I was one of the 31 names on the 1985 IEEE standard. If anyone thinks things are
awkward now,
try looking at the swamp we had beforehand.
What I believe IS useful is to provide examples and to explain them in tutorial
fashion.
We need to recognize that our computations have limitations. Most common
computing platforms
use IEEE binary arithmetic, but not all.
This was much more "in our face" when we used slide rules or
hand-crank calculators. I still
have slide rules and a Monroe "Portable" calculator -- 5 kg! It's
worth bringing them out every
so often and being thankful for the power and speed of modern computing, while
remembering to
watch for the cowpads of REAL and REAL*8 arithmetic.
JN
On 2022-02-01 22:45, Avi Gross via R-help wrote:> This is a discussion forum, Richard, and I welcome requests to clarify what
I wrote or to be corrected, especially when my words have been read with an
attempt to understand. I do get private responses too and some days i wonder if
I am not communicating the way people like!
>
> But let me repeat. The question we started with asked about R. My answer
applies to quite a few languages besides R and maybe just about all of them.
>
> I got private email insisting the numbers being added were not irrational
so why would they not be represented easily as a sum. I know my answers included
parts at various levels of abstraction as well as examples of cases when
Decimals notation for a number like 1/7 results in an infinite repeating
sequence. So, I think it wise to follow up with what binary looks like and why
hardly ANYTHING that looks reasonable is hard to represent exactly.
>
> Consider that binary means POWERS OF TWO. The sequence 1101 before a
decimal point means (starting from the right and heading left) that you have one
ONES and no TWOS and one FOURS and one EIGHTS. Powers of two ranging from 2 to
the zero power to two cubed. You can make any integer whatsoever using as long a
sequence of zeros and ones as you like. Compare this to decimal notation where
you use powers of ten and of course can use any of 0-9.
>
> But looking at fractional numbers, like 1/7 and 1/10, it gets hard and
inexact.
>
> Remember now we are in BINARY. Here are some fractions with everything not
shown to the right being zeros and thus not needed to be shown explicitly.
Starting with the decimal point, read this from left to right to see the powers
in the denominator rising so 1/2 then 1/4 then 1/8 ...:
>
> 0.0 would be 0.
> 0.1 would be 1/2
> 0.101 would be 1/2 + 1/8 or 5/8
> 0.11 would be 1/2 + 1/4 or 3/4
> 0.111 would be 1/2 + 1/4 + 1/8 or 7/8
>
> We are now using negative powers where 2 raised to the minus one power is
one over two raised to the plus one power, or 1/2 and so on. As you head to the
right you get to fairly small numbers like 1/2048 ...
>
> Every single binary fraction is thus a possibly infinite sum of negative
powers of two, or rather the reciprocals of those in positive terms.
>
> If you want to make 1/7, to some number of decimal places, it looks like
this up to some point where I stop:
>
> 0.00100100100100100101
>
> So no halves, no quarters, 1/8, no sixteenths, no thirty-seconds, 1/64, and
so on. But if you add all that up, and note the sequence was STOPPED before it
could continue further, you get this translated into decimal:
>
> 0.142857 55157470703125
>
> Recall 1/7 in decimal notation is
> 0.142857 142857142857142857...
>
> Note the divergence at the seventh digit after the decimal point. I left a
space to show where they diverge. If I used more binary digits, I can get as
close as I want but computers these days do not allow too many more digits
unless you use highly specialized programs. There are packages that give you
access such as "mpfr" but generally nothing can give you infinite
precision. R will not handle an infinite number of infinitesimals.
>
> The original problem that began our thread was about numbers like 0.1 and
0.2 and so on. In base ten, they look nice but I repeat in base 2 only powers of
TWO reign.
>
> 0.1 in base two is about 0.0001100110011001101
>
> that reads as 1/16 + 1/32 + 1/256 + 1/512 + ...
>
> If I convert the above segment, which I repeat was stopped short, I get
0.1000003814697265625 which is a tad over and had I taken the last 1 and changed
it to a zero as in 0.0001100110011001100 then we would have a bit under at
0.09999847412109375
>
> So the only way to write 0.1 exactly is to continue infinitely, again. Do
the analysis and understand why most rational numbers will not easily convert to
a small number of bits. But the advantages of computers doing operations in
binary are huge and need not be explained. You may THINK you are entering
numbers in decimal form but they rarely remain that way for long before they
simply become binary and often remain binary unless and until you ask to print
them out, usually in decimal.
>
> BTW, I used a random web site to do the above conversion calculations:
>
> https://www.rapidtables.com/convert/number/binary-to-decimal.html
>
> Since I am writing in plain text, I cannot show what it says in the box on
that page further down under Decimal Calculation Steps so I wonder what the rest
of this message looks like:
>
> (0.0001100110011001100)? = (0 ? 2?) + (0 ? 2??) + (0 ? 2??) + (0 ? 2??) +
(1 ? 2??) + (1 ? 2??) + (0 ? 2??) + (0 ? 2??) + (1 ? 2??) + (1 ? 2??) + (0 ?
2???) + (0 ? 2???) + (1 ? 2???) + (1 ? 2???) + (0 ? 2???) + (0 ? 2???) + (1 ?
2???) + (1 ? 2???) + (0 ? 2???) + (0 ? 2???) = (0.09999847412109375)??
>
> I think my part in this particular discussion can now finally come to an
end. R and everything else can be incomplete. Deal with it!
>
> -----Original Message-----
> From: Richard M. Heiberger <rmh at temple.edu>
> To: Avi Gross <avigross at verizon.net>
> Cc: nboeger at gmail.com <nboeger at gmail.com>; r-help at
r-project.org <r-help at r-project.org>
> Sent: Tue, Feb 1, 2022 9:04 pm
> Subject: Re: [External] [R] Funky calculations
>
>
> I apologize if my tone came across wrong.? I enjoy reading your comments on
this list.
>
> My goal was to describe what the IEEE and R interpret "careful
coding" to be.
>
>
>> On Feb 01, 2022, at 20:42, Avi Gross <avigross at verizon.net>
wrote:
>>
>> Richard,
>>
>> I think it was fairly clear I was explaining how people do arithmetic
manually and often truncate or round to some number of decimal places. I said
nothing about what R does or what the IEEE standards say and I do not
particularly care when making MY point.
>>
>> My point is that humans before computers also had trouble writing down
any decimals that continue indefinitely. It cannot be expected computer versions
of arithmetic can do much better. Different people can opt to do the calculation
with the same or different numbers of digits ad when compared to each other they
may not match.
>>
>> I do care what it does in my programs, of course. My goal here was to
explain to someone that the anomaly found was not really an anomaly and that
careful coding may be required in these situations.
>>
>>
>> -----Original Message-----
>> From: Richard M. Heiberger <rmh at temple.edu>
>> To: Avi Gross <avigross at verizon.net>
>> Cc: Nathan Boeger <nboeger at gmail.com>; r-help at r-project.org
<r-help at r-project.org>
>> Sent: Tue, Feb 1, 2022 2:44 pm
>> Subject: Re: [External] [R] Funky calculations
>>
>>
>> RShowDoc('FAQ')
>>
>>
>> then search for 7.31
>>
>>
>> This statement
>> "If you stop at a 5 or 7 or 8 and back up to the previous digit,
you round up. Else you leave the previous result alone."
>> is not quite right.? The recommendation in IEEE 754, and this is how R
does arithmetic, is to Round Even.
>>
>> I ilustrate here with decimal, even though R and other programs use
binary.
>>
>>> x <- c(1.4, 1.5, 1.6, 2.4, 2.5, 2.6, 3.4, 3.5, 3.6, 4.4, 4.5,
4.6)
>>> r <- round(x)
>>> cbind(x, r)
>> ? ? ? ? x r
>> [1,] 1.4 1
>> [2,] 1.5 2
>> [3,] 1.6 2
>> [4,] 2.4 2
>> [5,] 2.5 2
>> [6,] 2.6 3
>> [7,] 3.4 3
>> [8,] 3.5 4
>> [9,] 3.6 4
>> [10,] 4.4 4
>> [11,] 4.5 4
>> [12,] 4.6 5
>>>
>>
>> Numbers whose last digit is not 5 (when in decimal) round to the
nearest integer.
>> Numbers who last digit is 5 (1.5, 2.5, 3.5, 4.5 above)
>> round to the nearest EVEN integer.
>> Hence 1.5 and 3.5 round up to the even numbers 2 and 4.
>> 2.5 and 4.5 round down do the even numbers 2 and 4.
>>
>> This way the round ups and downs average out to 0.? If we always went
up from .5 we would have
>> an updrift over time.
>>
>> For even more detail click on the link in FAQ 7.31 to my appendix
>> https:// link.springer.com/content/pdf/bbm%3A978-1-4939-2122-5%2F1.pdf
>> and search for "Appendix G".
>>
>> Section G.5 explains Round to Even.
>> Sections G.6 onward illustrate specific examples, such as the one that
started this email thread.
>>
>> Rich
>>
>
>
>
>
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