[R] [External] Funky calculations

[Avi Gross]

JC,

Interesting. People often misunderstand standards and the need for them as well as the way they are a compromise by lots of people representing different interests. You say you were one of 31, but in my experience, many more people are involved than the ones who show up. I had some involvements with standards back in the nineties and sometimes the results made sense and sometimes we ended up with designing a camel with stripes while trying to design a horse.

The issue here that needs repeating is that some things are basically impossible and the best we can do is to go part-way and make it as sane as we can.

Binary is NOT a must but for now it is. Electronic circuits that work in two modes we call zero and one are not the only choices and we may someday work with more complex circuits. Some mechanical devices have been built that work in a decimal manner. But given what we are using, a standard that lets you work with numbers to a reasonable level of representation is the best we can do. As has been pointed out, mathematically, only selected real numbers can be represented exactly in base two, and even fewer if you limit the number of bits you can use.

So floating point can be used with caution. I have seen some problems where it was chosen to deliberately convert all the numbers used to integers by say changing from measuring in continuous ways in meters to measuring in millimeters with no fractional millimeters allowed. All calculations done (excluding standard division) could now be compared to others without ambiguity. But for most cases, this is not necessary or even possible.

-----Original Message-----
From: J C Nash <profjcnash using gmail.com>
To: r-help using r-project.org
Sent: Wed, Feb 2, 2022 8:35 am
Subject: Re: [R] [External] Funky calculations


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)₂ = (0 × 2⁰) + (0 × 2⁻¹) + (0 × 2⁻²) + (0 × 2⁻³) + (1 × 2⁻⁴) + (1 × 2⁻⁵) + (0 × 2⁻⁶) + (0 × 2⁻⁷) + (1 × 2⁻⁸) + (1 × 2⁻⁹) + (0 × 2⁻¹⁰) + (0 × 2⁻¹¹) + (1 × 2⁻¹²) + (1 × 2⁻¹³) + (0 × 2⁻¹⁴) + (0 × 2⁻¹⁵) + (1 × 2⁻¹⁶) + (1 × 2⁻¹⁷) + (0 × 2⁻¹⁸) + (0 × 2⁻¹⁹) = (0.09999847412109375)₁₀
> 
> 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 using temple.edu>
> To: Avi Gross <avigross using verizon.net>
> Cc: nboeger using gmail.com <nboeger using gmail.com>; r-help using r-project.org <r-help using 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 using 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 using temple.edu>
>> To: Avi Gross <avigross using verizon.net>
>> Cc: Nathan Boeger <nboeger using gmail.com>; r-help using r-project.org <r-help using 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