[shuffle] Stand back! I'm about to (try to) do math!

Especially with blends and large tree heights there was a problem with
the fuzzer where it would end up with enough undef shuffle elements in
enough parts of the tree that in a birthday-attack kind of way we ended
up regularly having large numbers of undef elements in the result. I was
seeing reasonably frequent cases of *all* results being undef which
prevents us from doing any correctness checking at all. While having
undef lanes is important, this was too much.

So I've tried to apply some math to the probabilities of having an undef
lane and balance them against the tree height. Please be gentle, I'm
really terrible at math. I probably made a bunch of amateur mistakes
here. Fixes, etc. are quite welcome. =D At least in running it some, it
seems to be producing more interesting (for correctness testing)
results.

git-svn-id: https://llvm.org/svn/llvm-project/llvm/trunk@215540 91177308-0d34-0410-b5e6-96231b3b80d8

utils/shuffle_fuzz.py

@@ -57,9 +57,46 @@ def main():
       'f32': 1 << 32, 'f64': 1 << 64}[element_type]
 
   shuffle_range = (2 * width) if args.blends else width
-  shuffle_indices = [-1] + range(shuffle_range)
 
-  shuffle_tree = [[[random.choice(shuffle_indices)
+  # Because undef (-1) saturates and is indistinguishable when testing the
+  # correctness of a shuffle, we want to bias our fuzz toward having a decent
+  # mixture of non-undef lanes in the end. With a deep shuffle tree, the
+  # probabilies aren't good so we need to bias things. The math here is that if
+  # we uniformly select between -1 and the other inputs, each element of the
+  # result will have the following probability of being undef:
+  #
+  #   1 - (shuffle_range/(shuffle_range+1))^max_shuffle_height
+  #
+  # More generally, for any probability P of selecting a defined element in
+  # a single shuffle, the end result is:
+  #
+  #   1 - P^max_shuffle_height
+  #
+  # The power of the shuffle height is the real problem, as we want:
+  #
+  #   1 - shuffle_range/(shuffle_range+1)
+  #
+  # So we bias the selection of undef at any given node based on the tree
+  # height. Below, let 'A' be 'len(shuffle_range)', 'C' be 'max_shuffle_height',
+  # and 'B' be the bias we use to compensate for
+  # C '((A+1)*A^(1/C))/(A*(A+1)^(1/C))':
+  #
+  #   1 - (B * A)/(A + 1)^C = 1 - A/(A + 1)
+  #
+  # So at each node we use:
+  #
+  #   1 - (B * A)/(A + 1)
+  # = 1 - ((A + 1) * A * A^(1/C))/(A * (A + 1) * (A + 1)^(1/C))
+  # = 1 - ((A + 1) * A^((C + 1)/C))/(A * (A + 1)^((C + 1)/C))
+  #
+  # This is the formula we use to select undef lanes in the shuffle.
+  A = float(shuffle_range)
+  C = float(args.max_shuffle_height)
+  undef_prob = 1.0 - (((A + 1.0) * pow(A, (C + 1.0)/C)) /
+                      (A * pow(A + 1.0, (C + 1.0)/C)))
+
+  shuffle_tree = [[[-1 if random.random() <= undef_prob
+                       else random.choice(range(shuffle_range))
                     for _ in itertools.repeat(None, width)]
                    for _ in itertools.repeat(None, args.max_shuffle_height - i)]
                   for i in xrange(args.max_shuffle_height)]