Re: [boost] Simulating log{10}(2)

From: Paul A Bristow (pbristow_at_[hidden])
Date: 2006-10-31 12:59:40

|  -----Original Message-----
|  From: boost-bounces_at_[hidden] 
|  [mailto:boost-bounces_at_[hidden]] On Behalf Of Daryle Walker
|  Sent: 31 October 2006 17:17
|  To: Boost mailing list
|  Subject: Re: [boost] Simulating log{10}(2)
|  
|  On 10/30/06 5:50 AM, "Paul A Bristow" 
|  <pbristow_at_[hidden]> wrote:
|  
|  >> -----Original Message-----
|  >> From: boost-bounces_at_[hidden]
|  >> [mailto:boost-bounces_at_[hidden]] On Behalf Of Daryle Walker
|  >> Sent: 29 October 2006 20:00
|  >> To: Boost mailing list
|  >> Subject: [boost] Simulating log{10}(2)
|  > <snip>
|  > 
|  >> The common approximation 0.301 differs from the actual 
|  expansion at a4 = 2.
|  >> The convergent there is 59/196.  The absolute differences 
|  between those
|  >> values and the actual value are:
|  >> 
|  >>    |log{10}(2) - 0.301| = 2.9995664e-5
|  >>   |log{10}(2) - 59/196| = 9.58750072e-6
|  >> 
|  >> The convergent is 3.1 times closer to the actual value 
|  than the common
|  >> approximation.  Using 0.30103 versus the convergent 
|  4004/13301 at the point
|  >> of differing a7 = 6 gives the convergent an improvement 
|  factor of 2.1.  The
|  >> convergents have smaller numerators and denominators than 
|  their common
|  >> approximations, making them easier to calculate without 
|  worry of overflow.
|  >> Conversely, if you know that you have more leeway in 
|  multiplications, then
|  >> using higher convergent fractions instead of the decimal 
|  expansion are a
|  >> better bet.
|  > 
|  > Well this is very interesting and I am sure you are right, 
|  but for the
|  > purpose of calculating the number of decimal digits, the 
|  approximation seems
|  > to me to be 'fit for purpose' (or good enough).
|  [TRUNCATE how the current approximations are used]
|  
|  But we have the option of doing better.  I'm wondering if we 
|  should change
|  our code to use the better approximating fractions.

Very probably would be better.  But I'm not the maintainer of lexical cast code, etc, merely an interefering bystander ;-)

Paul

---
Paul A Bristow
Prizet Farmhouse, Kendal, Cumbria UK LA8 8AB
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