Calculate large factorials on a HP48 (SysRPL)

02182014, 10:32 PM
Post: #1




Calculate large factorials on a HP48 (SysRPL)
It's nice to see that some of you want learn or want to have a look inside SysRPL. One idea was using SysRPL for speed up calculation speed, here's another approach using SysRPL. This is an example from my fund I wrote years ago.
What is 253! = 5.17346099264E499 And what is 254! = ! Error: Overflow The easy way to solve this, instead of multiplying the numbers, simply add the logarithm to the base 10 of each number. This is the UserRPL code doing this: Code:
But calculate 10! with this program and you see the disadvantage of this method: inaccuracy! A solution for this is using the internal data type "long real". This SysRPL program using the same algorithm like before. Because the logarithm to the base 10 don't exist for long real I use the natural logarithm function %%LN. Here's the source code for compiling with HPTOOLS: Code:
So back to the entire question, what is 254! = "1.31405909214E502". For single step execution of SysRPL programs on the HP48 I prefer Jazz. Hope you enjoy, Christoph 

02192014, 10:45 AM
Post: #2




RE: Calculate large factorials on a HP48 (SysRPL)
For both accuracy and speed, however, you'd be better off dividing by LN(10) only once, after the loop?
Cheers, Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE 

02192014, 11:45 AM
Post: #3




RE: Calculate large factorials on a HP48 (SysRPL)  
02192014, 01:21 PM
(This post was last modified: 02192014 01:21 PM by Gerson W. Barbosa.)
Post: #4




RE: Calculate large factorials on a HP48 (SysRPL)
(02192014 11:45 AM)walter b Wrote: Personally, I enjoy using the WP 34S for such problems: what is e.g. 5432! ? Returns: 3.022 553 598 4 E17 931. See the description of LNΓ on p. 102, IIRC. What is the last nonzero digit of 5432! ? My good old HP33C says it is 2, in about 30 seconds! :) (Program in message #80 here) 

02192014, 04:44 PM
Post: #5




RE: Calculate large factorials on a HP48 (SysRPL)
(02192014 10:45 AM)Werner Wrote: For both accuracy and speed, however, you'd be better off dividing by LN(10) only once, after the loop? I agree with the speed reason, but when I wrote the program years ago I got better results adding the LOG instead of the LN numbers. But your suggested modification is simple for own tests: Code: ... Christoph 

02202014, 06:22 PM
(This post was last modified: 02202014 06:24 PM by Dieter.)
Post: #6




RE: Calculate large factorials on a HP48 (SysRPL)
(02192014 11:45 AM)walter b Wrote: Personally, I enjoy using the WP 34S for such problems: what is e.g. 5432! ? Returns: 3.022 553 598 4 E17 931. See the description of LNΓ on p. 102, IIRC. Users without access to a ln Γ function may try one of the Stirling formulas instead. ;) BTW, a general caveat regarding methods based on lg n! or ln Γ: Since the exponent 17931 has five digits, at least five mantissa digits of the result are uncertain. On a WP34s in 16digit SP mode the valid digits are exactly those eleven you posted. Yes, there is a DP option. ;) Dieter 

02232014, 04:33 PM
(This post was last modified: 02232014 05:11 PM by Dieter.)
Post: #7




RE: Calculate large factorials on a HP48 (SysRPL)
(02182014 10:32 PM)Christoph Giesselink Wrote: A solution for this is using the internal data type "long real". This SysRPL program using the same algorithm like before. Because the logarithm to the base 10 don't exist for long real I use the natural logarithm function %%LN. So essentially you're calculating ln n! first and then n! is obtained from this. There is an easier and much faster method for this: simply use a Stirling approximation. With only two terms, the largest error in log_{10} n! is less than 1 unit in the 12th significant digit if n ≥ 70. Since n! can be calculated directly for n ≤ 253, we only have to consider n ≥ 254. If evaluated exactly (!), the absolute error of log_{10} n! is less than 3,3 E16, i.e. better than the available working precicion. So two terms are all we need: \(ln n! = n · ln n + \frac{1}{2}ln(2 n \pi)  n + \frac{1}{12 n}  \frac{1}{360 n^3} + ...\) The resulting accuracy is essentially limited by that of the internal 15digit functions and the order of addition: add the smallest terms first. Dieter 

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