## This page has been replaced

The information on this page is no longer being maintained.

The “calculators” referenced on this page have been replaced by an “analyzer” at IEEE-754.

## Old Contents

This page provides some reference material for the IEEE-754 floating-point standard. It is a companion page to the three calculator pages:

- Convert decimal numbers to IEEE-754 representation. Enter a decimal value and see how it would be encoded as a single-precision or double-precision IEEE-754 floating point number.
- Analyze a single-precision IEEE-754 value. Enter a 64-bit value in hexadecimal, and see the corresponding decimal value and the breakdown of all the fields.
- Analyze a double-precision IEEE-754 value. Enter a 64-bit value in hexadecimal, and see the corresponding decimal value and the breakdown of all the fields.

## What’s Here

## History of the Calculators

These calculators came into being in the fall semester of 1997 when I assigned my Computer Organization course an optional project to write a program that would print the values of the fields of IEEE-754 floating point numbers. One student, Quanfei Wen, decided to do the program as an interactive web page, using Javascript to do the calculations.

I put Quanfei’s pages up on my web site as a resource for other students taking my Computer Organization course, where it was found by Keven Brewer, who was working for Delco Electronics at the time. Kevin noticed some special cases that Quanfei’s calculators didn’t handle, and volunteered to work up the precise versions of the code that the calculators now use. Kevin also did a web search for information on the IEEE-754 standard, which is included in the Kevin’s Report section below. Kevin also prepared some web tables listing the parameters of the standard.

The calculators have turned out to be a popular resource on the web for academics and industry alike. I know of translations into Spanish and German, and they get over 10,000 hits a month on my web server.

In December 2004 I started reworking these pages to improve their appearance and conformance to web standards. The javascript code behind them has been moved to separate files, but interested users can download the code from the js/ directory. The original pages, with embedded javascript code, remain available at my old Computer Organization course web site.

## Some References

“In 1976 Intel began to design a floating-point co-processor for its i8086/8 and i432 microprocessors. Dr. John Palmer persuaded Intel that it needed an arithmetic standard to prevent different boxes with “Intel” on the outside from computing disparate results inside. At Stanford ten years earlier, Palmer had heard a visiting professor, William Kahan, analyze commercially significant arithmetics and assess how much their anomalies inflated the costs of reliable and portable numerical software. Kahan had also enhanced the numerical prowess of a successful line of Hewlett-Packard calculators. Palmer, now the manager of Intel’s floating-point effort, recruited Kahan as a consultant to help design the arithmetic for the i432 ( which died later ) and for the i8086/8’s upcoming i8087 coprocessor.” From An Interview with the Old Man of Floating-Point, a short, facinating read on the development of the IEEE-754 Floating-Point standard.

*The IEEE-754 Standard for Binary Floating-Point Arithmetic*
was published in 1985. You can order a copy of the standard from the
IEEE. The web site for the IEEE-754 is a good place to go for
links to information about IEEE-754 and floating-point in general.

A popular paper on the mathematical properties of floating-point numbers, including the IEEE-754 standard is What Every Computer Scientist Should Know About Floating-Point Arithmetic, which is also available from docs.sun.com

The Q4, 1999 issue of the Intel Technology Journal had an article, * IA-64 Floating-Point operations and the
IEEE standard for binary floating-point arithmetic,* by M. Cornea-Hasegan and B. Norin, which provides a
summary of the IEEE-754 standard and includes a table of ten different floating-point formats used by
Intel’s 64-bit microprocessors (the IA-64 architecture). Most of the article discusses featues of the
IA-64 floating-point instruction set. [2008-11-14: Link updated to point to PDF version of the article.]

A significant floating-point standard, which pre-dates the IEEE-754 standard, is the “hexadecimal
encoding” used on IBM mainframes. This format uses sixteen instead of two as the base to which the
exponent is raised. The IBM S/390 G5 processor was the first one to integrate IBM’s traditional
hexadecimal encoding and IEEE-754 in the same floating-point unit. It is described in the paper, *The S/390 G5 floating point unit* by
E. M. Schwarz and C. A. Krygowski, which appeared in the IBM Journal of Research and Development, vol. 43, No.
5/6, September/November 1999, pp 707-721.

## Kevin’s Report

The rest of the material on this page came from Kevin J. Brewer, who worked for Delco Electronics at the time he wrote it. In addition to the material below, Kevin greatly refined the JavaScript code for the IEEE-754 Calculator page originally written by a Queens College student, Quanfei Wen.

At the end of this page are Kevin’s Charts, which summarize the IEEE-754 single and double precision formats.

If you find a broken link in the material below, please let me know, especially if you know where the page
has moved. (Send mail to *vickery at babbage.cs.qc.edu* with “IEEE-754” in the Subject line.)
Where there are links that I know are broken in what follows, I put the broken link in [square brackets] and
preserved Kevin’s surrounding text.

Kevin suggested, “Scroll up and down from the locations cited below in order to learn other information about the IEEE-754 standard.”

The source which showed me that there were actually positive and negative NaNs and introduced me to a new special number, Indeterminate, was [ this page ] (link updated 2007-10-13). To find the table showing these NaNs and Indeterminate, use the Edit | Find... command on the string “the corresponding values”. Scroll up a little in order to take a look at the “Special Operations” table. And right above that table is the list of special numbers and their meanings.

The source which introduced me to the concepts of “signaling” and “quiet” NaNs was [ http://www.cas.american.edu/~studdard/classes/fall1995/4028201/notes/17oct95/I.html ]. To find the section on “signaling” and “quiet” NaNs, use the Edit | Find... command on the string “NaNs can be signaling or quiet”.

The source which allowed me to distinguish between “signaling” and “quiet” NaNs was [ this page ]. To find the section on NaNs and the encodings of other special numbers, use the Edit | Find... command on the string “The definition of NaNs”.

[ This source ] shows the mathematical equations which define the various IEEE-754 values and ranges.

The source which introduced me to IEEE-754’s four rounding modes and the guard, round, and “sticky” bits was [ this page ]. To find the section on rounding, use the Edit | Find... command on the string “four different rounding modes”.

Some sources on the Web claim that IEEE-754 specifies four floating-point formats in two groups, basic and extended, with a “single-precision” and a “double-precision” format in each of the two groups. To find this information, use the Edit | Find... command on the string “IEEE 754 specifies four” on [ http://www.cas.american.edu/~studdard/classes/fall1995/4028201/notes/17oct95/I.html ] and the Edit | Find... command on the string “The other two formats” on [ this page ].

Upon reading the IEEE-754 standard, one learns from “Table 1, Summary of Format Parameters” on page 9 that the extended formats are very loosely defined with unspecified exponential biases and only lower bounds for precisions and exponents, while the basic formats are specified exactly in terms of field widths and semantics. The extended formats are so loosely defined that particular implementations of these formats may be so different that numerical approximation routines using them could be non-portable.

Other sources on the Web claim that IEEE-754 specifies only three floating-point formats, “single-precision”, “double-precision”, and “quadruple-precision”. One source [ href="http://www.iac.tut.fi/usr/local/doc/Fortran-90/Version-2.0/f9a200_spd.txt"> ] shows the three IEEE-754 formats and their max and min values in DEC’s Fortran-90 documentation. To find the section on the three IEEE-754 formats, use the Edit | Find... command on the string “32-bit IEEE”. [ Another source ] shows the encodings of the special numbers and the number of bits in each field for each of the three IEEE-754 formats. To find the sections on the three IEEE-754 formats, use the Edit | Find... command on the string “For single-precision floating point numbers” and start scrolling down.

When comparing the format parameters of “extended double-precision” in IEEE-754’s Table 1 and those of the so-called “quadruple-precision”, one finds that the “quadruple-precision” format is simply a specific instance of the “extended double-precision” format. Similarly, one will note that “double-precision” is a specific instance of “extended single-precision”.

The 80-bit “extended-precision” format is used “internally” by the Intel 80x87 floating-point math “co-processor” in order to be able to shift operands back and forth without any loss of precision in the IEEE-754 64-bit (and 32-bit) format. To find this information, use the Edit | Find... command on the string “it also implements an “extended-precision” format” on [ http://www.cas.american.edu/~studdard/classes/fall1995/4028201/notes/17oct95/I.html ].

A source which describes the exponential bias of Intel’s 80-bit “extended-precision” format and its usage of the additional bits it contains relative to the “double-precision” format is [ webster.cs.ucr.edu ]. (Link updated 2005-05-03). To find this data, use the Edit | Find... command on the string “In order to help ensure accuracy”.

[ http://webster.cs.ucr.edu/Page_asm/ArtofAssembly/CH14/CH14-3.html ] states that Intel’s “extended-precision” format supports non-normalized numbers (values very close to zero whose most significant mantissa bit is not zero). To find this support information, use the Edit | Find... command on the string “Normalized values provide”.

When one compares these stated and implied format parameters of Intel’s “extended-precision” with those of “extended double-precision” in Table 1, one finds that the “extended-precision” format is a specific instance of the “extended double-precision” format, similarly to the “quadruple-precision” format.

No. | Parameter | Format | |||||
---|---|---|---|---|---|---|---|

Single |
Single Extended |
Double |
Double Extended |
Quadruple ^{ +} |
Extended ^{ #} |
||

© Copyright 1985 by The Institute of Electrical and Electronics Engineers, Inc | |||||||

^{+ }Although the “quadruple-precision” name and the
particular parameters of its format are not specified in the
IEEE-754 standard, it is a legally derived IEEE-754 format because
its parameters are specific subset elements within the bounds of
those specified for the “extended double-precision” format. |
|||||||

^{# }Like the “quadruple-precision” format, Intel’s
“extended-precision” format is a legal IEEE-754 format derived
from the “extended double-precision” format. |
|||||||

(1) |
p (precision,apparent mantissa width in bits) |
24 | ≥ 32 | 53 | ≥ 64 | 113 | 64 |

(2) |
Decimal digits of precisionp / log_{2}(10) |
7.22 | ≥ 9.63 | 15.95 | ≥ 19.26 | 34.01 | 19.26 |

(3) |
Mantissa’s MS-Bit | hidden bit | unspecified | hidden bit | unspecified | hidden bit | explicit bit |

(4) |
Actual mantissa width in bits | 23 | ≥ 31 | 52 | ≥ 63 | 112 | 64 |

(5) |
E_{max} |
+127 | ≥ +1023 | +1023 | ≥ +16383 | +16383 | +16383 |

(6) |
E_{min} |
-126 | ≤ -1022 | -1022 | ≤ -16382 | -16382 | -16382 |

(7) |
Exponent bias |
+127 | unspecified | +1023 | unspecified | +16383 | +16383 |

(8) |
Exponent width in bits | 8 | ≥ 11 | 11 | ≥ 15 | 15 | 15 |

(9) |
Sign width in bits | 1 | 1 | 1 | 1 | 1 | 1 |

(10) |
Format width in bits(9) + (8) + (4) |
32 | ≥ 43 | 64 | ≥ 79 | 128 | 80 |

(11) |
Range Magnitude Maximum 2 ^{Emax + 1} |
3.4028E+38 | ≥ 1.7976E+308 | 1.7976E+308 | ≥ 1.1897E+4932 | 1.1897E+4932 | 1.1897E+4932 |

(12) |
Range Magnitude Minimum 2 ^{Emin} |
1.1754E-38 | ≤ 2.2250E-308 | 2.2250E-308 | ≤ 3.3621E-4932 | 3.3621E-4932 | 3.3621E-4932 |

(13) |
Range Magnitude Minimum (Denormalized) 2 ^{Emin - (4)} |
1.4012E-45 | ≤ 1.0361E-317 | 4.9406E-324 | ≤ 3.6451E-4951 | 6.4751E-4966 | 1.8225E-4951 |

(14) |
FORTRAN Language Type | REAL*4 | REAL*8 | REAL*16 | REAL*10 | ||

(15) |
C Language Type | float | double | long double | long double |

Other sources on IEEE-754 include:

- [ http://spectra.eng.hawaii.edu/Courses/EE361.S95/Lectures/Lec38/lec38.3.html ]
- [ http://www.ece.uiuc.edu/~ece291/lecture/l11.html ]
- [ Carleton University ]
- [ Grinnell College ]
- [ http://cch.loria.fr/documentation/IEEE754/index.html#wkahan Papers on Floating-Point by William Kahan -- “The Father of IEEE-754” ] Broken link, but see Prof. Kahan’ home page.

## Kevin’s Summary Charts

### Storage Layout and Ranges of Floating-Point Numbers

IEEE-754 floating-point numbers require three component fields: the sign,
the exponent, and the mantissa. The exponential base is 2 and is never stored
in any way with the value in either the registers or memory (it is implied).
In order to allow the exponent and mantissa, when taken together, to vary
monotonically, the signed exponent is represented in excess-127 unsigned form
for single precision and excess-1023 for double precision. This excess-127
(or excess-1023) representation is indicated by the variable
“*e*” below.

Since IEEE-754 floating-point numbers are stored in a *signed
magnitude* form, the ranges and binary patterns of the positive and
negative numbers are symmetric about the midpoint of the entire range
of values (between the positive and negative zeros). As a result,
essentially any statement made in regard to the positive numbers is
also true of the negative numbers and vice versa.

The range of positive floating-point numbers is split into
*normalized* numbers (*normal* numbers) which preserve the
full precision of the mantissa, including the hidden bit, (24 bits for
single precision and 53 bits for double precision) and
*denormalized* numbers (*subnormal* numbers, so-called
unnormalized numbers) which have from 1 to 23 significant bits for
single precision and 1 to 52 bits for double precision.

The number line tables below, which show the layout for single (32-bit) and double (64-bit) precision floating-point numbers and their special values, were inspired by the table on [ this page ] (link updated 2007-10-13). To find the table on which these two are based, use the Edit | Find... command on the string “the corresponding values”. In their column headers, these tables indicate the number of bits in each field along with their bit ranges in square brackets.

The values shown in the **Decimal Range** column of the tables
are the end points of their respective ranges with the IEEE-754
round-to-nearest value mode applied. JavaScript uses IEEE-754 double
precision floating-point with round-to-nearest value mode to perform
all of its arithmetic operations including its input string to numeric
conversion routine. Therefore, by default, double (64-bit) precision
conversions are automatically rounded to values matching these
tables. In order for single (32-bit) precision conversions to be
rounded to values matching these tables, the user *must* click
the **Rounded** button on those pages where it is present.

### 32-bit Single Precision

Range Name | Sign (s)1 [31] |
Exponent (e)8 [30-23] |
Mantissa (m)23 [22-0] |
Hexadecimal Range | Range | Decimal Range^{ §} |
---|---|---|---|---|---|---|

Quiet -NaN |
1 | 11..11 | 11..11 : 10..01 |
FFFFFFFF: FFC00001 |
||

Indeterminate | 1 | 11..11 | 10..00 | FFC00000 |
||

Signaling -NaN |
1 | 11..11 | 01..11 : 00..01 |
FFBFFFFF: FF800001 |
||

-Infinity (Negative Overflow) |
1 | 11..11 | 00..00 | FF800000 |
< -(2-2^{-23}) × 2^{127} |
≤ -3.4028235677973365E+38 |

Negative Normalized-1. m × 2^{(e-127)} |
1 | 11..10 : 00..01 |
11..11 : 00..00 |
FF7FFFFF: 80800000 |
-(2-2^{-23}) × 2^{127}: -2 ^{-126} |
-3.4028234663852886E+38 : -1.1754943508222875E-38 |

Negative Denormalized-0. m × 2^{(-126)} |
1 | 00..00 | 11..11 : 00..01 |
807FFFFF: 80000001 |
-(1-2^{-23}) × 2^{-126}: -2 ^{-149}(-(1+2 ^{-52}) × 2^{-150})^{ *} |
-1.1754942106924411E-38 : -1.4012984643248170E-45 (-7.0064923216240862E-46) ^{ *} |

Negative Underflow | 1 | 00..00 | 00..00 | 80000000 |
-2^{-150}: < -0 |
-7.0064923216240861E-46 : < -0 |

-0 |
1 | 00..00 | 00..00 | 80000000 |
-0 | -0 |

+0 |
0 | 00..00 | 00..00 | 00000000 |
0 | 0 |

Positive Underflow | 0 | 00..00 | 00..00 | 00000000 |
> 0 : 2 ^{-150} |
> 0 : 7.0064923216240861E-46 |

Positive Denormalized0. m × 2^{(-126)} |
0 | 00..00 | 00..01 : 11..11 |
00000001: 007FFFFF |
((1+2^{-52}) × 2^{-150})^{ *}2 ^{-149}: (1-2 ^{-23}) × 2^{-126} |
(7.0064923216240862E-46)^{ *}1.4012984643248170E-45 : 1.1754942106924411E-38 |

Positive Normalized1. m × 2^{(e-127)} |
0 | 00..01 : 11..10 |
00..00 : 11..11 |
00800000: 7F7FFFFF |
2^{-126}: (2-2 ^{-23}) × 2^{127} |
1.1754943508222875E-38 : 3.4028234663852886E+38 |

+Infinity (Positive Overflow) |
0 | 11..11 | 00..00 | 7F800000 |
> (2-2^{-23}) × 2^{127} |
≥ 3.4028235677973365E+38 |

Signaling +NaN |
0 | 11..11 | 00..01 : 01..11 |
7F800001: 7FBFFFFF |
||

Quiet +NaN |
0 | 11..11 | 10..00 : 11..11 |
7FC00000: 7FFFFFFF |

### 64-bit Double Precision

Range Name | Sign (s)1 [63] |
Exponent (e)11 [62-52] |
Mantissa (m)52 [51-0] |
Hexadecimal Range | Range | Decimal Range^{ §} |
---|---|---|---|---|---|---|

Quiet -NaN |
1 | 11..11 | 11..11 : 10..01 |
FFFFFFFFFFFFFFFF: FFF8000000000001 |
||

Indeterminate | 1 | 11..11 | 10..00 | FFF8000000000000 |
||

Signaling -NaN |
1 | 11..11 | 01..11 : 00..01 |
FFF7FFFFFFFFFFFF: FFF0000000000001 |
||

-Infinity (Negative Overflow) |
1 | 11..11 | 00..00 | FFF0000000000000 |
< -(2-2^{-52}) × 2^{1023} |
≤ -1.7976931348623158E+308 |

Negative Normalized-1. m × 2^{(e-1023)} |
1 | 11..10 : 00..01 |
11..11 : 00..00 |
FFEFFFFFFFFFFFFF: 8010000000000000 |
-(2-2^{-52}) × 2^{1023}: -2 ^{-1022} |
-1.7976931348623157E+308 : -2.2250738585072014E-308 |

Negative Denormalized-0. m × 2^{(-1022)} |
1 | 00..00 | 11..11 : 00..01 |
800FFFFFFFFFFFFF: 8000000000000001 |
-(1-2^{-52}) × 2^{-1022}: -2 ^{-1074}(-(1+2 ^{-52}) × 2^{-1075})^{ *} |
-2.2250738585072010E-308 : -4.9406564584124654E-324 (-2.4703282292062328E-324) ^{ *} |

Negative Underflow | 1 | 00..00 | 00..00 | 8000000000000000 |
-2^{-1075}: < -0 |
-2.4703282292062327E-324 : < -0 |

-0 |
1 | 00..00 | 00..00 | 8000000000000000 |
-0 | -0 |

+0 |
0 | 00..00 | 00..00 | 0000000000000000 |
0 | 0 |

Positive Underflow | 0 | 00..00 | 00..00 | 0000000000000000 |
> 0 : 2 ^{-1075} |
> 0 : 2.4703282292062327E-324 |

Positive Denormalized0. m × 2^{(-1022)} |
0 | 00..00 | 00..01 : 11..11 |
0000000000000001: 000FFFFFFFFFFFFF |
((1+2^{-52}) × 2^{-1075})^{ *}2 ^{-1074}: (1-2 ^{-52}) × 2^{-1022} |
(2.4703282292062328E-324)^{ *}4.9406564584124654E-324 : 2.2250738585072010E-308 |

Positive Normalized1. m × 2^{(e-1023)} |
0 | 00..01 : 11..10 |
00..00 : 11..11 |
0010000000000000: 7FEFFFFFFFFFFFFF |
2^{-1022}: (2-2 ^{-52}) × 2^{1023} |
2.2250738585072014E-308 : 1.7976931348623157E+308 |

+Infinity (Positive Overflow) |
0 | 11..11 | 00..00 | 7FF0000000000000 |
> (2-2^{-52}) × 2^{1023} |
≥ 1.7976931348623158E+308 |

Signaling +NaN |
0 | 11..11 | 00..01 : 01..11 |
7FF0000000000001: 7FF7FFFFFFFFFFFF |
||

Quiet +NaN |
0 | 11..11 | 10..00 : 11..11 |
7FF8000000000000: 7FFFFFFFFFFFFFFF |

^{§ }Your least significant digits may
differ.

^{* }The minimum magnitude values of denormalized
ranges are represented by a single significant bit (a bit whose value
is 1) at the right hand end of its format’s mantissa. For single
(32-bit) and double (64-bit) precision, these minimum range values are
1.4012984643248170E-45 and 4.9406564584124654E-324 respectively. The
values 7.0064923216240862E-46 and 2.4703282292062328E-324 are each a
little more than half of these minima. They are represented by one
significant bit to the right of their format’s storable mantissa and
another 1-bit spaced the double precision’s mantissa width to the
right of the first bit. Then, as a result of the IEEE-754
round-to-nearest value mode’s operation, these values are rounded to
the denormalized range minimum values.