<johnps11 <at> bigpond.com> writes:

The issue is almost certainly in atRandom:

Running

30 timesRepeat: [ | a | a:= (2 raisedTo: 57) atRandom. Transcript show: a ; show: ' ' ; show: a even ; cr ].

Gives me 30 falses.

The odds of that are less than one in a billion (assuming uniformly distributed integers).

This doesn't happen for 2^56 or lower, and does for all 2^n, n>= 57.

My guess is that it's either a hardware issue or something about Random. I'm not clever enough to understand PRNGs, so I'll leave it others to work out what the answer is, although I suspect that for more than 56 bits you need a PRNG that uses LargeIntegers and not Floats.

Nothing surprising once again. Float have a 53 bits integer mantissa m and and a biased exponent e: f := m * (2 raisedTo: e). m highBit = 53. The PRNG has a uniform distribution on [0,1). f < 1 means 100% of floats have e < -53. Let us divide the uniform intervals in two recursively:

50% have e = -54 (0.5<=f<1) 50% have e < -54 (f<0.5)

25% have e = -55 (0.25<=f<0.5) 25% have e < -55 (f<0.25) 75% have e >=-55 (f>=0.25)

12.5% have e = -56 (0.125<=f<0.25) 12.5% have e < -56 (f<0.125) 87.5% have e >=-56 (f>=0.125)

6.25% have e = -57 (0.0625<=f<0.125) 6.25% have e < -57 (f<0.0625) 92.75% have e >=-57 (f>=0.0625)

3.125% have e = -58 (0.03125<=f<0.0625) 3.125% have e < -58 (f<0.03125) 96.875% have e >=-58 (f>=0.03125)

So when you multiply f by (2 raisedTo: 57), that means (m bitShift: e+57), 87.5% have (e+57>0), thus are shifted at least one bit to the left, and you get 87.5% of even numbers. since algorithm truncate then add 1, you then get 87.5% of odd results

I suspect that due to division by 16r7FFFFFFF, last bit of mantissa m is often zero. In which case, you get 92.75% of odd results.

Or maybe last two bits, in which case you have 96.875% of odd results.

((1 to: 10000) count: [:i | (2 raisedTo: 57) atRandom odd]) / 10000.0 . -> 0.9692 Yes, probably last two bits of these floats are set to 0...

The limit is by no way (2 raisedTo: 57).

The true limit at which you won't get any even number ? well, smallest generated pseudo random float is 1/(2^31-1)

A little higher than 1.0 timesTwoPoser: -31. smallest := ((1.0 timesTwoPower: 31) - 1) reciprocal. smallest significandAsInteger printStringBase: 2. -> '10000000000000000000000000000001000000000000000000000'

The biased exponent is then (-31-52=-83). (smallest significandAsInteger asFloat timesTwoPower: smallest exponent-52) = smallest -> true

So, (2 raisedTo: 84) atRandom is guaranteed to be odd.

We conjectured that last two bits are zero, maybe (2 raisedTo: 82) atRandom is already 100% odd? (1 to: 10000000 by: 11) detect: [:e | ((e * smallest) significandAsInteger bitAnd: 2r11) > 0] ifNone: [nil] ->1048587 Hmm the conjecture is false...

Floats are tricky, but once you understand they are just rational with denominator being a power of two and inexact operations truncating the mantissa at 53 bits, the mystery vanishes a bit.

Nicolas