(HP71B) Modern GAMMA formula for Forth or ASM?

07292024, 03:46 PM
Post: #1




(HP71B) Modern GAMMA formula for Forth or ASM?
Hello,
Gamma implementation can be different:  HP41 RPN https://hp41programs.yolasite.com/gamma.php  to 7 digits.. https://rosettacode.org/wiki/Gamma_function#Forth (we see the use of constants in other assembler programming)  2006 year formula https://www.rskey.org/CMS/thelibrary/?v...icle&id=11 Has anybody something more like anno 2024? Thanks. HP71B 4TH/ASM/Multimod, HP41CV/X/Y & Nov64d, PILBOX, HPIL 821.62A & 64A & 66A, Deb11 64bPC & PI2 3 4 w/ ILPER, VIDEO80, V41 & EMU71, DM41X 

08032024, 05:24 PM
(This post was last modified: 08032024 05:34 PM by peacecalc.)
Post: #2




RE: (HP71B) Modern GAMMA formula for Forth or ASM?
Hi floppy,
for over 30 years (not a knowledge we have today 2024) I programmed the gammafunction for a computer with a 368 processor and a 387 coprocessor (only for real arguments). The interesting point is only that I used different expressions for different arguments: 1) for positive x \[x \ge 10: \Gamma(x) \approx \sqrt{\frac{2\pi}{x}}\cdot \exp\left(x*\ln(x) + (S_8  x)\right)\] This is the expression  Stirling, but the infinite sum is reduced to 8 terms (is possible because x => 10): \[S_8 = \sum_{k=1}^{8} \frac{B_{2k}}{2k(2k1)x^{2k1}}\] The B(2k) are the Bernoullinumbers and the sum ist calculated with the Hornerscheme: \[S_8 = (((((((c_8x^{2} + c_7)x^{2} + c_6)x^{2} + c_5)x^{2} + c_4)x^{2} + c_3)x^{2} + c_2)x^{2} + c_1)x^{1} \] The c(k) are the precalculated expressions: \[ c_k = \frac{B_{2k}}{2k(2k1)} \] If we want to calculate for smaller positiv arguments let's us say x = 7.32, my former program calculates the value for 10.32 and in a loop it calculates: \[ \Gamma(9.32) = \frac{\Gamma(10.32)}{9.32 } \qquad \Gamma(8.32) = \frac{\Gamma(9.32)}{8.32 } \qquad \Gamma(7.32) = \frac{\Gamma(8.32)}{7.32 }\] My test if this works with a good accuracy is: \[ \Gamma(0.5) = \sqrt{\pi} \] Unfortunately I didn't document how many digit were correct. 2) And the negative Numbers x < 0 (without the negativ integers) were calculated with the expression: \[ \Gamma(x) = \frac{\pi}{\sin(\pi\cdot x)\Gamma(1x)} \] Let us say we have x =  3.6 then we calculate: \[ \Gamma(3.6) = \frac{\pi}{\sin(\pi\cdot (3.6))\Gamma(4.6)} \] This all was only a self educational exercise for learning programming the coprocessor with some aspects. May be someone else remembers or knows a more actual procedure if he reads that. 

08032024, 06:41 PM
Post: #3




RE: (HP71B) Modern GAMMA formula for Forth or ASM?
Very interesting. Thanks.
And: using multiprocessing for math calculation (I have already done with python where separate calculation like Result = Series A / Series B where both series are independent) is something I had in my head for HP71 spreading tasks to other HP71 in a loop (just for fun; probably giving the task via HPIL is too slow but could simulate a parallel processing). HP71B 4TH/ASM/Multimod, HP41CV/X/Y & Nov64d, PILBOX, HPIL 821.62A & 64A & 66A, Deb11 64bPC & PI2 3 4 w/ ILPER, VIDEO80, V41 & EMU71, DM41X 

08042024, 11:48 PM
(This post was last modified: 08042024 11:58 PM by John Keith.)
Post: #4




RE: (HP71B) Modern GAMMA formula for Forth or ASM?
More information on Stirling's approximation can be found here and here. More details here under "Implementation". Assuming that a reasonable number of terms are sufficient for 12digit accuracy, it would be faster and easier to use inline constants for the coefficients in your code.
The HP 49 and 50 have the Gamma function for real and complex arguments. I haven't tried to examine the internal code myself but I would guess that it is rather involved. 

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