hp41programs

Random Number Generators for the HP-41
 

Overview
 

-Several pseudo random number generators are listed on this web page:
-"RNG1" "RNG2" "RNG3" work on every HP-41.
-"RNG4" & the M-Code routine RAND require a Time-Module.
-Finally, the last program is an attempt to play ( win? ) the lottery...
 

Program#1
 

-A well known RNG is given by the formula:  x_n+1 = FRC ( 9821 x_n + 0.211327 ) which provides 1 million random numbers.
-The following program gives 1,000,000,000  random numbers r  ( 0 <= r < 1 ). The formula  x_n+1 = FRC ( 9⁸ x_n + 0.236067977 ) is used.
-The coefficient 9⁸ = 43,046,721 may be replaced by  a   where  a = 1 ( mod 20 )
                     and  0.236067977  may be replaced by  b  where   b*10⁹ is not divisible by 2 or 5.
 
 

01  LBL "RNG1" 02  9 03  ENTER^ 04  ENTER^ 05  R^ 06  *

07  FRC             08  * 09  FRC 10  * 11  FRC 12  *

13  FRC             14  * 15  FRC 16  * 17  FRC 18  *

19  FRC             20  * 21  FRC 22  5 23  SQRT 24  +

25  FRC             26  END

 
( 35 bytes / SIZE 001 )
 
 

STACK	INPUTS	OUTPUTS
X	xn	xn+1

 
Example:

  0.2   XEQ "RNG1"  yields  0.436067977
                                   R/S    0.779021394   ... etc ...
 

Program#2
 

-"RNG2" provides 9,999,999,996 random numbers with the formula:  x_n+1 = ( 10⁵⁹ x_n ) MOD p   where p = 9,999,999,967 is the greatest prime < 10¹⁰
   x_n are integers between 0 and p ( exclusive ) which are then divided by p to be reduced to a number between 0 and 1.
-This routine works well because the MOD function gives exact results even when the operands are greater than 10¹⁰.
-Actually, the exponent 59 may be replaced by any integer m provided m is relatively prime to p-1 = 2*3*11*457*331543
  but I don't know what is the best choice.
-Unlike "RNG1" and other routines based upon the same type of formulae, the least significant digits don't go through any cycle of ten, one hundred and so on.
-Register R00 is used to store the different  x_n integers.
 
 

01  LBL "RNG2"  02  RCL 00  03   E59  04  *  05  10  06  10^X  07  33  08  -  09  MOD  10  STO 00  11  LASTX  12  /  13  END

 
( 26 bytes / SIZE 001 )
 
 

STACK	INPUTS	OUTPUTS
X	/	0 < r < 1

 
Example:

    1  STO 00
        XEQ "RNG2" gives  0.3129146797   ( and R00 = 3129146787 = 10⁵⁹ ( mod p ) )
                R/S                   0.6904570204    ( R00 = 6904570181 ) ... etc ...
 

-Actually if p is a prime,  ( Z/pZ-{0} ; * ) is a group
  and if a is an integer,  the number of distinct elements in the subset   { 1 ; a ; a² ; ....... ; a^k ; .... } ( mod p )   divides  p-1
-If  p-1 is the smallest positive integer q such that a^q = 1 ( mod p ) , then the sequence   a ; a² ; ....... ; a^k ; .... ; a^p-1 ( mod p )
  is a permutation of  1 ; 2 ; ...... ; p-1
-In particular, if  p = 2p' + 1  where p' is also a prime, and if a^p' is not equal to 1 ( mod p )  then  a  satisfies the required property.
-For instance,  p = 7,841,296,787 = 2*3,920,648,393 + 1              7,841,296,787 and 3,920,648,393 are primes and  -10²⁴ = 4,851,307,369  ( mod p )
  satisfies  (-10²⁴)^p' = -1  therefore the routine:

  E24
  *
  CHS
  7841296787
  MOD                  gives  7,841,296,786  random integers

-These ideas may be used to create your own RNG.
 

Program#3
 

-The following algorithm is given by Clifford Pickover in "Keys to Infinity" ( John Wiley & Sons )  ISBN 0-471-11857-5
 
 

01  LBL "RNG3"  02  LN  03   E2  04  *  05  1  06  MOD  07  END

 
( 17 bytes / SIZE 001 )
 
 

STACK	INPUTS	OUTPUTS
X	xn	xn+1

 
Example:

    0.1   XEQ "RNG3" produces  0.74149070
                   R/S                          0.09073404  ... etc ...
 

Program#4
 
 
 

01  LBL "RNG4"  02  DATE  03  TIME  04  +  05   E49  06  *  07  PI  08  MOD  09  LN1+X  10  R-D  11  FRC  12  END

 
( 25 bytes / SIZE 000 )
 
 

STACK	INPUTS	OUTPUTS
X	/	0 < r < 1

 
-I cannot give any example since the result depends on the instant you press R/S
 

M-Code Routine
 

-This M-Code routine uses the TIME module - or an HP41-CX
 

084  "D"
00E  "N"
001  "A"
012  "R"
130  LDI S&X
010  16d
270  RAMSLCT
130  LDI S&X
0FB 251d
3F0  PRPH SLCT
3E8  WRIT 15(e)
038  READDATA
1BC RCR 11
046  C=0 S&X
270  RAMSLCT
05E  C=0 MS
130  LDI S&X
041  65d
2A0  SETDEC
10E  A=C ALL
04E  C=0 ALL
35C  PT=12
1D0  LD@PT- 7
210  LD@PT- 8
110  LD@PT- 4
050  LD@PT- 1
090  LD@PT- 2
250  LD@PT- 9
190  LD@PT- 6
1D0 LD@PT- 7
210  LD@PT- 8
1D0  LD@PT- 7
044  CLRF 4
070  N=C ALL
171  ?NCXQ
064  195C
10E  A=C ALL
0B0  C=N ALL
261  ?NCXQ
060  1898
0E8  WRIT 3(X)
3E0  RTN

( 42 words / SIZE 000 )
 

-"RAND" uses no data register
-XEQ "RAND" returns a pseudo-random number ( between 0 and 1 ) in X-register
-The previous content of register X is overwritten.
-Execution time = 0.22 second
 

Winning the Lottery?
 

-If you need, for instance, 7 integers between 1 and 49 , you can use 7 random numbers between 0 and 1,
  multiply them by 49 , add 1 and take the integer part of the result.
-The small routine hereafter is another possibility, if you accept to calculate the integer part in your mind:
 
 
 

01  LBL "$$$"  02  R-D  03  49  04  MOD  05  1  06  +  07  END

 
( 16 bytes / SIZE 000 )
 
 

STACK	INPUTS	OUTPUTS
X	r	0 < N < 50

 
Example:

   41  XEQ "$$$"  yields   47.1270
              R/S                      6.1759
              R/S                    11.8535
              R/S                    43.1567
              R/S                    23.6963
              R/S                    35.6962
              R/S                    37.2418         suggesting     47-06-11-43-23-35-37
 

Notes:

1-I'm not a statistician and I can't assure all these RNGs would stand up to sophisticated tests, but one may use his imagination in devising variations.
2-If you win one million dollars thanks to one of these programs, I accept to share the jackpot...