I've been poring through .NET disassemblies and the GCC source code, but can't seem to find anywhere the actual implementation of sin() and other math functions... they always seem to be referencing something else.
Can anyone help me find them? I feel like it's unlikely that ALL hardware that C will run on supports trig functions in hardware, so there must be a software algorithm somewhere, right?
I'm aware of several ways that functions can be calculated, and have written my own routines to compute functions using taylor series for fun. I'm curious about how real, production languages do it, since all of my implementations are always several orders of magnitude slower, even though I think my algorithms are pretty clever (obviously they're not).
In GNU libm, the implementation of sin is system-dependent. Therefore you can find the implementation, for each platform, somewhere in the appropriate subdirectory of sysdeps.
One directory includes an implementation in C, contributed by IBM. Since October 2011, this is the code that actually runs when you call sin() on a typical x86-64 Linux system. It is apparently faster than the fsin assembly instruction. Source code: sysdeps/ieee754/dbl-64/s_sin.c, look for __sin (double x).
This code is very complex. No one software algorithm is as fast as possible and also accurate over the whole range of x values, so the library implements several different algorithms, and its first job is to look at x and decide which algorithm to use.
When x is very very close to 0, sin(x) == x is the right answer.
A bit further out, sin(x) uses the familiar Taylor series. However, this is only accurate near 0, so...
When the angle is more than about 7°, a different algorithm is used, computing Taylor-series approximations for both sin(x) and cos(x), then using values from a precomputed table to refine the approximation.
When |x| > 2, none of the above algorithms would work, so the code starts by computing some value closer to 0 that can be fed to sin or cos instead.
There's yet another branch to deal with x being a NaN or infinity.
This code uses some numerical hacks I've never seen before, though for all I know they might be well-known among floating-point experts. Sometimes a few lines of code would take several paragraphs to explain. For example, these two lines
double t = (x * hpinv + toint);
double xn = t - toint;
are used (sometimes) in reducing x to a value close to 0 that differs from x by a multiple of π/2, specifically xn × π/2. The way this is done without division or branching is rather clever. But there's no comment at all!
Older 32-bit versions of GCC/glibc used the fsin instruction, which is surprisingly inaccurate for some inputs. There's a fascinating blog post illustrating this with just 2 lines of code.
fdlibm's implementation of sin in pure C is much simpler than glibc's and is nicely commented. Source code: fdlibm/s_sin.c and fdlibm/k_sin.c
Functions like sine and cosine are implemented in microcode inside microprocessors. Intel chips, for example, have assembly instructions for these. A C compiler will generate code that calls these assembly instructions. (By contrast, a Java compiler will not. Java evaluates trig functions in software rather than hardware, and so it runs much slower.)
Chips do not use Taylor series to compute trig functions, at least not entirely. First of all they use CORDIC, but they may also use a short Taylor series to polish up the result of CORDIC or for special cases such as computing sine with high relative accuracy for very small angles. For more explanation, see this StackOverflow answer.
OK kiddies, time for the pros....
This is one of my biggest complaints with inexperienced software engineers. They come in calculating transcendental functions from scratch (using Taylor's series) as if nobody had ever done these calculations before in their lives. Not true. This is a well defined problem and has been approached thousands of times by very clever software and hardware engineers and has a well defined solution.
Basically, most of the transcendental functions use Chebyshev Polynomials to calculate them. As to which polynomials are used depends on the circumstances. First, the bible on this matter is a book called "Computer Approximations" by Hart and Cheney. In that book, you can decide if you have a hardware adder, multiplier, divider, etc, and decide which operations are fastest. e.g. If you had a really fast divider, the fastest way to calculate sine might be P1(x)/P2(x) where P1, P2 are Chebyshev polynomials. Without the fast divider, it might be just P(x), where P has much more terms than P1 or P2....so it'd be slower. So, first step is to determine your hardware and what it can do. Then you choose the appropriate combination of Chebyshev polynomials (is usually of the form cos(ax) = aP(x) for cosine for example, again where P is a Chebyshev polynomial). Then you decide what decimal precision you want. e.g. if you want 7 digits precision, you look that up in the appropriate table in the book I mentioned, and it will give you (for precision = 7.33) a number N = 4 and a polynomial number 3502. N is the order of the polynomial (so it's p4.x^4 + p3.x^3 + p2.x^2 + p1.x + p0), because N=4. Then you look up the actual value of the p4,p3,p2,p1,p0 values in the back of the book under 3502 (they'll be in floating point). Then you implement your algorithm in software in the form:
(((p4.x + p3).x + p2).x + p1).x + p0
....and this is how you'd calculate cosine to 7 decimal places on that hardware.
Note that most hardware implementations of transcendental operations in an FPU usually involve some microcode and operations like this (depends on the hardware).
Chebyshev polynomials are used for most transcendentals but not all. e.g. Square root is faster to use a double iteration of Newton raphson method using a lookup table first.
Again, that book "Computer Approximations" will tell you that.
If you plan on implmementing these functions, I'd recommend to anyone that they get a copy of that book. It really is the bible for these kinds of algorithms.
Note that there are bunches of alternative means for calculating these values like cordics, etc, but these tend to be best for specific algorithms where you only need low precision. To guarantee the precision every time, the chebyshev polynomials are the way to go. Like I said, well defined problem. Has been solved for 50 years now.....and thats how it's done.
Now, that being said, there are techniques whereby the Chebyshev polynomials can be used to get a single precision result with a low degree polynomial (like the example for cosine above). Then, there are other techniques to interpolate between values to increase the accuracy without having to go to a much larger polynomial, such as "Gal's Accurate Tables Method". This latter technique is what the post referring to the ACM literature is referring to. But ultimately, the Chebyshev Polynomials are what are used to get 90% of the way there.
Enjoy.
For sin specifically, using Taylor expansion would give you:
sin(x) := x - x^3/3! + x^5/5! - x^7/7! + ... (1)
you would keep adding terms until either the difference between them is lower than an accepted tolerance level or just for a finite amount of steps (faster, but less precise). An example would be something like:
float sin(float x)
{
float res=0, pow=x, fact=1;
for(int i=0; i<5; ++i)
{
res+=pow/fact;
pow*=-1*x*x;
fact*=(2*(i+1))*(2*(i+1)+1);
}
return res;
}
Note: (1) works because of the aproximation sin(x)=x for small angles. For bigger angles you need to calculate more and more terms to get acceptable results.
You can use a while argument and continue for a certain accuracy:
double sin (double x){
int i = 1;
double cur = x;
double acc = 1;
double fact= 1;
double pow = x;
while (fabs(acc) > .00000001 && i < 100){
fact *= ((2*i)*(2*i+1));
pow *= -1 * x*x;
acc = pow / fact;
cur += acc;
i++;
}
return cur;
}
Concerning trigonometric function like sin(), cos(),tan() there has been no mention, after 5 years, of an important aspect of high quality trig functions: Range reduction.
An early step in any of these functions is to reduce the angle, in radians, to a range of a 2*π interval. But π is irrational so simple reductions like x = remainder(x, 2*M_PI) introduce error as M_PI, or machine pi, is an approximation of π. So, how to do x = remainder(x, 2*π)?
Early libraries used extended precision or crafted programming to give quality results but still over a limited range of double. When a large value was requested like sin(pow(2,30)), the results were meaningless or 0.0 and maybe with an error flag set to something like TLOSS total loss of precision or PLOSS partial loss of precision.
Good range reduction of large values to an interval like -π to π is a challenging problem that rivals the challenges of the basic trig function, like sin(), itself.
A good report is Argument reduction for huge arguments: Good to the last bit (1992). It covers the issue well: discusses the need and how things were on various platforms (SPARC, PC, HP, 30+ other) and provides a solution algorithm the gives quality results for all double from -DBL_MAX to DBL_MAX.
If the original arguments are in degrees, yet may be of a large value, use fmod() first for improved precision. A good fmod() will introduce no error and so provide excellent range reduction.
// sin(degrees2radians(x))
sin(degrees2radians(fmod(x, 360.0))); // -360.0 < fmod(x,360) < +360.0
Various trig identities and remquo() offer even more improvement. Sample: sind()
Yes, there are software algorithms for calculating sin too. Basically, calculating these kind of stuff with a digital computer is usually done using numerical methods like approximating the Taylor series representing the function.
Numerical methods can approximate functions to an arbitrary amount of accuracy and since the amount of accuracy you have in a floating number is finite, they suit these tasks pretty well.
Use Taylor series and try to find relation between terms of the series so you don't calculate things again and again
Here is an example for cosinus:
double cosinus(double x, double prec)
{
double t, s ;
int p;
p = 0;
s = 1.0;
t = 1.0;
while(fabs(t/s) > prec)
{
p++;
t = (-t * x * x) / ((2 * p - 1) * (2 * p));
s += t;
}
return s;
}
using this we can get the new term of the sum using the already used one (we avoid the factorial and x2p)
It is a complex question. Intel-like CPU of the x86 family have a hardware implementation of the sin() function, but it is part of the x87 FPU and not used anymore in 64-bit mode (where SSE2 registers are used instead). In that mode, a software implementation is used.
There are several such implementations out there. One is in fdlibm and is used in Java. As far as I know, the glibc implementation contains parts of fdlibm, and other parts contributed by IBM.
Software implementations of transcendental functions such as sin() typically use approximations by polynomials, often obtained from Taylor series.
Chebyshev polynomials, as mentioned in another answer, are the polynomials where the largest difference between the function and the polynomial is as small as possible. That is an excellent start.
In some cases, the maximum error is not what you are interested in, but the maximum relative error. For example for the sine function, the error near x = 0 should be much smaller than for larger values; you want a small relative error. So you would calculate the Chebyshev polynomial for sin x / x, and multiply that polynomial by x.
Next you have to figure out how to evaluate the polynomial. You want to evaluate it in such a way that the intermediate values are small and therefore rounding errors are small. Otherwise the rounding errors might become a lot larger than errors in the polynomial. And with functions like the sine function, if you are careless then it may be possible that the result that you calculate for sin x is greater than the result for sin y even when x < y. So careful choice of the calculation order and calculation of upper bounds for the rounding error are needed.
For example, sin x = x - x^3/6 + x^5 / 120 - x^7 / 5040... If you calculate naively sin x = x * (1 - x^2/6 + x^4/120 - x^6/5040...), then that function in parentheses is decreasing, and it will happen that if y is the next larger number to x, then sometimes sin y will be smaller than sin x. Instead, calculate sin x = x - x^3 * (1/6 - x^2 / 120 + x^4/5040...) where this cannot happen.
When calculating Chebyshev polynomials, you usually need to round the coefficients to double precision, for example. But while a Chebyshev polynomial is optimal, the Chebyshev polynomial with coefficients rounded to double precision is not the optimal polynomial with double precision coefficients!
For example for sin (x), where you need coefficients for x, x^3, x^5, x^7 etc. you do the following: Calculate the best approximation of sin x with a polynomial (ax + bx^3 + cx^5 + dx^7) with higher than double precision, then round a to double precision, giving A. The difference between a and A would be quite large. Now calculate the best approximation of (sin x - Ax) with a polynomial (b x^3 + cx^5 + dx^7). You get different coefficients, because they adapt to the difference between a and A. Round b to double precision B. Then approximate (sin x - Ax - Bx^3) with a polynomial cx^5 + dx^7 and so on. You will get a polynomial that is almost as good as the original Chebyshev polynomial, but much better than Chebyshev rounded to double precision.
Next you should take into account the rounding errors in the choice of polynomial. You found a polynomial with minimum error in the polynomial ignoring rounding error, but you want to optimise polynomial plus rounding error. Once you have the Chebyshev polynomial, you can calculate bounds for the rounding error. Say f (x) is your function, P (x) is the polynomial, and E (x) is the rounding error. You don't want to optimise | f (x) - P (x) |, you want to optimise | f (x) - P (x) +/- E (x) |. You will get a slightly different polynomial that tries to keep the polynomial errors down where the rounding error is large, and relaxes the polynomial errors a bit where the rounding error is small.
All this will get you easily rounding errors of at most 0.55 times the last bit, where +,-,*,/ have rounding errors of at most 0.50 times the last bit.
The actual implementation of library functions is up to the specific compiler and/or library provider. Whether it's done in hardware or software, whether it's a Taylor expansion or not, etc., will vary.
I realize that's absolutely no help.
There's nothing like hitting the source and seeing how someone has actually done it in a library in common use; let's look at one C library implementation in particular. I chose uLibC.
Here's the sin function:
http://git.uclibc.org/uClibc/tree/libm/s_sin.c
which looks like it handles a few special cases, and then carries out some argument reduction to map the input to the range [-pi/4,pi/4], (splitting the argument into two parts, a big part and a tail) before calling
http://git.uclibc.org/uClibc/tree/libm/k_sin.c
which then operates on those two parts.
If there is no tail, an approximate answer is generated using a polynomial of degree 13.
If there is a tail, you get a small corrective addition based on the principle that sin(x+y) = sin(x) + sin'(x')y
They are typically implemented in software and will not use the corresponding hardware (that is, aseembly) calls in most cases. However, as Jason pointed out, these are implementation specific.
Note that these software routines are not part of the compiler sources, but will rather be found in the correspoding library such as the clib, or glibc for the GNU compiler. See http://www.gnu.org/software/libc/manual/html_mono/libc.html#Trig-Functions
If you want greater control, you should carefully evaluate what you need exactly. Some of the typical methods are interpolation of look-up tables, the assembly call (which is often slow), or other approximation schemes such as Newton-Raphson for square roots.
If you want an implementation in software, not hardware, the place to look for a definitive answer to this question is Chapter 5 of Numerical Recipes. My copy is in a box, so I can't give details, but the short version (if I remember this right) is that you take tan(theta/2) as your primitive operation and compute the others from there. The computation is done with a series approximation, but it's something that converges much more quickly than a Taylor series.
Sorry I can't rembember more without getting my hand on the book.
Whenever such a function is evaluated, then at some level there is most likely either:
A table of values which is interpolated (for fast, inaccurate applications - e.g. computer graphics)
The evaluation of a series that converges to the desired value --- probably not a taylor series, more likely something based on a fancy quadrature like Clenshaw-Curtis.
If there is no hardware support then the compiler probably uses the latter method, emitting only assembler code (with no debug symbols), rather than using a c library --- making it tricky for you to track the actual code down in your debugger.
If you want to look at the actual GNU implementation of those functions in C, check out the latest trunk of glibc. See the GNU C Library.
As many people pointed out, it is implementation dependent. But as far as I understand your question, you were interested in a real software implemetnation of math functions, but just didn't manage to find one. If this is the case then here you are:
Download glibc source code from http://ftp.gnu.org/gnu/glibc/
Look at file dosincos.c located in unpacked glibc root\sysdeps\ieee754\dbl-64 folder
Similarly you can find implementations of the rest of the math library, just look for the file with appropriate name
You may also have a look at the files with the .tbl extension, their contents is nothing more than huge tables of precomputed values of different functions in a binary form. That is why the implementation is so fast: instead of computing all the coefficients of whatever series they use they just do a quick lookup, which is much faster. BTW, they do use Tailor series to calculate sine and cosine.
I hope this helps.
I'll try to answer for the case of sin() in a C program, compiled with GCC's C compiler on a current x86 processor (let's say a Intel Core 2 Duo).
In the C language the Standard C Library includes common math functions, not included in the language itself (e.g. pow, sin and cos for power, sine, and cosine respectively). The headers of which are included in math.h.
Now on a GNU/Linux system, these libraries functions are provided by glibc (GNU libc or GNU C Library). But the GCC compiler wants you to link to the math library (libm.so) using the -lm compiler flag to enable usage of these math functions. I'm not sure why it isn't part of the standard C library. These would be a software version of the floating point functions, or "soft-float".
Aside: The reason for having the math functions separate is historic, and was merely intended to reduce the size of executable programs in very old Unix systems, possibly before shared libraries were available, as far as I know.
Now the compiler may optimize the standard C library function sin() (provided by libm.so) to be replaced with an call to a native instruction to your CPU/FPU's built-in sin() function, which exists as an FPU instruction (FSIN for x86/x87) on newer processors like the Core 2 series (this is correct pretty much as far back as the i486DX). This would depend on optimization flags passed to the gcc compiler. If the compiler was told to write code that would execute on any i386 or newer processor, it would not make such an optimization. The -mcpu=486 flag would inform the compiler that it was safe to make such an optimization.
Now if the program executed the software version of the sin() function, it would do so based on a CORDIC (COordinate Rotation DIgital Computer) or BKM algorithm, or more likely a table or power-series calculation which is commonly used now to calculate such transcendental functions. [Src: http://en.wikipedia.org/wiki/Cordic#Application]
Any recent (since 2.9x approx.) version of gcc also offers a built-in version of sin, __builtin_sin() that it will used to replace the standard call to the C library version, as an optimization.
I'm sure that is as clear as mud, but hopefully gives you more information than you were expecting, and lots of jumping off points to learn more yourself.
Don't use Taylor series. Chebyshev polynomials are both faster and more accurate, as pointed out by a couple of people above. Here is an implementation (originally from the ZX Spectrum ROM): https://albertveli.wordpress.com/2015/01/10/zx-sine/
Computing sine/cosine/tangent is actually very easy to do through code using the Taylor series. Writing one yourself takes like 5 seconds.
The whole process can be summed up with this equation here:
Here are some routines I wrote for C:
double _pow(double a, double b) {
double c = 1;
for (int i=0; i<b; i++)
c *= a;
return c;
}
double _fact(double x) {
double ret = 1;
for (int i=1; i<=x; i++)
ret *= i;
return ret;
}
double _sin(double x) {
double y = x;
double s = -1;
for (int i=3; i<=100; i+=2) {
y+=s*(_pow(x,i)/_fact(i));
s *= -1;
}
return y;
}
double _cos(double x) {
double y = 1;
double s = -1;
for (int i=2; i<=100; i+=2) {
y+=s*(_pow(x,i)/_fact(i));
s *= -1;
}
return y;
}
double _tan(double x) {
return (_sin(x)/_cos(x));
}
Improved version of code from Blindy's answer
#define EPSILON .0000000000001
// this is smallest effective threshold, at least on my OS (WSL ubuntu 18)
// possibly because factorial part turns 0 at some point
// and it happens faster then series element turns 0;
// validation was made against sin() from <math.h>
double ft_sin(double x)
{
int k = 2;
double r = x;
double acc = 1;
double den = 1;
double num = x;
// precision drops rapidly when x is not close to 0
// so move x to 0 as close as possible
while (x > PI)
x -= PI;
while (x < -PI)
x += PI;
if (x > PI / 2)
return (ft_sin(PI - x));
if (x < -PI / 2)
return (ft_sin(-PI - x));
// not using fabs for performance reasons
while (acc > EPSILON || acc < -EPSILON)
{
num *= -x * x;
den *= k * (k + 1);
acc = num / den;
r += acc;
k += 2;
}
return (r);
}
The essence of how it does this lies in this excerpt from Applied Numerical Analysis by Gerald Wheatley:
When your software program asks the computer to get a value of
or , have you wondered how it can get the
values if the most powerful functions it can compute are polynomials?
It doesnt look these up in tables and interpolate! Rather, the
computer approximates every function other than polynomials from some
polynomial that is tailored to give the values very accurately.
A few points to mention on the above is that some algorithms do infact interpolate from a table, albeit only for the first few iterations. Also note how it mentions that computers utilise approximating polynomials without specifying which type of approximating polynomial. As others in the thread have pointed out, Chebyshev polynomials are more efficient than Taylor polynomials in this case.
if you want sin then
__asm__ __volatile__("fsin" : "=t"(vsin) : "0"(xrads));
if you want cos then
__asm__ __volatile__("fcos" : "=t"(vcos) : "0"(xrads));
if you want sqrt then
__asm__ __volatile__("fsqrt" : "=t"(vsqrt) : "0"(value));
so why use inaccurate code when the machine instructions will do?
Since the function fsin for computing the sin(x) function under the x86 dates back to the Pentium era, and apparently it doesn't even use SSE registers, I was wondering if there is a newer and better set of instructions for computing trigonometric functions.
I'm used to code in C++ and do some asm optimizations, so anything that fits in a pipeline starting from C++, to C to asm will do for me.
Thanks.
I'm under Linux 64 bit for now, with gcc and clang ( even tough clang doesn't really offer any FPU related optimization AFAIK ).
EDIT
I have already implemented a sin function, it's usually 2 times faster then std::sin even with sse on.
My function is never slower then fsin, even tough fsin is usually more accurate, but considering that fsin never outperforms my sin implementation, I'll keep my sin for now, also my sin is totally portable where fsin is for x86 only.
I need this for real time computation, so I'll trade precision for speed, I think that I'll be fine with 4-5 decimals of precision .
no to a table based approach, I'm not using it, it screws up the cache, makes everything slower, no algorithm based on memory access or lookup tables please.
If you need an approximation of sine optimized for absolute accuracy over -π … π, use:
x * (1 + x * x * (-0.1661251158026961831813227851437597220432 + x * x * (8.03943560729777481878247432892823524338e-3 + x * x * -1.4941402004593877749503989396238510717e-4))
It can be implemented with:
float xx = x * x;
float s = x + (x * xx) * (-0.16612511580269618f + xx * (8.0394356072977748e-3f + xx * -1.49414020045938777495e-4f));
And perhaps optimized depending on the characteristics of your target architecture. Also, not noted in the linked blog post, if you are implementing this in assembly, do use the FMADD instruction. If implementing in C or C++, if you use, say, the fmaf() C99 standard function, make sure that FMADD is generated. The emulated version is much more expensive than a multiplication and an addition, because what fmaf() does is not exactly equivalent to multiplication followed by addition (so it would be incorrect to just implement it so).
The difference between sin(x) and the above polynomial between -π and π graphs so:
The polynomial is optimized to reduce the difference between it and sin(x) between -π and π, not just something that someone thought was a good idea.
If you only need the [-1 … 1] definition interval, then the polynomial can be made more accurate on that interval by ignoring the rest. Running the optimization algorithm again for this definition interval produces:
x * (1 + x * x * (-1.666659904470566774477504230733785739156e-1 + x * x *(8.329797530524482484880881032235130379746e-3 + x * x *(-1.928379009208489415662312713847811393721e-4)))
The absolute error graph:
If that is too accurate for you, it is possible to optimize a polynomial of lower degree for the same objective. Then the absolute error will be larger but you will save a multiplication or two.
If you're okay with an approximation (I'm assuming you are, if you're trying to beat hardware), you should take a look at Nick's sin implementation at DevMaster:
http://devmaster.net/posts/9648/fast-and-accurate-sine-cosine
He has two versions: a "fast & sloppy" method and a "slow & accurate" method. A couple replies down someone estimates the relative errors as 12% and 0.2% respectively. I've done an implementation myself, and find runtimes of 1/14 and 1/8 the hardware times on my machine.
Hope that helps!
PS: If you do this yourself, you can refactor the slow/accurate method to avoid a multiplication and improve slightly over Nick's version, but I don't remember exactly how...
We have some code that looks like this:
inline int calc_something(double x) {
if (x > 0.0) {
// do something
return 1;
} else {
// do something else
return 0;
}
}
Unfortunately, when using the flag /fp:fast, we get calc_something(0)==1 so we are clearly taking the wrong code path. This only happens when we use the method at multiple points in our code with different parameters, so I think there is some fishy optimization going on here from the compiler (Microsoft Visual Studio 2008, SP1).
Also, the above problem goes away when we change the interface to
inline int calc_something(const double& x) {
But I have no idea why this fixes the strange behaviour. Can anyone explane this behaviour? If I cannot understand what's going on we will have to remove the /fp:fastswitch, but this would make our application quite a bit slower.
I'm not familiar enough with FPUs to comment with any certainty, but my guess would be that the compiler is letting an existing value that it thinks should be equal to x sit in on that comparison. Maybe you go y = x + 20.; y = y - 20; y is already on the FP stack, so rather than load x the compiler just compares against y. But due to rounding errors, y isn't quite 0.0 like it is supposed to be, and you get the odd results you see.
For a better explanation: Why is cos(x) != cos(y) even though x == y? from the C++FAQ lite. This is part of what I'm trying to get across, I just couldn't remember where exactly I had read it until just now.
Changing to a const reference fixes this because the compiler is worried about aliasing. It forces a load from x because it can't assume its value hasn't changed at some point after creating y, and since x is actually exactly 0.0 [which is representable in every floating point format I'm familiar with] the rounding errors vanish.
I'm pretty sure MS provides a pragma that allows you to set the FP flags on a per-function basis. Or you could move this routine to a separate file and give that file custom flags. Either way, it could prevent your whole program from suffering just to keep that one routine happy.
what are the results of calc_something(0L), or calc_something(0.0f) ? It could be linked to the size of the types before casting. An integer is 4 bytes, a double is 8.
Have you tried looking at the asembled code, to see how the aforementioned conversion is done ?
Googling for 'fp fast', I found this post [social.msdn.microsoft.com]
As I've said in other question, compilers suck at generating floating point code. The article Dennis links to explains the problems well. Here's another: An MSDN article.
If the performance of the code is important, you can easily1 out-perform the compiler by writing your own assembler code. If your algoritm is vectorisable then you can make use of SIMD too (with a slight loss of precision though).
Assuming you understand the way the FPU works.
inline int calc_something(double x) will (probably) use an 80 bits register. inline int calc_something(const double& x) would store the double in memory, where it takes 64 bits. That at least explains the difference between the two.
However, I find your test quite fishy to begin with. The results of calc_something are extremely sensitive to rounding of its input. Your FP algorithms should be robust to rounding. calc_something(1.0-(1.0/3.0)*3) should be the same as calc_something(0.0).
I think the behavior is correct.
You never compare a floating point number up to less than the holding type's precision.
Something that comes from zero may be equal, greater or less than another zero.
See http://floating-point-gui.de/
Sometimes I see and have used the following variation for a fast divide in C++ with floating point numbers.
// orig loop
double y = 44100.0;
for(int i=0; i<10000; ++i) {
double z = x / y;
}
// alternative
double y = 44100;
double y_div = 1.0 / y;
for(int i=0; i<10000; ++i) {
double z = x * y_div;
}
But someone hinted recently that this might not be the most accurate way.
Any thoughts?
On just about every CPU, a floating point divide is several times as expensive as a floating point multiply, so multiplying by the inverse of your divisor is a good optimization. The downside is that there is a possibility that you will lose a very small portion of accuracy on certain processors - eg, on modern x86 processors, 64-bit float operations are actually internally computed using 80 bits when using the default FPU mode, and storing it off in a variable will cause those extra precision bits to be truncated according to your FPU rounding mode (which defaults to nearest). This only really matters if you are concatenating many float operations and have to worry about the error accumulation.
Wikipedia agrees that this can be faster. The linked article also contains several other fast division algorithms that might be of interest.
I would guess that any industrial-strength modern compiler will make that optimization for you if it is going to profit you at all.
Your original
// original loop:
double y = 44100.0;
for(int i=0; i<10000; ++i) {
double z = x / y;
}
can easily be optimized to
// haha:
double y = 44100.0;
double z = x / y;
and the performance is pretty nice. ;-)
EDIT: People keep voting this down, so here's the not so funny version:
If there were a general way to make division faster for all cases, don't you think compiler writers might have happened upon it by now? Of course they would have done. Also, some of the people doing FPU circuits aren't exactly stupid, either.
So the only way you're going to get better performance is to know what specific situation you have at hand and doing optimal code for that. Most likely this is a complete waste of your time, because your program is slow for some other reason such as performing math on loop invariants. Go find a better algorithm instead.
In your example using gcc the division with the options -O3 -ffast-math yields the same code as the multiplication without -ffast-math. (In a testing environment with enough volatiles around that the loop is still there.)
So if you really want to optimise those divisions away and don’t care about the consequences, that’s the way to go. Multiplication seems to be roughly 15 times faster, btw.
multiplication is faster than division so the second method is faster. It might be slightly less accurate but unless you are doing hard core numerics the level of accuracy should be more than enough.
When processing audio, I prefer to use fixed point math instead. I suppose this depends on the level of precision you need. But, let's assume that 16.16 fixed point integers will do (meaning high 16 bits is whole number, low 16 is the fraction). Now, all calculation can be done as simple integer math:
unsigned int y = 44100 << 16;
unsigned int z = x / (y >> 16); // divisor must be the whole number portion
Or with macros to help:
#define FP_INT(x) (x << 16)
#define FP_MUL(x, y) (x * (y >> 16))
#define FP_DIV(x, y) (x / (y >> 16))
unsigned int y = FP_INT(44100);
unsigned int z = FP_MUL(x, y);
Audio, hunh? It's not just 44,100 divisions per second when you have, say, five tracks of audio running at once. Even a simple fader consumes cycles, after all. And that's just for a fairly bare-bones, minimal example -- what if you want to have, say, an eq and a compressor? Maybe a little reverb? Your total math budget, so to speak, gets eaten up quickly. It does make sense to wring out a little extra performance in those cases.
Profilers are good. Profilers are your friend. Profilers deserve blowjobs and pudding. But you already know where the main bottle neck is in audio work -- it's in the loop that processes samples, and the faster you can make that, the happier your users will be. Use everything you can! Multiply by reciprocals, shift bits whenever possible (exp(x*y) = exp (x)*exp(y), after all), use lookup tables, refer to variables by reference instead of values (less pushing/popping on the stack), refactor terms, and so forth. (If you're good, you'll laugh at these elementary optimizations.)
I presume from the original post that x is not a constant shown there but probably data from an array so x[i] is likely to be the source of the data and similarly for the output, it will be stored somewhere in memory.
I suggest that if the loop count really is 10,000 as in the original post that it will make little difference which you use as the whole loop won't even take a fraction of millisecond anyway on a modern cpu. If the loop count really is very much higher, perhaps 1,000,000 or more, then I would expect that the cost of memory access would likely make the faster operation completely irrelevent anyway as it will always be waiting for the data anyway.
I suggest trying both with your code and testing if it actually makes any significant difference in run time and if it doesn't then just write the straightforward division if that's what the algorithm needs.
here's the problem with doing it with a reciprocal, you still have to do the division before you can actually divide by Y. unless your only dividing by Y then i suppose this may be useful. this is not very practical since division is done in binary with similar algorithms.
I are looping 10,000 times simply to make the code take long enough to measure the time easily? Or do you have 10000 numbers to divide by the same number? If the former, put the "y_div = 1.0 / y;" inside the loop, because it's part of the operation.
If the latter, yes, floating point multiplication is generally faster than division. Don't change your code from the obvious to the arcane based on guesses, though. Benchmark first to find slow spots, and then optimize those (and take measurements before and after to make sure your idea actually causes an improvement)
On old CPUs like the 80286, floating point maths was abysmally slow and we employed lots of trickiness to speed things up.
On modern CPUs floating point maths is blindingly fast and optimising compilers can generally do wonders with fine-tuning things.
It is almost never worth the effort to employ little micro-optimisations like that.
Try to make your code simple and idiot-proof. Only of you find a real bottleneck (using a profiler) would you think of optimisations in your floating point calculations.