[Python] Mandelbrot set Coloring Error Around Period-2 Bulb (not colormap related)

I wrote some code to render the Mandelbrot set with continuous coloring, but the bulbs on the period-2 blub are not colored correctly. regions that shout should not escape are colored as though the escape after 100 iterations, which happens to be my iteration limit.

What i've tried:
1. Increasing resolution and max iteration count
2. Increasing precision (np.float32 -> np.float64). Unfortunately, I could not finish execution because of the time complexity.

import numpy as np
import matplotlib.pyplot as plt

def mandelbrot(cmax, width, height, maxiter):
real = np.linspace(-2, 0.5, width, dtype=np.float32)
imag = np.linspace(0, cmax.imag, height // 2 + 1, dtype=np.float32)
c_real, c_imag = np.meshgrid(real, imag)

output = np.zeros((height // 2 + 1, width), dtype=np.uint16)
z_real = np.zeros_like(c_real)
z_imag = np.zeros_like(c_imag)
mask = np.ones_like(c_real, dtype=bool)

p = np.sqrt((c_real - 0.25)**2 + c_imag**2)
cardioid = c_real < p - 2 * p**2 + 0.25
period2_bulb = (c_real + 1)**2 + c_imag**2 < 0.0625
mask[cardioid | period2_bulb] = False
output[~mask] = maxiter - 1

epsilon = 1e-10
for i in range(maxiter):

zr2 = z_real[mask] ** 2
zi2 = z_imag[mask] ** 2

z_imag_new = 2 * z_real[mask] * z_imag[mask] + c_imag[mask]
z_real[mask] = zr2 - zi2 + c_real[mask]
z_imag[mask] = z_imag_new

diverged = zr2 + zi2 >= 4.0
abs_z = zr2 + zi2 + epsilon
output[mask] = i + 1 - np.log(np.log(abs_z)) / np.log(2)
mask[mask] = ~diverged

output[output == maxiter - 1] = 0

full_output = np.vstack([np.flipud(output), np.flipud(output[-2::-1, :])])
return full_output


width, height, maxiter = 12000, 9000, 100
cmax = complex(0, 1)
mandelbrot_set = mandelbrot(cmax, width, height, maxiter)

plt.figure(figsize=(16, 12))
plt.imshow(mandelbrot_set, extent=[-2, 0.5, -1, 1], cmap="inferno", interpolation="bilinear")
plt.colorbar(label="Iterations to Divergence")
plt.title("Mandelbrot Set")
plt.xlabel("Re(c)")
plt.ylabel("Im(c)")
plt.show()

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