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The '''IEEE Standard for Floating-Point Arithmetic''' ('''IEEE 754''') is a [[technical standard]]  for [[floating point|floating-point]] computation established in 1985 by the [[Institute of Electrical and Electronics Engineers]] (IEEE). Many hardware [[floating point unit]]s use the IEEE 754 standard. The current version, '''IEEE 754-2008''' published in August 2008, includes nearly all of the original [[IEEE 754-1985]] standard and the IEEE Standard for [[Radix]]-Independent Floating-Point Arithmetic ([[IEEE 854-1987]]). The international standard '''ISO/IEC/IEEE 60559:2011''' (with identical content to IEEE 754) has been approved for adoption through [[ISO/IEC JTC1|JTC1]]/SC 25 under the ISO/IEEE PSDO Agreement<ref>[http://grouper.ieee.org/groups/754/email/msg04167.html FW: ISO/IEC/IEEE 60559 (IEEE Std 754-2008)]</ref> and published.<ref>[http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=57469 ISO/IEC/IEEE 60559:2011 - Information technology - Microprocessor Systems - Floating-Point arithmetic]</ref>
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* ''arithmetic formats:'' sets of [[Binary code|binary]] and [[decimal]] floating-point data, which consist of finite numbers (including [[signed zero]]s and [[subnormal number]]s), [[infinity|infinities]], and special "not a number" values ([[NaN]]s)
 
* ''interchange formats:'' encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form
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* ''rounding rules:'' properties to be satisfied when rounding numbers during arithmetic and conversions
 
* ''operations:'' arithmetic and other operations on arithmetic formats
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* ''exception handling:'' indications of exceptional conditions (such as [[division by zero]], overflow, ''etc.'')
 
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The standard also includes extensive recommendations for advanced exception handling, additional operations (such as [[trigonometric functions]]), expression evaluation, and for achieving reproducible results.
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The standard is derived from and replaces [[IEEE 754-1985]], the previous version, following a seven-year [[IEEE 754 revision|revision process]], chaired by Dan Zuras and edited by [[Mike Cowlishaw]]. The binary formats in the original standard are included in the new standard along with three new basic formats (one binary and two decimal). To conform to the current standard, an implementation must implement at least one of the basic formats as both an arithmetic format and an interchange format.
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An IEEE 754 ''format'' is a "set of representations of numerical values and symbols". A format may also include how the set is encoded.


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* Finite numbers, which may be either base 2 (binary) or base 10 (decimal).  Each finite number is described by three integers: ''s''&nbsp;= a ''sign'' (zero or one), ''c''&nbsp;= a ''[[significand]]'' (or 'coefficient'), ''q''&nbsp;= an ''exponent''.  The numerical value of a finite number is <br>&nbsp; (−1)<sup>''s''</sup> × ''c'' × ''b''<sup>''q''</sup> <br>where ''b'' is the base (2 or 10).  For example, if the sign is 1 (indicating negative), the significand is 12345, the exponent is −3, and the base is 10, then the value of the number is −12.345.
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* Two kinds of [[NaN]]: a quiet NaN (qNaN) and a signaling NaN (sNaN). A NaN may carry a ''payload'' that is intended for diagnostic information indicating the source of the NaN.  The sign of a NaN has no meaning, but it may be predictable in some circumstances.
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The possible finite values that can be represented in a format are determined by the base (''b''), the number of digits in the significand (precision, ''p''), and the exponent parameter ''emax'':
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* ''c'' must be an integer in the range zero through ''b''<sup>''p''</sup>−1 (''e.g.'', if ''b''=10 and ''p''=7 then c is 0 through 9999999)
 
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Hence (for the example parameters) the smallest non-zero positive number that can be represented is 1×10<sup>−101</sup> and the largest is 9999999×10<sup>90</sup> (9.999999×10<sup>96</sup>), and the full range of numbers is −9.999999×10<sup>96</sup> through 9.999999×10<sup>96</sup>. The numbers −''b''<sup>1−''emax''</sup> and ''b''<sup>1−''emax''</sup> (here, −1×10<sup>−95</sup> and 1×10<sup>−95</sup>) are the smallest (in magnitude) ''normal numbers''; non-zero numbers between these smallest numbers are called [[subnormal number]]s.
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Zero values are finite values with significand 0. These are [[signed zero]]s, the sign bit specifies if a zero is +0 (positive zero) or −0 (negative zero).
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=== Basic formats ===
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The standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits). The [[binary32]] and [[binary64]] formats are the ''single'' and ''double'' formats of [[IEEE 754-1985]]. A conforming implementation must fully implement at least one of the basic formats.
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The typical precision of the basic binary formats is one bit more than the width of its significand. The extra bit of precision comes from an implied (hidden) leading ''1'' bit. The typical floating point number will be normalized such that the most significant bit will be a one. If the leading bit is known to be one, then it need not be encoded in the interchange format.
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! Name !! Common name !! Base !! Digits !! E min !! E max !! Notes !! Decimal<br>digits !! Decimal<br>E max
 
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|[[Single precision floating-point format|binary32]]||Single precision||align="right"|2||align="right"|23+1||align="right"|−126||align="right"|+127|| ||align="right"|7.22||align="right"|38.23
 
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|[[Double precision floating-point format|binary64]]||Double precision||align="right"|2||align="right"|52+1||align="right"|−1022||align="right"|+1023|| ||align="right"|15.95||align="right"|307.95
 
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|[[Quadruple precision floating-point format|binary128]]||Quadruple precision||align="right"|2||align="right"|112+1||align="right"|−16382||align="right"|+16383|| ||align="right"|34.02||align="right"|4931.77
 
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|-
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|[[Decimal64 floating-point format|decimal64]]||||align="right"|10||align="right"|16||align="right"|−383||align="right"|+384|| ||align="right"|16||align="right"|384
|-
|[[Decimal128 floating-point format|decimal128]]||||align="right"|10||align="right"|34||align="right"|−6143||align="right"|+6144|| ||align="right"|34||align="right"|6144
|}


Decimal digits is ''digits'' × log<sub>10</sub> ''base'', this gives an approximate precision in decimal.
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Decimal E max is ''Emax'' × log<sub>10</sub> ''base'', this gives the maximum exponent in decimal.
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=== Extended and extendable precision formats ===
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The standard specifies extended and extendable precision formats, which are recommended for allowing a greater precision than that provided by the basic formats.<ref>{{Harvnb|IEEE 754|2008|loc=§3.7}}</ref> An extended precision format extends a basic format by using more precision and more exponent range. An extendable precision format allows the user to specify the precision and exponent range. An implementation may use whatever internal representation it chooses for such formats; all that needs to be defined are its parameters (''b'', ''p'', and ''emax'').  These parameters uniquely describe the set of finite numbers (combinations of sign, significand, and exponent for the given radix) that it can represent.


The standard does not require an implementation to support extended or extendable precision formats.
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The standard recommends that languages provide a method of specifying ''p'' and ''emax'' for each supported base ''b''.<ref>{{Harvnb|IEEE 754|2008|loc=§3.7}} states, "Language standards should define mechanisms supporting extendable precision for each supported radix."</ref><!-- extendable recommendation does not apply to "implementations". -->
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The standard recommends that languages and implementations support an extended format which has a greater precision than the largest basic format supported for each radix ''b''.<ref>{{Harvnb|IEEE 754|2008|loc=§3.7}} states, "Language standards or implementations should support an extended precision format that extends the widest basic format that is supported in that radix."</ref>
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For an extended format with a precision between two basic formats the exponent range must be as great as that of the next wider basic format. So for instance a 64-bit extended precision binary number must have an 'emax' of at least 16383. The [[x87]] [[Extended precision|80-bit extended format]] meets this requirement.
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=== Interchange formats ===
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Interchange formats are intended for the exchange of floating-point data using a fixed-length bit-string for a given format.
 
 
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For the exchange of binary floating-point numbers, interchange formats of length 16 bits, 32 bits, 64 bits, and any multiple of 32 bits ≥128 are defined.  The 16-bit format is intended for the exchange or storage of small numbers (''e.g.'', for graphics).
    
 
</ul>
The encoding scheme for these binary interchange formats is the same as that of IEEE 754-1985: a sign bit, followed by ''w'' exponent bits that describe the exponent offset by a ''bias'', and ''p''−1 bits that describe the significand.  The width of the exponent field for a ''k''-bit format is computed as ''w''&nbsp;=&nbsp;round(4&nbsp;log<sub>2</sub>(''k''))−13.  The existing 64- and 128-bit formats follow this rule, but the 16- and 32-bit formats have more exponent bits (5 and 8) than this formula would provide (3 and 7, respectively).
 
As with IEEE 754-1985, there is some flexibility in the encoding of signaling NaN.
 
For the exchange of decimal floating-point numbers, interchange formats of any multiple of 32 bits are defined.
 
The encoding scheme for the decimal interchange formats similarly encodes the sign, exponent, and significand, but two different bit-level representations are defined.  Interchange is complicated by the fact that some external indicator of the representation in use is required.  The two options allow the significand to be encoded as a compressed sequence of decimal digits (using [[densely packed decimal]]) or alternatively as a binary integer. The former is more convenient for direct hardware implementation of the standard, while the latter is more suited to software emulation on a binary computer.  In either case the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and special values (±zero, ±infinity, quiet NaNs, and signaling NaNs) have identical binary representations.
 
== Rounding rules ==
The standard defines five rounding rules. The first two round to a nearest value; the others are called ''[[directed rounding]]s'':
 
=== Roundings to nearest ===
* '''[[Rounding#Round_half_to_even|Round to nearest, ties to even]]'''&nbsp;– rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which occurs 50% of the time; this is the default for binary floating-point and the recommended default for decimal.
* '''[[Rounding#Round_half_away_from_zero|Round to nearest, ties away from zero]]'''&nbsp;– rounds to the nearest value; if the number falls midway it is rounded to the nearest value above (for positive numbers) or below (for negative numbers); this is intended as an option for decimal floating point.
 
=== Directed roundings ===
* '''Round toward 0'''&nbsp;– directed rounding towards zero (also known as ''truncation'').
* '''Round toward +∞'''&nbsp;– directed rounding towards positive infinity (also known as ''rounding up'' or ''ceiling'').
* '''Round toward −∞'''&nbsp;– directed rounding towards negative infinity (also known as ''rounding down'' or ''floor'').
 
{| class="wikitable"
|+ Rounding to integers using the IEEE 754 rules
! Rounding mode
! +11.5
! +12.5
! −11.5
! −12.5
|-
| to nearest, ties to even
| +12.0
| +12.0
| −12.0
| −12.0
|-
| to nearest, ties away from zero
| +12.0
| +13.0
| −12.0
| −13.0
|-
| toward 0
| +11.0
| +12.0
| −11.0
| −12.0
|-
| toward +∞
| +12.0
| +13.0
| −11.0
| −12.0
|-
| toward −∞
| +11.0
| +12.0
| −12.0
| −13.0
|}
 
== Operations ==
 
Required operations for a supported arithmetic format (including the basic formats) include:
 
* Arithmetic operations (add, subtract, multiply, divide, square root, [[fused multiply–add]], remainder, ''etc.'')
* Conversions (between formats, to and from strings, ''etc.'')
* Scaling and (for decimal) quantizing
* Copying and manipulating the sign (abs, negate, ''etc.'')
* Comparisons and total ordering
* Classification and testing for NaNs, ''etc.''
* Testing and setting flags
* Miscellaneous operations.
 
=== Total-ordering predicate ===
 
The standard provides a predicate ''totalOrder'' which defines a [[total order]]ing for all floating numbers for each format. The predicate agrees with the normal comparison operations when they say one floating point number is less than another. The normal comparison operations however treat NaNs as unordered and compare −0 and +0 as equal. The totalOrder predicate will order these cases, and it also distinguishes between different representations of NaNs and between the same decimal floating point number encoded in different ways.
 
== Exception handling ==
The standard defines five exceptions, each of which returns a default value and has a corresponding status flag that (except in certain cases of underflow) is raised when the exception occurs.  No other exception handling is required, but additional non-default alternatives are recommended (see below).
 
The five possible exceptions are:
* Invalid operation (''e.g.'', square root of a negative number) (returns qNaN by default).
* Division by zero (an operation on finite operands gives an exact infinite result, ''e.g.'', 1/0 or log(0)) (returns ±infinity by default).
* Overflow (a result is too large to be represented correctly) (returns ±infinity by default (for round-to-nearest mode)).
* Underflow (a result is very small (outside the normal range) and is inexact) (returns a denormalized value by default).
* Inexact (returns correctly rounded result by default).
 
These are the same five exceptions as were defined in IEEE 754-1985, but the ''division by zero'' exception has been extended to operations other than the division.
 
For decimal floating point, there are additional exceptions along with the above:<ref>[http://docs.python.org/library/decimal.html#signals 9.4. decimal — Decimal fixed point and floating point arithmetic — Python v2.7.3 documentation]</ref><ref>[http://speleotrove.com/decimal/daexcep.html Decimal Arithmetic - Exceptional conditions]</ref>
* Clamped (a result's exponent is too large for the destination format). By default, trailing zeros will be added to the coefficient to reduce the exponent to the largest usable value. If this is not possible (because this would cause the number of digits needed is more than the destination format) then overflow occurs.
* Rounded (a result's coefficient requires more digits than the destination format provides). The inexact is signaled if any non-zero digits are discarded.
 
Additionally, operations like quantize when either operand is infinite, or when the result does not fit the destination format, will also signal invalid operation exception.<ref>{{Harvnb|IEEE|2008|loc=&sect;7.2(h)}}</ref>
 
== Recommendations ==
 
=== Alternate exception handling ===
The standard recommends optional exception handling in various forms, including presubstitution of user-defined default values, and traps (exceptions that change the flow of control in some way) and other exception handling models which interrupt the flow, such as try/catch.  The traps and other exception mechanisms remain optional, as they were in IEEE 754-1985.
 
=== Recommended operations ===
Clause 9 in the standard recommends fifty operations, including logarithmic, exponential, and trigonometric functions, that language standards should define.<ref>{{Harvnb|IEEE 754|2008|loc=Clause 9}}</ref> These are all optional (none are required in order to conform to the standard). The operations include setting and accessing dynamic mode rounding direction,<ref>{{Harvnb|IEEE 754|2008|loc=§9.3}}</ref> and vector reduction operations such as sum, scaled product, and [[dot product]].<ref>{{Harvnb|IEEE 754|2008|loc=§9.4}}</ref> Conforming implementations must return correctly rounded results depending on the active rounding mode. The inexact exception need not be set correctly, however the other exceptions must be set as specified.
 
=== Expression evaluation ===
The standard recommends how language standards should specify the semantics of sequences of operations, and points out the subtleties of literal meanings and optimizations that change the value of a result. By contrast the previous [[IEEE 754-1985|1985]] version of the standard left aspects of the language interface unspecified, which led to inconsistent behaviour between compilers, or different optimization levels in a single compiler.
 
Programming languages should allow a user to specify a minimum precision for intermediate calculations of expressions for each radix. This is referred to as "preferredWidth" in the standard, and it should be possible to set this on a per block basis. Intermediate calculations within expressions should be calculated, and any temporaries saved, using the maximum of the width of the operands and the preferred width, if set. Thus for instance a compiler targeting [[x87]] floating point hardware should have a means of specifying that intermediate calculations must use [[Extended precision#IEEE 754 extended precision formats|doubled extended format]]. The stored value of a variable must always be used when evaluating subsequent expressions, rather than any precursor from before rounding and assigning to the variable.
 
=== Reproducibility ===
The IEEE 754-1985 allowed many variations in implementations (such as the encoding of some values and the detection of certain exceptions).  IEEE 754-2008 has tightened up many of these, but a few variations still remain (especially for binary formats). The reproducibility clause recommends that language standards should provide a means to write reproducible programs (''i.e.'', programs that will produce the same result in all implementations of a language), and describes what needs to be done to achieve reproducible results.
 
== Character representation ==
The standard requires operations to convert between basic formats and ''external character sequence'' formats.<ref>{{Harvnb|IEEE 754|2008|loc=§5.12}}</ref> Conversions to and from a decimal character format are required for all formats. Conversion to an external character sequence must be such that conversion back using round to even will recover the original number. There is no requirement to preserve the payload of a NaN or signaling NaN, and conversion from the external character sequence may turn a signaling NaN into a quiet NaN.
 
The original binary value will be preserved by converting to decimal and back again using:<ref>{{Harvnb|IEEE 754|2008|loc=§5.12.2}}</ref>
* 5 decimal digits for binary16
* 9 decimal digits for binary32
* 17 decimal digits for binary64
* 36 decimal digits for binary128
 
For other binary formats the required number of decimal digits is
 
:<math>1 +\lceil p\log_{10}(2)\rceil</math>
 
where ''p'' is the number of significant bits in the binary format, e.g. 24 bits for binary32.
 
(Note: as an implementation limit, correct rounding is only guaranteed for the number of decimal digits above plus 3 for the largest binary format supported. For instance if binary32 is the largest supported binary format supported, then a conversion from a decimal external sequence with 12 decimal digits is guaranteed to be correctly rounded when converted to binary32; but conversion of a sequence of 13 decimal digits is not; however the standard recommends that implementations impose no such limit.)
 
When using a decimal floating point format the decimal representation will be preserved using:
*7 decimal digits for decimal32
*16 decimal digits for decimal64
*34 decimal digits for decimal128
 
Algorithms, with code, for correctly rounded conversion from binary to decimal and decimal to binary are discussed in <ref>{{Cite document |first=David M. |last=Gay |title=Correctly rounded binary-decimal and decimal-binary conversions |series=Numerical Analysis Manuscript |id=90-10 |publisher=AT&T Laboratories |location=Murry Hill, NJ, USA |date=November 30, 1990 |url=http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.4049 |doi= |ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}</ref> and for testing in.<ref>{{Cite web |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.144.5889 |title=A Program for Testing IEEE Decimal–Binary Conversion |publisher= |series=Manuscript |first1=Vern |last1=Paxson |first2=William |last2=Kahn |author2-link=William Kahan |date=May 22, 1991 |accessdate=March 28, 2012 |doi= |ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}</ref>
 
== See also ==
* [[Coprocessor]]
* [[C99#IEEE 754 floating point support|C99]] for code examples demonstrating  access and use of IEEE 754 features.
* [[Floating point#IEEE 754: floating point in modern computers|Floating point]], for history, design rationale and example usage of IEEE-754 features.
* [[Half precision]]&nbsp;– [[Single precision]]&nbsp;– [[Double precision]]&nbsp;– [[Quadruple precision]]&nbsp;– [[Extended precision]].
* [[IBM System z9]], the first CPU to implement IEEE 754-2008 decimal arithmetic (using hardware microcode)
* [[IBM z10 (microprocessor)|IBM z10]] and [[IBM z196 (microprocessor)|IBM z196]], CPUs that implement IEEE 754-2008  decimal arithmetic fully in hardware
* [[ISO/IEC 10967]] Language independent arithmetic (LIA)
* [[Minifloat]], low-precision binary floating-point formats following IEEE 754 principles
* [[POWER6]] and [[POWER7]], CPUs that implement IEEE 754-2008  decimal arithmetic fully in hardware
* [[strictfp]], a keyword in the [[Java (programming language)|Java programming language]] that restricts arithmetic to IEEE 754 single and double precision to ensure reproducibility across common hardware platforms.
* [[Rounding#The table-maker's dilemma|The table-maker's dilemma]] for more about the correct rounding of functions.
 
== References ==
{{Reflist|30em}}
 
=== Standard ===
* {{Cite document |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=August 29, 2008 |publisher=IEEE |id=IEEE Std 754-2008 |url=http://ieeexplore.ieee.org/servlet/opac?punumber=4610933 |doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}
* [http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=57469 ISO/IEC/IEEE 60559:2011]
 
=== Secondary references ===
* [http://speleotrove.com/decimal Decimal floating-point] arithmetic, FAQs, bibliography, and links
* [http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm Comparing binary floats]
* [http://babbage.cs.qc.cuny.edu/IEEE-754.old/References.xhtml IEEE 754 Reference Material]
* [http://speleotrove.com/decimal/854mins.html IEEE 854-1987]&nbsp;– History and minutes
* [http://grouper.ieee.org/groups/754/reading.html Supplementary readings for IEEE 754]. Includes historical perspectives.
 
== Further reading ==
* {{cite journal
|    first = David
| last = Goldberg
|      title = What Every Computer Scientist Should Know About Floating-Point Arithmetic
|   journal = [[ACM Computing Surveys]]
|date=March 1991
|    volume = 23
|      issue = 1
|      pages = 5–48
|        doi = 10.1145/103162.103163
|        url = http://www.validlab.com/goldberg/paper.pdf
| accessdate =28 April 2008
|      quote =
|    ref = harv
}}
* {{cite journal
| first = Chris
| last = Hecker
|    authorlink = Chris Hecker
|      title = Let's Get To The (Floating) Point
|    journal = Game Developer Magazine
|date=February 1996
|      pages = 19–24
|      issn = 1073-922X
|        url = http://chrishecker.com/images/f/fb/Gdmfp.pdf
| accessdate =
|      quote =
|    ref = harv
}}
*{{cite journal
|    first = Charles
| last = Severance
|      title = IEEE 754: An Interview with William Kahan
|   journal = [[IEEE Computer]]
|date=March 1998
|    volume = 31
|      issue = 3
|      pages = 114–115
|        doi = 10.1109/MC.1998.660194
|        url = http://www.freecollab.com/dr-chuck/papers/columns/r3114.pdf
| accessdate =28 April 2008
|      quote =
|    ref = harv
}}
* {{cite journal
| first = Mike
| last = Cowlishaw
|    authorlink = Mike Cowlishaw
|      title = Decimal Floating-Point: Algorism for Computers
|    journal = Proceedings 16th IEEE Symposium on Computer Arithmetic
|date=June 2003
|      pages = 104–111
|      isbn = 0-7695-1894-X
|        url = http://www.ece.ucdavis.edu/acsel/arithmetic/arith16/papers/ARITH16_Cowlishaw.pdf
| accessdate =31 December 2008
|      quote =
|    publisher = IEEE Computer Society
|    location = Los Alamitos, Calif.
|    ref = harv
}}.  (Note: ''Algorism'' is not a misspelling of the title; see also [[algorism]].)
* {{cite journal
|    first = David
|  last = Monniaux
|      title = The pitfalls of verifying floating-point computations
|    journal = [[ACM Transactions on Programming Languages and Systems]]
|date=May 2008
|      pages = article #12
|    volume = 30
|      issue = 3
|        doi = 10.1145/1353445.1353446
|      issn = 0164-0925
|        url = http://hal.archives-ouvertes.fr/hal-00128124/en/
|    ref = harv}}: A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.
* {{cite book
|last1=Muller
|first1=Jean-Michel
|last2=Brisebarre
|first2=Nicolas
|last3=de Dinechin
|first3=Florent
|last4=Jeannerod
|first4=Claude-Pierre
|last5=Lefèvre
|first5=Vincent
|last6=Melquiond
|first6=Guillaume
|last7=Revol
|first7=Nathalie
|last8=Stehlé
|first8=Damien
|last9=Torres
|first9=Serge
|display-authors=9
|title=Handbook of Floating-Point Arithmetic
|year=2010
|publisher=Birkhäuser
|isbn=978-0-8176-4705-6}}
* {{cite book
|   first = Michael L.
| last = Overton
|  title = Numerical Computing with IEEE Floating Point Arithmetic
|  year = 2001
|  publisher = SIAM
}}
 
== External links ==
{{Commons category|IEEE 754}}
*[http://babbage.cs.qc.cuny.edu/IEEE-754/ Online IEEE 754 binary calculators]
 
{{IEEE standards}}
 
{{use dmy dates|date=January 2012}}
 
[[Category:Computer arithmetic]]
[[Category:IEEE standards]]

Latest revision as of 22:30, 26 February 2014

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