Benchmarking and Optimisation Q's

1) I've always wondered how at the most basic level, mathematical operations are handled, efficiently. I've looked through the math.h several times really carefully, trying to atleast understand or make out what each bit was doing. In general, I succeeded, but I didn't at all get how the code computed trignometric functions. I'm wondering whether it is a machine routine or what. I'd be thrilled to find out how it works

2) Another thing I've always wanted to check out was how much better a particular way of doing the program is over another method. I can perhaps reason out to a large extent, and make small little calculations of the number of calls require and all, but can't get any quantitative comparision. Also when programs are nearly the same, I don't know how to compare. So is there a nice little package that could print out the execution time and memory usage? (I'm not using too complex programs yet, so I generally compute the difference in time between the start and finish using time.h and all, but I'd like a better way).

Thanks for all the help.

1 Best bet I think would contact the makers of the compiler in question, they may be able to help you furrther. Look on their website for email or contact details.

2 The time a program takes to execute will vary on three things:

1) Does the user have to input - this takes time

2) size of the program - even the smallest program can take a while to execute. You may be able to create a while true loop that ends when the program finishes and reports back the seconds it took to complete the execution. Use <ctime> header for this

3} OS. Dependant on the operating system it is running on, porablility is still a slight issue regarding C/.C}} programs. if it is run on an older system, the procsessor may take extra time to complerw the tasks

These are only vaige answers, more people on the baord may have a better response

2. You are looking for profiling... google it and you will find many resources.

Ok, I'm using the gcc complier, but I've checked up the MSVC++8 too. I thought that something kinda basic like the math header would have been standardised to all compilers by now. But the question was absolutely in general.
Thanks for te quick replies

I believe trig functions and similar are done using a Taylor Series to approximate to a given precision.
http://en.wikipedia.org/wiki/Taylor_series

Note: I said believe, I could easily be wrong, and implementations can vary.

@ Dweia, I did feel the same, but I don't think that it would be very efficient. I've heard about an assembly code manner, but know no details (which is why I posted here).

You're not going to get much information on efficiency from looking at math.h.

Header files are interfaces, not implementations. Everyone has to provide the same interface. After that, the compiler / standard library writer has a free hand in deciding how to implement things, so long as the end result conforms to the interface.

If you want to read the implementation, then you typically have two choices.
1. Download the source code for glibc, if you're using GNU
2. Pay Microsoft lots of $$$ to get all the debug / symbol tools for your visual studio.

Quote:

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Originally Posted by Dweia

I believe trig functions and similar are done using a Taylor Series to approximate to a given precision.
http://en.wikipedia.org/wiki/Taylor_series

Note: I said believe, I could easily be wrong, and implementations can vary.

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Yep. The IA-32 instruction set already has instructions fsin, fcos, fptan, fpatan, and y*log_2(x) instructions, along with instructions that load constants such as log_2(e), pi, etc.

Ok I'm reviving this thread. I've gotten the glibc 2.4, and absolutely searched it head to toe. I looked into the math.h which refered all its prototypes the file mathcalls.h which contains macros of the form:
#define __MATHCALLX(function,suffix, args, attrib) \

eg: __MATHCALL (cos,, (_Mdouble_ __x));

Other than that, both files contain no other information. I've looked around the other files, but nothing of any use. Most of them refered to the ieee754 definitions, which I couldn't find or wasn't there. I'd really appreciate some help on how to find this stuff.

Is the code all in assembly? and is it integrated into the compiler?

I've also seen __builtin several times, giving me the above suspicion

Listen, code isn't in header files.

It is in source files, which probably means either math.c or math.asm, somewhere in the glibc SOURCE code distribution.

Thanks a lot salem, I found what I was looking for there (assembly). That's what I was asking before, if it were in assembly. I guess I didn't really understand the reply. Now I've got to understand it

As I suspected, the approximation isn't too taylor like:
for e^x:
/* How this works:

The input value, x, is written as

x = n * ln(2) + t/512 + delta[t] + x;

where:
- n is an integer, 127 >= n >= -150;
- t is an integer, 177 >= t >= -177
- delta is based on a table entry, delta[t] < 2^-28
- x is whatever is left, |x| < 2^-10

Then e^x is approximated as

e^x = 2^n ( e^(t/512 + delta[t])
+ ( e^(t/512 + delta[t])
* ( p(x + delta[t] + n * ln(2)) - delta ) ) )

where
- p(x) is a polynomial approximating e(x)-1;
- e^(t/512 + delta[t]) is obtained from a table.

The table used is the same one as for the double precision version;
since we have the table, we might as well use it.

It turns out to be faster to do calculations in double precision than
to perform an 'accurate table method' expf, because of the range reduction
overhead (compare exp2f).
*/

copied from the comments of ieee754 - e_expfx