### Table of Contents

# Random generator using XORshift

## Introduction

Prof. G. Marsaglia in 1995 published the well-known DIEHARD set of statistical tests for measuring the quality of random generators. In 2003 he published a novel way of generating random numbers, called the XORshift generator. It is fast and has good quality. It is also easy to implement and can be adapted to the needs of the user. The method works with any number of seeds and can be adapted to 16, 32 or 64 bits (or larger…)

## Principle of the Marsaglia algorithm

A random number is generated from a seed as follows:

- Take a copy of the seed
- Shift that copy left with a certain number of bits
- xor the original seed and the shifted seed to form the result of the first step
- Repeat these steps twice, each time with the output of the previous step as input, but with a right shift and finally with a left shift.

The actual number of bits used for the 3 shifts is critical. For a 32bit generator there are exactly 648 valid combinations. The favourite combination of Marsaglia was (13, 17, 5) which is used here.

For 16bit generators only a few valid combinations exist: (7, 9, 13) and (7, 9, 8).

For 64 bit generators 2200 valid combinations exist. Examples are: (24, 31, 35) or (19, 41, 21) and many others. See the link below for exact details

The Forth-code is probably easier to understand than this explanation…

### The algorithm in pseudo-code

variable seed - any value but 0x0 is acceptable as seed. Function: XORshift - get value from seed - make a copy of the value lshift copy with n bits xor with the original - repeat with the output of the previous cycle and a rshift with m bits - repeat with the output of the previous cycle and a lshift with o bits - return the result on stack - store the final result in the variable seed for a next cycle

### Generic Forth

\ 32bit version with 1 seed in a variable variable seed 2345 seed ! \ start the seed with any number but 0 : FORTHRANDOM1 ( address_seed -- rndm_val ) dup >r @ dup 13 lshift xor dup 17 rshift xor dup 5 lshift xor dup r> ! ;

## Adding a second (or more) seed

For the 32 bit version 1 seed is not enough to pass the DIEHARD quality tests, at least 2 seeds are needed. Adding an extra seed takes a few extra steps.

- first seed is input to the three cycles
- do the 3 cycles and return the final result
- update the seeds: second seed is copied into the first seed
- OR result of the 3 cycles with the value of the second seed

This way of adding more seeds can be done ad infinitum. The more seeds you add, the longer the wrap-to-zero period. With two 32 bit seeds, the wrap-to-zero period is 2^{64}-1 values.

### The algorithm with 2 seeds in pseudo-code

\ 2 seeds version variable seed1 variable seed2 Function: XORshift get value from seed1 make a copy of the value lshift copy with n bits xor with the original value repeat with the output of the previous cycle and a rshift with m bits repeat with the output of the previous cycle and a lshift with o bits return final the result on stack move value of seed2 to variable seed1 OR the final result with the value in seed2 for a next cycle

### Generic Forth version

\ 32 bit version with 2 seeds in values: 2345 value SEED0 6789 value SEED1 : FORTHRANDOM2 ( -- u ) seed0 \ put seed0 on stack seed1 to seed0 \ move value in seed1 to seed0 dup 13 lshift xor \ do three XORs of seed0 dup 17 rshift xor \ with shifted copies of itself dup 5 lshift xor dup seed1 xor to seed1 \ XOR the new random value with ; \ the old seed1 and update seed1

#### For 16 bit Forths:

The only change to the code are the 3 shift-factors. Here (7, 9, 13) are used. You can also use (7, 9, 8).

2345 value SEED0 6789 value SEED1 : FORTHRANDOM16 ( -- u ) \ for 16b Forth seed0 seed1 to seed0 dup 7 lshift xor dup 9 rshift xor dup 13 lshift xor dup seed1 xor to seed1 ;

#### Implementations:

#### A few points to note:

There is no limit to the number of seeds. If you want to use a thousand seeds, you can. The method functions fine with that. In that case it would be more efficient to put the seeds in a table and read and write to the table with two pointers. But it is hard to imagine a use-case where there is a need for more than 256 bits of seeds, so for instance eight 32 bit seeds. This gives a wrap-to-zero period of 2^{256}-1. Even if you generate 1 bilion values per second, the universe would cease to exist before the generator wraps to the start.

It is good practise to pre-load the generator after re-seeding by generating dummy random numbers. For each seed you use, generate at least 4 dummy numbers. So with 2 seeds 8 dummy numbers would be the minimum. This pre-loading ensures that in all cases a good quality stream of numbers is generated.

At least one seed must have one of the bits set for the generator to work.

#### Finally a handy word:

\ CHOOSE - limits the output of a random-generator to a range between 0 and u1 in a correct way. : CHOOSE ( u1 - u2 ) random um* nip ;

### Links:

- DIEHARD, Description of the Diehard tests
- George Marsaglia, XORShift Random Number Generators, Journal of Statistical Software 2003.
- Chapter 70 of the Egel-project, shows some other random-generators and a graphical test based on DIEHARD to show the effect of low-quality generators.