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pow is a very wasteful function for this purpose. A low hanging fruit
would be simply to replace with exp2f, and that does yield some speedup.
However, there are 2 drawbacks of this:
1. It does not exploit the integer nature of the argument.
2. (minor) Some platforms lack a proper exp2f routine, making benefits available
only to non broken libm.
3. exp2f does not solve the same issue that plagues pow, namely terrible
worst case performance. This is a fundamental issue known as the
"table-maker's dilemma" recognized by Prof. Kahan himself and
subsequently elaborated and researched by many others. All this is clear from benchmarks below.
This exploits the IEEE-754 format to get very good performance even in
the worst case for integer powers of 2. This solves all the issues noted
above. Function tested with clang usan over [-1000, 1000] (beyond range of
relevance for this, which is [-255, 255]), patch itself with FATE.
Benchmarks obtained on x86-64, Haswell, GNU-Linux via 10^5 iterations of
the pow call, START/STOP, and command ffplay ~/samples/jpeg2000/chiens_dcinema2K.mxf.
Low number of runs also given to prove the point about worst case:
pow:
216270 decicycles in pow, 1 runs, 0 skips
110175 decicycles in pow, 2 runs, 0 skips
56085 decicycles in pow, 4 runs, 0 skips
29013 decicycles in pow, 8 runs, 0 skips
15472 decicycles in pow, 16 runs, 0 skips
8689 decicycles in pow, 32 runs, 0 skips
5295 decicycles in pow, 64 runs, 0 skips
3599 decicycles in pow, 128 runs, 0 skips
2748 decicycles in pow, 256 runs, 0 skips
2304 decicycles in pow, 511 runs, 1 skips
2072 decicycles in pow, 1022 runs, 2 skips
1963 decicycles in pow, 2044 runs, 4 skips
1894 decicycles in pow, 4091 runs, 5 skips
1860 decicycles in pow, 8184 runs, 8 skips
exp2f:
134140 decicycles in pow, 1 runs, 0 skips
68110 decicycles in pow, 2 runs, 0 skips
34530 decicycles in pow, 4 runs, 0 skips
17677 decicycles in pow, 8 runs, 0 skips
9175 decicycles in pow, 16 runs, 0 skips
4931 decicycles in pow, 32 runs, 0 skips
2808 decicycles in pow, 64 runs, 0 skips
1747 decicycles in pow, 128 runs, 0 skips
1208 decicycles in pow, 256 runs, 0 skips
952 decicycles in pow, 512 runs, 0 skips
822 decicycles in pow, 1024 runs, 0 skips
765 decicycles in pow, 2047 runs, 1 skips
722 decicycles in pow, 4094 runs, 2 skips
693 decicycles in pow, 8190 runs, 2 skips
exp2fi:
2740 decicycles in pow, 1 runs, 0 skips
1530 decicycles in pow, 2 runs, 0 skips
955 decicycles in pow, 4 runs, 0 skips
622 decicycles in pow, 8 runs, 0 skips
477 decicycles in pow, 16 runs, 0 skips
368 decicycles in pow, 32 runs, 0 skips
317 decicycles in pow, 64 runs, 0 skips
291 decicycles in pow, 128 runs, 0 skips
277 decicycles in pow, 256 runs, 0 skips
268 decicycles in pow, 512 runs, 0 skips
265 decicycles in pow, 1024 runs, 0 skips
263 decicycles in pow, 2048 runs, 0 skips
263 decicycles in pow, 4095 runs, 1 skips
260 decicycles in pow, 8191 runs, 1 skips
Reviewed-by: Michael Niedermayer <michael@niedermayer.cc>
Signed-off-by: Ganesh Ajjanagadde <gajjanagadde@gmail.com>