Roger, Very nice! I'll go over Gustafson's Unums 2.0 slides again, to see if I can figure out how to extend the lookup tables to a larger domain.
Skip On Mon, May 16, 2016 at 1:07 AM, roger stokes <[email protected]> wrote: > Hi Skip > First thoughts on Gustafson's book: > > It is very impressive indeed: there is much of value and interest in it. > I'm glad I bought it. > > It predates Unums 2.0 and Unums 2.0 is certainly a major advance. > Hence I'm pretty sure there will be a revised version of the book, maybe > substantially different, > but even so I'm glad to have this one. > > It is impressive, not just for the content but for the presentation, and > even more so > because Gustafson says that what we see is the output of a Mathematica > notebook. > Until recently I would have said that for us J programmers there was > nothing comparable to > Mathematica notebooks. However, there was a recent mention on one of our > forums about > Jupyter notebooks: I mean to check this out. > > Unums 2.0 I think is beautiful. It has a kind of compelling rightness. I > have put on my > website a J script for modelling arithmetic in Unums 2.0. Have just made a > tiny beginning: > can do addition and multiplication on 2-bit unums and addition on 3-bit > unums. This may not seem much but > anything more is just a matter of writing the lookup tables. The J code is > very small and simple. > > The script is at: www.learningj.com/unums2.ijs > > Regards, > Roger > > > On Fri, May 13, 2016 at 6:57 PM, Skip Cave <[email protected]> > wrote: > > > Great Roger! I'll be interested to see what you think about the book, and > > Gustafson's proposed new binary number format. I haven't received my hard > > copy of his book as yet. > > On May 12, 2016 11:28 AM, "roger stokes" <[email protected]> > wrote: > > > > > Many thanks Skip for posting the reviews. Having read your post, I > > > couldn't hold out any longer and bought the Kindle version. > > > > > > Regards, > > > Roger > > > On May 12, 2016 8:22 AM, "Skip Cave" <[email protected]> wrote: > > > > > > > Below are a few reviews of John Gustafson's book "The End of Error" > on > > > > Amazon books. I've ordered a copy of the book. I wanted a hard copy, > > but > > > is > > > > back-ordered, and takes 3-4 weeks to ship. However, one can order a > > > Kindle > > > > electronic copy, and get it immediately. > > > > > > > > Also, this book answers Raul's question about worked examples. > > Gustafson > > > > provides a full Mathematica implementation of Unums, so one can > explore > > > all > > > > the ramifications of this new numerical representation. > > > > > > > > Skip > > > > > > > > <<<>>> > > > > > > > > "This book is revolutionary. That is the only way to describe it. I > > have > > > > been a professional computer science researcher for almost 40 years, > > and > > > > only once or twice before have I seen a book that is destined to make > > > such > > > > a profound change in the way we think about computation. It is hard > to > > > > imagine that after 70 years or so of computer arithmetic that there > is > > > > anything new to say about it, but this book reinvents the subject > from > > > the > > > > ground up, from the very notion of finite precision numbers to their > > > > bit-level representation, through the basic arithmetic operations, > the > > > > calculation of elementary functions, all the way to the fundamental > > > methods > > > > of numerical analysis, including completely new approaches to > > expression > > > > calculation, root finding, and the solution of differential > equations. > > On > > > > every page from the beginning to the end of the book there are > > surprises > > > > that just astonished me, making me re-think material that I thought > had > > > > been settled for decades. > > > > > > > > The methods described in this book are profoundly different from all > > > > previous treatments of numerical methods. Unum arithmetic is an > > extension > > > > of floating point arithmetic, but mathematically much cleaner. It > never > > > > does rounding, so there is no rounding error. It handles what in > > floating > > > > point arithmetic is called "overflow" and "underflow" in a far more > > > natural > > > > and correct way that makes them normal rather than exceptional. It > also > > > > handles exceptional values (NaN, +infinity, -infinity) cleanly and > > > > consistently. Those contributions alone would have been a profound > > > > contribution. But the book does much more. > > > > > > > > One of the reasons I think the book is revolutionary is that > unum-based > > > > numerical methods can effortlessly provide provable bounds on the > error > > > in > > > > numerical computation, something that is very rare for methods based > on > > > > floating point calculations. And the bounds are generally as tight as > > > > possible (or as tight as you want them), rather than the useless or > > > trivial > > > > bounds as often happens with floating point methods or even interval > > > > arithmetic methods. > > > > > > > > Another reason I consider the book revolutionary is that many of the > > > > unum-based methods are cleanly parallelizable, even for problems that > > are > > > > normally considered to be unavoidably sequential. This was completely > > > > unexpected. > > > > > > > > A third reason is that in most cases unum arithmetic uses fewer bits, > > and > > > > thus less power, storage, and bandwidth (the most precious resources > in > > > > today’s computers) than the comparable floating point calculation. It > > > hard > > > > to believe that we get this advantage in addition to all of the > others, > > > but > > > > it is amply demonstrated in the book. Doing efficient unum arithmetic > > > takes > > > > more logic (e.g. transistors) than comparable floating point > arithmetic > > > > does, but as the author points out, transistors are so cheap today > that > > > > that hardly matters, especially when compared to the other benefits. > > > > > > > > Some of the broader themes of the book are counter > > > > > > > > intuitive to people like me > > > > with > > > > advanced conventional training, so that I have to re-think > everything I > > > > “knew” before. For example, the discussion of just what it means to > > > “solve” > > > > an equation numerically is extraordinarily thought provoking. Another > > > > example is the author’s extended discussion of how calculus is not > the > > > best > > > > inspiration for computational numerical methods, even for problems > that > > > > would seem to absolutely require calculus-based thinking, such as the > > > > solution of ordinary differential equations. > > > > > > > > Not only is the content of the book brilliant, but so is the > > > presentation. > > > > The text is so well written, a mix of clarity, precision, and reader > > > > friendliness that it is a pure pleasure to read, rather then the > dense > > > > struggle that mathematical textbooks usually require of the reader. > But > > > in > > > > addition, almost every page has full color graphics and diagrams that > > are > > > > completely compelling in their ability to clearly communicate the > > ideas. > > > I > > > > cannot think of any technical book I have ever seen that is so > > > beautifully > > > > illustrated all the way through. > > > > > > > > I should add that I read the Kindle edition on an iPad, and for once > > > Amazon > > > > did not screw up the presentation of a technical book, at least for > > this > > > > platform. It is beautifully produced, in full color and detail, and > > with > > > > all of the fonts and graphics reproduced perfectly. > > > > > > > > Dr. Gustafson has also provided a Mathematica implementation of unums > > and > > > > of the many numerical methods discussed in the book. Let us hope that > > in > > > > the next few years there will be implementations in other languages, > > > > followed by hardware implementations. Over time there should be unum > > > > arithmetic units alongside of floating point arithmetic units on > every > > > CPU > > > > and GPU chip, and in the long run unums should replace floating point > > > > entirely. The case the author makes for this is overwhelming. > > > > > > > > If you are at all interested in computer arithmetic or numerical > > methods, > > > > read this book. It is destined to be a classic. > > > > " > > > > David Jefferson > > > > < > http://www.amazon.com/gp/pdp/profile/A3E5GY7K9GGYD0/ref=cm_cr_dp_pdp> > > > on > > > > April 18, 2015 > > > > > > > > Other > > > > Review > > > > s (Amazon) > > > > > > > > "The author of the present book believes that it is time to > supplement > > > the > > > > century-old floating point arithmetic with something better: unum > > > > arithmetic. The book covers various operations with unum arithmetic > and > > > > topics like polynomial evaluation, solving equations, two-body > problem, > > > > etc. The appendices give a glossary of unum functions, ubox > functions, > > > and > > > > some algorithm listings." > > > > ―*Zentralblatt MATH* 1320 > > > > > > > > "This book is an extraordinary reinvention of computer arithmetic and > > > > elementary numerical methods from the ground up. Unum arithmetic is > an > > > > extension of floating point in which it is also possible to represent > > the > > > > open intervals *between* two floating point numbers. This leads to > > > > arithmetic that is algebraically much cleaner, without rounding > error, > > > > overflow underflow, or negative zero, and with clean and consistent > > > > treatment of positive and negative infinity and NaN. These changes > are > > > not > > > > just marginal technical improvements. As the book fully demonstrates, > > > they > > > > lead to what can only be described as a radical re-foundation of > > > elementary > > > > numerical analysis, with new methods that are free of rounding error, > > > fully > > > > parallelizable, fully portable, easier for programmers to master, and > > > often > > > > more economical of memory, bandwidth, and power than comparable > > floating > > > > point methods. The book is exceptionally well written and produced > and > > is > > > > illustrated on every page with full-color diagrams that perfectly > > > > communicate the material. Anyone interested in computer arithmetic or > > > > numerical methods must read this book. It is surely destined to be a > > > > classic." > > > > ―David Jefferson, Center for Advanced Scientific Computing, Lawrence > > > > Livermore National Laboratory > > > > > > > > "John Gustafson’s book *The End of Error* presents the ideas of > > computer > > > > arithmetic in a very easy-to-read and understandable form. While the > > > title > > > > is provocative, the content provides an illuminating discussion of > the > > > > issues. The examples are engaging, well thought out, and simple to > > > follow." > > > > ―Jack Dongarra, University Distinguished Professor, University of > > > Tennessee > > > > > > > > "John Gustafson presents a bold and brilliant proposal for a > > > revolutionary > > > > number representation system, unum, for scientific (and potentially > all > > > > other) computers. Unum’s main advantage is that computing with these > > > > numbers gives scientists the correct answer all the time. Gustafson > is > > > able > > > > to show that the universal number, or unum, encompasses all standard > > > > floating-point formats as well as fixed-point and exact integer > > > arithmetic. > > > > The book is a call to action for the next stage: implementation and > > > testing > > > > that would lead to wide-scale adoption." > > > > ―Gordon Bell, Researcher Emeritus, Microsoft Research > > > > > > > > "Reading more and more in [John Gustafson’s] book became a big > > surprise. > > > I > > > > had not expected such an elaborate and sound piece of work. It is > hard > > to > > > > believe that a single person could develop so many nice ideas and put > > > them > > > > together into a sketch of what perhaps might be the future of > > computing. > > > > Reading [this] book is fascinating." > > > > ―Ulrich Kulisch, Karlsruhe Institute of Technology, Germany > > > > > > > > Skip Cave > > > > Cave Consulting LLC > > > > > > > > On Fri, Apr 29, 2016 at 12:17 PM, Skip Cave <[email protected] > > > > > > wrote: > > > > > > > > > Interesting comment from John Gustafson on Google Groups: > > > > > <<<>>> > > > > > Incidentally, I've been challenged to a debate by William Kahan at > > the > > > > > ARITH23 conference, July 10-13 in San Jose, CA. (Kahn is the > designer > > > of > > > > > the IEEE floating point numerical format). > > > > > > > > > > Title: "The Great Debate: The End of Error?" > > > > > > > > > > Kahan has apparently prepared a 34-page response to my book > > > (Gustafson's > > > > > "The End Of Error" book) though I have not seen it and he will > > probably > > > > > spring all kinds of surprises on me. It should be a good show! > > > > > <<<>> > > > > > > > > > > Skip > > > > > On Apr 29, 2016 10:27 AM, "Skip Cave" <[email protected]> > > wrote: > > > > > > > > > > Here's the Google Group on unum computing: > > > > > > > > > > https://groups.google.com/forum/#!forum/unum-computing > > > > > > > > > > > > > > > Here's Gustafson's home page: > > > > > > > > > > http://www.johngustafson.net/index.html > > > > > > > > > > > > > > > Skip > > > > > > > > > > > > > > > Skip Cave > > > > > Cave Consulting LLC > > > > > > > > > > On Fri, Apr 29, 2016 at 5:35 AM, Pierpaolo Bernardi < > > > [email protected] > > > > > > > > > > wrote: > > > > > > > > > >> On Fri, Apr 29, 2016 at 7:41 AM, Skip Cave < > [email protected] > > > > > > > >> wrote: > > > > >> > Here's a much newer presentation by Gustafson that goes into the > > > > >> > implementation of Unums in much more detail. > > > > >> > > > > > >> > > > > > > > http://www.johngustafson.net/presentations/Unums2.0slides-withNotes.pdf > > > > >> > > > > >> This is about Unums 2.0, which is a completely different idea. The > > > > >> choice of naming them Unums 2.0 is unfortunate IMO. > > > > >> > > > > >> Unums 2.0 is a way more exoteric idea then Unums, and as far as I > > > > >> understand it is still in an embryonal stage. > > > > >> > > > > >> For interested people there's a google group about unums where > > > > >> Gustafson participates, and up to now has always replied to > > questions. > > > > >> > > ---------------------------------------------------------------------- > > > > >> For information about J forums see > > > http://www.jsoftware.com/forums.htm > > > > >> > > > > > > > > > > > > > > > ---------------------------------------------------------------------- > > > > For information about J forums see > http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
