Random Numbers for C: End, at last?

From: George Marsaglia <geo@stat.fsu.edu>
Subject: Random numbers for C: End, at last?
Newsgroups: sci.stat.math, sci.math, sci.math.num-analysis, sci.crypt, sci.physics.research, comp.os.msdos.djgpp
Date: Thu, 21 Jan 1999 03:08:52 GMT
Organization: Florida State University
Lines: 301

My offer of RNG's for C was an invitation to dance; I did not expect the Tarantella. I hope this post will stop the music, or at least slow it to a stately dance for language chauvinists and software police---under a different heading.

In response to a number of requests for good RNG's in C, and mindful of the desirability of having a variety of methods readily available, I offered several. They were implemented as in-line functions using the #define feature of C.

Numerous responses have led to improvements; the result is the listing below, with comments describing the generators.

I thank all the experts who contributed suggestions, either directly to me or as part of the numerous threads.

It seems necessary to use a (circular) table in order to get extremely long periods for some RNG's. Each new number is some combination of the previous r numbers, kept in the circular table. The circular table has to keep at least the last r, but possible more than r, numbers.

For speed, an 8-bit index seems best for accessing members of the table---at least for Fortran, where an 8-bit integer is readily available via integer*1, and arithmetic on the index is automatically mod 256 (least-absolute-residue).

Having little experience with C, I got out my little (but BIG) Kernighan and Ritchie book to see if there were an 8-bit integer type. I found none, but I did find char and unsigned char: one byte. Furthemore, K&R said arithmetic on characters was ok. That, and a study of the #define examples, led me to propose #define's for in-line generators LFIB4 and SWB, with monster periods. But it turned out that char arithmetic jumps "out of character", other than for simple cases such as c++ or c+=1. So, for safety, the index arithmetic below is kept in character by the UC definition.

Another improvement on the original version takes advantage of the comma operator, which, to my chagrin, I had not seen in K&R. It is there, but only with an example of (expression,expression). From the advice of contributors, I found that the comma operator allows (expression,...,expression,expression) with the last expression determining the value. That makes it much easier to create in-line functions via #define (see SHR3, LFIB4, SWB and FIB below).

The improved #define's are listed below, with a function to initialize the table and a main program that calls each of the in-line functions one million times and then compares the result to what I got with a DOS version of gcc. That main program can serve as a test to see if your system produces the same results as mine.

  |If you run the program below, your output|
  | should be  seven lines, each a 0 (zero).|

Some readers of the threads are not much interested in the philosophical aspects of computer languages, but want to know: what is the use of this stuff? Here are simple examples of the use of the in-line functions: Include the #define's in your program, with the accompanying static variable declarations, and a procedure, such as the example, for initializing the static variable (seeds) and the table.

Then any one of those in-line functions, inserted in a C expression, will provide a random 32-bit integer, or a random float if UNI or VNI is used. For example, KISS&255; would provide a random byte, while 5.+2.*UNI; would provide a random real (float) from 5 to 7. Or 1+MWC%10; would provide the proverbial "take a number from 1 to 10", (but with not quite, but virtually, equal probabilities). More generally, something such as 1+KISS%n; would provide a practical uniform random choice from 1 to n, if n is not too big.

A key point is: a wide variety of very fast, high- quality, easy-to-use RNG's are available by means of the nine in-line functions below, used individually or in combination.

The comments after the main test program describe the generators. These descriptions are much as in the first post, for those who missed them. Some of the generators (KISS, MWC, LFIB4) seem to pass all tests of randomness, particularly the DIEHARD battery of tests, and combining virtually any two or more of them should provide fast, reliable, long period generators. (CONG or FIB alone and CONG+FIB are suspect, but quite useful in combinations.)

Serious users of random numbers may want to run their simulations with several different generators, to see if they get consistent results. That should be easy to do.

Bonne chance,

George Marsaglia

The C code follows---------------------------------:

#include <stdio.h>
#define znew (z=36969*(z&65535)+(z>>16))
#define wnew (w=18000*(w&65535)+(w>>16))
#define MWC ((znew<<16)+wnew )
#define SHR3 (jsr^=(jsr<<17), jsr^=(jsr>>13), jsr^=(jsr<<5))
#define CONG (jcong=69069*jcong+1234567)
#define FIB ((b=a+b),(a=b-a))
#define KISS ((MWC^CONG)+SHR3)
#define LFIB4 (c++,t[c]=t[c]+t[UC(c+58)]+t[UC(c+119)]+t[UC(c+178)])
#define SWB (c++,bro=(x<y),t[c]=(x=t[UC(c+34)])-(y=t[UC(c+19)]+bro))
#define UNI (KISS*2.328306e-10)
#define VNI ((long) KISS)*4.656613e-10
#define UC (unsigned char) /*a cast operation*/
typedef unsigned long UL;
/* Global static variables: */
static UL z=362436069, w=521288629, jsr=123456789, jcong=380116160;
static UL a=224466889, b=7584631, t[256];
/* Use random seeds to reset z,w,jsr,jcong,a,b, and the table
static UL x=0,y=0,bro; static unsigned char c=0;
/* Example procedure to set the table, using KISS: */
void settable(UL i1,UL i2,UL i3,UL i4,UL i5, UL i6)
{ int i; z=i1;w=i2,jsr=i3; jcong=i4; a=i5; b=i6;
for(i=0;i<256;i=i+1) t[i]=KISS;
/* This is a test main program. It should compile and print 7
0's. */
int main(void){
int i; UL k;
for(i=1;i<1000001;i++){k=LFIB4;} printf("%u\n", k-1064612766U);
for(i=1;i<1000001;i++){k=SWB ;} printf("%u\n", k- 627749721U);
for(i=1;i<1000001;i++){k=KISS ;} printf("%u\n", k-1372460312U);
for(i=1;i<1000001;i++){k=CONG ;} printf("%u\n", k-1529210297U);
for(i=1;i<1000001;i++){k=SHR3 ;} printf("%u\n", k-2642725982U);
for(i=1;i<1000001;i++){k=MWC ;} printf("%u\n", k- 904977562U);
for(i=1;i<1000001;i++){k=FIB ;} printf("%u\n", k-3519793928U);
Write your own calling program and try one or more of
the above, singly or in combination, when you run a
simulation. You may want to change the simple 1-letter
names, to avoid conflict with your own choices. */

/* All that follows is comment, mostly from the initial post. You may want to remove it */

/* Any one of KISS, MWC, FIB, LFIB4, SWB, SHR3, or CONG can be used in an expression to provide a random 32-bit integer.

The KISS generator, (Keep It Simple Stupid), is designed to combine the two multiply-with-carry generators in MWC with the 3-shift register SHR3 and the congruential generator CONG, using addition and exclusive-or. Period about 2^123. It is one of my favorite generators.

The MWC generator concatenates two 16-bit multiply- with-carry generators, x(n)=36969x(n-1)+carry, y(n)=18000y(n-1)+carry mod 2^16, has period about 2^60 and seems to pass all tests of randomness. A favorite stand-alone generator---faster than KISS, which contains it.

FIB is the classical Fibonacci sequence x(n)=x(n-1)+x(n-2),but taken modulo 2^32. Its period is 3*2^31 if one of its two seeds is odd and not 1 mod 8. It has little worth as a RNG by itself, but provides a simple and fast component for use in combination generators.

SHR3 is a 3-shift-register generator with period 2^32-1. It uses y(n)=y(n-1)(I+L^17)(I+R^13)(I+L^5), with the y's viewed as binary vectors, L the 32x32 binary matrix that shifts a vector left 1, and R its transpose. SHR3 seems to pass all except those related to the binary rank test, since 32 successive values, as binary vectors, must be linearly independent, while 32 successive truly random 32-bit integers, viewed as binary vectors, will be linearly independent only about 29% of the time.

CONG is a congruential generator with the widely used 69069 multiplier: x(n)=69069x(n-1)+1234567. It has period 2^32. The leading half of its 32 bits seem to pass tests, but bits in the last half are too regular.

LFIB4 is an extension of what I have previously defined as a lagged Fibonacci generator: x(n)=x(n-r) op x(n-s), with the x's in a finite set over which there is a binary operation op, such as +,- on integers mod 2^32, * on odd such integers, exclusive-or(xor) on binary vectors. Except for those using multiplication, lagged Fibonacci generators fail various tests of randomness, unless the lags are very long. (See SWB below). To see if more than two lags would serve to overcome the problems of 2-lag generators using +,- or xor, I have developed the 4-lag generator LFIB4 using addition: x(n)=x(n-256)+x(n-179)+x(n-119)+x(n-55) mod 2^32. Its period is 2^31*(2^256-1), about 2^287, and it seems to pass all tests---in particular, those of the kind for which 2-lag generators using +,-,xor seem to fail. For even more confidence in its suitability, LFIB4 can be combined with KISS, with a resulting period of about 2^410: just use (KISS+LFIB4) in any C expression.

SWB is a subtract-with-borrow generator that I developed to give a simple method for producing extremely long periods: x(n)=x(n-222)-x(n-237)- borrow mod 2^32. The 'borrow' is 0, or set to 1 if computing x(n-1) caused overflow in 32-bit integer arithmetic. This generator has a very long period, 2^7098(2^480-1), about 2^7578. It seems to pass all tests of randomness, except for the Birthday Spacings test, which it fails badly, as do all lagged Fibonacci generators using +,- or xor. I would suggest combining SWB with KISS, MWC, SHR3, or CONG. KISS+SWB has period >2^7700 and is highly recommended. Subtract-with-borrow has the same local behaviour as lagged Fibonacci using +,-,xor---the borrow merely provides a much longer period. SWB fails the birthday spacings test, as do all lagged Fibonacci and other generators that merely combine two previous values by means of =,- or xor. Those failures are for a particular case: m=512 birthdays in a year of n=2^24 days. There are choices of m and n for which lags >1000 will also fail the test. A reasonable precaution is to always combine a 2-lag Fibonacci or SWB generator with another kind of generator, unless the generator uses *, for which a very satisfactory sequence of odd 32-bit integers results.

The classical Fibonacci sequence mod 2^32 from FIB fails several tests. It is not suitable for use by itself, but is quite suitable for combining with other generators.

The last half of the bits of CONG are too regular, and it fails tests for which those bits play a significant role. CONG+FIB will also have too much regularity in trailing bits, as each does. But keep in mind that it is a rare application for which the trailing bits play a significant role. CONG is one of the most widely used generators of the last 30 years, as it was the system generator for VAX and was incorporated in several popular software packages, all seemingly without complaint.

Finally, because many simulations call for uniform random variables in 0<x<1 or -1<x<1, I use #define statements that permit inclusion of such variates directly in expressions: using UNI will provide a uniform random real (float) in (0,1), while VNI will provide one in (-1,1).

All of these: MWC, SHR3, CONG, KISS, LFIB4, SWB, FIB UNI and VNI, permit direct insertion of the desired random quantity into an expression, avoiding the time and space costs of a function call. I call these in-line-define functions. To use them, static variables z,w,jsr,jcong,a and b should be assigned seed values other than their initial values. If LFIB4 or SWB are used, the static table t[256] must be initialized.

A note on timing: It is difficult to provide exact time costs for inclusion of one of these in-line- define functions in an expression. Times may differ widely for different compilers, as the C operations may be deeply nested and tricky. I suggest these rough comparisons, based on averaging ten runs of a routine that is essentially a long loop: for(i=1;i<10000000;i++) L=KISS; then with KISS replaced with SHR3, CONG,... or KISS+SWB, etc. The times on my home PC, a Pentium 300MHz, in nanoseconds: FIB 49;LFIB4 77;SWB 80;CONG 80;SHR3 84;MWC 93;KISS 157; VNI 417;UNI 450; */