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Re: [Maxima-discuss] a fun demo for variable precision arithmetic

From: Przemek Klosowski
Subject: Re: [Maxima-discuss] a fun demo for variable precision arithmetic
Date: Fri, 4 Dec 2015 12:27:38 -0500
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.3.0

On 12/03/2015 12:12 AM, Richard Fateman wrote:
   From S. Rump's investigations..

   a  "polynomial" that evaluates the same in IEEE single or double 
precision, about 1.1726   but is wrong in both

This is a great example! comes in handy when discussing precision and accuracy in FP calculations. I have often seen people who do numerical simulations believing that double-precision solves the accuracy problems. I have written the following to some of my friends; please forgive the parts that are obvious to people on this list who do this stuff for living:

Over the years I discussed with all of you the pitfalls of precision and accuracy in floating point calculations. I just ran into a wonderful illustration how tricky this business is, so I am sharing it with you.

As you know, each of several available FP formats is precise to a certain number of significant digits: six for 32-bit single precision, fifteen for 64-bit double precision, etc. What this means is that, of course, a result obtained in single precision FP arithmetic cannot be more accurate that the format's precision, i.e. 0.001%---but often the errors accumulate and the overall accuracy is further lost due to rounding, faulty normalization and other effects. If we didn't have this accuracy loss, we could do our calculations with an actually existing half-precision FP numbers that have 3 significant digits of precision, and are very space-efficient (16 bits) and fast. Unfortunately, even though most of our measurements would be accurately represented within this 0.1% precision, the loss of accuracy in most calculations makes this format impractical.

Normally, we deal with the accuracy loss by accepting the price of calculating in higher precision: it's virtually impossible to get inaccurate result when we start from the double precision's 0.00000000000001%, right? Unfortunately, it turns out that it IS possible, even if rarely---and we can correct for that by doing something similar to Richardson's extrapolation ( ): obtain the result in single precision, and then in double precision. If the result is the same, we might be assured that we're not losing accuracy. If they are NOT the same, we could conclude that the double precision is the right one, or, if we're in an exacting mood, we could actually estimate the result for infinite precision.

Unfortunately, math is a harsh mistress and this doesn't always work. Here's an example of a simple polynomial (you don't need any muti-hour iterative calculations) where the accuracy loss is so severe that we're getting a wrong result for both single and double precision.

Doubly unfortunately, the wrong result is THE SAME for both, so it turns out that you can't even rely onĀ  comparing multiple precisions.

  here's where to evaluate it, and here's the right answer...

  float(rp(77617, 33096)); evaluates to -0.82739605994682

But changing the floating point precision gives different answers...

  for fpprec:10 step 5 thru 50 do print (rp(77617.0b0, 33096.0b0));

google will find you more info about Rump's polynomial..

using interval arithmetic with double-floats tells you that the correct 
answer is
  in  [- 2.7363628282260744E+22, 2.458240415520526E+22]

which is why it might make sense to use more precision.

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