import enum import time  import numba as nb import numpy as np import matplotlib from PIL import Image, PyAccess   # Amount of times to print the total progress PROGRESS_STEPS: int = 20  # Number of pixels (width*height) and aspect ratio (width/height) RESOLUTION: int = 1920*1080 ASPECT_RATIO: float = 1920/1080  # Value of the center pixel CENTER: complex = 0 + 0j  # For testing: -4.6875 + 2.63671875j  # Value range of the real part (width of the horizontal axis) RE_RANGE: float = 10  # For testing: 10/16  # Show grid lines for integer real and imaginary parts SHOW_GRID: bool = False GRID_COLOR: tuple[int, int, int] = (125, 125, 125)  # Matplotlib named colormap COLORMAP_NAME: str = 'inferno'  # Color of the interior of the fractal (convergent points) INTERIOR_COLOR: tuple[int, int, int] = (0, 0, 60) # Color for large divergence counts ERROR_COLOR: tuple[int, int, int] = (0, 0, 60)   # Plot range of the axes X_MIN = CENTER.real - RE_RANGE/2  # min Re(z) X_MAX = CENTER.real + RE_RANGE/2  # max Re(z) Y_MIN = CENTER.imag - RE_RANGE/(2*ASPECT_RATIO)  # min Im(z) Y_MAX = CENTER.imag + RE_RANGE/(2*ASPECT_RATIO)  # max Im(z)  x_range = X_MAX - X_MIN y_range = Y_MAX - Y_MIN pixels_per_unit = np.sqrt(RESOLUTION/(x_range*y_range))  # Width and height of the image in pixels WIDTH = round(pixels_per_unit*x_range) HEIGHT = round(pixels_per_unit*y_range)   # Maximum iterations for the divergence test MAX_ITER: int = 10**4  # recommended >= 10**3 # Minimum consecutive abs(r) decreases to declare linear divergence MIN_R_DROPS: int = 4  # recommended >= 2 # Minimum iterations to start checking for slow drift (unknown divergence) MIN_ITER_SLOW: int = 200  # recommended >= 100   # Max value of Re(z) and Im(z) such that the recursion doesn't overflow CUTOFF_RE = 7.564545572282618e+153 CUTOFF_IM = 112.10398935569289   # Precompute the colormap CMAP_LEN: int = 2000 cmap_mpl = matplotlib.colormaps[COLORMAP_NAME] # Start away from 0 (discard black values for the 'inferno' colormap) # Matplotlib's colormaps have 256 discrete color points n_cmap = round(256*0.85) CMAP = [cmap_mpl(k/256) for k in range(256 - n_cmap, 256)] # Interpolate x = np.linspace(0, 1, num=CMAP_LEN) xp = np.linspace(0, 1, num=n_cmap) c0, c1, c2 = tuple(np.interp(x, xp, [c[k] for c in CMAP]) for k in range(3)) CMAP = [] for x0, x1, x2 in zip(c0, c1, c2):     CMAP.append(tuple(round(255*x) for x in (x0, x1, x2)))   class DivType(enum.Enum):     """Divergence type."""      MAX_ITER = 0  # Maximum iterations reached     SLOW = 1  # Detected slow growth (maximum iterations will be reached)     CYCLE = 2  # Cycled back to the same value after 8 iterations     LINEAR = 3  # Detected linear divergence     CUTOFF_RE = 4  # Diverged by exceeding the real part cutoff     CUTOFF_IM = 5  # Diverged by exceeding the imaginary part cutoff   @nb.jit(nb.float64(nb.float64, nb.int64), nopython=True) def smooth(x, k=1):     """Recursive exponential smoothing function."""      y = np.expm1(np.pi*x)/np.expm1(np.pi)     if k <= 1:         return y     return smooth(y, np.fmin(6, k - 1))   @nb.jit(nb.float64(nb.float64), nopython=True) def get_delta_im(x):     """Get the fractional part of the smoothed divergence count for     imaginary part blow-up."""      nu = np.log(np.abs(x)/CUTOFF_IM)/(np.pi*CUTOFF_IM - np.log(CUTOFF_IM))     nu = np.fmax(0, np.fmin(nu, 1))     return smooth(1 - nu, 2)   @nb.jit(nb.float64(nb.float64, nb.float64), nopython=True) def get_delta_re(x, e):     """Get the fractional part of the smoothed divergence count for     real part blow-up."""      nu = np.log(np.abs(x)/CUTOFF_RE)/np.log1p(e)     nu = np.fmax(0, np.fmin(nu, 1))     return 1 - nu   @nb.jit(     nb.types.containers.Tuple((         nb.float64,         nb.types.EnumMember(DivType, nb.int64)     ))(nb.complex128),     nopython=True ) def divergence_count(z):     """Return a smoothed divergence count and the type of divergence."""      delta_im = -1     delta_re = -1     cycle = 0     r0 = -1     r_drops = 0  # Counter for number of consecutive times abs(r) decreases     a, b = z.real, z.imag     a_cycle, b_cycle = a, b     cutoff_re_squared = CUTOFF_RE*CUTOFF_RE      for k in range(MAX_ITER):          e = 0.5*np.exp(-np.pi*b)          cycle += 1         if cycle == 8:             cycle = 0             r = e*np.hypot(0.5 + a, b)/(1e-6 + np.abs(b))              if r < r0 < 0.5:                 r_drops += 1             else:                 r_drops = 0             # Stop early due to likely slow linear divergence             if r_drops >= MIN_R_DROPS:                 delta = 0.25*(CUTOFF_RE - a)                 return k + delta, DivType.LINEAR              # Detected slow growth (maximum iterations will be reached)             if ((k >= MIN_ITER_SLOW) and (r0 <= r)                     and (r + (r - r0)*(MAX_ITER - k) < 8*0.05)):                 delta = 0.25*(CUTOFF_RE - a)                 return k + delta, DivType.SLOW             r0 = r              # Cycled back to the same value after 8 iterations             if (a - a_cycle)**2 + (b - b_cycle)**2 < 1e-16:                 return k, DivType.CYCLE             a_cycle = a             b_cycle = b          a0 = a         # b0 = b         s = np.sin(np.pi*a)         c = np.cos(np.pi*a)         # Equivalent to:         # z = 0.25 + z - (0.25 + 0.5*z)*np.exp(np.pi*z*1j)         # where z = a + b*1j         a += e*(b*s - (0.5 + a)*c) + 0.25         b -= e*(b*c + (0.5 + a0)*s)          if b < -CUTOFF_IM:             delta_im = get_delta_im(-b)         if a*a + b*b > cutoff_re_squared:             delta_re = get_delta_re(np.hypot(a, b), e)         # Diverged by exceeding a cutoff         if delta_im >= 0 or delta_re >= 0:             if delta_re < 0 or delta_im <= delta_re:                 return k + delta_im, DivType.CUTOFF_IM             else:                 return k + delta_re, DivType.CUTOFF_RE      # Maximum iterations reached     return -1, DivType.MAX_ITER   @nb.jit(nb.complex128(nb.float64, nb.float64), nopython=True) def pixel_to_z(a, b):     re = X_MIN + (X_MAX - X_MIN)*a/WIDTH     im = Y_MAX - (Y_MAX - Y_MIN)*b/HEIGHT     return re + 1j*im   @nb.jit(nb.float64(nb.float64), nopython=True) def cyclic_map(g):     """A continuous function that cycles back and forth between 0 and 1."""      # This can be any continuous function.     # Log scale removes high-frequency color cycles.     # freq_div = 1     # g = np.log1p(np.fmax(0, g/freq_div)) - np.log1p(1/freq_div)      # Beyond this value for float64, decimals are truncated     if g >= 2**51:         return -1      # Normalize and cycle     # g += 0.5  # phase from 0 to 1     return 1 - np.abs(2*(g - np.floor(g)) - 1)   def get_pixel(px, py):     z = pixel_to_z(px, py)     dc, div_type = divergence_count(z)     match div_type:         case DivType.MAX_ITER