Hyperreal doesn't have any f64 mode. All math done with hyperreals is at infinite precision using a Rational of two BigUInts and a recursive real Computable. Real provides a cohesive interface over both allowing for easy scalar math. Computables are handled symbolically through a set of deterministic reduction rules until approximation is required, to preserve precision and reduce complexity. Approximation only happens at explicit public API boundaries like .to_f64_lossy() not used except for IO.
Hyperreal gets performance back through caching observed facts about the numbers it's representing at creation, and through operations, and specializing dispatch for predicates and geometric operations. Using this approach throughout the stack allows us to avoid computing on the full representation or collapsing it into an approximation. Instead asking questions like "do we know if it's definitely zero, definitely not, or unknown?" or "is it rational?" or "does it have a known sign, or unknown?" and so on. Each question specializes dispatch further, and some eliminate the need for it entirely.
Asking questions using the cached facts is approximately as fast as computing with f64s. So we do that whenever possible throughout the stack. But then when you actually need to do the exact computation, hyperreal does that too, and can approximate it out to whatever precision you'd like. f32 and f64 being common, but others being supported as well. The downside is that calculating quickly with them requires this sort specialization, but the work's been done for the geometry functions.
I'll look into DoubleFloat and ErrorPropagatingFloat for benches. I should mention that numerica@128bit beat the other pure rust bignum crates I tested. The benchmarks are mostly just to give me an understanding of the performance shapes of the implementation choices of high precision numeric libraries alongside hyperreal.