World: r3wp

anyway, this is always a problem, when we try to generate numbers 
in an "exotic" interval, not in the usual [0.0 .. 1.0]

I'll just grab some food, then I look at your r-uni.

hmm, seems, that the former may still produce X when rounding?

In any case, this chat should be the basis for the docs that go with 
it. People aren't aware of how much thought goes into these things.

>> two**62 / two**53+1
== 512.0

So two**53+1 * 512 should equal two**62, but it doesn't:

>> two**62
== 4611686018427387904
>> two**53+1 * 512
== 4611686018427388416

Why is that?

(I wanted to check , if d // two**53+1 gave an even distribution, 
but there seem to be something wrong with calculations of large numbers.)

Why is that?
 - rounding

This should give an math overflow, I think:

>> to integer! 2 ** 62
== 4611686018427387904

Like this does in R2:

>> to integer! 2 ** 31
** Math Error: Math or number overflow

no, R3 uses 64-bit integers

no wait, it's 2 ** 63, that should give overflow.

yeah

But!? :-)

>> to integer! 2 ** 62 - 1
== 4611686018427387904
>> to integer! 2 ** 62
== 4611686018427387904

Isn't that a bug?

no, it is the precision limit of IEEE754

I see.

(I mean, it exceeds the precision limit)

Yes, I understand.

After you do:

d: d // two**53+1


Are we sure, the new d is even distributed? That every value, d can 
now take, is equal possible?

hmm, to make sure, I should use rejection

as follows:

rnd: func [
	{random range with rejection, not using last bits}
	value [integer!]
	/local r s
] [
	s: two-to-62 - (two-to-62 // value)
	until [
		r: random two-to-62
		r <= s
	]
	s: s / value
	r - (r // s) / s + 1
]

If all possible values of d is equal possible, and d now can have 
values from 0 to two**53, then

d / two**53

will return values from 0.0 to 1.0 both inclusive, and 

d / two**53 * x

will return values from 0.0 to x both inclusive. Right?

yes

...multiplication may still trigger rounding, though

It give same result as your first RANDOM:

d: to-decimal to-binary 1023
blk: []
insert/dup blk 0 1024
random/seed now

loop 1000000 [i: to-integer to-binary r-uni d blk/(i + 1): blk/(i 
+ 1) + 1]

>> print [blk/1 blk/512 blk/1024]
462 1036 481

that is caused by the multiplication rounding

yes

I like it better than the version, where max is changed to 0.0

another variant would be to generate the uniform deviates in the 
definite interval not allowing multiplication to "mess things"

Is r-uni better than the version, you gave earlier?

tt62: to integer! 2 ** 62
r: func [x [decimal!]] [(random tt62) - 1 / tt62 * x]

r-uni is integer, guaranteeing uniformity by using rejection

sorry, ignore my post

r-uni surely differs from r when X is equal to 1.0

(since in that case no additional rounding occurs)

yeah, r-uni is better, I think.

Nice talk. Time for me to move to other things...

bye

for random 1.0 you cannot find any irregularities, there aren't any


I think, there are. Decimals with a certain exponent are equal spaced, 
but there are many different exponents involved going from 0.0 to 
1.0. The first normalized decimal is:

>> to-decimal #{0010 0000 0000 0000}
== 2.2250738585072e-308

The number with the next exponent is:

>> to-decimal #{0020 0000 0000 0000}
== 4.4501477170144e-308

I can take the difference:


>> (to-decimal #{0020 0000 0000 0000}) - to-decimal #{0010 0000 0000 
0000}
== 2.2250738585072e-308

and see the difference double with every new exponent:


>> (to-decimal #{0030 0000 0000 0000}) - to-decimal #{0020 0000 0000 
0000}
== 4.4501477170144e-308

>> (to-decimal #{0040 0000 0000 0000}) - to-decimal #{0030 0000 0000 
0000}
== 8.90029543402881e-308

>> (to-decimal #{0050 0000 0000 0000}) - to-decimal #{0040 0000 0000 
0000}
== 1.78005908680576e-307

So doing
random 1.0

many times with the current random function will favorize 0.0 a great 
deal. The consequence is, 0.0 will come out many more times than 
the first possible numbers just above 0.0, and the mean value will 
be a lot lower than 0.5.

The space between possible decimals around 1.0 is:


>> (to-decimal #{3ff0 0000 0000 0000}) - to-decimal #{3fef ffff ffff 
ffff}
== 1.11022302462516e-16

The space between possible decimals around 0.0 is:

>> to-decimal #{0000 0000 0000 0001}
== 4.94065645841247e-324


That's a huge difference. So it'll give a strange picture, if converting 
the max output of random (1.0 in this case) to 0.0.

It's easier to illustrate it with an image: http://www.fys.ku.dk/~niclasen/rebol/random-dist.png

The x-axis is the possible IEEE 754 numbers going from 0.0 to 1.0. 
The y-axis is how many 'hits' ever possible number gets, when doing 
RANDOM 1.0. Every gray box holds the same amount of possible number, 
namely 2 ** 52. I use the color to illustrate the density of numbers. 
So the numbers lie closer together at 0.0 than at 1.0.


The destribution is of course flat linear, if the x-axis was steps 
of e.g. 0.001 or something. There is the same amount of hits between 
0.001 and 0.002 as between 0.998 and 0.999. It's just, that there 
are many more possible numbers around 0.001 than around 0.999 (because 
of how the standard IEEE 754 works).

It should be clear, that it's a bad idea to move the outcome giving 
1.0 to 0.0, as is done now with the current RANDOM function in R3.

I added another comment to ticket #1027 in curecode.

but, you did not take into account, that the spacing of 4.94065645841247e-324 
is not used

(by the implementation)

Oh! Yes, I didn't have that in mind. So the smallest result larger 
than zero from RANDOM 1 is:

>> tt62: to integer! 2 ** 62
>> 1 / tt62
== 2.16840434497101e-19

It's still smaller than 1.11022302462516e-16

Can RANDOM 1.0 produce a result equal to 1.0 - 1.11022302462516e-16 
?

Hm, this is not trivial! :-)

Yes, it can. If
random tt62
result in tt62 - 257

>> 1.0 - (tt62 - 257 / tt62)
== 1.11022302462516e-16

So the problem is there, just not as big as I first thought.

The first random function was:

tt62: to integer! 2 ** 62
r: func [x [decimal!]] [(random tt62) - 1 / tt62 * x]


and if (random tt62) - 1 result in tt62 - 257, the space between 
numbers are smaller than if (random tt62) - 1 result in 1. Hope I 
make sense.

yes. If the difference is detectable by a test, then we should change 
the implementation

I found some interesting observations in working with Random data 
recently.  I was mostly working with the RandomMilliondigit.bin file 
data that is used to test compression algorithms.  It exhibits a 
characteristic in that repetitious data is ascends in almost a 1-2 
ratio.

Additionally, the distribution of bits are tight.  Meaning that the 
distribution of 10, 01, 00, 10 bit sequences for example are very 
close to the same quantity.

For example, in set of data you might find how many occurences of 
3 '0 bit squences you have in the data.  Once you now the number 
of the occurences then for random data the 4 '0 bit sequences should 
be about half that and so on.