This reduces the 8316 loops your code requires to 197 which is almost 2 orders of magnitude faster tahn your original and 3.5x faster than grinders example that needs 686 loops. Only 4:1 wastage on the test is pretty good as semi brute force goes and certainly beats the original 166:1. All it does different is to limit the looping to the limited subset of possible cases....

my ( $loops, $cnt );
printf "%4d %4d %4d %4d %4d\n", 100,50,20,10,5;

for (my $e5=0; $e5<=100; $e5+=10 ) {
  for (my $e10=0; $e10<=100-$e5; $e10+=10 ) {
    for (my $e20=0; $e20<=100-$e10-$e5; $e20+=20 ) {
      for (my $e50=0; $e50<=100-$e20-$e10-$e5; $e50+=50 ) {
        for (my $e100=0; $e100<=100-$e50-$e20-$e10-$e5; $e100+=100 ) {
          $loops++;
          if ( $e5+$e10+$e20+$e50+$e100 == 100 ) {
            printf "%4d %4d %4d %4d %4d\n",$e100/100,$e50/50,$e20/20,$
+e10/10,$e5/5;
            $cnt++;
          }
        }
      }
    }
  }
}
print "\n$cnt possibilities in $loops loops\n";
__DATA__
 100   50   20   10    5
   1    0    0    0    0
   0    2    0    0    0
   0    0    5    0    0
   0    1    2    1    0
   0    0    4    2    0
   0    1    1    3    0
   0    0    3    4    0
   0    1    0    5    0
   0    0    2    6    0
   0    0    1    8    0
   0    0    0   10    0
   0    1    2    0    2
   0    0    4    1    2
   0    1    1    2    2
   0    0    3    3    2
   0    1    0    4    2
   0    0    2    5    2
   0    0    1    7    2
   0    0    0    9    2
   0    0    4    0    4
   0    1    1    1    4
   0    0    3    2    4
   0    1    0    3    4
   0    0    2    4    4
   0    0    1    6    4
   0    0    0    8    4
   0    1    1    0    6
   0    0    3    1    6
   0    1    0    2    6
   0    0    2    3    6
   0    0    1    5    6
   0    0    0    7    6
   0    0    3    0    8
   0    1    0    1    8
   0    0    2    2    8
   0    0    1    4    8
   0    0    0    6    8
   0    1    0    0   10
   0    0    2    1   10
   0    0    1    3   10
   0    0    0    5   10
   0    0    2    0   12
   0    0    1    2   12
   0    0    0    4   12
   0    0    1    1   14
   0    0    0    3   14
   0    0    1    0   16
   0    0    0    2   16
   0    0    0    1   18
   0    0    0    0   20

50 possibilities in 197 loops
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