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Testing out optimal magic numbers for the old fisqt function
Searching for Optimal Magic Number
The fast inverse square root was a function designed to speed up the 1/x^2 operation, primarily for use in computer graphics. Read more in the wikipedia article here.
The function definition I used in Python is:
> 1) # arithmetic with magic number
packed_i = struct.pack('i', i)
y = struct.unpack('f', packed_i)[0] # treat int's bytes as float
y = y * (threehalfs - (x2 * y * y)) # Newton's method
return y">def struct_isqrt(number, magic=0x5f3759df): threehalfs = 1.5 x2 = number * 0.5 y = number
packed_y = struct.pack('f', y) i = struct.unpack('i', packed_y)[0] # treat float's bytes as int i = magic - (i >> 1) # arithmetic with magic number packed_i = struct.pack('i', i) y = struct.unpack('f', packed_i)[0] # treat int's bytes as float
y = y * (threehalfs - (x2 * y * y)) # Newton's method return y
The file search_magic_num_fisqt.py contain the main algorithm.
It utlizies all available cores on the system for multiple processes to search as quick as possible. I implemented this to save a lot of time as 8 cores > 1 core searching.
Initial Results
For different values for 1/x^2, the optimal magic number varies as seen in the graph. The optimal number used in the original algorithm is just about in the middle of the y values.
Trying to train a neural net on this data
I couldn't think of an easy way to serialize this result, which is a continuous function, into a closed form equation. I tried some Tensorflow regression networks, as seen in nn_fisqty.py but the model just couldn't easily fit. I may come back to this later.
