
COMMIT TRANSACTION;
|
|
|
|
BEGIN TRANSACTION;
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (0,'NaN');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (1,'225561990715.277245515991117670624124484084762557459065170589803293759247930753528436379932442146759103295277479258327642314622036941865221478746258727236601688778946696303277607709407496616423493315166963938393760548678730128692212077086588682984700837334554241405763691119669847463520746595280034536307041368063462023793177898200220207765205127584303464304601759554817607633012272490650155253979182893585119965271975927569080191838676053084168631217591768468344106219831174026139608715965691941366334940196517120885214887008671956523579678156919416435031020452971977153991139145404842034138317592877675821045409772456977018293365238179815614004574330200783530118851005077771478448804470170641452481992602803877112958872108069738434946694089025321283178188028224338756015337492913115267362635647236447601252924834642796058');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (2,'9553718264533556311125292459627965006385666643531070061102266984368939757379.536714147420215784125170401370065894858487440153494392538261078415409784085960333028254155527328359894197540839556987826344995348426293585457768226283066583722499658006242709930685932246087653832230889613022921575445199055131152661556678809191264086381976922223866204038615136758192929883317207903579770917317641181652055458721731297347443662717939116561947785705140374908203404860090658919334137955075887697259604047657534191202566335372150375993361370075961180728155127447781364264047857624746079509591666068708743260905728661917791822925979235918475633100283148558978385583805341715868143937062092264994833222352433299015979561976964779350640064096690062929265992966564232453102431600199173711947391200249130712039686700111791790265309426741120465259677894665532560198051256215915373145226284270408649736509');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (3,'982718444846268846508445482774217796844461660819285525931206164100817251856409365450682.362683768066405322653747385034480250394145008573806022660379219602846285813744865438912887625784087005970975437905783802114553690522787857272953842288090141945268495451006273685577260054069522075046955466204804067271437138871789034722069934693546671607506851844248427950939791205412350536883779850165603116191193657054604569586553874805856647223849267039531773072343908345333155562072887754900969504551717514980465801806565999410206735831440712124661645970935112535081991606671600328471264697018198676317466846450405861359235297846597981143547119390922405594115478086038680663368675222949247096131378724350715530605691796680604309063173515781378545860473572389718345696107553363715518601596249508215455106779522851210398208919496668879040223859884166805448827948087400426315425231119801173387715922086154065273');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (4,'861542720105376650266753999919217194383259935058507531116.774511336660822591851369622743235084609149542494189385785321912210129989390054947787009383210009523204976629456268332186620016067379702483800883493431423160815760933380418976582725913410929214462739708321325884209636272001805871036779154087677637129248122540412937033791526383240502286607736226090213753913654673523613612439527815137888202973659987501649474772884055648603290154867585312925699571949539600328906295652872654314913539778815035321695215634102441494403825526533235061083947035338872599854931230001361227174477274708230470794066733245241594719912710139298949856243576688344051439047966427547889756037265151798639614843866387316916203238068277912991427278268083231579195846744438643659745041780103653332041031419793815914447232121937821142169172566753399257291244398531365781832297786941359729799400');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (5,'198021976607570296508.271597639984889464620426933601643322058775615235389194561064983706229795978402690473201671702614911129095149240715527556855309177671128442458698638704394974473956869419481315262823632891676087912529523219333012290621046361106033860210270638559271706082115529424772192777643046125905852037759566224116373416253787241195450409652089019290072319861181399387753223422998872180810295299831487867222464355713552301775702554189470264147325049133532522718679336524769566984150923939420759804463781082299907043016120177416779442865059261387111806785876531152192378576258351599534512031062777609734092707165605364139201322351960602280089186180302246827234844736393745487324460438448807241887783263546165171099497316415863122023114646876909575845860402164818094500541234974716577550807551946414081410743197768993152975501');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (6,'.000873076977206566818052116526263730226812004454463281371489634779519089200224205946321120805055212090024554381349223642352209212670470260295303361873760972918129853308169576675500721645609379420329169271088810484607337679253503247351324049221970104335289487989027621978310506220905131150125321713385148268584530413680037620544212746920563790371941626294733473967065607791756894237438288480748407449237446113996117912144587258434808327522518688617394025018756570740098795745692805352377041347367240475846033282850136270250633825482156304826383360291164928049344226886150285595932088884965511963310715773499733217615863523253012606066583814112265708693122563204149232245895551314975524172504103194858904869273185785182598234060315036187756490539352752560361560286717869643902435677448962235275054804452967413005');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (7,'176514565873872717825163931126806100435750.096278384530154766967061948052237623936423931849868926020451465515367348890410352640552194499619062823622476972850692557798609619250753020363520533767813563613425606228355802781302735485038377521515850536680425059519814786118919994914180918228654298075183514200191737597656810036850772127169441661576862538643715648802139886576391427423689320082366572297580054381937437005879583216745596935643579262248665490169331304003204939561361718554509909313409421397022626924406091551900222555950699170864234411017062042057683304265485826061096835531732950909546314722726990314852356462874701181085379772134121978510387397276859318242238150439474660772561390798432890789762504242822787017140808209820627435991445529404692793744568204608385843245177656436105160780897472099970336514833257055017279707999437302548655364559');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (8,'72941951052009383458167.300747500436981484566111756088702608000390737594784514635592222758882092500858797317505303492923829092720870826490477962201959426813271424853341826896270963213736922458746003100613943600855942721319226948714369219316345322636075285343544788982588956431405042577296229122673590336976893594798942025893296105815818487227300314490440902574022885833779324177053242170024559675073866612316965636832258283516275906085642459351367507561963945012828379111856700009391438637054015804558386733558956649061672420804826896303889067785497738203077050774825608647969196321506624991188638449047860249367840775936911749905927108478444112230174584693363226143549933224252679398881354887872642908328737917862751077365602631600279486028043329404269490375935308156815477700961014566228692743960491745353377403533037122586797765130');
|
|
|
|
INSERT INTO num_exp_power_10_ln VALUES (9,'661239032819374816.097553651299556484820492272269662685578275493609248662925676004753503494252951243895572437264999063878330704584509915845096232798927524470286655554736724913758600775591269525423912692080421094644542553026831758426157681271572808657664918053119324646138457659418857926209701677786068580819823633713337632456905824562235373422309621872998037966404189020165296080436871220718574009921789858751384547836431858428729570977259373272041837411903005303672798845573379758630607982213326716018594073712340609488043353995410508475153538231445235003980586600882223782814368245305160648543466496726973755388826656879616734762068443462618454921858705377028522664844761719759342490380417060255776725333319537746890406213693117052223545525717132695297770810635066731941724108167146710297146989770382041617889670713111888375717');
|
|
|
|
COMMIT TRANSACTION;
|
|
|
|
BEGIN TRANSACTION;
|
|
|
|
INSERT INTO num_data VALUES (0, '0');
|
|
|
|
INSERT INTO num_data VALUES (1, '85243.39540024977626076239847863600785982737155858270959890014613035727868293618673807776733416230953723818527101593495895350807775607346277892835514324320448949370623441059033804864158715021903312693889518990256881059434042443507529601095150710777634743301398926463888783847290873199395304998050753365215426971278237920063435565949203678024225270616295573678510929020831006146661747271783837653203039829647102027431761129518881525935216608429897041525858540380754759125150233053469999022855035');
|
|
|
|
INSERT INTO num_data VALUES (2, '-994877526002806872754342148749241.04353023451866590387054959174736129501310680280823383331007646306243540953499740615246583399296334239109936336446284803020643582102868247857009494139535009572740621288230740389545481395');
|
|
|
|
INSERT INTO num_data VALUES (3, '-60302029489319384367663884408085757480.2322712404088283093870869198708849258097125725036189625900174440196153781995220721511009855207486224837798752903681993777275846325950111427710563453217985216966456785944859989497422927661683538629473170704026975786513125842675604577233871570629808699803522400038975396500769162308448069085909755023233588510630417065084295051270219462289785473643946404281422516357503746700705970360169619852905053433235726497292406142332833');
|
|
|
|
INSERT INTO num_data VALUES (4, '5329378275943663322215245.24931765987630429629836382184742348353920297283690739124220773955591340709935970062776650204659187764581615597720798385015942389765692769739983054442503547211560297249686289665792078548480268091496050883021187158502798880896590227542729659940394038802461081290690995869705131152889309663639310553909874081663091069118126221594338242710530718836025225507189149221049928936955230868771875644038572888630664890573507822342998964954667474300944699078658989010257103569231493090050659723450626338923049035040974032671138430612839043269997482582763267536489504794826476836323549796385028155416935072959933315468068930689064483178204550825728947252440604703474049780550458442808479096492346910001692358508618202898514895453589357');
|
|
|
|
INSERT INTO num_data VALUES (5, '-652755630.43456071828048833552593835051449845484289562110789582081210403487973096161149072377955192388469356112505543620695003436531392789029513380101663750625024853263344909355177280161504414335005574882649025508632900995595004153086358670541462762210415346958050909878501048483523600711486406055424807840429541335391538322886495085448421556770991545781035298449067051916630343957356635391594362639819978677032855590055900561501350354631803808000307050416047072513406855040715556454205065332997338225626635780147287003130754254277103928406089109802521803537038957372612837169223905290912251006321930223154562110264217937');
|
|
|
|
INSERT INTO num_data VALUES (6, '0.0469370721950711508944806393077762204079964905145503836835397203739563036579760026190241480514409364');
|
|
|
|
INSERT INTO num_data VALUES (7, '-818934540071845742');
|
|
|
|
INSERT INTO num_data VALUES (8, '8496986223.64063724353165506167988570717591150432282238795309964705925283285060558038824227595710139960766584401003765241409149793494330798800');
|
|
|
|
INSERT INTO num_data VALUES (9, '054863480.34685027005508022756223282084742813020271603840941647350440860843570182437301045468670059279379903480024743452620396345637401505220786389930600883087012615993343976556472498552535317826554614696684732913955544753638726438705858481670766245958647367500212800073774509075408148134050353551558174813940258910304990570172170811882520915334358633');
|
|
|
|
COMMIT TRANSACTION;
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Create indices for faster checks
|
|
|
|
-- ******************************
|
|
|
|
|
|
|
|
CREATE UNIQUE INDEX num_exp_add_idx ON num_exp_add (id1, id2);
|
|
|
|
CREATE UNIQUE INDEX num_exp_sub_idx ON num_exp_sub (id1, id2);
|
|
|
|
CREATE UNIQUE INDEX num_exp_div_idx ON num_exp_div (id1, id2);
|
|
|
|
CREATE UNIQUE INDEX num_exp_mul_idx ON num_exp_mul (id1, id2);
|
|
|
|
CREATE UNIQUE INDEX num_exp_sqrt_idx ON num_exp_sqrt (id);
|
|
|
|
CREATE UNIQUE INDEX num_exp_ln_idx ON num_exp_ln (id);
|
|
|
|
CREATE UNIQUE INDEX num_exp_log10_idx ON num_exp_log10 (id);
|
|
|
|
CREATE UNIQUE INDEX num_exp_power_10_ln_idx ON num_exp_power_10_ln (id);
|
|
|
|
|
|
|
|
VACUUM ANALYZE num_exp_add;
|
|
|
|
VACUUM ANALYZE num_exp_sub;
|
|
|
|
VACUUM ANALYZE num_exp_div;
|
|
|
|
VACUUM ANALYZE num_exp_mul;
|
|
|
|
VACUUM ANALYZE num_exp_sqrt;
|
|
|
|
VACUUM ANALYZE num_exp_ln;
|
|
|
|
VACUUM ANALYZE num_exp_log10;
|
|
|
|
VACUUM ANALYZE num_exp_power_10_ln;
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Now check the behaviour of the NUMERIC type
|
|
|
|
-- ******************************
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Addition check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, t1.val + t2.val
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_add t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, round(t1.val + t2.val, 10)
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, round(t2.expected, 10) as expected
|
|
|
|
FROM num_result t1, num_exp_add t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != round(t2.expected, 10);
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Subtraction check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, t1.val - t2.val
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_sub t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, round(t1.val - t2.val, 40)
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, round(t2.expected, 40)
|
|
|
|
FROM num_result t1, num_exp_sub t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != round(t2.expected, 40);
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Multiply check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, t1.val * t2.val
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_mul t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, round(t1.val * t2.val, 30)
|
|
|
|
FROM num_data t1, num_data t2;
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, round(t2.expected, 30) as expected
|
|
|
|
FROM num_result t1, num_exp_mul t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != round(t2.expected, 30);
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Division check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, t1.val / t2.val
|
|
|
|
FROM num_data t1, num_data t2
|
|
|
|
WHERE t2.val != '0.0';
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_div t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT t1.id, t2.id, round(t1.val / t2.val, 80)
|
|
|
|
FROM num_data t1, num_data t2
|
|
|
|
WHERE t2.val != '0.0';
|
|
|
|
SELECT t1.id1, t1.id2, t1.result, round(t2.expected, 80) as expected
|
|
|
|
FROM num_result t1, num_exp_div t2
|
|
|
|
WHERE t1.id1 = t2.id1 AND t1.id2 = t2.id2
|
|
|
|
AND t1.result != round(t2.expected, 80);
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Square root check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT id, 0, SQRT(ABS(val))
|
|
|
|
FROM num_data;
|
|
|
|
SELECT t1.id1, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_sqrt t2
|
|
|
|
WHERE t1.id1 = t2.id
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Natural logarithm check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT id, 0, LN(ABS(val))
|
|
|
|
FROM num_data
|
|
|
|
WHERE val != '0.0';
|
|
|
|
SELECT t1.id1, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_ln t2
|
|
|
|
WHERE t1.id1 = t2.id
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
-- ******************************
|
|
|
|
-- * Logarithm base 10 check
|
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
|
|
|
INSERT INTO num_result SELECT id, 0, LOG('10'::numeric, ABS(val))
|
|
|
|
FROM num_data
|
|
|
|
WHERE val != '0.0';
|
|
|
|
SELECT t1.id1, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_log10 t2
|
|
|
|
WHERE t1.id1 = t2.id
|
|
|
|
AND t1.result != t2.expected;
|
|
|
|
|
|
|
|
-- ******************************
|
2000-03-30 09:13:20 +02:00
|
|
|
-- * POW(10, LN(value)) check
|
1999-06-10 19:49:32 +02:00
|
|
|
-- ******************************
|
|
|
|
DELETE FROM num_result;
|
2000-03-30 09:13:20 +02:00
|
|
|
INSERT INTO num_result SELECT id, 0, POW(numeric '10', LN(ABS(round(val,1000))))
|
1999-06-10 19:49:32 +02:00
|
|
|
FROM num_data
|
|
|
|
WHERE val != '0.0';
|
|
|
|
SELECT t1.id1, t1.result, t2.expected
|
|
|
|
FROM num_result t1, num_exp_power_10_ln t2
|
|
|
|
WHERE t1.id1 = t2.id
|
|
|
|
AND t1.result != t2.expected;
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
|
|
|
|
|
|
|
|
--
|
|
|
|
-- Test code path for raising to integer powers
|
|
|
|
--
|
|
|
|
|
|
|
|
-- base less than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of POW():
|
|
|
|
--
|
|
|
|
-- for p in {-20..20}
|
|
|
|
-- do
|
|
|
|
-- b="0.084738"
|
|
|
|
-- r=$(bc -ql <<< "scale=500 ; $b^$p" | head -n 1)
|
|
|
|
-- echo "($b, $p, $r),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(b, p, bc_result) AS (VALUES
|
Improve the accuracy of numeric power() for integer exponents.
This makes the choice of result scale of numeric power() for integer
exponents consistent with the choice for non-integer exponents, and
with the result scale of other numeric functions. Specifically, the
result scale will be at least as large as the scale of either input,
and sufficient to ensure that the result has at least 16 significant
digits.
Formerly, the result scale was based only on the scale of the first
input, without taking into account the weight of the result. For
results with negative weight, that could lead to results with very few
or even no non-zero significant digits (e.g., 10.0 ^ (-18) produced
0.0000000000000000).
Fix this by moving responsibility for the choice of result scale into
power_var_int(), which already has code to estimate the result weight.
Per report by Adrian Klaver and suggested fix by Tom Lane.
No back-patch -- arguably this is a bug fix, but one which is easy to
work around, so it doesn't seem worth the risk of changing query
results in stable branches.
Discussion: https://postgr.es/m/12a40226-70ac-3a3b-3d3a-fdaf9e32d312%40aklaver.com
2022-10-20 11:10:17 +02:00
|
|
|
(0.084738, -20, 2744326694304960114888.785913),
|
|
|
|
(0.084738, -19, 232548755422013710215.445941),
|
|
|
|
(0.084738, -18, 19705716436950597776.236458),
|
|
|
|
(0.084738, -17, 1669822999434319754.362725),
|
|
|
|
(0.084738, -16, 141497461326065387.345189),
|
|
|
|
(0.084738, -15, 11990211877848128.792857),
|
|
|
|
(0.084738, -14, 1016026574105094.737649),
|
|
|
|
(0.084738, -13, 86096059836517.517879),
|
|
|
|
(0.084738, -12, 7295607918426.821430),
|
|
|
|
(0.084738, -11, 618215223791.651994),
|
|
|
|
(0.084738, -10, 52386321633.657007),
|
|
|
|
(0.084738, -9, 4439112122.5928274),
|
|
|
|
(0.084738, -8, 376161483.04427101),
|
|
|
|
(0.084738, -7, 31875171.750205437),
|
|
|
|
(0.084738, -6, 2701038.3037689083),
|
|
|
|
(0.084738, -5, 228880.58378476975),
|
|
|
|
(0.084738, -4, 19394.882908753819),
|
|
|
|
(0.084738, -3, 1643.4835879219811),
|
|
|
|
(0.084738, -2, 139.26551227333284),
|
|
|
|
(0.084738, -1, 11.801080979017678),
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
(0.084738, 0, 1),
|
|
|
|
(0.084738, 1, .084738),
|
|
|
|
(0.084738, 2, .007180528644),
|
Improve the accuracy of numeric power() for integer exponents.
This makes the choice of result scale of numeric power() for integer
exponents consistent with the choice for non-integer exponents, and
with the result scale of other numeric functions. Specifically, the
result scale will be at least as large as the scale of either input,
and sufficient to ensure that the result has at least 16 significant
digits.
Formerly, the result scale was based only on the scale of the first
input, without taking into account the weight of the result. For
results with negative weight, that could lead to results with very few
or even no non-zero significant digits (e.g., 10.0 ^ (-18) produced
0.0000000000000000).
Fix this by moving responsibility for the choice of result scale into
power_var_int(), which already has code to estimate the result weight.
Per report by Adrian Klaver and suggested fix by Tom Lane.
No back-patch -- arguably this is a bug fix, but one which is easy to
work around, so it doesn't seem worth the risk of changing query
results in stable branches.
Discussion: https://postgr.es/m/12a40226-70ac-3a3b-3d3a-fdaf9e32d312%40aklaver.com
2022-10-20 11:10:17 +02:00
|
|
|
(0.084738, 3, .000608463636235272),
|
|
|
|
(0.084738, 4, .00005155999160730448),
|
|
|
|
(0.084738, 5, .000004369090568819767),
|
|
|
|
(0.084738, 6, .0000003702279966206494),
|
|
|
|
(0.084738, 7, .00000003137237997764059),
|
|
|
|
(0.084738, 8, .000000002658432734545308),
|
|
|
|
(0.084738, 9, .0000000002252702730599003),
|
|
|
|
(0.084738, 10, .00000000001908895239854983),
|
|
|
|
(0.084738, 11, .000000000001617559648348316),
|
|
|
|
(0.084738, 12, .0000000000001370687694817396),
|
|
|
|
(0.084738, 13, .00000000000001161493338834365),
|
|
|
|
(0.084738, 14, .0000000000000009842262254614642),
|
|
|
|
(0.084738, 15, .00000000000000008340136189315355),
|
|
|
|
(0.084738, 16, .000000000000000007067264604102046),
|
|
|
|
(0.084738, 17, .0000000000000000005988658680223991),
|
|
|
|
(0.084738, 18, .00000000000000000005074669592448206),
|
|
|
|
(0.084738, 19, .000000000000000000004300173519248761),
|
|
|
|
(0.084738, 20, .0000000000000000000003643881036741015))
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- base greater than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of POW():
|
|
|
|
--
|
|
|
|
-- for p in {-20..20}
|
|
|
|
-- do
|
|
|
|
-- b="37.821637"
|
|
|
|
-- r=$(bc -ql <<< "scale=500 ; $b^$p" | head -n 1)
|
|
|
|
-- echo "($b, $p, $r),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(b, p, bc_result) AS (VALUES
|
Improve the accuracy of numeric power() for integer exponents.
This makes the choice of result scale of numeric power() for integer
exponents consistent with the choice for non-integer exponents, and
with the result scale of other numeric functions. Specifically, the
result scale will be at least as large as the scale of either input,
and sufficient to ensure that the result has at least 16 significant
digits.
Formerly, the result scale was based only on the scale of the first
input, without taking into account the weight of the result. For
results with negative weight, that could lead to results with very few
or even no non-zero significant digits (e.g., 10.0 ^ (-18) produced
0.0000000000000000).
Fix this by moving responsibility for the choice of result scale into
power_var_int(), which already has code to estimate the result weight.
Per report by Adrian Klaver and suggested fix by Tom Lane.
No back-patch -- arguably this is a bug fix, but one which is easy to
work around, so it doesn't seem worth the risk of changing query
results in stable branches.
Discussion: https://postgr.es/m/12a40226-70ac-3a3b-3d3a-fdaf9e32d312%40aklaver.com
2022-10-20 11:10:17 +02:00
|
|
|
(37.821637, -20, .00000000000000000000000000000002787363175065101),
|
|
|
|
(37.821637, -19, .000000000000000000000000000001054226381944797),
|
|
|
|
(37.821637, -18, .00000000000000000000000000003987256753373947),
|
|
|
|
(37.821637, -17, .000000000000000000000000001508045775519079),
|
|
|
|
(37.821637, -16, .00000000000000000000000005703675990106610),
|
|
|
|
(37.821637, -15, .000000000000000000000002157223628634278),
|
|
|
|
(37.821637, -14, .00000000000000000000008158972901002847),
|
|
|
|
(37.821637, -13, .000000000000000000003085857113545666),
|
|
|
|
(37.821637, -12, .0000000000000000001167121675823920),
|
|
|
|
(37.821637, -11, .000000000000000004414245235784397),
|
|
|
|
(37.821637, -10, .0000000000000001669539809368169),
|
|
|
|
(37.821637, -9, .000000000000006314472862697207),
|
|
|
|
(37.821637, -8, .0000000000002388237004592846),
|
|
|
|
(37.821637, -7, .000000000009032703305767796),
|
|
|
|
(37.821637, -6, .0000000003416316255594496),
|
|
|
|
(37.821637, -5, .00000001292106732962942),
|
|
|
|
(37.821637, -4, .0000004886959181938034),
|
|
|
|
(37.821637, -3, .00001848327962130773),
|
|
|
|
(37.821637, -2, .0006990678924065984),
|
|
|
|
(37.821637, -1, .02643989206495742),
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
(37.821637, 0, 1),
|
|
|
|
(37.821637, 1, 37.821637),
|
|
|
|
(37.821637, 2, 1430.476225359769),
|
Improve the accuracy of numeric power() for integer exponents.
This makes the choice of result scale of numeric power() for integer
exponents consistent with the choice for non-integer exponents, and
with the result scale of other numeric functions. Specifically, the
result scale will be at least as large as the scale of either input,
and sufficient to ensure that the result has at least 16 significant
digits.
Formerly, the result scale was based only on the scale of the first
input, without taking into account the weight of the result. For
results with negative weight, that could lead to results with very few
or even no non-zero significant digits (e.g., 10.0 ^ (-18) produced
0.0000000000000000).
Fix this by moving responsibility for the choice of result scale into
power_var_int(), which already has code to estimate the result weight.
Per report by Adrian Klaver and suggested fix by Tom Lane.
No back-patch -- arguably this is a bug fix, but one which is easy to
work around, so it doesn't seem worth the risk of changing query
results in stable branches.
Discussion: https://postgr.es/m/12a40226-70ac-3a3b-3d3a-fdaf9e32d312%40aklaver.com
2022-10-20 11:10:17 +02:00
|
|
|
(37.821637, 3, 54102.952532687378),
|
|
|
|
(37.821637, 4, 2046262.2313195326),
|
|
|
|
(37.821637, 5, 77392987.319777394),
|
|
|
|
(37.821637, 6, 2927129472.7542235),
|
|
|
|
(37.821637, 7, 110708828370.511632),
|
|
|
|
(37.821637, 8, 4187189119324.792454),
|
|
|
|
(37.821637, 9, 158366346921451.985294),
|
|
|
|
(37.821637, 10, 5989674486279224.500736),
|
|
|
|
(37.821637, 11, 226539294168214309.708325),
|
|
|
|
(37.821637, 12, 8568086950266418559.993831),
|
|
|
|
(37.821637, 13, 324059074417413536066.149409),
|
|
|
|
(37.821637, 14, 12256444679171401239980.310926),
|
|
|
|
(37.821637, 15, 463558801566202198479885.206986),
|
|
|
|
(37.821637, 16, 17532552720991931019508170.100286),
|
|
|
|
(37.821637, 17, 663109844696719094948877928.067252),
|
|
|
|
(37.821637, 18, 25079899837245684700124994552.671731),
|
|
|
|
(37.821637, 19, 948562867640665366544581398598.127577),
|
|
|
|
(37.821637, 20, 35876200451584291931921101974730.690104))
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
--
|
|
|
|
-- Tests for raising to non-integer powers
|
|
|
|
--
|
|
|
|
|
|
|
|
-- base less than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of POW():
|
|
|
|
--
|
|
|
|
-- for n in {-20..20}
|
|
|
|
-- do
|
|
|
|
-- b="0.06933247"
|
|
|
|
-- p="$n.342987"
|
|
|
|
-- r=$(bc -ql <<< "scale=500 ; e($p*l($b))" | head -n 1)
|
|
|
|
-- echo "($b, $p, $r),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(b, p, bc_result) AS (VALUES
|
|
|
|
(0.06933247, -20.342987, 379149253615977128356318.39406340),
|
|
|
|
(0.06933247, -19.342987, 26287354251852125772450.59436685),
|
|
|
|
(0.06933247, -18.342987, 1822567200045909954554.65766042),
|
|
|
|
(0.06933247, -17.342987, 126363085720167050546.86216560),
|
|
|
|
(0.06933247, -16.342987, 8761064849800910427.02880469),
|
|
|
|
(0.06933247, -15.342987, 607426265866876128.15466179),
|
|
|
|
(0.06933247, -14.342987, 42114363355427213.14899924),
|
|
|
|
(0.06933247, -13.342987, 2919892833909256.59283660),
|
|
|
|
(0.06933247, -12.342987, 202443382310228.51544515),
|
|
|
|
(0.06933247, -11.342987, 14035899730722.44924025),
|
|
|
|
(0.06933247, -10.342987, 973143597003.32229028),
|
|
|
|
(0.06933247, -9.342987, 67470449244.92493259),
|
|
|
|
(0.06933247, -8.342987, 4677892898.16028054),
|
|
|
|
(0.06933247, -7.342987, 324329869.02491071),
|
|
|
|
(0.06933247, -6.342987, 22486590.914273551),
|
|
|
|
(0.06933247, -5.342987, 1559050.8899661435),
|
|
|
|
(0.06933247, -4.342987, 108092.84905705095),
|
|
|
|
(0.06933247, -3.342987, 7494.3442144625131),
|
|
|
|
(0.06933247, -2.342987, 519.60139541889576),
|
|
|
|
(0.06933247, -1.342987, 36.025248159838727),
|
|
|
|
(0.06933247, 0.342987, .40036522320023350),
|
|
|
|
(0.06933247, 1.342987, .02775830982657349),
|
|
|
|
(0.06933247, 2.342987, .001924552183301612),
|
|
|
|
(0.06933247, 3.342987, .0001334339565121935),
|
|
|
|
(0.06933247, 4.342987, .000009251305786862961),
|
|
|
|
(0.06933247, 5.342987, .0000006414158809285026),
|
|
|
|
(0.06933247, 6.342987, .00000004447094732199898),
|
|
|
|
(0.06933247, 7.342987, .000000003083280621074075),
|
|
|
|
(0.06933247, 8.342987, .0000000002137714611621997),
|
|
|
|
(0.06933247, 9.342987, .00000000001482130341788437),
|
|
|
|
(0.06933247, 10.342987, .000000000001027597574581366),
|
|
|
|
(0.06933247, 11.342987, .00000000000007124587801173530),
|
|
|
|
(0.06933247, 12.342987, .000000000000004939652699872298),
|
|
|
|
(0.06933247, 13.342987, .0000000000000003424783226243151),
|
|
|
|
(0.06933247, 14.342987, .00000000000000002374486802900065),
|
|
|
|
(0.06933247, 15.342987, .000000000000000001646290350274646),
|
|
|
|
(0.06933247, 16.342987, .0000000000000000001141413763217064),
|
|
|
|
(0.06933247, 17.342987, .000000000000000000007913703549583420),
|
|
|
|
(0.06933247, 18.342987, .0000000000000000000005486766139403860),
|
|
|
|
(0.06933247, 19.342987, .00000000000000000000003804110487572339),
|
|
|
|
(0.06933247, 20.342987, .000000000000000000000002637483762562946))
|
|
|
|
SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- base greater than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of POW():
|
|
|
|
--
|
|
|
|
-- for n in {-20..20}
|
|
|
|
-- do
|
|
|
|
-- b="27.234987"
|
|
|
|
-- p="$n.230957"
|
|
|
|
-- r=$(bc -ql <<< "scale=500 ; e($p*l($b))" | head -n 1)
|
|
|
|
-- echo "($b, $p, $r),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(b, p, bc_result) AS (VALUES
|
|
|
|
(27.234987, -20.230957, .000000000000000000000000000009247064512095633),
|
|
|
|
(27.234987, -19.230957, .0000000000000000000000000002518436817750859),
|
|
|
|
(27.234987, -18.230957, .000000000000000000000000006858959399176602),
|
|
|
|
(27.234987, -17.230957, .0000000000000000000000001868036700701026),
|
|
|
|
(27.234987, -16.230957, .000000000000000000000005087595525911532),
|
|
|
|
(27.234987, -15.230957, .0000000000000000000001385605980094587),
|
|
|
|
(27.234987, -14.230957, .000000000000000000003773696085499835),
|
|
|
|
(27.234987, -13.230957, .0000000000000000001027765638305389),
|
|
|
|
(27.234987, -12.230957, .000000000000000002799118379829397),
|
|
|
|
(27.234987, -11.230957, .00000000000000007623395268611469),
|
|
|
|
(27.234987, -10.230957, .000000000000002076230710364949),
|
|
|
|
(27.234987, -9.230957, .00000000000005654611640579014),
|
|
|
|
(27.234987, -8.230957, .000000000001540032745212181),
|
|
|
|
(27.234987, -7.230957, .00000000004194277179542807),
|
|
|
|
(27.234987, -6.230957, .000000001142310844592450),
|
|
|
|
(27.234987, -5.230957, .00000003111082100243440),
|
|
|
|
(27.234987, -4.230957, .0000008473028055606278),
|
|
|
|
(27.234987, -3.230957, .00002307628089450723),
|
|
|
|
(27.234987, -2.230957, .0006284822101702527),
|
|
|
|
(27.234987, -1.230957, .01711670482371810),
|
|
|
|
(27.234987, 0.230957, 2.1451253063142300),
|
|
|
|
(27.234987, 1.230957, 58.422459830839071),
|
|
|
|
(27.234987, 2.230957, 1591.1349340009243),
|
|
|
|
(27.234987, 3.230957, 43334.539242761031),
|
|
|
|
(27.234987, 4.230957, 1180215.6129275865),
|
|
|
|
(27.234987, 5.230957, 32143156.875279851),
|
|
|
|
(27.234987, 6.230957, 875418459.63720737),
|
|
|
|
(27.234987, 7.230957, 23842010367.779367),
|
|
|
|
(27.234987, 8.230957, 649336842420.336290),
|
|
|
|
(27.234987, 9.230957, 17684680461938.907402),
|
|
|
|
(27.234987, 10.230957, 481642042480060.137900),
|
|
|
|
(27.234987, 11.230957, 13117514765597885.614921),
|
|
|
|
(27.234987, 12.230957, 357255344113366461.949871),
|
|
|
|
(27.234987, 13.230957, 9729844652608062117.440722),
|
|
|
|
(27.234987, 14.230957, 264992192625800087863.690528),
|
|
|
|
(27.234987, 15.230957, 7217058921265161257566.469315),
|
|
|
|
(27.234987, 16.230957, 196556505898890690402726.443417),
|
|
|
|
(27.234987, 17.230957, 5353213882921711267539279.451015),
|
|
|
|
(27.234987, 18.230957, 145794710509592328389185797.837767),
|
|
|
|
(27.234987, 19.230957, 3970717045397510438979206144.696206),
|
|
|
|
(27.234987, 20.230957, 108142427112079606637962972621.121293))
|
|
|
|
SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
|
|
|
|
|
2016-05-05 12:16:17 +02:00
|
|
|
-- Inputs close to overflow
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 2700 and truncated to 4 decimal
|
|
|
|
-- places.
|
|
|
|
|
|
|
|
WITH t(b, p, bc_result) AS (VALUES
|
|
|
|
(0.12, -2829.8369, 58463948950011752465280493160293790845494328939320966633018493248607815580903065923369555885857984675501574162389726507612128133630191173383130639968378879506624785786843501848666498440326970769604109017960864573408272864266102690849952650095786874354625921641729880352858506454246180842452983243549491658464046163869265572232996388827878976066830374513768599285647145439771472435206769249126377164951470622827631950210853282324510655982757098065657709137845327135766013147354253426364240746381620690117663724329288646510198895137275207992825719846135857839292915100523542874885080351683587865157015032404901182924720371819942957083390475846809517968191151435281268695782594904484795360890092607679215675240583291240729468370895035823777914792823688291214492607109455017754453939895630226174304357121900605689015734289765672740769194115142607443713769825894380064727556869268488695795705030158832909348803019429370973064732712469794182891757241046263341655894972953512257981661670321890336672832647028099324621932563236459127918144141230217523147304565594514812518826936144181257723061181656522095236928347413997136815409159361412494284201481609684892562646522086577634100783077813105675590737823924220663206479031113753135119759722725207724879578900186075841393115040465401462266086907464970054073340036852442184414587772177753008511913377364966775792477387717262694468450099866775550614257191941835797445874557362115814601886902749237439492398087966544817154173072811937702110580330775581851211123491341435883319798273456296794954514173820352334127081705706502510709179711510240917772628308487366740741280043704807717608366220401933596364641284631036907635403895053036499618723044314773148779735006542501244942039455169872946018271985844759209768927953340447524637670938413827595013338859796135512187473850161303598087634723542727044978083220970836296653305188470017342167913572166172051819741354902582606590658382067039498769674611071582171914886494269818475850690414812481252963932223686078322390396586222238852602472958831686564971334200490182175112490433364675164900946902818404704835106260174052265784055642968397240262737313737007322288203637798365320295080314524864099419556398713380156353062937736280885716820226469419928595465390700629307079710611273715705695938635644841913194091407807776191951797748706106000922803167645881087385311847268311361092838264814899353459146959869764278464187826798546290981492648723002412475976344071283321798061003719251864595518596639432393032991023409676558943539937377229130132816883146259468718344018277257037013406135980469482324577407154032999045733141275895.3432),
|
|
|
|
(1.2, 32908.8896, 58463467728170833376633133695001863276259293590926929026251227859007891876739460057725441400966420577009060860805883032969522911803372870882799865787473726926215148161529632590083389287080925059682489116446754279752928005457087175157581627230586554364417068189211136840990661174760199073702207450133797324318403866058202372178813998850887986769280847189341565507156189065295823921162851958925352114220880236114784962150135485415106748467247897246441194126125699204912883449386043559785865023459356275014504597646990160571664166410683323036984805434677654413174177920726210827006973855410386789516533036723888687725436216478665958434776205940192130053647653715221076841771578099896259902368829351569726536927952661429685419815305418450230567773264738536471211804481206474781470237730069753206249915908804615495060673071058534441654604668770343616386612119048579369195201590008082689834456232255266932976831478404670192731621439902738547169253818323045451045749609624500171633897705543164388470746657118050314064066768449450440405619135824055131398727045420324382226572368236570500391463795989258779677208133531636928003546809249007993065200108076924439703799231711400266122025052209803513232429907231051873161206025860851056337427740362763618748092029386371493898291580557004812947013231371383576580415676519066503391905962989205397824064923920045371823949776899815750413244195402085917098964452866825666226141169411712884994564949174271056284898570445214367063763956186792886147126466387576513166370247576466566827375268334148320298849218878848928271566491769458471357076035396330179659440244425914213309776100351793665960978678576150833311810944729586040624059867137538839913141142139636023129691775489034134511666020819676247950267220131499463010350308195762769192775344260909521732256844149916046793599150786757764962585268686580124987490115873389726527572428003433405659445349155536369077209682951123806333170190998931670309088422483075609203671527331975811507450670132060984691061148836994322505371265263690017938762760088575875666254883673433331627055180154954694693433502522592907190906966067656027637884202418119121728966267936832338377284832958974299187166554160783467156478554899314000348357280306042140481751668215838656488457943830180819301102535170705017482946779698265096226184239631924271857062033454725540956591929965181603262502135610768915716020374362368495244256420143645126927013882334008435586481691725030031204304273292938132599127402133470745819213047706793887965197191137237066440328777206799072470374264316425913530947082957300047105685634407092811630672103242089966046839626911122.7149))
|
|
|
|
SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
|
|
|
|
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
--
|
|
|
|
-- Tests for EXP()
|
|
|
|
--
|
|
|
|
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of EXP():
|
|
|
|
--
|
|
|
|
-- for n in {-20..20}
|
|
|
|
-- do
|
|
|
|
-- x="$n.29837"
|
|
|
|
-- r=$(bc -ql <<< "scale=500 ; e($x)" | head -n 1)
|
|
|
|
-- echo "($x, $r),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
(-20.29837, .000000001529431101152222),
|
|
|
|
(-19.29837, .000000004157424770142192),
|
|
|
|
(-18.29837, .00000001130105220586304),
|
|
|
|
(-17.29837, .00000003071944485366452),
|
|
|
|
(-16.29837, .00000008350410872606600),
|
|
|
|
(-15.29837, .0000002269877013517336),
|
|
|
|
(-14.29837, .0000006170165438681061),
|
|
|
|
(-13.29837, .000001677224859055276),
|
|
|
|
(-12.29837, .000004559169856609741),
|
|
|
|
(-11.29837, .00001239310857408049),
|
|
|
|
(-10.29837, .00003368796183504298),
|
|
|
|
(-9.29837, .00009157337449401917),
|
|
|
|
(-8.29837, .0002489222398577673),
|
|
|
|
(-7.29837, .0006766408013046928),
|
|
|
|
(-6.29837, .001839300394580514),
|
|
|
|
(-5.29837, .004999736839665763),
|
|
|
|
(-4.29837, .01359069379834070),
|
|
|
|
(-3.29837, .03694333598818056),
|
|
|
|
(-2.29837, .1004223988993283),
|
|
|
|
(-1.29837, .2729763820983097),
|
|
|
|
(0.29837, 1.3476603299656679),
|
|
|
|
(1.29837, 3.6633205858807959),
|
|
|
|
(2.29837, 9.9579377804197108),
|
|
|
|
(3.29837, 27.068481317440698),
|
|
|
|
(4.29837, 73.579760889182206),
|
|
|
|
(5.29837, 200.01052696742555),
|
|
|
|
(6.29837, 543.68498095607070),
|
|
|
|
(7.29837, 1477.8890041389891),
|
|
|
|
(8.29837, 4017.3188244304487),
|
|
|
|
(9.29837, 10920.204759575742),
|
|
|
|
(10.29837, 29684.194161006717),
|
|
|
|
(11.29837, 80690.005580314652),
|
|
|
|
(12.29837, 219338.17590722828),
|
|
|
|
(13.29837, 596222.97785597218),
|
|
|
|
(14.29837, 1620702.0864156289),
|
|
|
|
(15.29837, 4405525.0308492653),
|
|
|
|
(16.29837, 11975458.636179032),
|
|
|
|
(17.29837, 32552671.598188404),
|
|
|
|
(18.29837, 88487335.673150406),
|
|
|
|
(19.29837, 240533516.60908059),
|
|
|
|
(20.29837, 653837887.33381570))
|
|
|
|
SELECT x, bc_result, exp(x), exp(x)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
--
|
|
|
|
-- Tests for LN()
|
|
|
|
--
|
|
|
|
|
|
|
|
-- input very small
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(10^-$p)" | head -n 1)
|
|
|
|
-- echo "('1.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('1.0e-1', -2.3025850929940457),
|
|
|
|
('1.0e-2', -4.6051701859880914),
|
|
|
|
('1.0e-3', -6.9077552789821371),
|
|
|
|
('1.0e-4', -9.2103403719761827),
|
|
|
|
('1.0e-5', -11.512925464970228),
|
|
|
|
('1.0e-6', -13.815510557964274),
|
|
|
|
('1.0e-7', -16.118095650958320),
|
|
|
|
('1.0e-8', -18.420680743952365),
|
|
|
|
('1.0e-9', -20.723265836946411),
|
|
|
|
('1.0e-10', -23.025850929940457),
|
|
|
|
('1.0e-11', -25.328436022934503),
|
|
|
|
('1.0e-12', -27.631021115928548),
|
|
|
|
('1.0e-13', -29.933606208922594),
|
|
|
|
('1.0e-14', -32.236191301916640),
|
|
|
|
('1.0e-15', -34.5387763949106853),
|
|
|
|
('1.0e-16', -36.84136148790473094),
|
|
|
|
('1.0e-17', -39.143946580898776628),
|
|
|
|
('1.0e-18', -41.4465316738928223123),
|
|
|
|
('1.0e-19', -43.74911676688686799634),
|
|
|
|
('1.0e-20', -46.051701859880913680360),
|
|
|
|
('1.0e-21', -48.3542869528749593643778),
|
|
|
|
('1.0e-22', -50.65687204586900504839581),
|
|
|
|
('1.0e-23', -52.959457138863050732413803),
|
|
|
|
('1.0e-24', -55.2620422318570964164317949),
|
|
|
|
('1.0e-25', -57.56462732485114210044978637),
|
|
|
|
('1.0e-26', -59.867212417845187784467777822),
|
|
|
|
('1.0e-27', -62.1697975108392334684857692765),
|
|
|
|
('1.0e-28', -64.47238260383327915250376073116),
|
|
|
|
('1.0e-29', -66.774967696827324836521752185847),
|
|
|
|
('1.0e-30', -69.0775527898213705205397436405309),
|
|
|
|
('1.0e-31', -71.38013788281541620455773509521529),
|
|
|
|
('1.0e-32', -73.682722975809461888575726549899655),
|
|
|
|
('1.0e-33', -75.9853080688035075725937180045840189),
|
|
|
|
('1.0e-34', -78.28789316179755325661170945926838306),
|
|
|
|
('1.0e-35', -80.590478254791598940629700913952747266),
|
|
|
|
('1.0e-36', -82.8930633477856446246476923686371114736),
|
|
|
|
('1.0e-37', -85.19564844077969030866568382332147568124),
|
|
|
|
('1.0e-38', -87.498233533773735992683675278005839888842),
|
|
|
|
('1.0e-39', -89.8008186267677816767016667326902040964430),
|
|
|
|
('1.0e-40', -92.10340371976182736071965818737456830404406))
|
|
|
|
SELECT x, bc_result, ln(x::numeric), ln(x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very close to but smaller than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(1-10^-$p)" | head -n 1)
|
|
|
|
-- echo "('1.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('1.0e-1', -.10536051565782630),
|
|
|
|
('1.0e-2', -.010050335853501441),
|
|
|
|
('1.0e-3', -.0010005003335835335),
|
|
|
|
('1.0e-4', -.00010000500033335834),
|
|
|
|
('1.0e-5', -.000010000050000333336),
|
|
|
|
('1.0e-6', -.0000010000005000003333),
|
|
|
|
('1.0e-7', -.00000010000000500000033),
|
|
|
|
('1.0e-8', -.000000010000000050000000),
|
|
|
|
('1.0e-9', -.0000000010000000005000000),
|
|
|
|
('1.0e-10', -.00000000010000000000500000),
|
|
|
|
('1.0e-11', -.000000000010000000000050000),
|
|
|
|
('1.0e-12', -.0000000000010000000000005000),
|
|
|
|
('1.0e-13', -.00000000000010000000000000500),
|
|
|
|
('1.0e-14', -.000000000000010000000000000050),
|
|
|
|
('1.0e-15', -.0000000000000010000000000000005),
|
|
|
|
('1.0e-16', -.00000000000000010000000000000001),
|
|
|
|
('1.0e-17', -.000000000000000010000000000000000),
|
|
|
|
('1.0e-18', -.0000000000000000010000000000000000),
|
|
|
|
('1.0e-19', -.00000000000000000010000000000000000),
|
|
|
|
('1.0e-20', -.000000000000000000010000000000000000),
|
|
|
|
('1.0e-21', -.0000000000000000000010000000000000000),
|
|
|
|
('1.0e-22', -.00000000000000000000010000000000000000),
|
|
|
|
('1.0e-23', -.000000000000000000000010000000000000000),
|
|
|
|
('1.0e-24', -.0000000000000000000000010000000000000000),
|
|
|
|
('1.0e-25', -.00000000000000000000000010000000000000000),
|
|
|
|
('1.0e-26', -.000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-27', -.0000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-28', -.00000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-29', -.000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-30', -.0000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-31', -.00000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-32', -.000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-33', -.0000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-34', -.00000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-35', -.000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-36', -.0000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-37', -.00000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-38', -.000000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-39', -.0000000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-40', -.00000000000000000000000000000000000000010000000000000000))
|
|
|
|
SELECT '1-'||x, bc_result, ln(1.0-x::numeric), ln(1.0-x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very close to but larger than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(1+10^-$p)" | head -n 1)
|
|
|
|
-- echo "('1.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('1.0e-1', .09531017980432486),
|
|
|
|
('1.0e-2', .009950330853168083),
|
|
|
|
('1.0e-3', .0009995003330835332),
|
|
|
|
('1.0e-4', .00009999500033330834),
|
|
|
|
('1.0e-5', .000009999950000333331),
|
|
|
|
('1.0e-6', .0000009999995000003333),
|
|
|
|
('1.0e-7', .00000009999999500000033),
|
|
|
|
('1.0e-8', .000000009999999950000000),
|
|
|
|
('1.0e-9', .0000000009999999995000000),
|
|
|
|
('1.0e-10', .00000000009999999999500000),
|
|
|
|
('1.0e-11', .000000000009999999999950000),
|
|
|
|
('1.0e-12', .0000000000009999999999995000),
|
|
|
|
('1.0e-13', .00000000000009999999999999500),
|
|
|
|
('1.0e-14', .000000000000009999999999999950),
|
|
|
|
('1.0e-15', .0000000000000009999999999999995),
|
|
|
|
('1.0e-16', .00000000000000010000000000000000),
|
|
|
|
('1.0e-17', .000000000000000010000000000000000),
|
|
|
|
('1.0e-18', .0000000000000000010000000000000000),
|
|
|
|
('1.0e-19', .00000000000000000010000000000000000),
|
|
|
|
('1.0e-20', .000000000000000000010000000000000000),
|
|
|
|
('1.0e-21', .0000000000000000000010000000000000000),
|
|
|
|
('1.0e-22', .00000000000000000000010000000000000000),
|
|
|
|
('1.0e-23', .000000000000000000000010000000000000000),
|
|
|
|
('1.0e-24', .0000000000000000000000010000000000000000),
|
|
|
|
('1.0e-25', .00000000000000000000000010000000000000000),
|
|
|
|
('1.0e-26', .000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-27', .0000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-28', .00000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-29', .000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-30', .0000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-31', .00000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-32', .000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-33', .0000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-34', .00000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-35', .000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-36', .0000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-37', .00000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-38', .000000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-39', .0000000000000000000000000000000000000010000000000000000),
|
|
|
|
('1.0e-40', .00000000000000000000000000000000000000010000000000000000))
|
|
|
|
SELECT '1+'||x, bc_result, ln(1.0+x::numeric), ln(1.0+x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very large
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(10^$p)" | head -n 1)
|
|
|
|
-- echo "('1.0e$p', $l),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('1.0e1', 2.3025850929940457),
|
|
|
|
('1.0e2', 4.6051701859880914),
|
|
|
|
('1.0e3', 6.9077552789821371),
|
|
|
|
('1.0e4', 9.2103403719761827),
|
|
|
|
('1.0e5', 11.512925464970228),
|
|
|
|
('1.0e6', 13.815510557964274),
|
|
|
|
('1.0e7', 16.118095650958320),
|
|
|
|
('1.0e8', 18.420680743952365),
|
|
|
|
('1.0e9', 20.723265836946411),
|
|
|
|
('1.0e10', 23.025850929940457),
|
|
|
|
('1.0e11', 25.328436022934503),
|
|
|
|
('1.0e12', 27.631021115928548),
|
|
|
|
('1.0e13', 29.933606208922594),
|
|
|
|
('1.0e14', 32.236191301916640),
|
|
|
|
('1.0e15', 34.538776394910685),
|
|
|
|
('1.0e16', 36.841361487904731),
|
|
|
|
('1.0e17', 39.143946580898777),
|
|
|
|
('1.0e18', 41.446531673892822),
|
|
|
|
('1.0e19', 43.749116766886868),
|
|
|
|
('1.0e20', 46.051701859880914),
|
|
|
|
('1.0e21', 48.354286952874959),
|
|
|
|
('1.0e22', 50.656872045869005),
|
|
|
|
('1.0e23', 52.959457138863051),
|
|
|
|
('1.0e24', 55.262042231857096),
|
|
|
|
('1.0e25', 57.564627324851142),
|
|
|
|
('1.0e26', 59.867212417845188),
|
|
|
|
('1.0e27', 62.169797510839233),
|
|
|
|
('1.0e28', 64.472382603833279),
|
|
|
|
('1.0e29', 66.774967696827325),
|
|
|
|
('1.0e30', 69.077552789821371),
|
|
|
|
('1.0e31', 71.380137882815416),
|
|
|
|
('1.0e32', 73.682722975809462),
|
|
|
|
('1.0e33', 75.985308068803508),
|
|
|
|
('1.0e34', 78.287893161797553),
|
|
|
|
('1.0e35', 80.590478254791599),
|
|
|
|
('1.0e36', 82.893063347785645),
|
|
|
|
('1.0e37', 85.195648440779690),
|
|
|
|
('1.0e38', 87.498233533773736),
|
|
|
|
('1.0e39', 89.800818626767782),
|
|
|
|
('1.0e40', 92.103403719761827))
|
|
|
|
SELECT x, bc_result, ln(x::numeric), ln(x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input huge
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 1000 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..10}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=1000 ; l(10^${p}00)" | head -n 1)
|
|
|
|
-- echo "('1.0e${p}00', $l),"
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('1.0e100', 230.25850929940457),
|
|
|
|
('1.0e200', 460.51701859880914),
|
|
|
|
('1.0e300', 690.77552789821371),
|
|
|
|
('1.0e400', 921.03403719761827),
|
|
|
|
('1.0e500', 1151.2925464970228),
|
|
|
|
('1.0e600', 1381.5510557964274),
|
|
|
|
('1.0e700', 1611.8095650958320),
|
|
|
|
('1.0e800', 1842.0680743952365),
|
|
|
|
('1.0e900', 2072.3265836946411),
|
|
|
|
('1.0e1000', 2302.5850929940457))
|
|
|
|
SELECT x, bc_result, ln(x::numeric), ln(x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
2020-03-01 15:49:25 +01:00
|
|
|
-- inputs with 1000 decimal places
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 2000 and rounded to 1000 decimal
|
|
|
|
-- places
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
(484990182159328900690402236933516249572671683638747490717351807610531884491845416923860371219625151551889257298200816555016472471293780254009492949585031653913930735918829139712249577547959394351523545901788627247613322896296041868431769047433229466634098452564756860190085118463828382895145244362033728480588969626012192733802377468089120757046364393407262957242230928854711898925295251902007136232994524624903257456111389508582206404271734668422903183500589303866613158037169610592539145461637447957948521714058034772237111009429638870236361143304703683377693378577075353794118557951847394763531830696578809001981568860219578880229402696449243344235099860421846016326538272155937175661905904288335499593232232926636205909086901191153907183842087577811871344870731324067822883041265129394268082883745408414994.8967939438561591657171240282983703914075472645212002662497023142663831371447287624846942598424990784971781730103682951722370983277124599054059027055336437808366784501932987082321905202623642371063626378290734289114618092750984153422293450048717769065428713836637664433167768445609659527458911187829232316677137895259433038764404970599325009178297626038331436654541552998098529141205301472138026818453893127265938030066392881979113522757891639646670670272542401773230506961559808927249585675430838495658225557294666522469887436551840596777627408780618586500922973500018513068499587683746133637919751545157547095670767246977244726331271787622126889459658539988980096764323712767863722912919120929339399753431689512753214200090670880647731689804555417871258907716687575767185444541243606329768784843125926070743277339790277626515824924290352180761378846035233155198504033292692893297993698953705472933411199778880561376633444249703838589180474329586470353212010427945060694794274109764269805332803290229,
|
|
|
|
1864.3702986939570026328504202935192533137907736189919154633800554877738455118081651650863235106905871352085850240570561347180517240105510505203972860921397909573687877993477806728098306202020229409548306695695574102950949468160529713610952021974630774784174851619325758380143625473386495586347322798415543385655090746985183329114860118551572428921774322172798724455202876781611633419444058398798142214904998877857425038669920064728855823072107227506485770367799671977282350083029452784747350395161797215115525867416898416360638482342253129160308632504217096916335590470843180746834864303790913372081974355613359678634194879425862536147988835528973291020680020540866655622823550861337486588647231688134992810403147262346312159819432914207194632564009749236609081399504118359354620598232725290537215007867979331582119891661859015726276335168158288396939655310210558566592649049602925182137256134162660116182293851038854455437841571331011002023088829768308520393956515509475418031437505751407687618234418262),
|
|
|
|
(87190145885430429849953615409019208993240447426362428988181639909267773304254748257120061524000254226856815085523676417146197197996896030672521334101413071112068202429835905642444187493717977611730127126387257253646790849384975208460867137315507010888782632024640844766297185244443116696943912406389670302370461137850160539373600494054874979342373255280815156048999900951842673141766630630919020492255966628630634124452614590400422133958133100159154995520080124736657520969784129924799670552560034302960877087853678350801769339861812435411200669026902417951572668727488315537985378304242438181615160041688723201917323705450185975141141262578884689500612295576288125956289035673242989906973367691922065122033180281670221390667818909912035903387888639331486823729897326624516015340.0330856710565117793999512551468220085711713631167607285185762046751452975325645379302403715842570486302993296501788672462090620871511446272026693318239212657949496275318383141403236705902077406660768573015707706831878445598837931116223956945944726162551477136715847593742032488181481888084716920605114101902724395659898621880016853548602514706686907951229872573180602614761229992106144727082722940736406782659562775289407005631298246624198606031298081220736931229256511054595028182057216042683060059115371651410352645266000330509331097811566633211452233019461903115970558624057877018778178814946285827512359903934291318219271464841957435711594154280905473802599888081783098187210283997106131616471807951265003903143099667366508222327805543948921694362089860577380749774036318574113007382111997454202845559941557812813566442364810680529092880773126707073967537693927177460459341763934709686530005721141046645111784404932103241501569571235364365556796422998363930810983452790309019295181282099408260156,
|
|
|
|
1793.5767085750017553306932533574391150814202249805881581227430032600579405884415934520704053351781361105595296647510475380766428668443641914861849764330704062323054023252886955844207807229267936432730818329225450152491146839618683772020068682795388746108876393249306737841247788224204701299467519965182171772253974884845661168860422489046657965359832930382114760565628765599962013955588754803194908990025689040598990346417563277021386852342928910383706995866844541160576254266641602065102228267316550706943783591722246885978355472097314691737807509436806788803362444745551013400341861820755594413819894154786253014501454443272120342005711761286524843010157182464200556865694401941794983935172457481497909987740544409272349152397774548604845897687504977786762391359552407068124283290504752932824699865504970420939586707791994870941813718246825616335675307740641350673558328821461530563823677144691877374809441673507467507447891562257806191361453045937798278733402269265623588493124129181374135958668436774),
|
|
|
|
(93936642222690597390233191619858485419795942047468396309991947772747208870873993801669373075421461116465960407843923269693395211616591453397070258466704654943689268224479477016161636938138334729982904232438440955361656138189836032891825113139184685132178764873033678116450665758561650355252211196676137179184043639278410827092182700922151290703747496962700158844772453483316974221113826173404445159281421213715669245417896170368554410830320000019029956317336703559699859949692222685614036912057150632902650913831404804982509990655560731349634628713944739168096272097122388116038119844786988276635032016787352796502360718569977397214936366251320294621522016.6483354941025384161536675750898007896744690911429670830432784905421638721478353275821072200938900938046264210604940707974410950770029535636602548377806284157951164875821446035013896786653932045182167021839184824627082391478016195098055107001433336586881395912782883663046617432598969149948351689103230162742769845955320418573803127107923535948653168889411316007796459064267436246637115946581149511513369842911210359447262641996566147462977170742544980481275049898092152042927981394239266559286915303786701737610786594006685748456635797125029722684151298695274097006242412384086302106763844070230264910503179385988626477852818174114043927841085089058972074427820150462261941575665882880501074676800316585217150509780489224388148722603385921057007086785238310735038314861960410473809826927329368597558806004392175746233568789445929554890241140656324160187253042639339549705859147930476532359840809944163908006480881926041259363654863689570520534301207043189181147254153307163555433328278834311658232337,
|
|
|
|
1510.4332713542154696529645934345554302578243896764921637693542962119938599884313210100957753316832762996428481801312323020427109678979117469716796746760060470871840325255146954580681101106876674367471955788143763250819168311353856748872452260808797135108102729064040463343792765872545182299889360257515315869180266759715933989413256377582681707188367254513700731642913479683031478361835565783219287780434673712341147656477670848734998849030451414278832848680301511646182446524915091598080243532068451726548537866633622180283865668708517173065893429240665300584705585310049892047293928733753369421499719516009692095913169665213597158441636480707309244604139865130782756488091268094213446272360006907802989573582755585110277620911226015342778471352130366770729972784317323917141031824334355639769512749560550167491709646539950725523461943580211843652293561678342656010571108219244870234329176123205423872844099992204896411752620881541000940129833754169391528449211839693800724450201835161044717173715867437))
|
|
|
|
SELECT trim_scale(ln(x::numeric)-bc_result) AS diff FROM t;
|
|
|
|
|
Improve type numeric's calculations for ln(), log(), exp(), pow().
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
2015-11-14 20:55:38 +01:00
|
|
|
--
|
|
|
|
-- Tests for LOG() (base 10)
|
|
|
|
--
|
|
|
|
|
|
|
|
-- input very small, exact result known
|
|
|
|
WITH t(x) AS (SELECT '1e-'||n FROM generate_series(1, 100) g(n))
|
|
|
|
SELECT x, log(x::numeric) FROM t;
|
|
|
|
|
|
|
|
-- input very small, non-exact results
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..50..7}
|
|
|
|
-- do
|
|
|
|
-- for d in {9..1..3}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l($d*10^-$p) / l(10)" | head -n 1)
|
|
|
|
-- echo "('${d}.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('9.0e-1', -.04575749056067513),
|
|
|
|
('6.0e-1', -.2218487496163564),
|
|
|
|
('3.0e-1', -.5228787452803376),
|
|
|
|
('9.0e-8', -7.045757490560675),
|
|
|
|
('6.0e-8', -7.221848749616356),
|
|
|
|
('3.0e-8', -7.522878745280338),
|
|
|
|
('9.0e-15', -14.0457574905606751),
|
|
|
|
('6.0e-15', -14.2218487496163564),
|
|
|
|
('3.0e-15', -14.5228787452803376),
|
|
|
|
('9.0e-22', -21.04575749056067512540994),
|
|
|
|
('6.0e-22', -21.22184874961635636749123),
|
|
|
|
('3.0e-22', -21.52287874528033756270497),
|
|
|
|
('9.0e-29', -28.045757490560675125409944193490),
|
|
|
|
('6.0e-29', -28.221848749616356367491233202020),
|
|
|
|
('3.0e-29', -28.522878745280337562704972096745),
|
|
|
|
('9.0e-36', -35.0457574905606751254099441934897693816),
|
|
|
|
('6.0e-36', -35.2218487496163563674912332020203916640),
|
|
|
|
('3.0e-36', -35.5228787452803375627049720967448846908),
|
|
|
|
('9.0e-43', -42.04575749056067512540994419348976938159974227),
|
|
|
|
('6.0e-43', -42.22184874961635636749123320202039166403168125),
|
|
|
|
('3.0e-43', -42.52287874528033756270497209674488469079987114),
|
|
|
|
('9.0e-50', -49.045757490560675125409944193489769381599742271618608),
|
|
|
|
('6.0e-50', -49.221848749616356367491233202020391664031681254347196),
|
|
|
|
('3.0e-50', -49.522878745280337562704972096744884690799871135809304))
|
|
|
|
SELECT x, bc_result, log(x::numeric), log(x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very close to but smaller than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40..7}
|
|
|
|
-- do
|
|
|
|
-- for d in {9..1..3}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(1-$d*10^-$p) / l(10)" | head -n 1)
|
|
|
|
-- echo "('${d}.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('9.0e-1', -1.0000000000000000),
|
|
|
|
('6.0e-1', -.3979400086720376),
|
|
|
|
('3.0e-1', -.1549019599857432),
|
|
|
|
('9.0e-8', -.000000039086505130185422),
|
|
|
|
('6.0e-8', -.000000026057669695925208),
|
|
|
|
('3.0e-8', -.000000013028834652530076),
|
|
|
|
('9.0e-15', -.0000000000000039086503371292840),
|
|
|
|
('6.0e-15', -.0000000000000026057668914195188),
|
|
|
|
('3.0e-15', -.0000000000000013028834457097574),
|
|
|
|
('9.0e-22', -.00000000000000000000039086503371292664),
|
|
|
|
('6.0e-22', -.00000000000000000000026057668914195110),
|
|
|
|
('3.0e-22', -.00000000000000000000013028834457097555),
|
|
|
|
('9.0e-29', -.000000000000000000000000000039086503371292664),
|
|
|
|
('6.0e-29', -.000000000000000000000000000026057668914195110),
|
|
|
|
('3.0e-29', -.000000000000000000000000000013028834457097555),
|
|
|
|
('9.0e-36', -.0000000000000000000000000000000000039086503371292664),
|
|
|
|
('6.0e-36', -.0000000000000000000000000000000000026057668914195110),
|
|
|
|
('3.0e-36', -.0000000000000000000000000000000000013028834457097555))
|
|
|
|
SELECT '1-'||x, bc_result, log(1.0-x::numeric), log(1.0-x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very close to but larger than 1
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {1..40..7}
|
|
|
|
-- do
|
|
|
|
-- for d in {9..1..3}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l(1+$d*10^-$p) / l(10)" | head -n 1)
|
|
|
|
-- echo "('${d}.0e-$p', $l),"
|
|
|
|
-- done
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('9.0e-1', .2787536009528290),
|
|
|
|
('6.0e-1', .2041199826559248),
|
|
|
|
('3.0e-1', .1139433523068368),
|
|
|
|
('9.0e-8', .000000039086501612400118),
|
|
|
|
('6.0e-8', .000000026057668132465074),
|
|
|
|
('3.0e-8', .000000013028834261665042),
|
|
|
|
('9.0e-15', .0000000000000039086503371292489),
|
|
|
|
('6.0e-15', .0000000000000026057668914195031),
|
|
|
|
('3.0e-15', .0000000000000013028834457097535),
|
|
|
|
('9.0e-22', .00000000000000000000039086503371292664),
|
|
|
|
('6.0e-22', .00000000000000000000026057668914195110),
|
|
|
|
('3.0e-22', .00000000000000000000013028834457097555),
|
|
|
|
('9.0e-29', .000000000000000000000000000039086503371292664),
|
|
|
|
('6.0e-29', .000000000000000000000000000026057668914195110),
|
|
|
|
('3.0e-29', .000000000000000000000000000013028834457097555),
|
|
|
|
('9.0e-36', .0000000000000000000000000000000000039086503371292664),
|
|
|
|
('6.0e-36', .0000000000000000000000000000000000026057668914195110),
|
|
|
|
('3.0e-36', .0000000000000000000000000000000000013028834457097555))
|
|
|
|
SELECT '1+'||x, bc_result, log(1.0+x::numeric), log(1.0+x::numeric)-bc_result AS diff FROM t;
|
|
|
|
|
|
|
|
-- input very large, exact result known
|
|
|
|
WITH t(x) AS (SELECT '1e'||n FROM generate_series(1, 100) g(n))
|
|
|
|
SELECT x, log(x::numeric) FROM t;
|
|
|
|
|
|
|
|
-- input very large, non-exact results
|
|
|
|
--
|
|
|
|
-- bc(1) results computed with a scale of 500 and truncated using the script
|
|
|
|
-- below, and then rounded by hand to match the precision of LN():
|
|
|
|
--
|
|
|
|
-- for p in {10..50..7}
|
|
|
|
-- do
|
|
|
|
-- for d in {2..9..3}
|
|
|
|
-- do
|
|
|
|
-- l=$(bc -ql <<< "scale=500 ; l($d*10^$p) / l(10)" | head -n 1)
|
|
|
|
-- echo "('${d}.0e$p', $l),"
|
|
|
|
-- done
|
|
|
|
-- done
|
|
|
|
|
|
|
|
WITH t(x, bc_result) AS (VALUES
|
|
|
|
('2.0e10', 10.301029995663981),
|
|
|
|
('5.0e10', 10.698970004336019),
|
|
|
|
('8.0e10', 10.903089986991944),
|
|
|
|
('2.0e17', 17.301029995663981),
|
|
|
|
('5.0e17', 17.698970004336019),
|
|
|
|
('8.0e17', 17.903089986991944),
|
|
|
|
('2.0e24', 24.301029995663981),
|
|
|
|
('5.0e24', 24.698970004336019),
|
|
|
|
('8.0e24', 24.903089986991944),
|
|
|
|
('2.0e31', 31.301029995663981),
|
|
|
|
('5.0e31', 31.698970004336019),
|
|
|
|
('8.0e31', 31.903089986991944),
|
|
|
|
('2.0e38', 38.301029995663981),
|
|
|
|
('5.0e38', 38.698970004336019),
|
|
|
|
('8.0e38', 38.903089986991944),
|
|
|
|
('2.0e45', 45.30102999566398),
|
|
|
|
('5.0e45', 45.69897000433602),
|
|
|
|
('8.0e45', 45.90308998699194))
|
|
|
|
SELECT x, bc_result, log(x::numeric), log(x::numeric)-bc_result AS diff FROM t;
|