altering Newton’s method

The usual scheme for Newton’s method is z_{n+1} = z_{n} - \frac{f(z_{n})}{f'(z_{n})}. But what if we just choose some random g(z) for \frac{f(z_{n})}{f'(z_{n})}?

Set g(z) = \frac{\cos(e^{z})}{e^{\sin(z)}}
Now, as far as differential equations go, solving
\frac{f(z)}{f'(z)} = \frac{\cos(e^{z})}{e^{\sin(z)}}
is pretty tricky.

But I can still generate a fractal from it:

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