| -----Original Message----- | From: boost-bounces@lists.boost.org | [mailto:boost-bounces@lists.boost.org] 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@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. 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 +44 1539561830 & SMS, Mobile +44 7714 330204 & SMS pbristow@hetp.u-net.com