       Re: converting exact numbers to binary fixed-point representation

• To: mathgroup at smc.vnet.net
• Subject: [mg58460] Re: [mg58430] converting exact numbers to binary fixed-point representation
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sun, 3 Jul 2005 03:57:14 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Look at the functions RealDigits and FromDigits

(x=Sin[2*Pi*131/8191])//N//InputForm

0.10031897494050919

d=RealDigits[x,10,17]

{{1,0,0,3,1,8,9,7,4,9,4,0,5,0,9,1,9},0}

FromDigits[d,10]//N//InputForm

0.1003189749405092

b=RealDigits[x,2,16]

{{1,1,0,0,1,1,0,1,0,1,1,1,0,1,0,0},-3}

FromDigits[b,2]//N//InputForm

0.10031890869140625

Bob Hanlon

>
> From: Torsten Coym <torsten.coym at eas.iis.fraunhofer.de>
To: mathgroup at smc.vnet.net
> Date: 2005/07/02 Sat AM 04:06:34 EDT
> Subject: [mg58460] [mg58430] converting exact numbers to binary fixed-point
representation
>
> Hi group,
>
>
> what I want to achieve is to represent the exact value of an irrational
> number, say Sin[2*Pi*131/8191], as a binary fixed-point number having 16
> fractional (plus one sign bit) bits.
>
> First, I thought of converting to floating-point value and then
> converting to fixed-point using:
>
>
> Floor[N[Sin[2*Pi*(131/8191)]]*2^16]
>
> Now I'm worried about the precision of this conversion. The piece of
> code above truncates all fractional bits that occur after the left shift
> operation. The following two intermediate results (I changed to 4 bits
> for simplicity here) 1101,000...1 and 1100,111...1 will end up in two
> different code words 1101 and 1100, respectively.
> Though both values might be equally close to the exact value, the second
> would give the wrong solution. So how can I ensure, that *rounding* the
> exact value to a floating-point number will never lead to such a case,
> that eventually spoils my 16 bit representation?
>
> Is there a standard way to solve this problem?
> Is this a problem at all or am I worried too much?
>
> Any explanation is welcome.
>
> Torsten
>
>

```

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