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Re: Mathematica 8: first impressions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114044] Re: Mathematica 8: first impressions
*From*: blamm64 <blamm64 at charter.net>
*Date*: Mon, 22 Nov 2010 07:35:28 -0500 (EST)
*References*: <ic34r0$5rv$1@smc.vnet.net> <ic8ag0$83s$1@smc.vnet.net> <ic9ldn$nun$1@smc.vnet.net>
On Nov 20, 6:25 pm, Joseph Gwinn <joegw... at comcast.net> wrote:
> > but
> > by now I think Mathematica, if installed on a 64-bit system, should
> > have $MachinePrecision of 31.8.... , or something close to that on a
> > 64-bit system.
>
> Umm, 15.9 decimal digits *is* double precision floating point, which is
> what is meant by a machine real.
>
> An IEEE single (32-bit) float has a 24-bit mantissa (including the
> hidden bit), so its machine precision is Log[10,2^24]= 7.22 decimal
> digits.
>
> An IEEE double (64-bit) float (called a double in C/C++) has a 53-bit
> mantissa (including the hidden bit), so its Log[10,2^53]= 15.95 decimal
> digits.
>
> To achieve 31.8 decimal digits would require use of IEEE quad (128-bit)
> floats, for 34 decimal digits. The only computer I know of that offere=
d
> 128-bit floating-point arithmetic in hardware was the DEC VAX series, at
> least the larger members.
>
> A recent application where this kind of precision was truly needed was
> doing the ray-tracing of the laser beams in LIGO (Laser Interferometer
> Gravitational Wave Observatory), where the interferometer beams are
> kilometers long. To get 1/100 wavelength resolution using 1064 nanometer
> light over 20 Km total path requires precision of at least 10.64nm/20km=
=
> Log[10,5.32*10-13]= 12.3 digits.
>
> They did use VAXen for this, at least in the early days. More recent
> articles show them using Optica (which is a Mathematica package
> <http://www.opticasoftware.com/>), possibly in Mathematica's multiprecisi=
on mode.
>
> While integers can achieve higher precisions for a given size (because
> no space is spent on exponents), integers are pretty awkward to use
> unless one knows in advance what the allowed range of values is. For
> the record, Log[10,2^32]= 9.63 digits and Log[10,2^64]= 19.27 digits.
>
> Joe Gwinn
>
> Ref: <http://en.wikipedia.org/wiki/IEEE_754-2008>
Okay, thanks, I stand corrected. Could have done without the "Umm"
sarcasm though.
-Brian L.
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