Gamma function: Difference between revisions
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gamma( 70)= 1.711224524e98, 1.711224524281e98, 1.711224524281e98, 7.57303907062e-29, 1.709188578191e98
</pre>
=={{header|Arturo}}==
Line 703 ⟶ 673:
</pre>
=={{header|
==={{header|ANSI BASIC}}===
{{trans|BBC BASIC}} - Lanczos method.
{{works with|Decimal BASIC}}
<syntaxhighlight lang="basic">100 DECLARE EXTERNAL FUNCTION FNlngamma
110
120 DEF FNgamma(z) = EXP(FNlngamma(z))
130
140 FOR x = 0.1 TO 2.05 STEP 0.1
150 PRINT USING$("#.#",x), USING$("##.############", FNgamma(x))
160 NEXT x
170 END
180
190 EXTERNAL FUNCTION FNlngamma(z)
200 DIM lz(0 TO 6)
210 RESTORE
220 MAT READ lz
230 DATA 1.000000000190015, 76.18009172947146, -86.50532032941677, 24.01409824083091, -1.231739572450155, 0.0012086509738662, -0.000005395239385
240 IF z < 0.5 THEN
250 LET FNlngamma = LOG(PI / SIN(PI * z)) - FNlngamma(1.0 - z)
260 EXIT FUNCTION
270 END IF
280 LET z = z - 1.0
290 LET b = z + 5.5
300 LET a = lz(0)
310 FOR i = 1 TO 6
320 LET a = a + lz(i) / (z + i)
330 NEXT i
340 LET FNlngamma = (LOG(SQR(2*PI)) + LOG(a) - b) + LOG(b) * (z+0.5)
350 END FUNCTION</syntaxhighlight>
{{out}}
<pre>
.1 9.513507698670
.2 4.590843712000
.3 2.991568987689
.4 2.218159543760
.5 1.772453850902
.6 1.489192248811
.7 1.298055332647
.8 1.164229713725
.9 1.068628702119
1.0 1.000000000000
1.1 .951350769867
1.2 .918168742400
1.3 .897470696306
1.4 .887263817503
1.5 .886226925453
1.6 .893515349288
1.7 .908638732853
1.8 .931383770980
1.9 .961765831907
2.0 1.000000000000
</pre>
==={{header|BASIC256}}===
{{trans|FreeBASIC}} - Stirling method.
Line 737 ⟶ 761:
end function</syntaxhighlight>
==={{header|BBC BASIC}}===
{{works with|BBC BASIC for Windows}}
Uses the Lanczos approximation.
Line 1,260 ⟶ 1,284:
140,0 9,615723196940235E238
170,0 4,269068009004271E304</pre>
=={{header|EasyLang}}==
{{trans|AWK}}
<syntaxhighlight>
e = 2.718281828459
func stirling x .
return sqrt (2 * pi / x) * pow (x / e) x
.
print " X Stirling"
for i to 20
d = i / 10
numfmt 2 4
write d & " "
numfmt 3 4
print stirling d
.
</syntaxhighlight>
=={{header|Elixir}}==
Line 1,354 ⟶ 1,396:
90 : 1.65079551625067E+136
100 : 9.33262154536104E+155
</pre>
{{Trans|C#}}
The C# version can be translated to F# to support complex numbers:
<syntaxhighlight lang="f sharp">
open System.Numerics
open System
let rec gamma (z: Complex) =
let mutable z = z
let lanczosCoefficients = [| 676.520368121885; -1259.1392167224; 771.323428777653; -176.615029162141; 12.5073432786869; -0.13857109526572; 9.98436957801957E-06; 1.50563273514931E-07 |]
if z.Real < 0.5 then
Math.PI / (sin (Math.PI * z) * gamma (1.0 - z))
else
let mutable x = Complex.One
z <- z - 1.0
for i = 0 to lanczosCoefficients.Length - 1 do
x <- x + lanczosCoefficients.[i] / (z + Complex(i, 0) + 1.0)
let t = z + float lanczosCoefficients.Length - 0.5
sqrt (2.0 * Math.PI) * (t ** (z + 0.5)) * exp (-t) * x
Seq.iter (fun i -> printfn "Gamma(%f) = %A" i (gamma (Complex(i, 0)))) [ 0 .. 100 ]
Seq.iter2 (fun i j -> printfn "Gamma(%f + i%f) = %A" i j (gamma (Complex(i, j)))) [ 0 .. 100 ] [ 0 .. 100 ]
</syntaxhighlight>
{{out}}
<pre>
Gamma(0.000000) = (NaN, NaN)
Gamma(1.000000) = (1,0000000000000049, 0)
Gamma(2.000000) = (1,0000000000000115, 0)
Gamma(3.000000) = (2,0000000000000386, 0)
Gamma(4.000000) = (6,000000000000169, 0)
Gamma(5.000000) = (24,00000000000084, 0)
Gamma(6.000000) = (120,00000000000514, 0)
Gamma(7.000000) = (720,0000000000364, 0)
Gamma(8.000000) = (5040,000000000289, 0)
Gamma(9.000000) = (40320,000000002554, 0)
Gamma(10.000000) = (362880,00000002526, 0)
Gamma(11.000000) = (3628800,0000002664, 0)
Gamma(12.000000) = (39916800,000003114, 0)
Gamma(13.000000) = (479001600,00004023, 0)
Gamma(14.000000) = (6227020800,000546, 0)
Gamma(15.000000) = (87178291200,00801, 0)
Gamma(16.000000) = (1307674368000,1267, 0)
Gamma(17.000000) = (20922789888002,08, 0)
Gamma(18.000000) = (355687428096034,7, 0)
Gamma(19.000000) = (6402373705728658, 0)
Gamma(20.000000) = (1,2164510040884514E+17, 0)
Gamma(21.000000) = (2,4329020081769083E+18, 0)
Gamma(22.000000) = (5,109094217171516E+19, 0)
Gamma(23.000000) = (1,1240007277777331E+21, 0)
Gamma(24.000000) = (2,5852016738887927E+22, 0)
Gamma(25.000000) = (6,20448401733312E+23, 0)
Gamma(26.000000) = (1,5511210043332811E+25, 0)
Gamma(27.000000) = (4,032914611266542E+26, 0)
Gamma(28.000000) = (1,0888869450419632E+28, 0)
Gamma(29.000000) = (3,048883446117507E+29, 0)
Gamma(30.000000) = (8,841761993740793E+30, 0)
Gamma(31.000000) = (2,652528598122235E+32, 0)
Gamma(32.000000) = (8,222838654178949E+33, 0)
Gamma(33.000000) = (2,631308369337265E+35, 0)
Gamma(34.000000) = (8,683317618812963E+36, 0)
Gamma(35.000000) = (2,9523279903964145E+38, 0)
Gamma(36.000000) = (1,0333147966387422E+40, 0)
Gamma(37.000000) = (3,719933267899472E+41, 0)
Gamma(38.000000) = (1,3763753091228064E+43, 0)
Gamma(39.000000) = (5,230226174666675E+44, 0)
Gamma(40.000000) = (2,0397882081200028E+46, 0)
Gamma(41.000000) = (8,159152832480012E+47, 0)
Gamma(42.000000) = (3,3452526613168034E+49, 0)
Gamma(43.000000) = (1,4050061177530564E+51, 0)
Gamma(44.000000) = (6,041526306338149E+52, 0)
Gamma(45.000000) = (2,658271574788788E+54, 0)
Gamma(46.000000) = (1,1962222086549537E+56, 0)
Gamma(47.000000) = (5,502622159812779E+57, 0)
Gamma(48.000000) = (2,586232415112008E+59, 0)
Gamma(49.000000) = (1,2413915592537631E+61, 0)
Gamma(50.000000) = (6,082818640343433E+62, 0)
Gamma(51.000000) = (3,04140932017172E+64, 0)
Gamma(52.000000) = (1,551118753287575E+66, 0)
Gamma(53.000000) = (8,065817517095389E+67, 0)
Gamma(54.000000) = (4,27488328406056E+69, 0)
Gamma(55.000000) = (2,308436973392699E+71, 0)
Gamma(56.000000) = (1,2696403353659833E+73, 0)
Gamma(57.000000) = (7,109985878049497E+74, 0)
Gamma(58.000000) = (4,0526919504882125E+76, 0)
Gamma(59.000000) = (2,350561331283167E+78, 0)
Gamma(60.000000) = (1,386831185457067E+80, 0)
Gamma(61.000000) = (8,3209871127424E+81, 0)
Gamma(62.000000) = (5,075802138772854E+83, 0)
Gamma(63.000000) = (3,1469973260391715E+85, 0)
Gamma(64.000000) = (1,9826083154046777E+87, 0)
Gamma(65.000000) = (1,2688693218589942E+89, 0)
Gamma(66.000000) = (8,247650592083449E+90, 0)
Gamma(67.000000) = (5,443449390775078E+92, 0)
Gamma(68.000000) = (3,647111091819299E+94, 0)
Gamma(69.000000) = (2,48003554243712E+96, 0)
Gamma(70.000000) = (1,711224524281613E+98, 0)
Gamma(71.000000) = (1,1978571669971308E+100, 0)
Gamma(72.000000) = (8,504785885679606E+101, 0)
Gamma(73.000000) = (6,123445837689312E+103, 0)
Gamma(74.000000) = (4,470115461513189E+105, 0)
Gamma(75.000000) = (3,307885441519755E+107, 0)
Gamma(76.000000) = (2,4809140811398187E+109, 0)
Gamma(77.000000) = (1,8854947016662648E+111, 0)
Gamma(78.000000) = (1,451830920283022E+113, 0)
Gamma(79.000000) = (1,1324281178207572E+115, 0)
Gamma(80.000000) = (8,946182130783977E+116, 0)
Gamma(81.000000) = (7,1569457046271725E+118, 0)
Gamma(82.000000) = (5,797126020748008E+120, 0)
Gamma(83.000000) = (4,753643337013366E+122, 0)
Gamma(84.000000) = (3,945523969721089E+124, 0)
Gamma(85.000000) = (3,31424013456571E+126, 0)
Gamma(86.000000) = (2,8171041143808564E+128, 0)
Gamma(87.000000) = (2,4227095383675335E+130, 0)
Gamma(88.000000) = (2,1077572983797526E+132, 0)
Gamma(89.000000) = (1,8548264225741817E+134, 0)
Gamma(90.000000) = (1,650795516091023E+136, 0)
Gamma(91.000000) = (1,4857159644819212E+138, 0)
Gamma(92.000000) = (1,3520015276785438E+140, 0)
Gamma(93.000000) = (1,2438414054642616E+142, 0)
Gamma(94.000000) = (1,1567725070817618E+144, 0)
Gamma(95.000000) = (1,0873661566568553E+146, 0)
Gamma(96.000000) = (1,032997848824012E+148, 0)
Gamma(97.000000) = (9,916779348710516E+149, 0)
Gamma(98.000000) = (9,619275968249195E+151, 0)
Gamma(99.000000) = (9,42689044888421E+153, 0)
Gamma(100.000000) = (9,33262154439535E+155, 0)
Gamma(0.000000 + i0.000000) = (NaN, NaN)
Gamma(1.000000 + i1.000000) = (0,49801566811835923, -0,15494982830180806)
Gamma(2.000000 + i2.000000) = (0,11229424234632254, 0,3236128855019324)
Gamma(3.000000 + i3.000000) = (-0,4401134076370088, -0,0636372431263299)
Gamma(4.000000 + i4.000000) = (0,7058649325913451, -0,49673908399741584)
Gamma(5.000000 + i5.000000) = (-0,9743952418053669, 2,0066898827226805)
Gamma(6.000000 + i6.000000) = (1,0560845455210948, -7,123931816061554)
Gamma(7.000000 + i7.000000) = (-0,26095668519941206, 27,88827411508434)
Gamma(8.000000 + i8.000000) = (1,8442848156317595, -125,96060801752867)
Gamma(9.000000 + i9.000000) = (-94,00399991734474, 643,3621714431141)
Gamma(10.000000 + i10.000000) = (1423,851941789479, -3496,081973308168)
Gamma(11.000000 + i11.000000) = (-16211,00700465313, 18168,810510285286)
Gamma(12.000000 + i12.000000) = (158471,8890918886, -68793,30331463458)
Gamma(13.000000 + i13.000000) = (-1329505,1052081874, -142199,12520863872)
Gamma(14.000000 + i14.000000) = (8576976,67312178, 7218722,503716219)
Gamma(15.000000 + i15.000000) = (-20768001,573587183, -99064686,32101583)
Gamma(16.000000 + i16.000000) = (-490395650,85195, 847486174,9207268)
Gamma(17.000000 + i17.000000) = (9782747798,66319, -2523864726,357996)
Gamma(18.000000 + i18.000000) = (-91408144092,80728, -62548333665,79536)
Gamma(19.000000 + i19.000000) = (80368797570,63837, 1283152922013,1064)
Gamma(20.000000 + i20.000000) = (12322153606702,379, -9813622771583,531)
Gamma(21.000000 + i21.000000) = (-191651224429571,5, -67416801166719,305)
Gamma(22.000000 + i22.000000) = (476610838765573,75, 2709614130691551,5)
Gamma(23.000000 + i23.000000) = (31282423285710508, -23340622982977492)
Gamma(24.000000 + i24.000000) = (-5,062346571412891E+17, -2,8075410996386413E+17)
Gamma(25.000000 + i25.000000) = (-1,1135374386470528E+18, 8,889271476011264E+18)
Gamma(26.000000 + i26.000000) = (1,4103207063357242E+20, -3,1111347801608966E+19)
Gamma(27.000000 + i27.000000) = (-1,20394153792971E+21, -2,100813378567903E+21)
Gamma(28.000000 + i28.000000) = (-2,9772583255514483E+22, 2,9845666640271078E+22)
Gamma(29.000000 + i29.000000) = (6,474322395366405E+23, 4,002110222682179E+23)
Gamma(30.000000 + i30.000000) = (4,982468347052982E+24, -1,3332730971666784E+25)
Gamma(31.000000 + i31.000000) = (-2,701233953257487E+26, -5,3335652308619844E+25)
Gamma(32.000000 + i32.000000) = (-3,5481927258667466E+26, 5,492417944693437E+27)
Gamma(33.000000 + i33.000000) = (1,13499763334942E+29, -3,936060260026878E+27)
Gamma(34.000000 + i34.000000) = (-2,497617380231244E+29, -2,4036706229026682E+30)
Gamma(35.000000 + i35.000000) = (-5,244047476964302E+31, 7,550247174479453E+30)
Gamma(36.000000 + i36.000000) = (1,8326953464053433E+32, 1,1815797667494789E+33)
Gamma(37.000000 + i37.000000) = (2,7496330950708185E+34, -3,7903032739968576E+33)
Gamma(38.000000 + i38.000000) = (-6,153702881069039E+34, -6,593476804299637E+35)
Gamma(39.000000 + i39.000000) = (-1,622188754583588E+37, 3,702408035066972E+35)
Gamma(40.000000 + i40.000000) = (-2,9787072201603815E+37, 4,069596487794187E+38)
Gamma(41.000000 + i41.000000) = (1,0327223226588694E+40, 2,028448840265614E+39)
Gamma(42.000000 + i42.000000) = (9,258591867044077E+40, -2,623834405446389E+41)
Gamma(43.000000 + i43.000000) = (-6,5825696143104E+42, -3,6674407207212435E+42)
Gamma(44.000000 + i44.000000) = (-1,3470021394042014E+44, 1,5970917674147318E+44)
Gamma(45.000000 + i45.000000) = (3,611191294816286E+45, 4,700641025433376E+45)
Gamma(46.000000 + i46.000000) = (1,572050028597055E+47, -6,978410499639601E+46)
Gamma(47.000000 + i47.000000) = (-8,064316457005957E+47, -5,037500759309252E+48)
Gamma(48.000000 + i48.000000) = (-1,5346852327090097E+50, -1,8750204416673013E+49)
Gamma(49.000000 + i49.000000) = (-1,963123049571723E+51, 4,3640542056764855E+51)
Gamma(50.000000 + i50.000000) = (1,112141672863102E+53, 1,0242389193853564E+53)
Gamma(51.000000 + i51.000000) = (4,3124175139534486E+54, -2,2723736598857867E+54)
Gamma(52.000000 + i52.000000) = (-1,9794169465380243E+55, -1,5907071359978385E+56)
Gamma(53.000000 + i53.000000) = (-5,198770735397539E+57, -1,364053254830618E+57)
Gamma(54.000000 + i54.000000) = (-1,120153853109903E+59, 1,4557034579325553E+59)
Gamma(55.000000 + i55.000000) = (3,0502206279504673E+60, 5,6213787579421754E+60)
Gamma(56.000000 + i56.000000) = (2,263901598439912E+62, -1,3869357395249425E+61)
Gamma(57.000000 + i57.000000) = (3,1337184675942048E+63, -7,566798379913671E+63)
Gamma(58.000000 + i58.000000) = (-1,9547779277051327E+65, -2,289059342042218E+65)
Gamma(59.000000 + i59.000000) = (-1,0990589957539216E+67, 2,43674092519427E+66)
Gamma(60.000000 + i60.000000) = (-1,2138821648921022E+68, 4,1070828497360523E+68)
Gamma(61.000000 + i61.000000) = (1,1429060333301634E+70, 1,199617402383565E+70)
Gamma(62.000000 + i62.000000) = (6,356301051435098E+71, -1,4385777048397157E+71)
Gamma(63.000000 + i63.000000) = (8,496122222382356E+72, -2,4629448359526944E+73)
Gamma(64.000000 + i64.000000) = (-6,555666234065589E+74, -8,308825832655404E+74)
Gamma(65.000000 + i65.000000) = (-4,35298630131942E+76, 3,566503620394996E+75)
Gamma(66.000000 + i66.000000) = (-9,093720755985274E+77, 1,5886817012950856E+78)
Gamma(67.000000 + i67.000000) = (3,276704637172729E+79, 7,067540192000104E+79)
Gamma(68.000000 + i68.000000) = (3,3000031716552424E+81, 6,606403155114497E+80)
Gamma(69.000000 + i69.000000) = (1,1027558271573558E+83, -9,8053446601259E+82)
Gamma(70.000000 + i70.000000) = (-4,322638823338879E+83, -6,551055369074648E+84)
Gamma(71.000000 + i71.000000) = (-2,4300476213174805E+86, -1,695901195606269E+86)
Gamma(72.000000 + i72.000000) = (-1,3066738432408389E+88, 3,647419995967721E+87)
Gamma(73.000000 + i73.000000) = (-2,631035116210543E+89, 5,722329713478048E+89)
Gamma(74.000000 + i74.000000) = (1,2168690154152985E+91, 2,7033309001464033E+91)
Gamma(75.000000 + i75.000000) = (1,3482397421660745E+93, 4,28037329878204E+92)
Gamma(76.000000 + i76.000000) = (5,937687180258459E+94, -3,3970662984923895E+94)
Gamma(77.000000 + i77.000000) = (7,877518886977203E+95, -3,258429077036584E+96)
Gamma(78.000000 + i78.000000) = (-8,893580405683355E+97, -1,4068641589865653E+98)
Gamma(79.000000 + i79.000000) = (-8,171120040126758E+99, -1,8182361788970335E+99)
Gamma(80.000000 + i80.000000) = (-3,620672062418723E+101, 2,252383956928435E+101)
Gamma(81.000000 + i81.000000) = (-5,470323981863164E+102, 2,130475015794634E+103)
Gamma(82.000000 + i82.000000) = (5,5003853678128614E+104, 1,0085770500872239E+105)
Gamma(83.000000 + i83.000000) = (5,740274728712029E+106, 1,986132310865518E+106)
Gamma(84.000000 + i84.000000) = (3,002634263016605E+108, -1,245707037705857E+108)
Gamma(85.000000 + i85.000000) = (7,865153067057636E+109, -1,575289481058232E+110)
Gamma(86.000000 + i86.000000) = (-2,2903303039804873E+111, -9,374401780068583E+111)
Gamma(87.000000 + i87.000000) = (-4,291880277570836E+113, -3,196185984412844E+113)
Gamma(88.000000 + i88.000000) = (-2,9995707183640408E+115, 1,1845722184191957E+114)
Gamma(89.000000 + i89.000000) = (-1,2943245318203042E+117, 1,1073023439932073E+117)
Gamma(90.000000 + i90.000000) = (-2,0007723042777158E+118, 9,568042990491017E+118)
Gamma(91.000000 + i91.000000) = (2,4147687416625443E+120, 5,132960890087842E+120)
Gamma(92.000000 + i92.000000) = (2,9270673985442265E+122, 1,5846423875074053E+122)
Gamma(93.000000 + i93.000000) = (1,9583971561168918E+124, -2,5166936071920716E+123)
Gamma(94.000000 + i94.000000) = (8,735240498823332E+125, -7,993043748530299E+125)
Gamma(95.000000 + i95.000000) = (1,6159775690644545E+127, -6,9922194609237035E+127)
Gamma(96.000000 + i96.000000) = (-1,569012926564151E+129, -4,10649302643008E+129)
Gamma(97.000000 + i97.000000) = (-2,2080981691082815E+131, -1,5902953016632018E+131)
Gamma(98.000000 + i98.000000) = (-1,6995881449793942E+133, -8,982249136857376E+131)
Gamma(99.000000 + i99.000000) = (-9,394719018633069E+134, 5,234766160326565E+134)
Gamma(100.000000 + i100.000000) = (-3,3597454530316526E+136, 5,986962556433683E+136)
</pre>
Line 2,046 ⟶ 2,321:
<pre>2.220446049250313e-16</pre>
=={{header|Koka}}==
Based on OCaml implementation
<syntaxhighlight lang="koka">
import std/num/float64
fun gamma-lanczos(x)
val g = 7.0
// Coefficients used by the GNU Scientific Library
val c = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]
fun ag(z: float64, d: int)
if d == 0 then c[0].unjust + ag(z, 1)
elif d < 8 then c[d].unjust / (z + d.float64) + ag(z, d.inc)
else c[d].unjust / (z + d.float64)
fun gamma(z)
val z' = z - 1.0
val p = z' + g + 0.5
sqrt(2.0 * pi) * pow(p, (z' + 0.5)) * exp(0.0 - p) * ag(z', 0)
gamma(x)
val e = exp(1.0)
fun gamma-stirling(x)
sqrt(2.0 * pi / x) * pow(x / e, x)
fun gamma-stirling2(x')
// Extended Stirling method seen in Abramowitz and Stegun
val d = [1.0/12.0, 1.0/288.0, -139.0/51840.0, -571.0/2488320.0]
fun corr(z, x, n)
if n < d.length - 1 then d[n].unjust / x + corr(z, x*z, n.inc)
else d[n].unjust / x
fun gamma(z)
gamma-stirling(z)*(1.0 + corr(z, z, 0))
gamma(x')
fun mirror(gma, z)
if z > 0.5 then gma(z) else pi / sin(pi * z) / gma(1.0 - z)
fun main()
println("z\tLanczos\t\t\tStirling\t\tStirling2")
for(1, 20) fn(i)
val z = i.float64 / 10.0
println(z.show(1) ++ "\t" ++ mirror(gamma-lanczos, z).show ++ "\t" ++
mirror(gamma-stirling, z).show ++ "\t" ++ mirror(gamma-stirling2, z).show)
for(1, 7) fn(i)
val z = 10.0 * i.float64
println(z.show ++ "\t" ++ gamma-lanczos(z).show ++ "\t" ++
gamma-stirling(z).show ++ "\t" ++ gamma-stirling2(z).show)
</syntaxhighlight>
{{out}}
<pre>
z Lanczos Stirling Stirling2
0.1 9.5135076986687359 10.405084329555955 9.5210418318004439
0.2 4.5908437119988017 5.0739927535371763 4.596862295030256
0.3 2.9915689876875904 3.3503395433773222 2.9984402802949961
0.4 2.218159543757686 2.5270578096699556 2.2277588907113128
0.5 1.7724538509055157 2.0663656770612464 1.7883901437260497
0.6 1.4891922488128184 1.3071588574483559 1.4827753636029286
0.7 1.2980553326475577 1.1590532921139201 1.2950806801024195
0.8 1.1642297137253037 1.0533709684256085 1.1627054102439229
0.9 1.068628702119319 0.97706150787769541 1.0677830813298756
1.0 1.0000000000000002 0.92213700889578909 0.99949946853364036
1.1 0.95135076986687361 0.88348995316870382 0.95103799705518899
1.2 0.91816874239976076 0.85775533539659088 0.91796405783487933
1.3 0.89747069630627774 0.84267825944839203 0.8973312868034562
1.4 0.88726381750307537 0.8367445486370817 0.88716548484542823
1.5 0.88622692545275827 0.83895655252649626 0.88615538430170204
1.6 0.89351534928769061 0.8486932421525738 0.89346184003019224
1.7 0.90863873285329122 0.86562147179388405 0.90859770150945562
1.8 0.93138377098024272 0.8896396352879945 0.93135158986107858
1.9 0.96176583190738729 0.92084272189422933 0.96174006762796482
2.0 1.0000000000000002 0.95950217574449159 0.99997898067003532
10 362880.00000000105 359869.56187410367 362879.99717458693
20 1.2164510040883245e+17 1.2113934233805675e+17 1.2164510037907557e+17
30 8.841761993739658e+30 8.8172365307655063e+30 8.8417619934546387e+30
40 2.0397882081197221e+46 2.0355431612365591e+46 2.0397882081041343e+46
50 6.0828186403425409e+62 6.0726891878763362e+62 6.0828186403274418e+62
60 1.3868311854568534e+80 1.3849063858294502e+80 1.3868311854555093e+80
70 1.7112245242813438e+98 1.7091885781910795e+98 1.711224524280615e+98
</pre>
=={{header|Kotlin}}==
<syntaxhighlight lang="scala">// version 1.0.6
Line 2,108 ⟶ 2,465:
2.00 0.959502175744492 1.000000000000000
</pre>
=={{header|Lambdatalk}}==
Following Javascript, with Lanczos approximation.
<syntaxhighlight lang="Scheme">
{def gamma.p
{A.new 0.99999999999980993
676.5203681218851
-1259.1392167224028
771.32342877765313
-176.61502916214059
12.507343278686905
-0.13857109526572012
9.9843695780195716e-6
1.5056327351493116e-7
}}
-> gamma.p
{def gamma.rec
{lambda {:x :a :i}
{if {< :i {A.length {gamma.p}}}
then {gamma.rec :x
{+ :a {/ {A.get :i {gamma.p}} {+ :x :i}} }
{+ :i 1}}
else :a
}}}
-> gamma.rec
{def gamma
{lambda {:x}
{if {< :x 0.5}
then {/ {PI}
{* {sin {* {PI} :x}}
{gamma {- 1 :x}}}}
else {let { {:x {- :x 1}}
{:t {+ {- :x 1} 7 0.5}}
} {* {sqrt {* 2 {PI}}}
{pow :t {+ :x 0.5}}
{exp -:t}
{gamma.rec :x {A.first {gamma.p}} 1}}
}}}}
-> gamma
{S.map {lambda {:i}
{div} Γ(:i) = {gamma :i}}
{S.serie -5.5 5.5 0.5}}
Γ(-5.5) = 0.010912654781909836
Γ(-5) = -42755084646679.17
Γ(-4.5) = -0.06001960130050417
Γ(-4) = 267219279041745.34
Γ(-3.5) = 0.27008820585226917
Γ(-3) = -1425169488222640
Γ(-2.5) = -0.9453087204829418
Γ(-2) = 6413262697001885
Γ(-1.5) = 2.363271801207352
Γ(-1) = -25653050788007544
Γ(-0.5) = -3.5449077018110295
Γ(0) = Infinity
Γ(0.5) = 1.7724538509055159
Γ(1) = 0.9999999999999998
Γ(1.5) = 0.8862269254527586
Γ(2) = 1.0000000000000002
Γ(2.5) = 1.3293403881791384
Γ(3) = 2.000000000000001
Γ(3.5) = 3.3233509704478426
Γ(4) = 6.000000000000007
Γ(4.5) = 11.631728396567446
Γ(5) = 23.999999999999996
Γ(5.5) = 52.34277778455358
</syntaxhighlight>
=={{header|Limbo}}==
Line 3,606 ⟶ 4,035:
{
z <- z - 1
x <- p[1]
for (i in 1:8) {
x <- x+p[i+1]/(z+i)
}
tt <- z + g + 0.5
sqrt(2*pi) * tt^(z+0.5) * exp(-tt) * x
Line 3,721 ⟶ 4,153:
Note: The Taylor series isn't much good above values of <big>'''6½'''</big>.
Already on modest values of x (say > 3) you loose precision. See below for a solution.
<syntaxhighlight lang="rexx">/*REXX program calculates GAMMA using the Taylor series coefficients; ≈80 decimal digits*/
/*The GAMMA function symbol is the Greek capital letter: Γ */
Line 3,893 ⟶ 4,326:
3.3233509704478425511840640312646472177454 3.1415926535897932384626433832795028841972
6
</pre>
This program also has some minor issues. The number of steps, as well as e and pi, are hardcoded. This limits the precision to about 80 digits. See below for a solution.
===A generic Gamma function===
Below program calculates the Gamma function for any (real) element x (except integers < 1) in the specified precision (but over 100 digits it quickly becomes slow).
Improvements over above programs:
Closed formulas added for all half integers and positive integers (much faster than series). Argument reduction (mapping to interval 0.5...1.5) added to the Lanczos solution. Results are now also for larger values of x accurate to about 60 digits. Dynamic determination of digits and iterations added to the Spouge solution. It's now slightly faster on low x values and gives correct results using > 87 digits. Depending on the parameters, the program selects the optimal method for calculating Gamma.
As before, all needed functions are included in this version, so the entry is quite long... Scroll down to see the extra routines.
<syntaxhighlight lang="rexx" style="overflow: scroll; height: 220em">
parse version version; say version; glob. = ''
say 'Gamma function in arbitrary precision'
say
say '(Half)integers formulas'
w = '-99.5 -10.5 -5.5 -2.5 -1.5 -0.5 0.5 1 1.5 2 2.5 5 5.5 10 10.5 99 99.5'
numeric digits 100
do i = 1 to words(w)
x = word(w,i); call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Formulas' format(x,4,1) r '('e 'seconds)'
end
say
say 'Lanczos (max 60 digits) vs Spouge (no limit) vs Stirling (no limit) approximation'
w = '-12.8 -6.4 -3.2 -1.6 -0.8 -0.4 -0.2 -0.1 0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8'
do i = 1 to words(w)
x = word(w,i)
numeric digits 60
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
numeric digits 61
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Spouge ' format(x,4,1) r '('e 'seconds)'
if x > 0 then do
call time('r'); r = Stirling(x); e = format(time('e'),,3)
say 'Stirling' format(x,4,1) r '('e 'seconds)'
end
end
say
say 'Same for a bigger number'
w = '-99.9 99.9'
do i = 1 to words(w)
x = word(w,i)
numeric digits 60
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
numeric digits 100
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Spouge ' format(x,4,1) r '('e 'seconds)'
if x > 0 then do
call time('r'); r = Stirling(x); e = format(time('e'),,3)
say 'Stirling' format(x,4,1) r '('e 'seconds)'
end
end
exit
Gamma:
/* Gamma */
procedure expose glob.
arg x
/* Validity */
if x < 1 & Whole(x) then
return 'X'
/* Formulas for negative and positive (half)integers */
if x < 0 then do
if Half(x) then do
numeric digits Digits()+2
i = Abs(Floor(x)); y = (-1)**i*2**(2*i)*Fact(i)*Sqrt(Pi())/Fact(2*i)
numeric digits Digits()-2
return y+0
end
end
if x > 0 then do
if Whole(x) then
return Fact(x-1)
if Half(x) then do
numeric digits Digits()+2
i = Floor(x); y = Fact(2*i)*Sqrt(Pi())/(2**(2*i)*Fact(i))
numeric digits Digits()-2
return y+0
end
end
p = Digits()
if p < 61 then do
/* Lanczos with predefined coefficients */
/* Map negative x to positive x */
if x < 0 then
return Pi()/(Gamma(1-x)*Sin(Pi()*x))
/* Argument reduction to interval (0.5,1.5) */
numeric digits p+2
c = Trunc(x); x = x-c
if x < 0.5 then do
x = x+1; c = c-1
end
/* Series coefficients 1/Gamma(x) in 80 digits Fransen & Wrigge */
c.1 = 1.00000000000000000000000000000000000000000000000000000000000000000000000000000000
c.2 = 0.57721566490153286060651209008240243104215933593992359880576723488486772677766467
c.3 = -0.65587807152025388107701951514539048127976638047858434729236244568387083835372210
c.4 = -0.04200263503409523552900393487542981871139450040110609352206581297618009687597599
c.5 = 0.16653861138229148950170079510210523571778150224717434057046890317899386605647425
c.6 = -0.04219773455554433674820830128918739130165268418982248637691887327545901118558900
c.7 = -0.00962197152787697356211492167234819897536294225211300210513886262731167351446074
c.8 = 0.00721894324666309954239501034044657270990480088023831800109478117362259497415854
c.9 = -0.00116516759185906511211397108401838866680933379538405744340750527562002584816653
c.10 = -0.00021524167411495097281572996305364780647824192337833875035026748908563946371678
c.11 = 0.00012805028238811618615319862632816432339489209969367721490054583804120355204347
c.12 = -0.00002013485478078823865568939142102181838229483329797911526116267090822918618897
c.13 = -0.00000125049348214267065734535947383309224232265562115395981534992315749121245561
c.14 = 0.00000113302723198169588237412962033074494332400483862107565429550539546040842730
c.15 = -0.00000020563384169776071034501541300205728365125790262933794534683172533245680371
c.16 = 0.00000000611609510448141581786249868285534286727586571971232086732402927723507435
c.17 = 0.00000000500200764446922293005566504805999130304461274249448171895337887737472132
c.18 = -0.00000000118127457048702014458812656543650557773875950493258759096189263169643391
c.19 = 0.00000000010434267116911005104915403323122501914007098231258121210871073927347588
c.20 = 0.00000000000778226343990507125404993731136077722606808618139293881943550732692987
c.21 = -0.00000000000369680561864220570818781587808576623657096345136099513648454655443000
c.22 = 0.00000000000051003702874544759790154813228632318027268860697076321173501048565735
c.23 = -0.00000000000002058326053566506783222429544855237419746091080810147188058196444349
c.24 = -0.00000000000000534812253942301798237001731872793994898971547812068211168095493211
c.25 = 0.00000000000000122677862823826079015889384662242242816545575045632136601135999606
c.26 = -0.00000000000000011812593016974587695137645868422978312115572918048478798375081233
c.27 = 0.00000000000000000118669225475160033257977724292867407108849407966482711074006109
c.28 = 0.00000000000000000141238065531803178155580394756670903708635075033452562564122263
c.29 = -0.00000000000000000022987456844353702065924785806336992602845059314190367014889830
c.30 = 0.00000000000000000001714406321927337433383963370267257066812656062517433174649858
c.31 = 0.00000000000000000000013373517304936931148647813951222680228750594717618947898583
c.32 = -0.00000000000000000000020542335517666727893250253513557337960820379352387364127301
c.33 = 0.00000000000000000000002736030048607999844831509904330982014865311695836363370165
c.34 = -0.00000000000000000000000173235644591051663905742845156477979906974910879499841377
c.35 = -0.00000000000000000000000002360619024499287287343450735427531007926413552145370486
c.36 = 0.00000000000000000000000001864982941717294430718413161878666898945868429073668232
c.37 = -0.00000000000000000000000000221809562420719720439971691362686037973177950067567580
c.38 = 0.00000000000000000000000000012977819749479936688244144863305941656194998646391332
c.39 = 0.00000000000000000000000000000118069747496652840622274541550997151855968463784158
c.40 = -0.00000000000000000000000000000112458434927708809029365467426143951211941179558301
c.41 = 0.00000000000000000000000000000012770851751408662039902066777511246477487720656005
c.42 = -0.00000000000000000000000000000000739145116961514082346128933010855282371056899245
c.43 = 0.00000000000000000000000000000000001134750257554215760954165259469306393008612196
c.44 = 0.00000000000000000000000000000000004639134641058722029944804907952228463057968680
c.45 = -0.00000000000000000000000000000000000534733681843919887507741819670989332090488591
c.46 = 0.00000000000000000000000000000000000032079959236133526228612372790827943910901464
c.47 = -0.00000000000000000000000000000000000000444582973655075688210159035212464363740144
c.48 = -0.00000000000000000000000000000000000000131117451888198871290105849438992219023663
c.49 = 0.00000000000000000000000000000000000000016470333525438138868182593279063941453996
c.50 = -0.00000000000000000000000000000000000000001056233178503581218600561071538285049997
c.51 = 0.00000000000000000000000000000000000000000026784429826430494783549630718908519485
c.52 = 0.00000000000000000000000000000000000000000002424715494851782689673032938370921241
/* Series expansion */
x = x-1; s = 0
do k = 52 by -1 to 1
s = s*x+c.k
end
y = 1/s
/* Undo reduction */
if c = -1 then
y = y/x
else do
do i = 1 to c
y = (x+i)*y
end
end
end
else do
x = x-1
/* Spouge */
/* Estimate digits and iterations */
q = Floor(p*1.5); a = Floor(p*1.3)
numeric digits q
/* Series */
s = 0
do k = 1 to a-1
s = s+((-1)**(k-1)*Power(a-k,k-0.5)*Exp(a-k))/(Fact(k-1)*(x+k))
end
s = s+Sqrt(2*Pi()); y = Power(x+a,x+0.5)*Exp(-a-x)*s
end
/* Normalize */
numeric digits p
return y+0
Stirling:
/* Sterling */
procedure expose glob.
arg x
return sqrt(2*pi()/x) * power(x/e(),x)
E:
/* Euler number */
procedure expose glob.
p = Digits()
/* In memory? */
if glob.e.p <> '' then
return glob.e.p
if p < 101 then
/* Fast value */
glob.e.p = 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516643+0
else do
numeric digits Digits()+2
/* Taylor series */
y = 2; t = 1; v = y
do n = 2
t = t/n; y = y+t
if y = v then
leave
v = y
end
numeric digits Digits()-2
glob.e.p = y+0
end
return glob.e.p
Exp:
/* Exponential e^x */
procedure expose glob.
arg x
numeric digits Digits()+2
/* Fast values */
if Whole(x) then
return E()**x
/* Argument reduction */
i = x%1
if Abs(x-i) > 0.5 then
i = i+Sign(x)
/* Taylor series */
x = x-i; y = 1; t = 1; v = y
do n = 1
t = (t*x)/n; y = y+t
if y = v then
leave
v = y
end
/* Inverse reduction */
y = y*e()**i
numeric digits Digits()-2
return y+0
Fact:
/* Factorial n! */
procedure expose glob.
arg x
/* Validity */
if \ Whole(x) then
return 'X'
if x < 0 then
return 'X'
/* Current in memory? */
if glob.fact.x <> '' then
return glob.fact.x
w = x-1
/* Previous in memory? */
if glob.fact.w = '' then do
/* Loop cf definition */
y = 1
do n = 2 to x
y = y*n
end
glob.fact.x = y
end
else
/* Multiply */
glob.fact.x = glob.fact.w*x
return glob.fact.x
Floor:
/* Floor */
procedure expose glob.
arg x
/* Formula */
if Whole(x) then
return x
else
return Trunc(x)-(x<0)
Frac:
/* Fractional part */
procedure expose glob.
arg x
/* Formula */
return x-x%1
Half:
/* Is a number half integer? */
procedure expose glob.
arg x
/* Formula */
return (Frac(Abs(x))=0.5)
Ln:
/* Natural logarithm base e */
procedure expose glob.
arg x
/* Validity */
if x <= 0 then
return 'X'
/* Fast values */
if x = 1 then
return 0
p = Digits()
/* In memory? */
if glob.ln.x.p <> '' then
return glob.ln.x.p
/* Precalculated values */
if x = 2 & p < 101 then do
glob.ln.x.p = Ln2()
return glob.ln.x.p
end
if x = 4 & p < 101 then do
glob.ln.x.p = Ln4()
return glob.ln.x.p
end
if x = 8 & p < 101 then do
glob.ln.x.p = Ln8()
return glob.ln.x.p
end
if x = 10 & p < 101 then do
glob.ln.x.p = Ln10()
return glob.ln.x.p
end
numeric digits p+2
/* Argument reduction */
z = x; i = 0; e = 1/E()
if z < 0.5 then do
y = 1/z
do while y > 1.5
y = y*e; i = i-1
end
z = 1/y
end
if z > 1.5 then do
do while z > 1.5
z = z*e; i = i+1
end
end
/* Taylor series */
q = (z-1)/(z+1); f = q; y = q; v = q; q = q*q
do n = 3 by 2
f = f*q; y = y+f/n
if y = v then
leave
v = y
end
numeric digits p
/* Inverse reduction */
glob.ln.x.p = 2*y+i
return glob.ln.x.p
Power:
/* Power function x^y */
procedure expose glob.
arg x,y
/* Validity */
if x < 0 then
return 'X'
/* Fast values */
if x = 0 then
return 0
if y = 0 then
return 1
/* Fast formula */
if Whole(y) then
return x**y
/* Formulas */
if Abs(y//1) = 0.5 then
return Sqrt(x)**Sign(y)*x**(y%1)
else
return Exp(y*Ln(x))
Sin:
/* Sine */
procedure expose glob.
arg x
numeric digits Digits()+2
/* Argument reduction */
u = Pi(); x = x//(2*u)
if Abs(x) > u then
x = x-Sign(x)*2*u
/* Taylor series */
t = x; y = x; x = x*x; v = y
do n = 2 by 2
t = -t*x/(n*(n+1)); y = y+t
if y = v then
leave
v = y
end
numeric digits Digits()-2
return y+0
Sqrt:
/* Square root x^(1/2) */
procedure expose glob.
arg x
/* Validity */
if x < 0 then
return 'X'
/* Fast values */
if x = 0 then
return 0
if x = 1 then
return 1
p = Digits()
/* Predefined values */
if x = 2 & p < 101 then
return Sqrt2()
if x = 3 & p < 101 then
return Sqrt3()
if x = 5 & p < 101 then
return Sqrt5()
numeric digits p+2
/* Argument reduction to [0,100) */
i = Xpon(x); i = (i-(i<0))%2; x = x/100**i
/* First guess 1 digit accurate */
t = '2.5 6.5 12.5 20.5 30.5 42.5 56.5 72.5 90.5 100'
do y = 1 until word(t,y) > x
end
/* Dynamic precision */
d = Digits()
do n = 1 while d > 2
d.n = d; d = d%2+1
end
d.n = 2
/* Method Heron */
do k = n to 1 by -1
numeric digits d.k
y = (y+x/y)*0.5
end
numeric digits p
return y*10**i
Whole:
/* Is a number integer? */
procedure expose glob.
arg x
/* Formula */
return Datatype(x,'w')
Xpon:
/* Exponent */
procedure expose glob.
arg x
/* Formula */
if x = 0 then
return 0
else
return Right(x*1E+99999,6)-99999
Ln2:
/* Natural log of 2 */
procedure expose glob.
/* Fast value */
y = 0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
return y+0
Ln4:
/* Natural log of 4 */
procedure expose glob.
/* Fast value */
y = 1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375
return y+0
Ln8:
/* Natural log of 8 */
procedure expose glob.
/* Fast value */
y = 2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063
return y+0
Ln10:
/* Natural log of 10 */
procedure expose glob.
/* Fast value */
y = 2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599720508959830
return y+0
Pi:
/* Pi */
procedure expose glob.
p = Digits()
/* In memory? */
if glob.pi.p <> '' then
return glob.pi.p
if p < 101 then
/* Fast value */
glob.pi.p = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211707+0
else do
numeric digits Digits()+2
if p < 201 then do
/* Method Chudnovsky series */
y = 0
do n = 0
v = y; y = y + Fact(6*n)*(13591409+545140134*n)/(Fact(3*n)*Fact(n)**3*-640320**(3*n))
if y = v then
leave
end
y = 4270934400/(Sqrt(10005)*y)
end
else do
/* Method Agmean series */
y = 0.25; a = 1; g = Sqrt(0.5); n = 1
do until a = v
v = a
x = (a+g)*0.5; g = Sqrt(a*g)
y = y-n*(x-a)**2; n = n+n; a = x
end
y = a*a/y
end
numeric digits Digits()-2
glob.pi.p = y+0
end
return glob.pi.p
Sqrt2:
/* Square root of 2 */
procedure expose glob.
/* Fast value */
y = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
return y+0
Sqrt3:
/* Square root of 3 */
procedure expose glob.
/* Fast value */
y = 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
return y+0
Sqrt5:
/* Square root of 5 */
procedure expose glob.
/* Fast value */
y = 2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
return y+0
</syntaxhighlight>
{{out|Output:}}
<pre style="font-size:80%">
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024
Gamma function in arbitrary precision
(Half)integers formulas (100 digits)
Formulas -99.5 3.370459273906717035419140191178165368212858243169804822385940657239537725368041400224226278392502618E-157 (0.003 seconds)
Formulas -10.5 -2.640121820547716316246385325311240439682468432522587656059168154777653141232089673307782521306919765E-7 (0.000 seconds)
Formulas -5.5 0.01091265478190986298673234429377905644050439299584730891829569009625650045347052040480970101310888490 (0.001 seconds)
Formulas -2.5 -0.9453087204829418812256893244486107641586930432652731350473641545882193517818838300666403502605571549 (0.000 seconds)
Formulas -1.5 2.363271801207354703064223311121526910396732608163182837618410386470548379454709575166600875651392887 (0.000 seconds)
Formulas -0.5 -3.544907701811032054596334966682290365595098912244774256427615579705822569182064362749901313477089331 (0.000 seconds)
Formulas 0.5 1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665 (0.000 seconds)
Formulas 1.0 1 (0.000 seconds)
Formulas 1.5 0.8862269254527580136490837416705725913987747280611935641069038949264556422955160906874753283692723327 (0.001 seconds)
Formulas 2.0 1 (0.000 seconds)
Formulas 2.5 1.329340388179137020473625612505858887098162092091790346160355842389683463443274136031212992553908499 (0.000 seconds)
Formulas 5.0 24 (0.000 seconds)
Formulas 5.5 52.34277778455352018114900849241819367949013237611424488006401129409378637307891910622901158181014715 (0.000 seconds)
Formulas 10.0 362880 (0.000 seconds)
Formulas 10.5 1133278.388948785567334574165588892475560298308275159776608723414529483390056004153717630538727607291 (0.000 seconds)
Formulas 99.0 9.426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729113E+153 (0.001 seconds)
Formulas 99.5 9.367802114655996591305637999137598430056780942876744878408158310966911153528806387157595583300106562E+154 (0.001 seconds)
Lanczos (60 digits) vs Spouge (61 digits) vs Stirling (61 digits) approximations
Lanczos -12.8 -1.44241809348812392704183496823417171228485786984071442421769E-9 (0.001 seconds)
Spouge -12.8 -1.442418093488123927041834968234171712284857869840714424217876E-9 (0.061 seconds)
Lanczos -6.4 -0.00214311842968855616971089012135323947575979728834356196087007 (0.001 seconds)
Spouge -6.4 -0.002143118429688556169710890121353239475759797288343561960870050 (0.062 seconds)
Lanczos -3.2 0.689056412005979742919224040168358601528149721171394266681422 (0.001 seconds)
Spouge -3.2 0.6890564120059797429192240401683586015281497211713942666814006 (0.061 seconds)
Lanczos -1.6 2.31058285808092523235318127282049942788600677267861414816871 (0.001 seconds)
Spouge -1.6 2.310582858080925232353181272820499427886006772678614148168727 (0.063 seconds)
Lanczos -0.8 -5.73855463999850381650594784491144000429263749786675377223601 (0.001 seconds)
Spouge -0.8 -5.738554639998503816505947844911440004292637497866753772236066 (0.063 seconds)
Lanczos -0.4 -3.72298062203204275598583347080335570330149759689981183834669 (0.001 seconds)
Spouge -0.4 -3.722980622032042755985833470803355703301497596899811838346699 (0.062 seconds)
Lanczos -0.2 -5.82114856862651686818160469134229346570980884445593876492447 (0.001 seconds)
Spouge -0.2 -5.821148568626516868181604691342293465709808844455938764924472 (0.062 seconds)
Lanczos -0.1 -10.6862870211931935489730533569448077816983878506097317904937 (0.001 seconds)
Spouge -0.1 -10.68628702119319354897305335694480778169838785060973179049371 (0.066 seconds)
Lanczos 0.1 9.51350769866873183629248717726540219255057862608837734305000 (0.000 seconds)
Spouge 0.1 9.513507698668731836292487177265402192550578626088377343050001 (0.063 seconds)
Stirling 0.1 5.697187148977169278607259516105973271247103273209125657453591 (0.003 seconds)
Lanczos 0.2 4.59084371199880305320475827592915200343410999829340301778885 (0.000 seconds)
Spouge 0.2 4.590843711998803053204758275929152003434109998293403017788853 (0.062 seconds)
Stirling 0.2 3.325998424022392556831252338044446631141936051050822874561565 (0.003 seconds)
Lanczos 0.4 2.21815954375768822305905402190767945077056650177146958224198 (0.000 seconds)
Spouge 0.4 2.218159543757688223059054021907679450770566501771469582241978 (0.063 seconds)
Stirling 0.4 1.841476335936235407774504215721792671787348602355998395147546 (0.003 seconds)
Lanczos 0.8 1.16422971372530337363632093826845869314196176889118775298489 (0.000 seconds)
Spouge 0.8 1.164229713725303373636320938268458693141961768891187752984894 (0.059 seconds)
Stirling 0.8 1.053370968425608533997094555812234627247042644411690511716880 (0.003 seconds)
Lanczos 1.6 0.893515349287690261436600032992805368792359423255954841203208 (0.000 seconds)
Spouge 1.6 0.8935153492876902614366000329928053687923594232559548412032077 (0.064 seconds)
Stirling 1.6 0.8486932421525737351870671884548739160413838607490704857265386 (0.003 seconds)
Lanczos 3.2 2.42396547993536801209211236969059225781321007909891679339251 (0.000 seconds)
Spouge 3.2 2.423965479935368012092112369690592257813210079098916793392514 (0.069 seconds)
Stirling 3.2 2.361851203186240411720280147420856619690317833030951977155556 (0.003 seconds)
Lanczos 6.4 240.833779983445938793193921422181728753410453111118999895588 (0.000 seconds)
Spouge 6.4 240.8337799834459387931939214221817287534104531111189998955875 (0.060 seconds)
Stirling 6.4 237.7207525902404787072183649425603774456984237805478743551168 (0.002 seconds)
Lanczos 12.8 289487660.334241583580309266192984495005040185968050409506654 (0.000 seconds)
Spouge 12.8 289487660.3342415835803092661929844950050401859680504095066538 (0.059 seconds)
Stirling 12.8 287609477.0875058075487675889481380625378498188858381901124308 (0.003 seconds)
Same for a bigger number
Lanczos -99.9 1.72726520939328005075554286550876635176440408500391962571907E-157 (0.000 seconds)
Spouge -99.9 1.727265209393280050755542865508766351764404085003870332156550733790282578981487471194267631617799447E-157 (0.274 seconds)
Lanczos 99.9 5.89173215164436165675854187059394609460949390669692020855752E+155 (0.000 seconds)
Spouge 99.9 5.891732151644361656758541870593946094609493906696920208557519945524581875362460658552892944423816715E+155 (0.266 seconds)
Stirling 99.9 5.886819525828908144161960074977041612792659661747365945060453973672990129231211036627906746201731112E+155 (0.010 seconds)
</pre>
Line 3,926 ⟶ 4,960:
:<math>RecGamma(a) = \frac{1}{\Gamma(a)}</math>;
:<math>Pochhammer(a,x) = \frac{\Gamma(a+x)}{\Gamma(x)}</math>.
=={{header|RPL}}==
{{trans|Ada}}
≪ 1 -
{ 1.00000000000000000000 0.57721566490153286061 -0.65587807152025388108
-0.04200263503409523553 0.16653861138229148950 -0.04219773455554433675
-0.00962197152787697356 0.00721894324666309954 -0.00116516759185906511
-0.00021524167411495097 0.00012805028238811619 -0.00002013485478078824
-0.00000125049348214267 0.00000113302723198170 -0.00000020563384169776
0.00000000611609510448 0.00000000500200764447 -0.00000000118127457049
0.00000000010434267117 0.00000000000778226344 -0.00000000000369680562
0.00000000000051003703 -0.00000000000002058326 -0.00000000000000534812
0.00000000000000122678 -0.00000000000000011813 0.00000000000000000119
0.00000000000000000141 -0.00000000000000000023 0.00000000000000000002 }
→ y a
≪ a DUP SIZE GET
a SIZE 1 - 1 '''FOR''' n
y * a n GET +
-1 '''STEP'''
INV
≫ ≫ '<span style="color:blue>GAMMA</span>' STO
.3 <span style="color:blue>GAMMA</span>
The built-in FACT instruction is obviously based on a similar Taylor formula, since it returns same results:
.3 1 - FACT
{{out}}
<pre>
2: 2.99156898769
1: 2.99156898769
</pre>
=={{header|Ruby}}==
Line 3,971 ⟶ 5,035:
2.0
2.7781584804376647
</pre>
=={{header|Rust}}==
====Stirling====
{{trans|AWK}}
<syntaxhighlight lang="rust">
use std::f64::consts;
fn main() {
let gamma = |x: f64| { assert_ne!(x, 0.0); (2.0*consts::PI/x).sqrt() * (x * (x/consts::E).ln()).exp()};
(1..=20).for_each(|x| {
let x = f64::from(x) / 10.0;
println!("{:.02} => {:.10}", x, gamma(x));
});
}
</syntaxhighlight>
{{out}}
<pre>
0.10 => 5.6971871490
0.20 => 3.3259984240
0.30 => 2.3625300363
0.40 => 1.8414763359
0.50 => 1.5203469011
0.60 => 1.3071588574
0.70 => 1.1590532921
0.80 => 1.0533709684
0.90 => 0.9770615079
1.00 => 0.9221370089
1.10 => 0.8834899532
1.20 => 0.8577553354
1.30 => 0.8426782594
1.40 => 0.8367445486
1.50 => 0.8389565525
1.60 => 0.8486932422
1.70 => 0.8656214718
1.80 => 0.8896396353
1.90 => 0.9208427219
2.00 => 0.9595021757
</pre>
Line 4,731 ⟶ 5,835:
1.945403
2.709764</pre>
===Lanczos===
{{trans|Haskell}}
The Haskell version calculates the natural log of the gamma function, which is why the function is called <code>gammaln</code>; we correct that here by calling <code>exp</code>:
<syntaxhighlight lang="txrlisp">(defun gamma (x)
(let* ((cof #(76.18009172947146 -86.50532032941677
24.01409824083091 -1.231739572450155
0.001208650973866179 -0.000005395239384953))
(ser0 1.000000000190015)
(x55 (+ x 5.5))
(tmp (- x55 (* (+ x 0.5) (log x55))))
(ser (+ ser0 (sum [mapcar / cof (succ x)]))))
(exp (- (log (/ (* 2.5066282746310005 ser) x)) tmp))))
(each ((i (rlist 0.1..1.0..0.1 2..10)))
(put-line (pic "##.# ######.######" i (gamma i))))</syntaxhighlight>
{{out}}
<pre> 0.1 9.513508
0.2 4.590844
0.3 2.991569
0.4 2.218160
0.5 1.772454
0.6 1.489192
0.7 1.298055
0.8 1.164230
0.9 1.068629
1.0 1.000000
2.0 1.000000
3.0 2.000000
4.0 6.000000
5.0 24.000000
6.0 120.000000
7.0 720.000000
8.0 5040.000000
9.0 40320.000000
10.0 362880.000000
</pre>
From Wikipedia Python code. Output is identical to above.
<syntaxhighlight lang="txrlisp">(defun gamma (x)
(if (< x 0.5)
(/ %pi%
(* (sin (* %pi% x))
(gamma (- 1 x))))
(let* ((cof #(676.5203681218851 -1259.1392167224028
771.32342877765313 -176.61502916214059
12.507343278686905 -0.13857109526572012
9.9843695780195716e-6 1.5056327351493116e-7))
(ser0 0.99999999999980993)
(z (pred x))
(tmp (+ z (len cof) -0.5))
(ser (+ ser0 (sum [mapcar / cof (succ z)]))))
(* (sqrt (* 2 %pi%))
(expt tmp (+ z 0.5))
(exp (- tmp))
ser))))
(each ((i (rlist 0.1..1.0..0.1 2..10)))
(put-line (pic "##.# ######.######" i (gamma i))))</syntaxhighlight>
=={{header|Visual FoxPro}}==
Line 4,862 ⟶ 6,031:
{{libheader|Wren-math}}
The ''gamma'' method in the Math class is based on the Lanczos approximation.
<syntaxhighlight lang="
import "./math" for Math
var stirling = Fn.new { |x| (2 * Num.pi / x).sqrt * (x / Math.e).pow(x) }
Line 4,870 ⟶ 6,039:
for (i in 1..20) {
var d = i / 10
Fmt.print("$4.2f\t$16.14f\t$16.14f", d, stirling.call(d), Math.gamma(d))
}</syntaxhighlight>
Line 4,881 ⟶ 6,048:
0.10 5.69718714897717 9.51350769866875
0.20 3.32599842402239 4.59084371199881
0.30 2.36253003626962 2.
0.40 1.84147633593624 2.21815954375769
0.50 1.52034690106628 1.77245385090552
Line 4,899 ⟶ 6,066:
1.90 0.92084272189423 0.96176583190739
2.00 0.95950217574449 1.00000000000000
</pre>
=={{header|XPL0}}==
{{trans|Ada}}
<syntaxhighlight lang "XPL0">function real Gamma (X);
real X, A, Y, Sum;
integer N;
begin
A \constant array (0..29) of Long_Float\ :=
[ 1.00000_00000_00000_00000,
0.57721_56649_01532_86061,
-0.65587_80715_20253_88108,
-0.04200_26350_34095_23553,
0.16653_86113_82291_48950,
-0.04219_77345_55544_33675,
-0.00962_19715_27876_97356,
0.00721_89432_46663_09954,
-0.00116_51675_91859_06511,
-0.00021_52416_74114_95097,
0.00012_80502_82388_11619,
-0.00002_01348_54780_78824,
-0.00000_12504_93482_14267,
0.00000_11330_27231_98170,
-0.00000_02056_33841_69776,
0.00000_00061_16095_10448,
0.00000_00050_02007_64447,
-0.00000_00011_81274_57049,
0.00000_00001_04342_67117,
0.00000_00000_07782_26344,
-0.00000_00000_03696_80562,
0.00000_00000_00510_03703,
-0.00000_00000_00020_58326,
-0.00000_00000_00005_34812,
0.00000_00000_00001_22678,
-0.00000_00000_00000_11813,
0.00000_00000_00000_00119,
0.00000_00000_00000_00141,
-0.00000_00000_00000_00023,
0.00000_00000_00000_00002
];
Y := X - 1.0;
Sum := A (29);
for N:= 29-1 downto 0 do
Sum := Sum * Y + A (N);
return 1.0 / Sum;
end \Gamma\;
\Test program:
integer I;
begin
Format(0, 14);
for I:= 1 to 10 do
[RlOut(0, Gamma (Float (I) / 3.0)); CrLf(0)];
end</syntaxhighlight>
{{out}}
<pre>
2.67893853470775E+000
1.35411793942640E+000
1.00000000000000E+000
8.92979511569249E-001
9.02745292950934E-001
1.00000000000000E+000
1.19063934875900E+000
1.50457548825154E+000
1.99999999999397E+000
2.77815847933857E+000
</pre>
| |||