Re: Can't get Mathematica to evaluate correctly a

*To*: mathgroup at smc.vnet.net*Subject*: [mg111618] Re: Can't get Mathematica to evaluate correctly a*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sat, 7 Aug 2010 06:22:04 -0400 (EDT)

On 8/6/10 at 6:56 AM, toniivas at gmail.com (Toni Ivas) wrote: >Still something is strange with Mathematica 7.0 as my output shows: <results/code snipped> >After first evaluating without forcing with N[*,prec] Mathematica >gives wrong answer You are missing the point. The fact you get a result like In[13]:= 173746*Sin[10^22] + 94228*Log[17.1] - 78487*Exp[0.42] Out[13]= 2.91038*10^-11 has essentially nothing to do with Mathematica. This type of result will occur in *any* software capable of doing the same computation. The result is a direct consequence of using floating point arithmetic. The exact value you get depends on details of the FPU hardware you are using. This is a wrong answer with respect to the true mathematical answer using infinite precision numbers. But it is not wrong answer with respect to floating point arithmetic. It is inherent to the way floating point arithmetic works on any computer. All three of the terms in the sum have values on the order of 10^5. You are looking for an answer on the order of 10^-12. You simply cannot achieve the accuracy you are asking for in any valid computation when you start with numbers with only 16 digit precision. And in Mathematica, whenever you explicitly type a decimal point in a number without explicitly specifying the precision of that number you will be using machine precision numbers that have approximately 16 digit precision. The advantage of machine precision arithmetic is it is all done in hardware and is the fastest possible computation A result like: In[14]:= N[ 173746*Sin[10^22] + 94228*Log[171/10] - 78487*Exp[42/100], 16] Out[14]= -1.341818957829620*10^-12 has a lot to do with Mathematica. This result takes advantage of Mathematica's ability to do arbitrary precision arithmetic something many other computational software packages cannot easily do. Mathematica gives you a choice. You can choose speed an accept whatever accuracy is possible using your hardware. Or you can sacrifice speed and choose whatever accuracy you want. It is up to you to choose wisely.