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From: Liu Jin (cpp_at_[hidden])
Date: 20050530 01:49:11
>>>>> "Paul" == Paul A Bristow <pbristow_at_[hidden]> writes:
>  Original Message  From:
> boostbounces_at_[hidden] 
> [mailto:boostbounces_at_[hidden]] On Behalf Of Paul A
> Bristow  Sent: 17 May 2005 15:23  To: boost_at_[hidden] 
> 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 whizzkids 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_at_[hidden]
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