The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard established by the Institute of Electrical and Electronics Engineers (IEEE) and the most widely used standard for floating-point computation, followed by many hardware (CPU and FPU) and software implementations. Many computer languages allow or require that some or all arithmetic be carried out using IEEE 754 formats and operations. The current version is IEEE 754-2008, which was published in August 2008; it includes nearly all of the original IEEE 754-1985 (which was published in 1985) and the IEEE Standard for Radix-Independent Floating-Point Arithmetic (IEEE 854-1987). The international standard ISO/IEC/IEEE 60559:2011 has been approved for adoption through JTC1/SC 25 under the ISO/IEEE PSDO Agreement[1] and published.[2]
The standard defines
The standard also includes extensive recommendations for advanced exception handling, additional operations (such as trigonometric functions), expression evaluation, and for achieving reproducible results.
The standard is derived from and replaces IEEE 754-1985, the previous version, following a seven-year 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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Formats in IEEE 754 describe sets of floating-point data and encodings for interchanging them.
A given format comprises:
The possible finite values that can be represented in a given format are determined by the base (b), the number of digits in the significand (precision, p), and the exponent parameter emax:
Hence (for the example parameters) the smallest non-zero positive number that can be represented is 1×10−101 and the largest is 9999999×1090 (9.999999×1096), and the full range of numbers is −9.999999×1096 through 9.999999×1096. The numbers −b1−emax and b1−emax (here, −1×10−95 and 1×10−95) are the smallest (in magnitude) normal numbers; non-zero numbers between these smallest numbers are called subnormal numbers.
Zero values are finite values with significand 0. These are signed zeros, the sign bit specifies if a zero is +0 (positive zero) or −0 (negative zero).
The standard defines five basic formats, named using their base and the number of bits used to encode them. A conforming implementation must fully implement at least one of the basic formats. There are three binary floating-point basic formats (which can be encoded using 32, 64 or 128 bits) and two decimal floating-point basic formats (which can be encoded using 64 or 128 bits). The binary32 and binary64 formats are the single and double formats of IEEE 754-1985.
The precision of the binary formats is one greater than the width of its significand, because there is an implied (hidden) 1 bit.
| Name | Common name | Base | Digits | E min | E max | Notes | Decimal digits |
Decimal E max |
|---|---|---|---|---|---|---|---|---|
| binary16 | Half precision | 2 | 10+1 | −14 | +15 | storage, not basic | 3.31 | 4.51 |
| binary32 | Single precision | 2 | 23+1 | −126 | +127 | 7.22 | 38.23 | |
| binary64 | Double precision | 2 | 52+1 | −1022 | +1023 | 15.95 | 307.95 | |
| binary128 | Quadruple precision | 2 | 112+1 | −16382 | +16383 | 34.02 | 4931.77 | |
| decimal32 | 10 | 7 | −95 | +96 | storage, not basic | 7 | 96 | |
| decimal64 | 10 | 16 | −383 | +384 | 16 | 384 | ||
| decimal128 | 10 | 34 | −6143 | +6144 | 34 | 6144 |
Decimal digits is digits × log10 base, this gives the precision in decimal.
Decimal E max is Emax × log10 base, this gives the maximum exponent in decimal.
All the basic formats are available in both hardware and software implementations.
A format that is just to be used for arithmetic and other operations need not have an encoding associated with it (that is, an implementation can use whatever internal representation it chooses); 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) that it can represent.
Interchange formats are intended for the exchange of floating-point data using a fixed-length bit-string for a given format.
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).
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 = floor(4 log2(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 NaNs.
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 uses a more complex approach to allow the significand to be encoded as a compressed sequence of decimal digits (using Densely Packed Decimal) or as a binary integer. In either case the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and signaling NaNs have a unique encoding (and the same set of possible payloads).
The standard defines five rounding algorithms. The first two round to a nearest value; the others are called directed roundings:
Required operations for a supported arithmetic format (including the basic formats) include:
The standard provides a predicate totalOrder which defines a total ordering 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.
The standard defines five exceptions, each of which has a corresponding status flag that (except in certain cases of underflow) is raised when the exception occurs. No other action is required, but alternatives are recommended (see below).
The five possible exceptions are:
These are the same five exceptions as were defined in IEEE 754-1985.
The standard recommends optional exception handling in various forms, including 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.
A new clause in the standard recommends fifty operations, including log, power, and trigonometric functions, that language standards should define. These are all optional (none are required in order to conform to the standard). The operations include some on dynamic modes for attributes, and also a set of reduction operations (sum, scaled product, etc.). All are required to supply a correctly rounded result, but they do not have to detect or report inexactness.
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.
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.
The standard requires operations to convert between basic formats and external character sequence formats. 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:
For other binary formats the required number of decimal digits is
where p is the number of significant bits in the binary format, e.g. 24 bits for binary32.
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.
When using a decimal floating point format the decimal representation will be preserved using:
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