9999999999999999.0 – 9999999999999998.0

My university moved it from a 100 series course to a 200 series course but it's still being taught to ECE undergrads.

The problem is more that we don't have the tools to track and understand how the errors are altered as we do the math (e.g. how would you even begin to try representing catastrophic cancellation at compile-time) and doing the numerical error analysis on the abstract math itself is hard once the math gets complex let alone trying to figure it out after you've optimized the code for performance & tweaked the algorithms for real-world data/discrete space.

Now perhaps it could be possible to do it at runtime in some way but I suspect the performance of that is prohibitive to the point where arbitrary precision math or decimal numbers is going to be a better solution.

The point is that this reveals a common weakness in most programming languages. Not that floating point math has limits, but that this isn't well communicated to the user. One of the hallmarks of good programming language design is the "principle of least surprise" which things like funky floating point problems definitely fall into. Not everyone who uses programming languages, in fact very few of them, have taken numerical analysis, and many devs are not well versed in the weaknesses of floating point math. So much so that a very common way for devs to become acquainted with those limits and weaknesses is by simply blundering into them, unknowingly writing bugs, and then finding the hard way the sharp corners in the dark. This is not ideal.

Consider a similar example, pointers. Some languages (like C and C++) use pointers heavily and it's expected that devs using those languages will be experienced with them. However, pointers are very "sharp" tools and have to be used exceedingly carefully to avoid creating programs with major defects (crashes, memory leaks, vulnerabilities, etc.) They are so hard to get right that even software written by the best coders in the world commonly has major defects in it related to pointer use. This problem is so troubling to some that there are many languages (java, javascript, python, C#, rust, etc.) which have been designed to avoid a lot of the most difficult to use aspects of languages like C and C++, they use garbage collection for memory management, they discourage you from using pointers directly, and so on. However, even those languages do very little to protect the user from blundering into a mindfield of floating point math.

Consider, for example, simply this statement:

x = 9999999999999999.0

Seems rather straightforward, right? But it's not, it's a lie. Because in many languages the value of x won't be as above, it'll be (to one decimal digit precision) 10000000000000000.0 instead. Whereas the value of ....98.0 is the same as the double precision float representation to one decimal digit precision (thus the difference between the two comes out as 2.0 instead of 1.0). Now, maybe in a "the handle is also a knife" language like C this is fine, but we have so many languages which go to such extremes everywhere else to protect the user from hurting themselves except when it comes to floating point math. And here's a perfect case where the compiler, runtime, or IDE could toss an error or a warning. Here you have a perfect example of trying to tell the language something you want which it can't do for you in the way you've written, that sounds like an error to me. The string representation of this number implies that you want a precision of at least the 1's place in the decimal representation, and possibly down to tenths. If that's not possible, then it would be helpful for the toolchain you're using for development to tell you that's impossible as close to you doing it as possible, so that you know what's actually going on under the hood and the limitations involved.

Something which would also drive developers towards actually learning the limitations of floating point numbers closer to when they start using them in potentially dangerous ways than instead of having to learn by fumbling around and finding all the sharp edges in the dark. The sharp edges are known already, tools should help you find and avoid them not help new developers run into them again and again.

I majored in math and physics. In math, our version of the course was called "numerical analysis," and we covered those things. In physics, the behavior of floating point math was covered in one of our lab courses.

I don't know what's taught to CS majors, and as others have pointed out, programmers don't necessarily study CS.

I believe the issue is just that the pitfalls of floating point are not apparent without a certain level of math education.

But there may be one more pitfall, which is that those of us using FP regularly, also happen to be "scientific" or "exploratory" programmers who haven't learned a lot of formal software engineering discipline (including me). So we understand the math but might be more prone to making mistakes with it.

I do kind of like the idea of flagging any number that is potentially exposed to a FP issue. We all make mistakes. Displaying all floats in exponential notation by default would be a good enough warning to the wary. We only display them as decimals for readability.