Referenced here and here...Why would I use two's complement over an epsilon method? It seems like the epsilon method would be good enough for most cases.
Update: I'm purely looking for a theoretical reason why you'd use one over the other. I've always used the epsilon method.
Has anyone used the 2's complement comparison successfully? Why? Why Not?
the second link you reference mentions an article that has quite a long description of the issue:
http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm
but unless you are tweaking performance I would stick with epsilon so people can debug your code
The bits method might be faster. I say might because on modern (multicore, highly pipelined) processors it is often impossible to guess what is really faster.
Code the simplest most obviously correct implementation, then measure, then optomise.
In short, when comparing two floats with unknown origins, picking an epsilon that is valid is almost impossible.
For example:
What is a good epsilon when comparing distance in miles between Atlanta GA, Dallas TX and some place in Ohio?
What is a good epsilon when comparing distance in miles between my left foot, my right foot and the computer under my desk?
EDIT:
Ok, I'm getting a fair number of people not understanding why you wouldn't know what your epsilon is.
Back in the old days of lore, I wrote two programs that worked with NeverWinter Nights (a game made by BioWare). One of the programs took a binary model and converted it to ASCII. The other program took an ASCII model and compiled it into binary. One of the tests I wrote was to take all of BioWare's binary models, decompile them to ASCII and then back to binary. Then I compared my binary version with original one from BioWare. One of the problems during the comparison was dealing with some of the slight variances in floating point values. So instead of coming up with a bunch of different EPSILONS for each type of floating point number (vertex, normal, etc), I wanted to use something such as this twos compliment compare. Thus avoiding the whole multiple EPSILON issue.
The same type of issue can apply to any type of software that processes 3rd party data and then needs to validate their results with the original. In these cases you might not even know what the floating point values represent, you just have to compare them. We ran into this issue with our industrial automation software.
EDIT:
LOL, this has been voted up and down by different people.
I'll boil the problem down to this, given two arbitrary floating point numbers, how do you decide what epsilon to use? You can't.
How can you compare 1e23 and 1.0001e23 with an epsilon and still compare 1e-23 and 5.2e-23 using the same epsilon? Sure, you can do some dynamic epsilon tricks, but that is the whole point to the integer compare (which does NOT require the integers be exact).
The integer compare is able to compare two floats using an epsilon relative to the magnitude of the numbers.
EDIT
Steve, lets look at what you said in the comments:
"But you know what equality means to you... Hence, you should be able to find an appropriate epsilon".
Turn this statement around to say:
"If you know what equality means to you, then you should be able to find an appropriate epsilon."
The whole point to what I am trying to say is that there are applications where we don't know what equality means in the absolute sense, thus we have to resort to a relative compare which is what the integer version is trying to do.
When it comes to speed, follow these rules:
If you're not a very experienced developer, don't optimize.
If you are an experienced developer, don't optimize yet.
Do the easiest method.
Alex
Oskar's right. Don't screw with this unless you really, really need that performance.
And you don't. If you were in the situation that did, you wouldn't have needed to ask the question -- you'd already know. If you think you do, then you don't. Your performance problems lie elsewhere. Just use the readable version.
Using any method that compares bitwise will result in trouble when fractions are represented by approximations. All floating point numbers with fractions that are not denominated in powers of two (1/2, 1/4, 1/8, 1/65536, &c) are approximated. So, of course, are all irrational numbers.
float third = 1/3;
float two=2.0;
float another_two=third*6.0;
if(two != another_two)
print ("Approximation!\n");
The only time comparing bitwise would work is when you derive the floating point numbers exactly the same way or they are exact representations (whole numbers, fraction powers of two). Even then, there can be multiple representations of some numbers, though I have never seen this in a working system.
Related
Is there some clever and reliable way to print series of bits as an IEEE-754 without actually using a float type?
I have found a way to print fractions, which allows me to represent the float as a a fraction. However, I then came to realize that the exponent may range from -127 to 128 (after adjusting with bias), which may result in the multiplication mantissa * 2^128. The fraction method relies on representing the numerator as an integer, and I would require a really large integer to do this multiplication. I mean, I could use "custom" type to represent this large value (i.e. https://gmplib.org/), but I would prefer if to avoid this. If we multiplied by 10^x, I could simply adjust the decimal point and add some zeros, but sadly that's not the case either.
I have not been able to find anything that mentions any solution for this. Probably due to the fact that googling stuff like "print from
Why am I actually trying to do this?
I'm only doing this to get a better understanding of how floats (IEEE-754 in particular) work, and I find that it always help to do some practical example. So I thought "Hey, why not try to code it?". This has no practical application (that I know of)!
So, almost immediately after posting this, I finally succeded in finding the resources I've been looking for.
https://www.ryanjuckett.com/printing-floating-point-numbers/ talks about it, and references other relevant sources.
I am writing a simulation program that proceeds in discrete steps. The simulation consists of many nodes, each of which has a floating-point value associated with it that is re-calculated on every step. The result can be positive, negative or zero.
In the case where the result is zero or less something happens. So far this seems straightforward - I can just do something like this for each node:
if (value <= 0.0f) something_happens();
A problem has arisen, however, after some recent changes I made to the program in which I re-arranged the order in which certain calculations are done. In a perfect world the values would still come out the same after this re-arrangement, but because of the imprecision of floating point representation they come out very slightly different. Since the calculations for each step depend on the results of the previous step, these slight variations in the results can accumulate into larger variations as the simulation proceeds.
Here's a simple example program that demonstrates the phenomena I'm describing:
float f1 = 0.000001f, f2 = 0.000002f;
f1 += 0.000004f; // This part happens first here
f1 += (f2 * 0.000003f);
printf("%.16f\n", f1);
f1 = 0.000001f, f2 = 0.000002f;
f1 += (f2 * 0.000003f);
f1 += 0.000004f; // This time this happens second
printf("%.16f\n", f1);
The output of this program is
0.0000050000057854
0.0000050000062402
even though addition is commutative so both results should be the same. Note: I understand perfectly well why this is happening - that's not the issue. The problem is that these variations can mean that sometimes a value that used to come out negative on step N, triggering something_happens(), now may come out negative a step or two earlier or later, which can lead to very different overall simulation results because something_happens() has a large effect.
What I want to know is whether there is a good way to decide when something_happens() should be triggered that is not going to be affected by the tiny variations in calculation results that result from re-ordering operations so that the behavior of newer versions of my program will be consistent with the older versions.
The only solution I've so far been able to think of is to use some value epsilon like this:
if (value < epsilon) something_happens();
but because the tiny variations in the results accumulate over time I need to make epsilon quite large (relatively speaking) to ensure that the variations don't result in something_happens() being triggered on a different step. Is there a better way?
I've read this excellent article on floating point comparison, but I don't see how any of the comparison methods described could help me in this situation.
Note: Using integer values instead is not an option.
Edit the possibility of using doubles instead of floats has been raised. This wouldn't solve my problem since the variations would still be there, they'd just be of a smaller magnitude.
I've worked with simulation models for 2 years and the epsilon approach is the sanest way to compare your floats.
Generally, using suitable epsilon values is the way to go if you need to use floating point numbers. Here are a few things which may help:
If your values are in a known range you and you don't need divisions you may be able to scale the problem and use exact operations on integers. In general, the conditions don't apply.
A variation is to use rational numbers to do exact computations. This still has restrictions on the operations available and it typically has severe performance implications: you trade performance for accuracy.
The rounding mode can be changed. This can be use to compute an interval rather than an individual value (possibly with 3 values resulting from round up, round down, and round closest). Again, it won't work for everything but you may get an error estimate out of this.
Keeping track of the value and a number of operations (possible multiple counters) may also be used to estimate the current size of the error.
To possibly experiment with different numeric representations (float, double, interval, etc.) you might want to implement your simulation as templates parameterized for the numeric type.
There are many books written on estimating and minimizing errors when using floating point arithmetic. This is the topic of numerical mathematics.
Most cases I'm aware of experiment briefly with some of the methods mentioned above and conclude that the model is imprecise anyway and don't bother with the effort. Also, doing something else than using float may yield better result but is just too slow, even using double due to the doubled memory footprint and the smaller opportunity of using SIMD operations.
I recommend that you single step - preferably in assembly mode - through the calculations while doing the same arithmetic on a calculator. You should be able to determine which calculation orderings yield results of lesser quality than you expect and which that work. You will learn from this and probably write better-ordered calculations in the future.
In the end - given the examples of numbers you use - you will probably need to accept the fact that you won't be able to do equality comparisons.
As to the epsilon approach you usually need one epsilon for every possible exponent. For the single-precision floating point format you would need 256 single precision floating point values as the exponent is 8 bits wide. Some exponents will be the result of exceptions but for simplicity it is better to have a 256 member vector than to do a lot of testing as well.
One way to do this could be to determine your base epsilon in the case where the exponent is 0 i e the value to be compared against is in the range 1.0 <= x < 2.0. Preferably the epsilon should be chosen to be base 2 adapted i e a value that can be exactly represented in a single precision floating point format - that way you know exactly what you are testing against and won't have to think about rounding problems in the epsilon as well. For exponent -1 you would use your base epsilon divided by two, for -2 divided by 4 and so on. As you approach the lowest and the highest parts of the exponent range you gradually run out of precision - bit by bit - so you need to be aware that extreme values can cause the epsilon method to fail.
If it absolutely has to be floats then using an epsilon value may help but may not eliminate all problems. I would recommend using doubles for the spots in the code you know for sure will have variation.
Another way is to use floats to emulate doubles, there are many techniques out there and the most basic one is to use 2 floats and do a little bit of math to save most of the number in one float and the remainder in the other (saw a great guide on this, if I find it I'll link it).
Certainly you should be using doubles instead of floats. This will probably reduce the number of flipped nodes significantly.
Generally, using an epsilon threshold is only useful when you are comparing two floating-point number for equality, not when you are comparing them to see which is bigger. So (for most models, at least) using epsilon won't gain you anything at all -- it will just change the set of flipped nodes, it wont make that set smaller. If your model itself is chaotic, then it's chaotic.
Recently, I was curious how hash algorithms for floating points worked, so I looked at the source code for boost::hash_value. It turns out to be fairly complicated. The actual implementation loops over each digit in the radix and accumulates a hash value. Compared to the integer hash functions, it's much more involved.
My question is: why should a floating-point hash algorithm be any more complicated? Why not just hash the binary representation of the floating point value as if it was an integer?
Like:
std::size_t hash_value(float f)
{
return hash_value(*(reinterpret_cast<int*>(&f)));
}
I realize that float is not guaranteed to be the same size as int on all systems, but that sort of thing could be handled with a few template meta-programs to deduce an integral type that is the same size as float. So what is the advantage of introducing an entirely different hash function that specifically operates on floating point types?
Take a look at https://svn.boost.org/trac/boost/ticket/4038
In essence it boils down to two things:
Portability: when you take the binary representation of a float, then on some platform it could be possible that a float with a same value has multiple representations in binary. I don't know if there is actually a platform where such an issue exists, but with the complication of denormelized numbers, I'm not sure if this might actually happen.
the second issue is what you proposed, it might be that sizeof(float) does not equal sizeof(int).
I did not find anyone mentioning that the boost hash indeed avoids fewer collisions. Although I assume that separating the mantissa from the exponent might help, but the above link does not suggest that this was the driving design decision.
One reason not to just use the bit pattern is that some different bit patterns must be considered equals and thus have the same hashcode, namely
positive and negative zero
possibly denormalized numbers (I don't think this can occur with IEEE 754, but C allows other float representations).
possibly NANs (there are many, at least in IEEE 754. It actually requires NAN patterns to be considered unequal to themselves, which arguably means the cannot be meaningfully used in a hashtable)
Why are you wanting to hash floating point values? For the same reason that comparing floating point values for equality has a number of pitfalls, hashing them can have similar (negative) consequences.
However given that you really do want to do this, I suspect that the boost algorithm is complicated because when you take into account denormalized numbers different bit patterns can represent the same number (and should probably have the same hash). In IEEE 754 there are also both positive and negative 0 values that compare equal but have different bit patterns.
This probably wouldn't come up in the hashing if it wouldn't have come up otherwise in your algorithm but you still need to take care about signaling NaN values.
Additionally what would be the meaning of hashing +/- infinity and/or NaN? Specifically NaN can have many representations, should they all result in the same hash? Infinity seems to have just two representations so it seems like it would work out ok.
I imagine it's so that two machines with incompatible floating-point formats hash to the same value.
I am using math.h with GCC and GSL. I was wondering how to get this to evaluate?
I was hoping that the pow function would recognize pow(-1,1.2) as ((-1)^6)^(1/5). But it doesn't.
Does anybody know of a c++ library that will recognize these? Perhaps somebody has a decomposition routine they could share.
Mathematically, pow(-1, 1.2) is simply not defined. There are no powers with fractional exponents of negative numbers, and I hope there is no library that will simply return some arbitray value for such an expression. Would you also expect things like
pow(-1, 0.5) = ((-1)^2)^(1/4) = 1
which obviously isn't desirable.
Moreover, the floating point number 1.2 isn't even exactly equal to 6/5. The closest double precision number to 1.2 is
1.1999999999999999555910790149937383830547332763671875
Given this, what result would you expect now for pow(-1, 1.2)?
If you want to raise negative numbers to powers -- especially fractional powers -- use the cpow() method. You'll need to include <complex> to use it.
It seems like you're looking for pow(abs(x), y).
Explanation: you seem to be thinking in terms of
xy = (xN)(y/N)
If we choose that N === 2, then you have
(x2)y/2 = ((x2)1/2)y
But
(x2)1/2 = |x|
Substituting gives
|x|y
This is a stretch, because the above manipulations only work for non-negative x, but you're the one who chose to use that assumption.
Sounds like you want to perform a complex power (cpow()) and then take the magnitude (abs()) of that after.
>>> abs(cmath.exp(1.2*cmath.log(-1)))
1.0
>>> abs(cmath.exp(1.2*cmath.log(-293.2834)))
913.57662451612202
pow(a,b) is often thought of, defined as, and implemented as exp(log(a)*b) where log(a) is natural logarithm of a. log(a) is not defined for a<=0 in real numbers. So you need to either write a function with special case for negative a and integer b and/or b=1/(some_integer). It's easy to special-case for integer b, but for b=1/(some_integer) it's prone to round-off problems, like Sven Marnach pointed out.
Maybe for your domain pow(-a,b) should always be -pow(a,b)? But then you'd just implement such function, so I assume the question warrants more explanation .
Like duskwuff suggested, a much more robust and "mathematical" solution is to use complex functions log and exp, but it's much more "complex" (excuse my pun) than it seems on the surface (even though there's cpow function). And it'll be much slower if you have to compute a lot of pow()s.
Now there's an important catch with complex numbers that may or may not be relevant to your problem domain: when done right, the result of pow(a,b) is not one, but often a few complex numbers, but in the cases you care about, one of them will be complex number with nearly-zero imaginary part (it'll be non-zero due to roundoff errors) which you can simply ignore and/or not compute in your code.
To demonstrate it, consider what pow(-1,.5) is. It's a number X such that X^2==-1. Guess what? There are 2 such numbers: i and -i. Generally, pow(-1, 1/N) has exactly N solutions, although you're interested in only one of them.
If the imaginary part of all results of pow(a,b) is significant, it means you are passing wrong values. For single-precision floating point values in the range you describe, 1e-6*max(abs(a),abs(b)) would be a good starting point for defining the "significant enough" threshold. The extreme "wrong values" would be pow(-1,0.5) which would return 0 + 1i (0 in real part, 1 in imaginary part). Here the imaginary part is huge relative to the input and real part, so you know you screwed up your input values.
In any reasonable single-return-result implementation of cpow() , cpow(-1,0.3333) will probably return something like -1+0.000001i and ignore two other values with significant imaginary parts. So you can just take that real value and that's your answer.
Use std::complex. Without that, the roots of unity don't make much sense. With it they make a whole lot of sense.
I am checking to make sure a float is not zero. It is impossible for the float to become negative. So is it faster to do this float != 0.0f or this float > 0.0f?
Thanks.
Edit: Yes, I know this is micro-optimisation. But this is going to be called every time through my game loop, and I would like to know anyway.
There is not likely to be a detectable difference in performance.
Consider, for entertainment purposes only:
Only 2 floating point values compare equal to 0f: zero and negative zero, and they differ only at 1 bit. So circuitry/software emulation that tests whether the 31 non-sign bits are clear will do it.
The comparison >0f is slightly more complicated, since negative numbers and 0 result in false, positive numbers result in true, but NaNs (of both signs) also result in false, so it's slightly more than just checking the sign bit.
Depending on the floating point mode, either operation could cause a super-precise result in a floating point register to be rounded to 32 bit before comparison, so the score's even there.
If there was a difference at all, I'd sort of expect != to be faster, but I wouldn't really expect there to be a difference and I wouldn't be very surprised to be wrong on some particular implementation.
I assume that your proof that the value cannot be negative is not subject to floating point errors. For example, calculations along the lines of 1/2.0 - 1/3.0 - 1/6.0 or 0.4 - 0.2 - 0.2 can result in either positive or negative values if the errors happen to accumulate rather than cancelling, so presumably nothing like that is going on. About only real use of a floating-point test for equality with 0, is to test whether you have assigned a literal 0 to it. Or the result of some other calculation guaranteed to have result 0 in float, but that can be tricksy.
It is not possible to give a clear cut answer without knowing your platform and compiler. The C standard does not define how floats are implemented.
On some platforms, yes, on other platforms, no.
If in doubt, measure.
As far as I know, f != 0.0f will sometimes return true when you think it should be false.
To check whether a float number is non-zero, you should do Math.abs(f) > EPSILON, where EPSILON is the error you can tolerate.
Performance shouldn't be a big issue in this comparison.
This is almost certainly the sort of micro-optimization you shouldn't do until you have quantitative data showing that it's a problem. If you can prove it's a problem, you should figure out how to make your compiler show the machine instructions it's generating, then take that info and go to the data book for the processor you are using, and look up the number of clock cycles required for alternative implementations of the same logic. Then you should measure again to make sure you are seeing the benefits, if any.
If you don't have any data showing that's it's a performance problem stick with the implementation that most clearly and simply presents the logic of what you are trying to do.