I am having some trouble with IEEE floating point rules preventing compiler optimizations that seem obvious. For example,
char foo(float x) {
if (x == x)
return 1;
else
return 0;
}
cannot be optimized to just return 1 because NaN == NaN is false. Okay, fine, I guess.
However, I want to write such that the optimizer can actually fix stuff up for me. Are there mathematical identities that hold for all floats? For example, I would be willing to write !(x - x) if it meant the compiler could assume that it held all the time (though that also isn't the case).
I see some reference to such identities on the web, for example here, but I haven't found any organized information, including in a light scan of the IEEE 754 standard.
It'd also be fine if I could get the optimizer to assume isnormal(x) without generating additional code (in gcc or clang).
Clearly I'm not actually going to write (x == x) in my source code, but I have a function that's designed for inlining. The function may be declared as foo(float x, float y), but often x is 0, or y is 0, or x and y are both z, etc. The floats represent onscreen geometric coordinates. These are all cases where if I were coding by hand without use of the function I'd never distinguish between 0 and (x - x), I'd just hand-optimize stupid stuff away. So, I really don't care about the IEEE rules in what the compiler does after inlining my function, and I'd just as soon have the compiler ignore them. Rounding differences are also not very important since we're basically doing onscreen drawing.
I don't think -ffast-math is an option for me, because the function appears in a header file, and it is not appropriate that the .c files that use the function compile with -ffast-math.
Another reference that might be of some use for you is a really nice article on floating-point optimization in Game Programming Gems volume 2, by Yossarian King. You can read the article here. It discusses the IEEE format in quite detail, taking into account implementations and architecture, and provides many optimization tricks.
I think that you are always going to struggle to make computer floating-point-number arithmetic behave like mathematical real-number arithmetic, and suggest that you don't for any reason. I suggest that you are making a type error trying to compare the equality of 2 fp numbers. Since fp numbers are, in the overwhelming majority, approximations, you should accept this and use approximate-equality as your test.
Computer integers exist for equality testing of numerical values.
Well, that's what I think, you go ahead and fight the machine (well, all the machines actually) if you wish.
Now, to answer some parts of your question:
-- for every mathematical identity you are familiar with from real-number arithmetic, there are counter examples in the domain of floating-point numbers, whether IEEE or otherwise;
-- 'clever' programming almost always makes it more difficult for a compiler to optimise code than straightforward programming;
-- it seems that you are doing some graphics programming: in the end the coordinates of points in your conceptual space are going to be mapped to pixels on a screen; pixels always have integer coordinates; your translation from conceptual space to screen space defines your approximate-equality function
Regards
Mark
If you can assume that floating-point numbers used in this module will not be Inf/NaN, you can compile it with -ffinite-math-only (in GCC). This may "improve" the codegen for examples like the one you posted.
You could compare for bitwise equality. Although you might get bitten for some values that are equivalent but bitwise different, it will catch all those cases where you have a true equality as you mentioned. And I am not sure the compiler will recognize what you do and remove it when inlining (which I believe is what you are after), but that can easily be checked.
What happened when you tried it the obvious way and profiled it? or examined the generated asm?
If the function is inlined with values known at the call site, the optimizer has this information available. For example: foo(0, y).
You may be surprised at the work you don't have to do, but at the very least profiling or looking at what the compiler actually does with the code will give you more information and help you figure out where to proceed next.
That said, if you know certain things that the optimizer can't figure out itself, you can write multiple versions of the function, and specify the one you want to call. This is something of a hassle, but at least with inline functions they will all be specified together in one header. It's also quite a bit easier than the next step, which is using inline asm to do exactly what you want.
Related
Out of curiosity I know pow() returns double
Is there any particular reason why math.h has this API with double as argument and return value.
If I have a case where I need an int as the return value I would write my own API as shown below. Why does math.h doesn't have an API something like this and have double pow(double,double);
or Is there one for this? If the answer is too obvious let me know I will delete the question. I couldn't figure out the reason behind this.
int i = pow1(2,4);
int pow1(int i,int j)
{
int k = 1;
while(j--)
{
k = k*i;
}
return k;
}
I would consider the following arguments:
The C standard library is already quite large, and every extra function has a cost; providing a single pow function which can (at least in principle) treat both floating-points and integers means less functions, which is a good thing;
The int version is sufficiently simple (modulo overflows and exceptional cases) that someone may code their own if necessary, so it is not felt as an absolute necessity (there are several other similar functions which might also have been provided in the standard library, but probably for similar reasons have not been included);
How to deal with overflow? IEEE-754 already specifies how to deal with all those nasty exceptional cases for floating-points, but for integers you would have to choose yourself: use errno? Limit the input argument? Provide no guarantees about overflow whatsoever? Should there be a version for each integer type? Or simply one that returns long? If so, then surely people would complain about the lack of an int version, and a short version as well...
One of the arguably most useful cases of pow for integers is when the base is 2, in which case shifts might be more efficient. Trying to provide an efficient function that detects such cases (otherwise, people would start arguing whether one should use shifts or call ipow, and then more discussion would ensue...), and also when the base is 4, or 8, or 1, or 0, etc., becomes a real issue for the library developer.
In the end, this hypothetical ipow function would either have an extremely verbose and over-specified description (mentioning performance aspects), or it would lack visibility: should a developer reliably expect their ipow function to be optimized for special cases? The user might end up coding their own version to have a better understanding of its behavior, and since this is a relatively simple function, different implementations would abound.
All in all, from the point of view of the user, int pow() seems lacking, but from the point of view of the library developers and maintainers, it is yet another burden that they might not be willing to undertake, for a benefit which is arguably quite limited.
It is even possible that early versions of the library considered including it, then after some discussion realized that it had several shortcomings, and decided to remove it from the API.
Anol's answer is very on-point and I believe is the most correct assessment as to why int pow() is not a part of the standard library, however I would like to amend it with another possibility.
The most useful aspect of the pow function is the ability to correctly (at least as far as floating point arithmetic is concerned) perform exponentiation to non-integer powers. This type of behavior is non-trivial to implement using integer math, let alone floating point math. Rather than asking library implementors to write both an integer exponentiation routine and a floating point exponentiation routine, they decided to ask for the more useful of the two. It also helps that the x87 instruction set, as well as many other FPU instruction sets, provides built-in floating point exponentiation instructions to make the implementation on the software side trivial.
C also doesn't have any notion of exceptions, and no language-level global state that could expose things like CPU flags. This would make overflow detection in an int pow() implementation difficult to provide. In the case of double pow(), it can just return a non-signalling NaN or Infinity in the case of exceptional circumstances. However there is no notion of NaN or Infinity in the integer world.
In the case of an exception, the program could do one of the following:
Overflow Silently
Return INT_MIN or INT_MAX
Change an integer whose pointer was provided to the function by the caller
Change some global state like errno
Emit a SIGFPE
Provide a sister function int pow_overflow_check() that can be used to boilerplate the call
None of these are desirable. The first two options would lead to hard-to-diagnose logic errors in possibly unrelated areas of a program. The third one would provide an annoying and cumbersome barrier to the use of the function. The fourth one would ruin any sort of reentrancy guarantee. The fifth option would make the entire program grind to a halt. The sixth option would work well, but would also bloat the standard more, and make the function more cumbersome to use like in the third option.
Further, just like others have stated, exponentiation to an integer power, especially if the base is 2, is trivial to implement. A first year compsci student with a basic understanding of math should be able to formulate a correct implementation that could suit most use cases.
The original C language was aimed squarely at assembly programmers, and this is one of the symptoms of that. A very common theme in the language spec is that the language library should only provide things that are either impossible to implement in-language in a portable manner, or extremely non-trivial to implement purely in-language, especially when there are commonly available CPU/FPU instructions that can do it for you.
It's probably a decision to keep old standards simple. Although newer standards have included similar functions, there was never a function of your liking implemented using simple int argument or return values with the intent of preventing overflow issues as other answers here have presented. Quoting http://www.cplusplus.com/reference/cmath/pow/
C99
double pow (double base , double exponent);
float powf (float base , float exponent);
long double powl (long double base, long double exponent);
Quoting
Why isn't int pow(int base, int exponent) in the standard C++ libraries?
For example the C99 rationale document specifically carries forward
two of the C89 guiding principles which limit what can be added:
Keep the language small and simple.
Provide only one way to do an operation.
Credits to the original author of the best answer in that topic. That topic might answer your question in its entirety, I believe.
I wrote an Ansi C compiler for a friend's custom 16-bit stack-based CPU several years ago but I never got around to implementing all the data types. Now I would like to finish the job so I'm wondering if there are any math libraries out there that I can use to fill the gaps. I can handle 16-bit integer data types since they are native to the CPU and therefore I have all the math routines (ie. +, -, *, /, %) done for them. However, since his CPU does not handle floating point then I have to implement floats/doubles myself. I also have to implement the 8-bit and 32-bit data types (bother integer and floats/doubles). I'm pretty sure this has been done and redone many times and since I'm not particularly looking forward to recreating the wheel I would appreciate it if someone would point me at a library that can help me out.
Now I was looking at GMP but it seems to be overkill (library must be absolutely huge, not sure my custom compiler would be able to handle it) and it takes numbers in the form of strings which would be wasteful for obvious reasons. For example :
mpz_set_str(x, "7612058254738945", 10);
mpz_set_str(y, "9263591128439081", 10);
mpz_mul(result, x, y);
This seems simple enough, I like the api... but I would rather pass in an array rather than a string. For example, if I wanted to multiply two 32-bit longs together I would like to be able to pass it two arrays of size two where each array contains two 16-bit values that actually represent a 32-bit long and have the library place the output into an output array. If I needed floating point then I should be able to specify the precision as well.
This may seem like asking for too much but I'm asking in the hopes that someone has seen something like this.
Many thanks in advance!
Let's divide the answer.
8-bit arithmetic
This one is very easy. In fact, C already talks about this under the term "integer promotion". This means that if you have 8-bit data and you want to do an operation on them, you simply pad them with zero (or one if signed and negative) to make them 16-bit. Then you proceed with the normal 16-bit operation.
32-bit arithmetic
Note: so long as the standard is concerned, you don't really need to have 32-bit integers.
This could be a bit tricky, but it is still not worth using a library for. For each operation, you would need to take a look at how you learned to do them in elementary school in base 10, and then do the same in base 216 for 2 digit numbers (each digit being one 16-bit integer). Once you understand the analogy with simple base 10 math (and hence the algorithms), you would need to implement them in assembly of your CPU.
This basically means loading the most significant 16 bit on one register, and the least significant in another register. Then follow the algorithm for each operation and perform it. You would most likely need to get help from overflow and other flags.
Floating point arithmetic
Note: so long as the standard is concerned, you don't really need to conform to IEEE 754.
There are various libraries already written for software emulated floating points. You may find this gcc wiki page interesting:
GNU libc has a third implementation, soft-fp. (Variants of this are also used for Linux kernel math emulation on some targets.) soft-fp is used in glibc on PowerPC --without-fp to provide the same soft-float functions as in libgcc. It is also used on Alpha, SPARC and PowerPC to provide some ABI-specified floating-point functions (which in turn may get used by GCC); on PowerPC these are IEEE quad functions, not IBM long double ones.
Performance measurements with EEMBC indicate that soft-fp (as speeded up somewhat using ideas from ieeelib) is about 10-15% faster than fp-bit and ieeelib about 1% faster than soft-fp, testing on IBM PowerPC 405 and 440. These are geometric mean measurements across EEMBC; some tests are several times faster with soft-fp than with fp-bit if they make heavy use of floating point, while others don't make significant use of floating point. Depending on the particular test, either soft-fp or ieeelib may be faster; for example, soft-fp is somewhat faster on Whetstone.
One answer could be to take a look at the source code for glibc and see if you could salvage what you need.
I write C code that makes certain assumptions about the implementation, such as:
char is 8 bits.
signed integral types are two's complement.
>> on signed integers sign-extends.
integer division rounds negative quotients towards zero.
double is IEEE-754 doubles and can be type-punned to and from uint64_t with the expected result.
comparisons involving NaN always evaluate to false.
a null pointer is all zero bits.
all data pointers have the same representation, and can be converted to size_t and back again without information loss.
pointer arithmetic on char* is the same as ordinary arithmetic on size_t.
functions pointers can be cast to void* and back again without information loss.
Now, all of these are things that the C standard doesn't guarantee, so strictly speaking my code is non-portable. However, they happen to be true on the architectures and ABIs I'm currently targeting, and after careful consideration I've decided that the risk they will fail to hold on some architecture that I'll need to target in the future is acceptably low compared to the pragmatic benefits I derive from making the assumptions now.
The question is: how do I best document this decision? Many of my assumptions are made by practically everyone (non-octet chars? or sign-magnitude integers? on a future, commercially successful, architecture?). Others are more arguable -- the most risky probably being the one about function pointers. But if I just list everything I assume beyond what the standard gives me, the reader's eyes are just going to glaze over, and he may not notice the ones that actually matter.
So, is there some well-known set of assumptions about being a "somewhat orthodox" architecture that I can incorporate by reference, and then only document explicitly where I go beyond even that? (Effectively such a "profile" would define a new language that is a superset of C, but it might not acknowledge that in so many words -- and it may not be a pragmatically useful way to think of it either).
Clarification: I'm looking for a shorthand way to document my choices, not for a way to test automatically whether a given compiler matches my expectations. The latter is obviously useful too, but does not solve everything. For example, if a business partner contacts us saying, "we're making a device based on Google's new G2015 chip; will your software run on it?" -- then it would be nice to be able to answer "we haven't worked with that arch yet, but it shouldn't be a problem if it has a C compiler that satisfies such-and-such".
Clarify even more since somebody has voted to close as "not constructive": I'm not looking for discussion here, just for pointers to actual, existing, formal documents that can simplify my documentation by being incorporated by reference.
I would introduce a STATIC_ASSERT macro and put all your assumptions in such asserts.
Unfortunately, not only is there a lack of standards for a dialect of C that combines the extensions which have emerged as de facto standards during the 1990s (two's-complement, universally-ranked pointers, etc.) but compilers trends are moving in the opposite direction. Given the following requirements for a function:
* Accept int parameters x,y,z:
* Return 0 if x-y is computable as "int" and is less than Z
* Return 1 if x-y is computable as "int" and is not less than Z
* Return 0 or 1 if x-y is not computable */
The vast majority of compilers in the 1990s would have allowed:
int diffCompare(int x, int y, int z)
{ return (x-y) >= z; }
On some platforms, in cases where the difference between x-y was not computable as int, it would be faster to compute a "wrapped" two's-complement value of x-y and compare that, while on others it would be faster to perform the calculation using a type larger than int and compare that. By the late 1990s, however, nearly every C compiler would implement the above code to use one of whichever one of those approaches would have been more efficient on its hardware platform.
Since 2010, however, compiler writers seem to have taken the attitude that if computations overflow, compilers shouldn't perform the calculations in whatever fashion is normal for their platform and let what happens happens, nor should they recognizably trap (which would break some code, but could prevent certain kinds of errant program behavior), but instead they should overflows as an excuse to negate laws of time and causality. Consequently, even if a programmer would have been perfectly happy with any behavior a 1990s compiler would have produced, the programmer must replace the code with something like:
{ return ((long)x-y) >= z; }
which would greatly reduce efficiency on many platforms, or
{ return x+(INT_MAX+1U)-y >= z+(INT_MAX+1U); }
which requires specifying a bunch of calculations the programmer doesn't actually want in the hopes that the optimizer will omit them (using signed comparison to make them unnecessary), and would reduce efficiency on a number of platforms (especially DSPs) where the form using (long) would have been more efficient.
It would be helpful if there were standard profiles which would allow programmers to avoid the need for nasty horrible kludges like the above using INT_MAX+1U, but if trends continue they will become more and more necessary.
Most compiler documentation includes a section that describes the specific behavior of implementation-dependent features. Can you point to that section of the gcc or msvc docs to describe your assumptions?
You can write a header file "document.h" where you collect all your assumptions.
Then, in every file that you know that non-standard assumptions are made, you can #include such a file.
Perhaps "document.h" would not have real sentences at all, but only commented text and some macros.
// [T] DOCUMENT.H
//
#ifndef DOCUMENT_H
#define DOCUMENT_H
// [S] 1. Basic assumptions.
//
// If this file is included in a compilation unit it means that
// the following assumptions are made:
// [1] A char has 8 bits.
// [#]
#define MY_CHARBITSIZE 8
// [2] IEEE 754 doubles are addopted for type: double.
// ........
// [S] 2. Detailed information
//
#endif
The tags in brackets: [T] [S] [#] [1] [2] stand for:
* [T]: Document Title
* [S]: Section
* [#]: Print the following (non-commented) lines as a code-block.
* [1], [2]: Numbered items of a list.
Now, the idea here is to use the file "document.h" in a different way:
To parse the file in order to convert the comments in "document.h" to some printable document, or some basic HTML.
Thus, the tags [T] [S] [#] etc., are intended to be interpreted by a parser that convert any comment into an HTML line of text (for example), and generate <h1></h1>, <b></b> (or whatever you want), when a tag appears.
If you keep the parser as a simple and small program, this can give you a short hand to handle this kind of documentation.
I have to use C and Fortran together to do some simulations. In their course I use the same memory in both programming language parts, by defining a pointer in C to access memory allocated by Fortran.
The datatype of the problematic variable is
real(kind=8)
for Fortran, and
double
for C. The results of the same calculations now differ in the respective programming languages, and I need to directly compare them and get a zero. All calculations are done only with the above accuracies. The difference is always in the 13-14th digit.
What would be a good way to resolve this? Any compiler-flags? Just cut-off after some digits?
Many thanks!
Floating point is not perfectly accurate. Ever. Even cos(x) == cos(y) can be false if x == y.
So when doing your comparisons, take this into account, and allow the values to differ by some small epsilon value.
This is a problem with the inaccuracy with floating point numbers - they will be inaccurate and a certain place. You usually compare them either by rounding them to a digit that you know will be in the accurate area, or by providing an epsilon of appropiate value (small enough to not impact further calculations, and big enough to take care of the inaccuracy while comparing).
One thing you might check is to be sure that the FPU control word is the same in both cases. If it is set to 53-bit precision in one case and 64-bit in the other, it would likely produce different results. You can use the instructions fstcw and fldcw to read and load the control word value. Nonetheless, as others have mentioned, you should not depend on the accuracy being identical even if you can make it work in one situation.
Perfect portability is very difficult to achieve in floating point operations. Changing the order of the machine instructions might change the rounding. One compiler might keep values in registers, while another copy it to memory, which can change the precision. Currently the Fortran and C languages allow a certain amount of latitude. The IEEE module of Fortran 2008, when implemented, will allow requiring more specific and therefore more portable floating point computations.
Since you are compiling for an x86 architecture, it's likely that one of the compilers is maintaining intermediate values in floating point registers, which are 80 bits as opposed to the 64 bits of a C double.
For GCC, you can supply the -ffloat-store option to inhibit this optimisation. You may also need to change the code to explicitly store some intermediate results in double variables. Some experimentation is likely in order.
1) I've got many constants in my C algo.
2) my code works both in floating-point and fixed-point.
Right now, these constants are initialized by a function, float2fixed, whereby in floating-point it does nothing, while in fixed-point, it finds their fixed-point representation. For instance, 0.5f stays 0.5f if working in floating-point, whereas it uses the pow() routine and becomes 32768 if working in fixed-point and the fixed-point representation is Qx.16.
That's easy to maintain, but it takes a lot of time actually to compute these constants in fixed-point (pow is a floatin-point function). In C++, I'd use some meta-programming, so the compiler computes these values at compile-time, so there's no hit at run-time. But in C, thats not possible. Or is it? Anybody knows of such a trick? Is any compiler clever enough to do that?
Looking forward to any answers.
A
Rather than using (unsigned)(x*pow(2,16)) to do your fixed point conversion, write it as (unsigned)(0.5f * (1 << 16))
This should be an acceptable as a compile-time constant expression since it involves only builtin operators.
When using fixed-point, can you write a program that takes your floating point values and converts them into correct, constant initializers for the fixed point type, so you effectively add a step to the compilation that generates the fixed point values.
One advantage of this will be that you can then define and declare your constants with const so that they won't change at run-time - whereas with the initialization functions, of course, the values have to be modifiable because they are calculated once.
I mean write a simple program that can scan for formulaic lines that might read:
const double somename = 3.14159;
it would read that and generate:
const fixedpoint_t somename = { ...whatever is needed... };
You design the operation to make it easy to manage for both notations - so maybe your converter always reads the file and sometimes rewrites it.
datafile.c: datafile.constants converter
converter datafile.constants > datafile.c
In plain C, there's not much you can do. You need to do the conversion at some point, and the compiler doesn't give you any access to call interesting user-provided functions at compile time. Theoretically, you could try to coax the preprocessor to do it for you, but that's the quick road to total insanity (i.e. you'd have to implement pow() in macros, which is pretty hideous).
Some options I can think of:
Maintain a persistent cache on disk. At least then it'd only be slow once, though you still have to load it, make sure it's not corrupt, etc.
As mentioned in another comment, use template metaprogramming anyway and compile with a C++ compiler. Most C works just fine (arguably better) with a C++ compiler.
Hmm, I guess that's about all I can think of. Good luck.
Recent versions of GCC ( around 4.3 ) added the ability to use GMP and MPFR to do some compile-time optimisations by evaluating more complex functions that are constant. That approach leaves your code simple and portable, and trust the compiler to do the heavy lifting.
Of course, there are limits to what it can do, and it would be hard to know if it's optimizing a given instance without going and looking at the assembly. But it might be worth checking out. Here's a link to the description in the changelog