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From: Daryle Walker (darylew_at_[hidden])
Date: 2006-10-31 12:17:07

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.

Daryle Walker
Mac, Internet, and Video Game Junkie
darylew AT hotmail DOT com

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