I have a numerical computation problem which requires solving nonlinear equations (with long integers) in multiple precision.I tried an MPFR C++ wrapper from this link by Pavel:mpfr C++ wrapper by PavelThe wrapper can be downloaded here:mpfrc++-3.5.6.zipHowever, there is precision loss in the solution when handling very long integers (equations with small integers worked well).I tried three options as in the sample code below:to use the code immediately does not work with constant integer toolong error. below the two options worked with precision lossHow can I handle such equations in multiple precision without precision loss?Below is a sample of my nonlinear equations with the long integers:#include mpreal.hVectorXmp testfun126(VectorXmp x0){    VectorXmp value(3);    mpreal x,y,z;    x=x0(0);    y=x0(1);    z=x0(2);// Working option 1: write all the long integers into double ( but there are precision loss )value(0)=(256.*((256.*(-24610976415716501050652227.*x+256.*(-10153609683556422184100.+374519398571124540883.*y-4145573659500944095488.*z))*(29427736469514379027531261659072347.+58899562724319710108573382000184640.*y-1732944474195510410991057714955859184.*z))/pow(5042560366642267.*x-256.*(2446745837411900.+4901398098088043.*y-144207654645973248.*z),3)-(256.*(-308518681989548429992935348850261.+41445095210006425938788783390458.*y-1638970396838251453451269879637336.*z)*(-801790542801929135637671.-732048260009923946735424.*x+56975701334774517040256.*y-187552638032246240630656.*z))/pow(-3075770275504817.+198931044892562752.*x+14199788245258112.*y-1122852841901814912.*z,3)+(5*(-89303793175477833893354121208000.+6533090911353242906294143748495.*y-32276910383172707359896832089932.*z)*(-61468981380127448102256.-5328427636421850183140.*x+4647710007810227520885.*y-13344414478836548348450.*z))/pow(-46366672189358032.-18896234711237580.*x+3927118781169095.*y+14705346416259850.*z,3)-(3*(9101665097092871812176.+3063507166600182944940.*x+6945927557350563805665.*y+1052001549322007294950.*z)*(19493858980629008651267653094056.+93282964805436900100617577630195.*y+42271355681070699741325611572830.*z))/pow(46366672189358032.+18896234711237580.*x-3927118781169095.*y-14705346416259850.*z,3)-(4*(39553725461800043367392.+17203831108841472538824.*x+45483386678520344593037.*y+2703260049547565568088.*z)*(52830583937680669669892057655944.+303023948138837354463602341532495.*y+134962043561465977901954677856080.*z))/pow(92856945980914656.+51329763147513032.*x-8586501277743859.*y-56199770659759016.*z,3)-((22670037111266004087968.+12461845278544574559640.*x+39219302812923818032157.*y-46563087562792926056.*z)*(95973949246309465842551069546976.+723429769797021053206211106031819.*y+317530466286898645427564085427048.*z))/pow(50159316775994592.+36243094308305160.*x-4827156544231217.*y-52318895858217464.*z,3)-(80.*(4157117722725769078952.+4534359335248895646832.*x+26193979470458655189977.*y-2382852476120229696128.*z)*(205429639975670471114284923188348.+2095815907391732802212116237430935.*y+883539023887333564964405237094400.*z))/pow(45070329471431608.+130124049256651728.*x-5583613021604317.*y-387630670566282112.*z,3)-(16.*(9439334964924689507817.+17499514376929345709248.*x+187907876794815451253888.*y-21704870055089718153088.*z)*(943164674716649969807523653958385.+18130967224506023673179633045358720.*y+7486136216172114262568716503454336.*z))/pow(-3075770275504817.+198931044892562752.*x+14199788245258112.*y-1122852841901814912.*z,3)+(80.*(2304705299858575630109.*x-256.*(204828849006588248100.+19508530860149228990861.*y-2445924471668591306496.*z))*(-179928369646271075844345534739549.+3401432279430696137250330740801392.*y+12500875943051297916024009205116096.*z))/pow(5042560366642267.*x-256.*(2446745837411900.+4901398098088043.*y-144207654645973248.*z),3)+(80.*(-805507884940017483975376678503744.+52529278437993151034132605337909.*y-620040027953848498781390188900552.*z)*(-716026618045942942760.*x+243780804476456624597.*y-8*(408351630952413337484.+89777022692195474597.*z)))/pow(-50159316775994592.-36243094308305160.*x+4827156544231217.*y+52318895858217464.*z,3)+(768.*(61889933231497708820968.+30294916915069669525488.*x-4484037822343607626207.*y+13934625423713945278848.*z)*(16858970779944867265671037333379.*y-176.*(1546216290476124632111328928258.+3134171189636832381705249359145.*z)))/pow(45070329471431608.+130124049256651728.*x-5583613021604317.*y-387630670566282112.*z,3)-(40.*(1717566388539311579248.*x+7025931019459451548321.*y+48.*(46537098413809906919.-8301700878138964680.*z))*(3434616943638241443585000648954199.*y+320.*(1107265969195848092307625165761.+4643932844541992753284837619195.*z)))/pow(85141430232132048.+97951351741329392.*x-8855616621991191.*y-199920422688690560.*z,3)+(12.*(88457226224862447127008.+13504083955712971035976.*x-6622138801690554356387.*y+19322683651036147287512.*z)*(36451820000039413375829754767131.*y-8*(66864837166560711793644210325852.+35619205657210451197984743698883.*z)))/pow(92856945980914656.+51329763147513032.*x-8586501277743859.*y-56199770659759016.*z,3)+(512.*(45619694076424722199344.+14936846773318822792976.*x-3365788117861218576473.*y+10130491989577935272320.*z)*(12048859085295019197936041733505.*y-6*(32519187452933223586671104614156.+40471151781636260063426632487709.*z)))/pow(85141430232132048.+97951351741329392.*x-8855616621991191.*y-199920422688690560.*z,3)))/125.;value(1)=(128.*((32768.*(24610976415716501050652227.*x-256.*(-10153609683556422184100.+374519398571124540883.*y-4145573659500944095488.*z))*(98990697209366584150952278657452.+920305667567495470446459093752885.*x-65799721166407263195366683527104.*z))/pow(5042560366642267.*x-256.*(2446745837411900.+4901398098088043.*y-144207654645973248.*z),3)+(1024.*(-10864227594859409007678067839115.+566592725765813239786863532667460.*x-3214793226869529893757297514562848.*z)*(9439334964924689507817.+17499514376929345709248.*x+187907876794815451253888.*y-217048700550897181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69078952.+4534359335248895646832.*x+26193979470458655189977.*y-2382852476120229696128.*z))/pow(45070329471431608.+130124049256651728.*x-5583613021604317.*y-387630670566282112.*z,3)+(1600.*(100744894915663705876272277122960.+74302925512671884052557401907120.*x+343788061485767567210745697763531.*y)*(1717566388539311579248.*x+7025931019459451548321.*y+48.*(46537098413809906919.-8301700878138964680.*z)))/pow(85141430232132048.+97951351741329392.*x-8855616621991191.*y-199920422688690560.*z,3)+(16.*(72249495731635781189477972681776.+39691308285862330678445510678381.*x+125252403980353077736842003056195.*y)*(22670037111266004087968.+12461845278544574559640.*x+39219302812923818032157.*y-46563087562792926056.*z))/pow(50159316775994592.+36243094308305160.*x-4827156544231217.*y-52318895858217464.*z,3)+(640.*(505227745581172894057712966825000.+155010006988462124695347547225138.*x-39602745269819371968458588313429.*y)*(3266813047619306699872.+716026618045942942760.*x-243780804476456624597.*y+718216181537563796776.*z))/pow(50159316775994592.+36243094308305160.*x-4827156544231217.*y-52318895858217464.*z,3)+(2*(356681541401645116923690413208956.+126814067043212099223976834718490.*x+191427609122898840477329914007915.*y)*(9101665097092871812176.+3063507166600182944940.*x+6945927557350563805665.*y+1052001549322007294950.*z))/pow(46366672189358032.+18896234711237580.*x-3927118781169095.*y-14705346416259850.*z,3)+(8*(301993014170585471859024964195112.+134962043561465977901954677856080.*x+309197308873592242001670976702725.*y)*(39553725461800043367392.+17203831108841472538824.*x+45483386678520344593037.*y+2703260049547565568088.*z))/pow(92856945980914656.+51329763147513032.*x-8586501277743859.*y-56199770659759016.*z,3)+(128.*(4874430224431350455160317539284048.+1942615285518540483044478359410032.*x-372364031472286149332017066304111.*y)*(45619694076424722199344.+14936846773318822792976.*x-3365788117861218576473.*y+10130491989577935272320.*z))/pow(85141430232132048.+97951351741329392.*x-8855616621991191.*y-199920422688690560.*z,3)+((1486971442137244004077030949061728.+322769103831727073598968320899320.*x-117921178154991189983038750880755.*y)*(61468981380127448102256.+5328427636421850183140.*x-4647710007810227520885.*y+13344414478836548348450.*z))/pow(46366672189358032.+18896234711237580.*x-3927118781169095.*y-14705346416259850.*z,3)+(512.*(3005184872892536482128059816733656.+1654842388128247497540371661628560.*x-221674333970538545014500438709313.*y)*(61889933231497708820968.+30294916915069669525488.*x-4484037822343607626207.*y+13934625423713945278848.*z))/pow(45070329471431608.+130124049256651728.*x-5583613021604317.*y-387630670566282112.*z,3)+(192.*(137644881571986015841084811827840.+35619205657210451197984743698883.*x-10947203161441300431632175319262.*y)*(88457226224862447127008.+13504083955712971035976.*x-6622138801690554356387.*y+19322683651036147287512.*z))/pow(92856945980914656.+51329763147513032.*x-8586501277743859.*y-56199770659759016.*z,3)+(64.*(13728575451141247570683309821008705.+13111763174706011627610159037098688.*x-935548712435961241962462081828256.*y)*(801790542801929135637671.+732048260009923946735424.*x-56975701334774517040256.*y+187552638032246240630656.*z))/pow(-3075770275504817.+198931044892562752.*x+14199788245258112.*y-1122852841901814912.*z,3)))/125.;// This will work: all long interger being written into a string ( but there are precisiion loss)//value(0)=(256*((256*(-24610976415716501050652227*x+256*(-10153609683556422184100+374519398571124540883*y-4145573659500944095488*z))*(29427736469514379027531261659072347+58899562724319710108573382000184640*y-1732944474195510410991057714955859184*z))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)-(256*(-308518681989548429992935348850261+41445095210006425938788783390458*y-1638970396838251453451269879637336*z)*(-801790542801929135637671-732048260009923946735424*x+56975701334774517040256*y-187552638032246240630656*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)+(5*(-89303793175477833893354121208000+6533090911353242906294143748495*y-32276910383172707359896832089932*z)*(-61468981380127448102256-5328427636421850183140*x+4647710007810227520885*y-13344414478836548348450*z))/pow(-46366672189358032-18896234711237580*x+3927118781169095*y+14705346416259850*z,3)-(3*(9101665097092871812176+3063507166600182944940*x+6945927557350563805665*y+1052001549322007294950*z)*(19493858980629008651267653094056+93282964805436900100617577630195*y+42271355681070699741325611572830*z))/pow(46366672189358032+18896234711237580*x-3927118781169095*y-14705346416259850*z,3)-(4*(39553725461800043367392+17203831108841472538824*x+45483386678520344593037*y+2703260049547565568088*z)*(52830583937680669669892057655944+303023948138837354463602341532495*y+134962043561465977901954677856080*z))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)-((22670037111266004087968+12461845278544574559640*x+39219302812923818032157*y-46563087562792926056*z)*(95973949246309465842551069546976+723429769797021053206211106031819*y+317530466286898645427564085427048*z))/pow(50159316775994592+36243094308305160*x-4827156544231217*y-52318895858217464*z,3)-(80*(4157117722725769078952+4534359335248895646832*x+26193979470458655189977*y-2382852476120229696128*z)*(205429639975670471114284923188348+2095815907391732802212116237430935*y+883539023887333564964405237094400*z))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)-(16*(9439334964924689507817+17499514376929345709248*x+187907876794815451253888*y-21704870055089718153088*z)*(943164674716649969807523653958385+18130967224506023673179633045358720*y+7486136216172114262568716503454336*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)+(80*(2304705299858575630109*x-256*(204828849006588248100+19508530860149228990861*y-2445924471668591306496*z))*(-179928369646271075844345534739549+3401432279430696137250330740801392*y+12500875943051297916024009205116096*z))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)+(80*(-805507884940017483975376678503744+52529278437993151034132605337909*y-620040027953848498781390188900552*z)*(-716026618045942942760*x+243780804476456624597*y-8*(408351630952413337484+89777022692195474597*z)))/pow(-50159316775994592-36243094308305160*x+4827156544231217*y+52318895858217464*z,3)+(768*(61889933231497708820968+30294916915069669525488*x-4484037822343607626207*y+13934625423713945278848*z)*(16858970779944867265671037333379*y-176*(1546216290476124632111328928258+3134171189636832381705249359145*z)))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)-(40*(1717566388539311579248*x+7025931019459451548321*y+48*(46537098413809906919-8301700878138964680*z))*(3434616943638241443585000648954199*y+320*(1107265969195848092307625165761+4643932844541992753284837619195*z)))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)+(12*(88457226224862447127008+13504083955712971035976*x-6622138801690554356387*y+19322683651036147287512*z)*(36451820000039413375829754767131*y-8*(66864837166560711793644210325852+35619205657210451197984743698883*z)))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)+(512*(45619694076424722199344+14936846773318822792976*x-3365788117861218576473*y+10130491989577935272320*z)*(12048859085295019197936041733505*y-6*(32519187452933223586671104614156+40471151781636260063426632487709*z)))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)))/125;//value(1)=(128*((32768*(24610976415716501050652227*x-256*(-10153609683556422184100+374519398571124540883*y-4145573659500944095488*z))*(98990697209366584150952278657452+920305667567495470446459093752885*x-65799721166407263195366683527104*z))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)+(1024*(-10864227594859409007678067839115+566592725765813239786863532667460*x-3214793226869529893757297514562848*z)*(9439334964924689507817+17499514376929345709248*x+187907876794815451253888*y-21704870055089718153088*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)+(40*(2938923392457131154149055759247753+8383263629566931208848464949723740*x-24821520393182477390523323699174560*z)*(4157117722725769078952+4534359335248895646832*x+26193979470458655189977*y-2382852476120229696128*z))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)+(80*(1717566388539311579248*x+7025931019459451548321*y+48*(46537098413809906919-8301700878138964680*z))*(3017477155357435955713408172820441+3434616943638241443585000648954199*x-6875761229715351344214913955270620*z))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)+(2*(1013986939222028224203834326214704+723429769797021053206211106031819*x-1002019231842824621894736024449560*z)*(22670037111266004087968+12461845278544574559640*x+39219302812923818032157*y-46563087562792926056*z))/pow(50159316775994592+36243094308305160*x-4827156544231217*y-52318895858217464*z,3)+(2*(698833722744934775627393528218146+279848894416310700301852732890585*x-191427609122898840477329914007915*z)*(9101665097092871812176+3063507166600182944940*x+6945927557350563805665*y+1052001549322007294950*z))/pow(46366672189358032+18896234711237580*x-3927118781169095*y-14705346416259850*z,3)+(8*(557016173590538671691101855964863+303023948138837354463602341532495*x-309197308873592242001670976702725*z)*(39553725461800043367392+17203831108841472538824*x+45483386678520344593037*y+2703260049547565568088*z))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)-(128*(-57335208466953058729715954197164+96390872682360153583488333868040*x-372364031472286149332017066304111*z)*(45619694076424722199344+14936846773318822792976*x-3365788117861218576473*y+10130491989577935272320*z))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)-(5*(-5058036108182894712997605343704+13066181822706485812588287496990*x-23584235630998237996607750176151*z)*(61468981380127448102256+5328427636421850183140*x-4647710007810227520885*y+13344414478836548348450*z))/pow(46366672189358032+18896234711237580*x-3927118781169095*y-14705346416259850*z,3)-(256*(-35027435322808897803896166913833+101153824679669203594026224000274*x-443348667941077090029000877418626*z)*(61889933231497708820968+30294916915069669525488*x-4484037822343607626207*y+13934625423713945278848*z))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)-(24*(-23539469566855513950637813409344+36451820000039413375829754767131*x-87577625291530403453057402554096*z)*(88457226224862447127008+13504083955712971035976*x-6622138801690554356387*y+19322683651036147287512*z))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)-(112*(97743545586690977941666831119873+189463292388600804291605866927808*x-534599264249120709692835475330432*z)*(801790542801929135637671+732048260009923946735424*x-56975701334774517040256*y+187552638032246240630656*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)-(2560*(2304705299858575630109*x-256*(204828849006588248100+19508530860149228990861*y-2445924471668591306496*z))*(-29205293090710790323990469408790736+212589517464418508578145671300087*x+1750806894610755007047140949242022912*z))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)-(160*(3266813047619306699872+716026618045942942760*x-243780804476456624597*y+718216181537563796776*z)*(52529278437993151034132605337909*x-4*(8646336391489439377118003754263+39602745269819371968458588313429*z)))/pow(50159316775994592+36243094308305160*x-4827156544231217*y-52318895858217464*z,3)))/125;//value(2)=(128*((-24576*(3839508863935892182987929073642496+36103009879073133562313702394913733*x-87732961555209684260488911369472*y)*(24610976415716501050652227*x-256*(-10153609683556422184100+374519398571124540883*y-4145573659500944095488*z)))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)-(30720*(65108728870058843312625047943313*x-256*(4791937744017588738333042319232+569924119339438478856491194414721*y))*(2304705299858575630109*x-256*(204828849006588248100+19508530860149228990861*y-2445924471668591306496*z)))/pow(5042560366642267*x-256*(2446745837411900+4901398098088043*y-144207654645973248*z),3)+(256*(650985307933227267490679218098413+935767027021514282821089562931792*x+12859172907478119575029190058251392*y)*(9439334964924689507817+17499514376929345709248*x+187907876794815451253888*y-21704870055089718153088*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)+(1280*(114748411888321695540849692963124+110442377985916695620550654636800*x+775672512286952418453853865599205*y)*(4157117722725769078952+4534359335248895646832*x+26193979470458655189977*y-2382852476120229696128*z))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)+(1600*(100744894915663705876272277122960+74302925512671884052557401907120*x+343788061485767567210745697763531*y)*(1717566388539311579248*x+7025931019459451548321*y+48*(46537098413809906919-8301700878138964680*z)))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)+(16*(72249495731635781189477972681776+39691308285862330678445510678381*x+125252403980353077736842003056195*y)*(22670037111266004087968+12461845278544574559640*x+39219302812923818032157*y-46563087562792926056*z))/pow(50159316775994592+36243094308305160*x-4827156544231217*y-52318895858217464*z,3)+(640*(505227745581172894057712966825000+155010006988462124695347547225138*x-39602745269819371968458588313429*y)*(3266813047619306699872+716026618045942942760*x-243780804476456624597*y+718216181537563796776*z))/pow(50159316775994592+36243094308305160*x-4827156544231217*y-52318895858217464*z,3)+(2*(356681541401645116923690413208956+126814067043212099223976834718490*x+191427609122898840477329914007915*y)*(9101665097092871812176+3063507166600182944940*x+6945927557350563805665*y+1052001549322007294950*z))/pow(46366672189358032+18896234711237580*x-3927118781169095*y-14705346416259850*z,3)+(8*(301993014170585471859024964195112+134962043561465977901954677856080*x+309197308873592242001670976702725*y)*(39553725461800043367392+17203831108841472538824*x+45483386678520344593037*y+2703260049547565568088*z))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)+(128*(4874430224431350455160317539284048+1942615285518540483044478359410032*x-372364031472286149332017066304111*y)*(45619694076424722199344+14936846773318822792976*x-3365788117861218576473*y+10130491989577935272320*z))/pow(85141430232132048+97951351741329392*x-8855616621991191*y-199920422688690560*z,3)+((1486971442137244004077030949061728+322769103831727073598968320899320*x-117921178154991189983038750880755*y)*(61468981380127448102256+5328427636421850183140*x-4647710007810227520885*y+13344414478836548348450*z))/pow(46366672189358032+18896234711237580*x-3927118781169095*y-14705346416259850*z,3)+(512*(3005184872892536482128059816733656+1654842388128247497540371661628560*x-221674333970538545014500438709313*y)*(61889933231497708820968+30294916915069669525488*x-4484037822343607626207*y+13934625423713945278848*z))/pow(45070329471431608+130124049256651728*x-5583613021604317*y-387630670566282112*z,3)+(192*(137644881571986015841084811827840+35619205657210451197984743698883*x-10947203161441300431632175319262*y)*(88457226224862447127008+13504083955712971035976*x-6622138801690554356387*y+19322683651036147287512*z))/pow(92856945980914656+51329763147513032*x-8586501277743859*y-56199770659759016*z,3)+(64*(13728575451141247570683309821008705+13111763174706011627610159037098688*x-935548712435961241962462081828256*y)*(801790542801929135637671+732048260009923946735424*x-56975701334774517040256*y+187552638032246240630656*z))/pow(-3075770275504817+198931044892562752*x+14199788245258112*y-1122852841901814912*z,3)))/125; return value;}
How to avoid precision loss when handling very long integer constant involved multiple precision computation?
c++
Libraries like MPFR & GMP  [ http://gmplib.org/ ] are designed exactly for this requirement.  Native datatypes for C and C++ have defined, fixed limits, GMP, MPFR,and other bignum libraries have extensible limits on precision. Each handled in its own way.MPFR is also such a library, like GMP, it requires that it be installed.  It is designed to have a fixed precision (declared at the beginningand the limits are smaller than GMP), GMP limits can be changed during runtime.  MPFR has native (that is built-in) support for transcendental functions.  Which can be a win if you need them.However you choose to go, read the documentation, do not simply copy somebody else's code and expect good results. 