Binary integer decimal: Difference between revisions

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{{OrphanUse dmy dates|date=FebruaryMay 20092019|cs1-dates=y}}

{{floating-point}}

The draft [[IEEE 754r]] standard proposes an encoding format for decimal floating point numbers in which the significand and the exponent (and the payloads of [[NaN]]s) can be encoded in two ways, referred to in the draft as binary encoding and decimal encoding.<ref>{{cite web

{{Cleanup|reason=This article more describes DPD encoding and IEEE 754 encoding in general, rather than BID encoding which it should acc. it's title.|date=December 2024}}

| title = DRAFT Standard for Floating Point Arithmetic P754

The draft [[IEEE 754r754-2008]] standard proposes an encoding format forincludes decimal floating-point pointnumber numbersformats in which the [[significand]] and the exponent (and the payloads of [[NaN]]s) can be encoded in two ways, referred to in the draft as '''binary encoding''' and ''decimal encoding''.<ref>{{cite web

| date = 2006-10-04

| title = DRAFT Standard for Floating Point Arithmetic P754

| url = http://754r.ucbtest.org/drafts/archive/2006-10-04.pdf

| date = 2006-10-04

| accessdate = 2007-07-01 }}</ref>.

| url = http://754r.ucbtest.org/drafts/archive/2006-10-04.pdf

| accessdate = 2007-07-01 }}</ref>.

}}{{dead link|date=November 2016 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

<!-- Not sure why this next one is here? Proprietary/trade marks and propaganda articles not scientific, surely? Arith18 paper would be much better

The binary encoding format is referred to by Intel<ref>

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</ref> and others as Binary Integer Decimal (BID).

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Both formats break a number down into a sign bit ''s'', an exponent ''q'' (between ''q''_min and ''q''_max), and a ''p''-digit significand ''c'' (between 0 and 10^''p''−1). The value encoded is (&minus;1)^''s''×10^''q''×''c''. In both formats the range of possible values is identical, but they differ in how the significand ''c'' is represented. In the decimal encoding, it is encoded as a series of ''p'' decimal digits (using the [[densely packed decimal]] (DPD) encoding), while in the '''binary integer decimal''' ('''BID''') encoding, it is encoded as a binary number.

==Format==

Using the fact that 2¹⁰ = 1024 is only slightly more than 10³ = 1000, 3''n''-digit decimal numbers can be efficiently packed into 10''n'' binary bits. However, the IEEE formats have significands of 3''n''+1 digits, which would generally require 10''n''+4 binary bits to represent.

This would not be efficient, because only 10 of the 16 possible values of the additional four bits are needed. A more efficient encoding can be designed using the fact that the exponent range is of the form 3×2^''k'', so the exponent never starts with <code>11</code>. Using the Decimal32 encoding (with a significand of 3*2+1 decimal digits) as an example (<code>e</code> stands for exponent, <code>m</code> for mantissa, i.e. significand):

* If the significand starts with <code>0mmm</code>, omitting the leading 0 bit lets the significand fit into 23 bits:

s 00eeeeee (0)mmm mmmmmmmmmm mmmmmmmmmm

s 01eeeeee (0)mmm mmmmmmmmmm mmmmmmmmmm

s 10eeeeee (0)mmm mmmmmmmmmm mmmmmmmmmm

* If the significand starts with <code>100m</code>, omitting the leading 100 bits lets the significand fit into 21 bits. The exponent is shifted over 2 bits, and a <code>11</code> bit pair shows that this form is being used:

s 1100eeeeee (100)m mmmmmmmmmm mmmmmmmmmm

s 1101eeeeee (100)m mmmmmmmmmm mmmmmmmmmm

s 1110eeeeee (100)m mmmmmmmmmm mmmmmmmmmm

* Infinity, quiet [[NaN]] and signaling NaN use encodings beginning with <code>s 1111</code>:

s 11110 xxxxxxxxxxxxxxxxxxxxxxxxxx

s 111110 xxxxxxxxxxxxxxxxxxxxxxxxx

s 111111 xxxxxxxxxxxxxxxxxxxxxxxxx

The bits shown in parentheses are ''implicit'': they are not included in the 32 bits of the Decimal32 encoding, but are implied by the two bits after the sign bit.

The Decimal64 and Decimal128 encodings have larger exponent and significand fields, but operate in a similar fashion.

For the Decimal128 encoding, 113 bits of significand is actually enough to encode 34 decimal digits, and the second form is never actually required.

==Cohort==

A decimal floating -point number can be encoded in several ways, the different ways represent different precisions, for example 100.0 is encoded as 1000*101000×10^-1−1, while 100.00 is encoded as 10000*1010000×10^-2−2. The set of possible encodings of the same numerical value is called a ''cohort'' in the standard. If the result of a calculation is inexact the largest amount of significant data is preserved by selecting the cohort member with the largest integer that can be stored in the significand along with the required exponent.

==Range==

The proposed IEEE 754r standard limits the range of numbers to a significand of the form 10ⁿ&minus;1, where n is the number of whole decimal digits that can be stored in the bits available so that decimal rounding is effected correctly.

{| class="wikitable" style="text-align:right"

|-

|

! item

! 32 bit

! 64 bit

! 128 bit

|-

|! Storage bits

| 32

| 64

| 128

|-

|! Trailing Significand bits

| 20

| 50

| 110

|-

|! Significand bits

| 23/24

| 53/54

| 113

|-

! Significand digits

| Combination bit

| 7

| 16

| 34

|-

|! Combination bitbits

| 11

| 13

| 17

|-

|! Exponent bits

| 8

| 10

| 14

|-

|! Bias

| 101

| 398

| 6176

|-

|! Standard emax

| 96

| 384

| 6144

|-

|! Standard emin

| 95−95

| 383−383

| 6143−6143

|}

==Performance==

A binary encoding is inherently less efficient for conversions to or from decimal-encoded data, such as strings ([[ASCII]], [[Unicode]], etc.) and [[Binary-coded decimal|BCD]]. A binary encoding is therefore best chosen only when the data are binary rather than decimal. IBM has published some unverified performance data,.<ref>{{Cite availableweb|url=http://speleotrove.com/decimal/decperf.html|title at= Decimal Library Performance - 1.01}}</ref>

[http://speleotrove.com/decimal/decperf.html], however as both packages are available as open-source these figures could be verified indepedently.

==See also==

* [[IEEE 754r754]]

==References==

{{reflist}}

==Further reading==

* {{cite web |title=The Decimal Floating-Point Standard |author-first=John J. G. |author-last=Savard |date=2018 |orig-year=2007 |work=quadibloc |url=http://www.quadibloc.com/comp/cp020302.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180703002322/http://www.quadibloc.com/comp/cp020302.htm |archive-date=2018-07-03}}

[[Category:Computer arithmetic]]

[[Category:IEEE standards]]