what is the usage of \& and $expr->() - perl

Question

sub reduce(&#) {
my $expr = \&{shift #ARG};
my $result = shift #ARG;
while (scalar #ARG > 0) {
our $a = $result;
our $b = shift #ARG;
$result = $expr->();
}
return $result;
}
I cannot really understand some grammar in this code. Anyone can explain to me? like \& and $result = $expr->()

Answer 1

\&name returns a reference to the subroutine named name.
$code_ref->() calls the subroutine referenced by $code_ref.
$ perl -e'
sub f { CORE::say "Hi" }
my $code_ref = \&f;
$code_ref->();
'
Hi
In your case, shift #ARG returns a subroutine reference. \&{ $code_ref } simply returns the code ref. As such,
my $expr = \&{shift #ARG};
could have been written as
my $expr = shift #ARG;
Note that reduce's prototype allows it to be called as
reduce { ... } ...
but what is actually executed is
reduce( sub { ... }, ... )
Note that this version of reduce is buggy. You should use the one provided by List::Util.
local $a and local $b should have been used to avoid clobbering the values its caller might have in $a and $b.
This version of reduce expects its callback to have been compiled in the same package as reduce itself. Otherwise, the callback sub won't be able to simply use $a and $b.
Declaring the variables using our is actually completely useless in this version case since $a and $b are exempt from use strict; checks, and the undeclared use of $a and $b would access the very same package variables.

Answer 2

Having a look some List::Util::reduce() examples will probably help.
Let's take the first one:
$foo = reduce { $a > $b ? $a : $b } 1..10;
So reduce takes a BLOCK followed by a LIST, which the function signature declares: sub reduce(&#) {. The block in our case is the statement $a > $b ? $a : $b, while the list is 1..10. From the docs:
Reduces #list by calling "BLOCK" in a scalar context multiple times,
setting $a and $b each time. The first call will be with $a and $b set to
the first two elements of the list, subsequent calls will be done by
setting $a to the result of the previous call and $b to the next element
in the list.
Returns the result of the last call to the "BLOCK". If #list is empty then
"undef" is returned. If #list only contains one element then that element
is returned and "BLOCK" is not executed.
And now to an annotated version of the code:
$foo = reduce { $a > $b ? $a : $b } 1..10; # $foo will be set to 10
sub reduce(&#) {
# reduce() takes a BLOCK followed by a LIST
my $expr = \&{shift #ARG};
# $expr is now a subroutine reference, i.e.
# $expr = sub { $a > $b ? $a : $b };
# Start by setting $result to the first item in the list, 1
my $result = shift #ARG;
# While there are more items in the list...
while (scalar #ARG > 0) {
# Set $a to the current result
our $a = $result;
# Set $b to the next item in the list
our $b = shift #ARG;
# Set $result to the result of $a > $b ? $a : $b
$result = $expr->();
}
# List has now been reduced by the operation $a > $b ? $a : $b
return $result;
}