How to Calculate the 21! (21 Factorial) in Swift

How to calculate the 21! (21 factorial) in swift?

Unsigned 64 bit integer has a maximum value of 18,446,744,073,709,551,615. While 21! = 51,090,942,171,709,440,000. For this kind of case, you need a Big Integer type. I found a question about Big Integer in Swift. There's a library for Big Integer in that link.

BigInteger equivalent in Swift?

In Swift 3, how to calculate the factorial when the result becomes too high?

Here's an approach that will let you find very large factorials.

Represent large numbers as an array of digits. For instance 987 would be [9, 8, 7]. Multiplying that number by an integer n would require two steps.

1. Multiply each value in that array by n.
2. Perform a carry operation to return a result that is again single digits.

For example 987 * 2:

let arr = [9, 8, 7]
let arr2 = arr.map { $0 * 2 }
print(arr2)  // [18, 16, 14]

Now, perform the carry operation. Starting at the one's digit, 14 is too big, so keep the 4 and carry the 1. Add the 1 to 16 to get 17.

[18, 17, 4]

Repeat with the ten's place:

[19, 7, 4]

And then with the hundred's place:

[1, 9, 7, 4]

Finally, for printing, you could convert this back to a string:

let arr = [1, 9, 7, 4]
print(arr.map(String.init).joined())

1974

---

Applying that technique, here is a carryAll function that performs the carry operation, and a factorial that uses it to calculate very large factorials:

func carryAll(_ arr: [Int]) -> [Int] {
    var result = [Int]()

    var carry = 0
    for val in arr.reversed() {
        let total = val + carry
        let digit = total % 10
        carry = total / 10
        result.append(digit)
    }

    while carry > 0 {
        let digit = carry % 10
        carry = carry / 10
        result.append(digit)
    }

    return result.reversed()
}



func factorial(_ n: Int) -> String {
    var result = [1]
    for i in 2...n {
        result = result.map { $0 * i }
        result = carryAll(result)
    }

    return result.map(String.init).joined()
}

print(factorial(1000))

402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Factorial function produces wrong result for 21! and above

With long you can represent between -9,223,372,036,854,775,808 to +9,223,372,036,854,775,807 (source)so as soon the factorial pass that range you start to get errors.

How handle different threads in recursive function in swift?

My recommendation would be to split it into synchronous and asynchronous functions:

func factorial(of number: Double, completion: @escaping (Double) -> ()) {

    DispatchQueue.global(qos: .background).async {
        let result = factorial(of: number)
        DispatchQueue.main.sync { //I don't know what thread you're intending to use, so I picked main :)
            completion(result)
        }
    }
}

func factorial(of number: Double) -> Double {

    var result: Double = 0
    if number > 170 {
        return Double.infinity
    }
    if number != Double(Int(number)) || number < 1 {return 0}
    if number == 1 {
        return 1
    }

    result = number * factorial(of: number - 1)

    return result
}

This allows the meet of the work to happen in whichever way you want, without having to worry about potential threading issues.

Fast exact bigint factorial

I have a solution:

(4N!)=((2N!)^2) . mul(i=all primes<=4N) of [i^sum(j=1,2,3,4,5,...4N>=i^j) of [(4N/(i^j))%2]]

sub-terms of T2 are always prime^exponent where exponent can be computed on small integers like this:

for (e=0,j=N4;j;e+=j&1,j/=p);

where e is exponent, p is prime and N4 is 4*N

Code for the new equation:

// edit beg:
// Sorry, forget to copy sorted list of all primes up to max n here it is
// end of table is marked with 0
// Primes are in DWORDs so they only 4Byte per number
// so the table is very small compared with lookup table for the same max n!
// and also primes are needed for many other routines in bignum
// can compute n! for n <= max prime in table
DWORD _arithmetics_primes[]={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,0};
// edit end.

longnum fact(const DWORD &x)
    {
    if (x<=4)
        {
        if (x==4) return 24;
        if (x==3) return  6;
        if (x==2) return  2;
        if (x==1) return  1;
        if (x==0) return  1;
        }
    int N4,N2,p,i,j,e; longnum c,pp;
    N4=(x>>2)<<2;
    N2=N4>>1;
    c=fact(N2); c*=c;                 // c=((2N)!)^2;
    for (i=0;;i++)                    // c*= T2
        {
        p=_arithmetics_primes[i];
        if (!p) break;
        if (p>N4) break;
        for (e=0,j=N4;j;e+=j&1,j/=p);
        if (e)                        // c*=p^e
            {
            if (p==2) c<<=e;
            else for (pp=p;;)
                {
                if (int(e&1)) c*=pp;
                e>>=1; if (!e) break;
                pp*=pp;
                }
            }
        }
    for (i=N4+1;i<=x;i++) { c*=i; } c.round();
    return c;
    }

Here are rough time measurements for the first 128 factorials so you can estimate real complexity.

Fixed point 768.128 bits arithmetics ... 231.36 decimals.

[ 0.001 ms ] 1! = 1
[ 0.000 ms ] 2! = 2
[ 0.000 ms ] 3! = 6
[ 0.000 ms ] 4! = 24
[ 0.006 ms ] 5! = 120
[ 0.006 ms ] 6! = 720
[ 0.007 ms ] 7! = 5040
[ 0.005 ms ] 8! = 40320
[ 0.006 ms ] 9! = 362880
[ 0.007 ms ] 10! = 3628800
[ 0.008 ms ] 11! = 39916800
[ 0.012 ms ] 12! = 479001600
[ 0.013 ms ] 13! = 6227020800
[ 0.014 ms ] 14! = 87178291200
[ 0.016 ms ] 15! = 1307674368000
[ 0.014 ms ] 16! = 20922789888000
[ 0.015 ms ] 17! = 355687428096000
[ 0.017 ms ] 18! = 6402373705728000
[ 0.019 ms ] 19! = 121645100408832000
[ 0.016 ms ] 20! = 2432902008176640000
[ 0.017 ms ] 21! = 51090942171709440000
[ 0.019 ms ] 22! = 1124000727777607680000
[ 0.021 ms ] 23! = 25852016738884976640000
[ 0.023 ms ] 24! = 620448401733239439360000
[ 0.025 ms ] 25! = 15511210043330985984000000
[ 0.027 ms ] 26! = 403291461126605635584000000
[ 0.029 ms ] 27! = 10888869450418352160768000000
[ 0.032 ms ] 28! = 304888344611713860501504000000
[ 0.034 ms ] 29! = 8841761993739701954543616000000
[ 0.037 ms ] 30! = 265252859812191058636308480000000
[ 0.039 ms ] 31! = 8222838654177922817725562880000000
[ 0.034 ms ] 32! = 263130836933693530167218012160000000
[ 0.037 ms ] 33! = 8683317618811886495518194401280000000
[ 0.039 ms ] 34! = 295232799039604140847618609643520000000
[ 0.041 ms ] 35! = 10333147966386144929666651337523200000000
[ 0.039 ms ] 36! = 371993326789901217467999448150835200000000
[ 0.041 ms ] 37! = 13763753091226345046315979581580902400000000
[ 0.044 ms ] 38! = 523022617466601111760007224100074291200000000
[ 0.046 ms ] 39! = 20397882081197443358640281739902897356800000000
[ 0.041 ms ] 40! = 815915283247897734345611269596115894272000000000
[ 0.044 ms ] 41! = 33452526613163807108170062053440751665152000000000
[ 0.046 ms ] 42! = 1405006117752879898543142606244511569936384000000000
[ 0.049 ms ] 43! = 60415263063373835637355132068513997507264512000000000
[ 0.048 ms ] 44! = 2658271574788448768043625811014615890319638528000000000
[ 0.050 ms ] 45! = 119622220865480194561963161495657715064383733760000000000
[ 0.054 ms ] 46! = 5502622159812088949850305428800254892961651752960000000000
[ 0.056 ms ] 47! = 258623241511168180642964355153611979969197632389120000000000
[ 0.056 ms ] 48! = 12413915592536072670862289047373375038521486354677760000000000
[ 0.060 ms ] 49! = 608281864034267560872252163321295376887552831379210240000000000
[ 0.063 ms ] 50! = 30414093201713378043612608166064768844377641568960512000000000000
[ 0.066 ms ] 51! = 1551118753287382280224243016469303211063259720016986112000000000000
[ 0.065 ms ] 52! = 80658175170943878571660636856403766975289505440883277824000000000000
[ 0.069 ms ] 53! = 4274883284060025564298013753389399649690343788366813724672000000000000
[ 0.072 ms ] 54! = 230843697339241380472092742683027581083278564571807941132288000000000000
[ 0.076 ms ] 55! = 12696403353658275925965100847566516959580321051449436762275840000000000000
[ 0.077 ms ] 56! = 710998587804863451854045647463724949736497978881168458687447040000000000000
[ 0.162 ms ] 57! = 40526919504877216755680601905432322134980384796226602145184481280000000000000
[ 0.095 ms ] 58! = 2350561331282878571829474910515074683828862318181142924420699914240000000000000
[ 0.093 ms ] 59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000
[ 0.089 ms ] 60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000
[ 0.093 ms ] 61! = 507580213877224798800856812176625227226004528988036003099405939480985600000000000000
[ 0.098 ms ] 62! = 31469973260387937525653122354950764088012280797258232192163168247821107200000000000000
[ 0.096 ms ] 63! = 1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000
[ 0.090 ms ] 64! = 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000
[ 0.100 ms ] 65! = 8247650592082470666723170306785496252186258551345437492922123134388955774976000000000000000
[ 0.104 ms ] 66! = 544344939077443064003729240247842752644293064388798874532860126869671081148416000000000000000
[ 0.111 ms ] 67! = 36471110918188685288249859096605464427167635314049524593701628500267962436943872000000000000000
[ 0.100 ms ] 68! = 2480035542436830599600990418569171581047399201355367672371710738018221445712183296000000000000000
[ 0.121 ms ] 69! = 171122452428141311372468338881272839092270544893520369393648040923257279754140647424000000000000000
[ 0.109 ms ] 70! = 11978571669969891796072783721689098736458938142546425857555362864628009582789845319680000000000000000
[ 0.119 ms ] 71! = 850478588567862317521167644239926010288584608120796235886430763388588680378079017697280000000000000000
[ 0.104 ms ] 72! = 61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000
[ 0.124 ms ] 73! = 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000000
[ 0.113 ms ] 74! = 330788544151938641225953028221253782145683251820934971170611926835411235700971565459250872320000000000000000
[ 0.118 ms ] 75! = 24809140811395398091946477116594033660926243886570122837795894512655842677572867409443815424000000000000000000
[ 0.118 ms ] 76! = 1885494701666050254987932260861146558230394535379329335672487982961844043495537923117729972224000000000000000000
[ 0.123 ms ] 77! = 145183092028285869634070784086308284983740379224208358846781574688061991349156420080065207861248000000000000000000
[ 0.129 ms ] 78! = 11324281178206297831457521158732046228731749579488251990048962825668835325234200766245086213177344000000000000000000
[ 0.133 ms ] 79! = 894618213078297528685144171539831652069808216779571907213868063227837990693501860533361810841010176000000000000000000
[ 0.121 ms ] 80! = 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000
[ 0.119 ms ] 81! = 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000
[ 0.131 ms ] 82! = 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000
[ 0.150 ms ] 83! = 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000
[ 0.141 ms ] 84! = 3314240134565353266999387579130131288000666286242049487118846032383059131291716864129885722968716753156177920000000000000000000
[ 0.148 ms ] 85! = 281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000
[ 0.154 ms ] 86! = 24227095383672732381765523203441259715284870552429381750838764496720162249742450276789464634901319465571660595200000000000000000000
[ 0.163 ms ] 87! = 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000
[ 0.211 ms ] 88! = 185482642257398439114796845645546284380220968949399346684421580986889562184028199319100141244804501828416633516851200000000000000000000
[ 0.151 ms ] 89! = 16507955160908461081216919262453619309839666236496541854913520707833171034378509739399912570787600662729080382999756800000000000000000000
[ 0.157 ms ] 90! = 1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000
[ 0.166 ms ] 91! = 135200152767840296255166568759495142147586866476906677791741734597153670771559994765685283954750449427751168336768008192000000000000000000000
[ 0.161 ms ] 92! = 12438414054641307255475324325873553077577991715875414356840239582938137710983519518443046123837041347353107486982656753664000000000000000000000
[ 0.169 ms ] 93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467315215203289516844845303838996289387078090752000000000000000000000
[ 0.173 ms ] 94! = 108736615665674308027365285256786601004186803580182872307497374434045199869417927630229109214583415458560865651202385340530688000000000000000000000
[ 0.188 ms ] 95! = 10329978488239059262599702099394727095397746340117372869212250571234293987594703124871765375385424468563282236864226607350415360000000000000000000000
[ 0.181 ms ] 96! = 991677934870949689209571401541893801158183648651267795444376054838492222809091499987689476037000748982075094738965754305639874560000000000000000000000
[ 0.187 ms ] 97! = 96192759682482119853328425949563698712343813919172976158104477319333745612481875498805879175589072651261284189679678167647067832320000000000000000000000
[ 0.194 ms ] 98! = 9426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729119823605850588608460429412647567360000000000000000000000
[ 0.201 ms ] 99! = 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000
[ 0.185 ms ] 100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
[ 0.191 ms ] 101! = 9425947759838359420851623124482936749562312794702543768327889353416977599316221476503087861591808346911623490003549599583369706302603264000000000000000000000000
[ 0.202 ms ] 102! = 961446671503512660926865558697259548455355905059659464369444714048531715130254590603314961882364451384985595980362059157503710042865532928000000000000000000000000
[ 0.207 ms ] 103! = 99029007164861804075467152545817733490901658221144924830052805546998766658416222832141441073883538492653516385977292093222882134415149891584000000000000000000000000
[ 0.242 ms ] 104! = 10299016745145627623848583864765044283053772454999072182325491776887871732475287174542709871683888003235965704141638377695179741979175588724736000000000000000000000000
[ 0.210 ms ] 105! = 1081396758240290900504101305800329649720646107774902579144176636573226531909905153326984536526808240339776398934872029657993872907813436816097280000000000000000000000000
[ 0.215 ms ] 106! = 114628056373470835453434738414834942870388487424139673389282723476762012382449946252660360871841673476016298287096435143747350528228224302506311680000000000000000000000000
[ 0.221 ms ] 107! = 12265202031961379393517517010387338887131568154382945052653251412013535324922144249034658613287059061933743916719318560380966506520420000368175349760000000000000000000000000
[ 0.217 ms ] 108! = 1324641819451828974499891837121832599810209360673358065686551152497461815091591578895743130235002378688844343005686404521144382704205360039762937774080000000000000000000000000
[ 0.226 ms ] 109! = 144385958320249358220488210246279753379312820313396029159834075622223337844983482099636001195615259277084033387619818092804737714758384244334160217374720000000000000000000000000
[ 0.232 ms ] 110! = 15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
[ 0.240 ms ] 111! = 1762952551090244663872161047107075788761409536026565516041574063347346955087248316436555574598462315773196047662837978913145847497199871623320096254145331200000000000000000000000000
[ 0.213 ms ] 112! = 197450685722107402353682037275992488341277868034975337796656295094902858969771811440894224355027779366597957338237853638272334919686385621811850780464277094400000000000000000000000000
[ 0.231 ms ] 113! = 22311927486598136465966070212187151182564399087952213171022161345724023063584214692821047352118139068425569179220877461124773845924561575264739138192463311667200000000000000000000000000
[ 0.240 ms ] 114! = 2543559733472187557120132004189335234812341496026552301496526393412538629248600474981599398141467853800514886431180030568224218435400019580180261753940817530060800000000000000000000000000
[ 0.252 ms ] 115! = 292509369349301569068815180481773552003419272043053514672100535242441942363589054622883930786268803187059211939585703515345785120071002251720730101703194015956992000000000000000000000000000
[ 0.248 ms ] 116! = 33931086844518982011982560935885732032396635556994207701963662088123265314176330336254535971207181169698868584991941607780111073928236261199604691797570505851011072000000000000000000000000000
[ 0.598 ms ] 117! = 3969937160808720895401959629498630647790406360168322301129748464310422041758630649341780708631240196854767624444057168110272995649603642560353748940315749184568295424000000000000000000000000000
[ 0.259 ms ] 118! = 468452584975429065657431236280838416439267950499862031533310318788629800927518416622330123618486343228862579684398745837012213486653229822121742374957258403779058860032000000000000000000000000000
[ 0.261 ms ] 119! = 55745857612076058813234317117419771556272886109483581752463927935846946310374691578057284710599874844234646982443450754604453404911734348832487342619913750049708004343808000000000000000000000000000
[ 0.254 ms ] 120! = 6689502913449127057588118054090372586752746333138029810295671352301633557244962989366874165271984981308157637893214090552534408589408121859898481114389650005964960521256960000000000000000000000000000
[ 0.263 ms ] 121! = 809429852527344373968162284544935082997082306309701607045776233628497660426640521713391773997910182738287074185078904956856663439318382745047716214841147650721760223072092160000000000000000000000000000
[ 0.270 ms ] 122! = 98750442008336013624115798714482080125644041369783596059584700502676714572050143649033796427745042294071023050579626404736512939596842694895821378210620013388054747214795243520000000000000000000000000000
[ 0.281 ms ] 123! = 12146304367025329675766243241881295855454217088483382315328918161829235892362167668831156960612640202170735835221294047782591091570411651472186029519906261646730733907419814952960000000000000000000000000000
[ 0.290 ms ] 124! = 1506141741511140879795014161993280686076322918971939407100785852066825250652908790935063463115967385069171243567440461925041295354731044782551067660468376444194611004520057054167040000000000000000000000000000
[ 0.322 ms ] 125! = 188267717688892609974376770249160085759540364871492425887598231508353156331613598866882932889495923133646405445930057740630161919341380597818883457558547055524326375565007131770880000000000000000000000000000000
[ 0.303 ms ] 126! = 23721732428800468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000
[ 0.313 ms ] 127! = 3012660018457659544809977077527059692324164918673621799053346900596667207618480809067860692097713761984609779945772783965563851033300772326297773087851869982500270661791244122597621760000000000000000000000000000000
[ 0.307 ms ] 128! = 385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000
refernce     128! = 385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000

My measurements reveal that N! uses

• max of 2.2N fast low level long operations (+,-,<<,>>)
• slightly less than N/2 long multiplications, but most of them are convenient in size which speeds up the multiplication, so the measured times do not match the obvious O(N/2*O(multiplication)).
• After using analysis on the times the observed complexity best matches O(N^1.4) using Karatsuba as results are still way below NTT based multiplication, after that the complexity should be even better.

Also I have tried factorial as non recursive multiplication of primes only (similar to T2 term), but the results was much slower.

P.S.: Code posted in the question is also 100% working, but slower than new one (even if it uses fewer multiplications - because of more memory needed for recursion and not optimized multiplicants order).

C++ function to calculate factorial returns negative value

It is unwise to actually calculate factorials - they grow extremely fast. Generally, with combinatorial formulae it's a good idea to look for a way to re-order operations to keep intermediate results somewhat constrained.

For example, let's look at (2*n)!/(n!*n!). It can be rewritten as ((n+1)*(n+2)*...*(2*n)) / (1*2*...*n) == (n+1)/1 * (n+2)/2 * (n+3)/3 ... * (2*n)/n. By interleaving multiplication and division, the rate of growth of intermediate result is reduced.

So, something like this:

int f(int n) {
  int ret = 1;
  for (int i = 1; i <= n; ++i) {
    ret *= (n + i);
    ret /= i;
  }
  return ret;
}

Demo

Cannot calculate factorials bigger than 20! ! How to do so?

The limit on an unsigned long long is 18446744073709551615, or about 1.8e+19. 20! is about 2.4e+18, so within range, however 21! is about 5.1e+19, exceeding the maximum size of an unsigned long long.

You may find this helpful: Are there types bigger than long long int in C++?

Is there is some mathematical optimum base that would speed up factorial calculation?

If you're just trying to optimize running time for calculating factorial, and changing the base is the only parameter you're changing, then the optimum base will likely contain small factors. 60 might be a reasonable choice. If you want to experiment, I would try various bases of the form (2^a)(3^b)(5^c)

Improving the speed of multiplication is probably the best way performance. What algorithm are you using for multiplication? (school-book, Karatsuba, Toom-Cook, FFT, ...)

There are other factors to consider, too. If you will be converting the numbers to decimal frequently, then a base that is a power of 10 will make the conversion as fast as possible.

Many(*) years ago, I wrote a base-6 floating point library specifically to solve a problem with repeated multiplication/division by 2 and/or 3. But unless you are trying to solve a specific problem, I think you will be better served by optimizing your algorithms in general than by just trying to optimize factorial.

casevh

(*) I originally said "Several years ago" until I remembered the program ran for many days on a 12Mhz 80286.

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