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Re: [PBML] Why does Perl insist that 1 does not equal 1??

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  • Rob Biedenharn
    ... This is application dependent. For your example, checking within . 0001 would be more than enough. ... You re making things worse, not better. Using
    Message 1 of 8 , Nov 6, 2006
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      On Nov 6, 2006, at 6:06 PM, thisis_not_anapple wrote:

      > Rob, I looked at the document you linked to but it's a little over my
      > head. There's a reasonably good explanation on floating points in
      > Wikipedia (which also links to the same article) so I think I got the
      > gist of it.
      >
      > While, I was aware of the concept of floating point rounding errors, I
      > guess I always assumed (incorrectly) that it was only an issue for
      > complex calculations. I was also especially surprised to see that the
      > order of addition makes a difference. However, I suppose that makes
      > sense since changing the order of addition will change the
      > intermediate results, some of which may contribute differently to
      > roundoff errors.
      >
      > So is it fair to say, then, that one should NEVER do a direct
      > comparison on floating point numbers but ALWAYS check only that
      > they're the same to a given precision? If so, are there any accepted
      > best practises for doing the check and for how many digits of
      > precision to check for?

      This is application dependent. For your example, checking within .
      0001 would be more than enough.

      >
      > From what I gathered, there should be 16 significant digits (in
      > decimal) stored in a floating point value, but the 16th digit may be
      > wrong due to roundoff error so you should never trust a floating point
      > to equal a decimal to more than 15 digits. However, since roundoff
      > errors can accumulate during calculations, any number that's a result
      > of a calculation won't necessarily match it's mathematical result to
      > 15 digits.
      >
      > Anyway, here's an attempt to compare floating point numbers correctly
      > that still has me a little mystified:

      ...code removed...

      > So:
      > 1) When using 16 significant digits, my compf() function incorrectly
      > claims that $sum is greater than 1, even though the sprintf() result
      > indicates otherwise. Is this because '<' is the wrong operator when
      > comparing numbers represented as strings? If not, what have I
      > missed here?

      You're making things worse, not better. Using numeric less-than (<)
      causes the strings to be coerced back to number before being
      compared. You need 'le' if you want to compare strings (but be
      careful as ("10" le "2") is true).

      > 2) The sprintf function seems to return a value with 17 significant
      > digits. I thought floating point numbers only contain 16...

      You ask for 17 and it will try to give them to you (I don't know if
      you can trust them to have any accuracy when you get that precise.)

      As an exercise, I tried:
      $ perl -le 'print sprintf("%.100g", log(2));'
      0.69314718055994528622676398299518041312694549560546875
      $ perl -le 'print length sprintf("%.100g", log(2));'
      55

      So it apparently only gives a max of 53 digits on my system. And they
      are probably not correct after about 15:
      $ perl -MMath::Trig -le 'print sprintf("%.100g", Math::Trig::acos(-1));'
      3.141592653589793115997963468544185161590576171875
      see http://
      3.141592653589793238462643383279502884197169399375105820974944592.com/
      or http://en.wikipedia.org/wiki/Pi
      or http://www.research.att.com/~njas/sequences/A00796

      > 3) Why does the 'eq' operator return a true value when the '=='
      > returns a false? From my experimenting, I find that the print
      > statement rounds off floating point numbers to 15 digits (which makes
      > sense if it's trying to hide roundoff errors in the last digit). When
      > applying the 'eq' operator, does it also round off a floating point to
      > 15 digits prior to converting to a string?
      >
      > 4) Is there a more efficient version of compf() that everyone other
      > than me knows about? (Ok, I'm a little paranoid... ;) )
      >
      > Thanks for all your help!

      Here's what I used recently when dealing with percentages (numbers
      between 0 and 1.0) that had about 6 significant digits.

      my $epsilon = 1.0e-8;

      sub approx_eq {
      my ($a, $b, $tolerance) = @_;
      return abs($a - $b) < $tolerance;
      }
      sub approx_ge {
      my ($a, $b, $tolerance) = @_;
      return($a > $b or approx_eq(@_));
      }

      Used as:
      push @error_msgs, "percentages don't sum to 1.0 (sum=$sum)"
      unless (approx_eq($sum, 1.0, $epsilon));

      or as:
      if (approx_ge($count, $nth, $epsilon)) {
      ...
      }

      You could make the tolerance optional and use a default in the sub,
      too. In my case, there was actually a command-line option to force a
      different value for $epsilon.

      -Rob

      Rob Biedenharn http://agileconsultingllc.com
      Rob@...
    • thisis_not_anapple
      ... ... I see your point. I didn’t want to use the ‘le’ operator since I wanted a numerical value comparison not an a string comparison which
      Message 2 of 8 , Nov 9, 2006
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        --- In perl-beginner@yahoogroups.com, Rob Biedenharn <Rob@...> wrote:
        <snip>

        > You're making things worse, not better. Using numeric less-than (<)
        > causes the strings to be coerced back to number before being
        > compared. You need 'le' if you want to compare strings (but be
        > careful as ("10" le "2") is true).
        >

        I see your point. I didn’t want to use the ‘le’ operator since I
        wanted a numerical value comparison not an a string comparison which
        would fail in situations like the example you mentioned.

        In the example I gave, my function failed because after doing the
        initial string comparison for equality, it used the string in the ‘<’
        operation. Since one of the strings was ‘0.9999999999999999’ as a
        result of the sprintf operation, it got rounded up to ‘1’ when
        converted back to a number (presumably, due to roundoff error again).
        I modified the function to retain the original numerical values for
        use in the ‘<’ operation and now the function seems to work as expected:

        sub compf2 {
        my ($f1, $f2, $sig) = @_;
        my $s1 = sprintf("%.${sig}g", $f1);
        my $s2 = sprintf("%.${sig}g", $f2);
        if ($s1 eq $s2) {
        return 0; # equal
        } elsif ($f1 < $f2) {
        return -1; # less-than
        } else {
        return 1; # greater-than
        }
        }

        Doing some more experimentation I see that (at least on my system) the
        ‘print’ statement will round numbers to 15 significant digits and the
        same thing happens when numbers are automatically converted to
        strings. So using ‘eq’ instead of ‘==’ could be a quick way of
        checking for numerical equality to an accuracy of 15 significant
        digits. However, I suspect that’s too much accuracy to account for
        error accumulation from calculations and I’m not sure if this behavior
        is universal for all versions of Perl.

        > > 2) The sprintf function seems to return a value with 17 significant
        > > digits. I thought floating point numbers only contain 16...
        >
        > You ask for 17 and it will try to give them to you (I don't know if
        > you can trust them to have any accuracy when you get that precise.)
        >
        > As an exercise, I tried:
        > $ perl -le 'print sprintf("%.100g", log(2));'
        > 0.69314718055994528622676398299518041312694549560546875
        > $ perl -le 'print length sprintf("%.100g", log(2));'
        > 55
        >
        > So it apparently only gives a max of 53 digits on my system. And they
        > are probably not correct after about 15:
        > $ perl -MMath::Trig -le 'print sprintf("%.100g", Math::Trig::acos(-1));'
        > 3.141592653589793115997963468544185161590576171875
        > see http://
        > 3.141592653589793238462643383279502884197169399375105820974944592.com/
        > or http://en.wikipedia.org/wiki/Pi
        > or http://www.research.att.com/~njas/sequences/A00796
        >

        On my system, the sprintf function never returns a number with more
        than 17 significant digits. I tried it with the examples you gave.
        Typically the 17th digit is not accurate. In your pi example, above,
        although many additional digits are shown, it loses accuracy after the
        16th digit. I’m not clear on where the extra digits are pulled from. I
        suspect the differences in the maximum number of digits shown between
        are systems comes down to Perl version and/or OS. At first, I thought
        maybe on your system more memory is used to store the floating point,
        so it may have a higher accuracy. But if that were the case, pi should
        match its true value to far more digits than 16, which did not seem to
        be the case.

        In any case, it seems like nothing past 15 digits should ever be trusted…

        > Here's what I used recently when dealing with percentages (numbers
        > between 0 and 1.0) that had about 6 significant digits.
        >
        > my $epsilon = 1.0e-8;
        >
        > sub approx_eq {
        > my ($a, $b, $tolerance) = @_;
        > return abs($a - $b) < $tolerance;
        > }
        > sub approx_ge {
        > my ($a, $b, $tolerance) = @_;
        > return($a > $b or approx_eq(@_));
        > }
        >
        > Used as:
        > push @error_msgs, "percentages don't sum to 1.0 (sum=$sum)"
        > unless (approx_eq($sum, 1.0, $epsilon));
        >
        > or as:
        > if (approx_ge($count, $nth, $epsilon)) {
        > ...
        > }
        >
        > You could make the tolerance optional and use a default in the sub,
        > too. In my case, there was actually a command-line option to force a
        > different value for $epsilon.

        That approach seems reasonable when you have an expectation for what
        the magnitude of the numbers you’re comparing are. However, I’d like
        to have a function I can call for comparison that is independent of
        scale, which is why I like the idea of rounding to an arbitrary number
        of significant digits, which effectively gives you a percentage
        accuracy regardless of scale.

        I suppose, the same could be true of your approach, if you make
        $epsilon dependent on the other arguments. For instance calling
        something like: approx_eq($a, $b, $a*10**-8) would give you roughly
        the same precision, even if $a and $b were on the scale of 10-7. Or
        your functions could be modified to do this automatically. Something like:

        sub approx_eq {
        my ($a, $b, $tolerance) = @_;
        if (abs($a) > abs($b)) {
        $tolerance = abs($a) * 10**-$tolerance;
        } else {
        $tolerance = abs($b) * 10**-$tolerance;
        }
        return abs($a - $b) < $tolerance;
        }

        Finally, can you get into situations where the accuracy seems worse
        due to rounding effects? For instance if a calculation should
        theoretically give:
        0.1234999999999987 but due to roundoff error gives 0.1235000000000013.
        In this case if you round off the number to 4 to 14 significant digits
        they will be equal. But if you were to round off to only 3 significant
        digits, they wouldn’t. Although, I believe in this case comparing the
        absolute difference to a tolerance would always work while using the
        sprintf approach to round each number first would fail in rare cases.

        Is that right?
        Am I just over thinking this at this point?

        Thanks again!

        P.S. Sorry for the rambling…
      • Rob Biedenharn
        ... It s not roundoff error exactly. The problem is fundamentally that the computer is storing floating BINARY point values that represent floating DECIMAL
        Message 3 of 8 , Nov 10, 2006
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          On Nov 9, 2006, at 7:29 PM, thisis_not_anapple wrote:

          > --- In perl-beginner@yahoogroups.com, Rob Biedenharn <Rob@...> wrote:
          > <snip>
          >
          >> You're making things worse, not better. Using numeric less-than (<)
          >> causes the strings to be coerced back to number before being
          >> compared. You need 'le' if you want to compare strings (but be
          >> careful as ("10" le "2") is true).
          >>
          >
          > I see your point. I didn’t want to use the ‘le’ operator since I
          > wanted a numerical value comparison not an a string comparison which
          > would fail in situations like the example you mentioned.
          >
          > In the example I gave, my function failed because after doing the
          > initial string comparison for equality, it used the string in the â
          > €˜<’
          > operation. Since one of the strings was ‘0.9999999999999999’ as a
          > result of the sprintf operation, it got rounded up to ‘1’ when
          > converted back to a number (presumably, due to roundoff error again).

          It's not "roundoff" error exactly. The problem is fundamentally that
          the computer is storing floating BINARY point values that represent
          floating DECIMAL point numbers. As an exercise, let's look at how to
          represent 0.3 (decimal) as binary:

          n 2^n bit cummulative
          0 1. 0.
          -1 0.5 0
          -2 0.25 1 0.25000000000000000
          -3 0.125 0
          -4 0.0625 0
          -5 0.03125 1 0.28125000000000000
          -6 0.015625 1 0.29687500000000000
          -7 0.0078125 0
          -8 0.00390625 0
          -9 0.001953125 1 0.29882812500000000
          -10 0.0009765625 1 0.29980468750000000
          -11 0.00048828125 0
          -12 0.000244140625 0
          -13 0.0001220703125 1 0.29992675781250000
          -14 0.00006103515625 1 0.29998779296875000
          -15 0.000030517578125 0

          so that's 0.010011001100110... (Trust me that it goes on forever
          like this.)

          At some point the limit of bits is set and some values are never
          going to be represented exactly. (But it's interesting to think that
          a number like 0.296875 is represented more accurately that 0.3 within
          the computer.)

          > In any case, it seems like nothing past 15 digits should ever be
          > trusted…
          >
          > .... Although, I believe in this case comparing the
          > absolute difference to a tolerance would always work while using the
          > sprintf approach to round each number first would fail in rare cases.
          >
          > Is that right?
          > Am I just over thinking this at this point?
          >
          > Thanks again!
          >
          > P.S. Sorry for the rambling…

          If you need that kind of accuracy, you probably need to deal with a
          lot more theory. Everything you'd need is in the paper I first cited
          (or probably in the Wikipedia article cited by someone else). One
          thing that quickly gets into a danger area is doing operations on
          values with vastly different magnitudes.

          If you know the number of digits and the scale, you can use fixed
          point arithmetic where an integer represents something like the
          number of pennies or 1/256th nautical mile (like an air traffic
          control system I worked on years ago with no floating point
          coprocessor). Believe me, things get much more complicated when you
          have to keep track of the scale yourself.

          -Rob

          Rob Biedenharn http://agileconsultingllc.com
          Rob@...
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