What are the fastest algorithms for rendering the mandelbrot set?

Question

I've tried many algorithms for the rendering of the Mandelbrot set, inclusive of the naive escape time algorithm, as well as the optimized escape time algorithm. But, are there faster algorithms that are used to produce really deep zooms efficiently like the ones we see on YouTube. Also, I would love to get some ideas on how to increase my precision beyond the C/C++ double

Answer 1

Even High end CPU will be much slower in comparison to average GPU. You can get to real time rendering even with naive iteration algo on GPU. So using better algorithms on GPU could get to high zooms however for any decent algo you need:

• multi pass rendering as we can not self modify a texture on GPU
• high precision floating point as floats/doubles are not enough.

Here few related QAs:

• GLSL RT Mandelbrot
• Interior Distance Estimate algorithm for the Mandelbrot set
• I get infinitely small numbers for fractals
• Perturbation theory

which might get you kick started...

One way to speed up is you can use fractional escape like I did in the first link. It improves image quality while keeping max iteration low.

The second link will get you approximation of which parts of fractal are in and out and how far. Its not very accurate but can be used to avoid computing iterations for parts that are "outside for sure".

Next link will show you how to achieve better precision.

Last link is about Perturbation The idea is that you use high precision math only for some reference points and use that to compute its neighbor points with low precision math without loosing precision. Never used that however but looks promising.

And finally once you achieved fast rendering you might want to aim for this:

• How to adjust panning while zooming Mandelbrot set

Here a small example of 3* 64bit double used for single value in GLSL:

// high precision float (very slow)
dvec3 fnor(dvec3 a)
    {
    dvec3 c=a;
    if (abs(c.x)>1e-5){ c.y+=c.x; c.x=0.0; }
    if (abs(c.y)>1e+5){ c.z+=c.y; c.y=0.0; }
    return c;
    }
double fget(dvec3 a){ return a.x+a.y+a.z; }
dvec3 fset(double a){ return fnor(dvec3(a,0.0,0.0)); }
dvec3 fadd(dvec3 a,double b){ return fnor(a+fset(b)); }
dvec3 fsub(dvec3 a,double b){ return fnor(a-fset(b)); }
dvec3 fmul(dvec3 a,double b){ return fnor(a*b); }
dvec3 fadd(dvec3 a,dvec3 b){ return fnor(a+b); }
dvec3 fsub(dvec3 a,dvec3 b){ return fnor(a-b); }
dvec3 fmul(dvec3 a,dvec3 b)
    {
    dvec3 c;
    c =fnor(a*b.x);
    c+=fnor(a*b.y);
    c+=fnor(a*b.z);
    return fnor(c);
    }

so each hi precision value is dvec3 ... the thresholds in fnor can be changed to any ranges. You can convert this to vec3 and float ...

[Edit1] "fast" C++ example

Ok I wanted to try my new SSD1306 driver along with my AVR32 MCU to compute Mandelbrot So I can compare speed with this Arduino + 3D + Pong + Mandelbrot. I used AT32UC3A3256 with ~66MHz no FPU no GPU and 128x64x1bpp display. No external memory only internal 16+32+32 KByte. Naive Mandlebrot was way to slow (~2.5sec per frame) so I busted up something like this (taking advantage of that position and zoom of the view is sort of continuous):

1. reduce resolution by 2

   to make room for dithering as my output is just B&W

2. use variable max iteration n based on zoom

   On change of n invalidate last frame to enforce full recompute. I know this is slow but it happens only 3 times on transitions between zoom ranges.

   Scaling count from last frame is not looking good as its not linear.

   Its possible to use the last counts but for that it would be needed also to remember the complex variables used for iteration and that would take too much memory.

3. remember last frame and also which x,y screen coordinate mapped to which Mandelbrot coordinate.

4. On each frame compute the mapping between screen coordinates and Mandelbrot coordinates.

5. remap last frame to adjust to new position and zoom

   so simply look at the data from #3,#4 and if we have there the same positions in both last and actual frame (closer then half of pixel size), copy the pixels. And recompute the rest.

   This will hugely improve performance if your view is smooth (so position and zoom does not change a lot on per frame basis).

I know its a bit vague description so here a C++ code where you can infer all doubts:

//---------------------------------------------------------------------------
//--- Fast Mandelbrot set ver: 1.000 ----------------------------------------
//---------------------------------------------------------------------------
template<int xs,int ys,int sh> void mandelbrot_draw(float mx,float my,float zoom)
    {
    // xs,ys - screen resolution
    // sh    - log2(pixel_size) ... dithering pixel size
    // mx,my - Mandelbrot position (center of view) <-1.5,+0.5>,<-1.0,+1.0>
    // zoom  - zoom
    // ----------------
    // (previous/actual) frame
    static U8 p[xs>>sh][ys>>sh];            // intensities (raw Mandelbrot image)
    static int n0=0;                        // max iteraions
    static float px[(xs>>sh)+1]={-1000.0};  // pixel x position in Mandlebrot
    static float py[(ys>>sh)+1];            // pixel y position in Mandlebrot
    // temp variables
    U8 shd;                                 // just pattern for dithering
    int ix,iy,i,n,jx,jy,kx,ky,sz;           // index variables
    int nx=xs>>sh,ny=ys>>sh;                // real Mandelbrot resolution
    float fx,fy,fd;                         // floating Mandlebrot position and pixel step
    float x,y,xx,yy,q;                      // Mandelbrot iteration stuff (this need to be high precision)
    int   qx[xs>>sh],qy[ys>>sh];            // maping of pixels between last and actual frame
    float px0[xs>>sh],py0[ys>>sh];          // pixel position in Mandlebrot from last frame
    // init vars
         if (zoom<   10.0) n= 31;
    else if (zoom<  100.0) n= 63;
    else if (zoom< 1000.0) n=127;
    else                   n=255;
    sz=1<<sh;
    ix=xs; if (ix>ys) ix=ys; ix/=sz;
    fd=2.0/(float(ix-1)*zoom);
    mx-=float(xs>>(1+sh))*fd;
    my-=float(ys>>(1+sh))*fd;
    // init buffers
    if ((px[0]<-999.0)||(n0!=n))
        {
        n0=n;
        for (ix=0;ix<nx;ix++) px[ix]=-999.0; 
        for (iy=0;iy<ny;iy++) py[iy]=-999.0;
        for (ix=0;ix<nx;ix++)
         for (iy=0;iy<ny;iy++)
          p[ix][iy]=0;
        }
    // store old and compute new float positions of pixels in Mandelbrot to px[],py[],px0[],py0[]
    for (fx=mx,ix=0;ix<nx;ix++,fx+=fd){ px0[ix]=px[ix]; px[ix]=fx; qx[ix]=-1; }
    for (fy=my,iy=0;iy<ny;iy++,fy+=fd){ py0[iy]=py[iy]; py[iy]=fy; qy[iy]=-1; }
    // match old and new x coordinates to qx[]
    for (ix=0,jx=0;(ix<nx)&&(jx<nx);)
        {
        x=px[ix]; y=px0[jx];
        xx=(x-y)/fd; if (xx<0.0) xx=-xx;
        if (xx<=0.5){ qx[ix]=jx; px[ix]=y; }
        if (x<y) ix++; else jx++;
        }
    // match old and new y coordinates to qy[]
    for (ix=0,jx=0;(ix<ny)&&(jx<ny);)
        {
        x=py[ix]; y=py0[jx];
        xx=(x-y)/fd; if (xx<0.0) xx=-xx;
        if (xx<=0.5){ qy[ix]=jx; py[ix]=y; }
        if (x<y) ix++; else jx++;
        }
    // remap p[][] by qx[]
    for (ix=0,jx=nx-1;ix<nx;ix++,jx--)
        {
        i=qx[ix]; if ((i>=0)&&(i>=ix)) for (iy=0;iy<ny;iy++) p[ix][iy]=p[i][iy];
        i=qx[jx]; if ((i>=0)&&(i<=jx)) for (iy=0;iy<ny;iy++) p[jx][iy]=p[i][iy];
        }
    // remap p[][] by qy[]
    for (iy=0,jy=ny-1;iy<ny;iy++,jy--)
        {
        i=qy[iy]; if ((i>=0)&&(i>=iy)) for (ix=0;ix<nx;ix++) p[ix][iy]=p[ix][i];
        i=qy[jy]; if ((i>=0)&&(i<=jy)) for (ix=0;ix<nx;ix++) p[ix][jy]=p[ix][i];
        }
    // Mandelbrot
    for (iy=0,ky=0,fy=py[iy];iy<ny;iy++,ky+=sz,fy=py[iy]) if ((fy>=-1.0)&&(fy<=+1.0))
     for (ix=0,kx=0,fx=px[ix];ix<nx;ix++,kx+=sz,fx=px[ix]) if ((fx>=-1.5)&&(fx<=+0.5))
        {
        // invalid qx,qy ... recompute Mandelbrot
        if ((qx[ix]<0)||(qy[iy]<0))
            {
            for (x=0.0,y=0.0,xx=0.0,yy=0.0,i=0;(i<n)&&(xx+yy<4.0);i++)
                {
                q=xx-yy+fx;
                y=(2.0*x*y)+fy;
                x=q;
                xx=x*x;
                yy=y*y;
                }
            i=(16*i)/(n-1); if (i>16) i=16; if (i<0) i=0;
            i=16-i; p[ix][iy]=i;
            }
        // use stored intensity
        else i=p[ix][iy];
        // render point with intensity i coresponding to ix,iy position in map
        for (i<<=3                 ,jy=0;jy<sz;jy++)
         for (shd=shade8x8[i+(jy&7)],jx=0;jx<sz;jx++)
          lcd.pixel(kx+jx,ky+jy,shd&(1<<(jx&7)));
        }
    }
//---------------------------------------------------------------------------
//---------------------------------------------------------------------------
//---------------------------------------------------------------------------

the lcd and shade8x8 stuff can be found in the linked SSD1306 QA. However you can ignore it its just dithering and outputting a pixel so you can instead output the i directly (even without the scaling to <0..16>.

Here preview (on PC as I was lazy to connect camera ...):

so it 64x32 Mandelbrot pixels displayed as 128x64 dithered image. On my AVR32 is this may be 8x time faster than naive method (maybe 3-4fps)... The code might be more optimized however take in mind Mandelbrot is not the only stuff running as I have some ISR handlers on backround to handle the LCD and also my TTS engine based on this which I upgraded a lot since then and use for debugging of this (yes it can speek in parallel to rendering). Also I am low on memory as my 3D engine takes a lot ~11 KByte away (mostly depth buffer).

The preview was done with this code (inside timer):

static float zoom=1.0;
mandelbrot_draw<128,64,1>(+0.37,-0.1,zoom);
zoom*=1.02; if (zoom>100000) zoom=1.0;

Also for non AVR32 C++ environment use this:

//------------------------------------------------------------------------------------------
#ifndef _AVR32_compiler_h
#define _AVR32_compiler_h
//------------------------------------------------------------------------------------------
typedef int32_t  S32;
typedef int16_t  S16;
typedef int8_t   S8;
typedef uint32_t U32;
typedef uint16_t U16;
typedef uint8_t  U8;
//------------------------------------------------------------------------------------------
#endif
//------------------------------------------------------------------------------------------

[Edit2] higher float precision in GLSL

The main problem with Mandelbrot is that it need to add numbers with very big exponent difference. For +,- operations we need to align mantissa of both operands, add them as integer and normalize back to scientific notation. However if the exponent difference is big then the result mantissa need more bits than can fit into 32 bit float so only 24 most significant bits are preserved. This creates the rounding errors causing your pixelation. If you look at 32bit float in binary you will see this:

float a=1000.000,b=0.00001,c=a+b; 

//012345678901234567890123456789 ... just to easy count bits
a=1111101000b                                         // a=1000
b=         0.00000000000000001010011111000101101011b  // b=0.00000999999974737875
c=1111101000.00000000000000001010011111000101101011b  // not rounded result
c=1111101000.00000000000000b                          // c=1000 rounded to 24 bits of mantissa

Now the idea is to enlarge the number of mantissa bits. The easiest trick is to have 2 floats instead of one:

//012345678901234567890123 ... just to easy count bits
a=1111101000b                                         // a=1000
                       //012345678901234567890123 ... just to easy count     b=         0.0000000000000000101001111100010110101100b    // b=0.00000999999974737875
c=1111101000.00000000000000001010011111000101101011b  // not rounded result
c=1111101000.00000000000000b                          // c=1000 rounded to 24 
 +          .0000000000000000101001111100010110101100b
                           //012345678901234567890123 ... just to easy count bits

so some part of the result is in one float and the rest in the other... The more floats per single value we have the bigger the mantissa. However doing this on bit exact division of big mantissa into 24 bit chunks would be complicated and slow in GLSL (if even possible due to GLSL limitations). Instead we can select for each of the float some range of exponents (just like in example above).

So in the example we got 3 floats (vec3) per single (float) value. Each of the coordinates represent different range:

       abs(x) <= 1e-5
1e-5 < abs(y) <= 1e+5
1e+5 < abs(z) 

and value = (x+y+z) so we kind of have 3*24 bit mantissa however the ranges are not exactly matching 24 bits. For that the exponent range should be divided by:

log10(2^24)=7.2247198959355486851297334733878

instead of 10... for example something like this:

       abs(x) <= 1e-7
1e-7 < abs(y) <= 1e+0
1e+0 < abs(z) 

Also the ranges must be selected so they handle the ranges of values you use otherwise it would be for nothing. So if your numbers are <4 its pointless to have range >10^+5 So first you need to see what bounds of values you have, then dissect it to exponent ranges (as many as you have floats per value).

Beware some (but much less than native float) rounding still occurs !!!

Now doing operations on such numbers is slightly more complicated than on normal floats as you need to handle each value as bracketed sum of all components so:

(x0+y0+z0) + (x1+y1+z1) = (x0+x1 + y0+y1 + z0+z1)
(x0+y0+z0) - (x1+y1+z1) = (x0-x1 + y0-y1 + z0-z1)
(x0+y0+z0) * (x1+y1+z1) = x0*(x1+y1+z1) + y0*(x1+y1+z1) + z0*(x1+y1+z1)

And do not forget to normalize the values back to the defined ranges. Avoid adding small and big (abs) values so avoid x0+z0 etc ...

[Edit3] new win32 demo CPU vs. GPU

• win32 Mandelbrot demo 64bit floats

Both executables are preset to the same location and zoom to show when the doubles starts to round off. I had to upgrade slightly the way how px,py coordinates are computed as around 10^9 the y axis started to deviate at this location (The threshold still might be too big for other locations)

Here preview CPU vs. GPU for high zoom (n=1600):

RT GIF capture of CPU (n=62++, GIF 4x scaled down):

Answer 2

The optimized escape algorithm should be fast enough to draw the Mandelbrot set in real time. You can use multiple threads so that your implementation will be faster (this is very easy using OpenMP for example). You can also manually vectorize your code using SIMD instructions to make it even faster if needed. You could even run this directly on the GPU using either shaders and/or GPU computing frameworks (OpenCL or CUDA) if this is still not fast enough to you (although this is a bit complex to do efficiently). Finally you should tune the number of iterations so it is rather small.

Zooming should not have any direct impact on the performance. It just changes the input window of the computation. However, it does have an indirect impact since the actual number of iterations will change. Points outside the window should not be computed.

Double precision should also be enough for drawing Mandelbrot set correctly. But if you really want more precise calculation, you can use double-double precision which gives a quite good precision and not too bad performance. However, implementing double-double precision manually is a bit tricky and it is still significantly slower than using just double precision.

Answer 3

My fastest solutions avoid iterating over large areas of the same depth by following a contour boundary and filling. There is a penalty that it is possible to nip off small buds instead of going around them, but all-in-all a small price to pay for a quick zoom.

One possible efficiency is that if a zoom doubles the scale, you already have ¼ of the points.

For animation, I file each frame's values, doubling the scale each time, and interpolate the in-between frames on playback in real time, so the animation doubles once per second. The double type allows more than 50 key frames to be stored, giving an animation that lasts more than a minute (in and then back out).

The actual iteration is done by hand-crafted assembler, so one pixel is iterated entirely in the FPU.