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

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  • Paul Archer
    ... You got it. You need to think in terms of precision. For example, a simplistic way to do things, multiply everything by 10, add your four numbers up, use
    Message 1 of 8 , Nov 2, 2006
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      7:11pm, thisis_not_anapple wrote:

      >[massive snippage]
      > Is this some sort of floating point rounding issue?
      >
      You got it. You need to think in terms of precision. For example, a
      simplistic way to do things, multiply everything by 10, add your four
      numbers up, use int to make sure they're rounded off, and compare to 10
      instead of 1.

      Paul



      ----------------------
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      | $10,000 reward |
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      | Dead or Alive |
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    • thisis_not_anapple
      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
      Message 2 of 8 , Nov 6, 2006
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        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?

        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:

        The source:
        ======================================
        use strict;
        use warnings;

        my ($a, $b, $c, $d) = (.7, .1, .1, .1);
        my $sum = $a + $b + $c + $d;
        $\ = "\n";
        for my $sig (15..17) {
        print "\$sig = $sig";
        print "\$sum = $sum = ", sprintf("%.${sig}g", $sum);
        print "compf(\$sum,1,$sig) = ",compf($sum,1,$sig);
        print "\$sum eq 1 = ", ($sum eq 1),"\n";
        }

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

        The Output:
        ======================================
        $sig = 15
        $sum = 1 = 1
        compf($sum,1,15) = 0
        $sum eq 1 = 1

        $sig = 16
        $sum = 1 = 0.9999999999999999
        compf($sum,1,16) = 1
        $sum eq 1 = 1

        $sig = 17
        $sum = 1 = 0.99999999999999989
        compf($sum,1,17) = -1
        $sum eq 1 = 1

        ======================================

        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?

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

        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!
      • 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 3 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 4 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 5 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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