Problems with SKILL functions: fix, truncate, floor

Hi all,
I'm running into some problems with integer-skill functions.

I do the following directly in the CIW, it also won't work when I use
these functions in my pcells that I'm trying to get to work correctly:

Set the variables:

w=0.7
x=0.8

so (w+0.1)/x evaluates correctly to 1.0

fix((w+0.1)/x) evaluates falsly to 0,
wheras fix(1.0) gives correctly 1

same problems with truncate and floor.
What am I missing here?

Manuel

 

everybody thinks they find problems with the SKILL
integer functions.

in the case where you think (w+0.1)/x evaluates to 1.0,
i think you will actually find that
(w+0.1)/x == 1.0 evaluates to nil.

i.e., (w+0.1)/x prints as 1.0 but is acutally something
like .9999999999999999999999


don't be confused into thinking that just because
a number prints as 1.0 that it is identically equal
to 1.0

-jim

 

And to add to what Jim says - this is not unique to SKILL. Other
languages (C for example) will do exactly the same thing.

The confusion arises primarily because rounding errors occur because
floating point numbers are represented with a binary mantissa in IEEE
floating point formats. 0.1 (decimal) is an infinitely recurring
binary fraction. So whilst you would be quite expecting to see
rounding errors if you do manual decimal arithmetic, and represent
1/3 as 0.33333 to a finite number of digits, you don't expect that the
machine will have the same problem in its representation of 1/10.

In any number base, you can only represent numbers which are sums of
fractions of the factors of the base exactly. In decimal this means
1/10, 1/5, 1/2 and combinations and powers of these. In binary that
only means 1/2 (and so 1/4, 1/8, 1/16 etc). You cannot represent 1/10
as a sum of powers of 1/2 exactly.

So it's very easy to get rounding errors on numbers that would appear
to be easy to calculate as a human, as Jim outlined.

Regards,

Adrew.

 

Yep. Also, note that SKILL offers a function that C doesn't: round().
We often use this when we need to get a polygon point onto a
manufacturing grid.

If you truly need floor, truncate, equality, etc., it's often necessary
to fudge with an epsilon value. Unfortunately, there's no "silver
bullet" -- each floating-point operation can potentially add some error,
and the correct epsilon to use depends on the computation. Entire PhD
theses have been devoted to this subject.

A few Google searches on this topic for further reading. If your head
doesn't hurt after reading these, you might consider a career in
numerical analysis...

J. Demmel, "Basic Issues in Floating Point Arithmetic and Error
Analysis" (Berkeley CS267 lecture notes)
http://http.cs.berkeley.edu/~demmel/cs267/lecture21/lecture21.html

"Python Tutorial: B. Floating Point Arithmetic: Issues and Limitations"
http://docs.python.org/tut/node15.html

D. Goldberg, "What Every Computer Scientist Should Know About
Floating-Point Arithmetic."
http://docs.sun.com/source/806-3568/ncg_goldberg.html

 

actually COMMON LISP does offer lossless calculation in the
form of rational numbers. i.e., internally represented as
pairs of arbitrarily large integers (bignums). The calculations
can be expensive and memory intensive but you can really do the
equivalent of

a = 1 / 3
b = a * 3 ; returns 1 exactly

(setf a (/ 1 3)
b (* a 3))


I'm not sure why SKILL does not implement bignums and rationals.

-jim

 

I would like to remind us that 0.99999... == 1 should return true.
without using series convergence, an easy way to see it is :
x=0.99999...
10x-x
=9.9999... - 0.99999
=9
=9x
and 9x=x has x=1 for solution.

 

Er, this is rubbish. That's only true if there is an infinite sequence of 9's in the fractional
part, or you extend the number with another 9 when you multiply by 10.
Otherwise, you'll end up with a rounding error, or zero-extension error. For example,
imagine you were representing a number with 6 decimal significant digits:

x=0.999999
10x => 9.999990
10x-x => 8.999991 (which may well get rounded down to 8.99999)
and so it does not give 9 as the solution. Since 9x=8.999991 gives
x being 0.999999 your argument falls over...

Andrew.

 

Probably because SKILL is based on a flavour of lisp which pre-dates Common Lisp,
and nobody has ever asked for rational number support.

Andrew.

 

another intuitive way to see this is

1/9 = .1111111111...
2/9 = .2222222222...
3/9 = .3333333333...
4/9 = .4444444444...
....
8/9 = .8888888888...
9/9 = .9999999999...
but of course 9/9 is just another way of writing 1

-jim

 

There are various round() functions available with some C/C++ libraries. You
can also implement your own round (though not as good as a built-in
function):
* If number is positive, add 0.5 and convert to an integer
* If number is negative, subtract 0.5 and convert to an integer

If you want to round to, say, tenths, then multiply the floating-point
number by 10, round, and divide by 10 (using floating-point division).

Frank