$ perl  index_regex_bench.pl
           Rate Regex Index
Regex 5010020/s    --  -23%
Index 6544503/s   31%    --
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This is one of the code versions I used, the one with which I obtained the above timings:

#!/usr/bin/perl 
use Benchmark qw(:all); 
use strict; 
use warnings; 

my $sentence = "The quick brown for jumps over the lazy dog"; 
my $results = timethese(5000000, 
{ 
    'Index' => \&code1, 
    'Regex' => \&code2, 
}, 
'none' 
); 
cmpthese( $results ) ; 

#------ 
sub code1 { 
    my $val = index $sentence, "fox" ; 
} 
#------ 
sub code2 { 
    my $val = ($sentence =~ /fox/);
}
[download]

Using a precompiled regex does not improve the speed of RE search in this case, quite the contrary.