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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Axel Vogt 25 Mar '17

25 Mar '17

Kahan on that approach: https://people.eecs.berkeley.edu/~wkahan/ (scroll down to the bottom) > Subject: [math-fun] Beyond Floats: Next Gen Computer Arithmetic > FYI -- > > https://www.youtube.com/watch?v=aP0Y1uAA-2Y > > Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic > > John L. Gustafson, Natl Univ of Singapore > > https://web.stanford.edu/class/ee380/Abstracts/170201.html > > A new data type called a "posit" is designed for direct drop-in replacement for IEEE Standard 754 floats. Unlike unum arithmetic, posits do not require interval-type mathematics or variable size operands, and they round if an answer is inexact, much the way floats do. However, they provide compelling advantages over floats, including simpler hardware implementation that scales from as few as two-bit operands to thousands of bits. For any bit width, they have a larger dynamic range, higher accuracy, better closure under arithmetic operations, and simpler exception-handling. For example, posits never overflow to infinity or underflow to zero, and there is no "Not-a-Number" (NaN) value. Posits should take up less space to implement in silicon than an IEEE float of the same size. With fewer gate delays per operation as well as lower silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPs using similar hardware reso > urces. GPU accelerators, in particular, could do more arithmetic per watt and per dollar yet deliver superior answer quality. > > A series of comprehensive benchmarks compares how many decimals of accuracy can be produced for a set number of bits-per-value, using various number formats. Low-precision posits provide a better solution than "approximate computing" methods that try to tolerate decreases in answer quality. High-precision posits provide better answers (more correct decimals) than floats of the same size, suggesting that in some cases, a 32-bit posit may do a better job than a 64-bit float. In other words, posits beat floats at their own game. > > http://ee380.stanford.edu/Abstracts/170201-slides.pdf

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Re: [math-fun] [seqfan] Re: Is 4 a semi-Fibonacci number?
by David Wilson 25 Mar '17

25 Mar '17

Apologies, then, for being a day late. At any rate, I was curious as to the growth rate of (odd indexed elements) of sF, so I computed a few values of sF(2^k - 1). Looks to grow pretty fast, perhaps like n^(c log(n)). n sF(n) 1 1 3 2 7 5 15 16 31 69 63 430 127 4137 255 64436 511 1676353 1023 74555322 2047 5777029421 4095 792086153688 8191 194591768192733 16383 86534148901444102 32767 70244955881077121873 65535 104827174339054175240700 131071 289320796542222620694103961 262143 1484554051525798547483095566354 524287 14226732533670039175251226319995157 1048575 255658362329675830634139832019198358176 2097151 8646156383344527622677186248446917786683381 4194303 552073085216495053730117030388698424955408298334 8388607 66749153256722612438012197449644764555811889274235129 16777215 15322089889433495509821892166057427813245442064758958688836 33554431 6693576684186674844436583641024041055004958124882988723321728369 67108863 5577272394534357747969836555204334929788023962243060637883527988942186 134217727 8881483115234363964483793627146185513254764624755802869033890445863234590077 268435455 27080472748266980783930080876503920295698702610405257020769093391323735487516698472 536870911 158372376484792407948495752863706348514435815566310070102912509445802547425058287520342413 From: maximilian(a)hasler.fr [mailto:maximilian@hasler.fr] On Behalf Of M. F. Hasler Sent: Saturday, March 25, 2017 5:44 PM To: David Wilson Subject: Re: [seqfan] Re: Is 4 a semi-Fibonacci number? On Sat, Mar 25, 2017 at 12:09 AM, David Wilson <davidwwilson(a)comcast.net> wrote: Also, sF(n) is strictly increasing on odd n, so {sF(2n-1): n >= 1} is the ordered sequence of all range values of sF. 4 is not in that sequence. That's what I said a day ago... <http://list.seqfan.eu/pipermail/seqfan/2017-March/017390.html> ;-) M. > -----Original Message----- > From: SeqFan [mailto:seqfan-bounces@list.seqfan.eu] On Behalf Of > israel(a)math.ubc.ca > Sent: Thursday, March 23, 2017 11:24 AM > To: Sequence Fanatics Discussion list > Subject: [seqfan] Re: Is 4 a semi-Fibonacci number? > > The least n for which any value occurs must be odd, since sF(n) = sF(n/2) > for even n. So if you have ruled out sF(n) = 4 for odd n, it follows that 4 > can never occur. > > Cheers, > Robert > > On Mar 23 2017, Alonso Del Arte wrote:t > > >As you know, 4 is not a Fibonacci number. The Fibonacci function can be > >extended to all real numbers, but to get Fibonacci(x) = 4 requires x be > >what looks like a transcendental number. > > > >But could 4 be a semi-Fibonacci number? (A030067) The definition is sF(1) = > >1, sF(n) = sF(n/2) if n is even, sF(n) = sF(n - 1) + sF(n - 2) if n is odd. > > > >Couple of years ago, Roberg G Wilson v determined that 4 does not occur > >among the first million terms. It's not a rigorous proof, of course, but it > >does suggest that 4 never occurs. > > > >It's fairly easy to prove that sF(n) = 4 is impossible if n is odd. But I > >haven't been able to rule out sF(n) = 4 for n even. That would mean sF(n) = > >8, but the parity of n seems like it could be anything. I can visualize a > >whole tree but many of the branches of that tree might not even exist. > > > >Can anyone make a determination on this question, or is this another one > of > >those plausible but unproven conjectures? > > > >Al

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Dave Dyer 25 Mar '17

25 Mar '17

> >> https://www.youtube.com/watch?v=aP0Y1uAA-2Y >> >> Stanford Seminar: Beyond Floating Point: Next Generation Computer >> Arithmetic Sounds pretty solid to me. The unintended consequence if this were widely adopted as 'plug compatible' for IEEE floats, would be that data dumps which contained binary floats would be aliased. IEEE floats are so ubiquitous, everyone treats them as if they were as god-given as integers. It's dangerous to change them.

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Re: [math-fun] probability
by Dan Asimov 25 Mar '17

25 Mar '17

My favorite probability paradox is intransitive dice. In one of its simplest forms, three 6-sided dice A, B, C with A beating B, B beating C, and C beating A — each more than half the time. E.g., A: 2, 2, 4, 4, 9, 9 B: 1, 1, 6, 6, 8, 8 C: 3, 3, 5, 5, 7, 7. where the common probability of A>B, or of B>C, or of C>A, is 5/9. —Dan >From: Cris Moore <moore(a)santafe.edu> >Is there a good source for intro-level probability “paradoxes” that would give me an opportunity to pit cognitive biases against mathematical level-headedness? I have in mind things suitable for e.g. 6th-graders, things like: “I flip 8 coins. Which is more likely, that they come up HTTHHTTT or HHHHHHHH?”

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Re: [math-fun] probability
by Dan Asimov 25 Mar '17

25 Mar '17

----- . . . that something can be true of every subset but not of the whole set. ----- That's not quite how Simpson's paradox is usually stated. The Wikipedia article at https://en.wikipedia.org/wiki/Simpson's_paradox has some nice examples. —Dan -----Original Message----- >From: "Keith F. Lynch" <kfl(a)KeithLynch.net> >My favorite probability paradox is Simpson's paradox, i.e. that >something can be true of every subset but not of the whole set. >A surprising real-world example is that until recently US life >expectancy was going up, but that there were two groups for which >it was going down: Smokers and non-smokers. > >The explanation is that, roughly speaking, life expectancy was only >increasing because of people quitting smoking, and that all else was >a net loss, presumably because improvements in medical care were more >than offset by the increasing unaffordability of that care. > >Of course it's not really a paradox, just an example of how >mathematical intuition is often wrong.

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Re: [math-fun] probability
by Keith F. Lynch 25 Mar '17

25 Mar '17

My favorite probability paradox is Simpson's paradox, i.e. that something can be true of every subset but not of the whole set. A surprising real-world example is that until recently US life expectancy was going up, but that there were two groups for which it was going down: Smokers and non-smokers. The explanation is that, roughly speaking, life expectancy was only increasing because of people quitting smoking, and that all else was a net loss, presumably because improvements in medical care were more than offset by the increasing unaffordability of that care. Of course it's not really a paradox, just an example of how mathematical intuition is often wrong.

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Leo Broukhis 25 Mar '17

25 Mar '17

On Sat, 25 Mar 2017 06:42:50 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote > > > Were you able to figure out how the "regime" bits in posits work? > The "regime" term is a misnomer. The exponent uses a mixed unary-binary representation of integers with a fixed-sized binary tail (say, 3); 0 - 10 000 1 - 10 001 ... 7 - 10 111 8 - 110 000 9 - 110 001 ... 16 - 1110 000 ... 25 - 11110 001 ... 32 - 111110 000 ... -1 - 01 111 ... -8 - 01 000 -9 - 001 111 In other words, if the number starts with N 1s, its value is (N-1)*2^3 + tail, otherwise -N*2^3 + tail. Leo >

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Leo Broukhis 25 Mar '17

25 Mar '17

On Mon, 20 Mar 2017 14:18:16 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote: FYI -- > > https://www.youtube.com/watch?v=aP0Y1uAA-2Y > > Stanford Seminar: Beyond Floating Point: Next Generation Computer > Arithmetic > > John L. Gustafson, Natl Univ of Singapore > > https://web.stanford.edu/class/ee380/Abstracts/170201.html > > A new data type called a "posit" is designed for direct drop-in > replacement for IEEE Standard 754 floats. > I whipped up my own C++ implementation and verified that the formula on page 40 is indeed evaluated with better precision using 32-bit posits than 32-bit floats. However, using "posits" will require more care to avoid losing precision due to the order of magnitude of the intermediate results. I've plugged my Posit class into enquire.c ( http://homepages.cwi.nl/~steven/enquire.html) and paranoia.c ( http://www.netlib.org/paranoia/paranoia.c) The former gets very confused, claiming that some bits aren't used, and the latter finds several flaws, defects, and failures. Leo

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Henry Baker 25 Mar '17

25 Mar '17

Hi Leo: Were you able to figure out how the "regime" bits in posits work? At 11:21 PM 3/24/2017, Leo Broukhis wrote: >On Mon, 20 Mar 2017 14:18:16 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote: > >FYI -- > >> https://www.youtube.com/watch?v=aP0Y1uAA-2Y >> >> Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic >> >> John L. Gustafson, Natl Univ of Singapore >> >> https://web.stanford.edu/class/ee380/Abstracts/170201.html >> >> A new data type called a "posit" is designed for direct drop-in >> replacement for IEEE Standard 754 floats. > >I whipped up my own C++ implementation and verified that the formula on >page 40 is indeed evaluated with better precision using 32-bit posits than >32-bit floats. > >However, using "posits" will require more care to avoid losing precision >due to the order of magnitude of the intermediate results. > >I've plugged my Posit class into enquire.c ( >http://homepages.cwi.nl/~steven/enquire.html) and paranoia.c ( >http://www.netlib.org/paranoia/paranoia.c) The former gets very confused, >claiming that some bits aren't used, and the latter finds several flaws, >defects, and failures. > >Leo

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Re: [math-fun] Help with some math
by Bill Gosper 25 Mar '17

25 Mar '17

gosper.org/xx-2.png ries (@ -l6) failed to recognize c2(-2-) ~ -4.95208960443219142648286540385 - 1.88898604639119460912777031678 I --rwg On 2017-03-24 17:07, David Wilson wrote: > I presume you are talking about evaluating > > c2(x) = c(c(x)) - x^2 + 2 > > with > > c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2 > > If you start with |x| >= 2, you end up with > > c(x) = y^√2 + z^√2 > > where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well. > > If |x| < 2, however, we end up with > > c(x) = y^√2 + z^√2 > > Where y and z are complex numbers. It then seems that y^√2 and z^√2 > take on an infinitude of non-real complex values, none better than > another. So how do you compute y^√2 or z^√2 with y or z complex? Is > there some principle value? > > At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of > c2(x) on |x| < 2. > I would like to see a plot of c2(x) on the real line. > >> -----Original Message----- [Chopped]

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