Consistency: how to defeat the purpose of IEEE floating point

I don't know much about the design of IEEE floating point, except for the fact that a lot of knowledge and what they call "intellectual effort" went into it. I don't even know the requirements, and I suspect those were pretty detailed and complex (for example, the benefits of having a separate representation for +0 and -0 seem hard to grasp unless you know about the very specific and hairy examples in the complex plane). So I don't trust my own summary of the requirements very much. That said, here's the summary: the basic purpose of IEEE floating point is to give you results of the highest practically possible precision at each step of your computation.

I'm not going to claim this requirement is misguided, because I don't feel like arguing with people two orders of magnitude more competent than myself who have likely faced much tougher numerical problems than I've ever seen. What I will claim is that differences in numerical needs divide programmers into roughly three camps, and the highest-possible-precision approach hurts one of them really badly, and so has to be worked around in ways we'll discuss. The camps are:

  1. The huge camp of people who do businessy accounting. Those should work with integral types to get complete, deterministic, portable control over rounding and all that. Many of the clueless people in this camp represent 1 dollar and 10 cents as the floating point number 1.1. While they are likely a major driving force behind economical growth, I still think they deserve all the trouble they bring upon themselves.
  2. The tiny camp doing high-end scientific computing. Those are the people who can really appreciate the design of IEEE floating point and use its full power. It's great that humanity accidentally satisfied the needs of this small but really cool group, making great floating point hardware available everywhere through blind market forces. It's like having a built-in Stradivari in each home appliance. Yes, perhaps I exaggerate; I get that a lot.
  3. The sizable camp that deals with low-end to mid-range semi-scientific computing. You know, programs that have some geometry or physics or algebra in them. 99.99% of the code snippets in that realm work great with 64b floating point, without the author having invested any thought at all into "numerical analysis". 99% of the code works with 32b floats. When someone stumbles upon a piece of code in the 1% and actually witnesses fatal precision loss, everybody gathers to have a look as if they heard about a beautiful rainbow or a smoke suggesting a forest fire near the horizon.

The majority of people who use and actually should be using floating point are thus in camp 3. Those people don't care about precision anywhere near camp 2, nor do they know how to make the best of IEEE floating point in the very unlikely circumstances where their naive approach will actually fail to work. What they do care about though is consistency. It's important that things compute the same on all platforms. Perhaps more importantly for most, they should compute the same under different build settings, most notably debug and release mode, because otherwise you can't reproduce problems.

Side note: I don't believe in build modes; I usually debug production code in release mode. It's not just floating point that's inconsistent across modes – it's code snippets with behavior undefined by the language, buggy dependence on timing, optimizer bugs, conditional compilation, etc. Many other cans of worms. But the fact is that most people have trouble debugging optimized code, and nobody likes it, so it's nice to have the option to debug in debug mode, and to do that, you need things to reproduce there.

Also, comparing results of different build modes is one way to find worms from those other cans, like undefined behavior and optimizer bugs. Also, many changes you make are optimizations or refaptorings and you can check their sanity by making sure they didn't change the results of the previous version. As we'll see, IEEE FP won't give you even that, regardless of platforms and build modes. The bottom line is that if you're in camp 3, you want consistency, while the "only" things you can expect from IEEE FP is precision and speed. Sure, "only" should be put in quotes because it's a lot to get, it's just a real pity that fulfilling the smaller and more popular wish for consistency is somewhere between hard and impossible.

Some numerical analysts seem annoyed by the camp 3 whiners. To them I say: look, if IEEE FP wasn't the huge success that it is in the precision and speed departments, you wouldn't be hearing from us because we'd be busy struggling with those problems. What we're saying is the exact opposite of "IEEE FP sucks". It's "IEEE FP is so damn precise and fast that I'm happy with ALL of its many answers – the one in optimized x86 build, the one in debug PowerPC build, the one before I added a couple of local variables to that function and the one I got after that change. I just wish I consistently got ONE of these answers, any of them, but the same one." I think it's more flattering than insulting.

I've accumulated quite some experience in defeating the purpose of IEEE floating point and getting consistency at the (tiny, IMO) cost of precision and speed. I want to share this knowledge with humanity, with the hope of getting rewarded in the comments. The reward I'm after is a refutation of my current theory that you can only eliminate 95%-99% of the pain automatically and have to solve the rest manually each time it raises its ugly head.

The pain breakdown

I know three main sources of floating point inconsistency pain:

  1. Algebraic compiler optimizations
  2. "Complex" instructions like multiply-accumulate or sine
  3. x86-specific pain not available on any other platform; not that ~100% of non-embedded devices is a small market share for a pain.

The good news is that most pain comes from item 3 which can be more or less solved automatically. For the purpose of decision making ("should we invest energy into FP consistency or is it futile?"), I'd say that it's not futile and if you can cite actual benefits you'd get from consistency, than it's worth the (continuous) effort.

Disclaimer: I only discuss problems I know and to the extent of my understanding. I have no solid evidence that this understanding is complete. Perhaps the pain breakdown list should have item 4, and perhaps items 1 to 3 have more meat than I think. And while I tried to get the legal stuff right – which behavior conforms to IEEE 754, which conforms to C99, and which conforms to nothing but is still out there – I'm generally a weak tech lawyer and can be wrong. I can only give you the "worked on my 4 families of machines" kind of warranty.

Algebraic compiler optimizations

Compilers, or more specifically buggy optimization passes, assume that floating point numbers can be treated as a field – you know, associativity, distributivity, the works. This means that a+b+c can be computed in either the order implied by (a+b)+c or the one implied by a+(b+c). Adding actual parentheses in source code doesn't help you one bit. The compiler assumes associativity and may implement the computation in the order implied by regrouping your parentheses. Since each floating point operation loses precision on some of the possible inputs, the code generated by different optimizers or under different optimization settings may produce different results.

This could be extremely intimidating because you can't trust any FP expression with more than 2 inputs. However, I think that programming languages in general don't allow optimizers to make these assumptions, and in particular, the C standard doesn't (C99 § #13, didn't read it in the document but saw it cited in two sources). So this sort of optimization is actually a bug that will likely be fixed if reported, and at any given time, the number of these bugs in a particular compiler should be small.

I only recall one recurring algebraic optimization problem. Specifically, a*(b+1) tended to be compiled to a*b+a in release mode. The reason is that floating-point literal values like 1 are expensive; 1 becomes a hairy hexadecimal value that has to be loaded from a constant address at run time. So the optimizer was probably happy to optimize a literal away. I was always able to solve this problem by changing the spelling in the source code to a*b+a – the optimizer simply had less work to do, while the debug build saw no reason to make me miserable by undoing my regrouping back into a*(b+1).

This implies a general way of solving this sort of problem: find what the optimizer does by looking at the generated assembly, and do it yourself in the source code. This almost certainly guarantees that debug and release will work the same. With different compilers and platforms, the guarantee is less certain. The second optimizer may think that the optimization you copied from the first optimizer into your source code is brain-dead, and undo it and do a different optimization. However, that means you target two radically different optimizers, both of which are buggy and can't be fixed in the near future; how unlucky can you get?

The bottom line is that you rarely have to deal with this problem, and when it can't be solved with a bug report, you can look at the assembly and do the optimization in the source code yourself. If that fails because you have to use two very different and buggy compilers, use the shotgun described in the next item.

"Complex" instructions

Your target hardware can have instructions computing "non-trivial" expressions beyond a*b or a+b, such as a+=b*c or sin(x). The precision of the intermediate result b*c in a+=b*c may be higher than the size of an FP register would allow, had that result been actually stored in a register. IEEE and the C standard think it's great, because the single instruction generated from a+=b*c is both faster and more precise than the 2 instructions implementing it as d=b*c, a=a+d. Camp 3 people like myself don't think it's so great, because it happens in some build modes but not others, and across platforms the availability of these instruction varies, as does their precision.

AFAIK the "contraction" of a+=b*c is permitted by both the IEEE FP standard (which defines FP + and *) and the C standard (which defines FP types that can map to standards other than IEEE). On the other hand, sin(x), which also gets implemented in hardware these days, isn't addressed by either standard – to the same effect of making the optimization perfectly legitimate. So you can't solve this by reporting a bug the way you could with algebraic optimizations. The other way in which this is tougher is that tweaking the code according to the optimizer's wishes doesn't help much. AFAIK what can help is one of these two things:

  1. Ask the compiler to not generate these instructions. Sometimes there's an exact compiler option for that, like gcc's platform-specific -mno-fused-madd flag, or there's (a defined and actually implemented) pragma directive such as #pragma STDC FP_CONTRACT. Sometimes you don't have any of that, but you can lie to the compiler that you're using an older (compatible) revision of the processor architecture without the "complex" instructions. The latter is an all-or-nothing thing enabling/disabling lots of stuff, so it can degrade performance in many different ways; you have to check the cost.
  2. If compiler flags can't help, there's the shotgun approach I promised to discuss above, also useful for hypothetical cases of hard-to-work-around algebraic optimizations. Instead of helping the optimizer, we get in its way and make optimization impossible using separate compilation. For example, we can convert a+=b*c to a+=multiply_dammit(b,c); multiply_dammit is defined in a separate file. This makes it impossible for most optimizers to see the expression a+=b*c, and forces them to implement multiplication and addition separately. Modern compilers support link-time inlining and then they do optimize the result as a whole. But you can disable that option, and as a side effect speed up linkage a great deal; if that seriously hurts performance, your program is insane and you're a team of scary ravioli coders.

The trouble with the shotgun approach, aside from its ugliness, is that you can't afford to shoot at the performance-critical parts of your code that way. Let us hope that you'll never really have to choose between FP consistency and performance, as I've never had to date.


Intel is the birthplace of IEEE floating point, and the manufacturer of the most camp-3-painful and otherwise convoluted FP hardware. The pain comes, somewhat understandably, from a unique commitment to the IEEE FP philosophy – intermediate results should be as precise as possible; more on that in a moment. The "convoluted" part is consistent with the general insanity of the x86 instruction set. Specifically, the "old" (a.k.a "x87") floating point unit uses a stack architecture for addressing FP operands, which is pretty much the exact opposite of the compiler writer's dream target, but so is the rest of x86. The "new" floating point instructions in SSE don't have these problems, at the cost of creating the aesthetic/psychiatric problem of actually having two FP ISAs in the same processor.

Now, in our context we don't care about the FP stack thingie and all that, the only thing that matters is the consistency of precision. The "old" FP unit handles precision thusly. Precision of stuff written to memory is according to the number of bits of the variable, 'cause what else can it be. Precision of intermediate results in the "registers" (or the "FP stack" or whatever you call it) is defined according to the FPU control & status register, globally for all intermediate results in your program.

By default, it's 80 bits. This means that when you compute a*b+c*d and a,b,c,d are 32b floats, a*b and c*d are computed in 80b precision, and then their 80b sum is converted to a 32b result in memory (if a*b+c*d is indeed written to memory and isn't itself an "intermediate" result). Indeed, what's "intermediate" in the sense of not being written to memory and what isn't? That depends on:

  1. Debug/release build. If we have "float e=a*b+c*d", and e is only used once right in the next line, the optimizer probably won't see a point in writing it to memory. However, in a debug build there's a good reason to allocate it in memory, because if you single-step your program and you're already past the line that used e, you still might want to look at the value of e, so it's good that the compiler kept a copy of it for the debugger to find.
  2. The code "near" e=a*b+c*d according to the compiler's notion of proximity also affects its decisions. There are only so many registers, and sometimes you run out of them and have to store things in memory. This means that if you add or remove code near the line or in inline functions called near the line, the allocation of intermediate results may change.

Compilers could have an option asking them to hide this mess and give us consistent results. The problems with this are that (1) if you care about cross-platform/compiler consistency, then the availability of cross-mode consistency options in one compiler doesn't help with the other compiler and (2) for some reason, compilers apparently don't offer this option in a very good way. For example, MS C++ used to have a /fltconsistency switch but seems to have abandoned it in favor of an insane special-casing of the syntax float(a*b)+float(c*d) – that spelling forces consistency (although the C++ standard doesn't assign it a special meaning not included in the plain and sane a*b+c*d).

I'd guess they changed it because of the speed penalty it implies rather than the precision penalty as they say. I haven't heard about someone caring both about consistency and that level of precision, but I did hear that gcc's consistency-forcing -ffloat-store flag caused notable slowdowns. And the reason it did is implied by its name – AFAIK the only way to implement x86 FP consistency at compile time is to generate code storing FP values to memory to get rid of the extra precision bits. And -ffloat-store only affects named variables, not unnamed intermediate results (annoying, isn't it?), so /fltconsistency, assuming it actually gave you consistency of all results, should have been much slower. Anyway, the bottom line seems to be that you can't get much help from compilers here; deal with it yourself. Even Java gave up on its initial intent of getting consistent results on the x87 FPU and retreated to a cowardly strictfp scheme.

And the thing is, you never have to deal with it outside of x86 – all floating point units I've come across, including the ones specified by Intel's SSE and SSE2, simply compute 32b results from 32b inputs. People who decided to do it that way and rob us of quite some bits of precision have my deepest gratitude, because there's absolutely no good way to work around the generosity of the original x86 FPU designers and get consistent results. Here's what you can do:

  1. Leave the FP CSR configured to 80b precision. 32b and 64b intermediate results aren't really 32b and 64b. The fact that it's the default means that if you care about FP result consistency, intensive hair pulling during your first debugging sessions is an almost inevitable rite of passage.
  2. Set the FP CSR to 64b precision. If you only use 64b variables, debug==release and you're all set. If you have 32b floats though, then intermediate 32b results aren't really 32b. And usually you do have 32b floats.
  3. Set the FP CSR to 32b precision. debug==release, but you're far from "all set" because now your 64b results, intermediate or otherwise, are really 32b. Not only is this a stupid waste of bits, it is not unlikely to fail, too, because 32b isn't sufficient even for some of the problems encountered by camp 3. And of course it's not compatible with other platforms.
  4. Set the FP CSR to 64b precision during most of the program run, and temporarily set it to 32b in the parts of the program using 32b floats. I wouldn't go down that path; option 4 isn't really an option. I doubt that you use both 32b and 64b variables in a very thoughtful way and manage to have a clear boundary between them. If you depend on the ability of everyone to correctly maintain the CSR, then it sucks sucks sucks.

Side note: I sure as hell don't believe in "very special" "testing" build/running modes. For example, you could say that you have a special mode where you use option (3) and get 32b results, and use that mode to test debug==release or something. I think it's completely self-defeating, because the point of consistency is being able to reproduce a phenomenon X that happens in a mode which is actually important, in another mode where reproducing X is actually useful. Therefore, who needs consistency across inherently useless modes? We'd be defeating the purpose of defeating the purpose of IEEE floating point.

Therefore, if you don't have SSE, the only option is (2) – set the FP CSR to 64b and try to avoid 32b floats. On Linux, you can do it with:

#include <fpu_control.h>
fpu_control_t cw;
cw = (cw & ~_FPU_EXTENDED) | _FPU_DOUBLE;

Do it first thing in main(). If you use C++, you should do it first thing before main(), because people can use FP in constructors of global variables. This can be achieved by figuring out the compiler-specific translation unit initialization order, compiling your own C/C++ start-up library, overriding the entry point of a compiled start-up library using stuff like LD_PRELOAD, overwriting it in a statically linked program right there in the binary image, having a coding convention forcing to call FloatingPointSingleton::instance() before using FP, or shooting the people who like to do things before main(). It's a trade-off.

The situation is really even worse because the FPU CSR setting only affects mantissa precision but not the exponent range, so you never work with "real" 64b or 32b floats there. This matters in cases of huge numbers (overflow) and tiny numbers (double rounding of subnormals). But it's bad enough already, and camp 3 people don't really care about the extra horror; if you want those Halloween stories, you can find them here. The good news are that today, you are quite likely to have SSE2 and very likely to have SSE on your machine. So you can automatically sanitize all the mess as follows:

  1. If you have SSE2, use it and live happily ever after. SSE2 supports both 32b and 64b operations and the intermediate results are of the size of the operands. BTW, mixed expressions like a+b where a is float and b is double don't create consistency problems on any platform because the C standard specifies the rules for promotion precisely and portably (a will be promoted to double). The gcc way of using SSE2 for FP is -mfpmath=sse -msse2.
  2. If you only have SSE, use it for 32b floats which it does support (gcc: -mfpmath=sse -msse). 64b floats will go to the old FP unit, so you'll have to configure it to 64b intermediate results. Everything will work, the only annoying things being (1) the retained necessity to shoot the people having fun before main and (2) the slight differences in the semantics of control flags in the old FP and the SSE FP CSR, so if you configure your own policy, floats and doubles will not behave exactly the same. Neither problem is a very big deal.

Interestingly, SSE with its support for SIMD FP commands actually can make things worse in the standard-violating-algebraic-optimizations department. Specifically, Intel's compiler reportedly has (had?) an optimization which unrolls FP accumulation loops and reorders additions in order to utilize SIMD FP commands (gcc 4 does that, too, but only if you explicitly ask for trouble with -funsafe-math-optimizations or similar). But I wouldn't conclude anything from it, except that automatic vectorization, which is known to work only on the simplest of code snippets, actually doesn't work even on them.

Summary: use SSE2 or SSE, and if you can't, configure the FP CSR to use 64b intermediates and avoid 32b floats. Even the latter solution works passably in practice, as long as everybody is aware of it.

I think I covered everything I know except for things like long double, FP exceptions, etc. – and if you need that, you're not in camp 3; go away and hang out with your ivory tower numerical analyst friends. If you know a way to automate away more pain, I'll be grateful for every FP inconsistency debugging afternoon your advice will save me.

Happy Halloween!


#1 ljbirdwatcher on 10.31.08 at 2:31 pm

Beautiful and thought provoking, as usual. But… I assume consistency means getting identical bit patterns in FP operation results across different compilers and platforms. Now why is that goal desirable? Why care about a binary representation of a double being one way or another?

#2 Yossi Kreinin on 10.31.08 at 2:48 pm

Thanks…. "Bit pattern" is a loaded term because it may refer to the represented number or it may look beyond that at the actual bit string stored in memory (in which case you start talking about endianness, etc.). I assume we're talking about the represented number, so for example, the question is what's wrong with having computed x=1.0001 on one platform and 0.999 on another platform (build mode, compiler).

The problem is ultimately that x will look differently when printed, will yield a different value if converted to an int, and if it's compared with 1.0 in a conditional statement, a different code path will execute. This complicates testing because you can't compare outputs of different build settings or program versions. It also complicates debugging because you can't reproduce problems found in a hard-to-debug environment in an easy-to-debug environment.

Now, there's another problem – that of theoretical program correctness. That is, if you assume that (int)x is always between 1 and 10 and once in a lifetime x actually computes as 0.999 and you have an out-of-bounds access, than you're really screwed way beyond ease of testing and debugging.

However, this isn't the problem of camp 3. It's the problem of camp 2 who compute precise things using precise techniques. A camp 3 person should add a max(min((int)x,10),1) and be done with it. That's because a camp 3 person can't/doesn't want to/shouldn't be busy thinking about stuff like the behavior for subnormal numbers. The inherent precision of his data and algorithms is so lame that this effort is completely misplaced; it won't yield good results, and it costs much more than braindead sanitizing like the min/max thing.

The bottom line is that I want things to be the same because identical output eases testing and debugging, and not because I care about the exact bit pattern that will be computed identically.

#3 Anonymous on 10.31.08 at 3:15 pm

hur hur .. refaptoring!

#4 Yossi Kreinin on 10.31.08 at 3:28 pm

According to Google, the term was coined by Steve Yegge, and still has only one Google hit. This post is supposed to raise that number by 100%.

#5 anonymous on 10.31.08 at 3:34 pm

Google up math professor William Kahan at UC Berkeley. He was the guy who invented IEEE 754 arithmetic. You might ask him?

#6 ljbirdwatcher on 10.31.08 at 4:23 pm

I'd argue if you assume (int)x is always between 1 and 10 and once in a lifetime it actually computes as 0.999, you are screwed no matter if you have crosscompiler consistency in x, or not. Granted, it would not be a bad thing to have it, but even if you do, you are still screwed until you do the max(min) thing, or assume x is between 1-e and 10+e instead.

#7 Doug Merritt on 10.31.08 at 6:36 pm

I understand your desire for consistency, but there are some problems with your claims and assumptions.

Anonymous above recommend you check out Kahan, the father of IEEE floating point, and for anyone who has enough interest to read a blog entry like this, let alone write one, Kahan is in fact required reading — potentially up to reading everything on his site:

I read/re-read his publications every few years, and happened to have bookmarked some that I liked last month:

"Why Can I Debug some Numerical Programs that You Can't?"

"A Logarithm Too Clever by Half"

Also the must-read edited reprint of the paper What Every Computer Scientist Should Know About Floating-Point Arithmetic, by David Goldberg

Back to what I consider important misstatements, certainly IEEE tries to avoid unnecessary imprecision, and multiple things in its design show that, but that is different than what I consider to be an incorrect statement, that "to give you results of the highest practically possible precision at each step of your computation"

Nay nay. Your desire for consistency is similar to the goals of an expert numerical analyst like Kahan, but to a first approximation, *accuracy* is far more important than precision, and the two are not at all equivalent, yet programmers in general tend not to think about the difference.

Finally, most of the (interesting and valuable) concerns, approaches, and advice you present doesn't have anything to do with IEEE floating point. Almost all of your points have to do with with idiosyncracies of languages, compilers, and optimizers.

…possibly aside from your ending comments about fpu_control, which I merely skimmed and can't comment on due to the press of time.

Despite those critiques, thanks for your article. Any discussion that makes programmers more aware of any of the zillions of issues relating to floating point is a good public service.

#8 Doug Merritt on 10.31.08 at 6:50 pm

P.S. I forgot to mention…when I said "Almost all of your points have to do with with idiosyncracies of languages, compilers, and optimizers.", I meant to say that you would have similar issues with non-IEEE floating point, and pre-IEEE standard, there were many.

And when I mentioned the concern of experts like Kahan to have consistency, I meant to mention the (unfortunately not all that well-know fact) work by Kahan and his students and successors to create a C libm (best known as fdlibm — Freely Distributable Libm), which was "exactly" accurate, meaning that each function produced a result that differed from an infinite-precision result by no more than 1/2 of the final bit of precision.

In other words, Kahan with others under him produced a libm (first distributed with BSD Unix, IIRC) that was both as accurate as possible and had maximum precision possible, simultaneously.

What a tour de force!

Furthermore, they released that library for multiple floating point standards, not just IEEE, but also VAX and PDP 11 etc (IIRC).

That should make anyone stop and say "hmm!"

#9 Anonymous.1415926... on 10.31.08 at 7:40 pm

You lost me as soon as I found out your parentheses weren't done in Ada.

#10 Dan on 10.31.08 at 8:00 pm

The solution that has always worked for me when running tests on numerical code is to compare the results as numbers (rather than strings). If the difference between the expected result and the actual result is less than, say, 0.00001% then I consider it a pass.
This sounds a lot easier than messing with the code or worrying about rounding.

If floating-point error has any significant effect on the result, it would tend to indicate a poorly-conditioned numerical algorithm that should not be used anyway.

#11 Mattthew on 10.31.08 at 9:50 pm

You mispell "worms" as "warms" in two places.

#12 M on 10.31.08 at 10:27 pm

Actually, group two (high-end scientific calculations) also require consistency. Calculations need to be able to be repeated to obtain the same results. This inconsistency that you talk about is not an issue with IEEE floating points, it is an issue with the compilers you are using.

#13 Yossi Kreinin on 10.31.08 at 11:02 pm

To people saying that my problem is with software tools and not the FP standard: for those on x86, the biggest problem before SSE used to be the x87 FPU, which is almost impossible to get consistent results out of, and AFAIK IEEE 754 allows it to be built that way because it doesn't mind extra precision. Also, things like contraction are AFAIK explicitly supported by IEEE, without requiring a mode where contraction is forced to give results consistent to non-optimized mul-then-add implementation. Also, I think the 2008 revision of IEEE 754 has a reproducibility clause which discusses the way for hw/sw systems to provide the option of writing code generating reproducible results. So they do acknowledge this to be in their territory. I didn't talk about it because I tend to care more about de-facto standards than upcoming standards, but I thought it's worth mentioning now that you blame me for misattributing blame.

To people saying that algorithms shouldn't "depend" on small precision errors (where "depend" means "produce radically different results by taking a different code path or indexing into a different table entry", etc.) – for many, many problems it doesn't work that way and there's no reason to think of it like that. Think about problems without a "right" answer – for example, face recognition. Where does the region with the face start – at pixel 397 or 398? When a person in a crowd is obstructed by another person – should we lose the "target" at frame 55 or 56? And what if the vision algorithm is imperfect, as they all are, and you only find 97% of the things you should? "Noise" and algorithmic imperfection can make different examples appear in the false-negative 3%, so it's no longer +-1 in the pixel or frame number. Sure, you can and should implement automated tests measuring the quality of your code by defining tolerance criteria capable of saying "you're about that good" without relying on exact pixel numbers. However, many times what you care about isn't how good you are, but why you crash on input so and so in release build. Now, if debug build is only statistically similar to the release build, the crash will probably not reproduce on that input. Net result: FP inconsistency makes things hard to debug.

Now, the whole theme of "you're walking on a borderline and a slight numerical push can make you cross it" – it happens all the time, with examples much simpler than face recognition. It's wishful thinking to assume that "user-visible" program results shouldn't be affected by numerical noise or else the algorithm is "buggy".

Regarding W. Kahan, for whom I, um, Googled a lot: the fact that you can develop an insanely precise library on top of IEEE, and that you can fight and win the battle for consistent results on top of IEEE doesn't contradict what I say. What I say is that you can't do it without an extremely detailed and accurate numerical analysis. If you compute a trig function and you manage to reach the maximal possible precision your datatype supports, sure, you're precise and consistent. This the kind of thing camp 2 people can do and happen to be doing, although I'm not sure you'll always end up like that – if you could always reach the maximal possible datatype precision with IEEE, for every algorithm, it would be real numbers, and they aren't. But the thing is, camp 3 people deal with data and algorithms which aren't worth the effort, because you'll get lousy precision anyway. Those are the people who want consistency without the, um, enormous cost of detailed numerical analysis.

#14 Yossi Kreinin on 10.31.08 at 11:33 pm

To Doug Merritt: I only noticed your first comment now since WP cleverly put it into the moderation queue because of the links.

I'd be grateful if you explained me your argument regarding precision and accuracy in more detail, because I think the general issue of their difference is orthogonal to what I'm saying. In terms of this dichotomy, "accuracy" is related to algorithm design (an approximation of sine should actually approximate sine on all inputs – give a result close to the real value), and "precision" is related to algorithm implementation (the hw/sw combo running a sine approximation should compute the same thing each time you try it whether it's computing an accurate approximation or a crummy one; although it's not that trivial because one thing crummy algorithms do is overflow and the like, so the requirement of their designers to get precisely the same wrong answer across many systems may be "illegitimate" – that's why I think that the inability to control the exponent range on the x87 isn't a big deal).

Anyway, in terms of accuracy/precision, when I ask for "consistency", it's synonymous for "precision", actually the maximal possible precision – that is, "gimme the same answer everywhere". I do that based on the assumption that the accuracy of the basic operations of IEEE FP is good enough to be able to design decently accurate algorithms with very little effort in most cases (and that is true whether I use 64b regs or 80b regs). But in general, the question of analyzing the accuracy of algorithms implemented on top of IEEE 754 is out of my scope – it's the business of people designing them, and it was the business of the people designing IEEE 754 who cared about the ability to design specific algorithms accurately on top of the basic operations.

Now, the fact that I use "consistency of results" where you'd use "precision", and that I use "precision of operations" where you'd say "accuracy" is consistent with your claim that most programmers don't think in terms of accuracy/precision; I'm writing for these people, after all. I think my claims are still valid and readable though.

#15 undebugger on 11.01.08 at 1:31 am

refaptoring!!! it's a brillliant meme, didn't hear it before

(and the article is fine as always)

#16 Yossi Kreinin on 11.01.08 at 2:38 am

thanks, but as noted in a comment above – it's Steve Y's (there 1 Google hit for it right now.)

#17 Dan on 11.01.08 at 7:37 am

Your comment #13 finally made the point of this clear in my mind. The 'off-by-one-pixel' example in image processing is a great example of why one would need consistency.

Although, I would still argue that the 'relying-on-rounding-is-a-bug' argument still almost always applies to continuous problem spaces.

Thanks for the great article.

#18 Gabe on 11.01.08 at 11:18 am

It's definitely a pain point, but what's the solution? How are you going to standardize hardware and compiler implementations? If you managed to do that lots of people would get left out in the cold. Wishing for consistency in computers is a bit more realistic, but still in the same camp as biologists wishing for genetic equivalence in all members of a species. Sure it would solve a boat load of problems but it ain't gonna happen, and even if it did, the perfection would only exist until the first bug was found.

#19 Marcel Popescu on 11.01.08 at 11:25 am

Precision = how many digits you have in a number.
Accuracy = how close you are to the "real" value.

3.55555 is a more precise approximation of Pi than 3.1, but a less accurate one.

#20 Yossi Kreinin on 11.01.08 at 11:35 am

@Gabe: as I said in the article, AFAIK the specific problem of FP consistency is "almost solvable" on the hardware available today, so the state of things isn't bad, it's just that you don't get consistency by default.

@Marcel: the term "precision" is used to name many different things. "Precision" means "#digits" in the context of cout<<setprecision(20), it means "accuracy" as you defined it in a lot of informal discussions (including this article except for the comments mentioning the word "accuracy"), and in the context of "the difference between precision and accuracy", it means the distance between measurements/computations obtained from different trials:

So I don't think any of this is about the number of digits, which is a base-dependent and hardly interesting property.

#21 Marcel Popescu on 11.01.08 at 11:46 am

I was clarifying Doug Merritt's comment, which you seemed not to understand ("I’d be grateful if you explained me your argument regarding precision and accuracy in more detail…"). It seems obvious to me that he's using the same definition I am.

#22 Brooks Moses on 12.04.08 at 12:29 pm

A comment on this quote about compiler optimizations that alter precision: "So this sort of optimization is actually a bug that will likely be fixed if reported, and at any given time, the number of these bugs in a particular compiler should be small."

In many cases, this is not in fact a bug, but a case of using an optimization flag that tells the compiler to introduce these optimizations in the interest of greater speed of computation. In GCC, the flag is -ffast-math. I believe that in some commercial compilers, this sort of flag is on by default at higher levels of optimization.

Thus, if you rely on these optimizations being absent, you should read your compiler documentation carefully!

#23 Yossi Kreinin on 12.04.08 at 12:39 pm

I shouldn't; standard-compliant behavior ought to be the default.

Thanks for the info on -ffast-math though; I remember seeing it in some make output, but I haven't looked it up. Now I'll be sure to tear it out of every compiler option list I'll ever stumble upon.

Regarding the a*(b+1) => a*b+a transformation – I'm positive it was a bug; nobody would have stuffed -ffast-math into that Makefile.

#24 Yossi Kreinin on 12.04.08 at 12:41 pm

…Liked your (?) page at LtU.

#25 Websites tagged "embedded" on Postsaver on 12.26.08 at 8:17 am

[...] my channel flippin’ mind saved by Yaksah2008-12-14 – inspiration saved by frogfish2008-12-13 – Consistency: how to defeat the purpose of IEEE floating point saved by Jelloman182008-12-13 – as soon as the incarnational analogy is made, one’s indulgences [...]

#26 Simon on 09.24.09 at 5:24 am

Consistency is important. Code that is compiled on one system may not produce a consistent result on another system. I am talking about code that involves a large number of accumulated computations. If the program is run once, it produces a result, which we average over the length of the run. The second time we run the same code a new set of results emerge and the average can differ by 10%. The differences start out at the 15th significant figure and accumulate past the 8th and right down to the first figure. That means that the code behaves differently each time it is run, dependent on things like memory and stack. If the program is run on the system it is compiled on, it should always produce consistent results (but it will not). When it is run on the client's computer it definitely will not produce the same numbers – and that is serious if it is a weather prediction program.

#27 Glenn Fiedler on 02.25.10 at 11:09 pm

Brilliant article. And for those wondering why consistent floating point calculations are so important – many, MANY videogames rely on complete determinism and sending only inputs per-simulation from to keep players in sync, eg. most RTS, even some physics simulation games like "Little Big Planet", basically any situation in which the state of the system is too large to practically send over the network, determinism is key!

#28 Glenn Fiedler on 02.25.10 at 11:12 pm

Brilliant article. And for those wondering why consistent floating point calculations are so important – many, MANY videogames rely on complete determinism and sending only inputs per-simulation frame to keep each player's game simulation in sync. For example most RTS games use this technique (eg. "Starcraft", "Age of Empires") also some physics simulation games like “Little Big Planet” do too, and many sports games you play online as well . Basically, any situation where the state of the system is too large to practically send over the network, you'll find that determinism and reproducibility are key!

#29 Yossi Kreinin on 02.26.10 at 4:50 am

@Glenn: thanks; I hope my practical advice is, um, practical. These things are a bit hard to fully research (at least for me) – hard to know how many warts one left uncovered.

#30 Christer Ericson on 05.08.10 at 1:26 am

"Adding actual parentheses in source code doesn’t help you one bit. The compiler assumes associativity and may implement the computation in the order implied by regrouping your parentheses."

Utter and totally wrong! Read up on the C/C++ standards. A standard-complying compiler (and in this particular issue, they all are, unless you've opted out using some 'fast-math' compiler option) will not change the meaning of floating-point code.

E.g. see section 14 of the C (draft) standard. ("Rearrangement for floating-point expressions is often restricted because of limitations in
precision as well as range. The implementation cannot generally apply the mathematical associative rules
for addition or multiplication, nor the distributive rule, because of roundoff error, even in the absence of
overflow and underflow.")

See Annex F for even more detail. Same applies for C++.

#31 Yossi Kreinin on 05.08.10 at 11:13 pm

Um, I quote § in the paragraph right after the one you mention.

#32 Christer Ericson on 05.20.10 at 7:59 pm

Huh? I have no idea what you are trying to say with that reply.

Your original statement "The compiler assumes associativity" is unequivocally wrong. Floating-point arithmetic is NOT associative! Therefore no compiler "assumes associativity".

The bit where you say "Adding actual parentheses in source code doesn’t help you one bit." is also total nonsense. If I have explicitly added parenthesis to a floating-point expression to control the evaluation order, the compiler cannot remove these parentheses unless removing them leaves the expression exactly equivalent *under evaluation as a floating-point expression*. E.g. a compiler cannot rewrite "((x – y) + y)" into "x". So adding parenthesis DO matter!

You said "Compilers, or more specifically buggy optimization passes, assume that floating point numbers can be treated as a field – you know, associativity, distributivity, the works. This means that a+b+c can be computed in either the order implied by (a+b)+c or the one implied by a+(b+c)."

This too is complete nonsense. Floating-point numbers do not have associativity, and therefore cannot be treated as a field. The two floating-point expressions "(a+b)+c" and "a+(b+c)" are NOT equivalent!

I think you need to brush up on floating-point arithmetic a bit. See the Accuracy Problems section of the wikipedia page

for an example of how floating-point numbers are not associative.

#33 Yossi Kreinin on 05.22.10 at 11:10 am

I said "buggy optimization passes". If you've never run into a compiler bug of that sort, that's fine, and the fact that you know that such behavior is illegitimate and therefore is indicative of a compiler bug is also fine, but both add little to the discussion. The fact that I knew everything you just said is completely obvious from TFA to everyone who bothered to R it.

#34 JC on 11.05.12 at 4:35 pm

I know this article is 4 years old, but it takes me back 25 years when I worked on Fortran compilers for early 386/387s. We would dread the Gold* support call from a research scientist who would report his program worked perfectly in debug mode, but crashed in release mode. At Gold* support we knew we were about to do a full numerical analysis of the code to find where an 80 bit number had gone down one particular "if" branch of doom when its 64 bit debug alter-ego had gone down the "else" path of salvation.

25 years later I am the research scientist. I have the advantage of being forwarned and forearmed, so write code and tests that are tollarent of such eccentic behaviour. This is the general solution, as I often write complex/fast/precise algortithms which can I compare to easy/slow/naive versions. As the number and sequence of FP operations are quite different the answers will never be exactly the same even if the IEEE fp and the compiler were friendlier, and I don't care provided they match within my calculated tollerence.

#35 Yossi Kreinin on 11.05.12 at 10:05 pm

@JC: well, someone like you who is in the tiny minority of people who actually know what they're doing when they reach for floating point probably doesn't care about things like consistent behavior between debug and release…

#36 Amateur Gamedev on 04.27.13 at 12:31 pm

It is highly worth mentioning that this is an incredibly annoying problem for game developers. Eg. multiplayer real-time strategy games often depend on their simulations working precisely the same on all machines. If they don't, it is a synchronization error and the ongoing game is cancelled.

This means that for a multiplatform game, one is practically forced to use software floating or fixed point for everything that can affect the outcome of the game, diminishing the available performance an order of magnitude lower than a predictable hardware implementation would allow.

#37 Yossi Kreinin on 04.27.13 at 1:19 pm

I wrote the article a long time ago, but I think that if you're on x86, then you can ask for double precision instead of extended and then ask for SSE to be used for single precision and basically you get what Sun calls "single/double system" instead of an "extended-based system" in its manuals and it's deterministic enough; and you ask on all platforms that the compilers doesn't use fused MACs and such, and you don't use library functions like log or exp, and you're basically all set.

I'm not sure this covers it all but it sounds as if this is better than using emulation; maybe I miss something big making the use of hardware floating point hopeless.

Incidentally, I'm planning on writing a follow-up on this one; I've studied the issues in more detail, and I think I'm in a reasonably good position to argue with some of the ideas behind IEEE. I think this portability business is just botched up, for one thing (ironically this is pointed out in an appendix to the paper one of the commenters above linked to, the piece by Goldberg), and then nans, infs, denormals and rounding modes all have their very notable drawbacks which I think are worth discussing.

#38 Christian Plesner Hansen on 11.13.13 at 2:54 am

Great article. I've been looking into IEEE 754, trying to figure out how to support it in sane way in a high-level language. I've had a lot of the same thoughts along the way but this goes into much more detail. Looking forward to the follow-up.

#39 stephenmm on 08.12.14 at 7:01 pm

It seems to me that if you are doing things that are critical like this it might be better just to write it in inline assembly, no?

#40 Yossi Kreinin on 08.13.14 at 9:51 am

I dunno, I never prefer inline assembly to anything if there's an alternative. You can cross-compile it and you can't add printfs to it, for starters, which are big turnoffs for me…

#41 Tom Poole on 04.14.15 at 6:54 am

Precision, accuracy and consistency are ALL different things.

The standard meaning of precision isn't about consistency, but simply bit-depth, or number of decimal places/significant figures. I agree with your description of accuracy — actually approximating a sane representation of the real value, but I'd say that of the two, accuracy is actually closer in meaning to 'consistency' than 'precision' is.

PI = 3.141592222222234444444444101010101 is highly precise.
So is PI = 8.000000000000000000000000008.

But they are inconsistent and neither is accurate, to the extent that the second one looks like it was measured experimentally by someone living in a centrifuge with a tangential velocity approaching the speed of light… or by a drunk monkey.

So the term 'over-precise' is commonly used to refer to a number whose precision is not _justified_by_its_accuracy_. Why quote 7 more digits beyond the first inaccurate one?

A good example comes from our good friend Commander Data. He's regularly over-precise, most commonly by quoting durations to within far less than the time it takes to say the sentence.

I actually prefer some inconsistency in order to test the rigour of my algorithms. I do not suffer from errors showing up in a hard-to-debug environment, while not appearing in an easy-to-debug environment. If my algorithm isn't tolerant of error, I find that symptoms of some kind will show up on every platform. In other words I tend to agree with an earlier comment, "If floating-point error has any significant effect on the result, it would tend to indicate a poorly-conditioned numerical algorithm that should not be used anyway."

HOWEVER… I don't mean to answer a "How can we?" with a "What's the point?" I hate it when people do!

I can imagine situations where consistency would be highly desirable, for the debugging reasons you suggest. It IS an interesting problem, to try to achieve cross-platform consistency for every result. Applications in both chaos theory and rendered-on-the-fly machinima spring to mind. And especially chaotic machinima :)

What I AM saying is that I think it's desirable more rarely than you suggest, and that if you're spending many afternoons trying to achieve floating point consistency, then perhaps your algorithms aren't tolerant and rigorous enough, and you need to approach a lot of things with a fundamentally different mindset.

I also think there's a lot more cross-over between camps 2 and 3 than you've implied. The physics of 3D computer games will remain extremely flaky compared to Box2D, for a while yet, because of the deep computational problems that arise. But we will get there, thanks to the large, keen, oft-bespectacled crossover between camps 2 and 3!

Is developing a limited, performance optimised CAS library ivory tower stuff? With rational and surd data types, perhaps? Algebraic manipulation of multi-dimensional polynomial expressions? Of implicit equations based on said expressions? To many people these things might sound abstract and high-end, but to me it sounds like the underpinnings of a damn fine NURBS renderer, or a game physics engine to be reckoned with.

(I'm not talking Mathematica or TI-89 levels of versatility here. I mean the kind of limited, optimised CAS that helps you render one curved surface much more quickly than a million flat ones. Pixar's technique of tesselating curves into micro-polygons, as required for a particular camera position, has already gone way beyond the point where a limited CAS could be faster.)

I do like your article, and it's impressive that it was the Box2D FAQ that led me to it. It's thoughtful and thought provoking, reveals issues that hadn't occurred to me, and for me at least, it scares just the right amount with heavy programming jargon! Keeps me on my toes!

Generally, everyone here seems to know more about computer architecture than I do. I am a mathematician, naturally concerned with cosmic truths, and not naturally that interested in the parochial specifics of 21st century Earth computers. I try hard to correct this imbalance, so that I can earn money, and so I can use my mathematical and algorithmic cleverness to program graphics and physics. In general learning ability, I feel as overshadowed here as I would at a conference of chemical engineers. I wish I were better at what Ernest Rutherford derided as 'stamp collecting'; I'm a mental gymnast, but an ignoramus.

Thankyou for helping me to change that, here and there.


#42 Tom Poole on 04.14.15 at 8:04 am

JC is an interesting case of a thoughtful, intelligent commenter, who has somehow become a research scientist without being able to spell tolerance! What mad mixtures we are.

Hypocrisy? Nay sir! If my owne spelling of summe wordes seemeth strange, it be becauseth I amme fromme the olde worlde :)

Also, good comments there from Amateur Gamedev and Glenn Fiedler, (and probably others) about multiplayer gaming. I suppose that's a more normal thing to mention than my thing about rendered-on-the-fly machinima! I meant the term to include multiplayer gaming, but I'm not sure that's obvious.

Synchronisation problems are generally quite easy to do badly, and rarely done well. I can imagine a lot of developers tripping here, and even so-called 'AAA' games having problems with it. I still don't quite believe Amateur Gamedev, however, regarding "…one is practically forced to use software floating or fixed point for everything that can affect the outcome of the game, diminishing the available performance an order of magnitude lower…" Of course I like to think that I'M clever enough to get around it — and practically at that!

Sorry to be vague here with 'variations on the theme of' but it's a big subject:

Variations on the theme of one authoritative game server to correct erroneous values on less authoritative machines can surely do the trick; if algorithms are reasonably error-tolerant between synchronisations, corrections can surely be imperceptible. I'm speaking quite generally, and specifics of this approach would vary, going way beyond the scope of this conversation.

Here's a potentially good one to add (unless someone already said it, and I missed it…)

Has anyone experienced the pain of inconsistent floats with a distributed computing project? For instance, what about the reproducibility of a particularly significant result in scientific calculations? I can imagine even a Monte Carlo method storing/reporting any list of random numbers it used, or at least the list's seed. Thus, for instance, the algorithm producing a claimed low energy protein conformation could be independently reproduced and analysed for whatever reason; the same goes if we're rounding at random throughout a long calculation, to decrease systematic rounding errors, (an example of a difference between accuracy and consistency.) But such checks could be scuppered by inconsistent floats among various platforms. To my mind, that's potentially much harder to solve than the game synchronisation problem — if it's a problem that needs solving, that is. I don't know; I'm not a Rosetta@HOME developer!

How does all this CPU discussion pertain to GPUs? Are they more, or less, consistent in their floating point behaviour?

Fast, accurate emulation is another area that occurs to me as one that could do with hardware-based, yet consistent, floating point behaviour. Any experience of that, anyone?

And finally, sorry I can't be more help on the answer to how one achieves more consistency. Hopefully my questions are worthwhile though, and hopefully I've done something to justify my earlier point that it feels bad to answer ANY "How do we do this?" with a simple "Why?" or a "But you don't want to!" We already have a few of those answers! They're well justified and probably true as far as they go, but they're a bit unsatisfying all the same. Even the VERY wise cannot see all ends, Frodo!


#43 Tom Poole on 04.14.15 at 8:13 am

P.S. Another example of on-the-fly machinima would be multi-angle game replays, most commonly seen in racing games. The Super Monkey Ball games make small errors here, but only with banana collection so far as I've noticed. That's running on the same platform again of course, so the floating point pain in not the cause, but we can imagine a game saving replays, then having problems after a hardware or O/S upgrade, or problems with sharing such replays online. Perhaps consoles seem to do that sort of thing more, because it's more feasible with a consistent platform?

And for this problem, my 'variation on the theme' of an authoritative game server would be a pre-computed set of authoritative simulation frames.


#44 Chas Zizzo on 05.15.19 at 9:49 pm

I completely agree with your take on this subject and look forward to additional posts and comments here at Keep up the good work!

#45 resharper crack on 05.15.19 at 11:22 pm

bing bring me here. Thanks!

#46 vn hax on 05.16.19 at 1:52 pm

Cheers, great stuff, I enjoying.

#47 aimbot fortnite on 05.16.19 at 5:46 pm

I was looking at some of your articles on this site and I believe this internet site is really instructive! Keep on posting .

#48 nonsensediamond on 05.17.19 at 8:03 am

Some truly great stuff on this web site , appreciate it for contribution.

#49 red dead redemption 2 digital key resale on 05.17.19 at 4:33 pm

I have interest in this, danke.

#50 redline v3.0 on 05.17.19 at 7:40 pm

Enjoyed reading through this, very good stuff, thankyou .

#51 chaturbate hack cheat engine 2018 on 05.18.19 at 9:06 am

Intresting, will come back here later too.

#52 forza horizon 4 license key on 05.18.19 at 3:57 pm

Yeah bookmaking this wasn’t a risky decision outstanding post! .

#53 mining simulator 2019 on 05.19.19 at 8:01 am

I dugg some of you post as I thought they were very beneficial invaluable

#54 redline v3.0 on 05.21.19 at 8:11 am

Great, bing took me stright here. thanks btw for info. Cheers!

#55 free fire hack version unlimited diamond on 05.21.19 at 5:33 pm

Respect to website author , some wonderful entropy.

#56 nonsense diamond on 05.22.19 at 7:23 pm

I really enjoy examining on this website , it has got great stuff .

#57 krunker aimbot on 05.23.19 at 7:43 am

I have interest in this, cheers.

#58 bitcoin adder v.1.3.00 free download on 05.23.19 at 11:22 am

bing brought me here. Cheers!

#59 vn hax on 05.23.19 at 8:08 pm

Deference to op , some superb selective information .

#60 v9 on 05.24.19 at 8:53 am

I am glad to be one of the visitors on this great website (:, appreciate it for posting .

#61 ispoofer pogo activate seriale on 05.24.19 at 7:19 pm

I am not rattling great with English but I get hold this really easygoing to read .

#62 cheats for hempire game on 05.26.19 at 7:19 am

This is great!

#63 smart defrag 6.2 serial key on 05.26.19 at 4:38 pm

I consider something really special in this site.

#64 resetter epson l1110 on 05.26.19 at 7:27 pm

I truly enjoy looking through on this web site , it holds superb content .

#65 sims 4 seasons code free on 05.27.19 at 8:44 am

I am not rattling great with English but I get hold this really easygoing to read .

#66 rust hacks on 05.27.19 at 9:07 pm

You got yourself a new follower.

#67 strucid hacks on 05.28.19 at 11:25 am

Intresting, will come back here later too.

#68 expressvpn key on 05.28.19 at 8:26 pm

I was looking at some of your articles on this site and I believe this internet site is really instructive! Keep on posting .

#69 ispoofer key on 05.29.19 at 9:48 am

I have interest in this, cheers.

#70 aimbot free download fortnite on 05.29.19 at 1:48 pm

You got yourself a new rader.

#71 gamefly free trial on 05.29.19 at 1:53 pm

Pretty! This has been an extremely wonderful post. Many thanks for supplying this information.

#72 redline v3.0 on 05.29.19 at 6:13 pm

I must say, as a lot as I enjoyed reading what you had to say, I couldnt help but lose interest after a while.

#73 how to get help in windows 10 on 05.30.19 at 3:27 pm

Do you have any video of that? I'd care to find out
some additional information.

#74 gamefly free trial on 05.31.19 at 1:31 am

Hello it's me, I am also visiting this web site daily, this site is truly good and the people are
actually sharing good thoughts.

#75 xbox one mods free download on 05.31.19 at 2:03 pm

I have interest in this, xexe.

#76 fortnite aimbot download on 05.31.19 at 4:45 pm

Yeah bookmaking this wasn’t a risky decision outstanding post! .

#77 gamefly free trial on 05.31.19 at 9:21 pm

I think this is among the most vital information for
me. And i am glad reading your article. But should remark on some general
things, The website style is perfect, the articles is really
nice : D. Good job, cheers

#78 gamefly free trial on 06.01.19 at 3:08 am

Very good write-up. I absolutely appreciate this website.

#79 mpl pro on 06.01.19 at 7:19 pm

Deference to op , some superb selective information .

#80 hacks counter blox script on 06.02.19 at 7:32 am

Your website has proven useful to me.

#81 gamefly free trial on 06.02.19 at 2:57 pm

Pretty! This was an extremely wonderful post.
Thanks for supplying this info.

#82 gamefly free trial on 06.03.19 at 5:26 am

Please let me know if you're looking for a article writer for your site.
You have some really great articles and I think I would be a good asset.
If you ever want to take some of the load off,
I'd really like to write some articles for your blog in exchange for a link back to mine.
Please shoot me an e-mail if interested. Cheers!

#83 protosmasher download on 06.03.19 at 11:21 am

Found this on google and I’m happy I did. Well written site.

#84 gamefly free trial on 06.04.19 at 6:45 pm

Hi there to all, the contents existing at this
web page are genuinely amazing for people knowledge, well, keep up the good
work fellows.

#85 gamefly free trial on 06.05.19 at 8:39 am

Woah! I'm really enjoying the template/theme of this site.
It's simple, yet effective. A lot of times it's difficult to
get that "perfect balance" between usability and visual appeal.
I must say you've done a amazing job with this.
Also, the blog loads extremely fast for me on Safari.
Excellent Blog!

#86 FrankAmawn on 06.05.19 at 6:02 pm

Terms of Use commitment

Home Watch Now Video Gallery Everyday Heroes Heartland News Podcast Heartland Expert News Crime A Better Heartland National Gas Prices inspections Heartland Votes Election Results First Alert Weather Skycams Closings Sign up for weather emails Sports Heartland Football Friday Heartland Hoops Scores Business Business Break Community CNews [url=]charmdate review[/url] Calendar Health Southeast Health Programming Complete KFVS12, Heartland's CW [url=]charmdate review[/url] and Grit schedules CW Contests About Us KFVS12 Management Team KFVS12 sales force Meet the KFVS12 News Team Meet the First Alert Weather Team

This ("TOU") Has been updated to make it easier to read, To include additional features, Services and technique, And enhance several groups. If you carry on using our Services, tend to be bound by this revised TOU.

In addition to the t's and c's set forth in this TOU, Content uploaded by users on our websites (quite often called "Web signals"), Or other service providers, with this "ScribbleLive" Service are also foreclosures ScribbleLive's Terms of Use. Please see the separate Terms of Use for this specific repair on the ScribbleLive homepage.

This TOU governs your use of the several Web Channels, Mobile services and reasons ("apps") And each service that is owned, Operated or supplied by Gray Television, inc. Your use of the representation constitutes your binding consent to all such terms, health conditions and notices. This TOU also governs the completion and use of User Generated Content ("UGC"), consisting of text, pictures, sharp graphics, design, shows, Messages or other materials that you upload using the Apps.

PLEASE REVIEW THIS TOU rigorously. understand what AGREE TO BE BOUND BY THE TOU, don't use OUR SERVICES.

This TOU will remain in full force and effect as long as you are a user of the Services and in the eventuality of termination of any membership, internet service or feature, You it's still bound by certain obligations under this TOU; Please see segment 19 for details.

1. changing to TOU: We may change this TOU once in a while without advance notice to you. fot that reason, Please review this TOU frequently. When we replace the TOU, We will post the Effective Date and notice of such changes on our specified Services. if you can't accept the changes, Your only recourse is to stop using the Services and terminate your account(s) Or an account(s) in a flash. If you keep using the Services after we change the TOU, You accept all irregularities.

2. multiple Terms: We may provide even further terms, Such as rules of course activities and Services including, But not without reduction, prize draws, drawings, subscription email/text clubs and other interactive features and services ("excess Terms"). To the extent that there's a conflict between this TOU and the Additional Terms, the additional Terms shall govern.

3. rational Property Rights: All text, files, photographs, graphic, images, exchange bombs, Product or model names, blog titles, providing, User interfaces, Audio and/or video content and any other content provided on or through the assistance by GTI, gas stops, Third entity licensors (regarding UGC), Vendors and suppliers that give internal support to our Services (mutually "Operational companies"), And entrepreneurs, Sponsors or your business partners (each, "marketers"), Including the choice, co-ordination, And bouquet of any such content (with every, "GTI article content"), Are owned by or skilled to GTI, stops, publishers, Operational companies and/or our third party licensors. would likely (And you possibly will not permit or encourage anyone else to) Make offshoot works, widely perform, reproduce, submit, Transmit or link to GTI Content in or on every single third party website, Digital service or via any vehicle in any manner that is likely or meant to cause confusion about the owner or origin of GTI Content, Or is deceiving, Disparaging, intense, Or a detriment to GTI in our sole foresight. You may only use our Services and our GTI Content as expressly accepted in this TOU and for no other purpose. We may revoke permission to use GTI Content at any time and for any reason. GTI, stops, companies, Operational providers and/or licensors retain exclusive rights in any GTI Content that are not expressly provided in this TOU. And/or external laws, Whether for advertisement or non commercially produced use.


5. Links to Third Party Services or web sites: the requirements may contain links to external websites or services owned, run, Or managed by others (Collectively generally known as "Linked companies") instance "ScribbleLive, Although a Linked Website may feature a GTI or Station logo or similarity, Or a Linked rrnternet site is "framed" By a Web sales channel, Such third party websites are not under the domination over GTI or Stations. Nor is GTI or Stations responsible for the contents, jobs, Business practices or privacy policies of any Linked Website. We provide Linked Websites to you only as a freedom, And the inclusion of such links is not likely an endorsement by GTI or any of Stations in favor of any person offering any content, websites, Or products on the Linked web site. if you opt to access and use Linked Websites, you no doubt know, Acknowledge and agree that you just do so at your own risk.

6. ad servers: the inclusion of Advertisers' statements, voice messages, supplies, info, Or promotions on or through the Service is not an endorsement by GTI or any Stations in favor of same. Your communication and business dealings directly with, Or response in the promotions of Advertisers found on or through the Service, Including payment and delivery of your purchase of goods or services, And all other terms, states, Warranties or representations to do with such dealings, Are solely between you and the advertiser. you're confident of, Acknowledge and agree that GTI and Stations are not responsible or liable for any loss or damage of any sort incurred as a result of any such dealings or due to the inclusion of Advertisers on the Service.

7. Shopping and e-marketing: Our providers may offer various goods and companies for sale (each and every, "products and solutions"), Or direct or link to other domains that sell Products. We have no obligation or liability whatsoever for any Products you may obtain from or through other websites or web pages, Even should directed or linked to such a website or page through our Services. If a Product offered directly by GTI is listed at an incorrect price or with incorrect insight, We reserve the authority to refuse or cancel orders placed for that Product, irrespective of whether the order has been confirmed and even if your credit or debt card account has been charged (In which event we will issue a credit for your requirements for your order).

The creation or transmission of confirmation for an order does not necessarily follow final acceptance of your order, Nor does it create a binding proof of an offer to sell any Product. We reserve the right to accept or decline your order at all up until the time the Product is actually shipped to you. We will likely, without having having prior notice, Limit or reduce the quantity you ordered of any Product and we will notify you if we change your order. Products directly sold by our Services are restricted from commercial resale. we don't knowingly accept [url=][/url] orders from dealers, Exporters, retailers, recruits, shops or other similar persons or companies, And reserve the authority to refuse, Cancel or seek the return of any Products that are purchased in violation of the foregoing restrictions.

restricted Ads: Our Services may offer a classified ad service for the sale of goods and services ("catalogued Ads"). Sellers must certify that their Product is not on our banned Items List. Fees charged to post listings with special features or premium listings are non refundable if your jewelry is on the Prohibited Items List. We reserve the right to reject or remove any free classified, Even those that aren't included on the Prohibited Items List. natural meats, At our foresight, Deactivate the accounts of users who submit multiple classifieds we deem inappropriate. Sellers are entirely with regard to the content of their advertisements, communications with buyers and the delivery of all orders. We merely provide the forum and we are not involved in the actual transaction between clients. We have no control over the product quality, Safety or legality of publicized Products, Nor any matters relating to the operation. you're confident, Acknowledge and agree that we are not responsible or liable for any loss or damage of any kind arising out of or related to your use of Classified Ads.

8. forbidden Uses Generally: You are required to comply with all applicable law related to your use of the Services. We may provide additional criteria in Additional Terms or posted on our Services. As a condition of your use of the representation, You represent and warrant that you won't use the Services for any purpose that is unlawful or prohibited by this TOU.

#87 gamefly free trial on 06.06.19 at 10:20 pm

If some one desires to be updated with latest technologies then he must be visit this web
page and be up to date everyday.

#88 Calvinret on 06.10.19 at 6:26 am

LittleRadagast ough

LittleRadagast 11 difficulties processed 1 month your past

Dogs put on in reality completely grasp you may pounding worries these animals may people happen to be kicked earlier to. whenever you stop a dog which can be bouncing you, a few lift your leg to block / [url=][/url] made canine within your or to foot. my father languages like german Shepard would need to jump on me all this scrapes your canine [url=][/url] and also generates the message equipped with all the allow when you were a kick. Dogs don misconstrue it as being and doesn be on the lookout as bad to be able to.

Scolding dogs of the fact that behave as you format doesn go a long way likewise easily they aren't certified. IMO, hoping scold a acting up dog only will make puppy a good deal more safety individuals, As doggy is likely to either one picture for a moment you have on because of this dog or to pick up on your anxiety/fear.

have fun. I move [url=]latamdate review[/url] tend to be dog wearing park systems not including other dogs when major nuts dogs cause me to feel uneasy and simply accumulates dog upon it attempts to protect me.

Homitu 126 points presented 3 months the particular

I constantly pondered equipped with any one of these insane stunts (of your house details, snow-boarding, biking, just about anything which), the fact that hell would you give up apply to the current? it needs practicing so you may do it easily, although doing it easily equals you purchase your butt kicked in a horrible auto crash.

So figuring out how it over. really quite impressive, just one!

truly even if, proper after gazing your boyfriend's crashes, It wasn as bad as i dreamed in several ways,also you can. i assumed person in a literal sense stay hopeless, combined with no matter if you can create a dismantled the neck and throat effectively entirely kept approximately experience. But apparently minimal of achievable test pass which you have comparatively reliable take care of finished the method drive. talk about their experience abort and just fall beginning of the for.

#89 quest bars cheap on 06.14.19 at 5:39 pm

Appreciating the dedication you put into your blog and in depth information you present.
It's good to come across a blog every once in a while that
isn't the same out of date rehashed information. Fantastic read!
I've saved your site and I'm adding your RSS feeds
to my Google account.

#90 roblox script executor on 06.17.19 at 8:18 pm

Hello, i really think i will be back to your page

#91 proxo key on 06.19.19 at 4:44 pm

I was looking at some of your articles on this site and I believe this internet site is really instructive! Keep on posting .

#92 WalterTix on 06.20.19 at 6:46 am

Safe Online asian adult dating safe practices Tips

Online asian adult dating personal ads, And can be a very fun venue towards meeting that special someone. about the, you are required to still safety first. wood most online asian adult dating services do not pre screen the users? This means that they can not guarantee that the people you meet through the site safe enough to handle. which means, What makes you call the shots after being filtering.

do exercises to online asian adult dating less risky and more enjoyable:

Given actual e mail address, Personal url, identify, contact number, job, confront, And any other information that identifies you in your profile, Or the initial few e mails, What do you exchange with other members easy prey. Turn off your signature file and make sure it is lacking in [url=][/url] any identifying information when sending e mail to the members of your e mail.

In order for an idea of seen a person, this means that a tad safer, an application for a picture. The scanned image is not so much, So not much excuse may be, If you keep away.

Trusting in the online acquaintances easily can be that easy to bait you. If your gut feeling tells me that somebody is lying, There is a big chance that he is certainly lying. well, Do not go to warrant his act. Find someone that deserves trust. Online intimacy is very dangerous. Do not indulge yourself in the pursuit of such a mouse click. Logical thinking helps you to save a lot of trouble.

4. Call to start with, I do not drool

The ability of the person where the communication is easily measured in a phone call. whilst it might cost a little moolah, But it worth a chance of the subject itself as anti social. of course, To make things more secure, Use your cellular phone or a pay phone instead of using a wired phone. Unless you are completely happy with the believers, Do not enter your home phone number at the same time.

It is suggested that time and get to know the person wherever possible first before saying that there must be a face to face meeting. If it's only spouting or keying saccharine words you want to hear the person on the other end there is a questionable honesty. Do not feel like it is obliged to meet anyone still has the final word that is ready to meet the person or not. You are free to convince you, you have to something is wrong.

6. Psychoanalyze or towards the end of Psycho

Should be monitored for any signs of intense discouragement, rage, Or display hyped-up behavior. If the person tries to manipulate or pressure you, Stop the marketing. Here is another disturbing current expression of the bad behavior that, When personally seen, It should be made to communicate online friend:

Giving dubious info about the appearance, growing older, taste, job, spouse status, and more.

Refusal to talk with you by phone if you previously established online intimacy

Has frequently failed to answer direct questions directly, stop smoking,give up.

You are all too will abide by the statement

I ask that the travel placements

prior to meeting online friend, Tell someone reliable where you have and when you get back. Exit date name and phone number of a good idea. Bring your cellular phone. Do not post date pick you up at home. Their own method of travel and meet somewhere public. this date, Do not leave your drink alone. wedding ceremony date of [url=]Asiame.COM[/url] their leave as well.

to ensure online asian adult dating work, it is very important to be cautious. They have time to get to know the rest, Do not rush yourself or allow anyone to rush you to a single thing. The to be safe and dateless temporarly than sorry.

#93 vn hax pubg on 06.21.19 at 1:09 am

stays on topic and states valid points. Thank you.

#94 nonsense diamond download on 06.21.19 at 2:14 pm

Me enjoying, will read more. Cheers!

#95 badoo superpowers free on 06.23.19 at 11:23 pm

Respect to website author , some wonderful entropy.

#96 gx tool apk download on 06.24.19 at 9:12 pm

Deference to op , some superb selective information .

#97 geometry dash 2.11 download on 06.26.19 at 2:04 am

I love reading through and I believe this website got some genuinely utilitarian stuff on it! .

#98 Lakseolie hund on 06.26.19 at 2:27 am

Very informative blog article.Thanks Again. Keep writing.

#99 skisploit on 06.26.19 at 12:16 pm

This is good. Cheers!

#100 ispoofer key on 06.27.19 at 11:00 am

Respect to website author , some wonderful entropy.

#101 synapse x cracked on 06.28.19 at 2:30 am

I conceive this web site holds some real superb information for everyone : D.

#102 advanced systemcare 11.5 key on 06.28.19 at 5:10 pm

Thank You for this.

#103 how to get help in windows 10 on 06.29.19 at 4:18 am

Pretty section of content. I just stumbled upon your site
and in accession capital to assert that I get actually enjoyed
account your blog posts. Any way I will be subscribing to your augment and even I achievement you access consistently fast.

#104 cryptotab hack script free download 2019 on 06.29.19 at 11:06 am

Respect to website author , some wonderful entropy.

#105 cryptotab balance hack script v1.4 cracked by cryptechy03 on 06.29.19 at 5:32 pm

I really enjoy examining on this page , it has got interesting stuff .

#106 roblox asshurt on 07.01.19 at 1:08 pm

I am not rattling great with English but I get hold this really easygoing to read .

#107 cheat fortnite download no wirus on 07.01.19 at 11:58 pm

google brought me here. Thanks!

#108 vn hax download pc on 07.03.19 at 11:12 am

Enjoyed reading through this, very good stuff, thankyou .

#109 cyberhackid on 07.03.19 at 11:09 pm

Yeah bookmaking this wasn’t a risky decision outstanding post! .

#110 roblox prison life hack on 07.04.19 at 11:12 am

Hi, glad that i saw on this in google. Thanks!

#111 search engine optimization techniques on 07.04.19 at 1:44 pm

Parasite backlink SEO works well :)

#112 phantom forces aimbot on 07.04.19 at 11:12 pm

I like this website its a master peace ! Glad I found this on google .

#113 open dego on 07.05.19 at 11:50 am

I’m impressed, I have to admit. Genuinely rarely should i encounter a weblog that’s both educative and entertaining, and let me tell you, you may have hit the nail about the head. Your idea is outstanding; the problem is an element that insufficient persons are speaking intelligently about. I am delighted we came across this during my look for something with this.

#114 tom clancy's the division hacks on 07.05.19 at 11:50 pm

I have interest in this, xexe.

#115 synapse x roblox on 07.06.19 at 9:28 am

Me like, will read more. Thanks!

#116 Nolan Bazata on 07.06.19 at 10:19 am

The next phase of the labyrinth is to decode the order of the pyramid. This is your third secret tip! 517232125

#117 gx tool pubg uc hack on 07.06.19 at 3:26 pm

Hello, here from bing, i enjoyng this, will come back again.

#118 call of duty black ops 4 license key on 07.07.19 at 2:18 pm

Thanks for this web. I definitely agree with what you are saying.

#119 spyhunter key on 07.08.19 at 3:36 pm

I really enjoy examining on this web , it has got fine article .

#120 quest bars cheap 2019 coupon on 07.09.19 at 9:24 am

I think the admin of this site is in fact working hard for his website, for the reason that here every data is quality based data.

#121 fps unlocker download on 07.09.19 at 5:45 pm

I was looking at some of your articles on this site and I believe this internet site is really instructive! Keep on posting .

#122 quest bars cheap on 07.11.19 at 4:20 am

Pretty component to content. I just stumbled upon your blog and in accession capital
to assert that I acquire in fact enjoyed account your weblog posts.
Anyway I will be subscribing to your feeds and even I fulfillment you get admission to
consistently rapidly.

#123 suzy on 07.14.19 at 2:00 am

Thank you for the great read!

#124 porn adult show on 07.15.19 at 2:12 am

some great ideas this gave me!

#125 legalporno on 07.15.19 at 11:59 pm

great advice you give

#126 legalporn on 07.16.19 at 12:38 am

great advice you give

#127 how to get help in windows 10 on 07.17.19 at 3:45 am

That is really fascinating, You are a very professional blogger.
I've joined your feed and look ahead to in quest of more of your great post.

Additionally, I have shared your web site in my social

#128 amanda_hill on 07.19.19 at 1:47 am

amazing content thanks

#129 lucia_denville on 07.19.19 at 2:15 am

amazing content thanks

#130 Buy Drugs Online on 07.19.19 at 2:51 am

This blog is amazing! Thank you.

#131 plenty of fish dating site on 07.19.19 at 2:30 pm

Magnificent website. Lots of useful information here.
I'm sending it to some pals ans also sharing in delicious.
And certainly, thank you in your effort!

#132 how to get help in windows 10 on 07.20.19 at 5:32 pm

Hi, I do think this is a great website. I stumbledupon it ;
) I may revisit once again since I saved as a favorite it.
Money and freedom is the greatest way to change, may you be rich and continue
to help other people.

#133 how to get help in windows 10 on 07.21.19 at 12:26 pm

I all the time used to study article in news papers but now as I am a user of internet thus from now I
am using net for articles or reviews, thanks to web.

#134 prodigy games files on 07.21.19 at 8:27 pm

yahoo brought me here. Thanks!

#135 natalielise on 07.22.19 at 4:58 pm

I am actually thankful to the owner of this
web page who has shared this impressive paragraph at at this place.

natalielise pof

#136 on 07.23.19 at 8:02 pm

I like this article, some useful stuff on here : D.

#137 date clougar on 07.23.19 at 10:42 pm

I am 43 years old and a mother this helped me!

#138 dae cougar on 07.23.19 at 10:59 pm

I am 43 years old and a mother this helped me!

#139 date coughar on 07.23.19 at 11:27 pm

I am 43 years old and a mother this helped me!

#140 da6e cougar on 07.23.19 at 11:37 pm

I am 43 years old and a mother this helped me!

#141 date cougarf on 07.23.19 at 11:44 pm

I am 43 years old and a mother this helped me!

#142 forza horizon 2 key generator on 07.24.19 at 8:10 pm

I conceive you have mentioned some very interesting details , appreciate it for the post.

#143 plenty of fish dating site on 07.25.19 at 8:08 am

Greetings from Idaho! I'm bored to tears at work so I decided to
browse your website on my iphone during lunch break.
I love the info you provide here and can't wait to take a look when I get home.

I'm surprised at how fast your blog loaded on my cell phone ..
I'm not even using WIFI, just 3G .. Anyways, good site!

#144 ezfrags on 07.25.19 at 11:18 pm

Some truly fine goodies on this web site , appreciate it for contribution.

#145 skisploit on 07.27.19 at 12:36 am

I dugg some of you post as I thought they were very beneficial invaluable

#146 Gabrielle on 01.21.20 at 8:12 pm


#147 Gabrielle on 01.21.20 at 8:13 pm

je suis une grave bg qui veux je suis de tender tk bb

#148 kalid on 02.03.20 at 2:12 pm

I have never seen like this website