Gustafson re-invents APL/J's unary reciprocal notation as part of Unum 2.0... Slide 17 of Unums 2.0:
<<<>>> OK, this is a big one. Other than the SORN slide, this may be the most disruptive idea in this slide deck. I’m proposing a fundamental change to the way we commonly write and read numbers, one that conceivably could be taught in grade school from now on. The term “unary minus” means a minus sign that negates the value that follows, as opposed to the kind of minus we mean when we write x – y. We can abbreviate zero minus x as just, minus x. Leave out the zero. We could do the same thing for 1/x, and leave off the 1. Instead of saying “one over x” we would just say “over x”. We know that compilers can parse unary minus signs; they’ve been doing it for sixty years. It should not be at all difficult for them to parse the “over” sign, unary “/”. Notice that two unary minus signs cancel out. The negative of the negative of a number gives us back the positive. Similarly, the reciprocal of the reciprocal of a number gives us back the number. Usually you have to make an exception for dividing by zero, but we don’t with this system. The reciprocal of zero is ±∞, and the reciprocal of ±∞ is zero. As anyone who has ever used a pocket calculator knows, dividing x by y is not the same as multiplying x by 1/y. Like, if you divide 6 by 3 you will get exactly 2 on a calculator or a computer. But if you multiply 6 by the calculator version of 1/3, you will get a number close to 2, but not quite! Which can be pretty disastrous in even the simplest of computer programs, like when you are trying to use these small integers or fractions to decide how many times to execute a loop of code. We’ve long had the ability to negate numbers without any loss of accuracy or information. But reciprocation has always been sloppy, and you cannot take 1/x with floats and land exactly on another float unless you’re very lucky, so you experience rounding errors just with the simple act of asking for a reciprocal. Suppose going from 3 to 1/3 was as simple as writing “–3” for the negation of 3? You write “/3” and that is the answer. That is the way the computer stores it internally, and that is the way it displays the result for humans to read and understand. As I said, this probably has to be taught starting at the elementary school level at the time they introduce negative numbers, fractions, and decimals. They are just as hard to understand at first as fractions and decimals, and just as easy to understand after you get used to that notation. I sometimes say that “division is the ‘bad boy’ of arithmetic.” It’s the hardest one to learn in school and the slowest to execute on a computer. Remember the notorious Intel Pentium Divide Bug? That was the result of aggressive attempts to raise the speed of divides on a microprocessor, since divides are quite a bit slower than multiplies the way we do things now. But the system described here makes all four arithmetic operations equally simple, equally symmetric, equally fast. <<<>>> Skip 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
