G'day Athanasius,

"What am I doing wrong?"

I don't think you're doing anything wrong: this looks like a bug.

I have a different Perl version to you: "(v5.18.1) built for darwin-thread-multi-2level". I have the same (builtin) version of Math::BigFloat as you; you might like to check the versions of the other modules involved. Here's what I've got:

Math::BigFloat 1.998
Math::BigInt 1.9991
Math::BigInt::Calc 1.997
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[In the following code from various modules, I've removed a fair amount of comments and whitespace (and, in some cases, code); just leaving what seems relevant to this discussion. Also, when comparing your code with module code, I've replaced your $n with $YOUR_n to avoid ambiguity.]

The code where the error occurs is in Math::BigInt::Calc:

sub _len {
    my $cx = $_[1];
    (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));  # <--- line 1209
}
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So, $cx is NOT an arrayref; it's undef. _len() is called by _digit() (which is defined just below _len()):

sub _digit {
    my ($c,$x,$n) = @_;
    my $len = _len('',$x);
    ...
}
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So, $x here is assigned to $cx in _len(); therefore, $x is undef. _digit() is called by Math::BigInt::digit() — see below for how we get to this method.

Math::BigFloat doesn't have a digit() method; it does have:

@ISA = qw/Math::BigInt/;
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Math::BigInt has:

...
my $CALC = 'Math::BigInt::Calc';
...
sub digit {
    my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
    $n = $n->numify() if ref($n);
    $CALC->_digit($x->{value},$n||0);
}
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When you call "$YOUR_n->digit(-1)", $_[0] (i.e. $YOUR_n) is a reference and the first line evaluates as:

my ($self,$x,$n) = (undef, $YOUR_n, -1);
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$n, assigned from $_[1] (i.e. -1), is not a reference so "$n = $n->numify()" isn't executed (i.e. $n remains -1).

$x, assigned from $YOUR_n, has no 'value' hash key. This means the third line is evaluated as:

Math::BigInt::Calc->_digit(undef, -1);
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I suspect either the 'value' hash key should have been assigned earlier or a different hash key should have been used instead of 'value'. I haven't investigated this further.

Here's a short test demonstrating the issue:

#!/usr/bin/env perl

use 5.010;
use strict;
use warnings;

use Math::BigFloat;

my $n = Math::BigFloat->new(123);

say $n;
say 'REF $n: ', ref $n;

eval {
    say $n->digit(-1);
};
warn $@ if $@;

say '$n->{value} EXISTS: ', exists $n->{value} ? 'Yes' : 'No';
say '$n->{value} DEFINED: ', defined $n->{value} ? 'Yes' : 'No';

eval {
    $n->{value} = [123];
    say '$n->digit(-1): ', $n->digit(-1);
    say '$n->digit(1):  ', $n->digit(1);
    say '$n->digit():   ', $n->digit();
};
warn $@ if $@;
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Output:

123
REF $n: Math::BigFloat
Can't use an undefined value as an ARRAY reference at /Users/ken/perl5
+/perlbrew/perls/perl-5.18.1t/lib/5.18.1/Math/BigInt/Calc.pm line 1209
+.
$n->{value} EXISTS: No
$n->{value} DEFINED: No
$n->digit(-1): 1
$n->digit(1):  2
$n->digit():   3
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I'll leave you to verify that and, if correct, raise a bug report.