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Re: Mathematica 8: first impressions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg114104] Re: Mathematica 8: first impressions
  • From: Joseph Gwinn <joegwinn at comcast.net>
  • Date: Tue, 23 Nov 2010 06:02:40 -0500 (EST)
  • References: <ic34r0$5rv$1@smc.vnet.net> <ic8ag0$83s$1@smc.vnet.net> <ic9ldn$nun$1@smc.vnet.net> <icdo2b$680$1@smc.vnet.net>

In article <icdo2b$680$1 at smc.vnet.net>, blamm64 <blamm64 at charter.net> 
wrote:

> 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.

Well, the Umm is me trying to figure out a nice way to say it.  What 
approach would you have preferred?

Joe Gwinn


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