This file contains a random collection of the various hacks we did (or
did not dare to do) in order to get performance.  It is not intended
to be complete nor up to date, but it may be instructive for people to
read (see also my (Matteo Frigo's) slides on the web page).

1) Negative constants:

        a = 0.5 * b;               a = 0.5 * b;  
        c = 0.5 * d;	           c = -0.5 * d; 
        e = 1.0 + a;	 vs.       e = 1.0 + a;  
        f = 1.0 - c;	           f = 1.0 + c;  

   The code on the left is faster.  Guess why? The constants
   0.5 and -0.5 are stored in different memory locations and loaded
   separately.

2) Twiddle factors:

   For some reason that escapes me, every package I have seen
   (including FFTW 1.0) stores the twiddle factors in such
   a way that the codelets need to access memory in a random
   fashion.  It is much faster to just store the factors in the
   order they will be needed.  This increased spatial locality
   exploits caches and improves performance by about 30% for big
   arrays.

3) Loops:

   You may notice that the `twiddle' codelets contain a loop.
   This is the most logical choice, as opposed to moving the loop
   outside the codelet.  For example, one would expect that

      for (i = 0; i < n; ++i)
          foo();

   be slower than the case where the loop is inside foo().  This is
   usually the case, *except* for the ultrasparc.  FFTW would be 10%
   faster (more or less) if the loop were outside the codelet.
   Unfortunately, it would be slower everywhere else. If you want to
   play with this, the codelet generator can be easily hacked...

4) Array padding:

   (This is a disgusting hack that my programmer's ethic
   pervents me from releasing to the public).  

   On the IBM RS/6000, for big n, the following **** is *way* faster:

   - Take the input array
   - `inflate' it, that is, produce a bigger array in which every
     k-th element is unused (say k=512).  In other words, insert
     holes in the input.
   - execute fftw normally (properly hacked to keep the holes into
     account)
   - compact the array.

   With this hack, FFTW is sensibly faster than IBM's ESSL library.
   Sorry guys, I don't want to be responsible for releasing this
   monster (this works only on the RS/6000, fortunately).

5) Phase of moon:

   Don't take the numbers on the web page too seriously.  The
   architectural details of modern machines make performance
   *extremely* sensitive to little details such as where your code and
   data are placed in memory.  The ultimate example of brain damage is
   the Pentium Pro (what a surprise...), where we had a case in which
   adding a printf() statement to FFTW slowed down another completely
   unrelated fft program by a factor of 20.

   Our strategy to generate huge pieces of straight-line code should
   immunize FFTW sufficiently well against these problems, but you are
   still likely to observe weird things like: FFTW_ESTIMATE can be
   faster than FFTW_MEASURE.

   FFTW-2.0 will compute the phase of the moon in the planner and take
   it into account.

6) Estimator:

   The estimator tries to come up with a `reasonable' plan.
   Unfortunately, this concept is very machine-dependent.  The
   estimator in the release is tuned for the UltraSPARC.

   The following two parameters can be found in fftw/planner.c :

   #define NOTW_OPTIMAL_SIZE 32
   #define TWIDDLE_OPTIMAL_SIZE 12

   They say: use non-twiddle codelets of size close to 32, and twiddle
   codelets of size close to 12.  More or less, these are the most
   efficient pieces of code on the UltraSPARC.  Your mileage *will*
   vary.  Here are some suggestions for other machines.

   Pentium

   #define NOTW_OPTIMAL_SIZE 8   (or 16)
   #define TWIDDLE_OPTIMAL_SIZE 8

   Pentium Pro:

   #define NOTW_OPTIMAL_SIZE 32   (or 16)
   #define TWIDDLE_OPTIMAL_SIZE 2

   
   RS/6000:

   #define NOTW_OPTIMAL_SIZE 32
   #define TWIDDLE_OPTIMAL_SIZE 32

   The basic idea is: compute some plans for sizes you most care
   about, print the plan and plug in the numbers that appear more
   often (or some kind of average).  Note the dramatic difference
   between Pentium an Pentium Pro. (NO LONGER TRUE WITH --enable-i386-hacks)

7) Stack alignment:

   Pentium-type processors impose a huge performance penalty if double-
   precision values are not aligned to 8-byte boundaries in memory.
   (We have seen factors of 3 or more in tight loops.)  Unfortunately,
   the Intel ABI specifies that variables on the stack need only be aligned to
   4-byte boundaries.  Even more unfortunately, this convention is followed
   by Linux/x86 and gcc.

   To get around this, we wrote special macros (HACK_ALIGN_STACK_EVEN
   and HACK_ALIGN_STACK_ODD) that take effect when FFTW is compiled
   with gcc/egcs and 'configure --enable-i386-hacks' is used.  These
   macros are called before the computation-intensive "codelets"
   of FFTW.  They use the gcc __builtin_alloca function to examine the
   stack pointer and align the stack to an even or odd 4-byte boundary,
   depending upon how many values will be pushed on the stack to call
   the codelet.
