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.
Related
I am aware, that to compare two floating point values one needs to use some epsilon precision, as they are not exact. However, I wonder if there are edge cases, where I don't need that epsilon.
In particular, I would like to know if it is always safe to do something like this:
double foo(double x){
if (x < 0.0) return 0.0;
else return somethingelse(x); // somethingelse(x) != 0.0
}
int main(){
int x = -3.0;
if (foo(x) == 0.0) {
std::cout << "^- is this comparison ok?" << std::endl;
}
}
I know that there are better ways to write foo (e.g. returning a flag in addition), but I wonder if in general is it ok to assign 0.0 to a floating point variable and later compare it to 0.0.
Or more general, does the following comparison yield true always?
double x = 3.3;
double y = 3.3;
if (x == y) { std::cout << "is an epsilon required here?" << std::endl; }
When I tried it, it seems to work, but it might be that one should not rely on that.
Yes, in this example it is perfectly fine to check for == 0.0. This is not because 0.0 is special in any way, but because you only assign a value and compare it afterwards. You could also set it to 3.3 and compare for == 3.3, this would be fine too. You're storing a bit pattern, and comparing for that exact same bit pattern, as long as the values are not promoted to another type for doing the comparison.
However, calculation results that would mathematically equal zero would not always equal 0.0.
This Q/A has evolved to also include cases where different parts of the program are compiled by different compilers. The question does not mention this, my answer applies only when the same compiler is used for all relevant parts.
C++ 11 Standard,
§5.10 Equality operators
6 If both operands are of arithmetic or enumeration type, the usual
arithmetic conversions are performed on both operands; each of the
operators shall yield true if the specified relationship is true and
false if it is false.
The relationship is not defined further, so we have to use the common meaning of "equal".
§2.13.4 Floating literals
1 [...] If the scaled value is in the range of representable values
for its type, the result is the scaled value if representable, else
the larger or smaller representable value nearest the scaled value,
chosen in an implementation-defined manner. [...]
The compiler has to choose between exactly two values when converting a literal, when the value is not representable. If the same value is chosen for the same literal consistently, you are safe to compare values such as 3.3, because == means "equal".
Yes, if you return 0.0 you can compare it to 0.0; 0 is representable exactly as a floating-point value. If you return 3.3 you have to be a much more careful, since 3.3 is not exactly representable, so a conversion from double to float, for example, will produce a different value.
correction: 0 as a floating point value is not unique, but IEEE 754 defines the comparison 0.0==-0.0 to be true (any zero for that matter).
So with 0.0 this works - for every other number it does not. The literal 3.3 in one compilation unit (e.g. a library) and another (e.g. your application) might differ. The standard only requires the compiler to use the same rounding it would use at runtime - but different compilers / compiler settings might use different rounding.
It will work most of the time (for 0), but is very bad practice.
As long as you are using the same compiler with the same settings (e.g. one compilation unit) it will work because the literal 0.0 or 0.0f will translate to the same bit pattern every time. The representation of zero is not unique though. So if foo is declared in a library and your call to it in some application the same function might fail.
You can rescue this very case by using std::fpclassify to check whether the returned value represents a zero. For every finite (non-zero) value you will have to use an epsilon-comparison though unless you stay within one compilation unit and perform no operations on the values.
As written in both cases you are using identical constants in the same file fed to the same compiler. The string to float conversion the compiler uses should return the same bit pattern so these should not only be equal as in a plus or minus cases for zero thing but equal bit by bit.
Were you to have a constant which uses the operating systems C library to generate the bit pattern then have a string to f or something that can possibly use a different C library if the binary is transported to another computer than the one compiled on. You might have a problem.
Certainly if you compute 3.3 for one of the terms, runtime, and have the other 3.3 computed compile time again you can and will get failures on the equal comparisons. Some constants obviously are more likely to work than others.
Of course as written your 3.3 comparison is dead code and the compiler just removes it if optimizations are enabled.
You didnt specify the floating point format nor standard if any for that format you were interested in. Some formats have the +/- zero problem, some dont for example.
It is a common misconception that floating point values are "not exact". In fact each of them is perfectly exact (except, may be, some special cases as -0.0 or Inf) and equal to s·2e – (p – 1), where s, e, and p are significand, exponent, and precision correspondingly, each of them integer. E.g. in IEEE 754-2008 binary32 format (aka float32) p = 24 and 1 is represented as 0x800000·20 – 23. There are two things that are really not exact when you deal with floating point values:
Representation of a real value using a FP one. Obviously, not all real numbers can be represented using a given FP format, so they have to be somehow rounded. There are several rounding modes, but the most commonly used is the "Round to nearest, ties to even". If you always use the same rounding mode, which is almost certainly the case, the same real value is always represented with the same FP one. So you can be sure that if two real values are equal, their FP counterparts are exactly equal too (but not the reverse, obviously).
Operations with FP numbers are (mostly) inexact. So if you have some real-value function φ(ξ) implemented in the computer as a function of a FP argument f(x), and you want to compare its result with some "true" value y, you need to use some ε in comparison, because it is very hard (sometimes even impossible) to white a function giving exactly y. And the value of ε strongly depends on the nature of the FP operations involved, so in each particular case there may be different optimal value.
For more details see D. Goldberg. What Every Computer Scientist Should Know About Floating-Point Arithmetic, and J.-M. Muller et al. Handbook of Floating-Point Arithmetic. Both texts you can find in the Internet.
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
When comparing doubles for equality, we need to give a tolerance level, because floating-point computation might introduce errors. For example:
double x;
double y;
x = f();
y = g();
if (fabs(x-y)<epsilon) {
// they are equal!
} else {
// they are not!
}
However, if I simply assign a constant value, without any computation, do I still need to check the epsilon?
double x = 1;
double y = 1;
if (x==y) {
// they are equal!
} else {
// no they are not!
}
Is == comparison good enough? Or I need to do fabs(x-y)<epsilon again? Is it possible to introduce error in assigning? Am I too paranoid?
How about casting (double x = static_cast<double>(100))? Is that gonna introduce floating-point error as well?
I am using C++ on Linux, but if it differs by language, I would like to understand that as well.
Actually, it depends on the value and the implementation. The C++ standard (draft n3126) has this to say in 2.14.4 Floating literals:
If the scaled value is in the range of representable values for its type, the result is the scaled value if representable, else the larger or smaller representable value nearest the scaled value, chosen in an implementation-defined manner.
In other words, if the value is exactly representable (and 1 is, in IEEE754, as is 100 in your static cast), you get the value. Otherwise (such as with 0.1) you get an implementation-defined close match (a). Now I'd be very worried about an implementation that chose a different close match based on the same input token but it is possible.
(a) Actually, that paragraph can be read in two ways, either the implementation is free to choose either the closest higher or closest lower value regardless of which is actually the closest, or it must choose the closest to the desired value.
If the latter, it doesn't change this answer however since all you have to do is hardcode a floating point value exactly at the midpoint of two representable types and the implementation is once again free to choose either.
For example, it might alternate between the next higher and next lower for the same reason banker's rounding is applied - to reduce the cumulative errors.
No if you assign literals they should be the same :)
Also if you start with the same value and do the same operations, they should be the same.
Floating point values are non-exact, but the operations should produce consistent results :)
Both cases are ultimately subject to implementation defined representations.
Storage of floating point values and their representations take on may forms - load by address or constant? optimized out by fast math? what is the register width? is it stored in an SSE register? Many variations exist.
If you need precise behavior and portability, do not rely on this implementation defined behavior.
IEEE-754, which is a standard common implementations of floating point numbers abide to, requires floating-point operations to produce a result that is the nearest representable value to an infinitely-precise result. Thus the only imprecision that you will face is rounding after each operation you perform, as well as propagation of rounding errors from the operations performed earlier in the chain. Floats are not per se inexact. And by the way, epsilon can and should be computed, you can consult any numerics book on that.
Floating point numbers can represent integers precisely up to the length of their mantissa. So for example if you cast from an int to a double, it will always be exact, but for casting into into a float, it will no longer be exact for very large integers.
There is one major example of extensive usage of floating point numbers as a substitute for integers, it's the LUA scripting language, which has no integer built-in type, and floating-point numbers are used extensively for logic and flow control etc. The performance and storage penalty from using floating-point numbers turns out to be smaller than the penalty of resolving multiple types at run time and makes the implementation lighter. LUA has been extensively used not only on PC, but also on game consoles.
Now, many compilers have an optional switch that disables IEEE-754 compatibility. Then compromises are made. Denormalized numbers (very very small numbers where the exponent has reached smallest possible value) are often treated as zero, and approximations in implementation of power, logarithm, sqrt, and 1/(x^2) can be made, but addition/subtraction, comparison and multiplication should retain their properties for numbers which can be exactly represented.
The easy answer: For constants == is ok.
There are two exceptions which you should be aware of:
First exception:
0.0 == -0.0
There is a negative zero which compares equal for the IEEE 754 standard. This means
1/INFINITY == 1/-INFINITY which breaks f(x) == f(y) => x == y
Second exception:
NaN != NaN
This is a special caveat of NotaNumber which allows to find out if a number is a NaN
on systems which do not have a test function available (Yes, that happens).
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.
I have some math (in C++) which seems to be generating some very small, near zero, numbers (I suspect the trig function calls are my real problem), but I'd like to detect these cases so that I can study them in more detail.
I'm currently trying out the following, is it correct?
if ( std::abs(x) < DBL_MIN ) {
log_debug("detected small num, %Le, %Le", x, y);
}
Second, the nature of the mathematics is trigonometric in nature (aka using a lot of radian/degree conversions and sin/cos/tan calls, etc), what sort of transformations can I do to avoid mathematical errors?
Obviously for multiplications I can use a log transform - what else?
Contrary to widespread belief, DBL_MIN is not the smallest positive double value but the smallest positive normalized double value. Typically - for 64-bit ieee754 doubles - it's 2-1022, while the smallest positive double value is 2-1074. Therefore
I'm currently trying out the following, is it correct?
if ( std::abs(x) < DBL_MIN ) {
log_debug("detected small num, %Le, %Le", x, y);
}
may have an affirmative answer. The condition checks whether x is a denormalized (also called subnormal) number or ±0.0. Without knowing more about your specific situation, I cannot tell if that test is appropriate. Denormalized numbers can be legitimate results of calculations or the consequence of rounding where the correct result would be 0. It is also possible that rounding produces numbers of far greater magnitude than DBL_MIN when the mathematically correct result would be 0, so a much larger threshold could be sensible.
If x is a double, then one problem with this approach is that you can't distinguish between x being legitimately zero, and x being a positive value smaller than DBL_MIN. So this will work if you know x can never be legitimately zero, and you want to see when underflow occurs.
You could also try catching the SIGFPE signal, which will fire on a POSIX-compliant system any time there's a math error including floating-point underflow. See: http://en.wikipedia.org/wiki/SIGFPE
EDIT: To be clear, DBL_MIN is NOT the largest negative value that a double can hold, it is the smallest positive normalized value that a double can hold. So your approach is fine as long as the value can't be zero.
Another useful constant is DBL_EPSILON which is the smallest double value that can be added to 1.0 without getting 1.0 back. Note that this is a much larger value than DBL_MIN. But it may be useful to you since you're doing trigonometric functions that may tend toward 1 instead of tending toward 0.
Since you are using C++, the most idiomatic is to use std::numeric_limits from header <limits>.
For instance:
template <typename T>
bool is_close_to_zero(T x)
{
return std::abs(x) < std::numeric_limits<T>::epsilon();
}
The actual tolerance to be used heavily depends on your problem. Please complete your question with a concrete use case so that I can enhance my answer.
There is also std::numeric_limits<T>::min() and std::numeric_limits<T>::denorm_min() that may be useful. The first one is the smallest positive non-denormalized value of type T (equal to FLT/DBL/LDBL_MIN from <cfloat>), the second one is the smallest positive value of type T (no <cfloat> equivalent).
[You may find this document useful to read if you aren't at ease with floating point numbers representation.]
The first if check will actually only be true when your value is zero.
For your second question, you imply lots of conversions. Instead, pick one unit (deg or rad) and do all your computational operations in that unit. Then at the very end do a single conversion to the other value if you need to.