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a==b and FixedPoint

Can anyone tell me when two real numbers are Equal?  Experiments seem to
indicate that the ratio must be sufficiently close to 1, within 2^-44 on
the machines I use.

On the same topic, when does FixedPoint converge?  I get the feeling ==
is used.  This would explain why underflow occurs before convergence
when the limit is 0.

An approximate equality based on ratios makes sense to me, but it can
produce bizarre results when used in conjunction with a coprocessor that
uses other criteria. See below.

Jim Delany  ( jdelany at )

In[1]:= Log[2, .1^Precision[1.] ]
Out[1]= -53.1508
In[2]:= r=1; While[ 1. != 1.+r, r/=2]; Log[2, r]
Out[2]= -44
In[3]:= Log[2, -1.+FixedPoint[ Sqrt, 2.] ]
Out[3]= -43.5283
In[4]:= tf[x_]:= Cos[3x]==4Cos[x]^3-3Cos[x]
In[5]:= tf[2.^100]
Out[5]= True
In[6]:= tf[1.57]
Out[6]= False

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