Perl rounding error again

Question

The example to show the problem:

• having a number 105;
• divide with 1000 (result 0.105)
• rouded to 2 decimal places should be: 0.11

Answer 1

You're expecting a specific behaviour when the number is exactly 0.105, but floating point errors mean you can't expect a number to be exactly what you think it is.

105/1000 is a periodic number in binary just like 1/3 is periodic in decimal.

105/1000
       ____________________
= 0.00011010111000010100011 (bin)

~ 0.00011010111000010100011110101110000101000111101011100001 (bin)

= 0.10499999999999999611421941381195210851728916168212890625

0.1049999... is less than 0.105, so it rounds to 0.10.

But even if you had 0.105 exactly, that would still round to 0.10 since sprintf rounds half to even. A better test is 155/1000

155/1000
       ____________________
= 0.00100111101011100001010 (bin)

~ 0.0010011110101110000101000111101011100001010001111010111 (bin)

= 0.1549999999999999988897769753748434595763683319091796875

0.155 should round to 0.16, but it rounds to 0.15 due to floating point error.

$ perl -E'$_ = 155; say sprintf("%.2f", $_/1000);'
0.15

$ perl -E'$_ = 155; say sprintf("%.0f", $_/10)/100;'
0.16

The second one works because 5/10 isn't periodic, and therein lies the solution. As Sinan Unur said, you can correct the error by using sprintf. But you have to round to an integer if you don't want to lose your work.

$ perl -E'
   $_ = 155/1000;

   $_ *= 1000;                # Move decimal point past significant.
   $_ = sprintf("%.0f", $_);  # Fix floating-point error.
   $_ /= 10;                  # 5/10 is not periodic
   $_ = sprintf("%.0f", $_);  # Do our rounding.
   $_ /= 100;                 # Restore decimal point.

   say;
'
0.16

That will fix the rounding error, allowing sprintf to properly round half to even.

0.105  =>  0.10
0.115  =>  0.12
0.125  =>  0.12
0.135  =>  0.14
0.145  =>  0.14
0.155  =>  0.16
0.165  =>  0.16

If you want to round half up instead, you'll need to using something other than sprintf to do the final rounding. Or you could add s/5\z/6/; before the division by 10.

---

But that's complicated.

The first sentence of the answer is key. You're expecting a specific behaviour when the number is exactly 0.105, but floating point errors mean you can't expect a number to be exactly what you think it is. The solution is to introduce a tolerance. That's what rounding using sprintf does, but it's a blunt tool.

use strict;
use warnings;
use feature qw( say );

use POSIX qw( ceil floor );

sub round_half_up {
   my ($num, $places, $tol) = @_;

   my $mul = 1; $mul *= 10 for 1..$places;
   my $sign = $num >= 0 ? +1 : -1;

   my $scaled = $num * $sign * $mul;
   my $frac = $scaled - int($scaled);

   if ($sign >= 0) {
      if ($frac < 0.5-$tol) {
         return floor($scaled) / $mul;
      } else {
         return ceil($scaled) / $mul;
      }
   } else {
      if ($frac < 0.5+$tol) {
         return -floor($scaled) / $mul;
      } else {
         return -ceil($scaled) / $mul;
      }
   }
}

say sprintf '%5.2f', round_half_up( 0.10510000, 2, 0.00001);  #  0.11
say sprintf '%5.2f', round_half_up( 0.10500001, 2, 0.00001);  #  0.11  Within tol
say sprintf '%5.2f', round_half_up( 0.10500000, 2, 0.00001);  #  0.11  Within tol
say sprintf '%5.2f', round_half_up( 0.10499999, 2, 0.00001);  #  0.11  Within tol
say sprintf '%5.2f', round_half_up( 0.10410000, 2, 0.00001);  #  0.10
say sprintf '%5.2f', round_half_up(-0.10410000, 2, 0.00001);  # -0.10
say sprintf '%5.2f', round_half_up(-0.10499999, 2, 0.00001);  # -0.10  Within tol
say sprintf '%5.2f', round_half_up(-0.10500000, 2, 0.00001);  # -0.10  Within tol
say sprintf '%5.2f', round_half_up(-0.10500001, 2, 0.00001);  # -0.10  Within tol
say sprintf '%5.2f', round_half_up(-0.10510000, 2, 0.00001);  # -0.11

There's probably existing solutions that work along the same lines.