Gamma function

From Rosetta Code
Task
Gamma function
You are encouraged to solve this task according to the task description, using any language you may know.
Task

Implement one algorithm (or more) to compute the Gamma () function (in the real field only).

If your language has the function as built-in or you know a library which has it, compare your implementation's results with the results of the built-in/library function.

The Gamma function can be defined as:

This suggests a straightforward (but inefficient) way of computing the through numerical integration.


Better suggested methods:



360 Assembly

For maximum compatibility, this program uses only the basic instruction set. <lang 360asm>GAMMAT CSECT

        USING GAMMAT,R13

SAVEAR B STM-SAVEAR(R15)

        DC    17F'0'
        DC    CL8'GAMMAT'

STM STM R14,R12,12(R13)

        ST    R13,4(R15)
        ST    R15,8(R13)
        LR    R13,R15
  • ---- CODE
        LE    F4,=E'0'
        LH    R2,NI

LOOPI EQU *

        AE    F4,=E'1'         xi=xi+1
        LER   F0,F4
        DE    F0,=E'10'        x=xi/10
        STE   F0,X
        LE    F6,X
        SE    F6,=E'1'         xx=x-1.0
        LH    R4,NT
        BCTR  R4,0
        SLA   R4,2
        LE    F0,T(R4)         
        STE   F0,SUM           sum=t(nt)
        LH    R3,NT
        BCTR  R3,0
        SH    R4,=H'4'

LOOPJ CH R3,=H'1' for j=nt-1 downto 1

        BL    ENDLOOPJ
        LE    F0,SUM
        MER   F0,F6            sum*xx
        LE    F2,T(R4)         t(j)
        AER   F0,F2
        STE   F0,SUM           sum=sum*xx+t(j)
        BCTR  R3,0
        SH    R4,=H'4'
        B     LOOPJ

ENDLOOPJ EQU *

        LE    F0,=E'1'
        DE    F0,SUM
        STE   F0,GAMMA         gamma=1/sum
        LE    F0,X
        BAL   R14,CONVERT
        MVC   BUF(8),CONVERTM
        LE    F0,GAMMA
        BAL   R14,CONVERT
        MVC   BUF+9(13),CONVERTM
        WTO   MF=(E,WTOMSG)		  
        BCT   R2,LOOPI
  • ---- END CODE
        CNOP  0,4
        L     R13,4(0,R13)
        LM    R14,R12,12(R13)
        XR    R15,R15
        BR    R14
  • ---- DATA

NI DC H'30' NT DC AL2((TEND-T)/4) T DC E'1.00000000000000000000'

        DC    E'0.57721566490153286061'
        DC    E'-0.65587807152025388108'
        DC    E'-0.04200263503409523553'
        DC    E'0.16653861138229148950'
        DC    E'-0.04219773455554433675'
        DC    E'-0.00962197152787697356'
        DC    E'0.00721894324666309954'
        DC    E'-0.00116516759185906511'
        DC    E'-0.00021524167411495097'
        DC    E'0.00012805028238811619'
        DC    E'-0.00002013485478078824'
        DC    E'-0.00000125049348214267'
        DC    E'0.00000113302723198170'
        DC    E'-0.00000020563384169776'
        DC    E'0.00000000611609510448'
        DC    E'0.00000000500200764447'
        DC    E'-0.00000000118127457049'
        DC    E'0.00000000010434267117'
        DC    E'0.00000000000778226344'
        DC    E'-0.00000000000369680562'
        DC    E'0.00000000000051003703'
        DC    E'-0.00000000000002058326'
        DC    E'-0.00000000000000534812'
        DC    E'0.00000000000000122678'
        DC    E'-0.00000000000000011813'
        DC    E'0.00000000000000000119'
        DC    E'0.00000000000000000141'
        DC    E'-0.00000000000000000023'
        DC    E'0.00000000000000000002'

TEND DS 0E X DS E SUM DS E GAMMA DS E WTOMSG DS 0F

        DC    AL2(L'BUF),XL2'0000'

BUF DC CL80' '

  • Subroutine Convertion Float->Display

CONVERT CNOP 0,4

        ME    F0,CONVERTC
        STE   F0,CONVERTF
        MVI   CONVERTS,X'00'
        L     R9,CONVERTF
        CH    R9,=H'0'
        BZ    CONVERT7
        BNL   CONVERT1         is negative?
        MVI   CONVERTS,X'80'
        N     R9,=X'7FFFFFFF'  sign bit

CONVERT1 LR R8,R9

        N     R8,=X'00FFFFFF'
        BNZ   CONVERT2
        SR    R9,R9
        B     CONVERT7

CONVERT2 LR R8,R9

        N     R8,=X'FF000000'  characteristic
        SRL   R8,24
        CH    R8,=H'64'
        BH    CONVERT3
        SR    R9,R9
        B     CONVERT7

CONVERT3 CH R8,=H'72' 2**32

        BNH   CONVERT4
        L     R9,=X'7FFFFFFF'  biggest
        B     CONVERT6

CONVERT4 SR R8,R8

        SLDL  R8,8
        CH    R8,=H'72'
        BL    CONVERT5
        CH    R9,=H'0'
        BP    CONVERT5
        L     R9,=X'7FFFFFFF'
        B     CONVERT6

CONVERT5 SH R8,=H'72'

        LCR   R8,R8
        SLL   R8,2
        SRL   R9,0(R8)

CONVERT6 SR R8,R8

        IC    R8,CONVERTS
        CH    R8,=H'0'         sign bit set?
        BZ    CONVERT7
        LCR   R9,R9

CONVERT7 ST R9,CONVERTB

        CVD   R9,CONVERTP
        MVC   CONVERTD,=X'402020202120202020202020' 
        ED    CONVERTD,CONVERTP+2
        MVC   CONVERTM(6),CONVERTD 
        MVI   CONVERTM+6,C'.'
        MVC   CONVERTM+7(6),CONVERTD+6
        BR    R14

CONVERTC DC E'1E6' X'45F42400' CONVERTF DS F CONVERTB DS F CONVERTS DS X CONVERTM DS CL13 CONVERTD DS CL12 CONVERTP DS PL8

        EQUREGS
        EQUREGS REGS=FPR
        END   GAMMAT</lang>
Output:
     0.1      9.513504
     0.2      4.590844
     0.3      2.991569
     0.4      2.218160
     0.5      1.772453
     0.6      1.489192
     0.7      1.298056
     0.8      1.164229
     0.9      1.068628
     1.0      1.000000
     1.1      0.951350
     1.2      0.918168
     1.3      0.897470
     1.4      0.887263
     1.5      0.886227
     1.6      0.893515
     1.7      0.908638
     1.8      0.931383
     1.9      0.961766
     2.0      1.000000
     2.1      1.046486
     2.2      1.101803
     2.3      1.166712
     2.4      1.242169
     2.5      1.329341
     2.6      1.429626
     2.7      1.544686
     2.8      1.676492
     2.9      1.827354
     3.0      1.999999

Ada

The implementation uses Taylor series coefficients of Γ(x+1)-1, |x| < ∞. The coefficients are taken from Mathematical functions and their approximations by Yudell L. Luke. <lang ada>function Gamma (X : Long_Float) return Long_Float is

  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   : constant Long_Float := X - 1.0;
  Sum : Long_Float := A (A'Last);

begin

  for N in reverse A'First..A'Last - 1 loop
     Sum := Sum * Y + A (N);
  end loop;
  return 1.0 / Sum;

end Gamma;</lang> Test program: <lang ada>with Ada.Text_IO; use Ada.Text_IO; with Gamma;

procedure Test_Gamma is begin

  for I in 1..10 loop
     Put_Line (Long_Float'Image (Gamma (Long_Float (I) / 3.0)));
  end loop;

end Test_Gamma;</lang>

Output:
 2.67893853470775E+00
 1.35411793942640E+00
 1.00000000000000E+00
 8.92979511569249E-01
 9.02745292950934E-01
 1.00000000000000E+00
 1.19063934875900E+00
 1.50457548825154E+00
 1.99999999999397E+00
 2.77815847933858E+00

ALGOL 68

Translation of: C
- Stirling & Spouge methods.
Translation of: python
- Lanczos method.
Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386

<lang algol68># Coefficients used by the GNU Scientific Library # []LONG REAL p = ( LONG 0.99999 99999 99809 93,

                 LONG  676.52036 81218 851,    
                -LONG 1259.13921 67224 028, 
                 LONG  771.32342 87776 5313,  
                -LONG  176.61502 91621 4059,  
                 LONG   12.50734 32786 86905, 
                -LONG    0.13857 10952 65720 12,
                 LONG    9.98436 95780 19571 6e-6,
                 LONG    1.50563 27351 49311 6e-7);

PROC lanczos gamma = (LONG REAL in z)LONG REAL: (

 # Reflection formula #
 LONG REAL z := in z;
 IF z < LONG 0.5 THEN
   long pi / (long sin(long pi*z)*lanczos gamma(1-z))
 ELSE
   z -:= 1;
   LONG REAL x := p[1];
   FOR i TO UPB p - 1 DO x +:= p[i+1]/(z+i) OD;
   LONG REAL t = z + UPB p - LONG 1.5;
   long sqrt(2*long pi) * t**(z+LONG 0.5) * long exp(-t) * x
 FI

);

PROC taylor gamma = (LONG REAL x)LONG REAL: BEGIN # good for values between 0 and 1 #

   []LONG REAL a =
       ( LONG 1.00000 00000 00000 00000,
         LONG 0.57721 56649 01532 86061,
        -LONG 0.65587 80715 20253 88108,
        -LONG 0.04200 26350 34095 23553,
         LONG 0.16653 86113 82291 48950,
        -LONG 0.04219 77345 55544 33675,
        -LONG 0.00962 19715 27876 97356,
         LONG 0.00721 89432 46663 09954,
        -LONG 0.00116 51675 91859 06511,
        -LONG 0.00021 52416 74114 95097,
         LONG 0.00012 80502 82388 11619,
        -LONG 0.00002 01348 54780 78824,
        -LONG 0.00000 12504 93482 14267,
         LONG 0.00000 11330 27231 98170,
        -LONG 0.00000 02056 33841 69776,
         LONG 0.00000 00061 16095 10448,
         LONG 0.00000 00050 02007 64447,
        -LONG 0.00000 00011 81274 57049,
         LONG 0.00000 00001 04342 67117,
         LONG 0.00000 00000 07782 26344,
        -LONG 0.00000 00000 03696 80562,
         LONG 0.00000 00000 00510 03703,
        -LONG 0.00000 00000 00020 58326,
        -LONG 0.00000 00000 00005 34812,
         LONG 0.00000 00000 00001 22678,
        -LONG 0.00000 00000 00000 11813,
         LONG 0.00000 00000 00000 00119,
         LONG 0.00000 00000 00000 00141,
        -LONG 0.00000 00000 00000 00023,
         LONG 0.00000 00000 00000 00002
       );
   LONG REAL y = x - 1;
   LONG REAL sum := a [UPB a];
   FOR n FROM UPB a - 1 DOWNTO LWB a DO
      sum := sum * y + a [n]
   OD;
   1/sum

END # taylor gamma #;

LONG REAL long e = long exp(1);

PROC sterling gamma = (LONG REAL n)LONG REAL: ( # improves for values much greater then 1 #

 long sqrt(2*long pi/n)*(n/long e)**n

);

PROC factorial = (LONG INT n)LONG REAL: (

 IF n=0 OR n=1 THEN 1
 ELIF n=2 THEN 2
 ELSE n*factorial(n-1) FI

);

REF[]LONG REAL fm := NIL;

PROC sponge gamma = (LONG REAL x)LONG REAL: (

 INT a = 12; # alter to get required precision #
 REF []LONG REAL fm := NIL;
 LONG REAL res;

 IF fm :=: REF[]LONG REAL(NIL) THEN
   fm := HEAP[0:a-1]LONG REAL;
   fm[0] := long sqrt(LONG 2*long pi);
   FOR k TO a-1 DO
     fm[k] := (((k-1) MOD 2=0) | 1 | -1) * long exp(a-k) *

(a-k) **(k-LONG 0.5) / factorial(k-1)

   OD
 FI;
 res := fm[0];
 FOR k TO a-1 DO
   res +:= fm[k] / ( x + k )
 OD;
 res *:= long exp(-(x+a)) * (x+a)**(x + LONG 0.5);
 res/x

);

FORMAT real fmt = $g(-real width, real width - 2)$; FORMAT long real fmt16 = $g(-17, 17 - 2)$; # accurate to about 16 decimal places #

[]STRING methods = ("Genie", "Lanczos", "Sponge", "Taylor","Stirling");

printf(($11xg12xg12xg13xg13xgl$,methods));

FORMAT sample fmt = $"gamma("g(-3,1)")="f(real fmt)n(UPB methods-1)(", "f(long real fmt16))l$; FORMAT sqr sample fmt = $"gamma("g(-3,1)")**2="f(real fmt)n(UPB methods-1)(", "f(long real fmt16))l$; FORMAT sample exp fmt = $"gamma("g(-3)")="g(-15,11,0)n(UPB methods-1)(","g(-18,14,0))l$;

PROC sample = (LONG REAL x)[]LONG REAL:

 (gamma(SHORTEN x), lanczos gamma(x), sponge gamma(x), taylor gamma(x), sterling gamma(x));

FOR i FROM 1 TO 20 DO

 LONG REAL x = i / LONG 10;
 printf((sample fmt, x, sample(x)));
 IF i = 5 THEN # insert special case of a half #
   printf((sqr sample fmt,
           x, gamma(SHORTEN x)**2,  lanczos gamma(x)**2, sponge gamma(x)**2,
           taylor gamma(x)**2, sterling gamma(x)**2))
 FI

OD; FOR x FROM 10 BY 10 TO 70 DO

 printf((sample exp fmt, x, sample(x)))

OD</lang>

Output:
           Genie            Lanczos            Sponge             Taylor             Stirling
gamma(0.1)=9.5135076986687, 9.513507698668730, 9.513507698668731, 9.513509522249043, 5.697187148977169
gamma(0.2)=4.5908437119988, 4.590843711998802, 4.590843711998803, 4.590843743037192, 3.325998424022393
gamma(0.3)=2.9915689876876, 2.991568987687590, 2.991568987687590, 2.991568988322729, 2.362530036269620
gamma(0.4)=2.2181595437577, 2.218159543757688, 2.218159543757688, 2.218159543764845, 1.841476335936235
gamma(0.5)=1.7724538509055, 1.772453850905517, 1.772453850905516, 1.772453850905353, 1.520346901066281
gamma(0.5)**2=3.1415926535898, 3.141592653589795, 3.141592653589793, 3.141592653589216, 2.311454699581843
gamma(0.6)=1.4891922488128, 1.489192248812817, 1.489192248812817, 1.489192248812758, 1.307158857448356
gamma(0.7)=1.2980553326476, 1.298055332647558, 1.298055332647558, 1.298055332647558, 1.159053292113920
gamma(0.8)=1.1642297137253, 1.164229713725304, 1.164229713725303, 1.164229713725303, 1.053370968425609
gamma(0.9)=1.0686287021193, 1.068628702119320, 1.068628702119319, 1.068628702119319, 0.977061507877695
gamma(1.0)=1.0000000000000, 1.000000000000000, 1.000000000000000, 1.000000000000000, 0.922137008895789
gamma(1.1)=0.9513507698669, 0.951350769866873, 0.951350769866873, 0.951350769866873, 0.883489953168704
gamma(1.2)=0.9181687423998, 0.918168742399761, 0.918168742399760, 0.918168742399761, 0.857755335396591
gamma(1.3)=0.8974706963063, 0.897470696306277, 0.897470696306277, 0.897470696306277, 0.842678259448392
gamma(1.4)=0.8872638175031, 0.887263817503076, 0.887263817503075, 0.887263817503064, 0.836744548637082
gamma(1.5)=0.8862269254528, 0.886226925452758, 0.886226925452758, 0.886226925452919, 0.838956552526496
gamma(1.6)=0.8935153492877, 0.893515349287691, 0.893515349287690, 0.893515349288799, 0.848693242152574
gamma(1.7)=0.9086387328533, 0.908638732853291, 0.908638732853290, 0.908638732822421, 0.865621471793884
gamma(1.8)=0.9313837709802, 0.931383770980243, 0.931383770980242, 0.931383769950169, 0.889639635287994
gamma(1.9)=0.9617658319074, 0.961765831907388, 0.961765831907387, 0.961765815012982, 0.920842721894229
gamma(2.0)=1.0000000000000, 1.000000000000000, 0.999999999999999, 1.000000010045742, 0.959502175744492
gamma( 10)= 3.6288000000e5, 3.6288000000000e5, 3.6288000000000e5, 4.051218760300e-7, 3.5986956187410e5
gamma( 20)= 1.216451004e17, 1.216451004088e17, 1.216451004088e17, 1.07701514977e-18, 1.211393423381e17
gamma( 30)= 8.841761994e30, 8.841761993740e30, 8.841761993739e30, 7.98891286318e-23, 8.817236530765e30
gamma( 40)= 2.039788208e46, 2.039788208120e46, 2.039788208120e46, 6.97946184592e-25, 2.035543161237e46
gamma( 50)= 6.082818640e62, 6.082818640343e62, 6.082818640342e62, 1.81016585713e-26, 6.072689187876e62
gamma( 60)= 1.386831185e80, 1.386831185457e80, 1.386831185457e80, 9.27306839649e-28, 1.384906385829e80
gamma( 70)= 1.711224524e98, 1.711224524281e98, 1.711224524281e98, 7.57303907062e-29, 1.709188578191e98

ANSI Standard BASIC

Translation of: BBC Basic
- Lanczos method.

<lang ANSI Standard 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</lang>

AutoHotkey

Search autohotkey.com: function
Source: AutoHotkey forum by Laszlo <lang autohotkey>/* Here is the upper incomplete Gamma function. Omitting or setting the second parameter to 0 we get the (complete) Gamma function. The code is based on: "Computation of Special Functions" Zhang and Jin, John Wiley and Sons, 1996

  • /

SetFormat FloatFast, 0.9e

Loop 10

  MsgBox % GAMMA(A_Index/3) "`n" GAMMA(A_Index*10) 

GAMMA(a,x=0) {  ; upper incomplete gamma: Integral(t**(a-1)*e**-t, t = x..inf)

  If (a > 171 || x < 0) 
     Return 2.e308   ; overflow 
  xam := x > 0 ? -x+a*ln(x) : 0 
  If (xam > 700) 
     Return 2.e308   ; overflow 
  If (x > 1+a) {     ; no need for gamma(a) 
     t0 := 0, k := 60 
     Loop 60 
         t0 := (k-a)/(1+k/(x+t0)), --k 
     Return exp(xam) / (x+t0) 
  } 
  r := 1, ga := 1.0  ; compute ga = gamma(a) ... 
  If (a = round(a))  ; if integer: factorial 
     If (a > 0) 
        Loop % a-1 
           ga *= A_Index 
     Else            ; negative integer 
        ga := 1.7976931348623157e+308 ; Dmax 
  Else {             ; not integer 
     If (abs(a) > 1) { 
        z := abs(a) 
        m := floor(z) 
        Loop %m% 
            r *= (z-A_Index) 
        z -= m 
     } 
     Else 
        z := a 
     gr := (((((((((((((((((((((((       0.14e-14 
         *z - 0.54e-14)             *z - 0.206e-13)          *z + 0.51e-12) 
         *z - 0.36968e-11)          *z + 0.77823e-11)        *z + 0.1043427e-9) 
         *z - 0.11812746e-8)        *z + 0.50020075e-8)      *z + 0.6116095e-8) 
         *z - 0.2056338417e-6)      *z + 0.1133027232e-5)    *z - 0.12504934821e-5) 
         *z - 0.201348547807e-4)    *z + 0.1280502823882e-3) *z - 0.2152416741149e-3) 
         *z - 0.11651675918591e-2)  *z + 0.7218943246663e-2) *z - 0.9621971527877e-2) 
         *z - 0.421977345555443e-1) *z + 0.1665386113822915) *z - 0.420026350340952e-1) 
         *z - 0.6558780715202538)   *z + 0.5772156649015329) *z + 1 
     ga := 1.0/(gr*z) * r 
     If (a < -1) 
        ga := -3.1415926535897931/(a*ga*sin(3.1415926535897931*a)) 
  } 
  If (x = 0)         ; complete gamma requested 
     Return ga 
  s := 1/a           ; here x <= 1+a 
  r := s 
  Loop 60 { 
     r *= x/(a+A_Index) 
     s += r 
     If (abs(r/s) < 1.e-15) 
        break 
  } 
  Return ga - exp(xam)*s 

}

/* The 10 results shown: 2.678938535e+000 1.354117939e+000 1.0 8.929795115e-001 9.027452930e-001 3.628800000e+005 1.216451004e+017 8.841761994e+030 2.039788208e+046 6.082818640e+062

1.000000000e+000 1.190639348e+000 1.504575489e+000 2.000000000e+000 2.778158479e+000 1.386831185e+080 1.711224524e+098 8.946182131e+116 1.650795516e+136 9.332621544e+155

  • /</lang>

BBC BASIC

Uses the Lanczos approximation. <lang bbcbasic> *FLOAT64

     INSTALL @lib$+"FNUSING"
     
     FOR x = 0.1 TO 2.05 STEP 0.1
       PRINT FNusing("#.#",x), FNusing("##.############", FNgamma(x))
     NEXT
     END
     
     DEF FNgamma(z) = EXP(FNlngamma(z))
     
     DEF FNlngamma(z)
     LOCAL a, b, i%, lz()
     DIM lz(6)
     lz() = 1.000000000190015, 76.18009172947146, -86.50532032941677, \
     \ 24.01409824083091, -1.231739572450155, 0.0012086509738662, -0.000005395239385
     IF z < 0.5 THEN = LN(PI / SIN(PI * z)) - FNlngamma(1.0 - z)
     z -= 1.0
     b = z + 5.5
     a = lz(0)
     FOR i% = 1 TO 6
       a += lz(i%) / (z + i%)
     NEXT
     = (LNSQR(2*PI) + LN(a) - b) + LN(b) * (z+0.5)</lang>

Output:

0.1        9.513507698670
0.2        4.590843712000
0.3        2.991568987689
0.4        2.218159543760
0.5        1.772453850902
0.6        1.489192248811
0.7        1.298055332647
0.8        1.164229713725
0.9        1.068628702119
1.0        1.000000000000
1.1        0.951350769867
1.2        0.918168742400
1.3        0.897470696306
1.4        0.887263817503
1.5        0.886226925453
1.6        0.893515349288
1.7        0.908638732853
1.8        0.931383770980
1.9        0.961765831907
2.0        1.000000000000

C

This implements Stirling's approximation and Spouge's approximation.

<lang c>#include <stdio.h>

  1. include <stdlib.h>
  2. include <math.h>
  3. include <gsl/gsl_sf_gamma.h>
  4. ifndef M_PI
  5. define M_PI 3.14159265358979323846
  6. endif

/* very simple approximation */ double st_gamma(double x) {

 return sqrt(2.0*M_PI/x)*pow(x/M_E, x);

}

  1. define A 12

double sp_gamma(double z) {

 const int a = A;
 static double c_space[A];
 static double *c = NULL;
 int k;
 double accm;

 if ( c == NULL ) {
   double k1_factrl = 1.0; /* (k - 1)!*(-1)^k with 0!==1*/
   c = c_space;
   c[0] = sqrt(2.0*M_PI);
   for(k=1; k < a; k++) {
     c[k] = exp(a-k) * pow(a-k, k-0.5) / k1_factrl;

k1_factrl *= -k;

   }
 }
 accm = c[0];
 for(k=1; k < a; k++) {
   accm += c[k] / ( z + k );
 }
 accm *= exp(-(z+a)) * pow(z+a, z+0.5); /* Gamma(z+1) */
 return accm/z;

}

int main() {

 double x;


 printf("%15s%15s%15s%15s\n", "Stirling", "Spouge", "GSL", "libm");
 for(x=1.0; x <= 10.0; x+=1.0) {
   printf("%15.8lf%15.8lf%15.8lf%15.8lf\n", st_gamma(x/3.0), sp_gamma(x/3.0), 

gsl_sf_gamma(x/3.0), tgamma(x/3.0));

 }
 return 0;

}</lang>

Output:
       Stirling         Spouge            GSL           libm
     2.15697602     2.67893853     2.67893853     2.67893853
     1.20285073     1.35411794     1.35411794     1.35411794
     0.92213701     1.00000000     1.00000000     1.00000000
     0.83974270     0.89297951     0.89297951     0.89297951
     0.85919025     0.90274529     0.90274529     0.90274529
     0.95950218     1.00000000     1.00000000     1.00000000
     1.14910642     1.19063935     1.19063935     1.19063935
     1.45849038     1.50457549     1.50457549     1.50457549
     1.94540320     2.00000000     2.00000000     2.00000000
     2.70976382     2.77815848     2.77815848     2.77815848

C#

This is just rewritten from the Wikipedia Lanczos article. Works with complex numbers as well as reals. <lang csharp>using System; using System.Numerics;

static int g = 7; static double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7};

Complex Gamma(Complex z) {

   // Reflection formula
   if (z.Real < 0.5)

{

       return Math.PI / (Complex.Sin( Math.PI * z) * Gamma(1 - z));

}

   else

{

       z -= 1;
       Complex x = p[0];
       for (var i = 1; i < g + 2; i++)

{

           x += p[i]/(z+i);

}

       Complex t = z + g + 0.5;
       return Complex.Sqrt(2 * Math.PI) * (Complex.Pow(t, z + 0.5)) * Complex.Exp(-t) * x;

} } </lang>

Clojure

<lang clojure>(defn gamma

 "Returns Gamma(z + 1 = number) using Lanczos approximation."
 [number]
 (if (< number 0.5)
      (/ Math/PI (* (Math/sin (* Math/PI number))

(gamma (- 1 number))))

      (let [n (dec number)
     	     c [0.99999999999980993 676.5203681218851 -1259.1392167224028

771.32342877765313 -176.61502916214059 12.507343278686905 -0.13857109526572012 9.9843695780195716e-6 1.5056327351493116e-7]]

        (* (Math/sqrt (* 2 Math/PI))

(Math/pow (+ n 7 0.5) (+ n 0.5)) (Math/exp (- (+ n 7 0.5)))

           (+ (first c) 
              (apply + (map-indexed #(/ %2 (+ n %1 1)) (next c))))))))</lang>
Output:

<lang clojure>(map #(printf "%.1f %.4f\n" % (gamma %)) (map #(float (/ % 10)) (range 1 31)))</lang>

0.1 9.5135
0.2 4.5908
0.3 2.9916
0.4 2.2182
0.5 1.7725
0.6 1.4892
0.7 1.2981
0.8 1.1642
0.9 1.0686
1.0 1.0000
1.1 0.9514
1.2 0.9182
1.3 0.8975
1.4 0.8873
1.5 0.8862
1.6 0.8935
1.7 0.9086
1.8 0.9314
1.9 0.9618
2.0 1.0000
2.1 1.0465
2.2 1.1018
2.3 1.1667
2.4 1.2422
2.5 1.3293
2.6 1.4296
2.7 1.5447
2.8 1.6765
2.9 1.8274
3.0 2.0000

Common Lisp

<lang lisp>; Taylor series coefficients (defconstant tcoeff

 '( 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))
number of coefficients

(defconstant numcoeff (length tcoeff))

(defun gamma (x)

 (let ((y (- x 1.0))
       (sum (nth (- numcoeff 1) tcoeff)))
   (loop for i from (- numcoeff 2) downto 0 do 
         (setf sum (+ (* sum y) (nth i tcoeff))))
   (/ 1.0 sum)))

(loop for i from 1 to 10

  do (
    format t "~12,10f~%" (gamma (/ i 3.0))))</lang>
Produces:
2.6789380000
1.3541179000
1.0000000000
0.8929794500
0.9027453000
1.0000000000
1.1906393000
1.5045753000
1.9999995000
2.7781580000

Crystal

Taylor series vs Builtin Function

Translation of: Ruby

<lang ruby>def a

  [ 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 ]

end

def gamma(x)

 y = x.to_f - 1
 1.0 / a.reverse.reduce(0) { |sum, an| sum * y + an }

end

puts " Taylor Series Builtin Function" (1..27).each { |i| n = i/3.0; puts "gamma(%.2f) = %.14e  %.14e" % [n, gamma(n), Math.gamma(n)] }</lang>

Output:
               Taylor Series         Builtin Function
gamma(0.33) = 2.67893853470775e+00  2.67893853470775e+00
gamma(0.67) = 1.35411793942640e+00  1.35411793942640e+00
gamma(1.00) = 1.00000000000000e+00  1.00000000000000e+00
gamma(1.33) = 8.92979511569249e-01  8.92979511569249e-01
gamma(1.67) = 9.02745292950934e-01  9.02745292950934e-01
gamma(2.00) = 1.00000000000000e+00  1.00000000000000e+00
gamma(2.33) = 1.19063934875900e+00  1.19063934875900e+00
gamma(2.67) = 1.50457548825154e+00  1.50457548825156e+00
gamma(3.00) = 1.99999999999397e+00  2.00000000000000e+00
gamma(3.33) = 2.77815847933857e+00  2.77815848043766e+00
gamma(3.67) = 4.01220118377482e+00  4.01220130200415e+00
gamma(4.00) = 5.99999141007240e+00  6.00000000000000e+00
gamma(4.33) = 9.26006653812473e+00  9.26052826812554e+00
gamma(4.67) = 1.46918499266721e+01  1.47114047740152e+01
gamma(5.00) = 2.33327665969918e+01  2.40000000000000e+01
gamma(5.33) = 2.65211050660964e+01  4.01289558285440e+01
gamma(5.67) = 7.70471336505311e+00  6.86532222787377e+01
gamma(6.00) = 1.10934146590517e+00  1.20000000000000e+02
gamma(6.33) = 1.64621072447163e-01  2.14021097752235e+02
gamma(6.67) = 2.72102446536397e-02  3.89034926246180e+02
gamma(7.00) = 4.98014348954507e-03  7.20000000000000e+02
gamma(7.33) = 9.98845907123850e-04  1.35546695243082e+03
gamma(7.67) = 2.17513475446479e-04  2.59356617497454e+03
gamma(8.00) = 5.10217006678528e-05  5.04000000000000e+03
gamma(8.33) = 1.28035516395359e-05  9.94009098449270e+03
gamma(8.67) = 3.41689149138074e-06  1.98840073414714e+04
gamma(9.00) = 9.64721467591131e-07  4.03200000000000e+04

D

<lang d>import std.stdio, std.math, std.mathspecial;

real taylorGamma(in real x) pure nothrow @safe @nogc {

   static immutable real[30] table = [
    0x1p+0,                    0x1.2788cfc6fb618f4cp-1,
   -0x1.4fcf4026afa2dcecp-1,  -0x1.5815e8fa27047c8cp-5,
    0x1.5512320b43fbe5dep-3,  -0x1.59af103c340927bep-5,
   -0x1.3b4af28483e214e4p-7,   0x1.d919c527f60b195ap-8,
   -0x1.317112ce3a2a7bd2p-10, -0x1.c364fe6f1563ce9cp-13,
    0x1.0c8a78cd9f9d1a78p-13, -0x1.51ce8af47eabdfdcp-16,
   -0x1.4fad41fc34fbb2p-20,    0x1.302509dbc0de2c82p-20,
   -0x1.b9986666c225d1d4p-23,  0x1.a44b7ba22d628acap-28,
    0x1.57bc3fc384333fb2p-28, -0x1.44b4cedca388f7c6p-30,
    0x1.cae7675c18606c6p-34,   0x1.11d065bfaf06745ap-37,
   -0x1.0423bac8ca3faaa4p-38,  0x1.1f20151323cd0392p-41,
   -0x1.72cb88ea5ae6e778p-46, -0x1.815f72a05f16f348p-48,
    0x1.6198491a83bccbep-50,  -0x1.10613dde57a88bd6p-53,
    0x1.5e3fee81de0e9c84p-60,  0x1.a0dc770fb8a499b6p-60,
   -0x1.0f635344a29e9f8ep-62,  0x1.43d79a4b90ce8044p-66];
   immutable real y = x - 1.0L;
   real sm = table[$ - 1];
   foreach_reverse (immutable an; table[0 .. $ - 1])
       sm = sm * y + an;
   return 1.0L / sm;

}

real lanczosGamma(real z) pure nothrow @safe @nogc {

   // Coefficients used by the GNU Scientific Library.
   // http://en.wikipedia.org/wiki/Lanczos_approximation
   enum g = 7;
   static immutable real[9] table =
       [    0.99999_99999_99809_93,
          676.52036_81218_851,
        -1259.13921_67224_028,
          771.32342_87776_5313,
         -176.61502_91621_4059,
           12.50734_32786_86905,
           -0.13857_10952_65720_12,
            9.98436_95780_19571_6e-6,
            1.50563_27351_49311_6e-7];
   // Reflection formula.
   if (z < 0.5L) {
       return PI / (sin(PI * z) * lanczosGamma(1 - z));
   } else {
       z -= 1;
       real x = table[0];
       foreach (immutable i; 1 .. g + 2)
           x += table[i] / (z + i);
       immutable real t = z + g + 0.5L;
       return sqrt(2 * PI) * t ^^ (z + 0.5L) * exp(-t) * x;
   }

}

void main() {

   foreach (immutable i; 1 .. 11) {
       immutable real x = i / 3.0L;
       writefln("%f: %20.19e %20.19e %20.19e", x,
                x.taylorGamma, x.lanczosGamma, x.gamma);
   }

}</lang>

Output:
0.333333: 2.6789385347077476335e+00 2.6789385347077470551e+00 2.6789385347077476339e+00
0.666667: 1.3541179394264004169e+00 1.3541179394264007092e+00 1.3541179394264004170e+00
1.000000: 1.0000000000000000000e+00 1.0000000000000002126e+00 1.0000000000000000000e+00
1.333333: 8.9297951156924921124e-01 8.9297951156924947465e-01 8.9297951156924921132e-01
1.666667: 9.0274529295093361132e-01 9.0274529295093396555e-01 9.0274529295093361123e-01
2.000000: 1.0000000000000000000e+00 1.0000000000000004903e+00 1.0000000000000000000e+00
2.333333: 1.1906393487589989474e+00 1.1906393487589996490e+00 1.1906393487589989482e+00
2.666667: 1.5045754882515545787e+00 1.5045754882515570474e+00 1.5045754882515560190e+00
3.000000: 1.9999999999992207405e+00 2.0000000000000015575e+00 2.0000000000000000000e+00
3.333333: 2.7781584802531739378e+00 2.7781584804376666336e+00 2.7781584804376642124e+00

Elixir

Translation of: Ruby

<lang elixir>defmodule Gamma do

 @a [ 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 ]
    |> Enum.reverse
 def taylor(x) do
   y = x - 1
   1 / Enum.reduce(@a, 0, fn a,sum -> sum * y + a end)
 end

end

Enum.each(Enum.map(1..10, &(&1/3)), fn x ->

 :io.format "~f  ~18.15f~n", [x, Gamma.taylor(x)]

end)</lang>

Output:
0.333333   2.678938534707748
0.666667   1.354117939426401
1.000000   1.000000000000000
1.333333   0.892979511569249
1.666667   0.902745292950934
2.000000   1.000000000000000
2.333333   1.190639348758999
2.666667   1.504575488251540
3.000000   1.999999999993968
3.333333   2.778158479338573

Forth

Cristinel Mortici describes this method in Applied Mathematics Letters. "A substantial improvement of the Stirling formula". This algorithm is said to give about 7 good digits, but becomes more inaccurate close to zero. Therefore, a "shift" is performed to move the value returned into the more accurate domain. <lang forth>8 constant (gamma-shift)

(mortici) ( f1 -- f2)
 -1 s>f f+ 1 s>f
 fover 271828183e-8 f* 12 s>f f* f/
 fover 271828183e-8 f/ f+
 fover f** fswap
 628318530e-8 f* fsqrt f*             \ 2*pi
gamma ( f1 -- f2)
 fdup f0< >r fdup f0= r> or abort" Gamma less or equal to zero"
 fdup (gamma-shift) s>f f+ (mortici) fswap
 1 s>f (gamma-shift) 0 do fover i s>f f+ f* loop fswap fdrop f/
</lang>
0.1e gamma f. 9.51348888533932  ok
2e gamma f. 0.999999031674546  ok
10e gamma f. 362879.944850072  ok
70e gamma fe. 171.122444600510E96  ok

This is a word, based on a formula of Ramanujan's famous "lost notebook", which was rediscovered in 1976. His formula contained a constant, which had a value between 1/100 and 1/30. In 2008, E.A. Karatsuba described the function, which determines the value of this constant. Since it contains the gamma function itself, it can't be used in a word calculating the gamma function, so here it is emulated by two symmetrical sigmoidals. <lang forth>2 constant (gamma-shift) \ don't change this

                                      \ an approximation of the d(x) function
~d(x) ( f1 -- f2)
 fdup 10 s>f f<                       \ use first symmetrical sigmoidal
 if                                   \ for range 1-10
   -2705443e-8 fswap 2280802e-6 f/ 1428045e-6 f** 1 s>f f+ f/ 3187831e-8 f+
 else                                 \ use second symmetrical sigmoidal
   -29372563e-9 fswap 1841693e-6 f/ 1052779e-6 f** 1 s>f f+ f/ 3330828e-8 f+
 then 333333333e-10 fover f< if fdrop 1 s>f 30 s>f f/ then
\ perform some sane clipping to infinity
(ramanujan) ( f1 -- f2)
 fdup fdup f* 4 s>f f*                ( n 4n2)
 fover fover f* fdup f+ f+ fover f+   ( n 8n3+4n2+n)
 fover ~d(x) f+                       ( n 8n3+4n2+n+d[x])
 1 s>f 6 s>f f/ f**                   ( n 8n3+4n2+n+d[x]^1/6)
 fswap fdup 2.7182818284590452353e f/ ( 8n3+4n2+n+d[x]^1/6 n n/e)
 fswap f** f* pi fsqrt f*             ( f)
gamma ( f1 -- f2)
 fdup f0< >r fdup f0= r> or abort" Gamma less or equal to zero"
 fdup (gamma-shift) 1- s>f f+ (ramanujan) fswap
 1 s>f (gamma-shift) 0 do fover i s>f f+ f* loop fswap fdrop f/
</lang>
0.1e gamma f. 9.51351721918848  ok
2e gamma f. 0.999999966026125  ok
10e gamma f. 362879.999559333  ok
70e gamma fe. 171.122452428147E96  ok

Fortran

This code shows two methods: Numerical Integration through Simpson formula, and Lanczos approximation. The results of testing are printed altogether with the values given by the function gamma; this function is defined in the Fortran 2008 standard, and supported by GNU Fortran (and other vendors) as extension; if not present in your compiler, you can remove the last part of the print in order to get it compiled with any Fortran 95 compliant compiler.

Works with: Fortran version 2008
Works with: Fortran version 95 with extensions

<lang fortran>program ComputeGammaInt

 implicit none
 integer :: i
 write(*, "(3A15)") "Simpson", "Lanczos", "Builtin"
 do i=1, 10
    write(*, "(3F15.8)") my_gamma(i/3.0), lacz_gamma(i/3.0), gamma(i/3.0)
 end do

contains

 pure function intfuncgamma(x, y) result(z)
   real :: z
   real, intent(in) :: x, y
   
   z = x**(y-1.0) * exp(-x)
 end function intfuncgamma


 function my_gamma(a) result(g)
   real :: g
   real, intent(in) :: a
   real, parameter :: small = 1.0e-4
   integer, parameter :: points = 100000
   real :: infty, dx, p, sp(2, points), x
   integer :: i
   logical :: correction
   x = a
   correction = .false.
   ! value with x<1 gives \infty, so we use
   ! \Gamma(x+1) = x\Gamma(x)
   ! to avoid the problem
   if ( x < 1.0 ) then
      correction = .true.
      x = x + 1
   end if
   ! find a "reasonable" infinity...
   ! we compute this integral indeed
   ! \int_0^M dt t^{x-1} e^{-t}
   ! where M is such that M^{x-1} e^{-M} ≤ \epsilon
   infty = 1.0e4
   do while ( intfuncgamma(infty, x) > small )
      infty = infty * 10.0
   end do
   ! using simpson
   dx = infty/real(points)
   sp = 0.0
   forall(i=1:points/2-1) sp(1, 2*i) = intfuncgamma(2.0*(i)*dx, x)
   forall(i=1:points/2) sp(2, 2*i - 1) = intfuncgamma((2.0*(i)-1.0)*dx, x)
   g = (intfuncgamma(0.0, x) + 2.0*sum(sp(1,:)) + 4.0*sum(sp(2,:)) + &
        intfuncgamma(infty, x))*dx/3.0
   if ( correction ) g = g/a
 end function my_gamma


 recursive function lacz_gamma(a) result(g)
   real, intent(in) :: a
   real :: g
   real, parameter :: pi = 3.14159265358979324
   integer, parameter :: cg = 7
   ! these precomputed values are taken by the sample code in Wikipedia,
   ! and the sample itself takes them from the GNU Scientific Library
   real, dimension(0:8), parameter :: p = &
        (/ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, &
        771.32342877765313, -176.61502916214059, 12.507343278686905, &
        -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 /)
   real :: t, w, x
   integer :: i
   x = a
   if ( x < 0.5 ) then
      g = pi / ( sin(pi*x) * lacz_gamma(1.0-x) )
   else
      x = x - 1.0
      t = p(0)
      do i=1, cg+2
         t = t + p(i)/(x+real(i))
      end do
      w = x + real(cg) + 0.5
      g = sqrt(2.0*pi) * w**(x+0.5) * exp(-w) * t
   end if
 end function lacz_gamma

end program ComputeGammaInt</lang>

Output:
        Simpson        Lanczos        Builtin
     2.65968132     2.67893744     2.67893839
     1.35269761     1.35411859     1.35411787
     1.00000060     1.00000024     1.00000000
     0.88656044     0.89297968     0.89297950
     0.90179849     0.90274525     0.90274531
     0.99999803     1.00000036     1.00000000
     1.19070935     1.19063985     1.19063926
     1.50460517     1.50457609     1.50457561
     2.00000286     2.00000072     2.00000000
     2.77815390     2.77816010     2.77815843

FreeBASIC

Translation of: Java

<lang freebasic>' FB 1.05.0 Win64

Const pi = 3.1415926535897932 Const e = 2.7182818284590452

Function gammaStirling (x As Double) As Double

 Return Sqr(2.0 * pi / x) * ((x / e) ^ x)

End Function

Function gammaLanczos (x As Double) As Double

 Dim p(0 To 8) As Double = _ 
 { _
      0.99999999999980993, _ 
    676.5203681218851, _ 
  -1259.1392167224028, _			     	  
    771.32342877765313, _ 
   -176.61502916214059, _ 
     12.507343278686905, _
     -0.13857109526572012, _ 
      9.9843695780195716e-6, _
      1.5056327351493116e-7 _
 }

 Dim As Integer g = 7
 If x < 0.5 Then Return pi / (Sin(pi * x) * gammaLanczos(1-x))
 x -= 1
 Dim a As Double = p(0)
 Dim t As Double = x + g + 0.5
 
 For i As Integer = 1 To 8
   a += p(i) / (x + i)
 Next		 

 Return Sqr(2.0 * pi) * (t ^ (x + 0.5)) * Exp(-t) * a  

End Function

Print " x", " Stirling",, " Lanczos" Print For i As Integer = 1 To 20

  Dim As Double d = i / 10.0
  Print   Using "#.##"; d; 
  Print , Using "#.###############"; gammaStirling(d);
  Print , Using "#.###############"; gammaLanczos(d)

Next Print Print "Press any key to quit" Sleep</lang>

Output:
 x                Stirling                    Lanczos

0.10          5.697187148977170           9.513507698668738
0.20          3.325998424022393           4.590843711998803
0.30          2.362530036269620           2.991568987687590
0.40          1.841476335936235           2.218159543757687
0.50          1.520346901066281           1.772453850905516
0.60          1.307158857448356           1.489192248812818
0.70          1.159053292113920           1.298055332647558
0.80          1.053370968425609           1.164229713725303
0.90          0.977061507877695           1.068628702119319
1.00          0.922137008895789           1.000000000000000
1.10          0.883489953168704           0.951350769866874
1.20          0.857755335396591           0.918168742399761
1.30          0.842678259448392           0.897470696306278
1.40          0.836744548637082           0.887263817503076
1.50          0.838956552526496           0.886226925452759
1.60          0.848693242152574           0.893515349287691
1.70          0.865621471793884           0.908638732853291
1.80          0.889639635287995           0.931383770980243
1.90          0.920842721894229           0.961765831907388
2.00          0.959502175744492           1.000000000000000

F#

Solved using the Lanczos Coefficients described in Numerical Recipes (Press et al)

<lang F Sharp>

open System

let gamma z =

   let lanczosCoefficients = [76.18009172947146;-86.50532032941677;24.01409824083091;-1.231739572450155;0.1208650973866179e-2;-0.5395239384953e-5]
   let rec sumCoefficients acc i coefficients =
       match coefficients with
       | []   -> acc
       | h::t -> sumCoefficients (acc + (h/i)) (i+1.0) t
   let gamma = 5.0
   let x = z - 1.0
   Math.Pow(x + gamma + 0.5, x + 0.5) * Math.Exp( -(x + gamma + 0.5) ) * Math.Sqrt( 2.0 * Math.PI ) * sumCoefficients 1.000000000190015 (x + 1.0) lanczosCoefficients

seq { for i in 1 .. 20 do yield ((double)i/10.0) } |> Seq.iter ( fun v -> System.Console.WriteLine("{0} : {1}", v, gamma v ) ) seq { for i in 1 .. 10 do yield ((double)i*10.0) } |> Seq.iter ( fun v -> System.Console.WriteLine("{0} : {1}", v, gamma v ) )

</lang>

Output:
0.1 : 9.51350769855015
0.2 : 4.59084371196153
0.3 : 2.99156898767207
0.4 : 2.21815954375051
0.5 : 1.77245385090205
0.6 : 1.48919224881114
0.7 : 1.29805533264677
0.8 : 1.16422971372497
0.9 : 1.06862870211921
1 : 1
1.1 : 0.951350769866919
1.2 : 0.91816874239982
1.3 : 0.897470696306335
1.4 : 0.887263817503124
1.5 : 0.886226925452797
1.6 : 0.893515349287718
1.7 : 0.908638732853309
1.8 : 0.931383770980253
1.9 : 0.961765831907391
2 : 1
10 : 362880.000000085
20 : 1.21645100409886E+17
30 : 8.84176199395902E+30
40 : 2.03978820820436E+46
50 : 6.08281864068541E+62
60 : 1.38683118555266E+80
70 : 1.71122452441801E+98
80 : 8.94618213157899E+116
90 : 1.65079551625067E+136
100 : 9.33262154536104E+155

Go

<lang go>package main

import (

   "fmt"
   "math"

)

func main() {

   fmt.Println("    x               math.Gamma                 Lanczos7")
   for _, x := range []float64{-.5, .1, .5, 1, 1.5, 2, 3, 10, 140, 170} {
       fmt.Printf("%5.1f %24.16g %24.16g\n", x, math.Gamma(x), lanczos7(x))
   }

}

func lanczos7(z float64) float64 {

   t := z + 6.5
   x := .99999999999980993 +
       676.5203681218851/z -
       1259.1392167224028/(z+1) +
       771.32342877765313/(z+2) -
       176.61502916214059/(z+3) +
       12.507343278686905/(z+4) -
       .13857109526572012/(z+5) +
       9.9843695780195716e-6/(z+6) +
       1.5056327351493116e-7/(z+7)
   return math.Sqrt2 * math.SqrtPi * math.Pow(t, z-.5) * math.Exp(-t) * x

}</lang>

Output:
    x               math.Gamma                 Lanczos7
 -0.5       -3.544907701811032       -3.544907701811087
  0.1        9.513507698668732        9.513507698668752
  0.5        1.772453850905516        1.772453850905517
  1.0                        1                        1
  1.5       0.8862269254527579       0.8862269254527587
  2.0                        1                        1
  3.0                        2                        2
 10.0                   362880        362880.0000000015
140.0    9.61572319694107e+238   9.615723196940201e+238
170.0   4.269068009004746e+304                     +Inf

Groovy

Translation of: Ada

<lang groovy>a = [ 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].reverse()

def gamma = { 1.0 / a.inject(0) { sm, a_i -> sm * (it - 1) + a_i } }

(1..10).each{ printf("% 1.9e\n", gamma(it / 3.0)) } </lang>

Output:
  2.678938535e+00
  1.354117939e+00
  1.000000000e+00
  8.929795116e-01
  9.027452930e-01
  1.000000000e+00
  1.190639349e+00
  1.504575488e+00
  2.000000000e+00
  2.778158479e+00

Haskell

Based on HaskellWiki (compatible license):

The Gamma and Beta function as described in 'Numerical Recipes in C++', the approximation is taken from [Lanczos, C. 1964 SIAM Journal on Numerical Analysis, ser. B, vol. 1, pp. 86-96]

<lang haskell>cof :: [Double] cof =

 [ 76.18009172947146
 , -86.50532032941677
 , 24.01409824083091
 , -1.231739572450155
 , 0.001208650973866179
 , -0.000005395239384953
 ]

ser :: Double ser = 1.000000000190015

gammaln :: Double -> Double gammaln xx =

 let tmp_ = (xx + 5.5) - (xx + 0.5) * log (xx + 5.5)
     ser_ = ser + sum (zipWith (/) cof [xx + 1 ..])
 in -tmp_ + log (2.5066282746310005 * ser_ / xx)

main :: IO () main = mapM_ print $ gammaln <$> [0.1,0.2 .. 1.0]</lang>

Or equivalently, as a point-free applicative expression: <lang haskell>import Control.Applicative (liftA2)

cof :: [Double] cof =

 [ 76.18009172947146
 , -86.50532032941677
 , 24.01409824083091
 , -1.231739572450155
 , 0.001208650973866179
 , -0.000005395239384953
 ]

gammaln :: Double -> Double gammaln =

 ((+) . negate . liftA2 (-) (5.5 +) (liftA2 (*) (0.5 +) (log . (5.5 +)))) <*>
 (log .
  ((/) =<<
   (2.5066282746310007 *) .
   (1.000000000190015 +) . sum . zipWith (/) cof . enumFrom . (1 +)))

main :: IO () main = mapM_ print $ gammaln <$> [0.1,0.2 .. 1.0]</lang>

Output:
2.252712651734255
1.5240638224308496
1.09579799481814
0.7966778177018394
0.572364942924743
0.3982338580692666
0.2608672465316877
0.15205967839984869
6.637623973474716e-2
-4.440892098500626e-16

Icon and Unicon

This works in Unicon. Changing the !10 into (1 to 10) would enable it to work in Icon. <lang unicon>procedure main()

   every write(left(i := !10/10.0,5),gamma(.i))

end

procedure gamma(z) # Stirling's approximation

   return (2*&pi/z)^0.5 * (z/&e)^z

end</lang>

Output:
->gamma
0.1  5.69718714897717
0.2  3.325998424022393
0.3  2.36253003626962
0.4  1.841476335936235
0.5  1.520346901066281
0.6  1.307158857448356
0.7  1.15905329211392
0.8  1.053370968425609
0.9  0.9770615078776954
1.0  0.9221370088957891
->

J

This code shows the built-in method, which works for any value (positive, negative and complex numbers -- but note that negative integer arguments give infinite results). <lang j>gamma=: !@<:</lang> Note that <: subtracts one from a number. It's sort of like --lvalue in C, except it always accepts an "rvalue" as an argument (which means it does not modify that argument). And !value finds the factorial of value if value is a positive integer. This illustrates the close relationship between the factorial and gamma functions.

The following direct coding of the task comes from the Stirling's approximation essay on the J wiki: <lang j>sbase =: %:@(2p1&%) * %&1x1 ^ ] scorr =: 1 1r12 1r288 _139r51840 _571r2488320&p.@% stirlg=: sbase * scorr</lang> Checking against !@<: we can see that this approximation loses accuracy for small arguments <lang j> (,. stirlg ,. gamma) 10 1p1 1x1 1.5 1

    10   362880   362880

3.14159 2.28803 2.28804 2.71828 1.56746 1.56747

   1.5 0.886155 0.886227
     1 0.999499        1</lang>

(Column 1 is the argument, column 2 is the stirling approximation and column 3 uses the builtin support for gamma.)

Java

Implementation of Stirling's approximation and Lanczos approximation. <lang java>public class GammaFunction {

public double st_gamma(double x){ return Math.sqrt(2*Math.PI/x)*Math.pow((x/Math.E), x); }

public double la_gamma(double x){ double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7}; int g = 7; if(x < 0.5) return Math.PI / (Math.sin(Math.PI * x)*la_gamma(1-x));

x -= 1; double a = p[0]; double t = x+g+0.5; for(int i = 1; i < p.length; i++){ a += p[i]/(x+i); }

return Math.sqrt(2*Math.PI)*Math.pow(t, x+0.5)*Math.exp(-t)*a; }

public static void main(String[] args) { GammaFunction test = new GammaFunction(); System.out.println("Gamma \t\tStirling \t\tLanczos"); for(double i = 1; i <= 20; i += 1){ System.out.println("" + i/10.0 + "\t\t" + test.st_gamma(i/10.0) + "\t" + test.la_gamma(i/10.0)); } } }</lang>

Output:
Gamma 		Stirling 		Lanczos
0.1		5.697187148977169	9.513507698668734
0.2		3.3259984240223925	4.590843711998803
0.3		2.3625300362696198	2.9915689876875904
0.4		1.8414763359362354	2.218159543757687
0.5		1.5203469010662807	1.7724538509055159
0.6		1.307158857448356	1.489192248812818
0.7		1.15905329211392	1.2980553326475577
0.8		1.0533709684256085	1.1642297137253035
0.9		0.9770615078776954	1.0686287021193193
1.0		0.9221370088957891	0.9999999999999998
1.1		0.8834899531687038	0.9513507698668735
1.2		0.8577553353965909	0.9181687423997607
1.3		0.8426782594483921	0.8974706963062777
1.4		0.8367445486370817	0.8872638175030757
1.5		0.8389565525264963	0.8862269254527586
1.6		0.8486932421525738	0.8935153492876909
1.7		0.865621471793884	0.9086387328532916
1.8		0.8896396352879945	0.9313837709802425
1.9		0.9208427218942293	0.9617658319073877
2.0		0.9595021757444916	1.0000000000000002

JavaScript

Implementation of Lanczos approximation. <lang javascript>function gamma(x) {

   var p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
       771.32342877765313, -176.61502916214059, 12.507343278686905,
       -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
   ];
   var g = 7;
   if (x < 0.5) {
       return Math.PI / (Math.sin(Math.PI * x) * gamma(1 - x));
   }
   x -= 1;
   var a = p[0];
   var t = x + g + 0.5;
   for (var i = 1; i < p.length; i++) {
       a += p[i] / (x + i);
   }
   return Math.sqrt(2 * Math.PI) * Math.pow(t, x + 0.5) * Math.exp(-t) * a;

}</lang>

jq

Works with: jq version 1.4

Taylor Series

Translation of: Ada

<lang jq>def gamma:

 [
   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
 ] as $a
 | (. - 1) as $y
 | ($a|length) as $n
 | reduce range(2; 1+$n) as $an
     ($a[$n-1]; (. * $y) + $a[$n - $an])
 | 1 / . ;</lang>

Lanczos Approximation

<lang jq># for reals, but easily extended to complex values def gamma_by_lanczos:

 def pow(x): if x == 0 then 1 elif x == 1 then . else x * log | exp end;
 . as $x
 | ((1|atan) * 4) as $pi
 | if $x < 0.5 then $pi / ((($pi * $x) | sin) * ((1-$x)|gamma_by_lanczos ))
   else
     [   0.99999999999980993, 676.5203681218851,     -1259.1392167224028,
       771.32342877765313,   -176.61502916214059,       12.507343278686905,
        -0.13857109526572012,   9.9843695780195716e-6,   1.5056327351493116e-7] as $p
   | ($x - 1) as $x
   | ($x + 7.5) as $t
   |  reduce range(1; $p|length) as $i
         ($p[0]; . + ($p[$i]/($x + $i) ))
      * ((2*$pi) | sqrt) * ($t | pow($x+0.5)) * ((-$t)|exp) 
   end;</lang>

Stirling's Approximation

<lang jq>def gamma_by_stirling:

 def pow(x): if x == 0 then 1 elif x == 1 then . else x * log | exp end;
 ((1|atan) * 8) as $twopi
 | . as $x
 | (($twopi/$x) | sqrt) * ( ($x / (1|exp)) | pow($x));</lang>

Examples

Stirling's method produces poor results, so to save space, the examples contrast the Taylor series and Lanczos methods with built-in tgamma: <lang jq>def pad(n): tostring | . + (n - length) * " ";

" i: gamma lanczos tgamma", (range(1;11)

| . / 3.0
| "\(pad(18)): \(gamma|pad(18)) : \(gamma_by_lanczos|pad(18)) : \(tgamma)")</lang>
Output:

<lang sh>$ jq -M -r -n -f Gamma_function_Stirling.jq

                i:      gamma                lanczos              tgamma

0.3333333333333333: 2.6789385347077483 : 2.6789385347077483 : 2.678938534707748 0.6666666666666666: 1.3541179394264005 : 1.3541179394263998 : 1.3541179394264005 1  : 1  : 0.9999999999999998 : 1 1.3333333333333333: 0.8929795115692493 : 0.8929795115692494 : 0.8929795115692493 1.6666666666666667: 0.9027452929509336 : 0.9027452929509342 : 0.9027452929509336 2  : 1  : 1.0000000000000002 : 1 2.3333333333333335: 1.190639348758999  : 1.1906393487589995 : 1.190639348758999 2.6666666666666665: 1.5045754882515399 : 1.5045754882515576 : 1.5045754882515558 3  : 1.9999999999939684 : 2.0000000000000013 : 2 3.3333333333333335: 2.778158479338573  : 2.778158480437665  : 2.7781584804376647</lang>

Jsish

Translation of: Javascript

<lang javascript>#!/usr/bin/env jsish /* Gamma function, in Jsish, using the Lanczos approximation */ function gamma(x) {

   var p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
       771.32342877765313, -176.61502916214059, 12.507343278686905,
       -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
   ];

   var g = 7;
   if (x < 0.5) {
       return Math.PI / (Math.sin(Math.PI * x) * gamma(1 - x));
   }

   x -= 1;
   var a = p[0];
   var t = x + g + 0.5;
   for (var i = 1; i < p.length; i++) {
       a += p[i] / (x + i);
   }

   return Math.sqrt(2 * Math.PI) * Math.pow(t, x + 0.5) * Math.exp(-t) * a;

}

if (Interp.conf('unitTest')) {

   for (var i=-5.5; i <= 5.5; i += 0.5) {
       printf('%2.1f %+e\n', i, gamma(i));
   }

}

/*

!EXPECTSTART!

-5.5 +1.091265e-02 -5.0 -4.275508e+13 -4.5 -6.001960e-02 -4.0 +2.672193e+14 -3.5 +2.700882e-01 -3.0 -1.425169e+15 -2.5 -9.453087e-01 -2.0 +6.413263e+15 -1.5 +2.363272e+00 -1.0 -2.565305e+16 -0.5 -3.544908e+00 0.0 +inf 0.5 +1.772454e+00 1.0 +1.000000e+00 1.5 +8.862269e-01 2.0 +1.000000e+00 2.5 +1.329340e+00 3.0 +2.000000e+00 3.5 +3.323351e+00 4.0 +6.000000e+00 4.5 +1.163173e+01 5.0 +2.400000e+01 5.5 +5.234278e+01

!EXPECTEND!

  • /</lang>
Output:
prompt$ jsish --U gammaFunction.jsi
-5.5 +1.091265e-02
-5.0 -4.275508e+13
-4.5 -6.001960e-02
-4.0 +2.672193e+14
-3.5 +2.700882e-01
-3.0 -1.425169e+15
-2.5 -9.453087e-01
-2.0 +6.413263e+15
-1.5 +2.363272e+00
-1.0 -2.565305e+16
-0.5 -3.544908e+00
0.0 +inf
0.5 +1.772454e+00
1.0 +1.000000e+00
1.5 +8.862269e-01
2.0 +1.000000e+00
2.5 +1.329340e+00
3.0 +2.000000e+00
3.5 +3.323351e+00
4.0 +6.000000e+00
4.5 +1.163173e+01
5.0 +2.400000e+01
5.5 +5.234278e+01

prompt$ jsish -u gammaFunction.jsi
[PASS] gammaFunction.jsi

Julia

Works with: Julia version 0.6

Built-in function: <lang julia>@show gamma(1)</lang>

By adaptive Gauss-Kronrod integration: <lang julia>using QuadGK gammaquad(t::Float64) = first(quadgk(x -> x ^ (t - 1) * exp(-x), zero(t), Inf, reltol = 100eps(t))) @show gammaquad(1.0)</lang>

Output:
gamma(1) = 1.0
gammaquad(1.0) = 0.9999999999999999
Works with: Julia version 1.0

Library function: <lang julia>using SpecialFunctions gamma(1/2) - sqrt(pi)</lang>

Output:
2.220446049250313e-16

Kotlin

<lang scala>// version 1.0.6

fun gammaStirling(x: Double): Double = Math.sqrt(2.0 * Math.PI / x) * Math.pow(x / Math.E, x)

fun gammaLanczos(x: Double): Double {

   var xx = x
   val p = doubleArrayOf(
       0.99999999999980993, 
     676.5203681218851,
   -1259.1392167224028,			     	  
     771.32342877765313,
    -176.61502916214059,
      12.507343278686905,
      -0.13857109526572012,
       9.9843695780195716e-6,
       1.5056327351493116e-7
   )
   val g = 7
   if (xx < 0.5) return Math.PI / (Math.sin(Math.PI * xx) * gammaLanczos(1.0 - xx))
   xx--
   var a = p[0]
   val t = xx + g + 0.5
   for (i in 1 until p.size) a += p[i] / (xx + i)
   return Math.sqrt(2.0 * Math.PI) * Math.pow(t, xx + 0.5) * Math.exp(-t) * a

}

fun main(args: Array<String>) {

   println(" x\tStirling\t\tLanczos\n")
   for (i in 1 .. 20) {
       val d = i / 10.0
       print("%4.2f\t".format(d))
       print("%17.15f\t".format(gammaStirling(d)))
       println("%17.15f".format(gammaLanczos(d)))
   }

}</lang>

Output:
 x      Stirling                Lanczos

0.10    5.697187148977170       9.513507698668736
0.20    3.325998424022393       4.590843711998803
0.30    2.362530036269620       2.991568987687590
0.40    1.841476335936235       2.218159543757687
0.50    1.520346901066281       1.772453850905516
0.60    1.307158857448356       1.489192248812818
0.70    1.159053292113920       1.298055332647558
0.80    1.053370968425609       1.164229713725304
0.90    0.977061507877695       1.068628702119319
1.00    0.922137008895789       1.000000000000000
1.10    0.883489953168704       0.951350769866874
1.20    0.857755335396591       0.918168742399761
1.30    0.842678259448392       0.897470696306278
1.40    0.836744548637082       0.887263817503076
1.50    0.838956552526496       0.886226925452759
1.60    0.848693242152574       0.893515349287691
1.70    0.865621471793884       0.908638732853292
1.80    0.889639635287995       0.931383770980243
1.90    0.920842721894229       0.961765831907388
2.00    0.959502175744492       1.000000000000000

Limbo

Translation of: Go

A fairly straightforward port of the Go code. (It could almost have been done with sed). A few small differences are in the use of a tuple as a return value for the builtin gamma function, and we import a few functions from the math library so that we don't have to qualify them.

<lang Limbo>implement Lanczos7;

include "sys.m"; sys: Sys; include "draw.m"; include "math.m"; math: Math; lgamma, exp, pow, sqrt: import math;

Lanczos7: module { init: fn(nil: ref Draw->Context, nil: list of string); };

init(nil: ref Draw->Context, nil: list of string) { sys = load Sys Sys->PATH; math = load Math Math->PATH; # We ignore some floating point exceptions: math->FPcontrol(0, Math->OVFL|Math->UNFL); ns : list of real = -0.5 :: 0.1 :: 0.5 :: 1.0 :: 1.5 :: 2.0 :: 3.0 :: 10.0 :: 140.0 :: 170.0 :: nil;

sys->print("%5s %24s %24s\n", "x", "math->lgamma", "lanczos7"); while(ns != nil) { x := hd ns; ns = tl ns; # math->lgamma returns a tuple. (i, r) := lgamma(x); g := real i * exp(r); sys->print("%5.1f %24.16g %24.16g\n", x, g, lanczos7(x)); } }

lanczos7(z: real): real { t := z + 6.5; x := 0.99999999999980993 + 676.5203681218851/z - 1259.1392167224028/(z+1.0) + 771.32342877765313/(z+2.0) - 176.61502916214059/(z+3.0) + 12.507343278686905/(z+4.0) - 0.13857109526572012/(z+5.0) + 9.9843695780195716e-6/(z+6.0) + 1.5056327351493116e-7/(z+7.0); return sqrt(2.0) * sqrt(Math->Pi) * pow(t, z - 0.5) * exp(-t) * x; } </lang>

Output:
    x             math->lgamma                 lanczos7
 -0.5       -3.544907701811032       -3.544907701811089
  0.1        9.513507698668729         9.51350769866875
  0.5        1.772453850905516        1.772453850905516
  1.0                        1       0.9999999999999999
  1.5       0.8862269254527581       0.8862269254527587
  2.0                        1                        1
  3.0                        2        2.000000000000001
 10.0        362880.0000000005        362880.0000000015
140.0   9.615723196940553e+238   9.615723196940235e+238
170.0   4.269068009004526e+304                 Infinity

Lua

Uses the wp:Reciprocal gamma function to calculate small values. <lang lua>gamma, coeff, quad, qui, set = 0.577215664901, -0.65587807152056, -0.042002635033944, 0.16653861138228, -0.042197734555571 function recigamma(z)

 return z + gamma * z^2 + coeff * z^3 + quad * z^4 + qui * z^5 + set * z^6

end

function gammafunc(z)

 if z == 1 then return 1
 elseif math.abs(z) <= 0.5 then return 1 / recigamma(z)
 else return (z - 1) * gammafunc(z-1)
 end

end</lang>

M2000 Interpreter

<lang M2000 Interpreter> Module PrepareLambdaFunctions {

     Const e = 2.7182818284590452@
     Exp= Lambda e (x) -> e^x
     gammaStirling=lambda e (x As decimal)->Sqrt(2.0 * pi / x) * ((x / e) ^ x)
     Rad2Deg =Lambda pidivby180=pi/180 (RadAngle)->RadAngle / pidivby180
     Dim p(9)
     p(0)=0.99999999999980993@, 676.5203681218851@,   -1259.1392167224028@,  771.32342877765313@
     p(4)=-176.61502916214059@,  12.507343278686905@,  -0.13857109526572012@,  0.0000099843695780195716@
     p(8)=0.00000015056327351493116@
     gammaLanczos =Lambda p(), Rad2Deg, Exp (x As decimal) -> {
           Def Decimal a, t
           If x < 0.5 Then =pi / (Sin(Rad2Deg(pi * x)) *Lambda(1-x)) : Exit
           x -= 1@
           a=p(0)
           t = x + 7.5@
           For i= 1@ To 8@ {
                 a += p(i) / (x + i)
           }
            = Sqrt(2.0 * pi) * (t ^ (x + 0.5)) * Exp(-t) * a  
     }
     Push gammaStirling, gammaLanczos

} Call PrepareLambdaFunctions Read gammaLanczos, gammaStirling Font "Courier New" Form 120, 40 document doc$=" χ Stirling Lanczos"+{ } Print $(2,20),"x", "Stirling",@(55),"Lanczos", $(0) Print For d = 0.1 To 2 step 0.1

     Print   $("0.00"), d,
     Print  $("0.000000000000000"), gammaStirling(d),
     Print  $("0.0000000000000000000000000000"), gammaLanczos(d)
     doc$=format$("{0:-10}  {1:-30}   {2:-34}",str$(d,"0.00"), str$(gammaStirling(d),"0.000000000000000"), str$(gammaLanczos(d),"0.0000000000000000000000000000"))+{
     }

Next d Print $("") clipboard doc$ </lang>

    χ                        Stirling                     Lanczos
     0.10               5.697187148977170       9.5135076986687024462927178610
     0.20               3.325998424022390       4.5908437119987955107204909409
     0.30               2.362530036269620       2.9915689876875914865114179656
     0.40               1.841476335936240       2.2181595437576816416854441034
     0.50               1.520346901066280       1.7724538509055147387430498835
     0.60               1.307158857448360       1.4891922488128208508983507496
     0.70               1.159053292113920       1.2980553326475564892857625396
     0.80               1.053370968425610       1.1642297137253055422419914101
     0.90               0.977061507877695       1.0686287021193206646594133376
     1.00               0.922137008895789       1.0000000000000007024882980221
     1.10               0.883489953168704       0.9513507698668745807357371716
     1.20               0.857755335396591       0.9181687423997605348002977483
     1.30               0.842678259448392       0.8974706963062785326402091223
     1.40               0.836744548637082       0.8872638175030748314253582066
     1.50               0.838956552526496       0.8862269254527587632845492097
     1.60               0.848693242152574       0.8935153492876912865293624528
     1.70               0.865621471793884       0.9086387328532921150064803085
     1.80               0.889639635287995       0.9313837709802428420608295699
     1.90               0.920842721894229       0.9617658319073891431109375442
     2.00               0.959502175744492       1.0000000000000015609456469406

Maple

Built-in method that accepts any value. <lang Maple>GAMMA(17/2); GAMMA(7*I); M := Matrix(2, 3, 'fill' = -3.6); MTM:-gamma(M);</lang>

Output:
2027025*sqrt(Pi)*(1/256)
GAMMA(7*I)
Matrix(2, 3, [[.2468571430, .2468571430, .2468571430], [.2468571430, .2468571430, .2468571430]])

Mathematica

This code shows the built-in method, which works for any value (positive, negative and complex numbers). <lang mathematica>Gamma[x]</lang> Output integers and half-integers (a space is multiplication in Mathematica):

1/2	Sqrt[pi]
1	1
3/2	Sqrt[pi]/2
2	1
5/2	(3 Sqrt[pi])/4
3	2
7/2	(15 Sqrt[pi])/8
4	6
9/2	(105 Sqrt[pi])/16
5	24
11/2	(945 Sqrt[pi])/32
6	120
13/2	(10395 Sqrt[pi])/64
7	720
15/2	(135135 Sqrt[pi])/128
8	5040
17/2	(2027025 Sqrt[pi])/256
9	40320
19/2	(34459425 Sqrt[pi])/512
10	362880

Output approximate numbers:

0.1	9.51351
0.2	4.59084
0.3	2.99157
0.4	2.21816
0.5	1.77245
0.6	1.48919
0.7	1.29806
0.8	1.16423
0.9	1.06863
1.	1.

Output complex numbers:

I	-0.15495-0.498016 I
2 I	0.00990244-0.075952 I
3 I	0.0112987-0.00643092 I
4 I	0.00173011+0.00157627 I
5 I	-0.000271704+0.000339933 I

Maxima

<lang maxima>fpprec: 30$

gamma_coeff(n) := block([a: makelist(1, n)],

  a[2]: bfloat(%gamma),
  for k from 3 thru n do
     a[k]: bfloat((sum((-1)^j * zeta(j) * a[k - j], j, 2, k - 1) - a[2] * a[k - 1]) / (1 - k * a[1])),
  a)$
     

poleval(a, x) := block([y: 0],

  for k from length(a) thru 1 step -1 do
     y: y * x + a[k],
  y)$

gc: gamma_coeff(20)$

gamma_approx(x) := block([y: 1],

  while x > 2 do (x: x - 1, y: y * x),
  y / (poleval(gc, x - 1)))$

gamma_approx(12.3b0) - gamma(12.3b0); /* -9.25224705314470500985141176997b-15 */</lang>

Modula-3

Translation of: Ada

<lang modula3>MODULE Gamma EXPORTS Main;

FROM IO IMPORT Put; FROM Fmt IMPORT Extended, Style;

PROCEDURE Taylor(x: EXTENDED): EXTENDED =

 CONST a = ARRAY [0..29] OF EXTENDED {
   1.00000000000000000000X0, 0.57721566490153286061X0,
   -0.65587807152025388108X0, -0.04200263503409523553X0,
   0.16653861138229148950X0, -0.04219773455554433675X0,
   -0.00962197152787697356X0, 0.00721894324666309954X0,
   -0.00116516759185906511X0, -0.00021524167411495097X0,
   0.00012805028238811619X0, -0.00002013485478078824X0,
   -0.00000125049348214267X0, 0.00000113302723198170X0,
   -0.00000020563384169776X0, 0.00000000611609510448X0,
   0.00000000500200764447X0, -0.00000000118127457049X0,
   0.00000000010434267117X0, 0.00000000000778226344X0,
   -0.00000000000369680562X0, 0.00000000000051003703X0,
   -0.00000000000002058326X0, -0.00000000000000534812X0,
   0.00000000000000122678X0, -0.00000000000000011813X0,
   0.00000000000000000119X0, 0.00000000000000000141X0,
   -0.00000000000000000023X0, 0.00000000000000000002X0 };
 VAR y := x - 1.0X0;
     sum := a[LAST(a)];
 BEGIN
   FOR i := LAST(a) - 1 TO FIRST(a) BY -1 DO
     sum := sum * y + a[i];
   END;
   RETURN 1.0X0 / sum;
 END Taylor;
 

BEGIN

 FOR i := 1 TO 10 DO
   Put(Extended(Taylor(FLOAT(i, EXTENDED) / 3.0X0), style := Style.Sci) & "\n");
 END;

END Gamma.</lang>

Output:
 2.6789385347077490e+000
 1.3541179394264005e+000
 1.0000000000000000e+000
 8.9297951156924930e-001
 9.0274529295093360e-001
 1.0000000000000000e+000
 1.1906393487589992e+000
 1.5045754882515399e+000
 1.9999999999939684e+000
 2.7781584793385790e+000

МК-61/52

П9	9	П0	ИП9	ИП9	1	+	*	Вx	L0
05	1	+	П9	^	ln	1	-	*	ИП9
1	2	*	1/x	+	e^x	<->	/	2	пи
*	ИП9	/	КвКор	*	^	ВП	3	+	Вx
-	С/П

Nim

Translation of: Ada

<lang nim>const a = [

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 ]

proc gamma(x: float): float =

 let y = x.float - 1.0
 result = a[a.high]
 for n in countdown(high(a) - 1, low(a)):
   result = result * y + a[n]
 result = 1.0 / result

for i in 1..10:

 echo gamma(i.float / 3.0)</lang>
Output:
2.678938534707748
1.3541179394264
1.0
0.8929795115692493
0.9027452929509336
1.0
1.190639348758999
1.50457548825154
1.999999999993968
2.778158479338573

OCaml

<lang ocaml>let e = exp 1. let pi = 4. *. atan 1. let sqrttwopi = sqrt (2. *. pi)

module Lanczos = struct

 (* Lanczos method *)
 (* Coefficients used by the GNU Scientific Library *)
 let g = 7.
 let c = [|0.99999999999980993; 676.5203681218851; -1259.1392167224028;

771.32342877765313; -176.61502916214059; 12.507343278686905; -0.13857109526572012; 9.9843695780195716e-6; 1.5056327351493116e-7|]

 let rec ag z d =
   if d = 0 then c.(0) +. ag z 1
   else if d < 8 then c.(d) /. (z +. float d) +. ag z (succ d)
   else c.(d) /. (z +. float d)
 let gamma z =
   let z = z -. 1. in
   let p = z +. g +. 0.5 in
   sqrttwopi *. p ** (z +. 0.5) *. exp (-. p) *. ag z 0

end

module Stirling = struct

 (* Stirling method *)
 let gamma z =
   sqrttwopi /. sqrt z *. (z /. e) ** z

end

module Stirling2 = struct

 (* Extended Stirling method seen in Abramowitz and Stegun *)
 let d = [|1./.12.; 1./.288.; -139./.51840.; -571./.2488320.|]
 let rec corr z x n =
   if n < Array.length d - 1 then d.(n) /. x +. corr z (x *. z) (succ n)
   else d.(n) /. x
 let gamma z = Stirling.gamma z *. (1. +. corr z z 0)

end

let mirror gma z =

 if z > 0.5 then gma z
 else pi /. sin (pi *. z) /. gma (1. -. z)

let _ =

 Printf.printf "z\t\tLanczos\t\tStirling\tStirling2\n";
 for i = 1 to 20 do
   let z = float i /. 10. in
   Printf.printf "%-10.8g\t%10.8e\t%10.8e\t%10.8e\n"
   		  z 

(mirror Lanczos.gamma z) (mirror Stirling.gamma z) (mirror Stirling2.gamma z)

 done;
 for i = 1 to 7 do
   let z = 10. *. float i in
   Printf.printf "%-10.8g\t%10.8e\t%10.8e\t%10.8e\n"
   		  z

(Lanczos.gamma z) (Stirling.gamma z) (Stirling2.gamma z)

 done</lang>
Output:
z               Lanczos         Stirling        Stirling2
0.1             9.51350770e+00  1.04050843e+01  9.52104183e+00
0.2             4.59084371e+00  5.07399275e+00  4.59686230e+00
0.3             2.99156899e+00  3.35033954e+00  2.99844028e+00
0.4             2.21815954e+00  2.52705781e+00  2.22775889e+00
0.5             1.77245385e+00  2.06636568e+00  1.78839014e+00
0.6             1.48919225e+00  1.30715886e+00  1.48277536e+00
0.7             1.29805533e+00  1.15905329e+00  1.29508068e+00
0.8             1.16422971e+00  1.05337097e+00  1.16270541e+00
0.9             1.06862870e+00  9.77061508e-01  1.06778308e+00
1               1.00000000e+00  9.22137009e-01  9.99499469e-01
1.1             9.51350770e-01  8.83489953e-01  9.51037997e-01
1.2             9.18168742e-01  8.57755335e-01  9.17964058e-01
1.3             8.97470696e-01  8.42678259e-01  8.97331287e-01
1.4             8.87263818e-01  8.36744549e-01  8.87165485e-01
1.5             8.86226925e-01  8.38956553e-01  8.86155384e-01
1.6             8.93515349e-01  8.48693242e-01  8.93461840e-01
1.7             9.08638733e-01  8.65621472e-01  9.08597702e-01
1.8             9.31383771e-01  8.89639635e-01  9.31351590e-01
1.9             9.61765832e-01  9.20842722e-01  9.61740068e-01
2               1.00000000e+00  9.59502176e-01  9.99978981e-01
10              3.62880000e+05  3.59869562e+05  3.62879997e+05
20              1.21645100e+17  1.21139342e+17  1.21645100e+17
30              8.84176199e+30  8.81723653e+30  8.84176199e+30
40              2.03978821e+46  2.03554316e+46  2.03978821e+46
50              6.08281864e+62  6.07268919e+62  6.08281864e+62
60              1.38683119e+80  1.38490639e+80  1.38683119e+80
70              1.71122452e+98  1.70918858e+98  1.71122452e+98

Octave

<lang octave>function g = lacz_gamma(a, cg=7)

 p = [ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, \
       771.32342877765313, -176.61502916214059, 12.507343278686905, \
       -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 ];
 x=a;
 if ( x < 0.5 )
   g = pi / ( sin(pi*x) * lacz_gamma(1.0-x) );
 else
   x = x - 1.0;
   t = p(1);
   for i=1:(cg+1)
     t = t + p(i+1)/(x+double(i));
   endfor
   w = x + double(cg) + 0.5;
   g = sqrt(2.0*pi) * w**(x+0.5) * exp(-w) * t;
 endif

endfunction


for i = 1:10

 printf("%f %f\n", gamma(i/3.0), lacz_gamma(i/3.0));

endfor</lang>

Output:
2.678939 2.678939
1.354118 1.354118
1.000000 1.000000
0.892980 0.892980
0.902745 0.902745
1.000000 1.000000
1.190639 1.190639
1.504575 1.504575
2.000000 2.000000
2.778158 2.778158

Which suggests that the built-in gamma uses the same approximation.

Oforth

<lang oforth>import: math

[

  676.5203681218851,  -1259.1392167224028, 771.32342877765313, 
 -176.61502916214059, 12.507343278686905, -0.13857109526572012, 
  9.9843695780195716e-6, 1.5056327351493116e-7

] const: Gamma.Lanczos

gamma(z)

| i t |

  z 0.5 < ifTrue: [ Pi dup z * sin 1.0 z - gamma * / return ]
  z 1.0 - ->z
  0.99999999999980993 Gamma.Lanczos size loop: i [ i Gamma.Lanczos at z i + / + ]
  z Gamma.Lanczos size + 0.5 - ->t
  2 Pi * sqrt * 
  t z 0.5 + powf *
  t neg exp * ;</lang>
Output:
>20 seq apply(#[ 10.0 / dup . gamma .cr ])
0.1 9.51350769866874
0.2 4.5908437119988
0.3 2.99156898768759
0.4 2.21815954375769
0.5 1.77245385090552
0.6 1.48919224881282
0.7 1.29805533264756
0.8 1.1642297137253
0.9 1.06862870211932
1 1
1.1 0.951350769866874
1.2 0.918168742399761
1.3 0.897470696306277
1.4 0.887263817503076
1.5 0.886226925452759
1.6 0.893515349287691
1.7 0.908638732853292
1.8 0.931383770980243
1.9 0.961765831907388
2 1

PARI/GP

Built-in

<lang parigp>gamma(x)</lang>

Double-exponential integration

[[+oo],k] means that the function approaches as <lang parigp>Gamma(x)=intnum(t=0,[+oo,1],t^(x-1)/exp(t))</lang>

Romberg integration

<lang parigp>Gamma(x)=intnumromb(t=0,9,t^(x-1)/exp(t),0)+intnumromb(t=9,max(x,99)^9,t^(x-1)/exp(t),2)</lang>

Stirling approximation

<lang parigp>Stirling(x)=x--;sqrt(2*Pi*x)*(x/exp(1))^x</lang>

Perl

<lang perl>use strict; use warnings; use constant pi => 4*atan2(1, 1); use constant e => exp(1);

  1. Normally would be: use Math::MPFR
  2. but this will use it if it's installed and ignore otherwise

my $have_MPFR = eval { require Math::MPFR; Math::MPFR->import(); 1; };

sub Gamma {

   my $z = shift;
   my $method = shift // 'lanczos';
   if ($method eq 'lanczos') {
       use constant g => 9;
       $z <  .5 ?  pi / sin(pi * $z) / Gamma(1 - $z, $method) :
       sqrt(2* pi) *
       ($z + g - .5)**($z - .5) *
       exp(-($z + g - .5)) *
       do {
           my @coeff = qw{
           1.000000000000000174663
       5716.400188274341379136
     -14815.30426768413909044
      14291.49277657478554025
      -6348.160217641458813289
       1301.608286058321874105
       -108.1767053514369634679
          2.605696505611755827729
         -0.7423452510201416151527e-2
          0.5384136432509564062961e-7
         -0.4023533141268236372067e-8
           };
           my ($sum, $i) = (shift(@coeff), 0);
           $sum += $_ / ($z + $i++) for @coeff;
           $sum;
       }
   } elsif ($method eq 'taylor') {
       $z <  .5 ? Gamma($z+1, $method)/$z     :
       $z > 1.5 ? ($z-1)*Gamma($z-1, $method) :

do { my $s = 0; ($s *= $z-1) += $_ for qw{ 0.00000000000000000002 -0.00000000000000000023 0.00000000000000000141 0.00000000000000000119 -0.00000000000000011813 0.00000000000000122678 -0.00000000000000534812 -0.00000000000002058326 0.00000000000051003703 -0.00000000000369680562 0.00000000000778226344 0.00000000010434267117 -0.00000000118127457049 0.00000000500200764447 0.00000000611609510448 -0.00000020563384169776 0.00000113302723198170 -0.00000125049348214267 -0.00002013485478078824 0.00012805028238811619 -0.00021524167411495097 -0.00116516759185906511 0.00721894324666309954 -0.00962197152787697356 -0.04219773455554433675 0.16653861138229148950 -0.04200263503409523553 -0.65587807152025388108 0.57721566490153286061 1.00000000000000000000 }; 1/$s; }

   } elsif ($method eq 'stirling') {
       no warnings qw(recursion);
       $z < 100 ? Gamma($z + 1, $method)/$z :
       sqrt(2*pi*$z)*($z/e + 1/(12*e*$z))**$z / $z;
   } elsif ($method eq 'MPFR') {
       my $result = Math::MPFR->new();
       Math::MPFR::Rmpfr_gamma($result, Math::MPFR->new($z), 0);
       $result;
   } else { die "unknown method '$method'" }

}

for my $method (qw(MPFR lanczos taylor stirling)) {

   next if $method eq 'MPFR' && !$have_MPFR;
   printf "%10s: ", $method;
   print join(' ', map { sprintf "%.12f", Gamma($_/3, $method) } 1 .. 10);
   print "\n";

}</lang>

Output:
      MPFR: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
   lanczos: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
    taylor: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
  stirling: 2.678938532866 1.354117938504 0.999999999306 0.892979510955 0.902745292336 0.999999999306 1.190639347940 1.504575487227 1.999999998611 2.778158478527

Perl 6

<lang perl6>sub Γ(\z) {

   constant g = 9;
   z < .5 ?? pi/ sin(pi * z) / Γ(1 - z) !!
   sqrt(2*pi) *
   (z + g - 1/2)**(z - 1/2) *
   exp(-(z + g - 1/2)) *
   [+] <
       1.000000000000000174663
    5716.400188274341379136
  -14815.30426768413909044
   14291.49277657478554025
   -6348.160217641458813289
    1301.608286058321874105
    -108.1767053514369634679
       2.605696505611755827729
      -0.7423452510201416151527e-2
       0.5384136432509564062961e-7
      -0.4023533141268236372067e-8
   > Z* 1, |map 1/(z + *), 0..*

}

say Γ($_) for 1/3, 2/3 ... 10/3;</lang>

Output:
2.67893853470775
1.3541179394264
1
0.892979511569248
0.902745292950934
1
1.190639348759
1.50457548825155
2
2.77815848043766

Phix

Translation of: C

<lang Phix>sequence c = repeat(0,12)

function gamma(atom z)

   atom accm = c[1]
   if accm=0 then
       accm = sqrt(2*PI)
       c[1] = accm
       atom k1_factrl = 1  -- (k - 1)!*(-1)^k with 0!==1
       for k=2 to 12 do
           c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
           k1_factrl *= -(k-1)
       end for
   end if
   for k=2 to 12 do
       accm += c[k]/(z+k-1)
   end for
   accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1)
   return accm/z

end function

procedure sq(atom x, atom mul) atom p = x*mul

   printf(1,"%18.16g,%18.15g\n",{x,p*p})

end procedure

procedure si(atom x)

   printf(1,"%18.15f\n",{x})

end procedure

sq(gamma(-3/2),3/4) sq(gamma(-1/2),-1/2) sq(gamma(1/2),1) si(gamma(1)) sq(gamma(3/2),2) si(gamma(2)) sq(gamma(5/2),4/3) si(gamma(3)) sq(gamma(7/2),8/15) si(gamma(4))</lang>

Output:
 2.363271801207354,  3.14159265358979
-3.544907701811032,  3.14159265358979
 1.772453850905515,  3.14159265358979
 1.000000000000001
0.8862269254527643,  3.14159265358984
 1.000000000000010
 1.329340388179146,  3.14159265358984
 2.000000000000024
 3.323350970447942,  3.14159265358998
 6.000000000000175

PicoLisp

Translation of: Ada

<lang PicoLisp>(scl 28)

(de *A

  ~(flip
     (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 ) ) )

(de gamma (X)

  (let (Y (- X 1.0)  Sum (car *A))
     (for A (cdr *A)
        (setq Sum (+ A (*/ Sum Y 1.0))) )
     (*/ 1.0 1.0 Sum) ) )</lang>
Output:
: (for I (range 1 10)
   (prinl (round (gamma (*/ I 1.0 3)) 14)) )
2.67893853470775
1.35411793942640
1.00000000000000
0.89297951156925
0.90274529295093
1.00000000000000
1.19063934875900
1.50457548825154
1.99999999999397
2.77815847933858

PL/I

<lang PL/I>/* From Rosetta Fortran */ test: procedure options (main);

 declare i fixed binary;
 on underflow ;
 put skip list ('Lanczos', 'Builtin' );
 do i = 1 to 10;
    put skip list (lanczos_gamma(i/3.0q0), gamma(i/3.0q0) );
 end;


lanczos_gamma: procedure (a) returns (float (18)) recursive;

   declare a float (18);
   declare pi float (18) value (3.14159265358979324E0);
   declare cg fixed binary initial ( 7 );
   /* these precomputed values are taken by the sample code in Wikipedia, */
   /* and the sample itself takes them from the GNU Scientific Library */
   declare p(0:8) float (18) static initial
        ( 0.99999999999980993e0, 676.5203681218851e0, -1259.1392167224028e0,
        771.32342877765313e0, -176.61502916214059e0, 12.507343278686905e0,
        -0.13857109526572012e0, 9.9843695780195716e-6, 1.5056327351493116e-7 );
   declare ( t, w, x ) float (18);
   declare i fixed binary;
   x = a;
   if x < 0.5 then
      return ( pi / ( sin(pi*x) * lanczos_gamma(1.0-x) ) );
   else
      do;
         x = x - 1.0;
         t = p(0);
         do i = 1 to cg+2;
            t = t + p(i)/(x+i);
         end;
         w = x + float(cg) + 0.5;
         return ( sqrt(2*pi) * w**(x+0.5) * exp(-w) * t );
      end;
 end lanczos_gamma;

end test;</lang>

Output:
Lanczos                 Builtin 
 2.67893853470774706E+0000           2.678938534707747630E+0000 
 1.35411793942640071E+0000           1.354117939426400420E+0000 
 1.00000000000000021E+0000           1.000000000000000000E+0000 
 8.92979511569249470E-0001           8.929795115692492110E-0001 
 9.02745292950933961E-0001           9.027452929509336110E-0001 
 1.00000000000000048E+0000           1.000000000000000000E+0000 
 1.19063934875899964E+0000           1.190639348758998950E+0000 
 1.50457548825155704E+0000           1.504575488251556020E+0000 
 2.00000000000000154E+0000           2.000000000000000000E+0000 
 2.77815848043766660E+0000           2.778158480437664210E+0000 

PowerShell

I would download the Math.NET Numerics dll(s). Documentation and download at: http://cyber-defense.sans.org/blog/2015/06/27/powershell-for-math-net-numerics/comment-page-1/ <lang PowerShell> Add-Type -Path "C:\Program Files (x86)\Math\MathNet.Numerics.3.12.0\lib\net40\MathNet.Numerics.dll"

1..20 | ForEach-Object {[MathNet.Numerics.SpecialFunctions]::Gamma($_ / 10)} </lang>

Output:
9.51350769866874
4.5908437119988
2.99156898768759
2.21815954375769
1.77245385090552
1.48919224881282
1.29805533264756
1.1642297137253
1.06862870211932
1
0.951350769866874
0.918168742399759
0.897470696306277
0.887263817503075
0.88622692545276
0.89351534928769
0.908638732853289
0.931383770980245
0.961765831907388
1

PureBasic

Below is PureBasic code for:

  • Complete Gamma function
  • Natural Logarithm of the Complete Gamma function
  • Factorial function

<lang PureBasic>Procedure.d Gamma(x.d) ; AKJ 01-May-10

Complete Gamma function for x>0 and x<2 (approx)
Extended outside this range via
Gamma(x+1) = x*Gamma(x)
Based on http://rosettacode.org/wiki/Gamma_function [Ada]

Protected Dim A.d(28) A(0) = 1.0 A(1) = 0.5772156649015328606 A(2) =-0.6558780715202538811 A(3) =-0.0420026350340952355 A(4) = 0.1665386113822914895 A(5) =-0.0421977345555443368 ; was ...33675 A(6) =-0.0096219715278769736 A(7) = 0.0072189432466630995 A(8) =-0.0011651675918590651 A(9) =-0.0002152416741149510 A(10) = 0.0001280502823881162 A(11) =-0.0000201348547807882 A(12) =-0.0000012504934821427 A(13) = 0.0000011330272319817 A(14) =-0.0000002056338416978 A(15) = 0.0000000061160951045 A(16) = 0.0000000050020076445 A(17) =-0.0000000011812745705 A(18) = 0.0000000001043426712 A(19) = 0.0000000000077822634 A(20) =-0.0000000000036968056 A(21) = 0.0000000000005100370 A(22) =-0.0000000000000205833 A(23) =-0.0000000000000053481 A(24) = 0.0000000000000012268 A(25) =-0.0000000000000001181 A(26) = 0.0000000000000000012 A(27) = 0.0000000000000000014 A(28) =-0.0000000000000000002

A(29) = 0.00000000000000000002

Protected y.d, Prod.d, Sum.d, N If x<=0.0: ProcedureReturn 0.0: EndIf ; Error y = x-1.0: Prod = 1.0 While y>=1.0

 Prod*y: y-1.0 ; Recurse using Gamma(x+1) = x*Gamma(x)

Wend Sum= A(28) For N = 27 To 0 Step -1

 Sum*y+A(N)

Next N If Sum=0.0: ProcedureReturn Infinity(): EndIf ProcedureReturn Prod / Sum EndProcedure

Procedure.d GammLn(x.d) ; AKJ 01-May-10

Returns Ln(Gamma()) or 0 on error
Based on Numerical Recipes gamma.h

Protected j, tmp.d, y.d, ser.d Protected Dim cof.d(13) cof(0) = 57.1562356658629235 cof(1) = -59.5979603554754912 cof(2) = 14.1360979747417471 cof(3) = -0.491913816097620199 cof(4) = 0.339946499848118887e-4 cof(5) = 0.465236289270485756e-4 cof(6) = -0.983744753048795646e-4 cof(7) = 0.158088703224912494e-3 cof(8) = -0.210264441724104883e-3 cof(9) = 0.217439618115212643e-3 cof(10) = -0.164318106536763890e-3 cof(11) = 0.844182239838527433e-4 cof(12) = -0.261908384015814087e-4 cof(13) = 0.368991826595316234e-5 If x<=0: ProcedureReturn 0: EndIf ; Bad argument y = x tmp = x+5.2421875 tmp = (x+0.5)*Log(tmp)-tmp ser = 0.999999999999997092 For j=0 To 13

 y + 1: ser + cof(j)/y

Next j ProcedureReturn tmp+Log(2.5066282746310005*ser/x) EndProcedure

Procedure Factorial(x) ; AKJ 01-May-10

 ProcedureReturn Gamma(x+1)

EndProcedure</lang>

Examples

<lang PureBasic>Debug "Gamma()" For i = 12 To 15

 Debug StrD(i/3.0, 3)+"   "+StrD(Gamma(i/3.0))

Next i Debug "" Debug "Ln(Gamma(5.0)) = "+StrD(GammLn(5.0), 16) ; Ln(24) Debug "" Debug "Factorial 6 = "+StrD(Factorial(6), 0) ; 72</lang>

Output:
[Debug] Gamma():
[Debug] 4.000   6.0000000000
[Debug] 4.333   9.2605282681
[Debug] 4.667   14.7114047740
[Debug] 5.000   24.0000000000
[Debug] 
[Debug] Ln(Gamma(5.0)) = 3.1780538303479458
[Debug] 
[Debug] Factorial 6 = 720

Python

Procedural

Translation of: Ada

<lang python>_a = ( 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
      )

def gamma (x):

  y  = float(x) - 1.0;
  sm = _a[-1];
  for an in _a[-2::-1]:
     sm = sm * y + an;
  return 1.0 / sm;

if __name__ == '__main__':

   for i in range(1,11):
       print "  %20.14e" % gamma(i/3.0)

</lang>

Output:
  2.67893853470775e+00
  1.35411793942640e+00
  1.00000000000000e+00
  8.92979511569249e-01
  9.02745292950934e-01
  1.00000000000000e+00
  1.19063934875900e+00
  1.50457548825154e+00
  1.99999999999397e+00
  2.77815847933857e+00

Functional

In terms of fold/reduce:

Works with: Python version 3.7

<lang python>Gamma function

from functools import reduce


  1. gamma_ :: [Float] -> Float -> Float

def gamma_(tbl):

   Gamma function.
   def go(x):
       y = float(x) - 1.0
       return 1.0 / reduce(
           lambda a, x: a * y + x,
           tbl[-2::-1],
           tbl[-1]
       )
   return lambda x: go(x)


  1. TBL :: [Float]

TBL = [

   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

]


  1. TEST ----------------------------------------------------
  2. main :: IO()

def main():

   Gamma function over a range of values.
   gamma = gamma_(TBL)
   print(
       fTable(' i -> gamma(i/3):\n')(repr)(lambda x: "%0.7e" % x)(
           lambda x: gamma(x / 3.0)
       )(enumFromTo(1)(10))
   )


  1. GENERIC -------------------------------------------------
  1. enumFromTo :: (Int, Int) -> [Int]

def enumFromTo(m):

   Integer enumeration from m to n.
   return lambda n: list(range(m, 1 + n))


  1. FORMATTING -------------------------------------------------
  1. fTable :: String -> (a -> String) ->
  2. (b -> String) -> (a -> b) -> [a] -> String

def fTable(s):

   Heading -> x display function -> fx display function ->
                    f -> xs -> tabular string.
   
   def go(xShow, fxShow, f, xs):
       ys = [xShow(x) for x in xs]
       w = max(map(len, ys))
       return s + '\n' + '\n'.join(map(
           lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
           xs, ys
       ))
   return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
       xShow, fxShow, f, xs
   )


  1. MAIN ---

if __name__ == '__main__':

   main()</lang>
Output:
 i -> gamma(i/3):

 1 -> 2.6789385e+00
 2 -> 1.3541179e+00
 3 -> 1.0000000e+00
 4 -> 8.9297951e-01
 5 -> 9.0274529e-01
 6 -> 1.0000000e+00
 7 -> 1.1906393e+00
 8 -> 1.5045755e+00
 9 -> 2.0000000e+00
10 -> 2.7781585e+00

R

Lanczos' approximation is loosely converted from the Octave code.

Translation of: Octave

<lang r>stirling <- function(z) sqrt(2*pi/z) * (exp(-1)*z)^z

nemes <- function(z) sqrt(2*pi/z) * (exp(-1)*(z + (12*z - (10*z)^-1)^-1))^z

lanczos <- function(z) {

  if(length(z) > 1)
  {
     sapply(z, lanczos)
  } else
  {
    g <- 7
     p <- c(0.99999999999980993, 676.5203681218851, -1259.1392167224028,
       771.32342877765313, -176.61502916214059, 12.507343278686905,
       -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7)
     z <- as.complex(z) 
     if(Re(z) < 0.5) 
     {
        pi / (sin(pi*z) * lanczos(1-z)) 
     } else
     {
        z <- z - 1
        x <- p[1] + sum(p[-1]/seq.int(z+1, z+g+1))
        tt <- z + g + 0.5
        sqrt(2*pi) * tt^(z+0.5) * exp(-tt) * x
     }
  }   

}

spouge <- function(z, a=49) {

  if(length(z) > 1)
  {
     sapply(z, spouge)
  } else
  {
     z <- z-1
     k <- seq.int(1, a-1)
     ck <- rep(c(1, -1), len=a-1) / factorial(k-1) * (a-k)^(k-0.5) * exp(a-k)
     (z + a)^(z+0.5) * exp(-z - a) * (sqrt(2*pi) + sum(ck/(z+k)))
  }

}

  1. Checks

z <- (1:10)/3 all.equal(gamma(z), stirling(z)) # Mean relative difference: 0.07181942 all.equal(gamma(z), nemes(z)) # Mean relative difference: 0.003460549 all.equal(as.complex(gamma(z)), lanczos(z)) # TRUE all.equal(gamma(z), spouge(z)) # TRUE data.frame(z=z, stirling=stirling(z), nemes=nemes(z), lanczos=lanczos(z), spouge=spouge(z), builtin=gamma(z))</lang>

Output:
          z  stirling     nemes      lanczos    spouge   builtin
1  0.3333333 2.1569760 2.6290752 2.6789385+0i 2.6789385 2.6789385
2  0.6666667 1.2028507 1.3515736 1.3541179+0i 1.3541179 1.3541179
3  1.0000000 0.9221370 0.9996275 1.0000000+0i 1.0000000 1.0000000
4  1.3333333 0.8397427 0.8928835 0.8929795+0i 0.8929795 0.8929795
5  1.6666667 0.8591902 0.9027098 0.9027453+0i 0.9027453 0.9027453
6  2.0000000 0.9595022 0.9999831 1.0000000+0i 1.0000000 1.0000000
7  2.3333333 1.1491064 1.1906296 1.1906393+0i 1.1906393 1.1906393
8  2.6666667 1.4584904 1.5045690 1.5045755+0i 1.5045755 1.5045755
9  3.0000000 1.9454032 1.9999951 2.0000000+0i 2.0000000 2.0000000
10 3.3333333 2.7097638 2.7781544 2.7781585+0i 2.7781585 2.7781585

Racket

<lang Racket>#lang racket (define (gamma number)

 (if (> 1/2 number)
     (/ pi (* (sin (* pi number))
              (gamma (- 1.0 number))))
     (let ((n (sub1 number))
           (c '(0.99999999999980993 676.5203681218851 -1259.1392167224028
                771.32342877765313 -176.61502916214059 12.507343278686905

-0.13857109526572012 9.9843695780195716e-6 1.5056327351493116e-7)))

       (* (sqrt (* pi 2))
          (expt (+ n 7 0.5) (+ n 0.5))
          (exp (- (+ n 7 0.5)))
          (+ (car c)
             (apply +
               (for/list ((i (in-range (length (cdr c)))) (x (in-list (cdr c))))
                 (/ x (+ 1 n i)))))))))

(map gamma '(0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0))

->
'(9.513507698668736
4.590843711998802
2.9915689876875904
2.218159543757687
1.7724538509055159
1.489192248812818
1.2980553326475577
1.1642297137253037
1.068628702119319
1.0)</lang>

REXX

This version uses a Taylor series 80-digits coefficients with much more accuracy.
As a result, the gamma value for   ½   is now   25   decimal digits more accurate than the previous version
(which only used   20   digit coefficients).

Note:   The Taylor series isn't much good above values of   . <lang rexx>/*REXX program calculates GAMMA using the Taylor series coefficients; ≈80 decimal digits*/

                           /*The GAMMA function symbol is the Greek capital letter:  Γ */

numeric digits 90 /*be able to handle extended precision.*/ parse arg LO HI . /*allow specification of gamma arg/args*/

                                                /* [↓]  either show a range or a ···   */
       do j=word(LO 1, 1)  to word(HI LO 9, 1)  /*              ··· single gamma value.*/
       say 'gamma('j") ="  gamma(j)             /*compute gamma of J and display value.*/
       end   /*j*/                              /* [↑]  default LO is one;  HI is nine.*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gamma: procedure; parse arg x; xm=x-1; sum=0

                               /*coefficients thanks to: Arne Fransén & Staffan Wrigge.*/
#.1 =  1                       /* [↓]    #.2   is the   Euler-Mascheroni  constant.    */
#.2 =  0.57721566490153286060651209008240243104215933593992359880576723488486772677766467
#.3 = -0.65587807152025388107701951514539048127976638047858434729236244568387083835372210
#.4 = -0.04200263503409523552900393487542981871139450040110609352206581297618009687597599
#.5 =  0.16653861138229148950170079510210523571778150224717434057046890317899386605647425
#.6 = -0.04219773455554433674820830128918739130165268418982248637691887327545901118558900
#.7 = -0.00962197152787697356211492167234819897536294225211300210513886262731167351446074
#.8 =  0.00721894324666309954239501034044657270990480088023831800109478117362259497415854
#.9 = -0.00116516759185906511211397108401838866680933379538405744340750527562002584816653
  1. .10 = -0.00021524167411495097281572996305364780647824192337833875035026748908563946371678
  2. .11 = 0.00012805028238811618615319862632816432339489209969367721490054583804120355204347
  3. .12 = -0.00002013485478078823865568939142102181838229483329797911526116267090822918618897
  4. .13 = -0.00000125049348214267065734535947383309224232265562115395981534992315749121245561
  5. .14 = 0.00000113302723198169588237412962033074494332400483862107565429550539546040842730
  6. .15 = -0.00000020563384169776071034501541300205728365125790262933794534683172533245680371
  7. .16 = 0.00000000611609510448141581786249868285534286727586571971232086732402927723507435
  8. .17 = 0.00000000500200764446922293005566504805999130304461274249448171895337887737472132
  9. .18 = -0.00000000118127457048702014458812656543650557773875950493258759096189263169643391
  10. .19 = 0.00000000010434267116911005104915403323122501914007098231258121210871073927347588
  11. .20 = 0.00000000000778226343990507125404993731136077722606808618139293881943550732692987
  12. .21 = -0.00000000000369680561864220570818781587808576623657096345136099513648454655443000
  13. .22 = 0.00000000000051003702874544759790154813228632318027268860697076321173501048565735
  14. .23 = -0.00000000000002058326053566506783222429544855237419746091080810147188058196444349
  15. .24 = -0.00000000000000534812253942301798237001731872793994898971547812068211168095493211
  16. .25 = 0.00000000000000122677862823826079015889384662242242816545575045632136601135999606
  17. .26 = -0.00000000000000011812593016974587695137645868422978312115572918048478798375081233
  18. .27 = 0.00000000000000000118669225475160033257977724292867407108849407966482711074006109
  19. .28 = 0.00000000000000000141238065531803178155580394756670903708635075033452562564122263
  20. .29 = -0.00000000000000000022987456844353702065924785806336992602845059314190367014889830
  21. .30 = 0.00000000000000000001714406321927337433383963370267257066812656062517433174649858
  22. .31 = 0.00000000000000000000013373517304936931148647813951222680228750594717618947898583
  23. .32 = -0.00000000000000000000020542335517666727893250253513557337960820379352387364127301
  24. .33 = 0.00000000000000000000002736030048607999844831509904330982014865311695836363370165
  25. .34 = -0.00000000000000000000000173235644591051663905742845156477979906974910879499841377
  26. .35 = -0.00000000000000000000000002360619024499287287343450735427531007926413552145370486
  27. .36 = 0.00000000000000000000000001864982941717294430718413161878666898945868429073668232
  28. .37 = -0.00000000000000000000000000221809562420719720439971691362686037973177950067567580
  29. .38 = 0.00000000000000000000000000012977819749479936688244144863305941656194998646391332
  30. .39 = 0.00000000000000000000000000000118069747496652840622274541550997151855968463784158
  31. .40 = -0.00000000000000000000000000000112458434927708809029365467426143951211941179558301
  32. .41 = 0.00000000000000000000000000000012770851751408662039902066777511246477487720656005
  33. .42 = -0.00000000000000000000000000000000739145116961514082346128933010855282371056899245
  34. .43 = 0.00000000000000000000000000000000001134750257554215760954165259469306393008612196
  35. .44 = 0.00000000000000000000000000000000004639134641058722029944804907952228463057968680
  36. .45 = -0.00000000000000000000000000000000000534733681843919887507741819670989332090488591
  37. .46 = 0.00000000000000000000000000000000000032079959236133526228612372790827943910901464
  38. .47 = -0.00000000000000000000000000000000000000444582973655075688210159035212464363740144
  39. .48 = -0.00000000000000000000000000000000000000131117451888198871290105849438992219023663
  40. .49 = 0.00000000000000000000000000000000000000016470333525438138868182593279063941453996
  41. .50 = -0.00000000000000000000000000000000000000001056233178503581218600561071538285049997
  42. .51 = 0.00000000000000000000000000000000000000000026784429826430494783549630718908519485
  43. .52 = 0.00000000000000000000000000000000000000000002424715494851782689673032938370921241
  44. =52; do k=# by -1 for #
                               sum=sum*xm  +  #.k
                               end   /*k*/

return 1/sum</lang>

output   when using the input of:     0.5
gamma(0.5) = 1.77245385090551602729816748334114518279754945612238712821380509003635917689651032047826593


Note that:   Γ(½) = =

1.77245 38509 05516 02729 81674 83341 14518 27975 49456 12238 71282 13807 78985 29112 84591 03218 13749 50656 73854 46654 16226 82362 +

to 110 digits past the decimal point,   the vinculum (overbar) marks the   difference digit   from the computed value (by this REXX program) of   gamma(½).

Ring

<lang ring> decimals(3) gamma = 0.577 coeff = -0.655 quad = -0.042 qui = 0.166 set = -0.042

for i=1 to 10

   see gammafunc(i / 3.0) + nl

next

func recigamma z

    return z + gamma * pow(z,2) + coeff * pow(z,3) + quad * pow(z,4) + qui * pow(z,5) + set * pow(z,6)

func gammafunc z

    if z = 1 return 1
    but fabs(z) <= 0.5 return 1 / recigamma(z)
    else return (z - 1) * gammafunc(z-1) ok

</lang>

RLaB

RLaB through GSL has the following functions related to the Gamma function, namely, Gamma, GammaRegularizedC, LogGamma, RecGamma, and Pochhammer, where

, the Gamma function;
, the regularized Gamma function which is also known as the normalized incomplete Gamma function;
, which the GSL calls the complementary normalized Gamma function;
;
;
.

Ruby

Taylor series

Translation of: Ada

<lang ruby>$a = [ 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 ]

def gamma(x)

 y = Float(x) - 1
 1.0 / $a.reverse.inject {|sum, an| sum * y + an}

end

(1..10).each {|i| puts format("%.14e", gamma(i/3.0))}</lang>

Output:
2.67893853470775e+00
1.35411793942640e+00
1.00000000000000e+00
8.92979511569249e-01
9.02745292950934e-01
1.00000000000000e+00
1.19063934875900e+00
1.50457548825154e+00
1.99999999999397e+00
2.77815847933857e+00

Built in

<lang ruby>(1..10).each{|i| puts Math.gamma(i/3.0)}</lang>

Output:
2.678938534707748
1.3541179394264005
1.0
0.8929795115692493
0.9027452929509336
1.0
1.190639348758999
1.5045754882515558
2.0
2.7781584804376647

Scala

<lang scala>import java.util.Locale._

object Gamma {

 def stGamma(x:Double):Double=math.sqrt(2*math.Pi/x)*math.pow((x/math.E), x)
 
 def laGamma(x:Double):Double={
   val p=Seq(676.5203681218851, -1259.1392167224028, 771.32342877765313, 
            -176.61502916214059, 12.507343278686905, -0.13857109526572012,
               9.9843695780195716e-6, 1.5056327351493116e-7)
   if(x < 0.5) {
     math.Pi/(math.sin(math.Pi*x)*laGamma(1-x))
   } else {
     val x2=x-1
     val t=x2+7+0.5
     val a=p.zipWithIndex.foldLeft(0.99999999999980993)((r,v) => r+v._1/(x2+v._2+1))
     math.sqrt(2*math.Pi)*math.pow(t, x2+0.5)*math.exp(-t)*a
   }
 }
 
 def main(args: Array[String]): Unit = {
   println("Gamma    Stirling             Lanczos")
   for(x <- 0.1 to 2.0 by 0.1)
     println("%.1f  ->  %.16f   %.16f".formatLocal(ENGLISH, x, stGamma(x), laGamma(x)))
 }

}</lang>

Output:
Gamma    Stirling             Lanczos
0.1  ->  5.6971871489771690   9.5135076986687340
0.2  ->  3.3259984240223925   4.5908437119988030
0.3  ->  2.3625300362696198   2.9915689876875904
0.4  ->  1.8414763359362354   2.2181595437576870
0.5  ->  1.5203469010662807   1.7724538509055159
0.6  ->  1.3071588574483560   1.4891922488128180
0.7  ->  1.1590532921139200   1.2980553326475577
0.8  ->  1.0533709684256085   1.1642297137253035
0.9  ->  0.9770615078776956   1.0686287021193193
1.0  ->  0.9221370088957892   1.0000000000000002
1.1  ->  0.8834899531687038   0.9513507698668728
1.2  ->  0.8577553353965909   0.9181687423997607
1.3  ->  0.8426782594483921   0.8974706963062777
1.4  ->  0.8367445486370817   0.8872638175030760
1.5  ->  0.8389565525264964   0.8862269254527583
1.6  ->  0.8486932421525738   0.8935153492876904
1.7  ->  0.8656214717938840   0.9086387328532912
1.8  ->  0.8896396352879945   0.9313837709802430
1.9  ->  0.9208427218942294   0.9617658319073875
2.0  ->  0.9595021757444918   1.0000000000000010

Scheme

Translation of: Scala
for Lanczos and Stirling
Translation of: Ruby
for Taylor

<lang scheme> (import (scheme base)

       (scheme inexact)
       (scheme write))

(define PI 3.14159265358979323846264338327950) (define e 2.7182818284590452353602875)

(define gamma-lanczos

 (let ((p '(676.5203681218851 -1259.1392167224028 771.32342877765313 
            -176.61502916214059 12.507343278686905 -0.13857109526572012
            9.9843695780195716e-6 1.5056327351493116e-7)))
   (lambda (x)
     (if (< x 0.5)
       (/ PI (* (sin (* PI x)) (gamma-lanczos (- 1 x))))
       (let* ((x2 (- x 1))
              (t (+ x2 7 0.5))
              (a (do ((ps p (cdr ps))
                      (idx 0 (+ 1 idx))
                      (res 0.99999999999980993 (+ res 
                                                  (/ (car ps)
                                                     (+ x2 idx 1)))))
                   ((null? ps) res))))
         (* (sqrt (* 2 PI)) (expt t (+ x2 0.5)) (exp (- t)) a))))))

(define (gamma-stirling x)

 (* (sqrt (* 2 (/ PI x))) (expt (/ x e) x)))

(define gamma-taylor

 (let ((a (reverse
            '(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))))
   (lambda (x)
     (let ((y (- x 1)))
       (do ((as a (cdr as))
            (res 0 (+ (car as) (* res y))))
         ((null? as) (/ 1 res)))))))

(do ((i 0.1 (+ i 0.1)))

 ((> i 2.01) )
 (display (string-append "Gamma ("
                         (number->string i)
                         "): "
                         "\n --- Lanczos : "
                         (number->string (gamma-lanczos i))
                         "\n --- Stirling: "
                         (number->string (gamma-stirling i))
                         "\n --- Taylor  : "
                         (number->string (gamma-taylor i))
                         "\n")))

</lang>

Output:
Gamma (0.1): 
 --- Lanczos : 9.513507698668736
 --- Stirling: 5.69718714897717
 --- Taylor  : 9.513507698668734
Gamma (0.2): 
 --- Lanczos : 4.590843711998803
 --- Stirling: 3.3259984240223925
 --- Taylor  : 4.5908437119988035
Gamma (0.30000000000000004): 
 --- Lanczos : 2.9915689876875904
 --- Stirling: 2.3625300362696198
 --- Taylor  : 2.991568987687591
Gamma (0.4): 
 --- Lanczos : 2.218159543757687
 --- Stirling: 1.8414763359362354
 --- Taylor  : 2.2181595437576886
Gamma (0.5): 
 --- Lanczos : 1.7724538509055159
 --- Stirling: 1.5203469010662807
 --- Taylor  : 1.772453850905516
Gamma (0.6): 
 --- Lanczos : 1.489192248812818
 --- Stirling: 1.307158857448356
 --- Taylor  : 1.489192248812817
Gamma (0.7): 
 --- Lanczos : 1.2980553326475577
 --- Stirling: 1.15905329211392
 --- Taylor  : 1.298055332647558
Gamma (0.7999999999999999): 
 --- Lanczos : 1.1642297137253035
 --- Stirling: 1.0533709684256085
 --- Taylor  : 1.1642297137253033
Gamma (0.8999999999999999): 
 --- Lanczos : 1.0686287021193193
 --- Stirling: 0.9770615078776956
 --- Taylor  : 1.0686287021193195
Gamma (0.9999999999999999): 
 --- Lanczos : 1.0000000000000002
 --- Stirling: 0.9221370088957892
 --- Taylor  : 1.0000000000000002
Gamma (1.0999999999999999): 
 --- Lanczos : 0.9513507698668728
 --- Stirling: 0.8834899531687039
 --- Taylor  : 0.9513507698668733
Gamma (1.2): 
 --- Lanczos : 0.9181687423997607
 --- Stirling: 0.8577553353965909
 --- Taylor  : 0.9181687423997608
Gamma (1.3): 
 --- Lanczos : 0.8974706963062777
 --- Stirling: 0.842678259448392
 --- Taylor  : 0.8974706963062773
Gamma (1.4000000000000001): 
 --- Lanczos : 0.8872638175030759
 --- Stirling: 0.8367445486370818
 --- Taylor  : 0.8872638175030753
Gamma (1.5000000000000002): 
 --- Lanczos : 0.8862269254527583
 --- Stirling: 0.8389565525264964
 --- Taylor  : 0.886226925452758
Gamma (1.6000000000000003): 
 --- Lanczos : 0.8935153492876904
 --- Stirling: 0.8486932421525738
 --- Taylor  : 0.8935153492876904
Gamma (1.7000000000000004): 
 --- Lanczos : 0.9086387328532912
 --- Stirling: 0.865621471793884
 --- Taylor  : 0.9086387328532904
Gamma (1.8000000000000005): 
 --- Lanczos : 0.931383770980243
 --- Stirling: 0.8896396352879945
 --- Taylor  : 0.9313837709802427
Gamma (1.9000000000000006): 
 --- Lanczos : 0.9617658319073875
 --- Stirling: 0.9208427218942294
 --- Taylor  : 0.9617658319073876
Gamma (2.0000000000000004): 
 --- Lanczos : 1.000000000000001
 --- Stirling: 0.9595021757444918
 --- Taylor  : 1.0000000000000002

Scilab

<lang>function x=gammal(z) // Lanczos'

   lz=[  1.000000000190015 ..
         76.18009172947146  -86.50532032941677      24.01409824083091    ..
        -1.231739572450155   1.208650973866179e-3  -5.395239384953129e-6 ]
   if z < 0.5 then 
       x=%pi/sin(%pi*z)-gammal(1-z)
   else
       z=z-1.0
       b=z+5.5
       a=lz(1)
       for i=1:6
           a=a+(lz(i+1)/(z+i))
       end
       x=exp((log(sqrt(2*%pi))+log(a)-b)+log(b)*(z+0.5))
   end

endfunction

printf("%4s %-9s %-9s\n","x","gamma(x)","gammal(x)") for i=1:30

   x=i/10
   printf("%4.1f %9f %9f\n",x,gamma(x),gammal(x))

end</lang>

Output:
   x gamma(x)  gammal(x)
 0.1  9.097779  9.097779
 0.2  4.180567  4.180567
 0.3  2.585167  2.585167
 0.4  1.814074  1.814074
 0.5  1.772454  1.772454
 0.6  1.489192  1.489192
 0.7  1.298055  1.298055
 0.8  1.164230  1.164230
 0.9  1.068629  1.068629
 1.0  1.000000  1.000000
 1.1  0.951351  0.951351
 1.2  0.918169  0.918169
 1.3  0.897471  0.897471
 1.4  0.887264  0.887264
 1.5  0.886227  0.886227
 1.6  0.893515  0.893515
 1.7  0.908639  0.908639
 1.8  0.931384  0.931384
 1.9  0.961766  0.961766
 2.0  1.000000  1.000000
 2.1  1.046486  1.046486
 2.2  1.101802  1.101802
 2.3  1.166712  1.166712
 2.4  1.242169  1.242169
 2.5  1.329340  1.329340
 2.6  1.429625  1.429625
 2.7  1.544686  1.544686
 2.8  1.676491  1.676491
 2.9  1.827355  1.827355
 3.0  2.000000  2.000000

Seed7

Translation of: Ada

<lang seed7>$ include "seed7_05.s7i";

 include "float.s7i";

const func float: gamma (in float: X) is func

 result
   var float: result is 0.0;
 local
   const array float: A is [] (
        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);
   var float: Y is 0.0;
   var float: Sum is A[maxIdx(A)];
   var integer: N is 0;
 begin
   Y := X - 1.0;
   for N range pred(maxIdx(A)) downto minIdx(A) do
     Sum := Sum * Y + A[N];
   end for;
   result := 1.0 / Sum;
 end func;

const proc: main is func

 local
   var integer: I is 0;
 begin
   for I range 1 to 10 do
     writeln((gamma(flt(I) / 3.0)) digits 15);
   end for;
 end func;</lang>
Output:
2.678937911987305
1.354117870330811
1.000000000000000
0.892979443073273
0.902745306491852
1.000000000000000
1.190639257431030
1.504575252532959
1.999999523162842
2.778157949447632

Sidef

Translation of: Ruby

<lang ruby>var a = [ 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 ]

func gamma(x) {

   var y = (x - 1)
   1 / a.reverse.reduce {|sum, an| sum*y + an}

}

for i in 1..10 {

   say ("%.14e" % gamma(i/3))

}</lang>

Output:
2.67893853470775e+00
1.35411793942640e+00
1.00000000000000e+00
8.92979511569249e-01
9.02745292950934e-01
1.00000000000000e+00
1.19063934875900e+00
1.50457548825154e+00
1.99999999999397e+00
2.77815847933858e+00

Lanczos approximation: <lang ruby>func gamma(z) {

   var epsilon = 0.0000001
   func withinepsilon(x) {
       abs(x - abs(x)) <= epsilon
   }

 

   var p = [
       676.5203681218851,     -1259.1392167224028,
       771.32342877765313,    -176.61502916214059,
       12.507343278686905,    -0.13857109526572012,
       9.9843695780195716e-6,  1.5056327351493116e-7,
   ]

 

   var result = 0
   const pi = Num.pi

 

   if (z.re < 0.5) {
       result = (pi / (sin(pi * z) * gamma(1 - z)))
   }
   else {
       z -= 1
       var x = 0.99999999999980993

 

       p.each_kv { |i, v|
           x += v/(z + i + 1)
       }

 

       var t = (z + p.len - 0.5)
       result = (sqrt(pi*2) * t**(z+0.5) * exp(-t) * x)
   }

 

   withinepsilon(result.im) ? result.re : result

}   for i in 1..10 {

   say ("%.14e" % gamma(i/3))

}</lang>

Output:
2.67893853470774e+00
1.35411793942640e+00
1.00000000000000e+00
8.92979511569252e-01
9.02745292950931e-01
1.00000000000000e+00
1.19063934875900e+00
1.50457548825155e+00
2.00000000000000e+00
2.77815848043767e+00

A simpler implementation: <lang ruby>define ℯ = Num.e define τ = Num.tau   func Γ(t) {

   t < 20 ? (__FUNC__(t + 1) / t)
          : (sqrt(τ*t) * pow(t/ℯ + 1/(12*ℯ*t), t) / t)

}   for i in (1..10) {

   say ("%.14e" % Γ(i/3))

}</lang>

Output:
2.67893831294932e+00
1.35411783267848e+00
9.99999913007168e-01
8.92979437649773e-01
9.02745221785653e-01
9.99999913007168e-01
1.19063925019970e+00
1.50457536964275e+00
1.99999982601434e+00
2.77815825046596e+00

Stata

This implementation uses the Taylor expansion of 1/gamma(1+x). The coefficients were computed with Maxima (see the Maxima implementation above). The results are compared to Mata's gamma function for each real between 1/100 and 100, by steps of 1/100.

<lang stata>mata _gamma_coef = 1.0, 5.7721566490153286061e-1,-6.5587807152025388108e-1, -4.2002635034095235529e-2,1.665386113822914895e-1, -4.2197734555544336748e-2,-9.6219715278769735621e-3, 7.2189432466630995424e-3,-1.1651675918590651121e-3, -2.1524167411495097282e-4,1.2805028238811618615e-4, -2.0134854780788238656e-5,-1.2504934821426706573e-6, 1.1330272319816958824e-6,-2.0563384169776071035e-7, 6.1160951044814158179e-9,5.0020076444692229301e-9, -1.1812745704870201446e-9,1.0434267116911005105e-10, 7.782263439905071254e-12,-3.6968056186422057082e-12, 5.100370287454475979e-13,-2.0583260535665067832e-14, -5.3481225394230179824e-15

function gamma_(x_) { external _gamma_coef x = x_ for (y=1; x>2;) y = --x*y z = _gamma_coef[24] x-- for (i=23; i>=1; i--) z = z*x+_gamma_coef[i] return(y/z) }

function map(f,a) { n = rows(a) p = cols(a) b = J(n,p,.) for (i=1; i<=n; i++) { for (j=1; j<=p; j++) { b[i,j] = (*f)(a[i,j]) } } return(b) }

x=(1::10000)/100 u=map(&gamma(),x) v=map(&gamma_(),x) max(abs((u-v):/u)) end</lang>

Output

1.27664e-13

Tcl

Works with: Tcl version 8.5
Library: Tcllib (Package: math)
Library: Tcllib (Package: math::calculus)

<lang tcl>package require math package require math::calculus

  1. gamma(1) and gamma(1.5)

set f 1.0 set f2 [expr {sqrt(acos(-1.))/2.}]

for {set x 1.0} {$x <= 10.0} {set x [expr {$x + 0.5}]} {

   # method 1 - numerical integration, Romberg's method, special
   #            case for an improper integral
   set g1 [math::calculus::romberg  \
               [list apply {{x t} {expr {$t ** ($x-1) * exp(-$t)}}} $x] \
               0 1 -relerror 1e-8]
   set g2 [math::calculus::romberg_infinity \
               [list apply {{x t} {expr {$t ** ($x-1) * exp(-$t)}}} $x] \
               1 Inf -relerror 1e-8]
   set gamma [expr {[lindex $g1 0] + [lindex $g2 0]}]
   # method 2 - library function
   set libgamma [expr {exp([math::ln_Gamma $x])}]
   # method 3 - special forms for integer and half-integer arguments
   if {$x == entier($x)} {
       puts [format {%4.1f %13.6f %13.6f %13.6f} $x $gamma $libgamma $f]
       set f [expr $f * $x]
   } else {
       puts [format {%4.1f %13.6f %13.6f %13.6f} $x $gamma $libgamma $f2]
       set f2 [expr $f2 * $x]
   }

}</lang>

Output:
 1.0      1.000000      1.000000      1.000000
 1.5      0.886228      0.886227      0.886227
 2.0      1.000000      1.000000      1.000000
 2.5      1.329340      1.329340      1.329340
 3.0      2.000000      2.000000      2.000000
 3.5      3.323351      3.323351      3.323351
 4.0      6.000000      6.000000      6.000000
 4.5     11.631731     11.631728     11.631728
 5.0     24.000009     24.000000     24.000000
 5.5     52.342778     52.342778     52.342778
 6.0    120.000000    120.000000    120.000000
 6.5    287.885278    287.885278    287.885278
 7.0    720.000001    720.000000    720.000000
 7.5   1871.254311   1871.254305   1871.254306
 8.0   5040.000032   5039.999999   5040.000000
 8.5  14034.298267  14034.407291  14034.407293
 9.0  40320.000705  40319.999992  40320.000000
 9.5 119292.464880 119292.461971 119292.461995
10.0 362880.010950 362879.999927 362880.000000

TI-83 BASIC

There is an hidden Gamma function in TI-83. Factorial (!) is implemented in increments of 0.5 :

.5! -> .8862269255 

As far as Gamma(n)=(n-1)! , we have everything needed.

Stirling's approximation

<lang ti83b>for(I,1,10) I/2→X X^(X-1/2)e^(-X)√(2π)→Y Disp X,(X-1)!,Y Pause End</lang>

Output:

The output display for x=0.5 to 5 by 0.5 : x, (x-1)!, Y(x) . Y(x) is Stirling's approximation of Gamma.

        0.5
1.772453851   
1.520346901
          1
          1             
.9221370089
        1.5
.8862269255   
.8389565525
          2
          1             
.9595021757
        2.5
1.329340388 
1.285978615
          3
          2
1.945403197
        3.5
 3.32335097
3.245363748
          4
          6             
5.876543783
        4.5
 11.6317284    
11.41865156   
          5
         24            
23.60383359

Lanczos' approximation

<lang ti83b>for(I,1,10) I/2→X e^(ln((1.0 +76.18009173/(X+1) -86.50532033/(X+2) +24.01409824/(X+3) -1.231739572/(X+4) +1.208650974E-3/(X+5) -5.395239385E-6/(X+6) )√(2π)/X) +(X+.5)ln(X+5.5)-X-5.5)->Y Disp X,(X-1)!,Y Pause End</lang>

Output:

The output display for x=0.5 to 5 by 0.5 : x, (x-1)!, Y(x) . Y(x) is Lanczos's approximation of Gamma.

        0.5
1.772453851   
1.772453851   
          1
          1             
          1             
        1.5
.8862269255   
.8862269254   
          2
          1             
          1             
        2.5
1.329340388 
1.329340388 
          3
          2
          2
        3.5
 3.32335097
 3.32335097
          4
          6             
          6             
        4.5
 11.6317284    
 11.6317284    
          5
         24            
         24            

Visual FoxPro

Translation of BBC Basic but with OOP extensions. Also some ideas from Numerical Methods (Press et al). <lang vfp> LOCAL i As Integer, x As Double, o As lanczos CLOSE DATABASES ALL CLEAR CREATE CURSOR results (ZVal B(1), GamVal B(15)) INDEX ON zval TAG ZVal COLLATE "Machine" SET ORDER TO 0 o = CREATEOBJECT("lanczos") FOR i = 1 TO 20 x = i/10

   INSERT INTO results VALUES (x, o.Gamma(x))

ENDFOR UPDATE results SET GamVal = ROUND(GamVal, 0) WHERE ZVal = INT(ZVal)

  • !* This just creates the output text - it is not part of the algorithm

DO cursor2txt.prg WITH "results", .T.

DEFINE CLASS lanczos As Session

  1. DEFINE FPF 5.5
  2. DEFINE HALF 0.5
  3. DEFINE PY PI()

DIMENSION LanCoeff[7] nSize = 0

PROCEDURE Init DODEFAULT() WITH THIS

   .LanCoeff[1] = 1.000000000190015
   .LanCoeff[2] = 76.18009172947146
   .LanCoeff[3] = -86.50532032941677
   .LanCoeff[4] = 24.01409824083091
   .LanCoeff[5] = -1.231739572450155
   .LanCoeff[6] = 0.0012086509738662
   .LanCoeff[7] = -0.000005395239385
   .nSize = ALEN(.LanCoeff)

ENDWITH ENDPROC

FUNCTION Gamma(z) RETURN EXP(THIS.LogGamma(z)) ENDFUNC

FUNCTION LogGamma(z) LOCAL a As Double, b As Double, i As Integer, j As Integer, lg As Double IF z < 0.5

   lg = LOG(PY/SIN(PY*z)) - THIS.LogGamma(1 - z)

ELSE

   WITH THIS	

z = z - 1 b = z + FPF a = .LanCoeff[1] FOR i = 2 TO .nSize j = i - 1 a = a + .LanCoeff[i]/(z + j) ENDFOR lg = (LOG(SQRT(2*PY)) + LOG(a) - b) + LOG(b)*(z + HALF)

   ENDWITH	

ENDIF RETURN lg ENDFUNC

ENDDEFINE </lang>

Output:
zval	gamval
0.1	9.513507698669704
0.2	4.590843712000122
0.3	2.991568987689402
0.4	2.218159543760185
0.5	1.772453850902053
0.6	1.489192248811141
0.7	1.298055332646772
0.8	1.164229713724969
0.9	1.068628702119210
1.0	1.000000000000000
1.1	0.951350769866919
1.2	0.918168742399821
1.3	0.897470696306335
1.4	0.887263817503125
1.5	0.886226925452796
1.6	0.893515349287718
1.7	0.908638732853309
1.8	0.931383770980253
1.9	0.961765831907391
2.0	1.000000000000000

zkl

Translation of: D
but without a built in gamma function.

<lang zkl>fcn taylorGamma(x){

  var table = T(
    0x1p+0,                    0x1.2788cfc6fb618f4cp-1,
   -0x1.4fcf4026afa2dcecp-1,  -0x1.5815e8fa27047c8cp-5,
    0x1.5512320b43fbe5dep-3,  -0x1.59af103c340927bep-5,
   -0x1.3b4af28483e214e4p-7,   0x1.d919c527f60b195ap-8,
   -0x1.317112ce3a2a7bd2p-10, -0x1.c364fe6f1563ce9cp-13,
    0x1.0c8a78cd9f9d1a78p-13, -0x1.51ce8af47eabdfdcp-16,
   -0x1.4fad41fc34fbb2p-20,    0x1.302509dbc0de2c82p-20,
   -0x1.b9986666c225d1d4p-23,  0x1.a44b7ba22d628acap-28,
    0x1.57bc3fc384333fb2p-28, -0x1.44b4cedca388f7c6p-30,
    0x1.cae7675c18606c6p-34,   0x1.11d065bfaf06745ap-37,
   -0x1.0423bac8ca3faaa4p-38,  0x1.1f20151323cd0392p-41,
   -0x1.72cb88ea5ae6e778p-46, -0x1.815f72a05f16f348p-48,
    0x1.6198491a83bccbep-50,  -0x1.10613dde57a88bd6p-53,
    0x1.5e3fee81de0e9c84p-60,  0x1.a0dc770fb8a499b6p-60,
   -0x1.0f635344a29e9f8ep-62,  0x1.43d79a4b90ce8044p-66).reverse();

   y  := x.toFloat() - 1.0;
   sm := table[1,*].reduce('wrap(sm,an){ sm * y + an },table[0]);
   return(1.0 / sm);

}</lang> <lang zkl>fcn lanczosGamma(z) { z = z.toFloat();

   // Coefficients used by the GNU Scientific Library.
   // http://en.wikipedia.org/wiki/Lanczos_approximation
   const g = 7, PI = (0.0).pi;
   exp := (0.0).e.pow;
   var table = T(
            0.99999_99999_99809_93,
          676.52036_81218_851,
        -1259.13921_67224_028,
          771.32342_87776_5313,
         -176.61502_91621_4059,
           12.50734_32786_86905,
           -0.13857_10952_65720_12,
            9.98436_95780_19571_6e-6,
            1.50563_27351_49311_6e-7);

   // Reflection formula.
   if (z < 0.5) {
       return(PI / ((PI * z).sin() * lanczosGamma(1.0 - z)));
   } else {
       z -= 1;
       x := table[0];
       foreach i in ([1 .. g + 1]){
           x += table[i] / (z + i); }
       t := z + g + 0.5;
       return((2.0 * PI).sqrt() * t.pow(z + 0.5) * exp(-t) * x);
   }

}</lang>

Output:
foreach i in ([1.0 .. 10]) {
   x := i / 3.0;
   println("%f: %20.19e %20.19e %e".fmt( x,
	   a:=taylorGamma(x), b:=lanczosGamma(x),(a-b).abs()));
}
0.333333: 2.6789385347077483424e+00 2.6789385347077474542e+00 8.881784e-16
0.666667: 1.3541179394264004632e+00 1.3541179394264002411e+00 2.220446e-16
1.000000: 1.0000000000000000000e+00 1.0000000000000002220e+00 2.220446e-16
1.333333: 8.9297951156924926241e-01 8.9297951156924970650e-01 4.440892e-16
1.666667: 9.0274529295093364212e-01 9.0274529295093353110e-01 1.110223e-16
2.000000: 1.0000000000000000000e+00 1.0000000000000006661e+00 6.661338e-16
2.333333: 1.1906393487589990166e+00 1.1906393487589996827e+00 6.661338e-16
2.666667: 1.5045754882515545159e+00 1.5045754882515582906e+00 3.774758e-15
3.000000: 1.9999999999992210675e+00 2.0000000000000017764e+00 7.807088e-13
3.333333: 2.7781584802531797962e+00 2.7781584804376668885e+00 1.844871e-10