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Re: converting exact numbers to binary fixed-point representation

• To: mathgroup at smc.vnet.net
• Subject: [mg58471] Re: converting exact numbers to binary fixed-point representation
• From: Peter Pein <petsie at dordos.net>
• Date: Sun, 3 Jul 2005 03:57:31 -0400 (EDT)
• References: <da5iog$1u4$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Torsten Coym schrieb:
> 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
> 
Try RealDigits[Sin[2*Pi*131/8191],2,16]

-- 
Peter Pein
Berlin