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Re: python math problem

 
 
Dennis Lee Bieber
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      02-15-2013
On Fri, 15 Feb 2013 15:23:29 -0500, Kene Meniru <(E-Mail Removed)>
declaimed the following in gmane.comp.python.general:

> Joel Goldstick wrote:
>
> >
> > This is not a string, it is scientific notion for 1.53... times 10 to the
> > -15th power. Because of rounding errors caused by doing floating point
> > math on in binary, you get a very small number instead of 0.
> >

>
> I was just doing some testing and it was not equating to zero. Perhaps if I
> rounded it up I may be more successful.


Classical CompSci teachings when working with floating point numbers
is to NEVER compare for equality. Instead one should compare against
some epsilon:

if x == y: #don't do

if abs(x - y) < epsilon: #do this

In your situation, "y" would be 0.0, so the test condenses to

if abs(x) < epsilon:

and a possible epsilon may be 1.0E-14

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Wulfraed Dennis Lee Bieber AF6VN
http://www.velocityreviews.com/forums/(E-Mail Removed) HTTP://wlfraed.home.netcom.com/

 
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Steven D'Aprano
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      02-16-2013
Dennis Lee Bieber wrote:

> Classical CompSci teachings when working with floating point numbers
> is to NEVER compare for equality. Instead one should compare against
> some epsilon:


"Don't compare floats for equality" is reasonably good advice.

Adding "never" to that advice, especially when shouting as you do, moves it
into the category "superstition".

Consider:

- Python floating point integers are exact for entire range of -2**53
to 2**53, or about -9 million million to +9 million million; if you
are working with floats that have integral values in this range,
testing for equality is perfectly fine.

- If you work exclusively with fractional powers of two, such as 1/2,
1/4, 1/8, 1/16, etc. floats are typically exact.

- Testing against an epsilon raises as many problems as it solves:

+ What epsilon should I pick? How do I know if my epsilon is too small,
and therefore I'm rejecting values that I should accept, or too large,
and so I'm accepting values I should reject?

+ If my epsilon is too small, calculating "abs(x - y) <= epsilon" is
exactly equivalent to "x == y", only slower.

+ Should I test for absolute error, or relative error?

+ If relative error, how do I deal with values around zero where
division is likely to introduce excessive rounding error?

+ Not to mention the risk of dividing by zero.

- And how do I deal with INFs?

py> x = float('inf')
py> x == x
True
py> abs(x - x) <= 1e-14
False



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Steven

 
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Dennis Lee Bieber
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      02-16-2013
On Sat, 16 Feb 2013 15:49:12 +1100, Steven D'Aprano
<(E-Mail Removed)> declaimed the following in
gmane.comp.python.general:

<snip>
> Consider:
>
> - Python floating point integers are exact for entire range of -2**53
> to 2**53, or about -9 million million to +9 million million; if you
> are working with floats that have integral values in this range,
> testing for equality is perfectly fine.
>
> - If you work exclusively with fractional powers of two, such as 1/2,
> 1/4, 1/8, 1/16, etc. floats are typically exact.
>
> - Testing against an epsilon raises as many problems as it solves:
>
> + What epsilon should I pick? How do I know if my epsilon is too small,
> and therefore I'm rejecting values that I should accept, or too large,
> and so I'm accepting values I should reject?
>
> + If my epsilon is too small, calculating "abs(x - y) <= epsilon" is
> exactly equivalent to "x == y", only slower.
>
> + Should I test for absolute error, or relative error?
>
> + If relative error, how do I deal with values around zero where
> division is likely to introduce excessive rounding error?
>
> + Not to mention the risk of dividing by zero.
>
> - And how do I deal with INFs?


Fine...

The take-away then becomes: One must know how floating point is
implemented in the computer in use (granted, practically everything is
now using IEEE specifications vs my college mainframe with its radix-16
format); and one must be cognizant of their problem domain to analyze
when guards must be taken for comparison of equality.

Since my "real world" experience has been in applications which are
not integral or powers-of-two number-crunching then an epsilon
comparison is pretty much a requirement -- especially when
transcendental functions are involved. Yes, one has to then evaluate the
problem domain to determine "how close is close enough".

The recommendation to always use an epsilon comparison for floating
point equality is a short phrase, and should trigger the needed analysis
to determine what epsilon is suitable for that comparison.

Or should Python implement REXX's NUMERIC statement? There is a can
of worms (I'm not even sure Regina REXX implements it correctly --
unless it is rounding to "digits" before applying "fuzz")

/* */

do D = 3 to 6
numeric digits D
do F = 0 to 3
if D <> F then
do
numeric fuzz F
call compare
end
end
end
exit

compare:
say "Digits:" digits() "Fuzz:" fuzz()
say
say '12345 = 12346 ' (12345 = 12346)
say '12345 = 12356 ' (12345 = 12356)
say '12345 = 12335 ' (12345 = 12335)
say '1234 = 1235 ' (1234 = 1235)
say '123.45 = 123.46 ' (123.45 = 123.46)
say '123.45 = 123.56 ' (123.45 = 123.56)
say
say

E:\UserData\Wulfraed\My Documents>regina test.rx
Digits: 3 Fuzz: 0

12345 = 12346 1
12345 = 12356 0
12345 = 12335 1
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 3 Fuzz: 1

12345 = 12346 1
12345 = 12356 1
12345 = 12335 1
1234 = 1235 1
123.45 = 123.46 1
123.45 = 123.56 1


Digits: 3 Fuzz: 2

12345 = 12346 1
12345 = 12356 1
12345 = 12335 1
1234 = 1235 1
123.45 = 123.46 1
123.45 = 123.56 1


Digits: 4 Fuzz: 0

12345 = 12346 1
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 4 Fuzz: 1

12345 = 12346 1
12345 = 12356 0
12345 = 12335 1
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 4 Fuzz: 2

12345 = 12346 1
12345 = 12356 1
12345 = 12335 1
1234 = 1235 1
123.45 = 123.46 1
123.45 = 123.56 1


Digits: 4 Fuzz: 3

12345 = 12346 1
12345 = 12356 1
12345 = 12335 1
1234 = 1235 1
123.45 = 123.46 1
123.45 = 123.56 1


Digits: 5 Fuzz: 0

12345 = 12346 0
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 0
123.45 = 123.56 0


Digits: 5 Fuzz: 1

12345 = 12346 1
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 5 Fuzz: 2

12345 = 12346 1
12345 = 12356 0
12345 = 12335 1
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 5 Fuzz: 3

12345 = 12346 1
12345 = 12356 1
12345 = 12335 1
1234 = 1235 1
123.45 = 123.46 1
123.45 = 123.56 1


Digits: 6 Fuzz: 0

12345 = 12346 0
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 0
123.45 = 123.56 0


Digits: 6 Fuzz: 1

12345 = 12346 0
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 0
123.45 = 123.56 0


Digits: 6 Fuzz: 2

12345 = 12346 1
12345 = 12356 0
12345 = 12335 0
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0


Digits: 6 Fuzz: 3

12345 = 12346 1
12345 = 12356 0
12345 = 12335 1
1234 = 1235 0
123.45 = 123.46 1
123.45 = 123.56 0



E:\UserData\Wulfraed\My Documents>


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Wulfraed Dennis Lee Bieber AF6VN
(E-Mail Removed) HTTP://wlfraed.home.netcom.com/

 
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