12C+ Gamma

[Gerson W. Barbosa]

Gamma(x+1)   (0 <= x <= 39.4)

01 STO 0	24 RCL 2

02 .       	25 y^x

03 6		26 /

04 y^x		27 STO+ 3

05 1		28 1

06 9		29 STO+ 2

07 *		30 GTO 12

08 STO 1	31 RCL 3

09 0		32 2

10 STO 2	33 SQRTx

11 STO 3	34 LASTx

12 RCL 2	35 /

13 RCL 1	36 /

14 x<=y		37 2

15 GTO 31	38 ENTER

16 x<>y		39 LN

17 2		40 /

18 *		41 RCL 0

19 1		42 1

20 +		43 +

21 RCL 0 	44 y^x

22 y^x		45 /

23 2		46 GTO 00

12.12 R/S => 648,976,950.1 after 3.5 seconds (HP-12C+)

On the 15C, 

12.12 x!  => 648,976,950.9 

This is based on the following approximation I've come up with:

Gamma(x+1) ~ SUM(k=0,inf,(2*k+1)^x/2^k)/(sqrt2/2*(2/ln(2))^(x+1))

Obviously, this is not the proper way to compute the Gamma function. It's just an example of what can be done on the fast 12C+. Just for an idea, it would take about 7 minutes to compute Gamma(40) on a normal 12C using this method.

Gerson.

--------------

The approximation is being used reversely, that is, it is meant to approximate
this integer sequence, not the Gamma function:

an ~ n!/(2*sqrt2)*(2/ln(2))^(n+1) 

I found it when thinking about John Smitherman's math joke turned into a problem. The numbers 2, 6, 34, 294... (which showed as the problem was getting progressively more complex) ought to mean something... I couldn't find it at OEIS but I found 1, 3, 17, 147..., the halved sequence. Eventually I found out the original sequence is the row sums of Sierpinski-Pascal-MacMahon triangle, whatever this might be.
Next time someone starts a joke I'll try to just laugh, as everybody else :-)

I just intended to put a computationally intensive programming problem to the 12C+ to see how it performed. For computing Gamma function on the 12C+ or the normal 12C and 12c Platinum there isn't anything better than Egan Ford's program:

http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=100275

See message #6.

Edited: 26 June 2009, 10:30 a.m.

[Bart (UK)]

Just for fun I thought I'd try it on the new 35s to see how it did (considering it's slighly(?) old CPU).



12.12 XEQ is about 30.5s with an answer of 648,976,950.899 (FIX = ALL).

40 XEQ is about 1 min.

Bart

[Gerson W. Barbosa]

13 and 23 seconds, respectively, on the 33s which used to be my fastest RPN calculator. I wonder how nice it would be if they released a new 15C+. If it were as fast as the new 12C+, 60.5 CHS x! would take about 200 ms to display -1.527756e-97 instead of current 13 seconds.

Gerson.

[Bart (UK)]

A fast 15C would be a dream come true.



I tried a HP-20S implementation:

01 LBL A	23 2		45 1/x

02 STO 0	24 +		46 *

03 y^x		25 1		47 2

04 .		26 =		48 =

05 6		27 y^x		49 y^x

06 =		28 RCL 0 	50 (

07 *		29 =		51 RCL 0

08 1		30 2		52 +

09 9		31 y^x		53 1

10 =		32 RCL 2 	54 )

11 STO 1	33 =		55 =

12 0		34 1/x		56 *

13 STO 2	35 *		57 2

14 STO 3	36 LAST		58 SQRT

15 LBL B	37 =		59 /

16 RCL 2	38 STO+ 3 	60 2

17 INPUT	39 1		61 =

18 RCL 1	40 STO+ 2 	62 1/x

19 x<=y?	41 GTO B 	63 *

20 GTO C	42 LBL C 	64 RCL 3

21 RCL 2	43 2		65 =

22 *		44 LN		66 RTN

Compared to the original post, one can see the advantage of RPN here. Although this is probably not the most optimum.



12.12 XEQ A, 15.5s; 648,976,950.9

40 XEQ A, 30s



Considering the program is 50% longer, that makes the 20S 3x faster than the 35s. Possibly the advantage of simple data types?

Bart


edited to try and make listing more readable

Edited: 30 June 2009, 11:09 a.m.

[Raymund Heuvel]

HP-71B with Math Pack

a=time @ GAMMA(13.12); @ time-a

results in

648976950.903 .23

thus 0.23 seconds ....


The good old HP-71B ...


BR Raymund

Edited: 29 June 2009, 6:37 p.m.

[Gerson W. Barbosa]

Quote:

---

648976950.903 .23
thus 0.23 seconds ....

---

Haven't you missed a "1" before ".23"?
1.22 seconds on mine...

...and 2.04 seconds for GAMMA(-68.5).

Gerson.

[Raymund Heuvel]

a=time @ GAMMA(13.12); @ time-a
execution time between 0.22 up to 0.26


a=time @ GAMMA(-68.5); @ time-a
result = -1.527757e-97 execution time 1.01 (10 runs)





FYI VER$ => HP71:1BBBB HPIL:1A MATH:1A FTH:1A EDT:A KBD:B



I've tested it on three 71-B's with different memory sizes (16K up to 144K Ram), all about the same.


Maybe you have a Plug-In or Lex File causing a lot of Interrupts or Polls?

BR

Ray

[Marcus von Cube, Germany]

Mine must be on steroids:

.09 seconds for GAMMA(13.12).

It appears to be a very inaccurate measurement because several tries resulted in vastly disparate timing results, a few outcomes even with a negative sign.

[Gerson W. Barbosa]

I was doing a=time @ GAMMA(13.12) @ time-a (without semicollon).

Now I get 0.21 to 0.22 and 1.02 to 1.05 for GAMMA(-68.5).

1BBBB version also with a 32K RAM module and card reader. They have no effect on timing though.

Thanks!

Gerson.