Talk:IEEE 754

sortability

There is a recent edit noting that IEEE-754 values are sortable as sign-magnitude. I believe this is true for most sign-magnitude floating point formats, at least for normalized values when they can be unnormalized. (I am not sure about denormals, though.) The PDP-10 floating point format uses two's complement on the whole word for negative values, such that they are comparable using integer compare instructions. Not many processors supply a sign-magnitude compare operation, though. Gah4 (talk) 20:38, 10 November 2023 (UTC)

Do you mean that the PDP-10 two's complement also applied on the exponent field, i.e. changing the sign of the FP number would also change the encoding of the exponent? That's important to make the FP numbers comparable using integer compare instructions. — Vincent Lefèvre (talk) 00:33, 11 November 2023 (UTC)

Yes the whole word, including exponent. I suspect that hardware uncomplements it before using it. Maybe harder for humans to read, though. I am not sure what you mean by encoding of the exponent, but I believe that there can be carry into the exponent. Gah4 (talk) 03:14, 11 November 2023 (UTC)

It's really old

It lacks many details for professionals.

It lacks simplicity as well.

They can't decide even audience after that many years.

It is really important. Boh39083 (talk) 05:03, 19 November 2023 (UTC)

This is hard to make sense of. Can you elaborate a bit, with more context and more specific details? –jacobolus (t) 03:43, 20 November 2023 (UTC)

decimal exponent

I did a revert to a change on decimal exponent values. I believe it is right, because of the way they are defined, but I start this in case someone wants to discuss it, as I noted in the edit summary. Gah4 (talk) 17:20, 19 January 2024 (UTC)

The change was correct. The decimal exponent values got wrong in 1179557460 (but before that, there were already errors in some values for decimal64, which were introduced in the previous change 1179553657 by Nsmeds). I've corrected another value in 1197287634. — Vincent Lefèvre (talk) 22:17, 19 January 2024 (UTC)

OK, I am confused. What I see now isn't what I remember from the differences I saw before. There is always the question of the position of the decimal point, and I thought it was just that. In any case, we are discussing them, which is what I wanted. Gah4 (talk) 03:49, 20 January 2024 (UTC)

Concerning the position of the decimal point, this can make a difference of 15 or 16 in the exponent for decimal64, but here, this was a factor 2. — Vincent Lefèvre (talk) 13:53, 20 January 2024 (UTC)

Yes, I thought it was closer to 15, and now I see that it was a factor of 2. That is why I said I was confused about it. In any case, it is good to discuss here. Gah4 (talk) 23:22, 20 January 2024 (UTC)

I am sorry for not keeping in touch on this issue. There have been some other things keeping me occupied lately. Looking at the table as it stands today https://en.wikipedia.org/w/index.php?title=IEEE_754&oldid=1210204290 I am happy with the way it looks as regards the table on some useful properties of the IEEE-754 floating point formats. I went through my spreadsheet to compare FP formats and found a bug in the computations of estimated log10(MAXVAL) and have a public (LibreOffice) copy uploaded to my Google Drive share

I will update the table with adjusted values of the Decimal format log10(Maxval) estimates. I am not sure, but I think that is the issue you discussed in this thread? Nsmeds (talk) 17:13, 25 February 2024 (UTC)

This is what I think the best approx is to log10(MAXVAL)

${\alpha =1+maxexp},{\beta =maxexp-digits}$

$log_{10}(10^{\alpha }-10^{\beta })=\alpha +log_{10}(1-10^{\beta -\alpha })=\alpha +{{log_{10}(1-10^{\beta -\alpha })} \over {ln(10)}}=$

$=\lbrace {\alpha >\beta >0}\rbrace ={\alpha -{{10^{\beta -\alpha }} \over {ln(10)}}}+\Omega ({10^{2\cdot (\beta -\alpha )})}=$

$={\alpha -log_{10}(e)\cdot 10^{\beta -\alpha }}={\alpha -10\cdot log_{10}(e)\cdot 10^{\beta -\alpha -1}}\approx {\alpha -4.34\cdot 10^{\beta -\alpha -1}}$

$={(1+maxexp)-4.34\cdot 10^{-(1+digits)}}$ Nsmeds (talk) 18:44, 25 February 2024 (UTC)

Yes, if there are p digits, then the correction term is

$\ln(1-10^{-p})/\ln(10)\approx -10^{-p}/\ln(10)\approx -4.34\cdot 10^{-p-1}$

— Vincent Lefèvre (talk) 03:21, 26 February 2024 (UTC)

Thanks for confirming. If you want to (and if you have access to LibreOffice/OpenOffice and dare to enable my macros), have a look at the spreadsheet. Some of the formats there are not officially accepted and/or necessarily correctly described, but I find it illuminating to feed in various research formats and see what comes out. :-) Nsmeds (talk) 08:39, 26 February 2024 (UTC)

Introduction to History

@Nsmeds: Though I agree that an introduction in IEEE 754#History would be useful, there are several issues with what has been added in 1210457800, so that I'm going to revert this change (mainly because of the first point below):

• First, this should be an introduction to the history of the standardization of floating-point arithmetic, not a history on FP arithmetic (there is an article Floating-point arithmetic on this larger subject, with its own history). So, here, it should just be explained what was before IEEE 754 and why standardization was needed.
• In the added text, "non-enumerable" is pointless and misleading: The issue compared to integers is that the considered numbers cannot be represented exactly in general. This is the case even if you restrict to the computable real numbers (which form the subset of R that really matters in computing), which are enumerable (countable), but also just rational numbers.
• Binary representation is not always used.
• The last two sentences are not clear, but anyway, they are not related to the standardization (like the whole paragraph).
• The image is missing, but anyway, it is not related to the standardization either.
• Note that "mantissa" and "de-normal" are not the correct terms (there are also English and typographic mistakes).

BTW, in section History of Floating-point arithmetic, there is a paragraph on the standardization, which could serve as a basis:

Initially, computers used many different representations for floating-point numbers. The lack of standardization at the mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to the creation of the IEEE 754 standard once the 32-bit (or 64-bit) word had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the i8087 numerical coprocessor; Motorola, which was designing the 68000 around the same time, gave significant input as well.

— Vincent Lefèvre (talk) 01:54, 27 February 2024 (UTC)

@Vincent Lefèvre I agree with your comments. Hopefully there is someone with us able to volunteer a more suitable introductory paragraph? Nsmeds (talk) 12:50, 27 February 2024 (UTC)