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fractint-digest        Thursday, April 27 2000        Volume 01 : Number 471




----------------------------------------------------------------------

Date: Tue, 25 Apr 2000 20:55:49 -0400
From: Ron Barnett <[e-mail]>
Subject: RE: (fractint) Pretty mandel

Hi caren,
Actually I was living in New Joyzee during the Fractint/Fractal Frenzy II
times. I am now in the mountains of very northern New York. Are you still
running a book store (I think that is what you were doing back then). Check
out my website if you have a chance.
Ron Barnett
http://www.hiddendimension.com
A fractal art and music site
in the Adirondacks


On Tuesday, April 25, 2000 20:07, caren park [SMTP:[e-mail]] wrote:
> talk about a ghost from the past, son!  still living in oregon, or am i
> thinking of someone else in my senility?  :)
>
> ----- Original Message -----
> From: Ron Barnett <[e-mail]>
> To: <[e-mail]>
> Sent: Tuesday, April 25, 2000 4:50 PM
> Subject: RE: (fractint) Pretty mandel
>
>
> > Wow,
> > a ghost from the past! Welcome back caren.
> > Ron Barnett
> >
> > On Tuesday, April 25, 2000 13:15, caren park [SMTP:[e-mail]]
> wrote:
> > > was the PSP image (bottom of that web page) paint shop pro-enhanced,
or
> was
> > > this another feature of fractint 20 (or ultra-frac)?
> > > i liked it...  LOTS...
> > >
> > > ----- Original Message -----
> > > From: Sylvie Gallet <[e-mail]>
> > > To: <[e-mail]>
> > > Sent: Tuesday, April 25, 2000 9:21 AM
> > > Subject: (fractint) Pretty mandel
> > >
> > >
> > > Hi Lee,
> > >
> > > >> Very nice!
> > >
> > >   Thanks!
> > >
> > > >> You had time already to generate a very small thumbnail?<g>
> > >
> > >   Oh, no!  I generated it at 1600x1200 on my PXV 30K!!! ;-)
> > >
> > >   The 1600x1200 GIF along with some 800x600 JPGs are available at:
> > >
> > >     <http://www.fractalus.com/sylvie/mandel_e.htm>
> > >
> > >   Cheers,
> > >
> > >         - Sylvie
> > >
> > > --------------------------------------------------------------
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> > >
> > >
> > > __________________________________________________
> > > Do You Yahoo!?
> > > Talk to your friends online with Yahoo! Messenger.
> > > http://im.yahoo.com
> > >
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> Talk to your friends online with Yahoo! Messenger.
> http://im.yahoo.com
>
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------------------------------

Date: Tue, 25 Apr 2000 18:13:02 -0700
From: Guy Hammer <[e-mail]>
Subject: Blast from the past; (was: (fractint) Pretty mandel)

Ron writes:
>Actually I was living in New Joyzee during the Fractint/Fractal Frenzy II
>times. I am now in the mountains of very northern New York.

Caren may have been thinking of me. I was, and still am living in
NW Oregon. :-)

Regards, Guy

PS: Anyone heard from Bob Dodson (sp?) lately?


[e-mail] (Guy Hammer)
http://www.teleport.com/~guyh

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------------------------------

Date: Tue, 25 Apr 2000 21:30:38 -0400
From: Lee Skinner <[e-mail]>
Subject: (fractint) Pretty mandel

Hi Miguel,

>> PS : The web page says "4700 hs. on a P300", while the posted par said=

it was a P233....? <<

It was computed on a P300 at my office.  I made the par on my home comput=
er
(a P233) and I forgot to correct the CPU speed in the automated comment.

Lee

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------------------------------

Date: Tue, 25 Apr 2000 21:30:38 -0400
From: Lee Skinner <[e-mail]>
Subject: (fractint) Pretty mandel

Hi Miguel,

>> PS : The web page says "4700 hs. on a P300", while the posted par said=

it was a P233....? <<

It was computed on a P300 at my office.  I made the par on my home comput=
er
(a P233) and I forgot to correct the CPU speed in the automated comment.

Lee

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------------------------------

Date: Wed, 26 Apr 2000 05:21:29 +0200
From: Guy Marson <[e-mail]>
Subject: Re: (fractint) MandelbrotMix4 lakes

At 15:12 24/04/00 -0400, you wrote:
>Hi All,
>
>  Here's a couple of MandelbrotMix4 Lake example.  Both formulas are based
>on Jim Muth's MandelbrotMix4, the original parameters used in MMix4 (p1,
>p2, p3) are now hard-coded in the formula (renamed as p_p1,...), in order
>to make p2 and p3 available for the lake effect.
>

Oohhh.. thanks Sylvie, Jim! That's a b'ful playground for little animations
(and tests)!!

cheers,

Guy


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------------------------------

Date: Wed, 26 Apr 2000 00:56:54 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD, 26-04-00  (Worth a Diamond [5])

FOTD -- April 26, 2000  (Rating 5)

Fractal visionaries and enthusiasts:

I named today's FOTD "Worth a Diamond".  The rather vague
diamond shape around the electric blue minibrot in the
background inspired the name.  The formula that drew the fractal
in which I found the diamond is Z^(-1.2)-(Z^1.2)+C.

This parent fractal consists of two disconnected and quite
distorted Mandeloids.  I can tell just by looking at these
twisted figures that they are filled with interesting midgets.
In fact, I might devote an entire week to investigating this
particular two-part fractal.

The parameter file runs in just over one minute on a modest
Pentium.  This is probably the most simple way to view the
image.  But for those who would rather see the image already
rendered, it has been posted to:

          <alt.binaries.pictures.fractals>

and to:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

The fractal weather was quite unpleasant today, as a steady
chilly rain fell and the temperature hovered around 48F (9C).
These conditions kept the cats safely ensconced indoors all day,
their frustration building as the day progressed.  Luckily, a
good meal of their favorite food kept their frustration under
control.

My philosophical musings also grew as the day progressed, but
they never reached the level of being worthy of being made
public.  As usual, I'll try again tomorrow.  Until then, take
care, and enjoy today's fractal.


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

Worth_a_Diamond    { ; time=0:01:06.19, SF5 on a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=ident
  center-mag=+0.14392839195860990/+2.73449807098121600\
  /16024.72/1/10 params=-1/1.2/1/-1.2/0/0 float=y
  maxiter=1200 bailout=25 inside=0 logmap=22
  symmetry=none periodicity=10
  colors=000b6H<2>V6DT6CP6BM6AI69G68I89L99PA9VC9Y6AYDL\
  cJKfOF<3>qXG<6>V_bS_eO_h<3>C`t<2>I_b<5>QXoRWqTWs<2>X\
  UyYUzZTz_Tz<3>IQz<3>PTzRUzSUz<2>QezPdz<2>Ocz<5>bczdc\
  zgcz<3>pcz<3>Kcz<3>HczGczFcz<3>HczHczIczIczJczJcz<3>\
  CczBcz9cz<3>3cz<6>KczMczOczRczTczVcz<3>jczhczEcz9cz<\
  3>IczKczNcz<3>Vez<4>LgzJhzHhz<3>Ajz<3>ClzDmzDmz<3>Fm\
  zFmzFmzDmzBmz<2>smz<3>qmzqmzqmz<2>pmz<7>QmzMmzJmz<2>\
  9mz6mz5mz<7>4mz4mz3mz<3>3mz<5>PmzTmzWmz<3>imz<5>cmz<\
  3>_mz_mzRmz<6>`mzbmzcmz<3>hmz<3>Smz<7>Emzgmzdmz
  }

frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}

END 20.0 PAR-FORMULA FILE==================================


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------------------------------

Date: Wed, 26 Apr 2000 18:37:50 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: (fractint) sqr() function considered harmful.

I took some time out to do some speed tests of Fractint's formula parser -
in particular, how it handles various ways of representing integer powers
of a complex number. I was surprised by the results.

I made these tests on the standard v20 distribution, after booting to DOS
so that my timings wouldn't be blurred by Windows rearranging Fractint's
memory in the background (though Fractint may not necessarily notice, it
was just as easy to avoid the possibility as ignore it).

I evaluated c^n using floating-point arithmetic about two million times in
succession for random values of c, where n ranged from 0 to 16 using the
following formula as a template:

{z=0,c=rand:c=[c^n],|z|<4}

c^n was evaluated in three different ways.

First off was the obvious "c^n" construct itself, e.g., c=c^5.
Second was to use repeated multiplication, e.g., c=c*c*c*c*c
And third was to use sqr() as much as possible to avoid repeated
expressions: e.g, c^5=sqr(sqr(c))*c, c^7=sqr(sqr(c)*c)*c,
c^8=sqr(sqr(sqr(c))).

I wrote an autokey script to do the boring work of running all these tests,
saving each completed test as an image, then took the timing information
from the resulting series of gifs.

Anyhow, here are the timings:

 n  power     mult      sqr
 0  1:29.74   1:08.28   1:08.28
 1  1:44.36   1:03.22   1:03.17
 2  1:17.83   1:18.21   1:19.09
 3  1:17.67   1:33.43   1:34.36
 4  1:17.61   1:50.24   1:35.57
 5  1:17.55   2:05.17   1:50.95
 6  1:17.50   2:20.23   1:51.50
 7  1:17.45   2:35.87   2:08.80
 8  1:17.39   2:52.63   1:51.17
 9  1:17.39   3:07.07   2:10.29
10  1:17.33   3:23.67   2:11.77
11  1:17.39   3:37.07   2:26.32
12  1:17.39   3:50.97   2:10.89
13  1:17.39   4:12.00   2:27.03
14  1:17.39   4:21.94   2:24.62
15  1:17.33   4:36.65   2:40.66
16  1:17.34   4:55.88   2:07.43

When n is equal to zero or one, both repeated multiplication and repeated
squaring result in the same expressions, respectively "1" and "c". So the
similarity in their timings for these two values of n is inevitable.

The ^ operator has trouble with the special cases c^0=1 and c^1=c;
otherwise, it consistently gave times of just over 1 minute 17.5 seconds.

The fluctuations in the sqr() timings is to be expected, given the
variations in the number of operations required for each n.

But the surprise is that sqr() did so poorly against the others. Above n=3,
the power function was far and away the fastest, and repeated
multiplication equally obviously the slowest; but even for the case n=2 (in
which you would expect to to shine), sqr(c) was still noticeably _slower_
than c^2, or even c*c.

Of course, the timed difference between c^2 and sqr(c) is only a matter of
about a second here, but remember that this is over the course of just a
minute's work at a (trivial) image.

Going by the above timings, it appears that if you're raising a number z to
the power of some fixed integer n, the fastest way of doing it is to write
"z^n" when n is 2 or more, and simply to write "1" or "z" if n is zero or one.

Repeated multiplication appears to be pretty much guaranteed to be slower,
and even sqr(z) is slower than z^2.

These results appear to contradict the usual advice given - use sqr(z) when
you're squaring, and repeated multiplication instead of exponentiation for
small values of n. I would be interested in seeing similar timings made by
others to see just which code is faster than which.

Morgan L. Owens
"Good grief - you do two _million_ complex exponentiations and you're
complaining about a delay of _one second_??!"



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------------------------------

Date: Wed, 26 Apr 2000 19:28:23 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) Mime-Version: 1.0

At 14:32 25/04/2000 -0400, you wrote:
>Hi Morgan,
>
>>> Here's a little lsystem idea I thought I'd try out after looking at
>>> reptiles too long.
>
>>> Pentominoes{ ; What's the collective noun for reptiles?
>
>  Wow!!!  It's great!
>
>  Cheers,
>
>        - Sylvie
>
Thank you!

Morgan L. Owens
"Yes, boss ... Uh, would you believe me if I it's a Perl format string
generator...?"



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------------------------------

Date: Wed, 26 Apr 2000 22:23:01 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) minibrots / mandeloids (whatever-you-call-'ems)

At 17:44 25/04/2000 GMT, Rupert Millard wrote:
>Hello all,
>
>I find it fascinating that the 'mandelbrot shape' (I really *don't* know
>what to call it!) occurs in so many different formulae. I have tried looking
>to see what mandel, magnet1m, magnet2m, mandellamba, nova etc. have in
>common with each other but *I* can't see anything. <:-(
>
What they have in common with each other ... well, it's more a matter with
what they have in common with the standard Mandelbrot set.

Consider the graph of a function. Just a standard wiggly line on graph
paper - no need at this point to worry about complex-valued functions of a
complex variable :-).

You'll notice that a typical graph has humps and hollows and things like
that. Zoom in on one of these hollows and you can for the most part ignore
the rest - they could easily be different and the hollow you're looking at
could still look pretty much the same. This is a quick hint of what's
described in Beauty of Fractals as "universality".

Now broaden your view out into another dimension. Instead of just having a
bumpy line, have a bumpy surface. Peaks, valleys, passes and all sorts of
other topography. Different features in the landscape produce different
structures in the resulting fractal. (If you really want, you can try
imagining that height above sea level is actually a complex number - which
adds yet another whole dimension to your catalogue of possible features and
possible structures).

But again, one hollow looks pretty much like another.

In a (somewhat metaphorical) sense, the graph of z^2+c consists entirely of
one single hollow - a bowl shape.

So now you set your point wandering all over this surface, following
whatever orbit its function (magnet, nova or whatever) describes. Now and
then your point may find itself in a bowl. Now, since one bowl looks pretty
much like another your point will act pretty much the same in each one (a
distant peak has less influence than a bowl that's _right there_).
Depending on how closely you zoom in and how little you care about the
surrounding countryside, you can approximate the bowl your point finds
itself in with something that looks more like your generic standard-issue
z^2+c, and less like whatever wild construction you _really_ have. And we
know how points in bowls behave, don't we?

The distortions that inevitably creep into the Mandelbrot sets that pop up
all over (stretching, colliding and worse) all these other functions is a
consequence of the fact that the functions _aren't_ really z^2+c after all:
if two fairly large bowls are close together, for example, their
Mandelbrot-set wannabes might collide. And of course there are all the
other structures being created by all the other features of the surface.

In short, a Mandelbrot-set-like thing shows up whenever and whereever a
function (however wild) looks _locally_ like something that could be
approximated by z^2+c: what I've be describing as a "bowl".

(Could probably explain this better. Want to know where it could be improved.)

Morgan L. Owens
"Who admits to speaking in parables on occasion."





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------------------------------

Date: Wed, 26 Apr 2000 15:25:30 -0300
From: "Fliguer, Miguel" <[e-mail]>
Subject: RE: (fractint) minibrots / mandeloids (whatever-you-call-'ems)

>In short, a Mandelbrot-set-like thing shows up whenever and whereever a
>function (however wild) looks _locally_ like something that could be
>approximated by z^2+c: what I've be describing as a "bowl".

By the same token, a n-lobe-distorted-M-set (or a picture of
Mona Lisa) should show up whenever the above wild chaotic function
is calculated on an area where it can be approximated by the
n-lobe-distorted-M-set formula (or the formula for the picture
of Mona Lisa). Are you still with me ?

After years of complex plane exploring, I can say I found a
lot of standard M-sets on many different formulas, but not
many n-lobe-distorted-M-sets, and still not a single Mona Lisa ;-)

So I guess the simpler the formula (i.e. z*z+c ), the greater
the probability for the associated shape (i.e the standard M-set)
to appear in the middle of chaos.

The above statements may be pure nonsense, of course. Anyway,
I enjoyed writing them. Fetch me some aspirins, please.

Just my $0.02 pesos
Regards,
Miguel Fliguer
Franktal Gallery - Shut Up and Draw Yer Fractals
http://members.xoom.com/fliguer/franktal.html

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------------------------------

Date: Wed, 26 Apr 2000 20:37:49 +0100
From: "Iain G. Stirling" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful.

Morgan L. Owens wrote:

> I took some time out to do some speed tests of Fractint's formula parser -
> in particular, how it handles various ways of representing integer powers
> of a complex number. I was surprised by the results.
>

Morgan,

This type of analysis is what I find most interesting about computing,
so I quickly set about producing my own set of results.

I used the following (clumsy) procedure:

Step 1. Create formulas for each of the tests in a .FRM file, e.g:

exp0 {z=0,c=rand,c=c^0,|z|<4}
sqr3 {z=0,c=rand,c=sqr(c)*c,|z|<4}

Step 2. Create a .PAR file, each entry testing one of the formulas, e.g:

sqr3          {
  reset=2000 type=formula formulafile=test.frm formulaname=sqr3
  passes=1 center-mag=0/0/0.6666667 float=y cyclerange=1/1
  colors=00000e
  }

Step 3. Create a .BAT file, running each PAR entry and saving the
image, e.g:

rem fractint @test.par/sqr2 BATCH=y VIDEO=AF2 SAVENAME=sqr2
rem fractint @test.par/sqr3 BATCH=y VIDEO=AF2 SAVENAME=sqr3

I used video mode AF2 (1600 x 1200 x 2 colours diskvideo), which
tests each formula 1,920,000 times. I then read the times from the
images after restoring them in Fractint.

Here are my times:

n  power    mult        sqr
0  7:48.46  5:47.40  5:44.44
1  8:33.50  5:47.35  5:44.49
2  8:34.82  6:19.43  6:07.29
3  8:21.64  6:53.75  6:52.55

(I only ran to n=3, as I am supposed to be studying for my final exams
at the moment!)

So, this fits quite well with the conventional wisdom in this area...
This begs the question ... What is going on here?

I have been trying to think of reasons for this. Do you use an non-Intel
processor, which might give a different performance for some FP
instructions?

If I have time (possibly at the weekend), I'll continue these tests.

Iain Stirling.


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------------------------------

Date: Wed, 26 Apr 2000 16:00:07 -0400
From: Bill Jemison <[e-mail]>
Subject: (fractint) anyone remember an old lady that looked like me?

 <are you still working on fractint fractal music?  i *think*
 <that was you playing with that stuff in the Way Back When...

Yes! I should have a few pars ready shortly.

Bill

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------------------------------

Date: Wed, 26 Apr 2000 16:43:04 -0400
From: Sylvie Gallet <[e-mail]>
Subject: RE: (fractint) Pretty mandel

Hi Miguel,

>> If I understand it well, if I ever venture into the arbitrary precisio=
n
>> realm, I better switch to passes=3D1, ok ?

  Yes, and at least, with passes=3D1, you're sure to get the most accurat=
e
image.

>> What about difussion mode ?

  It seems much slower than passes=3D1.

>> And the hidden synchronous orbit mode ? (I tried to plot Lee's par wit=
h
>> this method, it finished in about 40 seconds but the image was just 4
>> purple squares, each in 1/4 of the screen)

  This mode doesn't work with arbitrary precision.  I tried it with Lee's=

par, and it just froze Fractint.

  Cheers,

        - Sylvie

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------------------------------

Date: Wed, 26 Apr 2000 16:43:06 -0400
From: Sylvie Gallet <[e-mail]>
Subject: Re: (fractint) MandelbrotMix4 lakes

Hi Guy,

>> Oohhh.. thanks Sylvie, Jim! That's a b'ful playground for little
>> animations (and tests)!!

  You're welcome!  Have fun!!!

  Cheers,

        - Sylvie

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Date: Wed, 26 Apr 2000 17:57:36 -0300
From: "Fliguer, Miguel" <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

Guy wrote :

>> Oohhh.. thanks Sylvie, Jim! That's a b'ful playground for little
>> animations (and tests)!!

Check out "Mandelbrot SunSet" at
http://members.xoom.com/fliguer/gallery5.html
for a 570K AVI which features the "lake" effect.

Regards,
Miguel


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Date: Wed, 26 Apr 2000 18:52:08 -0400
From: davides <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

At 05:57 PM 4/26/2000 -0300, you wrote:
>Check out "Mandelbrot SunSet" at
>http://members.xoom.com/fliguer/gallery5.html
>for a 570K AVI which features the "lake" effect.
>
>Regards,
>Miguel


Very nice.


[e-mail]
Back up my hard drive?
How do I put it in reverse?

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Date: Wed, 26 Apr 2000 20:23:13 -0400
From: Lee Skinner <[e-mail]>
Subject: (fractint) anyone remember an old lady that looked like me?

Bill,

>> Yes! I should have a few pars ready shortly. <<

GREAT!!  I haven't played a new music par in a long time.

Lee

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Date: Wed, 26 Apr 2000 23:18:44 -0400
From: Harry Bissell <[e-mail]>
Subject: Re: (fractint) Fractal Music : was old lady looked like me?

Yeah that reminds me... I've played a little bit with the music feature.
Did someone
have a way to save the sequences as MIDI files. Best would be direct to
a std. midi
file format I could run to a dedicated sequencer CPU  (its a '286 so
sound card is out...'eh?).

What are your favorite file types for fractal music...

H^) harry  (hear for eye AND ear candy....)



Bill Jemison wrote:

>  <are you still working on fractint fractal music?  i *think*
>  <that was you playing with that stuff in the Way Back When...
>
> Yes! I should have a few pars ready shortly.
>
> Bill
>
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Date: Thu, 27 Apr 2000 00:48:00 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD, 27-04-00  (Worth a Diamond [6])

FOTD -- April 27, 2000  (Rating 6)

Fractal visionaries and enthusiasts:

Today's fractal is a picture of another . . . you guessed it
. . . midget from the ever-changing land of MandelbrotMix4.
This particular midget exists and always has existed in
abstract form in the Mandeloid fractal created by the formula
0.95(Z^(-1.1)-Z^1.1)+C.  But this is the first time that this
midget, which has been hiding since before eternity, has been
seen by the eyes of observers, human or otherwise.

The midget is one of the more decorative ones that I have
dredged up in recent days.  My ability to ferret out these
elusive objects seems to rise and fall in irregular cycles.  In
the more complex formulas such as those calculated by the M-Mix4
formula, the position of the midgets is not always apparent.
Sometimes they appear in areas that seem totally hopeless.  To
find these most elusive midgets I rely on instinct -- I get a
feeling that a midget lies buried at a certain location, and as
often as not, it does.

Change the magnification of today's FOTD to 100, change the
logmap to 1, and observe the hopeless image.  I can give no
logical reason why I searched the center of that image for a
midget, but I did, and today's FOTD is the result.

I named the picture "Be a Good Sport" for no reason in
particular.  It just seemed like a nice name.  The 8 minutes it
takes to run the parameter file is enough time to make a
download the better choice.  The download may be done by going
to:

          <alt.binaries.pictures.fractals>

on Usenet, and looking for the "Fractal of the Day" subject
line, or by going to:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

which is Paul Lee's site on the Web.

The fractal weather today was mostly cloudy and chilly, with a
few sprinkles of rain.  The fractal cats chose to venture
outside only briefly in the temperature of 56F (13C).

Much to the sorrow of those waiting for enlightenment, the
philosophy languished today.  But hopefully tomorrow will bring
the light.  It must.  If I don't bring light tomorrow, who will?

Until then, take care, and did God create fractals, or did man
invent them?


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

Be_a_Good_Sport    { ; time=0:08:04.88 on a P200, SF5
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=ident passes=1
  center-mag=-0.70258058382572460/+1.84578902125106600\
  /1.159885e+011/1/-95.001/-0.006 params=-1/1.1/1/-1.1\
  /-0.05/1000 float=y maxiter=1200 bailout=25 inside=0
  logmap=142 symmetry=none periodicity=10
  colors=000MAk<3>PAhQAgRAfSAeTAdUAc<3>ZAe_Af`AfaAf<5>\
  gFfhGfiHf<3>mLfnMcoNZ<3>rbFrVAsK5PA4KA1oK2kY3ge4<3>T\
  P6PK7LF8IC8KH7LN7NR6OX6Q`6Rf5ck5hp4hu4mz4mz5mz6hx7bu\
  8Op8<3>HkCFjCEhD<2>9eF7dP5cZ1di<6>HZZJYYLXX<3>UUR`UP\
  dTOiSNmZL<2>umH<2>zmDwmErcFmKGhJHdII<10>J7SH6TF5U<3>\
  72X<11>WdcYgd_kd<3>gwf<3>UxmRxnOxp<3>Bxv<3>VgpZbncZm\
  hVkmRjtMk<14>LZUJ_TG`R<3>7dN1fL<6>TWMXUM`TM<3>oNM<11\
  >TYYRZZP__<3>HccFai<3>IiSJkOKmJLoFMs7<3>KwMKxQJyT<3>\
  IzgHzjHznHzq<7>TzqVzqWzq<3>azqhzy<6>3zA
  }

frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}

END 20.0 PAR-FORMULA FILE==================================


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Date: Thu, 27 Apr 2000 20:07:04 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful.

At 20:37 26/04/2000 +0100, you wrote:
>Morgan L. Owens wrote:
>
>> I took some time out to do some speed tests of Fractint's formula parser -
>> in particular, how it handles various ways of representing integer powers
>> of a complex number. I was surprised by the results.
>>
>
>Morgan,
>
>This type of analysis is what I find most interesting about computing,
>so I quickly set about producing my own set of results.
>
>I used the following (clumsy) procedure:
>
>...
>
Substantially it's the same approach as mine - in particular the formulae
are almost identical ("almost" - see below). Rather than create pars and
run them from a batch file, though, I used an autokey script with lines like:

"t"
ENTER
"power03"
ENTER
ENTER
CALCWAIT
"s"
CALCWAIT

to run Fractint in a single long run through all 51 formulae.

The video mode I used was 1600x1200x256, with passes=g (which was easier
than switching to a lower resolution) and a maxiter of 1023. That comes to
round about 2000000 iterations per image (since only something like one in
a thousand pixels would be evaluated).

>Here are my times:
>
>n  power    mult        sqr
>0  7:48.46  5:47.40  5:44.44
>1  8:33.50  5:47.35  5:44.49
>2  8:34.82  6:19.43  6:07.29
>3  8:21.64  6:53.75  6:52.55
>
>(I only ran to n=3, as I am supposed to be studying for my final exams
>at the moment!)
>
This looks a lot more like what I'd expected to see. The much higher times
overall than mine can be explained with a look at the formulae:

exp0 {z=0,c=rand,c=c^0,|z|<4}
sqr3 {z=0,c=rand,c=sqr(c)*c,|z|<4}

The second comma should be a colon; as things stand all four statements are
executed with every iteration, instead of only the last two.

In fact, mine run far slower than they should - the final test need only be
something guaranteed to be true: "1==1" (or even just "1", if the parser
works the way I think it does) would be a lot faster, and so make for more
accurate times.

>So, this fits quite well with the conventional wisdom in this area...
>This begs the question ... What is going on here?
>
>I have been trying to think of reasons for this. Do you use an non-Intel
>processor, which might give a different performance for some FP
>instructions?
>
Nope, it's a bog-standard Pentium 166.

This sort of result is what I half-suspected, which is why I hoped someone
else would run similar tests. I had got suspicious when some optimisations
I was making to a few of my formulae didn't appear to have any result. But
I surmised that conventional wisdom must have had some justification.

When I find another spare hour or two I'll fiddle with the experimental
conditions and make another run.

> If I have time (possibly at the weekend), I'll continue these tests.
>
I'll be interested in the results, in particular where the breakovers occur
- - how high does n have to be before powers are cheaper than multiple
multiplicatios?


Incidentally, how does Fractint actually time a run? By comparing wall time
at the start and end (or interruption) of a run, or does it use an internal
timer? Calibrated how?

Morgan L. Owens
"Spend the time now, save it later. I wish I could do that with money."



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Date: Thu, 27 Apr 2000 21:56:38 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: RE: (fractint) minibrots / mandeloids (whatever-you-call-'ems)

At 15:25 26/04/2000 -0300, Miguel Fliguer wrote:
>
>>In short, a Mandelbrot-set-like thing shows up whenever and whereever a
>>function (however wild) looks _locally_ like something that could be
>>approximated by z^2+c: what I've be describing as a "bowl".
>
I should have pointed out earlier that the _really_ important bits of the
landscape are the flat points - like those at the bottom of the bowl. These
so-called "critical points" are very sensitive to what goes on around them
- - jiggle them a tiniest bit and they're not flat any more! But since a bowl
_has_ to have a critical point (assuming the surface is smooth), then if
one is destroyed, another will appear to takes its place. The same thing
happens on the tops of humps of course, but our orbiting points avoid them.

>
>So I guess the simpler the formula (i.e. z*z+c ), the greater
>the probability for the associated shape (i.e the standard M-set)
>to appear in the middle of chaos.
>
In more complicated formulas, the stability of critical points is by no
means assured - they could split or merge even if the formula changes only
a tiny bit. z^2+c survives because it's only got one critical point and
while it can wander about it can't disappear.

z^3+c has one critical point also - at z=0+0i - but in its case the
slightest alteration to the formula (like adding 0.0000001z) causes this
solitary critical point to split into two (in this case into
+-1/30000000+0i.) Not a lot of difference, but play with these formulas and
see what happens to the cubic as the perturbation gets larger. The
perturbed Quadratic is just the M-set when p1=0. Note how well that stays
together as p1 increases. Oh, and that z=sqrt...whatever business? Well,
these things are supposed to start from a critical point, so...

perturbedCubic{
z=sqrt(-p1/3),c=pixel:
z=z^3+p1*z+c
|z|<=4
}

perturbedQuadratic{
z=-p1/2,c=pixel:
z=z^2+p1*z+c
|z|<=4
}


Morgan L. Owens
"Whew! And I did all that without once mentioning Morse's Theorem. Which is
not necessarily a Good Thing."



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Date: Thu, 27 Apr 2000 13:31:53 -0300
From: "Ricardo M. Forno" <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

Miguel:
After several attempts, I was unable to connect to your page.
Do you know what is going on?

- ----- Original Message -----
From: Fliguer, Miguel <[e-mail]>
To: <[e-mail]>
Sent: Wednesday, April 26, 2000 5:57 PM
Subject: RE: (fractint) MandelbrotMix4 lakes


> Guy wrote :
>
> >> Oohhh.. thanks Sylvie, Jim! That's a b'ful playground for little
> >> animations (and tests)!!
>
> Check out "Mandelbrot SunSet" at
> http://members.xoom.com/fliguer/gallery5.html
> for a 570K AVI which features the "lake" effect.
>
> Regards,
> Miguel
>
>
> --------------------------------------------------------------
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End of fractint-digest V1 #471
******************************