Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!ncar!asuvax!mcdphx!udc!chant!aglew From: aglew@urbana.mcd.mot.com (Andy-Krazy-Glew) Newsgroups: comp.arch Subject: IEEE FP denorms and Deming's Arithmetics With Variable Precision Message-ID:Date: 29 Sep 89 17:50:55 GMT Sender: aglew@urbana.mcd.mot.com Organization: Work: Motorola MCD, Urbana Design Center; School: University of Illinois at Urbana-Champaign Lines: 43 (Is this a good place for computer arithmetic questions?) I just read a paper by Deming in the 8th Computer Arithmetic Symposium, which basically deflates some of the proposed alternate arithmetics, such as level index (where a floating point number consists of three fields: pointer, exponent, and mantissa, and the pointer indicates how many bits are "borrowed" from the mantissa to expand the exponent and prevent overflow or underflow) or what I call the meta-exponential form, where a number is expressed as exp(exp(...(exp(m)))), and you store the number of exponentiations necessary to bring the value m within the desired range. Level index basically has a much larger range than normal FP, so overflow more rarely, while the meta-exponential form can be shown to be closed under +-*/ for a given number of bits in the representation. IE. both of these representations trade range for reduced relative precision at the extrema of the range. Deming shows how this tradeoff moves the complexity of coding reliable numerical software from avoiding overflow, to handling roundoff. IE. reduced precision makes rounding error analysis more complicated. QUESTION: Don't the same arguments apply to IEEE Floating Point with denormalized numbers? Ie. don't denormalized numbers complicate roundoff error analysis in the same way reduced precision complicates the other arithmetics? Deming suggests a sticky register which tracks the least relative precision ever used in the calculation of intermediate results, which will give you worst-case rounding error. Would such a register be worthwhile tracking the most extremely denormalized IEEE FP number encountered? Does anyone do this sort of thing? -- Andy "Krazy" Glew, Motorola MCD, aglew@urbana.mcd.mot.com 1101 E. University, Urbana, IL 61801, USA. {uunet!,}uiucuxc!udc!aglew My opinions are my own; I indicate my company only so that the reader may account for any possible bias I may have towards our products.