2

If I write a C program to say the value of LDBL_MAX the largest value for a long double,

#include <float.h>
#include <stdio.h>  
int main(void) {
  const long double max = LDBL_MAX;

  printf("%Lf\n", max);
  printf("%LG\n", max);
}

on my Linux laptops it prints

1189731495357231765021263853030970205169063322294624200440323733891737005522970722616410290336528882853545697807495577314427443153670288434198125573853743678673593200706973263201915918282961524365529510646791086614311790632169778838896134786560600399148753433211454911160088679845154866512852340149773037600009125479393966223151383622417838542743917838138717805889487540575168226347659235576974805113725649020884855222494791399377585026011773549180099796226026859508558883608159846900235645132346594476384939859276456284579661772930407806609229102715046085388087959327781622986827547830768080040150694942303411728957777100335714010559775242124057347007386251660110828379119623008469277200965153500208474470792443848545912886723000619085126472111951361467527633519562927597957250278002980795904193139603021470997035276467445530922022679656280991498232083329641241038509239184734786121921697210543484287048353408113042573002216421348917347174234800714880751002064390517234247656004721768096486107994943415703476320643558624207443504424380566136017608837478165389027809576975977286860071487028287955567141404632615832623602762896316173978484254486860609948270867968048078702511858930838546584223040908805996294594586201903766048446790926002225410530775901065760671347200125846406957030257138960983757998926954553052368560758683179223113639519468850880771872104705203957587480013143131444254943919940175753169339392366881856189129931729104252921236835159922322050998001677102784035360140829296398115122877768135706045789343535451696539561254048846447169786893211671087229088082778350518228857646062218739702851655083720992349483334435228984751232753726636066213902281264706234075352071724058665079518217303463782631353393706774901950197841690441824738063162828586857741432581165364040218402724913393320949219498422442730427019873044536620350262386957804682003601447291997123095530057206141866974852846856186514832715974481203121946751686379343096189615107330065552421485195201762858595091051839472502863871632494167613804996319791441870254302706758495192008837915169401581740046711477877201459644461175204059453504764721807975761111720846273639279600339670470037613374509553184150073796412605047923251661354841291884211340823015473304754067072818763503617332908005951896325207071673904547777129682265206225651439919376804400292380903112437912614776255964694221981375146967079446870358004392507659451618379811859392049544036114915310782251072691486979809240946772142727012404377187409216756613634938900451232351668146089322400697993176017805338191849981933008410985993938760292601390911414526003720284872132411955424282101831204216104467404621635336900583664606591156298764745525068145003932941404131495400677602951005962253022823003631473824681059648442441324864573137437595096416168048024129351876204668135636877532814675538798871771836512893947195335061885003267607354388673368002074387849657014576090349857571243045102038730494854256702479339322809110526041538528994849203991091946129912491633289917998094380337879522093131466946149705939664152375949285890960489916121944989986384837022486672249148924678410206183364627416969576307632480235587975245253737035433882960862753427740016333434055083537048507374544819754722228975281083020898682633020285259923084168054539687911418297629988964576482765287504562854924265165217750799516259669229114977788962356670956627138482018191348321687995863652637620978285070099337294396784639879024914514222742527006363942327998483976739987154418554201562244154926653014515504685489258620276085761837129763358761215382565129633538141663949516556000264159186554850057052611431952919918807954522394649627635630178580896692226406235382898535867595990647008385687123810329591926494846250768992258419305480763620215089022149220528069842018350840586938493815498909445461977893029113576516775406232278298314033473276603952231603422824717528181818844304880921321933550869873395861276073670866652375555675803171490108477320096424318780070008797346032906278943553743564448851907191616455141155761939399690767415156402826543664026760095087523945507341556135867933066031744720924446513532366647649735400851967040771103640538150073486891798364049570606189535005089840913826869535090066783324472578712196604415284924840041850932811908963634175739897166596000759487800619164094854338758520657116541072260996288150123144377944008749301944744330784388995701842710004808305012177123560622895076269042856800047718893158089358515593863176652948089031267747029662545110861548958395087796755464137944895960527975209874813839762578592105756284401759349324162148339565350189196811389091843795734703269406342890087805846940352453479398080674273236297887100867175802531561302356064878709259865288416350972529537091114317204887747405539054009425375424119317944175137064689643861517718849867010341532542385911089624710885385808688837777258648564145934262121086647588489260031762345960769508849149662444156604419552086811989770240.000000
1.18973E+4932

a 1 followed by 4,932 0s.

My confusion is that:

  1. Why is it that number, and why so large? I cannot tell how many bits are used to represent this number because it is too large for every language interpreter and website.

  2. How is this possible? Wikipedia tells us that long double is 80 bits on Intel, but there is no way to fit 4933 significant digits into 80 bits. The largest integer in 80 bits is 1208925819614629174706176.

This is gcc (Ubuntu 7.2.0-8ubuntu3.2) 7.2.0 / clang version 4.0.1-6 on Linux 4.13.0-38-generic #43-Ubuntu SMP Wed Mar 14 15:20:44 UTC 2018 x86_64 x86_64 x86_64 GNU/Linux.

The CPU is Intel® Core™ i3-3217U and the chipset is Mobile Intel Panther point HM77 in a Lenovo Ideapad but the result is the same on a HM68 Core i7 Toshiba.

  • there is no way to fit 4933 significant digits into 80 bits Why do you think there are 4933 significant digits in a long double? – Andrew Henle Apr 19 '18 at 14:49
  • The integer portion of your number is not "too large for every language interpreter and website". For example, GHCi (the Haskell interpreter) can work with it without complaint – Fox Apr 19 '18 at 15:32
  • @fox i forgot about GHCI, i tried Python and Factor and Pure and other sites and interpreters i thought had arbitrary precision – cat Apr 19 '18 at 18:45
3

You are probably used to thinking of numbers as some number times some power of 10. Floating point numbers are represented as some number times some power of 2. You can represent 102410 (100000000002) with just a single significant bit, even though it looks like it has three significant digits.

The number you listed:

1189731495357231765021263853030970205169063322294624200440323733891737005522970722616410290336528882853545697807495577314427443153670288434198125573853743678673593200706973263201915918282961524365529510646791086614311790632169778838896134786560600399148753433211454911160088679845154866512852340149773037600009125479393966223151383622417838542743917838138717805889487540575168226347659235576974805113725649020884855222494791399377585026011773549180099796226026859508558883608159846900235645132346594476384939859276456284579661772930407806609229102715046085388087959327781622986827547830768080040150694942303411728957777100335714010559775242124057347007386251660110828379119623008469277200965153500208474470792443848545912886723000619085126472111951361467527633519562927597957250278002980795904193139603021470997035276467445530922022679656280991498232083329641241038509239184734786121921697210543484287048353408113042573002216421348917347174234800714880751002064390517234247656004721768096486107994943415703476320643558624207443504424380566136017608837478165389027809576975977286860071487028287955567141404632615832623602762896316173978484254486860609948270867968048078702511858930838546584223040908805996294594586201903766048446790926002225410530775901065760671347200125846406957030257138960983757998926954553052368560758683179223113639519468850880771872104705203957587480013143131444254943919940175753169339392366881856189129931729104252921236835159922322050998001677102784035360140829296398115122877768135706045789343535451696539561254048846447169786893211671087229088082778350518228857646062218739702851655083720992349483334435228984751232753726636066213902281264706234075352071724058665079518217303463782631353393706774901950197841690441824738063162828586857741432581165364040218402724913393320949219498422442730427019873044536620350262386957804682003601447291997123095530057206141866974852846856186514832715974481203121946751686379343096189615107330065552421485195201762858595091051839472502863871632494167613804996319791441870254302706758495192008837915169401581740046711477877201459644461175204059453504764721807975761111720846273639279600339670470037613374509553184150073796412605047923251661354841291884211340823015473304754067072818763503617332908005951896325207071673904547777129682265206225651439919376804400292380903112437912614776255964694221981375146967079446870358004392507659451618379811859392049544036114915310782251072691486979809240946772142727012404377187409216756613634938900451232351668146089322400697993176017805338191849981933008410985993938760292601390911414526003720284872132411955424282101831204216104467404621635336900583664606591156298764745525068145003932941404131495400677602951005962253022823003631473824681059648442441324864573137437595096416168048024129351876204668135636877532814675538798871771836512893947195335061885003267607354388673368002074387849657014576090349857571243045102038730494854256702479339322809110526041538528994849203991091946129912491633289917998094380337879522093131466946149705939664152375949285890960489916121944989986384837022486672249148924678410206183364627416969576307632480235587975245253737035433882960862753427740016333434055083537048507374544819754722228975281083020898682633020285259923084168054539687911418297629988964576482765287504562854924265165217750799516259669229114977788962356670956627138482018191348321687995863652637620978285070099337294396784639879024914514222742527006363942327998483976739987154418554201562244154926653014515504685489258620276085761837129763358761215382565129633538141663949516556000264159186554850057052611431952919918807954522394649627635630178580896692226406235382898535867595990647008385687123810329591926494846250768992258419305480763620215089022149220528069842018350840586938493815498909445461977893029113576516775406232278298314033473276603952231603422824717528181818844304880921321933550869873395861276073670866652375555675803171490108477320096424318780070008797346032906278943553743564448851907191616455141155761939399690767415156402826543664026760095087523945507341556135867933066031744720924446513532366647649735400851967040771103640538150073486891798364049570606189535005089840913826869535090066783324472578712196604415284924840041850932811908963634175739897166596000759487800619164094854338758520657116541072260996288150123144377944008749301944744330784388995701842710004808305012177123560622895076269042856800047718893158089358515593863176652948089031267747029662545110861548958395087796755464137944895960527975209874813839762578592105756284401759349324162148339565350189196811389091843795734703269406342890087805846940352453479398080674273236297887100867175802531561302356064878709259865288416350972529537091114317204887747405539054009425375424119317944175137064689643861517718849867010341532542385911089624710885385808688837777258648564145934262121086647588489260031762345960769508849149662444156604419552086811989770240.000000

is equal to (264-1)*216320. Expanding only the mantissa, this is 18446744073709551615*216320, which is only 20 significant digits.

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