Cannot summate small float values

Cannot summate small float values

[tolkamp]

When I summate Float values smaller then 1.0E-6 then the summation is
not done.

Code Example:

X, Dx : Float;
X := 0.0;
Dx := 1.0E-7;
lwhile X <  1.0 loop
    X = X + Dx;
    Float_Io.Put(X, 3,9,0); New_Line;
end loop;

Re: Cannot summate small float values

[Niklas Holsti]

tolkamp wrote:
> When I summate Float values smaller then 1.0E-6 then the summation is
> not done.
> 
> Code Example:
> 
> X, Dx : Float;
> X := 0.0;
> Dx := 1.0E-7;
> lwhile X <  1.0 loop
>     X = X + Dx;
>     Float_Io.Put(X, 3,9,0); New_Line;
> end loop;

Certainly the addition is done. Your program (after some small syntactic 
corrections) prints:

   0.000000100
   0.000000200
   0.000000300
   0.000000400
   0.000000500
   0.000000600
   0.000000700

and so on. If your program prints out something else, please show the 
source code of your whole program, exactly as you compile and run it. 
Don't re-type it into your message.

However, when X approaches 1.0, at some point the addition of 1.0E-7 may 
be lost in round-off, since it is close to the precision limit of the 
Float type, relative to 1.0. On my system (Debian, Gnat) the X variable 
does reach 1.0 and the program stops.

What are you really trying to do? There are probably safer and more 
accurate ways of doing it.

Here is the program that I used:

with Ada.Text_IO;
with Ada.Float_Text_IO;

procedure Sums
is
    use Ada.Text_IO, Ada.Float_Text_IO;
    X, Dx : Float;
begin
    X := 0.0;
    Dx := 1.0E-7;
    while X <  1.0 loop
        X := X + Dx;
        Put(X, 3,9,0); New_Line;
    end loop;
end Sums;

-- 
Niklas Holsti
Tidorum Ltd
niklas holsti tidorum fi
       .      @       .

Re: Cannot summate small float values

[tolkamp]

On 20 nov, 14:49, Niklas Holsti <niklas.hol...@tidorum.invalid> wrote:
> tolkamp wrote:
> > When I summate Float values smaller then 1.0E-6 then the summation is
> > not done.
>
> > Code Example:
>
> > X, Dx : Float;
> > X := 0.0;
> > Dx := 1.0E-7;
> > lwhile X <  1.0 loop
> >     X = X + Dx;
> >     Float_Io.Put(X, 3,9,0); New_Line;
> > end loop;
>
> Certainly the addition is done. Your program (after some small syntactic
> corrections) prints:
>
>    0.000000100
>    0.000000200
>    0.000000300
>    0.000000400
>    0.000000500
>    0.000000600
>    0.000000700
>
> and so on. If your program prints out something else, please show the
> source code of your whole program, exactly as you compile and run it.
> Don't re-type it into your message.
>
> However, when X approaches 1.0, at some point the addition of 1.0E-7 may
> be lost in round-off, since it is close to the precision limit of the
> Float type, relative to 1.0. On my system (Debian, Gnat) the X variable
> does reach 1.0 and the program stops.
>
> What are you really trying to do? There are probably safer and more
> accurate ways of doing it.
>
> Here is the program that I used:
>
> with Ada.Text_IO;
> with Ada.Float_Text_IO;
>
> procedure Sums
> is
>     use Ada.Text_IO, Ada.Float_Text_IO;
>     X, Dx : Float;
> begin
>     X := 0.0;
>     Dx := 1.0E-7;
>     while X <  1.0 loop
>         X := X + Dx;
>         Put(X, 3,9,0); New_Line;
>     end loop;
> end Sums;
>
> --
> Niklas Holsti
> Tidorum Ltd
> niklas holsti tidorum fi
>        .      @       .

Thank you your reaction.
Using your procedure Sums I found out that when the start value of X <
0.24 the summation works correct with Dx = 1.0E-8
When start X > 0.25 the summation remains 0.250000000.

Re: Cannot summate small float values

[Niklas Holsti]

tolkamp wrote:
> On 20 nov, 14:49, Niklas Holsti <niklas.hol...@tidorum.invalid> wrote:
>> tolkamp wrote:
>>> When I summate Float values smaller then 1.0E-6 then the summation is
>>> not done.
>>> Code Example:
>>> X, Dx : Float;
>>> X := 0.0;
>>> Dx := 1.0E-7;
>>> lwhile X <  1.0 loop
>>>     X = X + Dx;
>>>     Float_Io.Put(X, 3,9,0); New_Line;
>>> end loop;
>> Certainly the addition is done. Your program (after some small syntactic
>> corrections) prints:
>>
>>    0.000000100
>>    0.000000200
>>    0.000000300
>>    0.000000400
>>    0.000000500
>>    0.000000600
>>    0.000000700
>>
>> and so on. If your program prints out something else, please show the
>> source code of your whole program, exactly as you compile and run it.
>> Don't re-type it into your message.
>>
>> However, when X approaches 1.0, at some point the addition of 1.0E-7 may
>> be lost in round-off, since it is close to the precision limit of the
>> Float type, relative to 1.0. On my system (Debian, Gnat) the X variable
>> does reach 1.0 and the program stops.
>>
>> What are you really trying to do? There are probably safer and more
>> accurate ways of doing it.
>>
>> Here is the program that I used:
>>
>> with Ada.Text_IO;
>> with Ada.Float_Text_IO;
>>
>> procedure Sums
>> is
>>     use Ada.Text_IO, Ada.Float_Text_IO;
>>     X, Dx : Float;
>> begin
>>     X := 0.0;
>>     Dx := 1.0E-7;
>>     while X <  1.0 loop
>>         X := X + Dx;
>>         Put(X, 3,9,0); New_Line;
>>     end loop;
>> end Sums;
>>
> 
> Thank you your reaction.
> Using your procedure Sums I found out that when the start value of X <
> 0.24 the summation works correct with Dx = 1.0E-8
> When start X > 0.25 the summation remains 0.250000000.

That is still correct behaviour, because floating point addition can 
behave like that when the two addends have very different magnitudes. If 
you need to compute the sum of a large set of floating-point numbers, 
you should use a floating-point type with more digits or use a smart 
summation algorithm like the one by Kahan,
http://en.wikipedia.org/wiki/Kahan_summation_algorithm

-- 
Niklas Holsti
Tidorum Ltd
niklas holsti tidorum fi
       .      @       .

Re: Cannot summate small float values

[Gautier write-only]

On 21 nov, 22:06, tolkamp wrote:

> Thank you your reaction.
> Using your procedure Sums I found out that when the start value of X <
> 0.24 the summation works correct with Dx = 1.0E-8
> When start X > 0.25 the summation remains 0.250000000.

Anyway, since Dx is constant, you should never add it over and over:
it will accumulate numerical errors (with very rare exceptions) even
if it looks to work correctly on a few iterations.
The right way is
  X:= N * Real(Dx);
  N:= N + 1;
where Real is your floating-point type.
Cheers
______________________________________________________________
Gautier's Ada programming -- http://gautiersblog.blogspot.com/
NB: follow the above link for a working e-mail address

Re: Cannot summate small float values

[Julian Leyh]

On 21 Nov., 22:06, tolkamp <f.tolk...@gmail.com> wrote:
> On 20 nov, 14:49, Niklas Holsti <niklas.hol...@tidorum.invalid> wrote:
>
>
>
> > tolkamp wrote:
> > > When I summate Float values smaller then 1.0E-6 then the summation is
> > > not done.
>
> > > Code Example:
>
> > > X, Dx : Float;
> > > X := 0.0;
> > > Dx := 1.0E-7;
> > > lwhile X <  1.0 loop
> > >     X = X + Dx;
> > >     Float_Io.Put(X, 3,9,0); New_Line;
> > > end loop;
>
> > Certainly the addition is done. Your program (after some small syntactic
> > corrections) prints:
>
> >    0.000000100
> >    0.000000200
> >    0.000000300
> >    0.000000400
> >    0.000000500
> >    0.000000600
> >    0.000000700
>
> > and so on. If your program prints out something else, please show the
> > source code of your whole program, exactly as you compile and run it.
> > Don't re-type it into your message.
>
> > However, when X approaches 1.0, at some point the addition of 1.0E-7 may
> > be lost in round-off, since it is close to the precision limit of the
> > Float type, relative to 1.0. On my system (Debian, Gnat) the X variable
> > does reach 1.0 and the program stops.
>
> > What are you really trying to do? There are probably safer and more
> > accurate ways of doing it.
>
> > Here is the program that I used:
>
> > with Ada.Text_IO;
> > with Ada.Float_Text_IO;
>
> > procedure Sums
> > is
> >     use Ada.Text_IO, Ada.Float_Text_IO;
> >     X, Dx : Float;
> > begin
> >     X := 0.0;
> >     Dx := 1.0E-7;
> >     while X <  1.0 loop
> >         X := X + Dx;
> >         Put(X, 3,9,0); New_Line;
> >     end loop;
> > end Sums;
>
> > --
> > Niklas Holsti
> > Tidorum Ltd
> > niklas holsti tidorum fi
> >        .      @       .
>
> Thank you your reaction.
> Using your procedure Sums I found out that when the start value of X <
> 0.24 the summation works correct with Dx = 1.0E-8
> When start X > 0.25 the summation remains 0.250000000.

This should be taught in basics of computer nummerics or processor
architecture. Have a look at IEEE 754 floating point arithmetic.

Re: Cannot summate small float values

[Adam Beneschan]

On Nov 21, 1:06 pm, tolkamp <f.tolk...@gmail.com> wrote:
> On 20 nov, 14:49, Niklas Holsti <niklas.hol...@tidorum.invalid> wrote:
>
>
>
>
>
> > tolkamp wrote:
> > > When I summate Float values smaller then 1.0E-6 then the summation is
> > > not done.
>
> > > Code Example:
>
> > > X, Dx : Float;
> > > X := 0.0;
> > > Dx := 1.0E-7;
> > > lwhile X <  1.0 loop
> > >     X = X + Dx;
> > >     Float_Io.Put(X, 3,9,0); New_Line;
> > > end loop;
>
> > Certainly the addition is done. Your program (after some small syntactic
> > corrections) prints:
>
> >    0.000000100
> >    0.000000200
> >    0.000000300
> >    0.000000400
> >    0.000000500
> >    0.000000600
> >    0.000000700
>
> > and so on. If your program prints out something else, please show the
> > source code of your whole program, exactly as you compile and run it.
> > Don't re-type it into your message.
>
> > However, when X approaches 1.0, at some point the addition of 1.0E-7 may
> > be lost in round-off, since it is close to the precision limit of the
> > Float type, relative to 1.0. On my system (Debian, Gnat) the X variable
> > does reach 1.0 and the program stops.
>
> > What are you really trying to do? There are probably safer and more
> > accurate ways of doing it.
>
> > Here is the program that I used:
>
> > with Ada.Text_IO;
> > with Ada.Float_Text_IO;
>
> > procedure Sums
> > is
> >     use Ada.Text_IO, Ada.Float_Text_IO;
> >     X, Dx : Float;
> > begin
> >     X := 0.0;
> >     Dx := 1.0E-7;
> >     while X <  1.0 loop
> >         X := X + Dx;
> >         Put(X, 3,9,0); New_Line;
> >     end loop;
> > end Sums;
>
> Thank you your reaction.
> Using your procedure Sums I found out that when the start value of X <
> 0.24 the summation works correct with Dx = 1.0E-8
> When start X > 0.25 the summation remains 0.250000000.

You seem to lack a fundamental understanding of how floating-point
works.

In a 32-bit float, 23 of those bits are the "mantissa"; the rest are
used for the sign and exponent.  If the 23 bits are bbbb---bbbb, then
the value represented by the 32-bit float is

  [possibly negative] 1.bbbb---bbbb * (2**exp)

where exp is the exponent (represented by the other bits of the
float).  The 1.bbbb---bbbb is in binary notation, so that the first
"b" represents 2**-1, the second is 2**-2, etc. and the last is
2**-23.

Since 2**-23 is about 1.192E-7, this means that the ratio between
1.0000----0000 and 1.0000----0001 will be 1 + 1.192E-7.  So if you
start with the number 0.25, the smallest number greater than 0.25 that
can be represented is 0.25 + (0.25 * 1.192E-7).  This last part is
2.98E-8, which is a lot more than the 1E-8 that you're trying to add,
which is why 1E-8 is too small to make a difference when added.

                               -- Adam