Re: Distinguishable From 1.0
- To: mathgroup at smc.vnet.net
- Subject: [mg47740] Re: [mg47719] Distinguishable From 1.0
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Sat, 24 Apr 2004 04:15:41 -0400 (EDT)
- References: <200404230630.CAA03243@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Harold Noffke wrote: > $Version "5.0 for Microsoft Windows [2000] (November 18, 2003)" > > MathGroup: > > According to MathBook documentation ... > > "$MachineEpsilon gives the smallest machine-precision number which can > be added to 1.0 to give a result that is distinguishable from 1.0." > > The results below suggest that becoming "... distinguishable from 1.0" > does not occur until slightly under 1.5 * $MachineEpsilon. > > Is there a misstatement here, or am I misunderstanding something about > being "... distinguishable from 1.0"? > > In[1]:= 1. + 1.0 * $MachineEpsilon === 1. > Out[1]= True > > In[2]:= 1. + 1.1 * $MachineEpsilon === 1. > Out[2]= True > > In[3]:= 1. + 1.2 * $MachineEpsilon === 1. > Out[3]= True > > In[4]:= 1. + 1.3 * $MachineEpsilon === 1. > Out[4]= True > > In[5]:= 1. + 1.4 * $MachineEpsilon === 1. > Out[5]= True > > In[6]:= 1. + 1.5 * $MachineEpsilon === 1. > Out[6]= False > > Regards, > Harold > Now take into account what SameQ does: "SameQ requires exact correspondence between expressions, except that it considers Real numbers equal if their difference is less than the uncertainty of either of them." A pair of machine numbers that differ only in the last bit are thus regarded as SameQ. Your 1. + 1.5 * $MachineEpsilon was the first value above that differed by more than one bit from 1. InputForm will give a clue that this is the case, but it may be discerned with more certainty by checking RealDigits[number,2]. Daniel Lichtblau Wolfram Research
- References:
- Distinguishable From 1.0
- From: Harold.Noffke@wpafb.af.mil (Harold Noffke)
- Distinguishable From 1.0