| -----Original Message----- | From: boost-bounces@lists.boost.org | [mailto:boost-bounces@lists.boost.org] On Behalf Of Paul A Bristow | Sent: 17 May 2005 15:23 | To: boost@lists.boost.org | Subject: RE: [boost] Re: Math constants I attach a short file with some samples of math constants together with their 'interval' upper and lower values, which (mostly) when compiled will be only a single least significant bit different (or the same of course). The upper and lower values contain a lot of decimal digits because they are 'exactly representable' for the floating point layout given. C++ Exactly representable values do not involve any rounding on input (to the appropriate floating point format). Of course this file does not compile - many duplicates - but when the appropriate pair of upper and lower are chosen, they should do. For the most common case of an IEEE double, // C++ Exactly representable interval values of pi for 2 radix & 53 significand bits. static const double pi_l = 3.141592653589793115997963468544185161590576171875; // lower static const double pi = 3.1415926535897932384626433832795028841971694007530293; // starting with this value. static const double pi_u = 3.141592653589793560087173318606801331043243408203125; // upper When used to initialise a double, v_d_l = 3.1415926535897931 == 0x400921fb54442d18 // lower v_d = 3.1415926535897931 == 0x400921fb54442d18 // 'middle' == lower v_d_u = 3.1415926535897936 == 0x400921fb54442d19 // upper The number of possibly significant digits is given by std::numeric_limits<double::digits() * 3010 /10000 and is 17 (std::numeric_limits<double::digits10() in contrast is the number of __guaranteed__ accurate decimal digits and is 15). Questions: 1 Are there many (any?) potential users for interval arithmetic computing (we have a Boost library). 2 Would they find the intervals of the constants updated in the Sandbox useful? 3 Can any MACRO whizz-kids suggest the _best_ way to provide only pairs (or trios) for the appropriate floating point format? 4 I can only calculate for radix == 2 (and can't be bothered to work out for radix 8, 10, 16, 42...?). If anyone wants these and can tell me how, please do so. 5 So one is really only chosing float, double, long double, or some UDT like NTL arbitrary precision quads on the basis of the number of significand bits, usually 24, 53, 64, 105, 106, 113, 127. For example, to produce the above definitions for 53 significand IEEE 754 X86 double. Thanks. Paul Paul A Bristow Prizet Farmhouse, Kendal, Cumbria UK LA8 8AB +44 1539 561830 +44 7714 330204 mailto: pbristow@hetp.u-net.com