import math
def log_factorial(n):
if n < 0:
raise ValueError("Invalid input")
elif n > 254:
x = n + 1
return (x - 0.5)*math.log(x) - x + 0.5*math.log(2*math.pi) + 1.0/(12.0*x)
else:
lf = [
0.000000000000000,
0.000000000000000,
0.693147180559945,
1.791759469228055,
3.178053830347946,
4.787491742782046,
6.579251212010101,
8.525161361065415,
10.604602902745251,
12.801827480081469,
15.104412573075516,
17.502307845873887,
19.987214495661885,
22.552163853123421,
25.191221182738683,
27.899271383840894,
30.671860106080675,
33.505073450136891,
36.395445208033053,
39.339884187199495,
42.335616460753485,
45.380138898476908,
48.471181351835227,
51.606675567764377,
54.784729398112319,
58.003605222980518,
61.261701761002001,
64.557538627006323,
67.889743137181526,
71.257038967168000,
74.658236348830158,
78.092223553315307,
81.557959456115029,
85.054467017581516,
88.580827542197682,
92.136175603687079,
95.719694542143202,
99.330612454787428,
102.968198614513810,
106.631760260643450,
110.320639714757390,
114.034211781461690,
117.771881399745060,
121.533081515438640,
125.317271149356880,
129.123933639127240,
132.952575035616290,
136.802722637326350,
140.673923648234250,
144.565743946344900,
148.477766951773020,
152.409592584497350,
156.360836303078800,
160.331128216630930,
164.320112263195170,
168.327445448427650,
172.352797139162820,
176.395848406997370,
180.456291417543780,
184.533828861449510,
188.628173423671600,
192.739047287844900,
196.866181672889980,
201.009316399281570,
205.168199482641200,
209.342586752536820,
213.532241494563270,
217.736934113954250,
221.956441819130360,
226.190548323727570,
230.439043565776930,
234.701723442818260,
238.978389561834350,
243.268849002982730,
247.572914096186910,
251.890402209723190,
256.221135550009480,
260.564940971863220,
264.921649798552780,
269.291097651019810,
273.673124285693690,
278.067573440366120,
282.474292687630400,
286.893133295426990,
291.323950094270290,
295.766601350760600,
300.220948647014100,
304.686856765668720,
309.164193580146900,
313.652829949878990,
318.152639620209300,
322.663499126726210,
327.185287703775200,
331.717887196928470,
336.261181979198450,
340.815058870798960,
345.379407062266860,
349.954118040770250,
354.539085519440790,
359.134205369575340,
363.739375555563470,
368.354496072404690,
372.979468885689020,
377.614197873918670,
382.258588773060010,
386.912549123217560,
391.575988217329610,
396.248817051791490,
400.930948278915760,
405.622296161144900,
410.322776526937280,
415.032306728249580,
419.750805599544780,
424.478193418257090,
429.214391866651570,
433.959323995014870,
438.712914186121170,
443.475088120918940,
448.245772745384610,
453.024896238496130,
457.812387981278110,
462.608178526874890,
467.412199571608080,
472.224383926980520,
477.044665492585580,
481.872979229887900,
486.709261136839360,
491.553448223298010,
496.405478487217580,
501.265290891579240,
506.132825342034830,
511.008022665236070,
515.890824587822520,
520.781173716044240,
525.679013515995050,
530.584288294433580,
535.496943180169520,
540.416924105997740,
545.344177791154950,
550.278651724285620,
555.220294146894960,
560.169054037273100,
565.124881094874350,
570.087725725134190,
575.057539024710200,
580.034272767130800,
585.017879388839220,
590.008311975617860,
595.005524249382010,
600.009470555327430,
605.020105849423770,
610.037385686238740,
615.061266207084940,
620.091704128477430,
625.128656730891070,
630.172081847810200,
635.221937855059760,
640.278183660408100,
645.340778693435030,
650.409682895655240,
655.484856710889060,
660.566261075873510,
665.653857411105950,
670.747607611912710,
675.847474039736880,
680.953419513637530,
686.065407301994010,
691.183401114410800,
696.307365093814040,
701.437263808737160,
706.573062245787470,
711.714725802289990,
716.862220279103440,
722.015511873601330,
727.174567172815840,
732.339353146739310,
737.509837141777440,
742.685986874351220,
747.867770424643370,
753.055156230484160,
758.248113081374300,
763.446610112640200,
768.650616799717000,
773.860102952558460,
779.075038710167410,
784.295394535245690,
789.521141208958970,
794.752249825813460,
799.988691788643450,
805.230438803703120,
810.477462875863580,
815.729736303910160,
820.987231675937890,
826.249921864842800,
831.517780023906310,
836.790779582469900,
842.068894241700490,
847.352097970438420,
852.640365001133090,
857.933669825857460,
863.231987192405430,
868.535292100464630,
873.843559797865740,
879.156765776907600,
884.474885770751830,
889.797895749890240,
895.125771918679900,
900.458490711945270,
905.796028791646340,
911.138363043611210,
916.485470574328820,
921.837328707804890,
927.193914982476710,
932.555207148186240,
937.921183163208070,
943.291821191335660,
948.667099599019820,
954.046996952560450,
959.431492015349480,
964.820563745165940,
970.214191291518320,
975.612353993036210,
981.015031374908400,
986.422203146368590,
991.833849198223450,
997.249949600427840,
1002.670484599700300,
1008.095434617181700,
1013.524780246136200,
1018.958502249690200,
1024.396581558613400,
1029.838999269135500,
1035.285736640801600,
1040.736775094367400,
1046.192096209724900,
1051.651681723869200,
1057.115513528895000,
1062.583573670030100,
1068.055844343701400,
1073.532307895632800,
1079.012946818975000,
1084.497743752465600,
1089.986681478622400,
1095.479742921962700,
1100.976911147256000,
1106.478169357800900,
1111.983500893733000,
1117.492889230361000,
1123.006317976526100,
1128.523770872990800,
1134.045231790853000,
1139.570684729984800,
1145.100113817496100,
1150.633503306223700,
1156.170837573242400,
]
return lf[n]