# Gamma function

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

Implement one algorithm (or more) to compute the Gamma (${\displaystyle \Gamma }$) 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:

${\displaystyle \Gamma (x)=\displaystyle \int _{0}^{\infty }t^{x-1}e^{-t}dt}$

This suggests a straightforward (but inefficient) way of computing the ${\displaystyle \Gamma }$ through numerical integration.

Better suggested methods:

## 11l

Translation of: Python
V _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
]
F gamma(x)
V y = x - 1.0
V sm = :_a.last
L(n) (:_a.len-2 .. 0).step(-1)
sm = sm * y + :_a[n]
R 1.0 / sm

L(i) 1..10
print(‘#.14’.format(gamma(i / 3.0)))
Output:
2.67893853470775
1.35411793942640
1.00000000000000
0.89297951156925
0.90274529295093
1.00000000000000
1.19063934875900
1.50457548825154
1.99999999999397
2.77815847933857


## 360 Assembly

For maximum compatibility, this program uses only the basic instruction set.

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
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



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.

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;


Test program:

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;

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
# 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
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


## Arturo

A: @[
1.00000000000000000000     0.57721566490153286061 neg 0.65587807152025388108
neg 0.04200263503409523553     0.16653861138229148950 neg 0.04219773455554433675
neg 0.00962197152787697356     0.00721894324666309954 neg 0.00116516759185906511
neg 0.00021524167411495097     0.00012805028238811619 neg 0.00002013485478078824
neg 0.00000125049348214267     0.00000113302723198170 neg 0.00000020563384169776
0.00000000611609510448     0.00000000500200764447 neg 0.00000000118127457049
0.00000000010434267117     0.00000000000778226344 neg 0.00000000000369680562
0.00000000000051003703 neg 0.00000000000002058326 neg 0.00000000000000534812
0.00000000000000122678 neg 0.00000000000000011813     0.00000000000000000119
0.00000000000000000141 neg 0.00000000000000000023     0.00000000000000000002
]

ourGamma: function [x][
y: x - 1
result: last A
loop ((size A)-1)..0 'n ->
result: (result*y) + get A n
result: 1 // result
return result
]

loop 1..10 'z [
v1: ourGamma z // 3
v2: gamma z // 3
print [
pad (to :string z)++" =>" 10
pad (to :string v1)++" ~" 30
pad (to :string v2)++" :" 30
]
]

Output:
      1 =>            2.678938534707748 ~            2.678938534707748 :          4.440892098500626e-16
2 =>              1.3541179394264 ~              1.3541179394264 :                            0.0
3 =>                          1.0 ~                          1.0 :                            0.0
4 =>           0.8929795115692493 ~           0.8929795115692493 :                            0.0
5 =>           0.9027452929509336 ~           0.9027452929509336 :                            0.0
6 =>                          1.0 ~                          1.0 :                            0.0
7 =>            1.190639348758999 ~            1.190639348758999 :                            0.0
8 =>            1.504575488251335 ~            1.504575488251556 :         -2.204902926905561e-13
9 =>            1.999999999908069 ~                          2.0 :         -9.193090733106146e-11
10 =>            2.778158462440097 ~            2.778158480437665 :         -1.799756743636749e-08

## AutoHotkey

Search autohotkey.com: function
Source: AutoHotkey forum by Laszlo

/*
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
*/


## AWK

# syntax: GAWK -f GAMMA_FUNCTION.AWK
BEGIN {
e = (1+1/100000)^100000
pi = atan2(0,-1)

print("X    Stirling")
for (i=1; i<=20; i++) {
d = i / 10
printf("%4.2f %9.5f\n",d,gamma_stirling(d))
}
exit(0)
}
function gamma_stirling(x) {
return sqrt(2*pi/x) * pow(x/e,x)
}
function pow(a,b) {
return exp(b*log(a))
}

Output:
X    Stirling
0.10   5.69719
0.20   3.32600
0.30   2.36253
0.40   1.84148
0.50   1.52035
0.60   1.30716
0.70   1.15906
0.80   1.05338
0.90   0.97707
1.00   0.92214
1.10   0.88349
1.20   0.85776
1.30   0.84268
1.40   0.83675
1.50   0.83896
1.60   0.84870
1.70   0.86563
1.80   0.88965
1.90   0.92085
2.00   0.95951


## BASIC

### ANSI BASIC

Translation of: BBC BASIC

- Lanczos method.

Works with: Decimal 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
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

Output:
 .1                      9.513507698670
.2                      4.590843712000
.3                      2.991568987689
.4                      2.218159543760
.5                      1.772453850902
.6                      1.489192248811
.7                      1.298055332647
.8                      1.164229713725
.9                      1.068628702119
1.0                      1.000000000000
1.1                       .951350769867
1.2                       .918168742400
1.3                       .897470696306
1.4                       .887263817503
1.5                       .886226925453
1.6                       .893515349288
1.7                       .908638732853
1.8                       .931383770980
1.9                       .961765831907
2.0                      1.000000000000


### BASIC256

Translation of: FreeBASIC

- Stirling method.

Translation of: Phix

- Lanczos method.

print " x       Stirling         Lanczos"
print
for i = 1 to 20
d = i / 10.0
print d;
print chr(9); ljust(string(gammaStirling(d)), 13, "0");
print chr(9); ljust(string(gammaLanczos(d)),  13, "0")
next i
end

function gammaStirling (x)
e = exp(1)	# e is not predefined in BASIC256
return sqr(2.0 * pi / x) * ((x / e) ^ x)
end function

function gammaLanczos (x)
dim p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7}

g = 7
if x < 0.5 then return pi / (sin(pi * x) * gammaLanczos(1-x))
x -= 1
a = p[0]
t = x + g + 0.5

for i = 1 to 8
a += p[i] / (x + i)
next i
return sqr(2.0 * pi) * (t ^ (x + 0.5)) * exp(-t) * a
end function

### BBC BASIC

Uses the Lanczos approximation.

      *FLOAT64
foreach_reverse (immutable an; table[0 .. $- 1]) sm = sm * y + an; return 1.0L / sm; } real lanczosGamma(real z) pure nothrow @safe @nogc { // Coefficients used by the GNU Scientific Library. // http://en.wikipedia.org/wiki/Lanczos_approximation enum g = 7; static immutable real[9] table = [ 0.99999_99999_99809_93, 676.52036_81218_851, -1259.13921_67224_028, 771.32342_87776_5313, -176.61502_91621_4059, 12.50734_32786_86905, -0.13857_10952_65720_12, 9.98436_95780_19571_6e-6, 1.50563_27351_49311_6e-7]; // Reflection formula. if (z < 0.5L) { return PI / (sin(PI * z) * lanczosGamma(1 - z)); } else { z -= 1; real x = table[0]; foreach (immutable i; 1 .. g + 2) x += table[i] / (z + i); immutable real t = z + g + 0.5L; return sqrt(2 * PI) * t ^^ (z + 0.5L) * exp(-t) * x; } } void main() { foreach (immutable i; 1 .. 11) { immutable real x = i / 3.0L; writefln("%f: %20.19e %20.19e %20.19e", x, x.taylorGamma, x.lanczosGamma, x.gamma); } }  Output: 0.333333: 2.6789385347077476335e+00 2.6789385347077470551e+00 2.6789385347077476339e+00 0.666667: 1.3541179394264004169e+00 1.3541179394264007092e+00 1.3541179394264004170e+00 1.000000: 1.0000000000000000000e+00 1.0000000000000002126e+00 1.0000000000000000000e+00 1.333333: 8.9297951156924921124e-01 8.9297951156924947465e-01 8.9297951156924921132e-01 1.666667: 9.0274529295093361132e-01 9.0274529295093396555e-01 9.0274529295093361123e-01 2.000000: 1.0000000000000000000e+00 1.0000000000000004903e+00 1.0000000000000000000e+00 2.333333: 1.1906393487589989474e+00 1.1906393487589996490e+00 1.1906393487589989482e+00 2.666667: 1.5045754882515545787e+00 1.5045754882515570474e+00 1.5045754882515560190e+00 3.000000: 1.9999999999992207405e+00 2.0000000000000015575e+00 2.0000000000000000000e+00 3.333333: 2.7781584802531739378e+00 2.7781584804376666336e+00 2.7781584804376642124e+00 ## Delphi Library: System.Math Translation of: Go program Gamma_function; {$APPTYPE CONSOLE}

uses
System.SysUtils,
System.Math;

function lanczos7(z: double): Double;
begin
var t := z + 6.5;
var x := 0.99999999999980993 + 676.5203681218851 / z - 1259.1392167224028 / (z
+ 1) + 771.32342877765313 / (z + 2) - 176.61502916214059 / (z + 3) +
12.507343278686905 / (z + 4) - 0.13857109526572012 / (z + 5) +
9.9843695780195716e-6 / (z + 6) + 1.5056327351493116e-7 / (z + 7);

Result := Sqrt(2) * Sqrt(pi) * Power(t, z - 0.5) * exp(-t) * x;
end;

begin
var xs: TArray<double> := [-0.5, 0.1, 0.5, 1, 1.5, 2, 3, 10, 140, 170];
writeln('    x              Lanczos7');
for var x in xs do
writeln(format('%5.1f %24.16g', [x, lanczos7(x)]));
end.

Output:
    x              Lanczos7
-0,5       -3,544907701811089
0,1        9,513507698668747
0,5        1,772453850905517
1,0                        1
1,5       0,8862269254527583
2,0                        1
3,0        2,000000000000002
10,0        362880,0000000007
140,0    9,615723196940235E238
170,0    4,269068009004271E304

## EasyLang

Translation of: AWK
e = 2.718281828459
func stirling x .
return sqrt (2 * pi / x) * pow (x / e) x
.
print " X    Stirling"
for i to 20
d = i / 10
numfmt 2 4
write d & "    "
numfmt 3 4
print stirling d
.

## Elixir

Translation of: Ruby
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)

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


## F#

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

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 ) )
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

Translation of: C#

The C# version can be translated to F# to support complex numbers:

open System.Numerics
open System

let rec gamma (z: Complex) =
let mutable z = z
let lanczosCoefficients = [| 676.520368121885; -1259.1392167224; 771.323428777653; -176.615029162141; 12.5073432786869; -0.13857109526572; 9.98436957801957E-06; 1.50563273514931E-07 |]

if z.Real < 0.5 then
Math.PI / (sin (Math.PI * z) * gamma (1.0 - z))
else
let mutable x = Complex.One
z <- z - 1.0

for i = 0 to lanczosCoefficients.Length - 1 do
x <- x + lanczosCoefficients.[i] / (z + Complex(i, 0) + 1.0)

let t = z + float lanczosCoefficients.Length - 0.5
sqrt (2.0 * Math.PI) * (t ** (z + 0.5)) * exp (-t) * x

Seq.iter (fun i -> printfn "Gamma(%f) = %A" i (gamma (Complex(i, 0)))) [ 0 .. 100 ]
Seq.iter2 (fun i j -> printfn "Gamma(%f + i%f) = %A" i j (gamma (Complex(i, j)))) [ 0 .. 100 ] [ 0 .. 100 ]
Output:
Gamma(0.000000) = (NaN, NaN)
Gamma(1.000000) = (1,0000000000000049, 0)
Gamma(2.000000) = (1,0000000000000115, 0)
Gamma(3.000000) = (2,0000000000000386, 0)
Gamma(4.000000) = (6,000000000000169, 0)
Gamma(5.000000) = (24,00000000000084, 0)
Gamma(6.000000) = (120,00000000000514, 0)
Gamma(7.000000) = (720,0000000000364, 0)
Gamma(8.000000) = (5040,000000000289, 0)
Gamma(9.000000) = (40320,000000002554, 0)
Gamma(10.000000) = (362880,00000002526, 0)
Gamma(11.000000) = (3628800,0000002664, 0)
Gamma(12.000000) = (39916800,000003114, 0)
Gamma(13.000000) = (479001600,00004023, 0)
Gamma(14.000000) = (6227020800,000546, 0)
Gamma(15.000000) = (87178291200,00801, 0)
Gamma(16.000000) = (1307674368000,1267, 0)
Gamma(17.000000) = (20922789888002,08, 0)
Gamma(18.000000) = (355687428096034,7, 0)
Gamma(19.000000) = (6402373705728658, 0)
Gamma(20.000000) = (1,2164510040884514E+17, 0)
Gamma(21.000000) = (2,4329020081769083E+18, 0)
Gamma(22.000000) = (5,109094217171516E+19, 0)
Gamma(23.000000) = (1,1240007277777331E+21, 0)
Gamma(24.000000) = (2,5852016738887927E+22, 0)
Gamma(25.000000) = (6,20448401733312E+23, 0)
Gamma(26.000000) = (1,5511210043332811E+25, 0)
Gamma(27.000000) = (4,032914611266542E+26, 0)
Gamma(28.000000) = (1,0888869450419632E+28, 0)
Gamma(29.000000) = (3,048883446117507E+29, 0)
Gamma(30.000000) = (8,841761993740793E+30, 0)
Gamma(31.000000) = (2,652528598122235E+32, 0)
Gamma(32.000000) = (8,222838654178949E+33, 0)
Gamma(33.000000) = (2,631308369337265E+35, 0)
Gamma(34.000000) = (8,683317618812963E+36, 0)
Gamma(35.000000) = (2,9523279903964145E+38, 0)
Gamma(36.000000) = (1,0333147966387422E+40, 0)
Gamma(37.000000) = (3,719933267899472E+41, 0)
Gamma(38.000000) = (1,3763753091228064E+43, 0)
Gamma(39.000000) = (5,230226174666675E+44, 0)
Gamma(40.000000) = (2,0397882081200028E+46, 0)
Gamma(41.000000) = (8,159152832480012E+47, 0)
Gamma(42.000000) = (3,3452526613168034E+49, 0)
Gamma(43.000000) = (1,4050061177530564E+51, 0)
Gamma(44.000000) = (6,041526306338149E+52, 0)
Gamma(45.000000) = (2,658271574788788E+54, 0)
Gamma(46.000000) = (1,1962222086549537E+56, 0)
Gamma(47.000000) = (5,502622159812779E+57, 0)
Gamma(48.000000) = (2,586232415112008E+59, 0)
Gamma(49.000000) = (1,2413915592537631E+61, 0)
Gamma(50.000000) = (6,082818640343433E+62, 0)
Gamma(51.000000) = (3,04140932017172E+64, 0)
Gamma(52.000000) = (1,551118753287575E+66, 0)
Gamma(53.000000) = (8,065817517095389E+67, 0)
Gamma(54.000000) = (4,27488328406056E+69, 0)
Gamma(55.000000) = (2,308436973392699E+71, 0)
Gamma(56.000000) = (1,2696403353659833E+73, 0)
Gamma(57.000000) = (7,109985878049497E+74, 0)
Gamma(58.000000) = (4,0526919504882125E+76, 0)
Gamma(59.000000) = (2,350561331283167E+78, 0)
Gamma(60.000000) = (1,386831185457067E+80, 0)
Gamma(61.000000) = (8,3209871127424E+81, 0)
Gamma(62.000000) = (5,075802138772854E+83, 0)
Gamma(63.000000) = (3,1469973260391715E+85, 0)
Gamma(64.000000) = (1,9826083154046777E+87, 0)
Gamma(65.000000) = (1,2688693218589942E+89, 0)
Gamma(66.000000) = (8,247650592083449E+90, 0)
Gamma(67.000000) = (5,443449390775078E+92, 0)
Gamma(68.000000) = (3,647111091819299E+94, 0)
Gamma(69.000000) = (2,48003554243712E+96, 0)
Gamma(70.000000) = (1,711224524281613E+98, 0)
Gamma(71.000000) = (1,1978571669971308E+100, 0)
Gamma(72.000000) = (8,504785885679606E+101, 0)
Gamma(73.000000) = (6,123445837689312E+103, 0)
Gamma(74.000000) = (4,470115461513189E+105, 0)
Gamma(75.000000) = (3,307885441519755E+107, 0)
Gamma(76.000000) = (2,4809140811398187E+109, 0)
Gamma(77.000000) = (1,8854947016662648E+111, 0)
Gamma(78.000000) = (1,451830920283022E+113, 0)
Gamma(79.000000) = (1,1324281178207572E+115, 0)
Gamma(80.000000) = (8,946182130783977E+116, 0)
Gamma(81.000000) = (7,1569457046271725E+118, 0)
Gamma(82.000000) = (5,797126020748008E+120, 0)
Gamma(83.000000) = (4,753643337013366E+122, 0)
Gamma(84.000000) = (3,945523969721089E+124, 0)
Gamma(85.000000) = (3,31424013456571E+126, 0)
Gamma(86.000000) = (2,8171041143808564E+128, 0)
Gamma(87.000000) = (2,4227095383675335E+130, 0)
Gamma(88.000000) = (2,1077572983797526E+132, 0)
Gamma(89.000000) = (1,8548264225741817E+134, 0)
Gamma(90.000000) = (1,650795516091023E+136, 0)
Gamma(91.000000) = (1,4857159644819212E+138, 0)
Gamma(92.000000) = (1,3520015276785438E+140, 0)
Gamma(93.000000) = (1,2438414054642616E+142, 0)
Gamma(94.000000) = (1,1567725070817618E+144, 0)
Gamma(95.000000) = (1,0873661566568553E+146, 0)
Gamma(96.000000) = (1,032997848824012E+148, 0)
Gamma(97.000000) = (9,916779348710516E+149, 0)
Gamma(98.000000) = (9,619275968249195E+151, 0)
Gamma(99.000000) = (9,42689044888421E+153, 0)
Gamma(100.000000) = (9,33262154439535E+155, 0)
Gamma(0.000000 + i0.000000) = (NaN, NaN)
Gamma(1.000000 + i1.000000) = (0,49801566811835923, -0,15494982830180806)
Gamma(2.000000 + i2.000000) = (0,11229424234632254, 0,3236128855019324)
Gamma(3.000000 + i3.000000) = (-0,4401134076370088, -0,0636372431263299)
Gamma(4.000000 + i4.000000) = (0,7058649325913451, -0,49673908399741584)
Gamma(5.000000 + i5.000000) = (-0,9743952418053669, 2,0066898827226805)
Gamma(6.000000 + i6.000000) = (1,0560845455210948, -7,123931816061554)
Gamma(7.000000 + i7.000000) = (-0,26095668519941206, 27,88827411508434)
Gamma(8.000000 + i8.000000) = (1,8442848156317595, -125,96060801752867)
Gamma(9.000000 + i9.000000) = (-94,00399991734474, 643,3621714431141)
Gamma(10.000000 + i10.000000) = (1423,851941789479, -3496,081973308168)
Gamma(11.000000 + i11.000000) = (-16211,00700465313, 18168,810510285286)
Gamma(12.000000 + i12.000000) = (158471,8890918886, -68793,30331463458)
Gamma(13.000000 + i13.000000) = (-1329505,1052081874, -142199,12520863872)
Gamma(14.000000 + i14.000000) = (8576976,67312178, 7218722,503716219)
Gamma(15.000000 + i15.000000) = (-20768001,573587183, -99064686,32101583)
Gamma(16.000000 + i16.000000) = (-490395650,85195, 847486174,9207268)
Gamma(17.000000 + i17.000000) = (9782747798,66319, -2523864726,357996)
Gamma(18.000000 + i18.000000) = (-91408144092,80728, -62548333665,79536)
Gamma(19.000000 + i19.000000) = (80368797570,63837, 1283152922013,1064)
Gamma(20.000000 + i20.000000) = (12322153606702,379, -9813622771583,531)
Gamma(21.000000 + i21.000000) = (-191651224429571,5, -67416801166719,305)
Gamma(22.000000 + i22.000000) = (476610838765573,75, 2709614130691551,5)
Gamma(23.000000 + i23.000000) = (31282423285710508, -23340622982977492)
Gamma(24.000000 + i24.000000) = (-5,062346571412891E+17, -2,8075410996386413E+17)
Gamma(25.000000 + i25.000000) = (-1,1135374386470528E+18, 8,889271476011264E+18)
Gamma(26.000000 + i26.000000) = (1,4103207063357242E+20, -3,1111347801608966E+19)
Gamma(27.000000 + i27.000000) = (-1,20394153792971E+21, -2,100813378567903E+21)
Gamma(28.000000 + i28.000000) = (-2,9772583255514483E+22, 2,9845666640271078E+22)
Gamma(29.000000 + i29.000000) = (6,474322395366405E+23, 4,002110222682179E+23)
Gamma(30.000000 + i30.000000) = (4,982468347052982E+24, -1,3332730971666784E+25)
Gamma(31.000000 + i31.000000) = (-2,701233953257487E+26, -5,3335652308619844E+25)
Gamma(32.000000 + i32.000000) = (-3,5481927258667466E+26, 5,492417944693437E+27)
Gamma(33.000000 + i33.000000) = (1,13499763334942E+29, -3,936060260026878E+27)
Gamma(34.000000 + i34.000000) = (-2,497617380231244E+29, -2,4036706229026682E+30)
Gamma(35.000000 + i35.000000) = (-5,244047476964302E+31, 7,550247174479453E+30)
Gamma(36.000000 + i36.000000) = (1,8326953464053433E+32, 1,1815797667494789E+33)
Gamma(37.000000 + i37.000000) = (2,7496330950708185E+34, -3,7903032739968576E+33)
Gamma(38.000000 + i38.000000) = (-6,153702881069039E+34, -6,593476804299637E+35)
Gamma(39.000000 + i39.000000) = (-1,622188754583588E+37, 3,702408035066972E+35)
Gamma(40.000000 + i40.000000) = (-2,9787072201603815E+37, 4,069596487794187E+38)
Gamma(41.000000 + i41.000000) = (1,0327223226588694E+40, 2,028448840265614E+39)
Gamma(42.000000 + i42.000000) = (9,258591867044077E+40, -2,623834405446389E+41)
Gamma(43.000000 + i43.000000) = (-6,5825696143104E+42, -3,6674407207212435E+42)
Gamma(44.000000 + i44.000000) = (-1,3470021394042014E+44, 1,5970917674147318E+44)
Gamma(45.000000 + i45.000000) = (3,611191294816286E+45, 4,700641025433376E+45)
Gamma(46.000000 + i46.000000) = (1,572050028597055E+47, -6,978410499639601E+46)
Gamma(47.000000 + i47.000000) = (-8,064316457005957E+47, -5,037500759309252E+48)
Gamma(48.000000 + i48.000000) = (-1,5346852327090097E+50, -1,8750204416673013E+49)
Gamma(49.000000 + i49.000000) = (-1,963123049571723E+51, 4,3640542056764855E+51)
Gamma(50.000000 + i50.000000) = (1,112141672863102E+53, 1,0242389193853564E+53)
Gamma(51.000000 + i51.000000) = (4,3124175139534486E+54, -2,2723736598857867E+54)
Gamma(52.000000 + i52.000000) = (-1,9794169465380243E+55, -1,5907071359978385E+56)
Gamma(53.000000 + i53.000000) = (-5,198770735397539E+57, -1,364053254830618E+57)
Gamma(54.000000 + i54.000000) = (-1,120153853109903E+59, 1,4557034579325553E+59)
Gamma(55.000000 + i55.000000) = (3,0502206279504673E+60, 5,6213787579421754E+60)
Gamma(56.000000 + i56.000000) = (2,263901598439912E+62, -1,3869357395249425E+61)
Gamma(57.000000 + i57.000000) = (3,1337184675942048E+63, -7,566798379913671E+63)
Gamma(58.000000 + i58.000000) = (-1,9547779277051327E+65, -2,289059342042218E+65)
Gamma(59.000000 + i59.000000) = (-1,0990589957539216E+67, 2,43674092519427E+66)
Gamma(60.000000 + i60.000000) = (-1,2138821648921022E+68, 4,1070828497360523E+68)
Gamma(61.000000 + i61.000000) = (1,1429060333301634E+70, 1,199617402383565E+70)
Gamma(62.000000 + i62.000000) = (6,356301051435098E+71, -1,4385777048397157E+71)
Gamma(63.000000 + i63.000000) = (8,496122222382356E+72, -2,4629448359526944E+73)
Gamma(64.000000 + i64.000000) = (-6,555666234065589E+74, -8,308825832655404E+74)
Gamma(65.000000 + i65.000000) = (-4,35298630131942E+76, 3,566503620394996E+75)
Gamma(66.000000 + i66.000000) = (-9,093720755985274E+77, 1,5886817012950856E+78)
Gamma(67.000000 + i67.000000) = (3,276704637172729E+79, 7,067540192000104E+79)
Gamma(68.000000 + i68.000000) = (3,3000031716552424E+81, 6,606403155114497E+80)
Gamma(69.000000 + i69.000000) = (1,1027558271573558E+83, -9,8053446601259E+82)
Gamma(70.000000 + i70.000000) = (-4,322638823338879E+83, -6,551055369074648E+84)
Gamma(71.000000 + i71.000000) = (-2,4300476213174805E+86, -1,695901195606269E+86)
Gamma(72.000000 + i72.000000) = (-1,3066738432408389E+88, 3,647419995967721E+87)
Gamma(73.000000 + i73.000000) = (-2,631035116210543E+89, 5,722329713478048E+89)
Gamma(74.000000 + i74.000000) = (1,2168690154152985E+91, 2,7033309001464033E+91)
Gamma(75.000000 + i75.000000) = (1,3482397421660745E+93, 4,28037329878204E+92)
Gamma(76.000000 + i76.000000) = (5,937687180258459E+94, -3,3970662984923895E+94)
Gamma(77.000000 + i77.000000) = (7,877518886977203E+95, -3,258429077036584E+96)
Gamma(78.000000 + i78.000000) = (-8,893580405683355E+97, -1,4068641589865653E+98)
Gamma(79.000000 + i79.000000) = (-8,171120040126758E+99, -1,8182361788970335E+99)
Gamma(80.000000 + i80.000000) = (-3,620672062418723E+101, 2,252383956928435E+101)
Gamma(81.000000 + i81.000000) = (-5,470323981863164E+102, 2,130475015794634E+103)
Gamma(82.000000 + i82.000000) = (5,5003853678128614E+104, 1,0085770500872239E+105)
Gamma(83.000000 + i83.000000) = (5,740274728712029E+106, 1,986132310865518E+106)
Gamma(84.000000 + i84.000000) = (3,002634263016605E+108, -1,245707037705857E+108)
Gamma(85.000000 + i85.000000) = (7,865153067057636E+109, -1,575289481058232E+110)
Gamma(86.000000 + i86.000000) = (-2,2903303039804873E+111, -9,374401780068583E+111)
Gamma(87.000000 + i87.000000) = (-4,291880277570836E+113, -3,196185984412844E+113)
Gamma(88.000000 + i88.000000) = (-2,9995707183640408E+115, 1,1845722184191957E+114)
Gamma(89.000000 + i89.000000) = (-1,2943245318203042E+117, 1,1073023439932073E+117)
Gamma(90.000000 + i90.000000) = (-2,0007723042777158E+118, 9,568042990491017E+118)
Gamma(91.000000 + i91.000000) = (2,4147687416625443E+120, 5,132960890087842E+120)
Gamma(92.000000 + i92.000000) = (2,9270673985442265E+122, 1,5846423875074053E+122)
Gamma(93.000000 + i93.000000) = (1,9583971561168918E+124, -2,5166936071920716E+123)
Gamma(94.000000 + i94.000000) = (8,735240498823332E+125, -7,993043748530299E+125)
Gamma(95.000000 + i95.000000) = (1,6159775690644545E+127, -6,9922194609237035E+127)
Gamma(96.000000 + i96.000000) = (-1,569012926564151E+129, -4,10649302643008E+129)
Gamma(97.000000 + i97.000000) = (-2,2080981691082815E+131, -1,5902953016632018E+131)
Gamma(98.000000 + i98.000000) = (-1,6995881449793942E+133, -8,982249136857376E+131)
Gamma(99.000000 + i99.000000) = (-9,394719018633069E+134, 5,234766160326565E+134)
Gamma(100.000000 + i100.000000) = (-3,3597454530316526E+136, 5,986962556433683E+136)


## Factor

! built in
USING: picomath prettyprint ;
0.1 gamma .  ! 9.513507698668723
2.0 gamma .  ! 1.0
10. gamma .  ! 362880.0


## 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.

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/
;

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.

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/
;

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
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

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
' 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
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


## 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
}

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

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)) }

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

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]
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]


Or equivalently, as a point-free applicative expression:

import Control.Applicative

cof :: [Double]
cof =
[ 76.18009172947146
, -86.50532032941677
, 24.01409824083091
, -1.231739572450155
, 0.001208650973866179
, -0.000005395239384953
]

gammaln :: Double -> Double
gammaln =
((+) . negate . (((-) . (5.5 +)) <*> (((*) . (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]

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.

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

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).

gamma=: !@<:


<: 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:

sbase =: %:@(2p1&%) * %&1x1 ^ ]
scorr =: 1 1r12 1r288 _139r51840 _571r2488320&p.@%
stirlg=: sbase * scorr


Checking against !@<: we can see that this approximation loses accuracy for small arguments

   (,. 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


(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.

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));
}
}
}

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.

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;
}


## jq

Works with: jq version 1.4

#### Taylor Series

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 / . ;

#### Lanczos Approximation

# 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
| . as $x | (($twopi/$x) | sqrt) * ( ($x / (1|exp)) | pow($x)); #### Examples Stirling's method produces poor results, so to save space, the examples contrast the Taylor series and Lanczos methods with built-in tgamma: 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)") Output: $ 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


## Jsish

Translation of: 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!=
*/

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:

@show gamma(1)


using QuadGK
gammaquad(t::Float64) = first(quadgk(x -> x ^ (t - 1) * exp(-x), zero(t), Inf, reltol = 100eps(t)))

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

Library function:

using SpecialFunctions
gamma(1/2) - sqrt(pi)

Output:
2.220446049250313e-16

## Koka

Based on OCaml implementation

import std/num/float64

fun gamma-lanczos(x)
val g = 7.0
// Coefficients used by the GNU Scientific Library
val c = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]
fun ag(z: float64, d: int)
if d == 0 then c[0].unjust + ag(z, 1)
elif d < 8 then c[d].unjust / (z + d.float64) + ag(z, d.inc)
else c[d].unjust / (z + d.float64)
fun gamma(z)
val z' = z - 1.0
val p = z' + g + 0.5
sqrt(2.0 * pi) * pow(p, (z' + 0.5)) * exp(0.0 - p) * ag(z', 0)
gamma(x)

val e = exp(1.0)
fun gamma-stirling(x)
sqrt(2.0 * pi / x) * pow(x / e, x)

fun gamma-stirling2(x')
// Extended Stirling method seen in Abramowitz and Stegun
val d = [1.0/12.0, 1.0/288.0, -139.0/51840.0, -571.0/2488320.0]
fun corr(z, x, n)
if n < d.length - 1 then d[n].unjust / x + corr(z, x*z, n.inc)
else d[n].unjust / x
fun gamma(z)
gamma-stirling(z)*(1.0 + corr(z, z, 0))
gamma(x')

fun mirror(gma, z)
if z > 0.5 then gma(z) else pi / sin(pi * z) / gma(1.0 - z)

fun main()
println("z\tLanczos\t\t\tStirling\t\tStirling2")
for(1, 20) fn(i)
val z = i.float64 / 10.0
println(z.show(1) ++ "\t" ++ mirror(gamma-lanczos, z).show ++ "\t" ++
mirror(gamma-stirling, z).show ++ "\t" ++ mirror(gamma-stirling2, z).show)
for(1, 7) fn(i)
val z = 10.0 * i.float64
println(z.show ++ "\t" ++ gamma-lanczos(z).show ++ "\t" ++
gamma-stirling(z).show ++ "\t" ++ gamma-stirling2(z).show)

Output:
z	Lanczos			Stirling		Stirling2
0.1	9.5135076986687359	10.405084329555955	9.5210418318004439
0.2	4.5908437119988017	5.0739927535371763	4.596862295030256
0.3	2.9915689876875904	3.3503395433773222	2.9984402802949961
0.4	2.218159543757686	2.5270578096699556	2.2277588907113128
0.5	1.7724538509055157	2.0663656770612464	1.7883901437260497
0.6	1.4891922488128184	1.3071588574483559	1.4827753636029286
0.7	1.2980553326475577	1.1590532921139201	1.2950806801024195
0.8	1.1642297137253037	1.0533709684256085	1.1627054102439229
0.9	1.068628702119319	0.97706150787769541	1.0677830813298756
1.0	1.0000000000000002	0.92213700889578909	0.99949946853364036
1.1	0.95135076986687361	0.88348995316870382	0.95103799705518899
1.2	0.91816874239976076	0.85775533539659088	0.91796405783487933
1.3	0.89747069630627774	0.84267825944839203	0.8973312868034562
1.4	0.88726381750307537	0.8367445486370817	0.88716548484542823
1.5	0.88622692545275827	0.83895655252649626	0.88615538430170204
1.6	0.89351534928769061	0.8486932421525738	0.89346184003019224
1.7	0.90863873285329122	0.86562147179388405	0.90859770150945562
1.8	0.93138377098024272	0.8896396352879945	0.93135158986107858
1.9	0.96176583190738729	0.92084272189422933	0.96174006762796482
2.0	1.0000000000000002	0.95950217574449159	0.99997898067003532
10	362880.00000000105	359869.56187410367	362879.99717458693
20	1.2164510040883245e+17	1.2113934233805675e+17	1.2164510037907557e+17
30	8.841761993739658e+30	8.8172365307655063e+30	8.8417619934546387e+30
40	2.0397882081197221e+46	2.0355431612365591e+46	2.0397882081041343e+46
50	6.0828186403425409e+62	6.0726891878763362e+62	6.0828186403274418e+62
60	1.3868311854568534e+80	1.3849063858294502e+80	1.3868311854555093e+80
70	1.7112245242813438e+98	1.7091885781910795e+98	1.711224524280615e+98


## Kotlin

// 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)))
}
}

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


## Lambdatalk

Following Javascript, with Lanczos approximation.

{def gamma.p
{A.new 0.99999999999980993
676.5203681218851
-1259.1392167224028
771.32342877765313
-176.61502916214059
12.507343278686905
-0.13857109526572012
9.9843695780195716e-6
1.5056327351493116e-7
}}
-> gamma.p

{def gamma.rec
{lambda {:x :a :i}
{if {< :i {A.length {gamma.p}}}
then {gamma.rec :x
{+ :a {/ {A.get :i {gamma.p}} {+ :x :i}} }
{+ :i 1}}
else :a
}}}
-> gamma.rec

{def gamma
{lambda {:x}
{if {< :x 0.5}
then {/ {PI}
{* {sin {* {PI} :x}}
{gamma {- 1 :x}}}}
else {let { {:x {- :x 1}}
{:t {+ {- :x 1} 7 0.5}}
} {* {sqrt {* 2 {PI}}}
{pow :t {+ :x 0.5}}
{exp -:t}
{gamma.rec :x {A.first {gamma.p}} 1}}
}}}}
-> gamma

{S.map {lambda {:i}
{div} Γ(:i) = {gamma :i}}
{S.serie -5.5 5.5 0.5}}

Γ(-5.5) = 0.010912654781909836
Γ(-5) = -42755084646679.17
Γ(-4.5) = -0.06001960130050417
Γ(-4) = 267219279041745.34
Γ(-3.5) = 0.27008820585226917
Γ(-3) = -1425169488222640
Γ(-2.5) = -0.9453087204829418
Γ(-2) = 6413262697001885
Γ(-1.5) = 2.363271801207352
Γ(-1) = -25653050788007544
Γ(-0.5) = -3.5449077018110295
Γ(0) = Infinity
Γ(0.5) = 1.7724538509055159
Γ(1) = 0.9999999999999998
Γ(1.5) = 0.8862269254527586
Γ(2) = 1.0000000000000002
Γ(2.5) = 1.3293403881791384
Γ(3) = 2.000000000000001
Γ(3.5) = 3.3233509704478426
Γ(4) = 6.000000000000007
Γ(4.5) = 11.631728396567446
Γ(5) = 23.999999999999996
Γ(5.5) = 52.34277778455358


## 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.

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)
{
# 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;
}

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.

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


## 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)
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
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$  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. GAMMA(17/2); GAMMA(7*I); M := Matrix(2, 3, 'fill' = -3.6); MTM:-gamma(M); Output: 2027025*sqrt(Pi)*(1/256) GAMMA(7*I) Matrix(2, 3, [[.2468571430, .2468571430, .2468571430], [.2468571430, .2468571430, .2468571430]]) ## Mathematica/Wolfram Language This code shows the built-in method, which works for any value (positive, negative and complex numbers). Gamma[x]  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 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 */


## 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


## Modula-3

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.
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


## Nim

The algorithm is a translation of that from the Ada solution. We have added a comparison with the gamma function provided by the “math” module from Nim standard library (which is, in fact, the C “tgamma” function).

import math, strformat

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 - 1
result = A[^1]
for n in countdown(A.high - 1, A.low):
result = result * y + A[n]
result = 1 / result

echo "Our gamma function     Nim gamma function      Difference"
echo "------------------     ------------------      ----------"
for i in 1..10:
let val1 = gamma(i.toFloat / 3)
let val2 = math.gamma(i.toFloat / 3)
echo &"{val1:18.16f}     {val2:18.16f}     {val1 - val2:11.4e}"

Output:
Our gamma function     Nim gamma function      Difference
------------------     ------------------      ----------
2.6789385347077483     2.6789385347077479      4.4409e-16
1.3541179394264005     1.3541179394264005      0.0000e+00
1.0000000000000000     1.0000000000000000      0.0000e+00
0.8929795115692493     0.8929795115692493      0.0000e+00
0.9027452929509336     0.9027452929509336      0.0000e+00
1.0000000000000000     1.0000000000000000      0.0000e+00
1.1906393487589990     1.1906393487589990      0.0000e+00
1.5045754882515399     1.5045754882515558     -1.5987e-14
1.9999999999939684     2.0000000000000000     -6.0316e-12
2.7781584793385732     2.7781584804376647     -1.0991e-09

## 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

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

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

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

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 * ;
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

gamma(x)

### Double-exponential integration

[[+oo],k] means that the function approaches ${\displaystyle +\infty }$ as ${\displaystyle \exp(-kx).}$

Gamma(x)=intnum(t=0,[+oo,1],t^(x-1)/exp(t))

### Romberg integration

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)

### Stirling approximation

Stirling(x)=x--;sqrt(2*Pi*x)*(x/exp(1))^x

## Pascal

A console application in Free Pascal, created with the Lazarus IDE.

Based on the algorithm for ln(Gamma(x)) (x > 0) in Press et al., Numerical Recipes, 3rd edition, pp. 256-7. For x >= 1/2, we simply take the exponential of their value; for x < 1/2 we calculate Gamma(1 - x) and use the reflection formula Gamma(x)*Gamma(1 - x) = pi/sin(pi*x).

program GammaTest;
{$mode objfpc}{$H+}
uses SysUtils;

function Gamma( x : extended) : extended;
const COF : array [0..14] of extended =
(  0.999999999999997092, // may as well include this in the array
57.1562356658629235,
-59.5979603554754912,
14.1360979747417471,
-0.491913816097620199,
0.339946499848118887e-4,
0.465236289270485756e-4,
-0.983744753048795646e-4,
0.158088703224912494e-3,
-0.210264441724104883e-3,
0.217439618115212643e-3,
-0.164318106536763890e-3,
0.844182239838527433e-4,
-0.261908384015814087e-4,
0.368991826595316234e-5);
const
K = 2.5066282746310005;
PI_OVER_K = PI / K;
var
j : integer;
tmp, w, ser : extended;
reflect : boolean;
begin
reflect := (x < 0.5);
if reflect then w := 1.0 - x else w := x;
tmp := w + 5.2421875;
tmp := (w + 0.5)*Ln(tmp) - tmp;
ser := COF[0];
for j := 1 to 14 do ser := ser + COF[j]/(w + j);
try
if reflect then
result := PI_OVER_K * w * Exp(-tmp) / (Sin(PI*x) * ser)
else
result := K * Exp(tmp) * ser / w;
except
raise SysUtils.Exception.CreateFmt(
'Gamma(%g) is undefined or out of floating-point range', [x]);
end;
end;

// Main routine for testing the Gamma function
var
x, k : extended;
begin
WriteLn( 'Is it seamless at x = 1/2 ?');
x := 0.49999999999999;
WriteLn( SysUtils.Format( 'Gamma(%g) = %g', [x, Gamma(x)]));
x := 0.50000000000001;
WriteLn( SysUtils.Format( 'Gamma(%g) = %g', [x, Gamma(x)]));
WriteLn( 'Test a few values:');
WriteLn( SysUtils.Format( 'Gamma(1)   = %g', [Gamma(1)]));
WriteLn( SysUtils.Format( 'Gamma(2)   = %g', [Gamma(2)]));
WriteLn( SysUtils.Format( 'Gamma(3)   = %g', [Gamma(3)]));
WriteLn( SysUtils.Format( 'Gamma(10)  = %g', [Gamma(10)]));
WriteLn( SysUtils.Format( 'Gamma(101) = %g', [Gamma(101)]));
WriteLn( '      100! = 9.33262154439442E157');
WriteLn( SysUtils.Format( 'Gamma(1/2) = %g', [Gamma(0.5)]));
WriteLn( SysUtils.Format( 'Sqrt(pi)   = %g', [Sqrt(PI)]));
WriteLn( SysUtils.Format( 'Gamma(-7/2)       =  %g', [Gamma(-3.5)]));
(*
Note here a bug or misfeature in Lazarus (doesn't occur in Delphi):
Putting (16.0/105.0)*Sqrt(PI) does not give the required precision.
We have to explicitly define the integers as extended floating-point.
*)
k := extended(16.0)/extended(105.0);
WriteLn( SysUtils.Format( ' (16/105)Sqrt(pi) =  %g', [k*Sqrt(PI)]));
WriteLn( SysUtils.Format( 'Gamma(-9/2)       = %g', [Gamma(-4.5)]));
k := extended(32.0)/extended(945.0);
WriteLn( SysUtils.Format( '-(32/945)Sqrt(pi) = %g', [-k*Sqrt(PI)]));
end.

Output:
Is it seamless at x = 1/2 ?
Gamma(0.49999999999999) = 1.77245385090555
Gamma(0.50000000000001) = 1.77245385090548
Test a few values:
Gamma(1)   = 1
Gamma(2)   = 1
Gamma(3)   = 2
Gamma(10)  = 362880
Gamma(101) = 9.33262154439452E157
100! = 9.33262154439442E157
Gamma(1/2) = 1.77245385090552
Sqrt(pi)   = 1.77245385090552
Gamma(-7/2)       =  0.270088205852269
(16/105)Sqrt(pi) =  0.270088205852269
Gamma(-9/2)       = -0.0600196013005043
-(32/945)Sqrt(pi) = -0.0600196013005042


## Perl

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

# Normally would be:  use Math::MPFR
# 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";
}

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

## Phix

with javascript_semantics
sequence c = repeat(0,12)
function spouge_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

function taylor_gamma(atom x)
-- (good for values between 0 and 1, apparently)
constant t = { 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 }
atom y = x-1,
s = t[$] for n=length(t)-1 to 1 by -1 do s = s*y + t[n] end for return 1/s end function function lanczos_gamma(atom z) if z<0.5 then return PI / (sin(PI*z)*lanczos_gamma(1-z)) end if -- use a lanczos approximation: atom x = 0.99999999999980993, t = z + 6.5; sequence p = { 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 } z -= 1 for i=1 to length(p) do x += p[i] / (z + i) end for return sqrt(2*PI) * power(t,z+0.5) * exp(-t) * x end function constant sqPI = sqrt(PI) procedure sq(sequence zm, string fmt="%19.16f") atom {z, mul} = zm atom e = sqPI/mul sequence s = {spouge_gamma(z), taylor_gamma(z), lanczos_gamma(z)}, error = sq_abs(sq_sub(s,e)) string t = join(s,", ",fmt:=fmt)&", " integer bdx = smallest(error,return_index:=true) atom best = s[bdx], p = s[bdx]*mul for i=1 to length(s) do -- (potentially mark >1) if s[i]=best then t[i*22-2..i*22-1] = "*," end if end for string es = sprintf(fmt,e) printf(1,"%5g: %s %s, %19.16f\n",{z,t,es,p*p}) end procedure printf(1," z ------ spouge ----- ----- taylor ------ ----- lanczos ----- ---- expected ----- %19.16f\n",PI) papply({{-3/2,3/4},{-1/2,-1/2},{1/2,1},{1,sqPI},{3/2,2},{2,sqPI},{5/2,4/3},{3,sqPI/2},{7/2,8/15},{4,sqPI/6}},sq) sq({0.001,sqPI/999.4237725},"%19.15f") sq({0.01,sqPI/99.43258512},"%19.16f") sq({0.1,sqPI/9.513507699},"%19.16f") sq({10,sqPI/362880},"%19.12f") sq({100,sqPI/9.332621544e155},"%19.13g") if machine_bits()=64 then sq({150,sqPI/3.808922638e260},"%19.13g") -- (fatal power overflow error on 32 bits) end if  Output: The closest to the expected result for each z (row) is marked with a trailing asterisk. The final column is the value of PI (to 16dp) we would get from that best/starred result.  z ------ spouge ----- ----- taylor ------ ----- lanczos ----- ---- expected ----- 3.1415926535897932 -1.5: 2.3632718012073547*, 2.3632718095606211, 2.3632718012073532, 2.3632718012073547, 3.1415926535897932 -0.5: -3.5449077018110320*, -3.5449077018110306, -3.5449077018110308, -3.5449077018110321, 3.1415926535897932 0.5: 1.7724538509055158, 1.7724538509055160*, 1.7724538509055166, 1.7724538509055160, 3.1415926535897932 1: 0.9999999999999998, 1.0000000000000000*, 1.0000000000000002, 1.0000000000000000, 3.1415926535897932 1.5: 0.8862269254527577, 0.8862269254527580*, 0.8862269254527583, 0.8862269254527580, 3.1415926535897932 2: 0.9999999999999994, 1.0000000000000000*, 1.0000000000000005, 1.0000000000000000, 3.1415926535897932 2.5: 1.3293403881791359, 1.3293403881791365*, 1.3293403881791379, 1.3293403881791370, 3.1415926535897906 3: 1.9999999999999978, 1.9999999999939679, 2.0000000000000016*, 2.0000000000000000, 3.1415926535897981 3.5: 3.3233509704478376, 3.3233509583896768, 3.3233509704478456*, 3.3233509704478426, 3.1415926535897990 4: 5.9999999999999884, 5.9999914100724727, 6.0000000000000063*, 6.0000000000000000, 3.1415926535897998 0.001: 999.423772484595421*, 999.423772484595404, 999.423772484595254, 999.423772500000000, 3.1415926534929476 0.01: 99.4325851191505990, 99.4325851191506035*, 99.4325851191505828, 99.4325851200000000, 3.1415926535361195 0.1: 9.5135076986687313, 9.5135076986687318*, 9.5135076986687300, 9.5135076990000000, 3.1415926533710076 10: 362879.999999996094, 0.000000029163, 362880.000000000725*, 362880.000000000000, 3.1415926535898058 100: 9.332621544394e+155, 7.510232292979e-39, 9.332621544394e+155*, 9.332621544e+155, 3.1415926538549222 150: 3.80892263763e+260*, 5.128102530869e-44, 3.80892263763e+260, 3.808922638e+260, 3.1415926529801383  ### mpfr version Above translated to mpfr, but spouge only since there's not much point transferring inherent inaccuracies in taylor/lanczos constants, and compared against the builtin. Library: Phix/mpfr without javascript_semantics -- (no mpfr_exp(), mpfr_gamma() in pwa/p2js) constant dp = 30 string fmt = "%5s: %33s, %33s, %32s\n" requires("1.0.2") -- (mpfr_get_fixed(maxlen), mpfr_gamma) include mpfr.e mpfr_set_default_precision(-87) -- 87 decimal places. sequence mc = mpfr_inits(40) function mpfr_spouge_gamma(mpfr z) mpfr accm = mc[1] if mpfr_cmp_si(accm,0)=0 then -- mc[1] := sqrt(2*PI) mpfr_const_pi(accm) mpfr_mul_si(accm,accm,2) mpfr_sqrt(accm,accm) -- k1_factrl = (k - 1)!*(-1)^k with 0!==1 mpfr k1_factrl = mpfr_init(1), tmk = mpfr_init(), p = mpfr_init() for k=2 to length(mc) do -- mc[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl mpfr_set_si(tmk,length(mc)+1-k) mpfr_exp(mc[k],tmk) mpfr_set_d(p,k-1.5) mpfr_pow(p,tmk,p) mpfr_div(p,p,k1_factrl) mpfr_mul(mc[k],mc[k],p) -- k1_factrl *= -(k-1) mpfr_mul_si(k1_factrl,k1_factrl,-(k-1)) end for end if accm = mpfr_init_set(accm) for k=2 to length(mc) do -- accm += mc[k]/(z+k-1) mpfr ck = mpfr_init_set(mc[k]), zk = mpfr_init_set(z) mpfr_add_si(zk,zk,k-1) mpfr_div(ck,ck,zk) mpfr_add(accm,accm,ck) end for -- atom zc = z+length(mc) -- accm *= exp(-zc)*power(zc,z+0.5) -- Gamma(z+1) mpfr p = mpfr_init_set(z), ez = mpfr_init(), zh = mpfr_init(0.5) mpfr_add_si(p,p,length(mc)) mpfr_neg(ez,p) mpfr_exp(ez,ez) mpfr_add(zh,zh,z) mpfr_pow(p,p,zh) mpfr_mul(accm,accm,ez) mpfr_mul(accm,accm,p) -- return accm/z mpfr_div(accm,accm,z) return accm end function constant mPI = mpfr_init(), mqPI = mpfr_init() mpfr_const_pi(mPI) string pistr = mpfr_get_fixed(mPI,dp) mpfr_sqrt(mqPI,mPI) function makempfr(object x) mpfr res if string(x) then x = split(x,'/') res = mpfr_init(x[1]) mpfr_div_si(res,res,to_integer(x[2])) elsif sequence(x) then {mpfr x1, object x2} = x res = mpfr_init_set(x1) if string(x2) then mpfr d = mpfr_init(x2) mpfr_div(res,res,d) else mpfr_div_d(res,res,x2) end if else res = mpfr_init(x) end if return res end function procedure mq(sequence zm, integer d=dp) mpfr {z, mul} = apply(zm,makempfr) mpfr s = mpfr_spouge_gamma(z) string t = mpfr_get_fixed(s,d,10,maxlen:=dp+2) mpfr e = mpfr_init() mpfr_gamma(e,z) mpfr p = mpfr_init_set(s) mpfr_mul(p,p,mul) mpfr_mul(p,p,p) string zs = mpfr_get_fixed(z,3), es = mpfr_get_fixed(e,d,10,maxlen:=dp+2), ps = mpfr_get_fixed(p,dp,10) printf(1,fmt,{zs,t,es,ps}) end procedure printf(1," z %s %s %s\n",{pad(" spouge ",dp+2,"BOTH",'-'),pad(" expected ",dp+2,"BOTH",'-'),pistr}) papply({{-1.5,0.75},{-0.5,-0.5},{0.5,1},{1,{mqPI,1}},{1.5,2},{2,{mqPI,1}},{2.5,"4/3"},{3,{mqPI,2}},{3.5,"8/15"},{4,{mqPI,6}}},mq) mq({"1/1000",{mqPI,"999.4237724845954661149822012996"}},28) mq({"1/100",{mqPI,"99.43258511915060371353298887051"}},29) mq({"1/10",{mqPI,"9.513507698668731836292487177265"}},30) mq({10,{mqPI,362880}},25) mq({100,{mqPI,"9.332621544394415268169923885627e155"}},0) mq({150,{mqPI,"3.80892263763056972698595524350735e260"}},0)  Output:  z ------------ spouge ------------ ----------- expected ----------- 3.141592653589793238462643383279 -1.5: 2.363271801207354703064223311121, 2.363271801207354703064223311121, 3.141592653589793238462643383279 -0.5: -3.544907701811032054596334966e0, -3.544907701811032054596334966e0, 3.141592653589793238462643383279 0.5: 1.772453850905516027298167483341, 1.772453850905516027298167483341, 3.141592653589793238462643383279 1: 1.000000000000000000000000000000, 1, 3.141592653589793238462643383279 1.5: 0.886226925452758013649083741670, 0.886226925452758013649083741670, 3.141592653589793238462643383279 2: 1.000000000000000000000000000000, 1, 3.141592653589793238462643383279 2.5: 1.329340388179137020473625612505, 1.329340388179137020473625612505, 3.141592653589793238462643383279 3: 2.000000000000000000000000000000, 2, 3.141592653589793238462643383279 3.5: 3.323350970447842551184064031264, 3.323350970447842551184064031264, 3.141592653589793238462643383279 4: 6.000000000000000000000000000000, 6, 3.141592653589793238462643383279 0.001: 999.4237724845954661149822012996, 999.4237724845954661149822012996, 3.141592653589793238462643383279 0.009: 99.43258511915060371353298887051, 99.43258511915060371353298887051, 3.141592653589793238462643383279 0.1: 9.513507698668731836292487177265, 9.513507698668731836292487177265, 3.141592653589793238462643383279 10: 362880.0000000000000000000000000, 362880, 3.141592653589793238462643383279 100: 9.33262154439441526816992388e155, 9.33262154439441526816992388e155, 3.141592653589793238462643383279 150: 3.80892263763056972698595524e260, 3.80892263763056972698595524e260, 3.141592653589793238462643383279  ## Phixmonti 0.577215664901 var gamma -0.65587807152056 var coeff -0.042002635033944 var quad 0.16653861138228 var qui -0.042197734555571 var theSet def recigamma var z /# n -- n #/ z 6 power theSet * z 5 power qui * z 4 power quad * z 3 power coeff * z 2 power gamma * z + + + + + enddef /# without var def recigamma dup 6 power theSet * swap dup 5 power qui * swap dup 4 power quad * swap dup 3 power coeff * swap dup 2 power gamma * swap + + + + + enddef #/ def gammafunc /# n -- n #/ dup 1 == if else dup abs 0.5 <= if recigamma 1 swap / else dup 1 - gammafunc swap 1 - * endif endif enddef 0.1 2.1 .1 3 tolist for dup print " = " print gammafunc print nl endfor ## PicoLisp Translation of: Ada (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) ) ) 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 /* 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; 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/ 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)}

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


## Prolog

This version matches Wolfram Alpha to within a few digits at the end, so the last few digits are a bit off. There's an early check to stop evaluating coefficients once the desired accuracy is reached. Removing this check does not improve accuracy vs. Wolfram Alpha.

gamma_coefficients(
[ 1.00000000000000000000000,  0.57721566490153286060651, -0.65587807152025388107701,
-0.04200263503409523552900,  0.16653861138229148950170, -0.04219773455554433674820,
-0.00962197152787697356211,  0.00721894324666309954239, -0.00116516759185906511211,
-0.00021524167411495097281,  0.00012805028238811618615, -0.00002013485478078823865,
-0.00000125049348214267065,  0.00000113302723198169588, -0.00000020563384169776071,
0.00000000611609510448141,  0.00000000500200764446922, -0.00000000118127457048702,
0.00000000010434267116911,  0.00000000000778226343990, -0.00000000000369680561864,
0.00000000000051003702874, -0.00000000000002058326053, -0.00000000000000534812253,
0.00000000000000122677862, -0.00000000000000011812593,  0.00000000000000000118669,
0.00000000000000000141238, -0.00000000000000000022987,  0.00000000000000000001714
]).

tolerance(1e-17).

gamma(X, _) :- X =< 0.0, !, fail.
gamma(X, Y) :-
X < 1.0, small_gamma(X, Y), !.
gamma(1, 1) :- !.
gamma(1.0, 1) :- !.
gamma(X, Y) :-
X1 is X - 1,
gamma(X1, Y1),
Y is X1 * Y1.

small_gamma(X, Y) :-
gamma_coefficients(Cs),
recip_gamma(X, 1.0, Cs, 1.0, 0.0, Y0),
Y is 1 / Y0.

recip_gamma(_, _, [], _, Y, Y) :- !.
recip_gamma(_, _, [], X0, X1, Y) :- tolerance(Tol), abs(X1 - X0) < Tol, Y = X1, !. % early exit
recip_gamma(X, PrevPow, [C|Cs], _, X1, Y) :-
Power is PrevPow * X,
X2 is X1 + C*Power,
recip_gamma(X, Power, Cs, X1, X2, Y).

Output:
% see how close gamma(0.5) is to the square root of pi.
?- gamma(0.5,X), Y is sqrt(pi), Err is abs(X - Y).
X = 1.772453850905516,
Y = 1.7724538509055159,
Err = 2.220446049250313e-16.

?- gamma(1.5,X).
X = 0.886226925452758.

?- gamma(4.9,X).
X = 20.667385961857857.

?- gamma(5,X).
X = 24.

?- gamma(5.01,X).
X = 24.364473447872836.

?- gamma(6.9,X).
X = 597.4941281573107.

?- gamma(6.95,X).
X = 655.7662628554252.

?- gamma(7,X).
X = 720.

% 100!
?- gamma(101.0,X).
X = 9.33262154439441e+157.

% Note when passed integer, gamma(101) returns full big int precision
?- gamma(101,X).
X = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000.

?- gamma(100.98,X).
X = 8.510619261391532e+157.


## PureBasic

Below is PureBasic code for:

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

_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)

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
'''Gamma function'''

from functools import reduce

# 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)

# 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
]

# TEST ----------------------------------------------------
# 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))
)

# GENERIC -------------------------------------------------

# enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
'''Integer enumeration from m to n.'''
return lambda n: list(range(m, 1 + n))

# FORMATTING -------------------------------------------------

# fTable :: String -> (a -> String) ->
#                     (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
)

# MAIN ---
if __name__ == '__main__':
main()

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
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]
for (i in 1:8) {
x <- x+p[i+1]/(z+i)
}
tt <- z + g + 0.5
sqrt(2*pi) * tt^(z+0.5) * exp(-tt) * x
}
}
}

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)))
}
}

# 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))

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
(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)


## Raku

(formerly Perl 6)

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;  Output: 2.67893853470775 1.3541179394264 1 0.892979511569248 0.902745292950934 1 1.190639348759 1.50457548825155 2 2.77815848043766 ## REXX ### Taylor series, 80-digit coefficients This version uses a Taylor series with 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 . Already on modest values of x (say > 3) you loose precision. See below for a solution. /*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 #.10 = -0.00021524167411495097281572996305364780647824192337833875035026748908563946371678 #.11 = 0.00012805028238811618615319862632816432339489209969367721490054583804120355204347 #.12 = -0.00002013485478078823865568939142102181838229483329797911526116267090822918618897 #.13 = -0.00000125049348214267065734535947383309224232265562115395981534992315749121245561 #.14 = 0.00000113302723198169588237412962033074494332400483862107565429550539546040842730 #.15 = -0.00000020563384169776071034501541300205728365125790262933794534683172533245680371 #.16 = 0.00000000611609510448141581786249868285534286727586571971232086732402927723507435 #.17 = 0.00000000500200764446922293005566504805999130304461274249448171895337887737472132 #.18 = -0.00000000118127457048702014458812656543650557773875950493258759096189263169643391 #.19 = 0.00000000010434267116911005104915403323122501914007098231258121210871073927347588 #.20 = 0.00000000000778226343990507125404993731136077722606808618139293881943550732692987 #.21 = -0.00000000000369680561864220570818781587808576623657096345136099513648454655443000 #.22 = 0.00000000000051003702874544759790154813228632318027268860697076321173501048565735 #.23 = -0.00000000000002058326053566506783222429544855237419746091080810147188058196444349 #.24 = -0.00000000000000534812253942301798237001731872793994898971547812068211168095493211 #.25 = 0.00000000000000122677862823826079015889384662242242816545575045632136601135999606 #.26 = -0.00000000000000011812593016974587695137645868422978312115572918048478798375081233 #.27 = 0.00000000000000000118669225475160033257977724292867407108849407966482711074006109 #.28 = 0.00000000000000000141238065531803178155580394756670903708635075033452562564122263 #.29 = -0.00000000000000000022987456844353702065924785806336992602845059314190367014889830 #.30 = 0.00000000000000000001714406321927337433383963370267257066812656062517433174649858 #.31 = 0.00000000000000000000013373517304936931148647813951222680228750594717618947898583 #.32 = -0.00000000000000000000020542335517666727893250253513557337960820379352387364127301 #.33 = 0.00000000000000000000002736030048607999844831509904330982014865311695836363370165 #.34 = -0.00000000000000000000000173235644591051663905742845156477979906974910879499841377 #.35 = -0.00000000000000000000000002360619024499287287343450735427531007926413552145370486 #.36 = 0.00000000000000000000000001864982941717294430718413161878666898945868429073668232 #.37 = -0.00000000000000000000000000221809562420719720439971691362686037973177950067567580 #.38 = 0.00000000000000000000000000012977819749479936688244144863305941656194998646391332 #.39 = 0.00000000000000000000000000000118069747496652840622274541550997151855968463784158 #.40 = -0.00000000000000000000000000000112458434927708809029365467426143951211941179558301 #.41 = 0.00000000000000000000000000000012770851751408662039902066777511246477487720656005 #.42 = -0.00000000000000000000000000000000739145116961514082346128933010855282371056899245 #.43 = 0.00000000000000000000000000000000001134750257554215760954165259469306393008612196 #.44 = 0.00000000000000000000000000000000004639134641058722029944804907952228463057968680 #.45 = -0.00000000000000000000000000000000000534733681843919887507741819670989332090488591 #.46 = 0.00000000000000000000000000000000000032079959236133526228612372790827943910901464 #.47 = -0.00000000000000000000000000000000000000444582973655075688210159035212464363740144 #.48 = -0.00000000000000000000000000000000000000131117451888198871290105849438992219023663 #.49 = 0.00000000000000000000000000000000000000016470333525438138868182593279063941453996 #.50 = -0.00000000000000000000000000000000000000001056233178503581218600561071538285049997 #.51 = 0.00000000000000000000000000000000000000000026784429826430494783549630718908519485 #.52 = 0.00000000000000000000000000000000000000000002424715494851782689673032938370921241 #=52; do k=# by -1 for # sum=sum*xm + #.k end /*k*/ return 1/sum  output when using the input of: 0.5 gamma(0.5) = 1.77245385090551602729816748334114518279754945612238712821380509003635917689651032047826593  Note that: Γ(½) = ${\displaystyle \pi }$ = 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). ### Spouge's approximation, using 87 digit coefficients Translation of: Phix Translation of: C This REXX version is a translation of Phix but with more (decimal digits) precision and more steps. Many of the "normal" high-level mathematical functions aren't available in REXX, so some of them (RYO) are included here. /*REXX program calculates the gamma function using Spouge's approximation with 87 digits*/ e=2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138 numeric digits length(e) - length(.) /*use the number of decimal digits in E*/ c.= 0 # = 40 /*#: the number of steps in GAMMA func*/ call sq gamma(-3/2), 3/4 call sq gamma(-1/2), -1/2 call sq gamma( 1/2), 1 call si gamma( 1 ) call sq gamma( 3/2), 2 call si gamma( 2 ) call sq gamma( 5/2), 4/3 call si gamma( 3 ) call sq gamma( 7/2), 8/15 call si gamma( 4 ) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gamma: procedure expose c. e #; parse arg z; #p= # + 1 accm = c.1 if accm==0 then do; accm= sqrt( 2*pi() ) c.1 = accm kfact = 1 do k=2 to # c.k= exp(#p-k) * pow(#p-k, k-1.5) / kfact kfact = kfact * -(k-1) end /*k*/ end do j=2 to #; accm = accm + c.j / (z+j-1) end /*k*/ return (accm * exp(-(z+#)) * pow(z+#, z+0.5) ) / z /*──────────────────────────────────────────────────────────────────────────────────────*/ pi: return 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348 fmt: parse arg n,p,a; _= format(n,p,a); L= length(_); return left( strip0(_), L) isInt: return datatype(arg(1), 'W') /*is the argument an integer? */ sq: procedure expose #; parse arg x,mu; say fmt(x,9,#) fmt((x*mu)**2,9,#); return si: procedure expose #; parse arg x; say fmt(x,9,#); return strip0: procedure; arg _; if pos(., _)\==0 then _= strip(strip(_,'T',0),'T',.); return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ exp: procedure expose e; arg x; ix= x%1; if abs(x-ix)>.5 then ix=ix+sign(x); x= x-ix; z=1 _=1; w=1; do j=1; _= _*x/j; z= (z+_)/1; if z==w then leave; w=z end /*j*/; if z\==0 then z= e**ix * z; return z /*──────────────────────────────────────────────────────────────────────────────────────*/ ln: procedure; parse arg x; call e; ig= x>1.5; is= 1-2*(ig\==1); ii= 0; xx= x do while ig & xx>1.5 | \ig & xx<.5; _=e do k=-1; iz=xx*_**-is; if k>=0&(ig&iz<1|\ig&iz>.5) then leave; _=_*_; izz=iz; end xx= izz; ii= ii+is*2**k; end /*while*/; x= x*e**-ii-1; z=0; _= -1; p=z do k=1; _=-_*x; z=z+_/k; if z=p then leave; p=z; end; /*k*/; return z+ii /*──────────────────────────────────────────────────────────────────────────────────────*/ pow: procedure; parse arg x,y; if y=0 then return 1; if x=0 then return 0 if isInt(y) then return x**y; if isInt(1/y) then return root(x, 1/y) if abs(y//1)=.5 then return sqrt(x)**sign(y)*x**(y%1); return exp( y*ln(x) ) /*──────────────────────────────────────────────────────────────────────────────────────*/ root: procedure; parse arg x 1 ox,y 1 oy; if x=0 | y=1 then return x/1 if \isInt(y) then return$pow(x, 1/y)
if y==2  then return sqrt(x); if y==-2  then return 1/sqrt(x); return rooti(x,y)/1
/*──────────────────────────────────────────────────────────────────────────────────────*/
rooti:  x=abs(x); y=abs(y); a= digits() + 5;  m= y-1;  d= 5
parse value format(x,2,1,,0) 'E0'  with  ? 'E' _ .;   g= (?/y'E'_ % y) + (x>1)
do until d==a;   d=min(d+d, a);  numeric digits d;  o=0
do until o=g;  o=g;  g= format((m*g**y+x)/y/g**m,,d-2);  end;  end
_= g*sign(ox);  if oy<0  then _= 1/_;                                     return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x;  if x=0  then return 0;  d=digits();  numeric digits;  h=d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g *.5'e'_ %2
do j=0  while h>9;        m.j=h;                 h=h%2+1;          end  /*j*/
do k=j+5  to 0  by -1;    numeric digits m.k;    g=(g+x/g)*.5;     end  /*k*/
numeric digits d;     return g/1

output   when using the default input:
        2.3632718012073547030642233111215269103967         3.1415926535897932384626433832795028841972
-3.5449077018110320545963349666822903655951         3.1415926535897932384626433832795028841972
1.7724538509055160272981674833411451827975         3.1415926535897932384626433832795028841972
1
0.8862269254527580136490837416705725913988         3.1415926535897932384626433832795028841972
1
1.3293403881791370204736256125058588870982         3.1415926535897932384626433832795028841972
2
3.3233509704478425511840640312646472177454         3.1415926535897932384626433832795028841972
6


This program also has some minor issues. The number of steps, as well as e and pi, are hardcoded. This limits the precision to about 80 digits. See below for a solution.

### A generic Gamma function

Below program calculates the Gamma function for any (real) element x (except integers < 1) in the specified precision (but over 100 digits it quickly becomes slow). Improvements over above programs:

Closed formulas added for all half integers and positive integers (much faster than series). Argument reduction (mapping to interval 0.5...1.5) added to the Lanczos solution. Results are now also for larger values of x accurate to about 60 digits. Dynamic determination of digits and iterations added to the Spouge solution. It's now slightly faster on low x values and gives correct results using > 87 digits. Depending on the parameters, the program selects the optimal method for calculating Gamma.

As before, all needed functions are included in this version, so the entry is quite long... Scroll down to see the extra routines.

parse version version; say version; glob. = ''
say 'Gamma function in arbitrary precision'
say
say '(Half)integers formulas'
w = '-99.5 -10.5 -5.5 -2.5 -1.5 -0.5 0.5 1 1.5 2 2.5 5 5.5 10 10.5 99 99.5'
numeric digits 100
do i = 1 to words(w)
x = word(w,i); call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Formulas' format(x,4,1) r '('e 'seconds)'
end
say
say 'Lanczos (max 60 digits) vs Spouge (no limit) vs Stirling (no limit) approximation'
w = '-12.8 -6.4 -3.2 -1.6 -0.8 -0.4 -0.2 -0.1 0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8'
do i = 1 to words(w)
x = word(w,i)
numeric digits 60
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
numeric digits 61
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Spouge  ' format(x,4,1) r '('e 'seconds)'
if x > 0 then do
call time('r'); r = Stirling(x); e = format(time('e'),,3)
say 'Stirling' format(x,4,1) r '('e 'seconds)'
end
end
say
say 'Same for a bigger number'
w = '-99.9 99.9'
do i = 1 to words(w)
x = word(w,i)
numeric digits 60
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
numeric digits 100
call time('r'); r = Gamma(x); e = format(time('e'),,3)
say 'Spouge  ' format(x,4,1) r '('e 'seconds)'
if x > 0 then do
call time('r'); r = Stirling(x); e = format(time('e'),,3)
say 'Stirling' format(x,4,1) r '('e 'seconds)'
end
end
exit

Gamma:
/* Gamma */
procedure expose glob.
arg x
/* Validity */
if x < 1 & Whole(x) then
return 'X'
/* Formulas for negative and positive (half)integers */
if x < 0 then do
if Half(x) then do
numeric digits Digits()+2
i = Abs(Floor(x)); y = (-1)**i*2**(2*i)*Fact(i)*Sqrt(Pi())/Fact(2*i)
numeric digits Digits()-2
return y+0
end
end
if x > 0 then do
if Whole(x) then
return Fact(x-1)
if Half(x) then do
numeric digits Digits()+2
i = Floor(x); y = Fact(2*i)*Sqrt(Pi())/(2**(2*i)*Fact(i))
numeric digits Digits()-2
return y+0
end
end
p = Digits()
if p < 61 then do
/* Lanczos with predefined coefficients */
/* Map negative x to positive x */
if x < 0 then
return Pi()/(Gamma(1-x)*Sin(Pi()*x))
/* Argument reduction to interval (0.5,1.5) */
numeric digits p+2
c = Trunc(x); x = x-c
if x < 0.5 then do
x = x+1; c = c-1
end
/* Series coefficients 1/Gamma(x) in 80 digits Fransen & Wrigge */
c.1 =  1.00000000000000000000000000000000000000000000000000000000000000000000000000000000
c.2 =  0.57721566490153286060651209008240243104215933593992359880576723488486772677766467
c.3 = -0.65587807152025388107701951514539048127976638047858434729236244568387083835372210
c.4 = -0.04200263503409523552900393487542981871139450040110609352206581297618009687597599
c.5 =  0.16653861138229148950170079510210523571778150224717434057046890317899386605647425
c.6 = -0.04219773455554433674820830128918739130165268418982248637691887327545901118558900
c.7 = -0.00962197152787697356211492167234819897536294225211300210513886262731167351446074
c.8 =  0.00721894324666309954239501034044657270990480088023831800109478117362259497415854
c.9 = -0.00116516759185906511211397108401838866680933379538405744340750527562002584816653
c.10 = -0.00021524167411495097281572996305364780647824192337833875035026748908563946371678
c.11 =  0.00012805028238811618615319862632816432339489209969367721490054583804120355204347
c.12 = -0.00002013485478078823865568939142102181838229483329797911526116267090822918618897
c.13 = -0.00000125049348214267065734535947383309224232265562115395981534992315749121245561
c.14 =  0.00000113302723198169588237412962033074494332400483862107565429550539546040842730
c.15 = -0.00000020563384169776071034501541300205728365125790262933794534683172533245680371
c.16 =  0.00000000611609510448141581786249868285534286727586571971232086732402927723507435
c.17 =  0.00000000500200764446922293005566504805999130304461274249448171895337887737472132
c.18 = -0.00000000118127457048702014458812656543650557773875950493258759096189263169643391
c.19 =  0.00000000010434267116911005104915403323122501914007098231258121210871073927347588
c.20 =  0.00000000000778226343990507125404993731136077722606808618139293881943550732692987
c.21 = -0.00000000000369680561864220570818781587808576623657096345136099513648454655443000
c.22 =  0.00000000000051003702874544759790154813228632318027268860697076321173501048565735
c.23 = -0.00000000000002058326053566506783222429544855237419746091080810147188058196444349
c.24 = -0.00000000000000534812253942301798237001731872793994898971547812068211168095493211
c.25 =  0.00000000000000122677862823826079015889384662242242816545575045632136601135999606
c.26 = -0.00000000000000011812593016974587695137645868422978312115572918048478798375081233
c.27 =  0.00000000000000000118669225475160033257977724292867407108849407966482711074006109
c.28 =  0.00000000000000000141238065531803178155580394756670903708635075033452562564122263
c.29 = -0.00000000000000000022987456844353702065924785806336992602845059314190367014889830
c.30 =  0.00000000000000000001714406321927337433383963370267257066812656062517433174649858
c.31 =  0.00000000000000000000013373517304936931148647813951222680228750594717618947898583
c.32 = -0.00000000000000000000020542335517666727893250253513557337960820379352387364127301
c.33 =  0.00000000000000000000002736030048607999844831509904330982014865311695836363370165
c.34 = -0.00000000000000000000000173235644591051663905742845156477979906974910879499841377
c.35 = -0.00000000000000000000000002360619024499287287343450735427531007926413552145370486
c.36 =  0.00000000000000000000000001864982941717294430718413161878666898945868429073668232
c.37 = -0.00000000000000000000000000221809562420719720439971691362686037973177950067567580
c.38 =  0.00000000000000000000000000012977819749479936688244144863305941656194998646391332
c.39 =  0.00000000000000000000000000000118069747496652840622274541550997151855968463784158
c.40 = -0.00000000000000000000000000000112458434927708809029365467426143951211941179558301
c.41 =  0.00000000000000000000000000000012770851751408662039902066777511246477487720656005
c.42 = -0.00000000000000000000000000000000739145116961514082346128933010855282371056899245
c.43 =  0.00000000000000000000000000000000001134750257554215760954165259469306393008612196
c.44 =  0.00000000000000000000000000000000004639134641058722029944804907952228463057968680
c.45 = -0.00000000000000000000000000000000000534733681843919887507741819670989332090488591
c.46 =  0.00000000000000000000000000000000000032079959236133526228612372790827943910901464
c.47 = -0.00000000000000000000000000000000000000444582973655075688210159035212464363740144
c.48 = -0.00000000000000000000000000000000000000131117451888198871290105849438992219023663
c.49 =  0.00000000000000000000000000000000000000016470333525438138868182593279063941453996
c.50 = -0.00000000000000000000000000000000000000001056233178503581218600561071538285049997
c.51 =  0.00000000000000000000000000000000000000000026784429826430494783549630718908519485
c.52 =  0.00000000000000000000000000000000000000000002424715494851782689673032938370921241
/* Series expansion */
x = x-1; s = 0
do k = 52 by -1 to 1
s = s*x+c.k
end
y = 1/s
/* Undo reduction */
if c = -1 then
y = y/x
else do
do i = 1 to c
y = (x+i)*y
end
end
end
else do
x = x-1
/* Spouge */
/* Estimate digits and iterations */
q = Floor(p*1.5); a = Floor(p*1.3)
numeric digits q
/* Series */
s = 0
do k = 1 to a-1
s = s+((-1)**(k-1)*Power(a-k,k-0.5)*Exp(a-k))/(Fact(k-1)*(x+k))
end
s = s+Sqrt(2*Pi()); y = Power(x+a,x+0.5)*Exp(-a-x)*s
end
/* Normalize */
numeric digits p
return y+0

Stirling:
/* Sterling */
procedure expose glob.
arg x
return sqrt(2*pi()/x) * power(x/e(),x)

E:
/* Euler number */
procedure expose glob.
p = Digits()
/* In memory? */
if glob.e.p <> '' then
return glob.e.p
if p < 101 then
/* Fast value */
glob.e.p = 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516643+0
else do
numeric digits Digits()+2
/* Taylor series */
y = 2; t = 1; v = y
do n = 2
t = t/n; y = y+t
if y = v then
leave
v = y
end
numeric digits Digits()-2
glob.e.p = y+0
end
return glob.e.p

Exp:
/* Exponential e^x */
procedure expose glob.
arg x
numeric digits Digits()+2
/* Fast values */
if Whole(x) then
return E()**x
/* Argument reduction */
i = x%1
if Abs(x-i) > 0.5 then
i = i+Sign(x)
/* Taylor series */
x = x-i; y = 1; t = 1; v = y
do n = 1
t = (t*x)/n; y = y+t
if y = v then
leave
v = y
end
/* Inverse reduction */
y = y*e()**i
numeric digits Digits()-2
return y+0

Fact:
/* Factorial n! */
procedure expose glob.
arg x
/* Validity */
if \ Whole(x) then
return 'X'
if x < 0 then
return 'X'
/* Current in memory? */
if glob.fact.x <> '' then
return glob.fact.x
w = x-1
/* Previous in memory? */
if glob.fact.w = '' then do
/* Loop cf definition */
y = 1
do n = 2 to x
y = y*n
end
glob.fact.x = y
end
else
/* Multiply */
glob.fact.x = glob.fact.w*x
return glob.fact.x

Floor:
/* Floor */
procedure expose glob.
arg x
/* Formula */
if Whole(x) then
return x
else
return Trunc(x)-(x<0)

Frac:
/* Fractional part */
procedure expose glob.
arg x
/* Formula */
return x-x%1

Half:
/* Is a number half integer? */
procedure expose glob.
arg x
/* Formula */
return (Frac(Abs(x))=0.5)

Ln:
/* Natural logarithm base e */
procedure expose glob.
arg x
/* Validity */
if x <= 0 then
return 'X'
/* Fast values */
if x = 1 then
return 0
p = Digits()
/* In memory? */
if glob.ln.x.p <> '' then
return glob.ln.x.p
/* Precalculated values */
if x = 2 & p < 101 then do
glob.ln.x.p = Ln2()
return glob.ln.x.p
end
if x = 4 & p < 101 then do
glob.ln.x.p = Ln4()
return glob.ln.x.p
end
if x = 8 & p < 101 then do
glob.ln.x.p = Ln8()
return glob.ln.x.p
end
if x = 10 & p < 101 then do
glob.ln.x.p = Ln10()
return glob.ln.x.p
end
numeric digits p+2
/* Argument reduction */
z = x; i = 0; e = 1/E()
if z < 0.5 then do
y = 1/z
do while y > 1.5
y = y*e; i = i-1
end
z = 1/y
end
if z > 1.5 then do
do while z > 1.5
z = z*e; i = i+1
end
end
/* Taylor series */
q = (z-1)/(z+1); f = q; y = q; v = q; q = q*q
do n = 3 by 2
f = f*q; y = y+f/n
if y = v then
leave
v = y
end
numeric digits p
/* Inverse reduction */
glob.ln.x.p = 2*y+i
return glob.ln.x.p

Power:
/* Power function x^y */
procedure expose glob.
arg x,y
/* Validity */
if x < 0 then
return 'X'
/* Fast values */
if x = 0 then
return 0
if y = 0 then
return 1
/* Fast formula */
if Whole(y) then
return x**y
/* Formulas */
if Abs(y//1) = 0.5 then
return Sqrt(x)**Sign(y)*x**(y%1)
else
return Exp(y*Ln(x))

Sin:
/* Sine */
procedure expose glob.
arg x
numeric digits Digits()+2
/* Argument reduction */
u = Pi(); x = x//(2*u)
if Abs(x) > u then
x = x-Sign(x)*2*u
/* Taylor series */
t = x; y = x; x = x*x; v = y
do n = 2 by 2
t = -t*x/(n*(n+1)); y = y+t
if y = v then
leave
v = y
end
numeric digits Digits()-2
return y+0

Sqrt:
/* Square root x^(1/2) */
procedure expose glob.
arg x
/* Validity */
if x < 0 then
return 'X'
/* Fast values */
if x = 0 then
return 0
if x = 1 then
return 1
p = Digits()
/* Predefined values */
if x = 2 & p < 101 then
return Sqrt2()
if x = 3 & p < 101 then
return Sqrt3()
if x = 5 & p < 101 then
return Sqrt5()
numeric digits p+2
/* Argument reduction to [0,100) */
i = Xpon(x); i = (i-(i<0))%2; x = x/100**i
/* First guess 1 digit accurate */
t = '2.5 6.5 12.5 20.5 30.5 42.5 56.5 72.5 90.5 100'
do y = 1 until word(t,y) > x
end
/* Dynamic precision */
d = Digits()
do n = 1 while d > 2
d.n = d; d = d%2+1
end
d.n = 2
/* Method Heron */
do k = n to 1 by -1
numeric digits d.k
y = (y+x/y)*0.5
end
numeric digits p
return y*10**i

Whole:
/* Is a number integer? */
procedure expose glob.
arg x
/* Formula */
return Datatype(x,'w')

Xpon:
/* Exponent */
procedure expose glob.
arg x
/* Formula */
if x = 0 then
return 0
else
return Right(x*1E+99999,6)-99999

Ln2:
/* Natural log of 2 */
procedure expose glob.
/* Fast value */
y = 0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
return y+0

Ln4:
/* Natural log of 4 */
procedure expose glob.
/* Fast value */
y = 1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375
return y+0

Ln8:
/* Natural log of 8 */
procedure expose glob.
/* Fast value */
y = 2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063
return y+0

Ln10:
/* Natural log of 10 */
procedure expose glob.
/* Fast value */
y = 2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599720508959830
return y+0

Pi:
/* Pi */
procedure expose glob.
p = Digits()
/* In memory? */
if glob.pi.p <> '' then
return glob.pi.p
if p < 101 then
/* Fast value */
glob.pi.p = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211707+0
else do
numeric digits Digits()+2
if p < 201 then do
/* Method Chudnovsky series */
y = 0
do n = 0
v = y; y = y + Fact(6*n)*(13591409+545140134*n)/(Fact(3*n)*Fact(n)**3*-640320**(3*n))
if y = v then
leave
end
y = 4270934400/(Sqrt(10005)*y)
end
else do
/* Method Agmean series */
y = 0.25; a = 1; g = Sqrt(0.5); n = 1
do until a = v
v = a
x = (a+g)*0.5; g = Sqrt(a*g)
y = y-n*(x-a)**2; n = n+n; a = x
end
y = a*a/y
end
numeric digits Digits()-2
glob.pi.p = y+0
end
return glob.pi.p

Sqrt2:
/* Square root of 2 */
procedure expose glob.
/* Fast value */
y = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
return y+0

Sqrt3:
/* Square root of 3 */
procedure expose glob.
/* Fast value */
y = 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
return y+0

Sqrt5:
/* Square root of 5 */
procedure expose glob.
/* Fast value */
y = 2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
return y+0

Output
:
REXX-Regina_3.9.6(MT) 5.00 29 Apr 2024
Gamma function in arbitrary precision

(Half)integers formulas (100 digits)
Formulas  -99.5 3.370459273906717035419140191178165368212858243169804822385940657239537725368041400224226278392502618E-157 (0.003 seconds)
Formulas  -10.5 -2.640121820547716316246385325311240439682468432522587656059168154777653141232089673307782521306919765E-7 (0.000 seconds)
Formulas   -5.5 0.01091265478190986298673234429377905644050439299584730891829569009625650045347052040480970101310888490 (0.001 seconds)
Formulas   -2.5 -0.9453087204829418812256893244486107641586930432652731350473641545882193517818838300666403502605571549 (0.000 seconds)
Formulas   -1.5 2.363271801207354703064223311121526910396732608163182837618410386470548379454709575166600875651392887 (0.000 seconds)
Formulas   -0.5 -3.544907701811032054596334966682290365595098912244774256427615579705822569182064362749901313477089331 (0.000 seconds)
Formulas    0.5 1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665 (0.000 seconds)
Formulas    1.0 1 (0.000 seconds)
Formulas    1.5 0.8862269254527580136490837416705725913987747280611935641069038949264556422955160906874753283692723327 (0.001 seconds)
Formulas    2.0 1 (0.000 seconds)
Formulas    2.5 1.329340388179137020473625612505858887098162092091790346160355842389683463443274136031212992553908499 (0.000 seconds)
Formulas    5.0 24 (0.000 seconds)
Formulas    5.5 52.34277778455352018114900849241819367949013237611424488006401129409378637307891910622901158181014715 (0.000 seconds)
Formulas   10.0 362880 (0.000 seconds)
Formulas   10.5 1133278.388948785567334574165588892475560298308275159776608723414529483390056004153717630538727607291 (0.000 seconds)
Formulas   99.0 9.426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729113E+153 (0.001 seconds)
Formulas   99.5 9.367802114655996591305637999137598430056780942876744878408158310966911153528806387157595583300106562E+154 (0.001 seconds)

Lanczos (60 digits) vs Spouge (61 digits) vs Stirling (61 digits) approximations
Lanczos   -12.8 -1.44241809348812392704183496823417171228485786984071442421769E-9 (0.001 seconds)
Spouge    -12.8 -1.442418093488123927041834968234171712284857869840714424217876E-9 (0.061 seconds)
Lanczos    -6.4 -0.00214311842968855616971089012135323947575979728834356196087007 (0.001 seconds)
Spouge     -6.4 -0.002143118429688556169710890121353239475759797288343561960870050 (0.062 seconds)
Lanczos    -3.2 0.689056412005979742919224040168358601528149721171394266681422 (0.001 seconds)
Spouge     -3.2 0.6890564120059797429192240401683586015281497211713942666814006 (0.061 seconds)
Lanczos    -1.6 2.31058285808092523235318127282049942788600677267861414816871 (0.001 seconds)
Spouge     -1.6 2.310582858080925232353181272820499427886006772678614148168727 (0.063 seconds)
Lanczos    -0.8 -5.73855463999850381650594784491144000429263749786675377223601 (0.001 seconds)
Spouge     -0.8 -5.738554639998503816505947844911440004292637497866753772236066 (0.063 seconds)
Lanczos    -0.4 -3.72298062203204275598583347080335570330149759689981183834669 (0.001 seconds)
Spouge     -0.4 -3.722980622032042755985833470803355703301497596899811838346699 (0.062 seconds)
Lanczos    -0.2 -5.82114856862651686818160469134229346570980884445593876492447 (0.001 seconds)
Spouge     -0.2 -5.821148568626516868181604691342293465709808844455938764924472 (0.062 seconds)
Lanczos    -0.1 -10.6862870211931935489730533569448077816983878506097317904937 (0.001 seconds)
Spouge     -0.1 -10.68628702119319354897305335694480778169838785060973179049371 (0.066 seconds)
Lanczos     0.1 9.51350769866873183629248717726540219255057862608837734305000 (0.000 seconds)
Spouge      0.1 9.513507698668731836292487177265402192550578626088377343050001 (0.063 seconds)
Stirling    0.1 5.697187148977169278607259516105973271247103273209125657453591 (0.003 seconds)
Lanczos     0.2 4.59084371199880305320475827592915200343410999829340301778885 (0.000 seconds)
Spouge      0.2 4.590843711998803053204758275929152003434109998293403017788853 (0.062 seconds)
Stirling    0.2 3.325998424022392556831252338044446631141936051050822874561565 (0.003 seconds)
Lanczos     0.4 2.21815954375768822305905402190767945077056650177146958224198 (0.000 seconds)
Spouge      0.4 2.218159543757688223059054021907679450770566501771469582241978 (0.063 seconds)
Stirling    0.4 1.841476335936235407774504215721792671787348602355998395147546 (0.003 seconds)
Lanczos     0.8 1.16422971372530337363632093826845869314196176889118775298489 (0.000 seconds)
Spouge      0.8 1.164229713725303373636320938268458693141961768891187752984894 (0.059 seconds)
Stirling    0.8 1.053370968425608533997094555812234627247042644411690511716880 (0.003 seconds)
Lanczos     1.6 0.893515349287690261436600032992805368792359423255954841203208 (0.000 seconds)
Spouge      1.6 0.8935153492876902614366000329928053687923594232559548412032077 (0.064 seconds)
Stirling    1.6 0.8486932421525737351870671884548739160413838607490704857265386 (0.003 seconds)
Lanczos     3.2 2.42396547993536801209211236969059225781321007909891679339251 (0.000 seconds)
Spouge      3.2 2.423965479935368012092112369690592257813210079098916793392514 (0.069 seconds)
Stirling    3.2 2.361851203186240411720280147420856619690317833030951977155556 (0.003 seconds)
Lanczos     6.4 240.833779983445938793193921422181728753410453111118999895588 (0.000 seconds)
Spouge      6.4 240.8337799834459387931939214221817287534104531111189998955875 (0.060 seconds)
Stirling    6.4 237.7207525902404787072183649425603774456984237805478743551168 (0.002 seconds)
Lanczos    12.8 289487660.334241583580309266192984495005040185968050409506654 (0.000 seconds)
Spouge     12.8 289487660.3342415835803092661929844950050401859680504095066538 (0.059 seconds)
Stirling   12.8 287609477.0875058075487675889481380625378498188858381901124308 (0.003 seconds)

Same for a bigger number
Lanczos   -99.9 1.72726520939328005075554286550876635176440408500391962571907E-157 (0.000 seconds)
Spouge    -99.9 1.727265209393280050755542865508766351764404085003870332156550733790282578981487471194267631617799447E-157 (0.274 seconds)
Lanczos    99.9 5.89173215164436165675854187059394609460949390669692020855752E+155 (0.000 seconds)
Spouge     99.9 5.891732151644361656758541870593946094609493906696920208557519945524581875362460658552892944423816715E+155 (0.266 seconds)
Stirling   99.9 5.886819525828908144161960074977041612792659661747365945060453973672990129231211036627906746201731112E+155 (0.010 seconds)


## Ring

decimals(3)
gamma = 0.577
coeff = -0.655
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

## RLaB

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

${\displaystyle Gamma(a)=\Gamma (a)}$, the Gamma function;
${\displaystyle Gamma(a,x)={\frac {1}{\Gamma (a)}}\int _{x}^{\infty }dt\,t^{a-1}\,exp(-t)}$, the regularized Gamma function which is also known as the normalized incomplete Gamma function;
${\displaystyle GammaRegularizedC(a,x)={\frac {1}{\Gamma (a)}}\int _{0}^{x}dt\,t^{a-1}\,exp(-t)}$, which the GSL calls the complementary normalized Gamma function;
${\displaystyle LogGamma(a)=\ln \Gamma (a)}$;
${\displaystyle RecGamma(a)={\frac {1}{\Gamma (a)}}}$;
${\displaystyle Pochhammer(a,x)={\frac {\Gamma (a+x)}{\Gamma (x)}}}$.

## RPL

≪ 1 -
{ 1.00000000000000000000  0.57721566490153286061  -0.65587807152025388108
-0.04200263503409523553  0.16653861138229148950  -0.04219773455554433675
-0.00962197152787697356  0.00721894324666309954  -0.00116516759185906511
-0.00021524167411495097  0.00012805028238811619  -0.00002013485478078824
-0.00000125049348214267  0.00000113302723198170  -0.00000020563384169776
0.00000000611609510448  0.00000000500200764447  -0.00000000118127457049
0.00000000010434267117  0.00000000000778226344  -0.00000000000369680562
0.00000000000051003703  -0.00000000000002058326  -0.00000000000000534812
0.00000000000000122678  -0.00000000000000011813  0.00000000000000000119
0.00000000000000000141  -0.00000000000000000023  0.00000000000000000002 }
→ y a
≪ a DUP SIZE GET
a SIZE 1 - 1 FOR n
y * a n GET +
-1 STEP
INV
≫ ≫ 'GAMMA' STO

.3 GAMMA


The built-in FACT instruction is obviously based on a similar Taylor formula, since it returns same results:

.3 1 - FACT

Output:
2: 2.99156898769
1: 2.99156898769


## Ruby

#### Taylor series

$a = [ 1.00000_00000_00000_00000, 0.57721_56649_01532_86061, -0.65587_80715_20253_88108, -0.04200_26350_34095_23553, 0.16653_86113_82291_48950, -0.04219_77345_55544_33675, -0.00962_19715_27876_97356, 0.00721_89432_46663_09954, -0.00116_51675_91859_06511, -0.00021_52416_74114_95097, 0.00012_80502_82388_11619, -0.00002_01348_54780_78824, -0.00000_12504_93482_14267, 0.00000_11330_27231_98170, -0.00000_02056_33841_69776, 0.00000_00061_16095_10448, 0.00000_00050_02007_64447, -0.00000_00011_81274_57049, 0.00000_00001_04342_67117, 0.00000_00000_07782_26344, -0.00000_00000_03696_80562, 0.00000_00000_00510_03703, -0.00000_00000_00020_58326, -0.00000_00000_00005_34812, 0.00000_00000_00001_22678, -0.00000_00000_00000_11813, 0.00000_00000_00000_00119, 0.00000_00000_00000_00141, -0.00000_00000_00000_00023, 0.00000_00000_00000_00002 ] def gamma(x) y = Float(x) - 1 1.0 /$a.reverse.inject {|sum, an| sum * y + an}
end

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

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

(1..10).each{|i| puts Math.gamma(i/3.0)}

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


## Rust

#### Stirling

Translation of: AWK
use std::f64::consts;

fn main() {
let gamma = |x: f64| { assert_ne!(x, 0.0); (2.0*consts::PI/x).sqrt() * (x * (x/consts::E).ln()).exp()};
(1..=20).for_each(|x| {
let x = f64::from(x) / 10.0;
println!("{:.02} => {:.10}", x, gamma(x));
});
}

Output:
0.10 => 5.6971871490
0.20 => 3.3259984240
0.30 => 2.3625300363
0.40 => 1.8414763359
0.50 => 1.5203469011
0.60 => 1.3071588574
0.70 => 1.1590532921
0.80 => 1.0533709684
0.90 => 0.9770615079
1.00 => 0.9221370089
1.10 => 0.8834899532
1.20 => 0.8577553354
1.30 => 0.8426782594
1.40 => 0.8367445486
1.50 => 0.8389565525
1.60 => 0.8486932422
1.70 => 0.8656214718
1.80 => 0.8896396353
1.90 => 0.9208427219
2.00 => 0.9595021757


## 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)))
}
}

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

(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")))

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

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

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


$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; 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 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)) }  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: 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)) }  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: 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)) }  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 Python and mpmath (see below). The results are compared to Mata's gamma function for each real between 1/100 and 100, by steps of 1/100. mata _gamma_coef = 1.0, 5.772156649015328606065121e-1, -6.558780715202538810770195e-1, -4.200263503409523552900393e-2, 1.665386113822914895017008e-1, -4.219773455554433674820830e-2, -9.621971527876973562114922e-3, 7.218943246663099542395010e-3, -1.165167591859065112113971e-3, -2.152416741149509728157300e-4, 1.280502823881161861531986e-4, -2.013485478078823865568939e-5, -1.250493482142670657345359e-6, 1.133027231981695882374130e-6, -2.056338416977607103450154e-7, 6.116095104481415817862499e-9, 5.002007644469222930055665e-9, -1.181274570487020144588127e-9, 1.04342671169110051049154e-10, 7.782263439905071254049937e-12, -3.696805618642205708187816e-12, 5.100370287454475979015481e-13, -2.05832605356650678322243e-14, -5.348122539423017982370017e-15, 1.226778628238260790158894e-15, -1.181259301697458769513765e-16, 1.186692254751600332579777e-18, 1.412380655318031781555804e-18, -2.298745684435370206592479e-19, 1.714406321927337433383963e-20 function gamma_(x_) { external _gamma_coef x = x_ y = 1 while (x<0.5) y = y/x++ while (x>1.5) y = --x*y z = _gamma_coef[30] x-- for (i=29; 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::1000)/100 u=map(&gamma(),x) v=map(&gamma_(),x) max(abs((v-u):/u)) end  Output 9.80341e-15 Here follows the Python program to compute coefficients. from mpmath import mp mp.dps = 50 def gamma_coef(n): a = [mp.mpf(1), mp.mpf(mp.euler)] for k in range(3, n + 1): s = sum((-1)**j * mp.zeta(j) * a[k - j - 1] for j in range(2, k)) a.append((s - a[1] * a[k - 2]) / (1 - k * a[0])) return a def horner(a, x): y = 0 for s in reversed(a): y = y * x + s return y gc = gamma_coef(30) def gamma_approx(x): y = mp.mpf(1) while x < 0.5: y /= x x += 1 while x > 1.5: x -= 1 y *= x return y / horner(gc, x - 1) for x in gc: print(mp.nstr(x, 25))  ## Tcl Works with: Tcl version 8.5 Library: Tcllib (Package: math) Library: Tcllib (Package: math::calculus) package require math package require math::calculus # 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]
}
}

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

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

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
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


## TXR

### Taylor Series

Separator commas in numeric tokens are supported only as of TXR 283.

(defun gamma (x)
(/ (rpoly (- x 1.0)
#( 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))))

(each ((i 1..11))
(put-line (pic "##.######" (gamma (/ i 3.0)))))
Output:
 2.678939
1.354118
1.000000
0.892980
0.902745
1.000000
1.190639
1.504575
2.000000
2.778158

### Stirling

(defun gamma (x)
(* (sqrt (/ (* 2 %pi%)
x))
(expt (/ x %e%) x)))

(each ((i 1..11))
(put-line (pic "##.######" (gamma (/ i 3.0)))))
Output:
 2.156976
1.202851
0.922137
0.839743
0.859190
0.959502
1.149106
1.458490
1.945403
2.709764

### Lanczos

The Haskell version calculates the natural log of the gamma function, which is why the function is called gammaln; we correct that here by calling exp:

(defun gamma (x)
(let* ((cof #(76.18009172947146 -86.50532032941677
24.01409824083091 -1.231739572450155
0.001208650973866179 -0.000005395239384953))
(ser0 1.000000000190015)
(x55 (+ x 5.5))
(tmp (- x55 (* (+ x 0.5) (log x55))))
(ser (+ ser0 (sum [mapcar / cof (succ x)]))))
(exp (- (log (/ (* 2.5066282746310005 ser) x)) tmp))))

(each ((i (rlist 0.1..1.0..0.1 2..10)))
(put-line (pic "##.# ######.######" i (gamma i))))
Output:
 0.1      9.513508
0.2      4.590844
0.3      2.991569
0.4      2.218160
0.5      1.772454
0.6      1.489192
0.7      1.298055
0.8      1.164230
0.9      1.068629
1.0      1.000000
2.0      1.000000
3.0      2.000000
4.0      6.000000
5.0     24.000000
6.0    120.000000
7.0    720.000000
8.0   5040.000000
9.0  40320.000000
10.0 362880.000000


From Wikipedia Python code. Output is identical to above.

(defun gamma (x)
(if (< x 0.5)
(/ %pi%
(* (sin (* %pi% x))
(gamma (- 1 x))))
(let* ((cof #(676.5203681218851 -1259.1392167224028
771.32342877765313 -176.61502916214059
12.507343278686905 -0.13857109526572012
9.9843695780195716e-6 1.5056327351493116e-7))
(ser0 0.99999999999980993)
(z (pred x))
(tmp (+ z (len cof) -0.5))
(ser (+ ser0 (sum [mapcar / cof (succ z)]))))
(* (sqrt (* 2 %pi%))
(expt tmp (+ z 0.5))
(exp (- tmp))
ser))))

(each ((i (rlist 0.1..1.0..0.1 2..10)))
(put-line (pic "##.# ######.######" i (gamma i))))

## Visual FoxPro

Translation of BBC Basic but with OOP extensions. Also some ideas from Numerical Methods (Press et al).

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
#DEFINE FPF 5.5
#DEFINE HALF 0.5
#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

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


## V (Vlang)

Translation of: go
import math
fn main() {
println("    x               math.Gamma                 Lanczos7")
for x in [-.5, .1, .5, 1, 1.5, 2, 3, 10, 140, 170] {
println("${x:5.1f}${math.gamma(x):24.16} ${lanczos7(x):24.16}") } } fn lanczos7(z f64) f64 { 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.sqrt_pi * math.pow(t, z-.5) * math.exp(-t) * x } 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 ## Wren Translation of: Kotlin Library: Wren-fmt Library: Wren-math The gamma method in the Math class is based on the Lanczos approximation. import "./fmt" for Fmt import "./math" for Math var stirling = Fn.new { |x| (2 * Num.pi / x).sqrt * (x / Math.e).pow(x) } System.print(" x\tStirling\t\tLanczos\n") for (i in 1..20) { var d = i / 10 Fmt.print("$4.2f\t$16.14f\t$16.14f", d, stirling.call(d), Math.gamma(d))
}

Output:
 x	Stirling		Lanczos

0.10	5.69718714897717	9.51350769866875
0.20	3.32599842402239	4.59084371199881
0.30	2.36253003626962	2.99156898768759
0.40	1.84147633593624	2.21815954375769
0.50	1.52034690106628	1.77245385090552
0.60	1.30715885744836	1.48919224881282
0.70	1.15905329211392	1.29805533264756
0.80	1.05337096842561	1.16422971372530
0.90	0.97706150787770	1.06862870211932
1.00	0.92213700889579	1.00000000000000
1.10	0.88348995316870	0.95135076986687
1.20	0.85775533539659	0.91816874239976
1.30	0.84267825944839	0.89747069630628
1.40	0.83674454863708	0.88726381750308
1.50	0.83895655252650	0.88622692545276
1.60	0.84869324215257	0.89351534928769
1.70	0.86562147179388	0.90863873285329
1.80	0.88963963528799	0.93138377098024
1.90	0.92084272189423	0.96176583190739
2.00	0.95950217574449	1.00000000000000


## XPL0

function real Gamma (X);
real X, A, Y, Sum;
integer N;
begin
A  \constant array (0..29) of Long_Float\ :=
[  1.00000_00000_00000_00000,
0.57721_56649_01532_86061,
-0.65587_80715_20253_88108,
-0.04200_26350_34095_23553,
0.16653_86113_82291_48950,
-0.04219_77345_55544_33675,
-0.00962_19715_27876_97356,
0.00721_89432_46663_09954,
-0.00116_51675_91859_06511,
-0.00021_52416_74114_95097,
0.00012_80502_82388_11619,
-0.00002_01348_54780_78824,
-0.00000_12504_93482_14267,
0.00000_11330_27231_98170,
-0.00000_02056_33841_69776,
0.00000_00061_16095_10448,
0.00000_00050_02007_64447,
-0.00000_00011_81274_57049,
0.00000_00001_04342_67117,
0.00000_00000_07782_26344,
-0.00000_00000_03696_80562,
0.00000_00000_00510_03703,
-0.00000_00000_00020_58326,
-0.00000_00000_00005_34812,
0.00000_00000_00001_22678,
-0.00000_00000_00000_11813,
0.00000_00000_00000_00119,
0.00000_00000_00000_00141,
-0.00000_00000_00000_00023,
0.00000_00000_00000_00002
];
Y   := X - 1.0;
Sum := A (29);
for N:= 29-1 downto 0 do
Sum := Sum * Y + A (N);
return 1.0 / Sum;
end \Gamma\;

\Test program:
integer I;
begin
Format(0, 14);
for I:= 1 to 10 do
[RlOut(0, Gamma (Float (I) / 3.0));  CrLf(0)];
end
Output:
 2.67893853470775E+000
1.35411793942640E+000
1.00000000000000E+000
8.92979511569249E-001
9.02745292950934E-001
1.00000000000000E+000
1.19063934875900E+000
1.50457548825154E+000
1.99999999999397E+000
2.77815847933857E+000


## Yabasic

Translation of: Phix
dim c(12)

sub gamma(z)
local accm, k, k1_factrl

accm = c(1)
if accm=0 then
accm = sqrt(2*PI)
c(1) = accm
k1_factrl = 1
for k=2 to 12
c(k) = exp(13-k)*(13-k)^(k-1.5)/k1_factrl
k1_factrl = k1_factrl * -(k-1)
next
end if
for k=2 to 12
accm = accm + c(k)/(z+k-1)
next
accm = accm * exp(-(z+12))*(z+12)^(z+0.5)
return accm/z
end sub

sub si(x)
print x using "%18.13f"
end sub

for i = 0.1 to 2.1 step .1
print i, " = "; : si(gamma(i))
next

## zkl

Translation of: D

but without a built in gamma function.

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.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);
}
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);
}
}
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
`