# Reliable Mandelbrot

Mathematically reliable images, compare with

# 1 Plain Iteration

When iterating $Z \to Z^2 + C$ , how much precision is necessary?

Suppose one says the image is reliable when the final $Z$ , with all the rounding errors along the way, is equal to an exact $Z$ for a $C_*$ that is not too different from the input $C$ (for example, they are within a pixel distance).

Call the forward error in $Z$ due to rounding, $\Delta Z$ , and calculate derivatives $\frac{dZ}{dZ_1}$ , then the condition that there is a $C_*$ near $C$ resolves to $\left|\Delta Z\right| < \left|\Delta C\right| \left|\frac{dZ}{dZ_1}\right|,$ where $\Delta C < \frac{1}{Zoom \times Height}$ .

Rounding $Z_n$ gives $(1 + \epsilon_n) Z_n$ , where $|\epsilon_n|$ is at most $2^{-Precision}$ . The forward error compounds like: $\Delta Z_{n+1} = 2 (1 + \epsilon_n) Z_n \Delta Z_n + (\Delta Z_n)^2$ Assuming $\Delta Z$ is small (and $Z_n$ is large), ignore the squared term, to get $\Delta Z_n = \Delta Z_1 \left(\prod_k 2 Z_k (1 + \epsilon_k)\right) = \Delta Z_1 \frac{dZ_n}{dZ_1} \prod_k(1 + \epsilon_k)$ and as $Z_1 = C$ all its error is from rounding, so $\Delta Z_1 \approx 2^{-Precision}.$ Altogether in the worst case $(1 + 2^{-Precision})^{Iterations} 2^{-Precision} < \Delta C = \frac{1}{Zoom \times Height}$ which logs to $Iterations \times \log(1 + 2^{-Precision}) - Precision \log(2) < -\log (Zoom \times Height)$ Now $2^{-Precision}$ is small, so $\log(1 + 2^{-Precision}) \approx 2^{-Precision}$ which gives $Iterations \times 2^{-Precision} - Precision \log 2 < -\log (Zoom \times Height)$ and then by WolframAlpha, $Precision > \frac{W\left(\frac{Iterations}{Zoom \times Height}\right) + \log(Zoom \times Height)}{\log(2)}$ where $W(z)$ is the inverse function of $z \to z e^z$ . As $W$ grows very slowly, the commonly used approximation $Precision = \log_2(Zoom \times Height) + K$ is generally reasonable even when $K$ is quite small.

# 2 Perturbed Iteration

When iterating $Z \to Z^2 + C$ at one point (with plenty of precision) then rounding each $Z$ to a working precision and iterating differences of nearby points $z \to 2 Z z + z^2 + c$ , how much precision is necessary?

I have not found a way to compute this other than as a per-pixel status, which tells if the pixel is inaccurate.

Compute: $\Delta z_{n+1} = \Delta z_n \left(2 |Z_n| \left(1 + 2^{-Precision}\right) + 2 |z_n| + \Delta z_n\right) + 2 |Z_n| |z_n| 2^{-Precision}$ Then the pixel is inaccurate when: $\Delta z_n \times Zoom \times Height > \frac{dz_n}{dz_1}.$

Ideally it would be better to know beforehand how much precision is required for every pixel.