I've been doing some symbolica-like things recently in the https://github.com/timschmidt/hyperreal ecosystem. Not a full CAS, just enough symbolic math to maintain precision through the calculations.
Nice, I will check this out in more detail later. I had a quick look at the benchmarks and it looks like you compare f64 hyperreal with numericas 128 bit implementation, which will fall back to using arb-prec GMP. There is also F64(simply wrapping around f64), and now DoubleFloat with 106 bits precision, which should be much faster. There is also the ErrorPropagatingFloat wrapper that may be of interest.
For simple numerical operations, using an entire Symbolica Atom will introduce a large amount of overhead. It should only be used if the expression contains symbols as well. But perhaps I misunderstood the point of the benchmark?
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.
Thanks for the clarification! Hyperreal sounds very useful for zero testing (at the moment I use ErrorPropagatingFloat for this, but it is fickle), I will play around with this in the near future.
Yes, it should be useful for that. Hyperreal's trig and approximate functions performance is also stellar. Perhaps the biggest compromise in terms of the math supported by hyperreals at the moment is that although Rational equality can be exactly tested, Computable equality is currently structural. So it's possible to end up with two mathematically equivalent Computables which aren't structurally equal. Because it's not a full CAS.
It's still possible to approximate them both, and test them against each other, but since the whole architecture is built to reduce, avoid, and cache approximations because they're expensive, it's not the default.
In the end the zero test problem is undecidable for reasonably complicated expressions, so sadly there is no guarantee that you can rewrite one Computable into another even if they evaluate to the same. For polynomials you can do finite field evaluation tests to prove equality with a likelihood bound of your choosing. That may be interesting for hyperreal too.
Using dashes twice like that is valid. It's a bit like parentheses, to frame a tangential statement between them, but with emphasis instead of quietly. See: https://en.wikipedia.org/wiki/Dash
I use that construction in my totally human writing often enough. Some of us missed a few English classes it seems.
> Are you seriously going to claim that this is not LLM generated?
Do you see me making that claim? My comment seems to be about grammar. Do you always jump to conclusions?
> I made no comment as to the validity of the construction.
See:
> I mean look at this sentence which randomly contains the " - " pattern twice in a row
They're called parenthetical dashes. They're not random. And it's one pattern, not two. You'll find it used with parentheses (obviously) and commas as well as dashes and perhaps even other punctuation[1].
As to whether or not the post was written by AI, I don't care either way. That seems to be something you care about. But you shouldn't base those conclusions on the use of parenthetical dashes.
> I’m not a fan of “think of the children“ arguments
Yet you're making one.
> the Internet cannot actually be a complete free for all
Yet in many important ways, it is.
As much as publishers would like to shut down Scihub, it exists. The Pirate Bay famously persists. Nation states with entirely opposed legal systems connect and interoperate to at least some degree.
The OP said: "Extremely depraved things are not the only thing to use freedom of speech for, and freely speaking can result in all kinds of repressions."
Which is objectively true.
You're throwing reporters, political dissidents, whistleblowers, minority groups, and just regular people who don't appreciate the Stasi in with the child pornographers which some might take as an insult and offense.
What kind of criminal does Phil Zimmermann look like to you? We had this argument already in the 90s.
People lump them together because of an anti-technology reputation, but I don't think most Amish would have trucked with Luddites. Amish tend to avoid actively participating in popular social movements, and oppose violence and property destruction.
An excellent distinction to make. Life however often says "Why not both? And 11 more you'd have never thought of. And one that seems impossible just for fun."
If it's possible, and it can force a function up a gradient, life is almost certainly doing it somewhere.
SpaceX has consistently launched ~90% of the mass to orbit for the whole planet Earth over the last several years[1][2]. There's no one else who could more credibly make such a claim.
They seem to have constructed a rocket with 10x the payload to LEO of the one they used to put those 10k satellites in orbit, and even demonstrated payload deployment. So I'd say 100k looks do-able for them today.
10x that seems aspirational, but not comically so. Folks hate Musk, but that seems to cause them to not see the engineering going on in front of them.
> They seem to have constructed a rocket with 10x the payload to LEO of the one they used to put those 10k satellites in orbit
They seem to have constructed a rocket that consistently gets heavier and more complex and more expensive and farthrt behind schedule and hasn't demonstrated specified payload.
I checked the publicly released stats over Starship's development, and this is what I found: compared with the initial ~5,000 t / ~73.5 MN concept, the latest V3-class Starship/Super Heavy is trending toward roughly 35%+ more loaded propellant mass and about 40% more maximum liftoff thrust if you use the FAA’s ~103 MN figure. Payload capability has also moved upward from the early 100+ t reusable LEO baseline to SpaceX’s current public claim of up to 150 t fully reusable and 250 t expendable.
Starship has never met claimed specs and capabilities. It is so far behind schedule that it won't meet specs in time to remain relevant. Which is a generous way of saying it never will.
Agreed. They're already stretching starship. And there's long been talk of a wider version yet. Starship is already pretty impressive considering it's just about exactly the size of Sea Dragon.
While true, this is insufficient to make the new claim credible. If the proposed satellites only weighed 100kg and remain on orbit for 3 years, to keep a million up requires:
They've been approved for 44 Starship launches from Kennedy Space Center in Florida, and are aiming for 160 total launches in 2026. They've recently purchased a giant tract of land in Louisana to build a third starport. 222/year is looking doable.
At this point, 160 Starship launches in 2026 would be close to every weekday.
They already have three launch sites for Falcon and can't do 200.
(Also see edit, my first post relied on Apple's autocomplete for maths and it used a short ton, plus point about these numbers corresponding to a mere 100 kg per satellite).
The 160 launches figure includes falcons. Seems like Starship fuels and flight tests faster than Falcon though. And if they manage to reuse second stages, then that eliminates a significant manufacturing bottleneck.
If you're counting Falcons, you are making my point for me: even with those, on three launch sites, they still can't get close to the minimum for an extremely small, to the point of being unreasonable, target satellite mass.
Further, until they actually do solve upper stage reuse, it is an "if" which can kill the economics of the vehicle itself, let alone reach the eventual potential cost reductions necessary for space based data centres to be worthwhile.
I don't see any reason a non-renewable Starship upper stage would kill the economics of the vehicle. No one else has a renewable upper stage yet, so there's no competition in that space until someone else does. Stoke have an interesting design but it hasn't flown yet and is only about the size of Falcon.
If they do manage to reuse the upper stage, then they should have no problem exceeding falcon launch cadence. Starship is much easier to build than Falcon. Welding is simpler and less expensive than the carbon composites used on Falcon upper stages.
The competition isn't other launch providers, it's not going to space at all.
According to Google, the price threshold to make space make more sensible than building on the ground is $200/kg: https://arxiv.org/pdf/2511.19468
Without full reusability, the estimated cost for Starship to LEO is kinda hard to find (necessarily, given the design isn't yet finalised), Wikipedia says $100m/launch in expendable mode, and the SpaceX website* says 250 metric tonnes in expendable mode, which is $100e6/250 metric tonnes = $400/kg.
They can form new associations between concepts via their input prompts and thinking text. That is a form of learning. Just not very durable. I liken it to https://en.wikipedia.org/wiki/Anterograde_amnesia
I hear you. I think we are already seeing some middle ground with agentic systems using RAG, skills.md files, etc. It's a sort of disassociated card catalog memory. An engineer's notebook. Not the integrated, correlated, pre-processed set of relationships in the model. How to go backward from the notebook -> model cheaply without tanking performance is definitely one of those billion dollar questions.
a little glib, but there is in fact long term learning. It's just that you are not the one mentoring- the models go to intensive OpenAI/Anthropic/Google school for a quarter or half a year and come back (hopefully) improved. You just hope they're getting a good education. Certainly it's a very prestigious one.
Benchmarks against Symbolica and numerica here: https://github.com/timschmidt/hyperlattice/blob/main/benchma...
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