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fractint-digest         Sunday, April 30 2000         Volume 01 : Number 472




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

Date: Thu, 27 Apr 2000 17:00:43 +0100
From: "Iain G. Stirling" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful.

Morgan L. Owens wrote:

> 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.
> >>
> >

<snip>

>
> >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.

This might account for the difference we see. With my formulae, c is
reinitialised, and so takes on a fresh magnitude every iteration. With
your formulae, c is repeatedly diminshed, through 150 iterations
(assuming you have left the default maxiter value). Surely at some
point, the value is diminshed until it cannot be represented in floating
point any more, and it just takes the value zero. Then, perhaps the
particular function (*, ^ or sqr()) has an optimisation for the zero case,
allowing it perform very well? (This is just a hunch, mind you.)

<snip>

> 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?
>

I have decided to take a break from studying (well-earned, I assure
you) to continue my testing:

 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
 4  8:08.23   7:36.43  6:53.92
 5  8:00.59   8:22.08  7:38.73
 6  7:55.55   9:03.76  7:43.02
 7  7:52.08   9:48.80  8:35.86
 8  7:49.45 10:55.59  7:46.27
 9  7:47.36 12:09.47  8:45.15
10 7:53.95 13:39.32  8:53.05

So it can be seen that beyond n=4, multiplication is worst, and powers
and sqr fight over it, with sqr being successful on n=5,6 and 8.

What I will probably do next (on my next study break) is add that colon
into the formulae and get a new set of results, which I presume will
compare with yours. If you change the colon for a comma in your
formulae, will your times agree with mine above...?

For reference, I have a P133 and I am running Fractint from DOS, so
the times we get should be quite similar for identical tests.

Iain Stirling.


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

Date: Thu, 27 Apr 2000 15:10:31 -0300
From: "Fliguer, Miguel" <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

It works from here.

You can try using xoom's numerical ip address :

http://206.132.163.167/fliguer/franktal.html

(You can't see the graphics in this way)

If you can connect to the page, that means your
browser can't resolve xoom's address because of a local
DNS problem.

How's your book selling ? ;-)

- -----Mensaje original-----
De: Ricardo M. Forno [mailto:[e-mail]]
Enviado el: jueves 27 de abril de 2000 13:32
Para: [e-mail]
Asunto: 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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------------------------------

Date: Thu, 27 Apr 2000 21:30:02 +0100
From: "Iain G. Stirling" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful.

Iain G. Stirling wrote:

> Morgan L. Owens wrote:
>
> > 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.
> > >>
> > >
>
> <snip>
>
> >
> > >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.
>
> This might account for the difference we see. With my formulae, c is
> reinitialised, and so takes on a fresh magnitude every iteration. With
> your formulae, c is repeatedly diminshed, through 150 iterations
> (assuming you have left the default maxiter value). Surely at some
> point, the value is diminshed until it cannot be represented in floating
> point any more, and it just takes the value zero. Then, perhaps the
> particular function (*, ^ or sqr()) has an optimisation for the zero case,
> allowing it perform very well? (This is just a hunch, mind you.)

OOPS! I have just noticed that my _e-mail_ was the only place that the
second comma had been introduced - a simple typo. on my part. My
.FRM files all use the colon in the second position, and hence match
your formulae. So, the above reasoning (which might be sound) is
entirely unrelated to the testing we are performing.

Therefore it must surely be connected with the fact that you are using
passes=g whereas I am using passes=1.

Consequently my next tests will be with passes=g...

Sorry for the confusion,

Iain Stirling.


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

Date: Thu, 27 Apr 2000 23:17:48 +0200
From: Guy Marson <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

At 17:57 26/04/00 -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
>

yeh, very nice!

cheers,

Guy


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

Date: Thu, 27 Apr 2000 17:43:16 -0300
From: "Ricardo M. Forno" <[e-mail]>
Subject: RE: (fractint) MandelbrotMix4 lakes

I already was able to connect to your page at the end. Thanks a lot. Very
beautiful fractals, especially the "glasses".
My book is not selling very well. I gave it no charge to some
"personalities" in the production of TV shows, etc.
- ----- Original Message -----
From: Fliguer, Miguel <[e-mail]>
To: <[e-mail]>
Sent: Thursday, April 27, 2000 3:10 PM
Subject: RE: (fractint) MandelbrotMix4 lakes


> It works from here.
>
> You can try using xoom's numerical ip address :
>
> http://206.132.163.167/fliguer/franktal.html
>
> (You can't see the graphics in this way)
>
> If you can connect to the page, that means your
> browser can't resolve xoom's address because of a local
> DNS problem.
>
> How's your book selling ? ;-)
>
> -----Mensaje original-----
> De: Ricardo M. Forno [mailto:[e-mail]]
> Enviado el: jueves 27 de abril de 2000 13:32
> Para: [e-mail]
> Asunto: 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
> >
> >
> > --------------------------------------------------------------
> > Thanks for using Fractint, The Fractals and Fractint Discussion List
> > Post Message:   [e-mail]
> > Get Commands:   [e-mail] "help"
> > Administrator:  [e-mail]
> > Unsubscribe:    [e-mail] "unsubscribe fractint"
>
>
> --------------------------------------------------------------
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>
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------------------------------

Date: Fri, 28 Apr 2000 16:49:10 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful.

At 17:00 27/04/2000 +0100, Iain G, Stirling wrote:
>>
>> The second comma should be a colon; as things stand all four statements are
>> executed with every iteration, instead of only the last two.
>
>This might account for the difference we see. With my formulae, c is
>reinitialised, and so takes on a fresh magnitude every iteration. With
>your formulae, c is repeatedly diminshed, through 150 iterations
>(assuming you have left the default maxiter value). Surely at some
>point, the value is diminshed until it cannot be represented in floating
>point any more, and it just takes the value zero. Then, perhaps the
>particular function (*, ^ or sqr()) has an optimisation for the zero case,
>allowing it perform very well? (This is just a hunch, mind you.)
>
By that argument, * and sqr() should work really well, as ^ doesn't get the
opportunity to see that the expression is approaching zero (so to speak).
But the vanishing towards zero does sound like something that could affect
timings. I'll do that :<->, swap you suggest on the weekend.

>
>So it can be seen that beyond n=4, multiplication is worst, and powers
>and sqr fight over it, with sqr being successful on n=5,6 and 8.
>
sqr would be at its best when n is a power of 2. I might try comparing ^
and sqr for these values of n and see just how much of a fight sqr puts up
(assuming my timings start making sense).

Morgan L. Owens
"It's not an expense - it's an investment."


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

Date: Fri, 28 Apr 2000 16:44:48 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: RE: (fractint) sqr() function considered harmful.

At 15:50 27/04/2000, Robin wrote:
>As an aside while benchmarkings are being discussed:
>
>>then took the timing information
>>from the resulting series of gifs.
>
>If you invoke fractint with the option "debugflag=1"
>(or debugflag=any odd number     actually)
>it will write a text file with time stamps for all fractals generated with
>sub-second precision.
>
I really should look at that .doc file one day :-). This will certainly
make my monitor happy! Hey, that means I don't even need to save any gifs!

Morgan L. Owens
"clickclickclickclick"



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

Date: Fri, 28 Apr 2000 01:20:26 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD, 28-04-00  (Red Dwarf [5])

FOTD -- April 28, 2000  (Rating 5)

Fractal visionaries and enthusiasts:

I named today's average fractal "Red Dwarf", though perhaps
"Green Dwarf" or "Lilac Dwarf" would be more appropriate.  (No,
it is not named after a sci-fi TV show.)  The illustrated midget
is actually a scene in the fractal that results when the
expression 0.95Z-Z^101+C is iterated.  These baby Mandelbrots
are called midgets.  I have never heard one referred to as a
dwarf, though the name would be equally correct.

Today's parameter file renders in a little more than two minutes
on an average Pentium.  Downloading the image from:

          <alt.binaries.pictures.fractals>

or from:

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

is still the faster way of viewing the scene.

The cold, clammy, dreary and drizzly weather kept the cats
indoors all day.  The temperature of 45F (7C) kept the house
heat going in what is normally a warm time of year.

As can be gathered from the terseness of the discussion, the
philosophy went nowhere today -- perhaps because my particular
philosophy has nowhere it could go, but more likely because I
was simply too busy.  After all, anyone who spends as much time
searching for fractals as I do can't have that much time left
for deep thinking.

Until next time, take care, and think about it.


Jim Muth
[e-mail]


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

Red_Dwarf          { ; time=0:02:22.45, SF5 on a p200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=ident passes=t
  center-mag=+0.10939053115958090/+0.00013913783754621\
  /7534114/1/154.999 params=-1/101/0.95/1/0/0 float=y
  maxiter=1800 bailout=25 inside=0 logmap=45
  symmetry=none periodicity=10
  colors=00049K044034034014<3>004004013041081163364446\
  646848939B3BC1BE1CG1EH0GJ0GK0HM0JN0JT1PX4T<2>jEgnHkr\
  KqvNtzQzzTzzWzzXzrWzYUnCTb0SQ<2>1XY6_`9`cEbfHcgMfj<3\
  >_mtcnwfoyckrbgk`ce__YYWTMsnKvqKysKzvPKgTMkYMnbNrfNt\
  jPynPzrSwvTszUozXm<2>z`bz0zz0zz0rz0j<2>z8MY0zT0zP3z<\
  2>9Hz4Mz0Qz4Wz8_sBcmEgfJkYMoSPsKSwCUz4Yy6`w6ct6<2>ko\
  6nm6rk6ti6wg6ze6zc6z`6z_6z`CzbHzbMwcQteUreYqfbnffkgj\
  iinfircjvbjzcizcgzcfzefzeezeczebzfbzf`zf_zfYz<3>gWz`\
  YrW`jQbcKeWEfN8iG1j60m00n00q0<3>0v00w00y0<3>6z08z19z\
  1Bz3Cz3Ez4Ez4<3>Ee4E`4Gc6He6Hf8Jg8Ki9Kk9MmBNnBNoCPqC\
  QsEQtESvGTwGTyGYzJczKgzMmzN<2>zzSzzTzzUzzSqzPizNmzCo\
  z1rz0Wzt`zofzjkzeqz_vzUzzPzzJzzEzzJzzNyzQvzUrzXnz`jz\
  egzgczk_znWzrTztSzvSzvSzwSzwSzy<2>Szz<2>CzzEzzEzyEzt\
  EzqEzmEzgEzcEz_GzWGzSGzMGzHSzTKzNCzH
  }

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: Fri, 28 Apr 2000 10:23:53 -0400
From: Tom Conally <[e-mail]>
Subject: (fractint) Kaleidoscope

Hi Gang,
My first new par in a while.
This par when color-cycled looks as much like a kaleidoscope as any I've
seen.

kaleidoscope       { ; Kaleidoscope (c)Tom Conally, 4/2000
                  ; cycle color for kaleidoscope effect
                     ; t=26.04sec on p400
                     ; Version 2000 Patchlevel 9
                      reset=2000 type=julia(fn||fn) function=exp/sqr passes=1
  center-mag=+0.31781005849828920/-1.32276153613428900/8.573609/1.0309
  params=0/0/8 float=y maxiter=1000 bailout=128 bailoutest=manh
  fillcolor=1 outside=real logmap=yes decomp=32
  colors=WWWXNM<3>XjKWpJYEN<11>mgYnjZol_<3>uwcXDN<12>KWMJXMIZM<3>EdLXCN<9>\
  OIJNJJNKJ<3>JMHJMGING<3>EQEZCO<4>jBYlBZoA`<3>y9hQELJGIBJFXCN<6>aIHaJGbKF\
  <3>dNBeNAfO9<2>hR6XEM<3>ZOIZQH_TGWCN<14>CENAEN9EN<3>3FOXEN<2>XKNWDNXMN<3\
  >WUMWWMWYMV_LLVKAPJ<10>eHChGCkGB<3>wC8<4>aIPXJTTKW<3>AQj<4>AiW<2>AuN<4>C\
  eQCbQC_R<3>EOTFLUFIVGEVGBW<3>KLKMNHNQEOSBPV8RY4<13>RhfRiiRik<3>SmwkbK<12\
  >qYkrYmrXp<3>uVy<3>umqtrnvov<2>jos
  }


Tom Conally

In every boomerang there is a perfect throw.
Your life, Grasshopper, is to practice
till you find that throw
and become one with that boomerang!
http://fly.to/boomerangs  " Flying Frog Boomerangs"
http://www.vstore.com/vstorehardware/flyingfrogtools/  "Flying Frog Tools"
http://www.netpath.net/~conally   My greater internet index

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

Date: Fri, 28 Apr 2000 20:05 0000
From: [e-mail]
Subject: RE: RE: (fractint) sqr() function considered harmful.

Hi Morgan,

>I really should look at that .doc file one day :-). This will certainly
>make my monitor happy! Hey, that means I don't even need to save
>any gifs!

Well if you really want to check out the effect of using different ways of
raising to powers you could  save those gifs and then use debugflag=50
to detect any differences in the images... though I guesss that the file
size would be a good indicater too, you'd probably get differeces in the
size as a few pixels difference would compress differently.

If you do get differences then passes=g would be out as a valid
benchmarking method too... Gets more compicated the more you think
about it :-)

Cheers,
        Robin.

Then again, how do you tell which one's the right one?


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

Date: Sat, 29 Apr 2000 09:47:40 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD, 29-04-00  (Sugar-Plum Midget [6])

FOTD -- April 29, 2000  (Rating 6)

Fractal visionaries and enthusiasts:

The discussion is a little late again today because I once again
took a moment to relax after the day's work, and next thing I
knew it was 7:30am.  But all is not lost, for while dozing I
dreamed of sugar-plums.  Now I'm not sure that I would recognize
a sugar-plum if I saw one.  All I know about them is that
they're found around nutcrackers.  And how I know that the
things I dreamed of were sugar-plums when I wouldn't recognize
one is a secret that I will never reveal.

(Actually, it might not be a bad idea to start writing my FOTD
discussion in the morning, when I'm fresh and the ideas are
plentiful, rather than late in the evening, when I'm groggy and
want only a nice long rest.)

I might have given the name "Sugar-Plum Midget" to another FOTD
several years ago.  The name seems vaguely familiar, but I don't
remember for certain.  Regardless, today's picture has been
given the name because it reminds me of a midget surrounded by
sugar plums.

The formula behind all the candy is 1.4(Z^129)+1.05(Z)+(1/C), an
expression that appears rather boring at first glance.  But then
Z^2+C also appears boring and too simple to do anything, and
look what *it* started.

I'm getting interested in mixing high and low powers of Z, so
more images of this type are likely in the days to come.  The
parameter file takes over 22 minutes to render on an average
Pentium.  For relief, the image has been posted for over 12
hours to:

          <alt.binaries.pictures.fractals>

It may also be found on Paul Lee's web site at:

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

The fractal weather was cloudy and cool, with some sun in the
afternoon, but not nearly enough to lure the fractal cats
outdoors.  The fractal temperature reached 60F (15.5C for the
rest of the world) -- too cool to bring out the cats.  As a
result, the cats spent the day indoors.

The fractal philosophy list needs another kick in the pants,
which I might give it with a controversial statement or two in
the near future.  But for today, we've come to the end of the
line.  Until next time, take care, and enjoy the peace that only
fractals bring.


Jim Muth
[e-mail]


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

Sugar-Plum_Midget  { ; time=0:22:20.31, SF5 on a P200
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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: Sun, 30 Apr 2000 00:20:04 +0200
From: "J.P. Louvet" <[e-mail]>
Subject: (fractint) Re: [fractal-art] Fractal art FAQ

Terry and all other fractal lovers,

What you write is exactly what I think. I can write broad lines about some
programs (being more experimented with some of them) but I have said several
times that this attempt to have a fractal art FAQ needed a collective
participation to be useful and objective. Unfortunately I have received only
very few contributions and this goal doesn't seem to interest many people. I
think that it is not a question of time when I see, for example the
overabundant exchange of parameters and formulae in the UF list and, to a
less extent in the Fractint list.

Well, I will post also this reply in these 2 lists.

I have some material to do a new version of the draft and I will take my
responsibility, Tim, about the text describing the programs. But this effort
will be more useful if I have more contributory feedbacks.
Waiting for that I acknowledge in advance those who will accept to help the
fractal community, specially the beginners (note that I don't say "those who
will help  me" because this is not what is important).
 -------------------------------------------------------------------------
 J.P. Louvet                | Phone :  [phone]
 IUT Universite Bordeaux I  |
 33405 Talence CEDEX France | email : [e-mail]
 -------------------------------------------------------------------------
- ----- Original Message -----
From: "Terry W. Gintz" <[e-mail]>
To: <[e-mail]>
Sent: Saturday, April 29, 2000 11:22 PM
Subject: Re: [fractal-art] Fractal art FAQ


> Tim,
>
> I agree entirely with this.  I also have great respect for Jean-Pierre's
> views.  He's always been able to point out weaknesses in my programs
without
> 'inflaming my stys.'  He's also provided insightful suggestions for
> improvement.  If he can't write a fair description for each fractal
program
> as it now exists, then I don't know who could.  But he's also quite right
> when he asks for "the opinions of many people."   Jean-Pierre hasn't used
> every program around, or to the same depth, so some omissions/inaccuracies
> are unavoidable, unless...  If J-P is to act as a moderator, so to speak,
he
> should have all the pertinent information about each program(in the
broadest
> scope of the FAQ), and what the authors/users see as strengths and
> weaknesses.  Finally, everyone should realize the license and
responsibility
> that Jean-Pierre takes in writing this version of the FAQ, and limit their
> criticisms to constructive, rather than ***** feedback.
>
> Terry W. Gintz



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

Date: Sun, 30 Apr 2000 01:00:39 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD, 30-04-00  (Double-Minus Brot [4])

FOTD -- April 30, 2000  (Rating 4)

Fractal visionaries and enthusiasts:

Today I trimmed the overgrown fractal lawn, giving myself a
stiff back and a great excuse to do nothing but explore fractal
land for the rest of the day.  My fractal theme of the day was
all negative parameters and no fractional powers.  I started
with -1(Z^(-2))-2(1/Z)+(1/C).  I named the picture "Double-
Minus Brot" because of the two negative exponents.

It's another of those simple formulae that promise little but
deliver much.  In fact, the only change I made was to raise the
bailout radius to 1400, which broke the fractal into interesting
segments, separated by curious holes.

The rating for this fractal effort can only be an honest
somewhat-below-average 4, not quite worth the 15 minutes
required to run the parameter file, but at least worth the
minimal effort of downloading the pre-rendered image from Usenet
at:

          <alt.binaries.pictures.fractals>

or from the WWW at:

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

The fractal weather today was unexpectedly bright and warm, with
sunny skies and a temperature of 68F (20C), which brought out
the little bit of friskiness left in the fractal cats.

The philosophy is cooking right on schedule, so stay in touch
with the philofractal list all you frustrated fractal philosophy
fans for some fantastic philosophizing in the fairly near future.

Right now it's time for my Saturday evening spell in front of
the TV, watching some junky sci-fi.  I think it's Dr. Who this
evening in "Meglos".  That's the one with the dodecahedron god
and the cactus Doctor.  I'll return tomorrow, same time same
venue.  Until then, take care, and keep a stiff upper fractal.


Jim Muth
[e-mail]


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

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  }

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: Sun, 30 Apr 2000 18:53:17 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful?

At 16:49 28/04/2000 +1200, I wrote:
> I'll do that :<->, swap you suggest on the weekend.
>
The formulae I ended up with were things like:

sqr05{z=0:c=sqr(sqr(pixel))*pixel,|z|<=4}

I switched to pixel instead of rand so that I didn't have the RNG slowing
things down. It also meant that I didn't need to explicitly initialise c to
anything.

The running conditions are as for my first experiment: a DOS boot,
1600x1200x256 resolution, 1023 iterations and passes=g (with passes=1 I'd
be iterating something like a thousand times as many pixels). In all,
roughly two million iterations per formula. (I'm only rough because
passes=g is more thorough than one pixel per 1024-pixel block around the
edges.)

My new timings (in seconds are)

 n  power  mult   sqr
 0  15.16  14.22  14.23
 1  14.83  14.50  14.50
 2  14.72  14.83  14.83
 3  38.34  16.37  16.26
 4  38.56  17.52  15.87
 5  38.50  18.51  17.30
 6  38.45  20.05  17.30
 7  38.50  21.26  18.35
 8  38.50  21.20  16.75
 9  38.50  22.30  18.34
10  38.51  23.07  18.18
11  38.50  26.20  18.95
12  38.56  27.19  18.29
13  38.56  25.98  19.34
14  38.55  26.86  19.00
15  38.61  27.84  20.44
16  38.56  32.08  17.96

Which does look a lot more sensible.

One thing to notice is that repeated multiplication performs much better
here against exponentiation: even z^16 appears to run slightly slower than
z*z*z*z*z*z*z*z*z*z*z*z*z*z*z*z! Exponentiation is pretty much constant
around 38 1/2 seconds, increasing only slowly, though treating n=0, 1, 2 as
special cases. z*z also appears to be converted to sqr(z) when it appears.

However, z*z*z and sqr(z)*z are slightly different. This suggests that
Fractint doesn't spot the "z*z" in the cube expression; the extra time to
look up 'z' _three_ times instead of two makes for an extra tenth of a
second (out of two million iterations?! What are you complaining about?)

Of the three methods, repeated squaring appears to be the fastest approach
to writing hardwired integer powers; instead of saying z^9 or
z*z*z*z*z*z*z*z*z, sqr(sqr(sqr(z)))*z offers clear speed advantages.

These results are in conflict with those of Iain Stirling's, which say that
repeated multiplication is slower than the other two, which are on a par
with each other. I'm wondering if this may be a result of my switching to
using pixel as the source for my test data instead of randomly-generated
numbers?


But in the quest for maximum speed in user formulae, the story doesn't end
here. Consider this alternative method for writing z^15:

z=sqr(sqr(z))*z,z=sqr(z)*z

This saves an extra multiplication of the more straightforward repeated
squaring method, which gives

z=sqr(sqr(sqr(z)*z)*z)*z

And if, as one often does, wish to calculate a number of powers of a
variable, then it makes sense to reuse results as much as possible, rather
than calculate them fresh each time; if you've calculated w=sqr(z)*z
already, it makes more sense to calculate z^4 as w*z instead of
sqr(sqr(z)). The ultimate in this field is still an open problem - in
mathematics, not just in Fractint.

Morgan L. Owens
"Embarras de richesses"



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

Date: Sun, 30 Apr 2000 17:40:36 +0100
From: "Iain G. Stirling" <[e-mail]>
Subject: Re: (fractint) sqr() function considered harmful?

Morgan L. Owens wrote:

> At 16:49 28/04/2000 +1200, I wrote:
> > I'll do that :<->, swap you suggest on the weekend.
> >
> The formulae I ended up with were things like:
>
> sqr05{z=0:c=sqr(sqr(pixel))*pixel,|z|<=4}
>
> I switched to pixel instead of rand so that I didn't have the RNG slowing
> things down. It also meant that I didn't need to explicitly initialise c to
> anything.
>
> The running conditions are as for my first experiment: a DOS boot,
> 1600x1200x256 resolution, 1023 iterations and passes=g (with passes=1 I'd
> be iterating something like a thousand times as many pixels). In all,
> roughly two million iterations per formula. (I'm only rough because
> passes=g is more thorough than one pixel per 1024-pixel block around the
> edges.)
>
> My new timings (in seconds are)
>
>  n  power  mult   sqr
>  0  15.16  14.22  14.23
>  1  14.83  14.50  14.50
>  2  14.72  14.83  14.83
>  3  38.34  16.37  16.26
>  4  38.56  17.52  15.87
>  5  38.50  18.51  17.30
>  6  38.45  20.05  17.30
>  7  38.50  21.26  18.35
>  8  38.50  21.20  16.75
>  9  38.50  22.30  18.34
> 10  38.51  23.07  18.18
> 11  38.50  26.20  18.95
> 12  38.56  27.19  18.29
> 13  38.56  25.98  19.34
> 14  38.55  26.86  19.00
> 15  38.61  27.84  20.44
> 16  38.56  32.08  17.96
>
> Which does look a lot more sensible.
>
> One thing to notice is that repeated multiplication performs much better
> here against exponentiation: even z^16 appears to run slightly slower than
> z*z*z*z*z*z*z*z*z*z*z*z*z*z*z*z! Exponentiation is pretty much constant
> around 38 1/2 seconds, increasing only slowly, though treating n=0, 1, 2 as
> special cases. z*z also appears to be converted to sqr(z) when it appears.
>
> However, z*z*z and sqr(z)*z are slightly different. This suggests that
> Fractint doesn't spot the "z*z" in the cube expression; the extra time to
> look up 'z' _three_ times instead of two makes for an extra tenth of a
> second (out of two million iterations?! What are you complaining about?)
>
> Of the three methods, repeated squaring appears to be the fastest approach
> to writing hardwired integer powers; instead of saying z^9 or
> z*z*z*z*z*z*z*z*z, sqr(sqr(sqr(z)))*z offers clear speed advantages.
>
> These results are in conflict with those of Iain Stirling's, which say that
> repeated multiplication is slower than the other two, which are on a par
> with each other. I'm wondering if this may be a result of my switching to
> using pixel as the source for my test data instead of randomly-generated
> numbers?

I'm sure that modification could have a large effect. That is the true
difficulty of this issue - it really depends on precisely what is being
calculated.

 > But in the quest for maximum speed in user formulae, the story doesn't end
> here. Consider this alternative method for writing z^15:
>
> z=sqr(sqr(z))*z,z=sqr(z)*z
>
> This saves an extra multiplication of the more straightforward repeated
> squaring method, which gives
>
> z=sqr(sqr(sqr(z)*z)*z)*z
>
> And if, as one often does, wish to calculate a number of powers of a
> variable, then it makes sense to reuse results as much as possible, rather
> than calculate them fresh each time; if you've calculated w=sqr(z)*z
> already, it makes more sense to calculate z^4 as w*z instead of
> sqr(sqr(z)). The ultimate in this field is still an open problem - in
> mathematics, not just in Fractint.

Indeed. I think what we have seen here is that the conventional
wisdom of sqr being best is perhaps generally true, but your original
tests results show that there _is_ a case where it is not true. However,
that case may be degenerate and not apply in a realistic formula...

On the other hand, the huge difference between this latest set of
results and the two previous sets shows that there is no way to predict
what is going to happen.

Maybe it would be interesting as a final experiment to take some very
realistic cases, such as a well-known formula (using various powers of
z) with a good .PAR creating an image of it, and play about with it?

I will probably give this a go at some point.

Iain Stirling.



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End of fractint-digest V1 #472
******************************