It's not about the bugs, it's about designing the layout of the program in types first (ie, laying out all of the data structures required) such that the actual coding of the functionality is fairly trivial. This is known as type driven development: https://blog.ploeh.dk/2015/08/10/type-driven-development/
At work, I find type hints useful as basically enforced documentation and as a weak sort of test, but few type systems offer decent basic support for the sort of things you would need to do type driven programming in scientific/numerical work. Things like making sure matrices have compatible dimensions, handling units, and constraining the range of a numerical variable would be a solid minimum.
I've read that F# has units, Ada and Pascal have ranges as types (my understanding is these are runtime enforced mostly), Rust will land const generics that might be useful for matrix type stuff some time soon. Does any language support all 3 of these things well together? Do you basically need fully dependent types for this?
Obviously, with discipline you can work to enforce all these things at runtime, but I'd like it if there was a language that made all 3 of these things straightforward.
I suspect C++ still comes the closest to what you’re asking for today, at least among mainstream programming languages.
Matrix dimensions are certainly doable, for example, because templates representing mathematical types like matrices and vectors can be parametrised by integers defining their dimension(s) as well as the type of an individual element.
You can also use template wizardry to write libraries like mp-units¹ or units² that provide explicit representations for numerical values with units. You can even get fancy with user-defined literals so you can write things like 0.5_m and have a suitably-typed value created (though that particular trick does get less useful once you need arbitrary compound units like kg·m·s⁻²).
Both of those are fairly well-defined problems, and the available solutions do provide a good degree of static checking at compile time.
IMHO, the range question is the trickiest one of your three examples, because in real mathematical code there are so many different things you might want to constrain. You could define a parametrised type representing open or closed ranges of integers between X and Y easily enough, but how far down the rabbit hole do you go? Fractional values with attached precision/error metadata? The 572 specific varieties of matrix that get defined in a linear algebra textbook, and which variety you get back when you compute a product of any two of them?
I'd be happy for just ranges on floats being quick and easy to specify even if the checking is at runtime (which it seems like it almost will have to be). I can imagine how to attach precision error/metadata when I need it with custom types as long as operator overloading is supported. I think similarly for specialized matrices, normal user defined types and operator overloading gets tolerably far. Although I can understand how different languages may be better or worse at it. Multiple dispatch might be more convenient than single dispatch, operator overloading is way more convenient than not having operator overloading, etc.
A lot of my frustration it is that the ergonomics of these things tend to be not great even when they are available. Or the different pieces (units, shape checking, ranges) don't necessarily compose together easily because they end up as 3 separate libraries or something.
Crystal certainly supports that kind of typing, and being able to restrict bounds based on dynamic elements recently landed in GCC making it simple in plain C as well.
If x is of type T, what type do you want (x - x) to be?
That's a hard one because it depends on what sort of details you let into types and maybe even on the specific type T. Not saying what I'm asking for is easy! Units and shape would be preserved in all cases I can think of. But with subranges (x - x) may have a super-type of x... or if the type system is very clever the type of (x - x) maybe be narrowed to a value :p
And then there's a subtlety where units might be preserved, but x may be "absolute" where as (x - x) is relative and you can do operations with relative units you can't with absolute units and vice versa. Like the difference between x being a position on a map and delta_x being movement from a position. You can subtract two positions on a map in a standard mathematical sense but not add them.