On 10/30/06 5:50 AM, "Paul A Bristow" <pbristow@hetp.u-net.com> wrote:
-----Original Message----- From: boost-bounces@lists.boost.org [mailto:boost-bounces@lists.boost.org] 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