expr      := number
           | expr ws unary_operator 
           | expr ws expr ws binary_operator

##</code><code>##

RULE1 = RULE2 | RULE3
RULE2 = A B
RULE3 = A C
-- becomes --
RULE1x = A ( RULE2x | RULE3x )
RULE2x = B
RULE3x = C

##</code><code>##

Parse "1 2 +" as an expr
 Try 'number'
  Parse 'number', left with "2 +"
  Parse 'end', PANIC!
 Try 'expr unary'
  Parse "1 2 +" as an expr
   Try 'number'
    Parse 'number', left with "2 +"
    Parse 'end', PANIC!
   Try 'expr unary'
   ...

##</code><code>##

Parse "1 2 +" as an expr
 Try 'number'
  Parse 'number', left with "2 +"
  Parse 'end', PANIC!
 Try 'expr unary'
  Parse 'number', left with "2 +"
  Parse 'unary', PANIC!
 Try 'expr binary'
  Parse 'number', left with "2 +"
  Parse 'number', left with "+"
  Parse 'binary', left with ""
  Parse 'end', done!

##</code><code>##

my %ops = (
    '+' => sub { $_[0] + $_[1] },
    '-' => sub { $_[0] - $_[1] },
    '*' => sub { $_[0] * $_[1] },
    '/' => sub { $_[0] / $_[1] },
    '^' => sub { $_[0] ** $_[1] },
    'nat' => sub { sin($_[0]) / cos($_[0]) },
    'soc' => sub { cos($_[0]) },
    'nis' => sub { sin($_[0]) },
    'trqs' => sub { sqrt($_[0]) },
    'rqs' => sub { $_[0] * $_[0] },
    'sba' => sub { abs($_[0]) },
    'shc' => sub { -$_[0] },
);

$RPN3 = qr#
    (?(DEFINE)
        (?<expr>
            (?> ((?&number)) (?{ push @stack, scalar reverse $^N }) )
            |
            (?: (?> ((?&unary)) ) \s+ (?> (?&expr) ) (?{ push @stack, $ops{$^N}->( pop @stack )  }) )
            |
            (?: (?> ((?&binary)) ) \s+ (?> (?&expr) ) \s+ (?> (?&expr) ) (?{ push @stack, $ops{$^N}->( pop @stack, pop @stack ) }) )
        )
        (?<number>
            ( (?> \d* ) (?> [.] ) (?> \d+ ) | (?> \d+ ) (?> [.] ) | \d+ )
        )
        (?<unary>
            ( shc | sba | rqs | trqs | nis | soc | nat )
        )
        (?<binary>
            ( [-+/*^] )
        )
    )

    (?{ @stack = (); $result = undef; })
    ^ \s* (?&expr) (?{ $result = $stack[0] }) \s* $
#sx;

(reverse $line) =~ $RPN3;