Do Developers Understand IEEE Floating Point? [pdf] 116 points by matt_d 3 months ago | hide | past | web | favorite | 100 comments

 If you're like me and struggled to understand the traditional textbook floating point formula explanation, Fabien Sanglard (of doom/wolf3d black book fame) wrote a page that finally made sense for me: http://fabiensanglard.net/floating_point_visually_explained/...
 Can I ask an embarassing question to verify I'm not misunderstanding that page? Does this mean that floats are less precise the larger they are? You have as many bits to describe the "position of the number" between 0-1 as you do between 2048-4096
 This is precisely what the term floating point means. The decimal point "floats" right or left depending on magnitude due to the mantissa-exponent representation. The opposite is fixed point where you have a fixed number of bits reserved for the whole and fractional parts. A 32-bit floating point can represent a vastly greater range of magnitudes than a 32-bit fixed point, with a vastly greater precision for values near zero, but at the expense of absolute precision at large magnitudes.
 At the computer floating point was developed, people understood it implicitly, because it was just an automated version of their current tools: log tables and slide rules.And likewise they knew better than to use floating-point where it doesn't work.
 There are a couple of good images on this page which make clear the fact that floating point position 'degrades' in discrete steps as the number gets larger. It's not quite the same as counting in logarithmic terms, which is shown for comparison.
 They are less precise in the absolute sense (the absolute error is larger) but equally so in the relative sense (the relative error is the same).Well, essentially, of course.
 Yes, and some larger numbers are not representable at all due to gaps. E.g 16777217 is not representable in single precision IEEE 754. There are handy tables here for the range and precision values: https://en.wikipedia.org/wiki/IEEE_754-1985
 Yeah and thats why you often have to recale the problem before applying numerical methods
 > You have as many bits to describe the "position of the number" between 0-1 as you do between 2048-4096You mean 0.5-10-1 is about a quarter of the numbers, just as many as 1-inf
 The author's presentation of the traditional explanation makes sense only if M is visualized as binary, while E remains decimal. But the reader coming to the subject for the first time is in no position to guess this, and would need a clue like subscript notation: B_2, E_10.Traditional explanation done well:
 A one pager explanation I wrote after it finally clicked for me -- floating point is just binary scientific notation! Not sure why more places don't explain it like this. https://github.com/dasl-/floating-point-numbers#floating-poi...
 With some important algebraic things, like -0 and 0 are distinct numbers that are equal to each other, and 1/0 is infinity (depending on exception mode), but 1/-0 is -infinity, and all of the different kinds of NaNs, etc.
 Plugging my own explanation because HN seemed to like it:
 Although understanding floating point is important, it is often ignored in Data Science schools.The title reminds me a friend of mine who attends a data science master course without computer background. He asked me for help for his numpy homework on Bayesian Probability theory. The number he got was so small that he got weird result. I immedieatly thought it was because float64 couldn't cover that number so I told him to use longdouble dtype (it's 80 bits in Intel I think) and the problem solved.Later he told me even the professor didn't expect such case when announcing the homework and many students couldn't finish the hw on time because of this.
 Would the log-sum-exp trick have helped by any chance? https://en.wikipedia.org/wiki/LogSumExp
 I didn't know this (I'm not in ML/Data Science) but that looks very helpful. I looked through his learning materials but I have never seen this one though.
 Yeah I only happened to know because of ancourse I took that taught some ML. I mentioned it because, later on, I even came across a situation where it would've been directly applicable, but for some reason the thought that it might solve the problem escaped me at that moment, so I asked and someone else had to remind me of that fact... so I feel it's probably not the easiest thing to recognize when it's happening.
 > I immedieatly thought it was because float64 couldn't cover that number so I told him to use longdouble dtype (it's 80 bits in Intel I think) and the problem solved.This is a very dangerous habit. If 64 bits isn't enough you are generally using the wrong representation.
 That's a difficulty that exists for the simulations, optimization problems and various algorithms. Some need so many bits that they don't work at all.
 No such thing as “Data science” - just a fancy term for trade school computer skills. Thats why.
 > No such thing as “Data science” - just a fancy term for trade school computer skills. Thats why.Relevant xkcd: https://xkcd.com/435/Or as my friend put it: everything is just a trivial special case of either mathematics or physics. ;-)
 True :) I know the strip. tbh I just felt like trolling the insecure “data hackers” crowd. Looking at downvotes: I think it worked.
 Fun xkcd, but mathematics does not belong on a continuum of the scientific disciplines, all of which seek to describe nature."Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"https://en.m.wikipedia.org/wiki/Leopold_KroneckerVerbose, highly literal translation: "Loving God created the absolute integers; all else is the work of humanity."
 according to this document the people who implemented java did not:"How Java’s Floating-Point Hurts Everyone Everywhere"https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdfp.s.: it's by William Kahan, who the primary architect behind the IEEE 754-1985 standard for floating-point computation (and its radix-independent follow-on, IEEE 854)
 Would be great to see similar critiques for Python and Go. Found only this for Matlab, on Dr. Kahan's page:
 no language has a good support for IEEE754. The only doable thing is using some weird macros in C/C++, but the rest of the languages ignore the issue altogether. And LLVM too.
 C/C++ is not required to use ieee754 as it's float format, so your code would always run under that cloud.Fortran, D, Factor, and SBCL likely have better support.But anything where a compiler can reorder things is suspect, if you need exact behavior, since even simple register spilling causes trouble (Intel), among other things compilers may do.
 AFAIK C and C++ compilers are not allowed to re-order floating-point arithmetic precisely for that reason. Also, given that all cpus support SSE/Neon/etc. now and I haven't seen a compiler use the 387 FPU, I suspect register spilling shouldn't be an issue these days. Of course, legacy/exotic platforms will deviate from the norm in fun ways.
 >AFAIK C and C++ compilers are not allowed to re-order floating-point arithmetic precisely for that reason.You don't have to reorder to make bugs. Simply having a compiler replace (a-b)+b with a is a bug. Simply flushing an internal 80-bit floating register to a 64 bit memory place, then loading it, causes differences in computations.Note the C++ spec specifically states it does not require an underlying representation or even any certain level of accuracy (6.7.1.8). So many of these bugs are allowed.The spec does not even require that compile time evaluation of an expression must match a runtime evaluation of the same expression (8.20.6). When you hit that in code you're going to be surprised.C/C++ is not required to round-trip floats/doubles when printed and parsed. stream formatting rounding modes are allowed (and are) not exact, leading to different behavior on formatting the same number on different systems.Dealing with these led to compiler implementation defined behavior added due to market pressures, making C/C++ numerics less of a mess, not anything in the spec.C/C++ has notoriously been bad at all this, which is why the history of it is littered with such bugs, errors, undefined behaviors, and dozens of compiler switches to try and mitigate such behavior.For example, many compilers notoriously replaced high-precision Kahan summation with low-precision regular summation because optimizations tell the compiler that such things are allowed.The state of the art in C/C++ is now passable, but it is still flawed, and has been historically very bad.And all this met the C/C++ spec of the time. Because if the underlying representation does not require IEEE 754, then it cannot require only transformations for IEEE 754 without making other representations behave badly.
 Not developers. This survey included mostly PhD students (73), faculty (49), but only 23 software engineers. The rest are others. Only 1/4 see software engineering as their main role according to the paper.A PhD student is very different from a professional developer with years of real-world experience in industry. It's a completely different job.The title is highly misleading.
 It's OK, not misleading at all, and the abstract spells out what they're studying and the motivation behind it clearly.A "developer" is just someone who writes software and not necessarily a "professional developer" with years of experience in a particular kind of job like what you have in mind.Moreover, I would even expect that the vast majority of "professional software engineers" who make a living strictly writing software without any particular domain in focus, might actually fare worse in floating point gotchas than scientific domain workers who are "software engineering dilettantes" but nonetheless do computation as their job.
 Maybe. The paper even mentions this, sort of contradicting your expectation:>Perhaps not surprisingly, the most predictive factor is simply Contributed Codebase Size, the effect of which is shown in Figure 16.>There is a gain of only about 2/15 compared to those who reported a codebase where floating point was not intrinsic or where they were not involved.The title would be better suited if they included a higher number of HPC/ML/etc software engineers (post PhD) that work on larger scale projects in industry/labs.Specifically, I disagree with this assessment:>We believe that the combination of our recruitment process and the resulting background of the participants illustrated here suggest that our sample is a good representative of software developers who write code for, and in support of, science and engineering applications.That's just outright ignoring all science and engineering applications outside of academia/academic research.
 `````` > The title would be better suited if they included a higher number of HPC/ML/etc software engineers (post PhD) that work on larger scale projects in industry/labs. `````` If someone needs to be a "post PhD" software engineer working on large scale computation projects in industry to be considered "a developer" whose problems with floating point are worth investigating and dealing with, then we have a problem of definition.Their premise is that if folks in academia are having a hard time with it, then perhaps the training and tooling around floating point needs some reconsideration. That seems very reasonable to me.
 >Their premise is that if folks in academia are having a hard time with it, then perhaps the training and tooling around floating point needs some reconsiderationThere's been 50+ years of this reconsideration. Some problems are simply hard, and perhaps nothing will make it easier than to simply learn the nuances.For some domains, there are some solutions, like BigInt to avoid overflow, but each comes with other tradeoffs.For general floating-point, it's mathematically quite likely there is no better tradeoff. You cannot stuff infinite values into finite space.
 >That's just outright ignoring all science and engineering applications outside of academia/academic research.What % of scientific code do you think is written outside the areas this paper surveyed? Having been on both sides, I find that the majority of numerical code, especially anything more complex than simple math (which is almost always written poorly outside academic areas), is written/developed by academics, and the results adopted as libs in engineering.
 I'd wager the vast majority of "professional developers with years of real-world experience in industry" have a very superficial understanding of IEEE floating point behavior beyond "sometimes you get inaccurate results due to something-something-rounding-errors".
 I certainly am one of them :)
 I agree that there are definitely a lot of differences between developers in an academic environment to the ones in a corporate environment. I would expect that this counts in favor of the title and not against it. How many of the "professional developers" do you think even know(or remember) what "IEEE floating point" is?
 I once encountered an old database using floats as its primary keys. This was in a fairly popular piece of manufacturing software from the 90s.
 I sometimes, in Python, cautiously use floating point numbers as dictionary keys. I do this when what I mean is "this exact floating-point number" and not "the result of an equivalent computation." It's almost never a good idea, because it's really easy to trip yourself up by assuming the latter when only the former actually holds.
 And even the former apparently isn't guaranteed on all hardware. I found this gem in the Java documentation, I don't know which processors actually do this:>"Arithmetic operations on signaling NaNs turn them into quiet NaNs with a different, but often similar, bit pattern. However, on some processors merely copying a signaling NaN also performs that conversion. In particular, copying a signaling NaN to return it to the calling method may perform this conversion."
 If you're putting floats in a dict, you can't use NaNs anyway, since they aren't equal to each other.
 This is the stuff of nightmares.
 You could hardly avoid this if you, say, memoized some function that has floating-point inputs.
 Indeed, and for that reason I'd be very cautious when attempting to memoize a function with floating-point inputs.
 What type of data would that be? I once did this when I did my own primitive version of a geospatial database in the 90s.
 Useful in case you want multiple entries with an id of approximately 1?
 Potentially, might be useful if you want small fixed-length keys that are fast to compare, and also you want to be able to insert new items between existing ones.
 There was this article posted a while ago that used a Stern-Brocot tree and/or floats for ordering: https://news.ycombinator.com/item?id=16635440It doesn't work in the pathological case though as any float has a fixed number of bits, and I see the article has been amended with a note.
 I doubt most developers really understand how floating points work ( as in how it is represented in memory and why and the mantissa, bias, etc ). Most understand that it is just an approximation and go on about their day.
 When using floating point numbers, that's usually as far as you need to know. That and the range of safe integers for langauges like JS that only do float. (Though now there's BigInt at stage 3)
 Yes with http://0.30000000000000004.com/
 This is a pet subject of mine since as an FPGA developer a huge amount of my time is spent focusing on bit-precision and Fixed-Point and reduced precision floating point implementations of algorithms. In fact one of the interview questions I ask focuses on how to do different mathematical operations in floating point using only logical/integer operations.On the one hand I find the results unsurprising - even people I know to have worked a lot on numerics often have only a rudimental understanding of the intricacies of corner case behaviour for floating point, and yes that absolutely manifests in hitting a wall when something curious goes wrong. Mostly this results in head banging until you find the piece of code going wrong is the numerical piece and then very quickly you start looking at possible floating point gremlins.Having said that though this paper seems to have a very academic view of what HPC is. Even for people designing HPC systems numerical optimizations are rarely a huge chunk of their job so it's probably not actually that important - I think the fact few people have a good understanding is a reflection of the fact it's not often necessary.Finally while we're on the topic: Does anyone know a good tool that can allow me to write an equation, specific the precision of inputs/outputs and get the required precision of all the operators? I know Matlab have the DSP toolbox but it has some serious limitations, I'm still in search of something fantastic.
 Herbie is not an exact match for your need it might be useful nevertheless : https://herbie.uwplse.org/
 I understand floating point representation, but beyond that, it is a bit murky. I find floating point arithmetic to be very hard to really implement. For example, a basic question I have involves scaling: suppose I want to perform an operation involving large numbers like 13000.01 + 13000.01 + 13000.01. Do I get better precision results if I do 0.01 + 0.01 + 0.01 in floating point, then do the 3*13000 in integer arithmetic, and add the results?
 A more general way to do that is Kahan summation[1], where you keep track of the rounding errors made so far.
 Significand * 2^exponent so yes. Another way to think of it more naturally is just remember it's a floating point in a fixed number of bits. Since base doesn't really matter for theory you can just think of a decimal point instead of binary. If you move it over to put more digits/information on the left side then you have to take digits/information from the right side.If you were going to do this though I'd just do 2 ints acting as a single fixed point. Less glue logic and likely more precision. A float using all of the bits would likely be better than int + float as well depending how you configured it.
 It's interesting to point out that you can compute with exact real numbers. But this is of course going to be less efficient than using something crude and fast like floating point numbers. There's also an argument that imprecision from your physical sensors is probably going to be more significant than the rounding errors that come with floating point.Apart from greater speed, floating point numbers don't really have any advantages over exact real numbers. Problems like the inability to compute discontinuous functions over the real numbers manifest as even worse problems over the floating point numbers.I'm also a believer that anything beautiful is probably going to end up useful. I think exact real computation (as described in Type Two Computability Theory) is one such thing. Automatic differentiation was another thing I found beautiful, before I found out about its applications to neural networks.
 Type-two theory has a huge disadvantage: it elevates the type of all operations by a level. For example, real numbers become functions. Hence addition becomes a type 2 functional. Integration becomes type 3. It is very hard to reason about the efficiency (and computability, come to think of it) of type 3 functionals. There's some esoteric example involving the fan functional [1].The complexity theory of type-2 functionals is a theory which is natural to ask for, but after results by Cook, Bellantoni, Kapron, Seth, Pezzoli and others, has come to an impasse and now seems abandoned.
 They effectively change the structure of your code. Every time you do an operation on a real number, it essentially launches a new process that runs indefinitely in the background. I would file that under speed problems. To control the speed penalty, you need to make deep changes to your code. I think Andrej Bauer came to this conclusion.Regarding computability, it's not so bad. Basically a continuous operator is usually computable. Example: Integration is continuous, hence (likely, and in this case actually) computable.
 > floating point numbers don't really have any advantages over exact real numbers.With floating point numbers, you can solve quadratic equations, calculate sin/cos, calculate log/exp. These things have lots of applications.
 I never said that they don't have applications. And you can do all those things with real numbers. What's your point?
 > you can do all those things with real numbers.No you can't.> What's your point?Floats really have lots of advantages over exact real numbers. Being able to compute transcendental functions is one of them. There're others.They're freaking fast, my cheap laptop with integrated GPU can do almost 1 TFlops. They're fixed size i.e. you can handle them without dynamic memory allocations.
 I think you may be misunderstanding what the other guy means by exact real numbers: not rationals (for which indeed you can't do things like square roots) but real numbers expressed, say, as functions that deliver arbitrarily close rational approximations. With these you can compute e.g. square roots just fine.They have disadvantages of their own, though: for instance, y if two of these are equal then in general you can't determine that they are in finite time.
 > if two of these are equal then in general you can't determine that they are in finite time.This is arguably a problem with floating point as well. Two computations F and G which ought to produce the same result can end up differing because of rounding errors. So equality testing is not advisable to do over floating point. Also, inequality of real numbers is still semi-decidable, which is useful.People keep bringing up these problems around not being able to compute discontinuous functions, but those problems still manifest themselves over the floating point numbers, in the sense that a small rounding error in the input can drastically affect the output. This obviously isn't a problem over the rational numbers, but a lot of operations like square-root are not possible over the rational numbers.
 Symbolic computations are even slower than rational numbers. If you only need a couple numbers, you’ll probably be OK, but if you need a billion of them, you’ll have problems.While both rationals and symbolics have their use, saying that floats don’t have any advantages over them is just wrong.
 You're claiming I said things I didn't say.Two things:1. I did acknowledge that floating point numbers are faster. I expressly stated that they do have some advantages over real numbers.2. I'm not talking about symbolic computation.
 > I did acknowledge that floating point numbers are fasterYes you did. And then you said it’s their only advantage. It’s not the only one.For example, in some environments, dynamic memory is undesirable. Especially since you mentioned physical sensors: many real-life sensors are not directly connected to PCs but instead are handled by MCU chips.> I'm not talking about symbolic computation.The numbers you’re talking about are pretty close to symbolics in many practical aspects. From Wikipedia article “Computable number”:equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus
 I didn't mention memory use and dynamic allocation because I thought it was obvious. I filed it under "faster".
 No, it's not just "faster", it makes the difference between possible and impossible. With a physical sensor connected to a small MCU, either you do floats, or you do nothing at all. There's simply no way for a machine like that to do math the way you're thinking.I see this all the time with developers who have zero embedded experience like you: you don't seem to understand that in many embedded environments, dynamic memory allocation is simply not possible. You allocate all memory at the start of the program, and that's it. So doing math the way you suggest is completely impossible.
 My point about floating point numbers having no advantages other than speed and memory usage was referring to the following: Some people think that you can decide equality of floating point numbers, while you can't decide equality of real numbers, but this advantage is an illusion because rounding errors can make numbers that should equal each other not equal each other. So ultimately it ends up being a pure efficiency vs accuracy trade-off, where efficiency includes things like speed and memory, and of course that could make it impractical in an embedded setting. I can't list out every single disadvantage of something being slower and less space-efficient, including in your pet interest, which is embedded programming. Is that not politically correct?And ultimately, I'm interested in beautiful things as well as practical things. I fully acknowledged that the idea here is impractical, though theoretically possible, with some very interesting theorems coming out of it. I don't want to always think like a hard-nosed practitioner. That's stupidly dogmatic.Const-me is complaining that I didn't mention his pet interest, which is GPGPU. And then he said some totally irrelevant and obvious stuff (to anyone who knows CS), just like you did. This is why we can't have nice things.
 GPGPUs ain’t exactly my interest, they, and embedded too, are just parts of my job.I’ve been programming for living for couple decades now, and I like writing efficient code. Not all ideas, however beautiful they might seem to some people, are good enough for actual software that people use to accomplish stuff. This idea is one of them. Spending thousands of CPU cycles to compute a=b*c is extremely inefficient in vast majority of cases. Even a hardware-based implementation won’t be efficient, see what happened to LISP machines.
 The beauty is in the eye of the beholder and is therefore highly subjective.Apparently, that thing that you find beautiful, I find impractical and therefore only applicable to narrow set of problems and environments.Speaking of environments, even on PC, I code for GPGPU as well. Due to extreme levels of parallelism, GPU-running code has almost no support for dynamic memory: there’re append/consume buffers there but they’re very limited and not actually dynamic, just a static buffer with thread-safe cursor over them.
 Also sometimes, you can’t do dynamic memory allocations because it’s explicitly forbidden by law: DO-178B, IEC 62304, etc.
 I have interviewed a lot of developers, and floating point precision is one of my common phone screen questions.My experience is; those who should know about it, do. Of course, a lot of apps or web developers don't. And I think that is okay.The bigger issue is code designed for float32 (for performance reasons) being assumed valid for float64 data. And in these cases the issue is always, what is the appropriate way to communicate to your user the intended use case of the function? When the writer is long gone, a user will not intuitively know. Whenever I have to use a 3rd party library (ahem, PCL). I peruse the code and grep for float variables... If I see too many, I simply assume I need to recenter my float64 to be near 0.0.Not everybody has the experience for this diligence, thus it appears as if developers don't understand.
 A lecture on IEEE floating point from CMU including most of the gotchas the pdf contains https://scs.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=...This is an active research area, ENS de Lyon has a lot of material available if interested implementing floating point on FPGAs, plus there is Vol 2 of The Art of Computer Programming where Knuth shows the reader how Babylonian mathematicians made extensive use of a sexagesimal (radix sixty) positional notation that was unique in that it was actually a floating point form of representation with exponents omitted.
 Even if you understand them... in a polyglot world, you're still going to get burned.It is so easy--especially when you're jumping from a NUMBER language like SQL or Scheme to a floating-point one like C or JavaScript--to backslide into boners like`````` for(i = 0.0; i < 1.0; i += 0.1) `````` which would be a non-boner with a more advanced number tower.edit: My first Clojure program was the last time this got me--LISP on the outside, Java floats/ints/bigints on the inside.
 Yeah, and that’s why we have int for counting.
 So if I'm marking an axis from 0 to 0.9 in 0.1 increments, I should obviously do the integer indirection dance?After all, every damn fool knows--even ones like myself who've implemented FP in assembly--that 0.1 + 0.1 + 0.1 = 0.30000000000000004.Silly me!
 >> So if I'm marking an axis from 0 to 0.9 in 0.1 increments, I should obviously do the integer indirection dance?Yes. Because you clearly don't understand what you're doing at all if you believe that equivalence relation is defined the same way for floating-points as it is defined for integers.Floating-points aren't numbers. They're more like confidence intervals - and at a certain limit a confidence interval behaves just like an integer. But it's only a limit.The fact that most users of programming languages don't want to keep track of that in their code only reinforces my belief that they must do exactly as the software guidelines tell them to.
 I know what floating point is. I've implemented it in hardware. I've implemented it in software. But you know what? Floating point is not the only way people have ever represented real numbers on the computer.In most SQLs, decimal fixednum types are predominant. Most LISPs have a number tower that automatically juggles fixnums, bignums, ratios, floats, and complex numbers. In the 8/16-bit days--before everything had a floating-point unit--it was not uncommon to store and calculate numbers directly in BCD. Two-digits to the byte, and all of the operations--including formatting--were very straightforward to implement. Many of the old business mainframes even have hardware instructions to pack/unpack/operate on BCD.> that most users of programming languages don't want to keep track of that...Why would I want to keep track of that when the language (or more accurately, the data type) will do it for me?I'm sorry man. The whole world isn't C and JavaScript (yet). That we have forced crud-app developers to resort to garbage like this[0]--in places where the performance and storage benefits of floating-point are irrelevant--is a stain on the profession.
 No need to do an integer redirection. i < 0.95 would have sufficed.
 Outside of science and engineering, I think fixed point BCD is otherwise preferable. It's better matched to the expectations we've all had since grade-school.Original example was about steps, not termination. With FP, each tick label has to be post-formatted with a precision specifier to get the desired behavior. With BCD, casting to a string is sufficient (and super-simple to implement).
 Obligatory link to the classic paper on the subject, from 1991: https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.h...
 How many software engineers had to study IEEE floating point in (any amount of) detail? Part of my formal education is in computer (hardware) engineering, and we studied hardware number representations and manipulations quite extensively in my microprocessors courses, so I developed some facility with the topic once upon a time.
 I've met plenty of coders who aren't even aware of the problem, and I spent 13 years maintaining and extending an accounting system written by people who couldn't tell a floating point issue from their ass on a good day.It's not going to happen much until we stop pushing floats as the standard solution for dealing with decimals. I'm pretty sure they are more commonly abused than used at this point, and most cases would be better off with fixpoints, rationals or bignums in increasing order of flexibility and complexity.Lisp and Perl6 get it more or less right. My own baby, Snigl [0], only supports integers and fixpoints so far. I might well add floats eventually, but I intend on hiding them well enough to not encourage anyone who isn't motivated.
 > (...) until we stop pushing floats as the standard solution for dealing with decimals.I'm curious, in that sentence, who is "we"? I cannot imagine a scenario where someone would be so technically involved to be aware of the difference between a floating-point and fixed-point decimal, but still decide to use floating-point in an accounting environment.Are you talking about project managers? Architects?
 If you Google how to manipulate decimal values, or you type `rate = 1.23`, you're almost always going to get floating point. `1.1 + 2.2 == 3.3` yields false in most commonly used programming languages, with the notable exception of SQL.That's what it means that we've established floats as the standard. You have to specifically choose not to use floats, look up your language's entirely non-standardized approach for representing exact decimals, and then maintain constant vigilance against entirely intuitive things like decimal literals.
 Sorry, that wasn't very clear.I knew well enough, but the system was already approaching a million lines of semi-working floating point workarounds by the time. And since it was all my crazy idea, I would have been held responsible for every single rounding error from that point on, had I managed to get the ball rolling at all. Life is too short and too precious.I was thinking language designers mainly; but user preferences are shaped by habits, which means that anything but floats will take more effort to communicate and popularize.edit: Observe how popular this comment is for even floating (take that!) the idea. Plenty of people have serious issues when it comes to floats, that much is obvious. I'm guessing its mostly avoiding change at any cost, so good luck to us all.
 Language designers, I would guess.
 Do you mean binary fixed point, or decimal? Because for money matters I'll take decimal floating point over binary fixed point.
 I strongly prefer decimal fixpoints as well, despite being slower; that's what I'm doing in Snigl.
 Crikey, floating point in accounting software :(Integers with implied decimals is the way to go.
 I used to tell myself that our company was the exception.But in 30 years, I have yet to come across a piece of corporate software that's not crap. Established companies, consultants, startups; all crap. Different kinds of crap, it's not all rounding errors; but none of it would survive in most open source projects.Because quality is only an issue as far as it increases short term profits, and that's not a very successful heuristic for writing good software.
 > Zero Divide By Zero:> This question asks whether in floating point 0.0/0.0 is a non-NaN value, which it is not. NaN generation is desirable here, since it will propagate to the output as a NaN and thus make the user suspicious.Horrible nonsense. What is desirable is an exception so that it's loudly confirmed to the user that something has screwed up. We don't want to lead the user on with subtle hints and suspicions.Even a dollar store calculator from 1985 gets this right by locking up with an E on he display, requiring a clear.
 This is configurable. Sometimes you want to see 99% of your results and know those were weird ones, and you can deal later.
 60% of the people who took the survey develop software to support their main role. So, 60% of the people are not really software engineers but coders by necessity: could this explain the results easily?
 60% of the time, developers understand floating point 100% of the time.

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