Quine multiple times

CJam, 28 26 bytes * 0.9 = 23.4

Thanks to Sp3000 for saving 2 bytes.

{`"_~"+rd_z26*,\g1|@%f=}_~

Test it here.

Explanation

{`"_~"+ e# Generalised quine framework. Leaves the source code on the stack.
  rd    e# Read input and parse as double.
  _z    e# Copy and get absolute value.
  26*   e# Multiply by 26, the length of the quine to get length N of the output.
  ,     e# Get a range [0 1 ... floor(N-1)].
  \g    e# Swap with other copy and computer signum to determine direction of string.
  1|    e# This might be zero though, so take bitwise OR with 1 to avoid an error.
  @%    e# Pull up source string. Reverse it if the signum was -1 (no-op otherwise).
  f=    e# The range we pushed earlier corresponds to the (cyclic) indices of the source
        e# which make up the desired result, so we map each index to the corresponding
        e# character to get the final result.
}_~

Vitsy, 34 * 0.9 = 30.6 Bytes

Thanks to @Sp3000 for pointing out a flaw in my code!

Woo. My Physics teacher reminded me I had power functions to help me with this. Go figure.

'r(;Vd3*V2^12/^DvV/1+(rvl1-*\[DO{]
'                                   Start recording as a string - this grabs everything and pushes it to the stack as a string.
 r                                  Reverse the stack order.
  (;                                If the top item is zero, exit the program.
    V                               Grab the input as a final global variable.
     d3*                            Push the character ' to the stack.
        V2^12/^                     Get the absolute value of the input value.
               Dv                   Duplicate and save in a temp variable.
                 V                  Push the global variable to the stack.
                  /                 Divide the top two items - this gets -1 or 1 depending on the polarity of the input.
                   1+(              If it's -1, do the next instruction. Otherwise, don't.
                      r             Reverse the stack
                       v            Push the temporary variable to the stack.
                        l1-*        Multiply by the length of the stack minus 1.
                            \[   ]  Repeat everything in brackets top item of the stack times.
                              DO{   Duplicate an item, pop it off the stack and output it, then move one item over in the stack.

Perl, 104 bytes - 10% = 93.6

perl -i-0.3 -e '$_=q{$_="\$_=q{$_};eval";$_=reverse if$^I<0;$n=abs$^I;print+($_ x$n).substr$_,0,y///c*($n-int$n)};eval'

102 bytes + 2 bytes for -i - 10% for not being a palindrome. Input is passed as the argument to -i (e.g. -0.3 above).

How it works

This solution is based on the following quine:

$_=q{print"\$_=q{$_};eval"};eval

This works as follows. First, set $_ to the string:

print"\$_=q{$_};eval"

Next, call eval, which works on $_ by default. This calls print with one argument, a string literal:

"\$_=q{$_};eval"

Since this string is double-quoted, variables are interpolated. After interpolating $_, the value of the string is:

\$_=q{print"\$_=q{$_};eval"};eval

When printed, this outputs:

$_=q{print"\$_=q{$_};eval"};eval

which is the source code of the program itself.

The nice thing about this quine is that you can embed arbitrary code inside the string to be eval'd.

---

Here's a breakdown of the full solution:

perl -i-0.3 -e'
    $_=q{                     # string to be eval'd
        $_="\$_=q{$_};eval";  # append beginning and end of quine so the
                              #  entire thing can be reversed if necessary
        $_=reverse if $^I<0;  # reverse if input < 0
        $n=abs $^I;           # set $n to absolute value of input
        print                 # print
            +($_ x $n)        # $_ repeated $n times
            .                 # concatenated with
            substr $_,        # substring of $_
                   0,         # starting at the beginning
                   y///c      # having length x, where x is length of $_
                   *          # multiplied by
                   ($n-int$n) # fractional part of $n
    };
    eval                      # eval $_
'