• April 23, 2021, 04:12:51 PM

### Author Topic:  hybrid formulas  (Read 1770 times)

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#### claude

• 3f
• Posts: 1839
##### hybrid formulas
« on: May 28, 2020, 08:47:30 PM »
working on experimental hybrid formula stuff.

controls: abs x, abs y, negate x, negate y, power, a_re, a_im; formula fragment is a(±|x|±i|y|)^p where the absolute values are optional

each line has two formula fragments, which can be combined with any of + - * /
this gives enough flexibility to recreate the redshift formulas and simonbrot/mothbrot

the enabled lines in each group are evaluated, then comes the + C

the groups alternate, allowing for hybrids - attached is mandelbrot / burning ship

very early experiments, maybe something for 2.15, assuming it works at deep zooms without catastrophic loss of significance (haven't tried yet)

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #1 on: June 08, 2020, 03:12:11 PM »
I'd love to see some more of these amazing fractal hybrids!
"quotquotI am lightning, the rain transformed."quotquot
- Raiden, Metal Gear Solid 4: Guns of the Patriots

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #2 on: June 08, 2020, 04:33:36 PM »
rendered with 'et', but should be possible to get similar results in next KF:

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #3 on: June 12, 2020, 11:21:21 PM »
added Newton-Raphson zooming (period detection using ball method; Newton iteration using dual numbers; size estimate using average power

still to do:

- "auto-skew at escape" feature for hybrid formulas
- a way to disable hybrid formula mode and switch back to regular formulas
- save/load hybrid formulas from parameter files
- derivatives calculations for analytic DE/slopes (probably using dual numbers)
- support for long double and floatexp (though without series approximation, such deep zooms won't be fun)
- automatic calculation of average formula degree / power for smooth colouring
- try to get moebius cylinder modifier working, or remove it if that proves impossible
- create examples, perhaps most of the 70-odd custom formulas can be replaced with hybrids (depends on benchmarks if the old formulas will go away, though)

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #4 on: June 23, 2020, 01:20:42 PM »
Using the hybrid system, I've accidentally stumbled into a very familiar fractal: the Simon0030-A fractal, Mandelbrot variant. However, when zooming over a thousand times, it disappears.

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #5 on: June 23, 2020, 01:48:42 PM »
At zooms less that 1e3, the first reference is hardcoded to 0+0i.  Beyond that the first reference is the center of the screen.  Usually glitch detection should kick in and find secondary references that are more appropriate.  What is the formula?  (The hybrids are not yet saved in the KFR/image metadata, so screenshots of the hybrid formula dialog might be best for now)

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #6 on: June 23, 2020, 01:52:47 PM »
Here is the formula for the Simon0030-A (Mandelbrot variant)

zi = zr * zi * 2.0 + JuliaI2;\
zr = zrsqr - zisqr + JuliaR2;\
zisqr = zi * zi;\
zrsqr = zr * zr;

Or:

simon0030Amand {
z = c = 0 - pixel:
z = (z * z + c) + c
|z| < 4
}

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #7 on: June 23, 2020, 02:07:30 PM »
afaict that formula is just Z^2 + C (top) or Z^2 + 2C (bottom) which would simply be a scaled Mandelbrot, but combined with Z_0 = C it would get distorted?  not sure...

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #8 on: June 23, 2020, 02:38:31 PM »
I've also noticed that some multibrots above power 10 disappear after 1e3. The power 11 Mandelbrot works, but the power 30 and 50 Mandelbrot sets disappear in some areas, for example after 1e3. But in some manners, the hybrid formula system brings back the arbitrary power Multibrots in Kalles Fraktaler. This time, the glitch correction works much better on multibrots above power 10.

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #9 on: June 23, 2020, 02:45:14 PM »
derivatives calculations for analytic DE/slopes (probably using dual numbers)
Implemented and working well.

Quote
support for long double and floatexp (though without series approximation, such deep zooms won't be fun)
Should be simple, just need to do it.

Quote
(depends on benchmarks if the old formulas will go away, though)
Currently the new hybrids are quite a bit vastly slower than the equivalent original formulas (around 25% 5x).  Needs improvement.  Not sure why they are so much slower...
« Last Edit: June 23, 2020, 04:06:24 PM by claude, Reason: benchmark technique was flawed »

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #10 on: June 23, 2020, 02:51:15 PM »
Here is a 30th power Mandelbrot, zoomed in around 134 million magnification (1.34e+008), with a gold color palette. The good news is that despite being much slower, it is explorable and the glitch correction works very well here.

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #11 on: June 23, 2020, 03:34:13 PM »
Here is the Simon0051-B fractal. Basically a two-group Mandelbrot/Buffalo hybrid.

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #12 on: June 26, 2020, 07:40:30 PM »
Here is a zoomed in image of a power 50 Mandelbrot rendered with Kalles Fraktaler using the hybrid system. The problem is that there are glitches that aren't recognized by the glitch correction algorithm. At least multibrots above power 10 have better display.

#### claude

• 3f
• Posts: 1839
##### Re: hybrid formulas
« Reply #13 on: June 26, 2020, 08:43:55 PM »
Oh, I still need to implement automatic reduction of glitch threshold at higher powers. Also "glitch low tolerance" probably has no effect on hybrids so far. Thanks for the report!

#### FractalAlex

• Fractal Frankfurter
• Posts: 541
• Experienced root-finding method expert
##### Re: hybrid formulas
« Reply #14 on: June 29, 2020, 01:56:20 PM »
I can't open my 64-bit build of Kalles Fraktaler 2.15 anymore on the Windows 7 laptop anymore. It used somewhat to work. It returned an error 0xc000001d. Can you help me? The following text here is in French, but you'll figure it out...
Code: [Select]
Windows ne peut pas accéder au fichier  pour une des raisons suivantes : un problème s’est produit avec la connexion réseau, le disque sur lequel le fichier est enregistré, ou les pilotes de stockage installés sur cet ordinateur, ou le disque est manquant. Windows a fermé le programme kf.exe en raison de cette erreur.Programme : kf.exeFichier : La valeur de l’erreur est affichée dans la section Données supplémentaires.Action utilisateur1. Ouvrez à nouveau le fichier. Cette situation peut résulter d’un problème temporaire qui se corrigera de lui-même à la prochaine exécution du programme.2. Si le fichier est toujours inaccessible et - Il se trouve sur le réseau : votre administrateur réseau devrait vérifier qu’il n’y a aucun problème avec le réseau et que le serveur peut être contacté. - Il se trouve sur un disque amovible, par exemple une disquette ou un CD-ROM : vérifiez que le disque est inséré correctement dans l’ordinateur.3. Vérifiez et réparez le système de fichiers en exécutant CHKDSK. Pour exécuter CHKDSK, cliquez sur Démarrer, Exécuter, entrez CMD puis cliquez sur OK. À l’invite de commandes, entrez CHKDSK /F et appuyez sur Entrée.4. Si le problème persiste, restaurez le fichier à partir d’une copie de sauvegarde.5. Déterminez si d’autres fichiers du même disque peuvent être ouverts. Si ce n’est pas le cas, le disque est peut-être endommagé. S’il s’agit d’un disque dur, contactez votre administrateur ou le distributeur de votre ordinateur pour obtenir une assistance supplémentaire.Données supplémentairesValeur de l’erreur : 00000000Type du disque : 0
Code: [Select]
Nom de l’application défaillante kf.exe, version : 0.0.0.0, horodatage : 0x5ef158a7Nom du module défaillant : kf.exe, version : 0.0.0.0, horodatage : 0x5ef158a7Code d’exception : 0xc000001dDécalage d’erreur : 0x000000000003e453ID du processus défaillant : 0x590Heure de début de l’application défaillante : 0x01d64e0bcd40dbbeChemin d’accès de l’application défaillante : E:\kf2-15\kf.exeChemin d’accès du module défaillant: E:\kf2-15\kf.exeID de rapport : 11935f43-b9ff-11ea-8e86-00262233b2d4The 32-bit build is working however...

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