Since & halves the number of 1s and | halves the number of 0s one would expect that the pattern for 1-X would just be the pattern for X with all the operators inverted: 

21 = (32 - 11) => ( ( R & R ) | R & R ) | R
27 = (32 - 5) => ( R | R & R ) | R | R

[download]

I see that 9 is also missing from your list.

I presume that the goal is to get exact ratios with the smallest number of operators? One way to explore this space is to create a Perl script to list the resulting spread for all the expressions with a certain number of operators

Update: I think this is about right (and if not then it should be fixable)

use strict;
use warnings;

sub operations
  {
    my($num_ops) = @_;

    #return (["M | M", 3 , 4], ["M & M", 1, 4]) 
    #                                if($num_ops == 1);
    return ["M", 1, 2] if($num_ops == 0);

    my @result;
    for(my $i=0;$i<$num_ops;$i++)
      {
        my @first = operations($i);
        my @second = operations($num_ops - $i - 1);
        foreach my $first_part (@first)
          {
            my($fp_expr,$fp_top,$fp_bot) = @{$first_part};
            foreach my $second_part (@second)
              {
                my($sp_expr,$sp_top,$sp_bot) = @{$second_part};
                # Two choices & or |
                # (NY + M(X-N))/(X*Y)
                # (X*Y - ((X-N)Y + (Y-M)N))/(X*Y)
                push @result,["($fp_expr) & ($sp_expr)",
                              $fp_bot*$sp_bot - 
                              ($fp_bot - $fp_top)*$sp_bot -
                              ($sp_bot - $sp_top)*$fp_top,
                              $fp_bot*$sp_bot],
                             ["($fp_expr) | ($sp_expr)",
                              $fp_top*$sp_bot +
                              $sp_top*($fp_bot - $fp_top),
                              $fp_bot*$sp_bot];
              }
          }
      }
    return @result;
  }

# Update: Edited the calling routine to provide easier to use results
my @got;

foreach my $count_expr (0..4)
  {
    foreach my $expr (operations($count_expr))
      {
        my($sp_expr,$top,$bot) = @{$expr};
        $top = $top * (32 / $bot);
        next if(defined $got[$top]);

        $sp_expr =~ s/\(M\)/M/g;
        $got[$top] = sprintf "%02d: $sp_expr\n",$top;
      }
  }

for(my $i=1;$i<32;$i++)
  {
    print $got[$i];
  }

[download]

Which gives results:

01: M & (M & (M & (M & M)))
02: M & (M & (M & M))
03: M & (M & (M & (M | M)))
04: M & (M & M)
05: M & (M & (M | (M & M)))
06: M & (M & (M | M))
07: M & (M & (M | (M | M)))
08: M & M
09: M & (M | (M & (M & M)))
10: M & (M | (M & M))
11: M & (M | (M & (M | M)))
12: M & (M | M)
13: M & (M | (M | (M & M)))
14: M & (M | (M | M))
15: M & (M | (M | (M | M)))
16: M
17: M | (M & (M & (M & M)))
18: M | (M & (M & M))
19: M | (M & (M & (M | M)))
20: M | (M & M)
21: M | (M & (M | (M & M)))
22: M | (M & (M | M))
23: M | (M & (M | (M | M)))
24: M | M
25: M | (M | (M & (M & M)))
26: M | (M | (M & M))
27: M | (M | (M & (M | M)))
28: M | (M | M)
29: M | (M | (M | (M & M)))
30: M | (M | (M | M))
31: M | (M | (M | (M | M)))

[download]

In reply to Re: Boolean math: Fill in the blanks. by hawtin
in thread Boolean math: Fill in the blanks. by BrowserUk

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