My Math Forum Precision Arithmetic: A New Floating-Point Arithmetic

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 May 23rd, 2010, 04:50 PM #1 Newbie   Joined: May 2010 Posts: 3 Thanks: 0 Precision Arithmetic: A New Floating-Point Arithmetic ABSTRACT: A new floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and its bounding range is much tighter. Generic standards and systematic methods for validating uncertainty-bearing arithmetics are discussed. The precision arithmetic is found to be better than interval arithmetic in uncertainty-tracking for linear algorithms. PDF: http://arxiv.org/ftp/cs/papers/0606/0606103.pdf Please comment. You can also email your comment to me Chengpu@gmail.com directly. I am thinking of start a source forge project on it.
 May 23rd, 2010, 07:38 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Precision Arithmetic: A New Floating-Point Arithmetic I think the topic is very interesting, something I've worked on before. I'm curious to see how well it performs in practice as well as in theory -- for example, calculating tan(120152108392) or (X ± x)(Y ± y)(Z ± z) for x,y,z uncorrelated and positive. *** On page 4 you write: "According to central limit theorem [4], most measured values are Gaussian distributed, which has no limited bounding range." But that's not what the central limit theorem says... On page 4 you also have: "If an bounding range is defined using a statistical rule on bounding leakage, such as the 6?-10^-6 rule for Gaussian distribution [4] (which says the bounding leakage is about 10^-6 for a bounding range of mean ± 6-fold of standard deviations), there is no guarantee that the calculation result will also obey the 6?-10^-6 rule using interval arithmetic, since interval arithmetic has no statistical ground." Can you give an example of where this fails? I would expect interval arithmetic to give far wider ranges, in general, than 6 standard deviations if that's what the data was given as. I have no other specific comments through page 14.
 May 25th, 2010, 08:25 PM #3 Newbie   Joined: May 2010 Posts: 3 Thanks: 0 Re: Precision Arithmetic: A New Floating-Point Arithmetic The Gaussian distribution is the de facto underline distribution and central limit theorem is one of the major reason. For example, if you collect data thru A/D, A/D holds signal by integrate signals in capacitor, thus averages signal in hardware. A/D are now the majority way to collect signal. The other way is thru event counter, whose result is usually either Posson distribution or binominal distribution, both of which can be approx well by Gaussian distribution. Of cause, I should have said that the Gaussian distribution is the de facto underline distribution. Thanks. In my result, Interval arithmetic always broadens the bounding range too fast, e.g, if you start with 6-sigma, after one or two calculations, it become 12-sigma or more. For FFT algorithm, independence arithmetic is the ideal one. You can see this effect clearly.
 June 2nd, 2010, 11:37 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Precision Arithmetic: A New Floating-Point Arithmetic Any progress on the SourceForge idea?

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