While going through this post at SO by the user #skrebbel who stated that the google testing framework does a good and fast job for comparing floats and doubles. So I wrote the following code to check the validity of the code and apparently it seems like I am missing something here , since I was expecting to enter the almost equal to section here this is my code
float left = 0.1234567;
float right= 0.1234566;
const FloatingPoint<float> lhs(left), rhs(right);
if (lhs.AlmostEquals(rhs))
{
std::cout << "EQUAL"; //Shouldnt it have entered here ?
}
Any suggetsions would be appreciated.
You can use
ASSERT_NEAR(val1, val2, abs_error);
where you can give the acceptable - your chosen one, like, say 0.0000001 - difference as abs_error, if the default one is too small, see here https://github.com/google/googletest/blob/master/googletest/docs/advanced.md#floating-point-comparison
Your left and right are not “almost equal” because they are too far apart, farther than the default tolerance of AlmostEquals. The code in one of the answers in the question you linked to shows a tolerance of 4 ULP, but your numbers are 14 ULP apart (using IEEE 754 32-bit binary and correctly rounding software). (An ULP is the minimum increment of the floating-point value. It is small for floating-point numbers of small magnitude and large for large numbers, so it is approximately relative to the magnitude of the numbers.)
You should never perform any floating-point comparison without understanding what errors may be in the values you are comparing and what comparison you are performing.
People often misstate that you cannot test floating-point values for equality. This is false; executing a == b is a perfect operation. It returns true if and only if a is equal to b (that is, a and b are numbers with exactly the same value). The actual problem is that they are trying to calculate a correct function given incorrect input. == is a function: It takes two inputs and returns a value. Obviously, if you give any function incorrect inputs, it may return an incorrect result. So the problem here is not floating-point comparison; it is incorrect inputs. You cannot generally calculate a sum, a product, a square root, a logarithm, or any other function correctly given incorrect input. Therefore, when using floating-point, you must design an algorithm to work with approximate values (or, in special cases, use great care to ensure no errors are introduced).
Often people try to work around errors in their floating-point values by accepting as equal numbers that are slightly different. This decreases false negatives (indications of inequality due to prior computing errors) at the expense of increasing false positives (indications of equality caused by lax acceptance). Whether this exchange of one kind of error for another is acceptable depends on the application. There is no general solution, which is why functions like AlmostEquals are generally bad.
The errors in floating-point values are the results of preceding operations and values. These errors can range from zero to infinity, depending on circumstances. Because of this, one should never simply accept the default tolerance of a function such as AlmostEquals. Instead, one should calculate the tolerance, which is specific to their applications, needs, and computations, and use that calculated tolerance (or not use a comparison at all).
Another problem is that functions such as AlmostEquals are often written using tolerances that are specified relative to the values being compared. However, the errors in the values may have been affected by intermediate values of vastly different magnitude, so the final error might be a function of data that is not present in the values being compared.
“Approximate” floating-point comparisons may be acceptable in code that is testing other code because most bugs are likely cause large errors, so a lax acceptance of equality will allow good code to continue but will report bugs in most bad code. However, even in this situation, you must set the expected result and the permitted error tolerance appropriately. The AlmostEquals code appears to hard-code the error tolerance.
(Not sure if this 100% applies to the original question but this is what I came for when I stumbled upon it)
There also exist ASSERT_FLOAT_EQ and EXPECT_FLOAT_EQ (or the corresponding versions for double) which you can use if you don't want to worry about tolerable errors yourself.
Docs: https://github.com/google/googletest/blob/master/docs/reference/assertions.md#floating-point-comparison-floating-point
Related
I have always had the problem of comparing double values for equality. There are functions around like some fuzzy_compare(double a, double b), but I often enough did not manage to find them in time. So I thought on building a wrapper class for double just for the comparison operator:
typedef union {
uint64_t i;
double d;
} number64;
bool Double::operator==(const double value) const {
number64 a, b;
a.d = this->value;
b.d = value;
if ((a.i & 0x8000000000000000) != (b.i & 0x8000000000000000)) {
if ((a.i & 0x7FFFFFFFFFFFFFFF) == 0 && (b.i & 0x7FFFFFFFFFFFFFFF) == 0)
return true;
return false;
}
if ((a.i & 0x7FF0000000000000) != (b.i & 0x7FF0000000000000))
return false;
uint64_t diff = (a.i & 0x000FFFFFFFFFFFF) - (b.i & 0x000FFFFFFFFFFFF) & 0x000FFFFFFFFFFFF;
return diff < 2; // 2 here is kind of some epsilon, but integer and independent of value range
}
The idea behind it is:
First, compare the sign. If it's different, the numbers are different. Except if all other bits are zero. That is comparing +0.0 with -0.0, which should be equal. Next, compare the exponent. If these are different, the numbers are different. Last, compare the mantissa. If the difference is low enough, the values are equal.
It seems to work, but just to be sure, I'd like a peer review. It could well be that I overlooked something.
And yes, this wrapper class needs all the operator overloading stuff. I skipped that because they're all trivial. The equality operator is the main purpose of this wrapper class.
This code has several problems:
Small values on different sides of zero always compare unequal, no matter how (not) far apart.
More importantly, -0.0 compares unequal with +epsilon but +0.0 compares equal with +epsilon (for some epsilon). That's really bad.
What about NaNs?
Values with different exponents compare unequal, even if one floating point "step" apart (e.g. the double before 1 compares unequal to 1, but the one after 1 compares equal...).
The last point could ironically be fixed by not distinguishing between exponent and mantissa: The binary representations of all positive floats are exactly in the order of their magnitude!
It appears that you want to just check whether two floats are a certain number of "steps" apart. If so, maybe this boost function might help. But I would also question whether that's actually reasonable:
Should the smallest positive non-denormal compare equal to zero? There are still many (denormal) floats between them. I doubt this is what you want.
If you operate on values that are expected to be of magnitude 1e16, then 1 should compare equal to 0, even though half of all positive doubles are between 0 and 1.
It is usually most practical to use a relative + absolute epsilon. But I think it will be most worthwhile to check out this article, which discusses the topic of comparing floats more extensively than I could fit into this answer:
https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/
To cite its conclusion:
Know what you’re doing
There is no silver bullet. You have to choose wisely.
If you are comparing against zero, then relative epsilons and ULPs based comparisons are usually meaningless. You’ll need to use an absolute epsilon, whose value might be some small multiple of FLT_EPSILON and the inputs to your calculation. Maybe.
If you are comparing against a non-zero number then relative epsilons or ULPs based comparisons are probably what you want. You’ll probably want some small multiple of FLT_EPSILON for your relative epsilon, or some small number of ULPs. An absolute epsilon could be used if you knew exactly what number you were comparing against.
If you are comparing two arbitrary numbers that could be zero or non-zero then you need the kitchen sink. Good luck and God speed.
Above all you need to understand what you are calculating, how stable the algorithms are, and what you should do if the error is larger than expected. Floating-point math can be stunningly accurate but you also need to understand what it is that you are actually calculating.
You store into one union member and then read from another. That causes aliasing problem (undefined behaviour) because the C++ language requires that objects of different types do not alias.
There are a few ways to remove the undefined behaviour:
Get rid of the union and just memcpy the double into uint64_t. The portable way.
Mark union member i type with [[gnu::may_alias]].
Insert a compiler memory barrier between storing into union member d and reading from member i.
Frame the question this way:
We have two numbers, a and b, that have been computed with floating-point arithmetic.
If they had been computed exactly with real-number mathematics, we would have a and b.
We want to compare a and b and get an answer that tells us whether a equals b.
In other words, you are trying to correct for errors that occurred while computing a and b. In general, that is impossible, of course, because we do not know what a and b are. We only have the approximations a and b.
The code you propose falls back to another strategy:
If a and b are close to each other, we will accept that a equals b. (In other words: If a is close to b, it is possible that a equals b, and the differences we have are only because of calculation errors, so we will accept that a equals b without further evidence.)
There are two problems with this strategy:
This strategy will incorrectly accept that a equals b even when it is not true, just because a and b are close.
We need to decide how close to require a and b to be.
Your code attempts to address the latter: It is establishing some tests about whether a and b are close enough. As others have pointed out, it is severely flawed:
It treats numbers as different if they have different signs, but floating-point arithmetic can cause a to be negative even if a is positive, and vice versa.
It treats numbers as different if they have different exponents, but floating-point arithmetic can cause a to have a different exponent from a.
It treats numbers as different if they differ by more than a fixed number of ULP (units of least precision), but floating-point arithmetic can, in general, cause a to differ from a by any amount.
It assumes an IEEE-754 format and needlessly uses aliasing with behavior not defined by the C++ standard.
The approach is fundamentally flawed because it needlessly fiddles with the floating-point representation. The actual way to determine from a and b whether a and b might be equal is to figure out, given a and b, what sets of values a and b have and whether there is any value in common in those sets.
In other words, given a, the value of a might be in some interval, (a−eal, a+ear) (that is, all the numbers from a minus some error on the left to a plus some error on the right), and, given b, the value of b might be in some interval, (b−ebl, b+ebr). If so, what you want to test is not some floating-point representation properties but whether the two intervals (a−eal, a+ear) and (b−ebl, b+ebr) overlap.
To do that, you need to know, or at least have bounds on, the errors eal, ear, ebl, and ebr. But those errors are not fixed by the floating-point format. They are not 2 ULP or 1 ULP or any number of ULP scaled by the exponent. They depend on how a and b were computed. In general, the errors can range from 0 to infinity, and they can also be NaN.
So, to test whether a and b might be equal, you need to analyze the floating-point arithmetic errors that could have occurred. In general, this is difficult. There is an entire field of mathematics for it, numerical analysis.
If you have computed bounds on the errors, then you can just compare the intervals using ordinary arithmetic. There is no need to take apart the floating-point representation and work with the bits. Just use the normal add, subtract, and comparison operations.
(The problem is actually more complicated than I allowed above. Given a computed value a, the potential values of a do not always lie in a single interval. They could be an arbitrary set of points.)
As I have written previously, there is no general solution for comparing numbers containing arithmetic errors: 0 1 2 3.
Once you figure out error bounds and write a test that returns true if a and b might be equal, you still have the problem that the test also accepts false negatives: It will return true even in cases where a and b are not equal. In other words, you have just replaced a program that is wrong because it rejects equality even though a and b would be equal with a program that is wrong in other cases because it accepts equality in cases where a and b are not equal. This is another reason there is no general solution: In some applications, accepting as equal numbers that are not equal is okay, at least for some situations. In other applications, that is not okay, and using a test like this will break the program.
I am porting some program from Matlab to C++ for efficiency. It is important for the output of both programs to be exactly the same (**).
I am facing different results for this operation:
std::sin(0.497418836818383950) = 0.477158760259608410 (C++)
sin(0.497418836818383950) = 0.47715876025960846000 (Matlab)
N[Sin[0.497418836818383950], 20] = 0.477158760259608433 (Mathematica)
So, as far as I know both C++ and Matlab are using IEEE754 defined double arithmetic. I think I have read somewhere that IEEE754 allows differents results in the last bit. Using mathematica to decide, seems like C++ is more close to the result. How can I force Matlab to compute the sin with precision to the last bit included, so that the results are the same?
In my program this behaviour leads to big errors because the numerical differential equation solver keeps increasing this error in the last bit. However I am not sure that C++ ported version is correct. I am guessing that even if the IEEE754 allows the last bit to be different, somehow guarantees that this error does not get bigger when using the result in more IEEE754 defined double operations (because otherwise, two different programs correct according to the IEEE754 standard could produce completely different outputs). So the other question is Am I right about this?
I would like get an answer to both bolded questions. Edit: The first question is being quite controversial, but is the less important, can someone comment about the second one?
Note: This is not an error in the printing, just in case you want to check, this is how I obtained these results:
http://i.imgur.com/cy5ToYy.png
Note (**): What I mean by this is that the final output, which are the results of some calculations showing some real numbers with 4 decimal places, need to be exactly the same. The error I talk about in the question gets bigger (because of more operations, each of one is different in Matlab and in C++) so the final differences are huge) (If you are curious enough to see how the difference start getting bigger, here is the full output [link soon], but this has nothing to do with the question)
Firstly, if your numerical method depends on the accuracy of sin to the last bit, then you probably need to use an arbitrary precision library, such as MPFR.
The IEEE754 2008 standard doesn't require that the functions be correctly rounded (it does "recommend" it though). Some C libms do provide correctly rounded trigonometric functions: I believe that the glibc libm does (typically used on most linux distributions), as does CRlibm. Most other modern libms will provide trig functions that are within 1 ulp (i.e. one of the two floating point values either side of the true value), often termed faithfully rounded, which is much quicker to compute.
None of those values you printed could actually arise as IEEE 64bit floating point values (even if rounded): the 3 nearest (printed to full precision) are:
0.477158760259608 405451814405751065351068973541259765625
0.477158760259608 46096296563700889237225055694580078125
0.477158760259608 516474116868266719393432140350341796875
The possible values you could want are:
The exact sin of the decimal .497418836818383950, which is
0.477158760259608 433132061388630377105954125778369485736356219...
(this appears to be what Mathematica gives).
The exact sin of the 64-bit float nearest .497418836818383950:
0.477158760259608 430531153841011107415427334794384396325832953...
In both cases, the first of the above list is the nearest (though only barely in the case of 1).
The sine of the double constant you wrote is about 0x1.e89c4e59427b173a8753edbcb95p-2, whose nearest double is 0x1.e89c4e59427b1p-2. To 20 decimal places, the two closest doubles are 0.47715876025960840545 and 0.47715876025960846096.
Perhaps Matlab is displaying a truncated value? (EDIT: I now see that the fourth-last digit is a 6, not a 0. Matlab is giving you a result that's still faithfully-rounded, but it's the farther of the two closest doubles to the desired result. And it's still printing out the wrong number.
I should also point out that Mathematica is probably trying to solve a different problem---compute the sine of the decimal number 0.497418836818383950 to 20 decimal places. You should not expect this to match either the C++ code's result or Matlab's result.
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.
Is there a general best practice strategy for dealing with floating point inaccuracy?
The project that I'm working on tried to solve them by wrapping everything in a Unit class which holds the floating point value and overloads the operators. Numbers are considered equal if they "close enough," comparisons like > or < are done by comparing with a slightly lower or higher value.
I understand the desire to encapsulate the logic of handling such floating point errors. But given that this project has had two different implementations (one based on the ratio of the numbers being compared and one based on the absolute difference) and I've been asked to look at the code because its not doing the right, the strategy seems to be a bad one.
So what is best the strategy for try to make sure you handle all of the floating point inaccuracy in a program?
You want to keep data as dumb as possible, generally. Behavior and the data are two concerns that should be kept separate.
The best way is to not have unit classes at all, in my opinion. If you have to have them, then avoid overloading operators unless it has to work one way all the time. Usually it doesn't, even if you think it does. As mentioned in the comments, it breaks strict weak ordering for instance.
I believe the sane way to handle it is to create some concrete comparators that aren't tied to anything else.
struct RatioCompare {
bool operator()(float lhs, float rhs) const;
};
struct EpsilonCompare {
bool operator()(float lhs, float rhs) const;
};
People writing algorithms can then use these in their containers or algorithms. This allows code reuse without demanding that anyone uses a specific strategy.
std::sort(prices.begin(), prices.end(), EpsilonCompare());
std::sort(prices.begin(), prices.end(), RatioCompare());
Usually people trying to overload operators to avoid these things will offer complaints about "good defaults", etc. If the compiler tells you immediately that there isn't a default, it's easy to fix. If a customer tells you that something isn't right somewhere in your million lines of price calculations, that is a little harder to track down. This can be especially dangerous if someone changed the default behavior at some point.
Check comparing floating point numbers and this post on deniweb and this on SO.
Both techniques are not good. See this article.
Google Test is a framework for writing C++ tests on a variety of platforms.
gtest.h contains the AlmostEquals function.
// Returns true iff this number is at most kMaxUlps ULP's away from
// rhs. In particular, this function:
//
// - returns false if either number is (or both are) NAN.
// - treats really large numbers as almost equal to infinity.
// - thinks +0.0 and -0.0 are 0 DLP's apart.
bool AlmostEquals(const FloatingPoint& rhs) const {
// The IEEE standard says that any comparison operation involving
// a NAN must return false.
if (is_nan() || rhs.is_nan()) return false;
return DistanceBetweenSignAndMagnitudeNumbers(u_.bits_, rhs.u_.bits_)
<= kMaxUlps;
}
Google implementation is good, fast and platform-independent.
A small documentation is here.
To me floating point errors are essentially those which on an x86 would lead to a floating point exception (assuming the coprocessor has that interrupt enabled). A special case is the "inexact" exception i e when the result was not exactly representable in the floating point format (such as when dividing 1 by 3). Newbies not yet at home in the floating-point world will expect exact results and will consider this case an error.
As I see it there are several strategies available.
Early data checking such that bad values are identified and handled
when they enter the software. This lessens the need for testing
during the floating operations themselves which should improve
performance.
Late data checking such that bad values are identified
immediately before they are used in actual floating point operations.
Should lead to lower performance.
Debugging with floating point
exception interrupts enabled. This is probably the fastest way to
gain a deeper understanding of floating point issues during the
development process.
to name just a few.
When I wrote a proprietary database engine over twenty years ago using an 80286 with an 80287 coprocessor I chose a form of late data checking and using x87 primitive operations. Since floating point operations were relatively slow I wanted to avoid doing floating point comparisons every time I loaded a value (some of which would cause exceptions). To achieve this my floating point (double precision) values were unions with unsigned integers such that I would test the floating point values using x86 operations before the x87 operations would be called upon. This was cumbersome but the integer operations were fast and when the floating point operations came into action the floating point value in question would be ready in the cache.
A typical C sequence (floating point division of two matrices) looked something like this:
// calculate source and destination pointers
type1=npx_load(src1pointer);
if (type1!=UNKNOWN) /* x87 stack contains negative, zero or positive value */
{
type2=npx_load(src2pointer);
if (!(type2==POSITIVE_NOT_0 || type2==NEGATIVE))
{
if (type2==ZERO) npx_pop();
npx_pop(); /* remove src1 value from stack since there won't be a division */
type1=UNKNOWN;
}
else npx_divide();
}
if (type1==UNKNOWN) npx_load_0(); /* x86 stack is empty so load zero */
npx_store(dstpointer); /* store either zero (from prev statement) or quotient as result */
npx_load would load value onto the top of the x87 stack providing it was valid. Otherwise the top of the stack would be empty. npx_pop simply removes the value currently at the top of the x87. BTW "npx" is an abbreviation for "Numeric Processor eXtenstion" as it was sometimes called.
The method chosen was my way of handling floating-point issues stemming from my own frustrating experiences at trying to get the coprocessor solution to behave in a predictable manner in an application.
For sure this solution led to overhead but a pure
*dstpointer = *src1pointer / *src2pointer;
was out of the question since it didn't contain any error handling. The extra cost of this error handling was more than made up for by how the pointers to the values were prepared. Also, the 99% case (both values valid) is quite fast so if the extra handling for the other cases is slower, so what?
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.