Deep zoom, perturbation and BLA
Deep zoom is less a question of magnification than of numerical separation: the renderer must still distinguish adjacent pixels after ordinary floating-point numbers have lost that ability.
The floating-point precision wall
Near the initial Mandelbrot view, a double-precision coordinate can represent far more detail than the display requires. As zoom grows, the coordinate step between neighboring pixels shrinks until several pixels map to the same representable value. Blocky or frozen detail at that point is a precision failure, not a property of the fractal.
A high-precision reference orbit
Instead of evaluating every pixel entirely with costly arbitrary-precision arithmetic, the renderer calculates one carefully chosen reference orbit at high precision. Nearby pixels are expressed as small deltas from that reference and can often be advanced with much faster arithmetic.
Perturbation formulas describe how those deltas evolve. If a delta stops being reliable, the renderer can rebase or fall back to a safer path rather than displaying rectangular artifacts.
Bilinear approximation skips stable work
BLA groups stretches of a reference orbit into reusable approximations. When the current delta lies inside the approximation’s validity range, many iterations can be advanced together. The acceleration is data-dependent: difficult boundary regions still receive the calculations they need.
Progressive quality at extreme depth
MandelKit renders coarse, useful feedback first and then replaces it with screen-matched detail. The final pass must reach the display’s actual pixel density; otherwise a mathematically correct deep zoom can still look as if it stopped at half or quarter resolution.