The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor.
The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…
> I would say modern c++ written by someone already familiar with rust will probably be structured in a way that's extremely easy to port because you end up modeling the borrow checker in your brain.
I can't stress out how much important this sentence is. I would even remove the "familiar with rust" part.
Anyone who still thinks it's good to use C/CPP on modern hardware where Rust support is available and good: please print the sentence above and post it all over your place.
Imagine you read a value from stdin and parse it as:
Maybe Int
So your program splits into two branches:
1. Nothing branch: you failed to obtain an Int.
There is no integer to use as an index, so you can’t even attempt a safe lookup into something like Vect n a.
2. Just i branch: you do have an Int called i.
But an Int is not automatically a valid index for Vect n a, because vectors are indexed by Fin n (a proof carrying “bounded natural”).
So inside the Just i branch, you refine further:
3. Try to turn the runtime integer i into a value of type Fin n.
There are two typical shapes of this step:
* Checked conversion returning Maybe (Fin n)
If the integer is in range, you get Just (fin : Fin n). Otherwise Nothing.
Checked conversion returning evidence (proof) that it’s in range
For example: produce k : Nat plus a proof like k < n (or LTE (S k) n), and then you can construct Fin n from that evidence.
(But it’s the same basically, you end up with a “Maybe LTE…”
Now if you also have a vector:
xs : Vect n a
… the n in Fin n and the n in Vect n a are the same n (that’s what “unifies” means here: the types line up), so you can do:
index fin xs : a
And crucially:
there is no branch in which you can call index without having constructed the Fin n first,
so out-of-bounds access is unrepresentable (it’s not “checked later”, it’s “cannot be expressed”).
And within _that_ branch of the program, you have a proof of Fin n.
Said differently: you don’t get “compile-time knowledge of i”; you get a compile-time guarantee that whatever value you ended up with satisfies a predicate.
Concretely: you run a runtime check i < n. _ONCE_
If it fails, you’re in a branch where you do not have Fin n.
If it succeeds, you construct fin : Fin n at runtime (it’s a value, you can’t get around that), but its type encodes the invariant “in bounds”/check was done somewhere in this branch.
As much as I agree with you. Iran is signatory of the NPT with all its consequences.
Instead of letting more countries develop these weapons, we should work on denuclearizing all countries, starting with the US and Russia and their insane arsenals! And maybe build a unified international legal framework for civilian nuclear developments and applications from energy to medical outside of the "security council's" ferule!
The next generation will include another processor to offload the inference from the RISC V processors used to offload inference from the host machine.
Though ignoring all the PL lessons for the sake of "pragmatism" is just plain ignorance.
Rust has a quite solid theory backing. Zig has nothing interesting really: it's "vibe coded" by someone who was doing game dev or something like that.