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Overview

Fractal rendering is computationally intensive, requiring millions of floating-point operations per frame. This guide explores the optimization strategies implemented in Fract’ol and techniques for balancing visual quality with performance.

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Key Performance Parameters

Three critical parameters defined in fractol_init.c:21-28 control the quality-performance tradeoff: 
void	data_init(t_fractol *fractol)
{
	fractol->escape_value = 4;
	fractol->iterations_definition = 30;
	fractol->shift_x = 0.0;
	fractol->shift_y = 0.0;
	fractol->zoom = 1;
}

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Iteration Limit (iterations_definition)

Default Value: 30 The iteration limit is the most impactful parameter for both quality and performance.
1
How It Affects Performance
2
Each pixel requires up to iterations_definition iterations:
3
while (i < fractol->iterations_definition)
{
	z = sum_complex(sqare_complex(z), c);
	if ((z.x * z.x) + (z.y * z.y) > fractol->escape_value)
		return;  // Early exit
	i++;
}
4
Computation cost:
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  • Single frame: 2000 × 1500 × iterations_definition operations
  • At 30 iterations: 90 million operations
  • At 100 iterations: 300 million operations
    • 6
    How It Affects Quality
    7
    Low iterations (10-30):
    8
  • Fast rendering
  • Lost detail near set boundaries
  • Blocky, less smooth gradients
  • Adequate for initial exploration
    • 9
    Medium iterations (50-100):
    10
  • Balanced quality/performance
  • Good detail in most regions
  • Smooth color transitions
  • Recommended for detailed viewing
    • 11
    High iterations (200+):
    12
  • Slow rendering
  • Maximum detail at deep zoom levels
  • Required for extreme magnifications
  • Necessary for publication-quality images
    • The optimal iteration count scales with zoom level. At 10x zoom, you may need 3x more iterations to maintain the same detail level.

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    Runtime Adjustment

    Users can dynamically adjust iterations using keyboard controls (events.c:38-41): 
    else if (keysym == XK_plus)
	fractol->iterations_definition += 5;
else if (keysym == XK_minus)
	fractol->iterations_definition -= 5;
    Each adjustment triggers a complete re-render via fractol_render(fractol).

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    Escape Value Optimization

    Default Value: 4 The escape value determines when we consider a point to have “escaped” to infinity: 
    if ((z.x * z.x) + (z.y * z.y) > fractol->escape_value)

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    Why 4 is Optimal

    For the quadratic map z = z² + c, if |z| > 2, the sequence is guaranteed to diverge to infinity. Since we compare the squared magnitude to avoid sqrt(), we use |z|² > 4.Using exactly 4 is optimal because:
    • Too low (< 4): False positives - some bounded points incorrectly marked as escaped
    • Too high (> 4): Wasted iterations checking points already proven to diverge
    Using squared magnitude instead of actual magnitude:Without optimization:
    if (sqrt(z.x * z.x + z.y * z.y) > 2.0)  // Expensive sqrt()
    Optimized version:
    if ((z.x * z.x) + (z.y * z.y) > 4)  // No sqrt() needed
    Performance gain: Approximately 30-40% faster rendering by eliminating 90 million sqrt() calls per frame.

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    Zoom and Precision Considerations

    Zoom is controlled via mouse wheel events (events.c:46-60): 
    int	mouse_handler(int button, int x, int y, t_fractol *fractol)
{
	if (button == Button4)
		fractol->zoom *= 0.95;  // Zoom in (5% closer)
	else if (button == Button5)
		fractol->zoom *= 1.05;  // Zoom out (5% farther)
	fractol_render(fractol);
	return (0);
}

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    Coordinate Calculation with Zoom

    From fractol_render.c:45-46: 
    z.x = (map(x, -2, +2, WIDTH) * fractol->zoom) + fractol->shift_x;
z.y = (map(y, +2, -2, HEIGHT) * fractol->zoom) + fractol->shift_y;
    As zoom decreases (zooming in):
    • Smaller coordinate range examined
    • Higher effective resolution
    • More iterations needed for same visual detail

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    Floating-Point Precision Limits

    C’s double type provides approximately 15-17 decimal digits of precision. At extreme zoom levels (zoom < 1e-13), you’ll encounter precision artifacts where the fractal appears pixelated or incorrect.
    Zoom thresholds:
    • zoom = 1.0: Default view, no precision issues
    • zoom = 0.01: 100x magnification, still precise
    • zoom = 1e-10: Near precision limits, may see artifacts
    • zoom < 1e-13: Significant precision degradation
    If you need deeper zoom capabilities, consider using arbitrary-precision libraries or specialized fractal software that implements perturbation theory.

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    Rendering Pipeline Efficiency

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    Sequential Pixel Processing

    The renderer processes pixels sequentially (fractol_render.c:63-81): 
    void	fractol_render(t_fractol *fractol)
{
	int	x;
	int	y;

	y = 0;
	while (y < HEIGHT)
	{
		x = 0;
		while (x < WIDTH)
		{
			handle_pixel(x, y, fractol);
			x++;
		}
		y++;
	}
	mlx_put_image_to_window(fractol->mlx_connection, fractol->mlx_window,
		fractol->img.img_ptr, 0, 0);
}

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    Memory Access Optimization

    Pixel data is written directly to the image buffer using pointer arithmetic (fractol_render.c:15-21): 
    static void	my_pixel_put(int x, int y, t_img *img, int color)
{
	int	offset;

	offset = (y * img->line_length) + (x * (img->bpp / 8));
	*(unsigned int *)(img->pixel_ptr + offset) = color;
}
    Why this is efficient:
    • Direct memory write, no function call overhead
    • Single calculation per pixel
    • Cache-friendly sequential access pattern
    mlx_put_image_to_window() is called once after all pixels are computed, minimizing expensive window operations.

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    Performance Optimization Strategies

    1
    Strategy 1: Adaptive Iteration Count
    2
    Start with low iterations for exploration, increase for detailed viewing:
    3
    // Quick preview
fractol->iterations_definition = 20;

// Detailed view
fractol->iterations_definition = 100;

// Deep zoom
fractol->iterations_definition = 300;
    4
    Strategy 2: Early Escape Detection
    5
    The algorithm already implements this optimally:
    6
    if ((z.x * z.x) + (z.y * z.y) > fractol->escape_value)
	return;  // Immediately exits loop
    7
    Points that escape quickly (most of the image) skip remaining iterations.
    8
    Strategy 3: Reduced Resolution for Real-Time Interaction
    9
    Modify fractol.h:24-25 for faster interaction:
    10
    // Default (high quality)
# define WIDTH   2000
# define HEIGHT  1500

// Fast preview
# define WIDTH   800
# define HEIGHT  600
    11
    Impact: 800×600 is 6.25× faster than 2000×1500
    12
    Strategy 4: Cache-Aware Color Computation
    13
    Color mapping is performed only for escaped points:
    14
    color = map(i, PSYCHEDELIC_LIME, PSYCHEDELIC_MINT,
            fractol->iterations_definition);
    15
    Points in the set (white) skip this calculation entirely.

    ​
    Benchmarking Different Settings

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    Typical Rendering Times

    These are approximate values on a modern CPU (Intel i5/i7 or equivalent). Actual times vary based on fractal region and hardware.
    Resolution: 2000×1500
    IterationsAvg TimeUse Case
    30~100msInitial exploration
    50~150msGeneral viewing
    100~300msDetailed examination
    200~600msDeep zoom
    500~1500msMaximum quality
    Resolution: 800×600
    IterationsAvg TimeUse Case
    30~15msSmooth real-time panning
    100~50msInteractive exploration

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    Computational Complexity

    Per-pixel complexity: 
    O(iterations_definition × operations_per_iteration)
    Per-frame complexity: 
    O(WIDTH × HEIGHT × iterations_definition)
    Operations per iteration:
    • 1× complex square: 3 multiplications
    • 1× complex add: 2 additions
    • 1× magnitude check: 3 operations (2 multiplies, 1 add, 1 compare)
    • Total: ~8 floating-point operations
    At 2000×1500 with 30 iterations: 720 million floating-point operations per frame!

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    Advanced Optimization Techniques

    The rendering loop is embarrassingly parallel - each pixel is independent. Implementing multi-threading could provide near-linear speedup based on CPU core count.Approach:
    • Divide screen into horizontal strips
    • Assign each strip to a separate thread
    • 8 cores could theoretically achieve 8× speedup
    Trade-off: Increased code complexity, thread synchronization overhead
    Single Instruction Multiple Data (SIMD) can process multiple pixels simultaneously using SSE/AVX instructions.Benefit: Process 4-8 pixels per CPU cycle Trade-off: Platform-specific code, complex implementation
    GPUs excel at parallel fractal rendering, capable of computing thousands of pixels simultaneously.Benefit: 100-1000× speedup possible Trade-off: Requires OpenCL/CUDA, different graphics framework (not MiniLibX)
    For deep zooms, perturbation theory allows calculating only the difference from a reference orbit, enabling arbitrary precision at standard double speed.Benefit: Deep zoom beyond float precision limits Trade-off: Complex algorithm, significant implementation effort

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    Quality vs Performance Guidelines

    For smooth interaction: 
    iterations_definition = 20-30
WIDTH = 800, HEIGHT = 600
    For balanced quality: 
    iterations_definition = 50-100
WIDTH = 1920, HEIGHT = 1080
    For maximum quality: 
    iterations_definition = 200-500
WIDTH = 3840, HEIGHT = 2160  // 4K
    At maximum quality settings, rendering a single frame may take several seconds. This is normal for fractal rendering and reflects the computational complexity of the mathematics.

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