[Scilab-users] Problems arising from truncation of %pi

[Stéphane Mottelet]

The nearest double-precision IEEE-754 binary floating-point number for 
the decimal number PI

3.141592653589793 23846264338327950288419716939937510582097494459230 
78164 06286 ....

is

3.141592653589793 115997963468544185161590576171875

It can be shown this way: its internal base 2 representation is (I added 
spaces to separate the sign, exponent and mantissa) is given in Scilab by

--> bitstring(%pi)
  ans  =

   "0 10000000000 1001001000011111101101010100010001000010110100011000"

i.e. a positive number, with exponent 2^(1024-1023) = 1 and mantissa

1.1001001000011111101101010100010001000010110100011000

i.e. the leading "1."  is implicit (normalized convention) and has to be 
added. Hence, the final numer, after mutiplication by 2^exponent, is

11.001001000011111101101010100010001000010110100011000

You won't be able to convert this back to decimal by using floating 
point arithmetic. There is an efficient converter at 
https://www.exploringbinary.com/binary-converter/ which gives the actual 
result

convert


S.

Le 08/01/2021 à 10:48, Jean-Yves Baudais a écrit :
> Hello,
>
> ----- Original Message -----
>> The function could be sinpi() or similar, with two arguments: the main
>> argument x and an integer argument n, being its result equivalent to
>>
>> sin(x - n*pi)
> So now the problem can be how these large numbers are obtained
> --> a=1e16+1
> --> a-1e16
> of course equals zero.
>
>> where pi is the exact value of pi
> Hum... What does "exact" mean in numerical calculus? (In symbolic one, it's simpler :-)
>
> --> format(25)
> --> %pi
> 3.141592653589793115998
>
> the 6 last digits are wrong (cf. https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/oeis.org/A000796) because of float64 format.
>
>> Probably it is not proper to refer to it as noise, since the difference
>> is deterministic as can be demonstrated by using the function nearfloat()
> Maybe noise is not the right word. But, the difference is fixed and %pi give all the time the same digit in format(25). It's a bit weird (up to my ignorance) to have these wrong deterministic digits... Hum, maybe I need to read "What Every Computer Scientist Should Know about Floating-Point Arithmetic", David Goldberg, ACM Computing Surveys, vol. 23, no  1, mars 1991 recommanded by https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/en.wikipedia.org/wiki/IEEE_754
>
> --Jean-Yves
>
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-- 
Stéphane Mottelet
Ingénieur de recherche
EA 4297 Transformations Intégrées de la Matière Renouvelable
Département Génie des Procédés Industriels
Sorbonne Universités - Université de Technologie de Compiègne
CS 60319, 60203 Compiègne cedex
Tel : +33(0)344234688
http://www.utc.fr/~mottelet

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