It looks like Maple also has problems with the last few digits. I
guess you need to evaluate to several digits beyond what you want to
ensure that the ones you want are accurate:
> a := ((2/(27+3*sqrt(69)))**(1/3) +((27+3*sqrt(69))/2)**(1/3)/
3)**30000;
30000
/ (1/3) (1/3)\
| (1/3) / 1 \ 1 /27 3 (1/2)\ |
|2 |--------------| + - |-- + - 69 | |
| | (1/2)| 3 \2 2 / |
\ \27 + 3 69 / /
> evalf(a, 6000);
5.043165960656654932150594692427574810776102746808935031678444677128757182264
\
862106103980765819340682753857825408213309478700354777871766946859877182545
\
765655157863751804030841080877201973602630453775992992526007874643437115478
\
340662656654242263167137970470876460028676595720882553729941413991192954257
\
293905515987341623164931728987789298548218169139563816353624128215994510253
\
193649036511102215152338143645279048930913371354240272588991921619868232355
\
282955810598732001078755504975632193024344987469732081489429513875050815089
\
544676150938253329884107629370887783660465530686857728553866459504826468144
\
393019990165743377364065922255679117090517971285490773119127788881441087366
\
597038621740186411069175749952023659419640302178959561265901520782425747790
\
756781859047252118783439795010489532686142903402653847407905146487735500216
\
875475845252225041904360870681078279227104457227362368796693241733784161166
\
959198410304677682964919409043870523002732003770237480443431032331691094261
\
441144380298348751920199059562790511888049750265002682883521913896858496289
\
686123578807667253851478579344449553914415375194372851217961223158428624231
\
899492121175279814486360858048056575120864817906982359465546791033345570306
\
144306889182376593558188718231183977511187240921162728629126713811608658037
\
408930071516876158725341127622968866605210362611399728215256794224599912431
\
852531227205445323135206226637942724364187507359956181911218068991795777326
\
796261608752343775804668422381361446090681588594231684449905631308569143981
\
611046414359764645058273105320423572778055041510761287441331660970400343561
\
555943556993103725670073628460112439243800517211794664815158651771479160942
\
653186737535761672899827891567658199185014614703946969879000550578218043674
\
277377855307965292573100704405035528657706565889113413367817192934796097513
\
499651992416721568858204193236978264956752167858467613016650514975083735547
\
048453768089171065791820462966374091485132898737129287935143459175813721830
\
821381768374444936696984757491555027227218191496737306894525429493301724974
\
763488028789233809728809898199416269816627876644539316701546354316106609944
\
787820220498246818854200290468467630982622988930679820858730060272235476735
\
450044805739148100924911019338705887449906544811444564378814239220572668180
\
278032797347013668175334692909590627108897158421082548983131341387676127885
\
490486578249126248836394203188792652324856135704773997090467104480778132423
\
835257758424180335745170573652705419795069195190885851098538390625498507467
\
309419611165780838698713310344086360548451070819222570296482527242703056995
\
074918767257414142822271890334717277643954342541859509907472906259527629621
\
846811719367901786024179895120064929613306517810763374494997293082055502432
\
076618464586027616119815676773736360189991491957175057346133494837930827478
\
266538261093888716461076377076699523221210638410121119502728028192147321213
\
391939138517117401805454360045347532235299958021923292027677884065299422286
\
086252105068056807793476844454728320229479895050738642250650424537785366800
\
824269336818816623176949556726361708670651772685686418825606961363993789380
\
923630615342066499106771216678307265280650651154307878220861870678168941097
\
431704060159606780139617770602048383255107304641825616786778667766641145888
\
295077288459709643465468572600850673752055883218340619680664771369547546505
\
826597221827875270380133508853607738471589700951963694066449868459559318968
\
227624570572944412896668467196250357904359310094303613350580572837195254247
\
505334922434012206891543167881708739008884991490588279135076804550682803394
\
454230822114888818737649635439941364151152126830053340927532304714488156842
\
955217262531239581967894186558682010795291845929630207675781253000000000000
\
000000000000000000000000000000000000000000000000000000000000000000000000000
\
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\
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\
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000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000
\
000000000000000000000000000000000000000000000000000000000000000000000000000
\
000000000000000000013089481144651747951236035489454929072510241720588227917
\
512706315775959573028069494819317749753909502880822522513852251922077408850
\
982456907715693706849952685092996714802140486783196224591724342709356982736
\
044367860439732958936561891275647223613057765646621093148215162220607946164
\
242694428625785134442405595081929385908408997293032313197351343122456184929
\
825694410538985099005649944281269485885003854631983177652291928247833224724
\
3663
73947007368525562887408463240932824486779007274030664120841078472914537396
10
> evalf(a, 3667);
5.043165960656654932150594692427574810776102746808935031678444677128757182264
\
862106103980765819340682753857825408213309478700354777871766946859877182545
\
765655157863751804030841080877201973602630453775992992526007874643437115478
\
340662656654242263167137970470876460028676595720882553729941413991192954257
\
293905515987341623164931728987789298548218169139563816353624128215994510253
\
193649036511102215152338143645279048930913371354240272588991921619868232355
\
282955810598732001078755504975632193024344987469732081489429513875050815089
\
544676150938253329884107629370887783660465530686857728553866459504826468144
\
393019990165743377364065922255679117090517971285490773119127788881441087366
\
597038621740186411069175749952023659419640302178959561265901520782425747790
\
756781859047252118783439795010489532686142903402653847407905146487735500216
\
875475845252225041904360870681078279227104457227362368796693241733784161166
\
959198410304677682964919409043870523002732003770237480443431032331691094261
\
441144380298348751920199059562790511888049750265002682883521913896858496289
\
686123578807667253851478579344449553914415375194372851217961223158428624231
\
899492121175279814486360858048056575120864817906982359465546791033345570306
\
144306889182376593558188718231183977511187240921162728629126713811608658037
\
408930071516876158725341127622968866605210362611399728215256794224599912431
\
852531227205445323135206226637942724364187507359956181911218068991795777326
\
796261608752343775804668422381361446090681588594231684449905631308569143981
\
611046414359764645058273105320423572778055041510761287441331660970400343561
\
555943556993103725670073628460112439243800517211794664815158651771479160942
\
653186737535761672899827891567658199185014614703946969879000550578218043674
\
277377855307965292573100704405035528657706565889113413367817192934796097513
\
499651992416721568858204193236978264956752167858467613016650514975083735547
\
048453768089171065791820462966374091485132898737129287935143459175813721830
\
821381768374444936696984757491555027227218191496737306894525429493301724974
\
763488028789233809728809898199416269816627876644539316701546354316106609944
\
787820220498246818854200290468467630982622988930679820858730060272235476735
\
450044805739148100924911019338705887449906544811444564378814239220572668180
\
278032797347013668175334692909590627108897158421082548983131341387676127885
\
490486578249126248836394203188792652324856135704773997090467104480778132423
\
835257758424180335745170573652705419795069195190885851098538390625498507467
\
309419611165780838698713310344086360548451070819222570296482527242703056995
\
074918767257414142822271890334717277643954342541859509907472906259527629621
\
846811719367901786024179895120064929613306517810763374494997293082055502432
\
076618464586027616119815676773736360189991491957175057346133494837930827478
\
266538261093888716461076377076699523221210638410121119502728028192147321213
\
391939138517117401805454360045347532235299958021923292027677884065299422286
\
086252105068056807793476844454728320229479895050738642250650424537785366800
\
824269336818816623176949556726361708670651772685686418825606961363993789380
\
923630615342066499106771216678307265280650651154307878220861870678168941097
\
431704060159606780139617770602048383255107304641825616786778667766641145888
\
295077288459709643465468572600850673752055883218340619680664771369547546505
\
826597221827875270380133508853607738471589700951963694066449868459559318968
\
227624570572944412896668467196250357904359310094303613350580572837195254247
\
505334922434012206891543167881708739008884991490588279135076804550682803394
\
454230822114888818737649635439941364151152126830053340927532304714488156842
\
3663
955217262531239581967894186558682010795291845929630207675781270946 10
> evalf(a, 3668);
5.043165960656654932150594692427574810776102746808935031678444677128757182264
\
862106103980765819340682753857825408213309478700354777871766946859877182545
\
765655157863751804030841080877201973602630453775992992526007874643437115478
\
340662656654242263167137970470876460028676595720882553729941413991192954257
\
293905515987341623164931728987789298548218169139563816353624128215994510253
\
193649036511102215152338143645279048930913371354240272588991921619868232355
\
282955810598732001078755504975632193024344987469732081489429513875050815089
\
544676150938253329884107629370887783660465530686857728553866459504826468144
\
393019990165743377364065922255679117090517971285490773119127788881441087366
\
597038621740186411069175749952023659419640302178959561265901520782425747790
\
756781859047252118783439795010489532686142903402653847407905146487735500216
\
875475845252225041904360870681078279227104457227362368796693241733784161166
\
959198410304677682964919409043870523002732003770237480443431032331691094261
\
441144380298348751920199059562790511888049750265002682883521913896858496289
\
686123578807667253851478579344449553914415375194372851217961223158428624231
\
899492121175279814486360858048056575120864817906982359465546791033345570306
\
144306889182376593558188718231183977511187240921162728629126713811608658037
\
408930071516876158725341127622968866605210362611399728215256794224599912431
\
852531227205445323135206226637942724364187507359956181911218068991795777326
\
796261608752343775804668422381361446090681588594231684449905631308569143981
\
611046414359764645058273105320423572778055041510761287441331660970400343561
\
555943556993103725670073628460112439243800517211794664815158651771479160942
\
653186737535761672899827891567658199185014614703946969879000550578218043674
\
277377855307965292573100704405035528657706565889113413367817192934796097513
\
499651992416721568858204193236978264956752167858467613016650514975083735547
\
048453768089171065791820462966374091485132898737129287935143459175813721830
\
821381768374444936696984757491555027227218191496737306894525429493301724974
\
763488028789233809728809898199416269816627876644539316701546354316106609944
\
787820220498246818854200290468467630982622988930679820858730060272235476735
\
450044805739148100924911019338705887449906544811444564378814239220572668180
\
278032797347013668175334692909590627108897158421082548983131341387676127885
\
490486578249126248836394203188792652324856135704773997090467104480778132423
\
835257758424180335745170573652705419795069195190885851098538390625498507467
\
309419611165780838698713310344086360548451070819222570296482527242703056995
\
074918767257414142822271890334717277643954342541859509907472906259527629621
\
846811719367901786024179895120064929613306517810763374494997293082055502432
\
076618464586027616119815676773736360189991491957175057346133494837930827478
\
266538261093888716461076377076699523221210638410121119502728028192147321213
\
391939138517117401805454360045347532235299958021923292027677884065299422286
\
086252105068056807793476844454728320229479895050738642250650424537785366800
\
824269336818816623176949556726361708670651772685686418825606961363993789380
\
923630615342066499106771216678307265280650651154307878220861870678168941097
\
431704060159606780139617770602048383255107304641825616786778667766641145888
\
295077288459709643465468572600850673752055883218340619680664771369547546505
\
826597221827875270380133508853607738471589700951963694066449868459559318968
\
227624570572944412896668467196250357904359310094303613350580572837195254247
\
505334922434012206891543167881708739008884991490588279135076804550682803394
\
454230822114888818737649635439941364151152126830053340927532304714488156842
\
3663
9552172625312395819678941865586820107952918459296302076757812481044
10
Aaron Meurer
On Jul 8, 2009, at 2:07 PM, Ondrej Certik wrote:
>
> Hi,
>
> while waiting for a cofee, I browsed an old Mathematica book from
> 1996, and let me assure you, we still have things to catch up on
> almost every page. :) But this caught my attention, try it in sympy:
>
> In [1]: e = ((2/(27+3*sqrt(69)))**(S(1)/3) +
> ((27+3*sqrt(69))/2)**(S(1)/3)/3)**30000
>
> In [2]: e
> Out[2]:
> 30000
> ⎛
> ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎞
> ⎜ ╱ ⎽⎽⎽⎽ ⎟
> ⎜ ╱ 3⋅╲╱ 69 ⎟
> ⎜ 3 ⎽⎽⎽ 3 ╱ 27/2 + ────────
> ⎟
> ⎜ ╲╱ 2 ╲╱ 2 ⎟
> ⎜────────────────── +
> ──────────────────────⎟
> ⎜ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
> 3 ⎟
> ⎜3 ╱ ⎽⎽⎽⎽ ⎟
> ⎝╲╱ 27 + 3⋅╲╱ 69 ⎠
>
> In [3]: e.n(6000)
> Out[3]:
> 504316596065665493215059469242757481077610274680893503167844467712875718226486
> 210610398076581934068275385782540821330947870035477787176694685987718254576565
> 515786375180403084108087720197360263045377599299252600787464343711547834066265
> 665424226316713797047087646002867659572088255372994141399119295425729390551598
> 734162316493172898778929854821816913956381635362412821599451025319364903651110
> 221515233814364527904893091337135424027258899192161986823235528295581059873200
> 107875550497563219302434498746973208148942951387505081508954467615093825332988
> 410762937088778366046553068685772855386645950482646814439301999016574337736406
> 592225567911709051797128549077311912778888144108736659703862174018641106917574
> 995202365941964030217895956126590152078242574779075678185904725211878343979501
> 048953268614290340265384740790514648773550021687547584525222504190436087068107
> 827922710445722736236879669324173378416116695919841030467768296491940904387052
> 300273200377023748044343103233169109426144114438029834875192019905956279051188
> 804975026500268288352191389685849628968612357880766725385147857934444955391441
> 537519437285121796122315842862423189949212117527981448636085804805657512086481
> 790698235946554679103334557030614430688918237659355818871823118397751118724092
> 116272862912671381160865803740893007151687615872534112762296886660521036261139
> 972821525679422459991243185253122720544532313520622663794272436418750735995618
> 191121806899179577732679626160875234377580466842238136144609068158859423168444
> 990563130856914398161104641435976464505827310532042357277805504151076128744133
> 166097040034356155594355699310372567007362846011243924380051721179466481515865
> 177147916094265318673753576167289982789156765819918501461470394696987900055057
> 821804367427737785530796529257310070440503552865770656588911341336781719293479
> 609751349965199241672156885820419323697826495675216785846761301665051497508373
> 554704845376808917106579182046296637409148513289873712928793514345917581372183
> 082138176837444493669698475749155502722721819149673730689452542949330172497476
> 348802878923380972880989819941626981662787664453931670154635431610660994478782
> 022049824681885420029046846763098262298893067982085873006027223547673545004480
> 573914810092491101933870588744990654481144456437881423922057266818027803279734
> 701366817533469290959062710889715842108254898313134138767612788549048657824912
> 624883639420318879265232485613570477399709046710448077813242383525775842418033
> 574517057365270541979506919519088585109853839062549850746730941961116578083869
> 871331034408636054845107081922257029648252724270305699507491876725741414282227
> 189033471727764395434254185950990747290625952762962184681171936790178602417989
> 512006492961330651781076337449499729308205550243207661846458602761611981567677
> 373636018999149195717505734613349483793082747826653826109388871646107637707669
> 952322121063841012111950272802819214732121339193913851711740180545436004534753
> 223529995802192329202767788406529942228608625210506805680779347684445472832022
> 947989505073864225065042453778536680082426933681881662317694955672636170867065
> 177268568641882560696136399378938092363061534206649910677121667830726528065065
> 115430787822086187067816894109743170406015960678013961777060204838325510730464
> 182561678677866776664114588829507728845970964346546857260085067375205588321834
> 061968066477136954754650582659722182787527038013350885360773847158970095196369
> 406644986845955931896822762457057294441289666846719625035790435931009430361335
> 058057283719525424750533492243401220689154316788170873900888499149058827913507
> 680455068280339445423082211488881873764963543994136415115212683005334092753230
> 4714488156842955217262531239581967894186558682010795291845929630207675781253.0
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000000130894811446517479512360354894549290725102
> 417205882279175127063157759595730280694948193177497539095028808225225138522519
> 220774088509824569077156937068499526850929967148021404867831962245917243427093
> 569827360443678604397329589365618912756472236130577656466210931482151622206079
> 461642426944286257851344424055950819293859084089972930323131973513431224561849
> 298256944105389850990056499442812694858850038546319831776522919282478332247247
> 3947007368525562887408463240932824486779007274030664120841078472914502641
>
>
>
> Look at the number of zeros, very nice number. Btw, the numbers are
> correct, or at least the same as in the Mathematica book. However, try
> this:
>
> In [4]: e.n(3667)
> Out[4]:
> 504316596065665493215059469242757481077610274680893503167844467712875718226486
> 210610398076581934068275385782540821330947870035477787176694685987718254576565
> 515786375180403084108087720197360263045377599299252600787464343711547834066265
> 665424226316713797047087646002867659572088255372994141399119295425729390551598
> 734162316493172898778929854821816913956381635362412821599451025319364903651110
> 221515233814364527904893091337135424027258899192161986823235528295581059873200
> 107875550497563219302434498746973208148942951387505081508954467615093825332988
> 410762937088778366046553068685772855386645950482646814439301999016574337736406
> 592225567911709051797128549077311912778888144108736659703862174018641106917574
> 995202365941964030217895956126590152078242574779075678185904725211878343979501
> 048953268614290340265384740790514648773550021687547584525222504190436087068107
> 827922710445722736236879669324173378416116695919841030467768296491940904387052
> 300273200377023748044343103233169109426144114438029834875192019905956279051188
> 804975026500268288352191389685849628968612357880766725385147857934444955391441
> 537519437285121796122315842862423189949212117527981448636085804805657512086481
> 790698235946554679103334557030614430688918237659355818871823118397751118724092
> 116272862912671381160865803740893007151687615872534112762296886660521036261139
> 972821525679422459991243185253122720544532313520622663794272436418750735995618
> 191121806899179577732679626160875234377580466842238136144609068158859423168444
> 990563130856914398161104641435976464505827310532042357277805504151076128744133
> 166097040034356155594355699310372567007362846011243924380051721179466481515865
> 177147916094265318673753576167289982789156765819918501461470394696987900055057
> 821804367427737785530796529257310070440503552865770656588911341336781719293479
> 609751349965199241672156885820419323697826495675216785846761301665051497508373
> 554704845376808917106579182046296637409148513289873712928793514345917581372183
> 082138176837444493669698475749155502722721819149673730689452542949330172497476
> 348802878923380972880989819941626981662787664453931670154635431610660994478782
> 022049824681885420029046846763098262298893067982085873006027223547673545004480
> 573914810092491101933870588744990654481144456437881423922057266818027803279734
> 701366817533469290959062710889715842108254898313134138767612788549048657824912
> 624883639420318879265232485613570477399709046710448077813242383525775842418033
> 574517057365270541979506919519088585109853839062549850746730941961116578083869
> 871331034408636054845107081922257029648252724270305699507491876725741414282227
> 189033471727764395434254185950990747290625952762962184681171936790178602417989
> 512006492961330651781076337449499729308205550243207661846458602761611981567677
> 373636018999149195717505734613349483793082747826653826109388871646107637707669
> 952322121063841012111950272802819214732121339193913851711740180545436004534753
> 223529995802192329202767788406529942228608625210506805680779347684445472832022
> 947989505073864225065042453778536680082426933681881662317694955672636170867065
> 177268568641882560696136399378938092363061534206649910677121667830726528065065
> 115430787822086187067816894109743170406015960678013961777060204838325510730464
> 182561678677866776664114588829507728845970964346546857260085067375205588321834
> 061968066477136954754650582659722182787527038013350885360773847158970095196369
> 406644986845955931896822762457057294441289666846719625035790435931009430361335
> 058057283719525424750533492243401220689154316788170873900888499149058827913507
> 680455068280339445423082211488881873764963543994136415115212683005334092753230
> 4714488156842955217262531239581967894186558682010795291845929630207675781252.9
> 99
>
>
> as you can see, the last 4 digits are wrong. Or not? If you try:
>
> In [6]: e.n(3668)
> [...]
> 4714488156842955217262531239581967894186558682010795291845929630207675781253.0
> 000
>
> so I am confused about those digits at the end. But apparently they
> can't be trusted.
>
> Ondrej
>
> >
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