Re: Output Precision Exploration
- To: mathgroup at smc.vnet.net
- Subject: [mg118426] Re: Output Precision Exploration
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Thu, 28 Apr 2011 06:32:46 -0400 (EDT)
Log[173.5/173.5] == 0
False
To see why this occurs use Trace
Trace[Log[173.5/173.5] == 0]
{{{{HoldForm[1/173.5], HoldForm[0.005763688760806916]},
HoldForm[173.5*0.005763688760806916], HoldForm[0.9999999999999999]},
HoldForm[Log[0.9999999999999999]], HoldForm[-1.1102230246251565*^-16]},
HoldForm[-1.1102230246251565*^-16 == 0], HoldForm[False]}
Which is to say that it evaluates the quotient x/y without any prior analysis to conclude that in this case it is x/x.
Use more precision
Log[173.5`20/173.5`20] == 0
True
Log[Rationalize[173.5/173.5]] == 0
True
Or define functions that compensate for failing to use appropriate precision
myLog[x_?NumericQ] := Log[Rationalize[x, 0]] // N
myLog[173.5/173.5] == 0
True
Bob Hanlon
---- Rafael Dunn <worthless.trash.junk at gmail.com> wrote:
=============
Mathematica 8.0.1.0, Mac OSX x86
In:=
Log[173.5/173.5]
Out:=
-1.11022*10^-16
I expect an output of exactly 0. Although 10^-16 is small, it turned
out to be the largest factor in a chemical equation I was attempting to
compute.
I discovered this is because Mathematica does not actually evaluate
173.5/173.5 = 1. The output is actually some number 0.9999999999...
However, for most decimal constants x/x produces an exact output of 1.
By entering a few decimals off the top of my head I also found 1733.5,
26.44, and 27.44 do not produce an output of 1 when divided by
themselves.
Why? I understand Mathematica's algorithms for working with decimals
must make approximations, but why is there so much variance among
decimal calculations? 173.49/173.49 = 1, while 173.5/173.5 != 1.
Furthermore, I find:
x=173.49999999999999
x/x = 173.5/173.5, with infinite precision. If you add or remove a
single 9 to the end of x, this ceases to be true.
Furthermore, this looks like a contradiction to me:
In:=
173.5/173.5 = 1
Log[1] = 0
Log[173.5/173.5] = 0
Out:=
True
True
False
I have learned a lot about Mathematica's precision and approximation through the help documentation, but I still can not explain this or see how I can expect Log[x/x] = 0 for the sake of calculations on the 10^-16 scale.