ISO/IEC 60559:2020
(Main)Information technology — Microprocessor Systems — Floating-Point arithmetic
Information technology — Microprocessor Systems — Floating-Point arithmetic
This standard specifies interchange and arithmetic formats and methods for binary and decimal floating-point arithmetic in computer programming environments. This standard specifies exception conditions and their default handling. An implementation of a floating-point system conforming to this standard may be realized entirely in software, entirely in hardware, or in any combination of software and hardware. For operations specified in the normative part of this standard, numerical results and exceptions are uniquely determined by the values of the input data, sequence of operations, and destination formats, all under user control.
Technologies de l'information — Systèmes de microprocesseurs — Arithmétique flottante
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ISO/IEC 60559
Edition 2.0 2020-05
™
IEEE Std 754
INTERNATIONAL
STANDARD
Floating-point arithmetic
ISO/IEC 60559:2020-05(en) IEEE Std 754-2019
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ISO/IEC 60559
Edition 2.0 2020-05
IEEE Std 754™
INTERNATIONAL
STANDARD
Floating-point arithmetic
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS 35.200 ISBN 978-2-8322-8178-9
Warning! Make sure that you obtained this publication from an authorized distributor.
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019
– 4 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
Contents
1. Overview.11
1.1 Scope.11
1.2 Purpose.11
1.3 Inclusions.11
1.4 Exclusions.11
1.5 Programming environment considerations.12
1.6 Word usage.12
2. Definitions, abbreviations, and acronyms.13
2.1 Definitions.13
2.2 Abbreviations and acronyms.15
3. Floating-point formats.16
3.1 Overview.16
3.2 Specification levels.17
3.3 Sets of floating-point data.17
3.4 Binary interchange format encodings.19
3.5 Decimal interchange format encodings.20
3.6 Interchange format parameters.23
3.7 Extended and extendable precisions.25
4. Attributes and rounding.26
4.1 Attribute specification.26
4.2 Dynamic modes for attributes.26
4.3 Rounding-direction attributes.27
5. Operations.29
5.1 Overview.29
5.2 Decimal exponent calculation.30
5.3 Homogeneous general-computational operations.31
5.4 formatOf general-computational operations.33
5.5 Quiet-computational operations.35
5.6 Signaling-computational operations.37
5.7 Non-computational operations.37
5.8 Details of conversions from floating-point to integer formats.39
5.9 Details of operations to round a floating-point datum to integral value.41
5.10 Details of totalOrder predicate.42
5.11 Details of comparison predicates.43
5.12 Details of conversion between floating-point data and external character sequences.44
6. Infinity, NaNs, and sign bit.48
6.1 Infinity arithmetic.48
6.2 Operations with NaNs.48
6.3 The sign bit.50
7. Exceptions and default exception handling.51
7.1 Overview: exceptions and flags.51
7.2 Invalid operation.52
7.3 Division by zero.53
7.4 Overflow.53
7.5 Underflow.53
7.6 Inexact.54
8. Alternate exception handling attributes.55
8.1 Overview.55
8.2 Resuming alternate exception handling attributes.55
8.3 Immediate and delayed alternate exception handling attributes.56
9
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019 – 5 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
9. Recommended operations.58
9.1 Conforming language- and implementation-defined operations.58
9.2 Additional mathematical operations.58
9.3 Dynamic mode operations.65
9.4 Reduction operations.66
9.5 Augmented arithmetic operations.68
9.6 Minimum and maximum operations.69
9.7 NaN payload operations.71
10. Expression evaluation.72
10.1 Expression evaluation rules.72
10.2 Assignments, parameters, and function values.72
10.3 preferredWidth attributes for expression evaluation.73
10.4 Literal meaning and value-changing optimizations.74
11. Reproducible floating-point results.75
Annex A (informative) Bibliography.77
Annex B (informative) Program debugging support.79
Annex C (informative) List of operations.81
Annex D (informative) IEEE list of participants.83
10
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019
– 6 –
FLOATING-POINT ARITHMETIC
FOREWORD
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019 – 7 –
International Standard ISO/IEC 60559/IEEE Std 754 has been processed through ISO/IEC
subcommittee 25: Interconnection of information technology equipment, of ISO/IEC joint
technical committee 1: Information technology, under the IEC/IEEE Dual Logo Agreement.
The text of this standard is based on the following documents:
IEEE Std FDIS Report on voting
754 (2019) JTC1-SC25/2933/FDIS JTC1-SC25/2936/RVD
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
The IEC Technical Committee and IEEE Technical Committee have decided that the contents
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• replaced by a revised edition, or
• amended.
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019
– 8 –
P754 2.63
IEEE Std 754™-2019
(Revision of IEEE Std 754-2008)
IEEE Standard for Floating-Point
Arithmetic
Sponsor
Microprocessor Standards Committee
of the
IEEE Computer Society
Approved 13 June 2019
IEEE-SA Standards Board
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019 – 9 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
Abstract: This standard specifies interchange and arithmetic formats and methods for binary and
decimal floating-point arithmetic in computer programming environments. This standard specifies
exception conditions and their default handling. An implementation of a floating-point system
conforming to this standard may be realized entirely in software, entirely in hardware, or in any
combination of software and hardware. For operations specified in the normative part of this
standard, numerical results and exceptions are uniquely determined by the values of the input
data, sequence of operations, and destination formats, all under user control.
Keywords: arithmetic, binary, computer, decimal, exponent, floating-point, format, IEEE 754™,
interchange, NaN, number, rounding, significand, subnormal.
2
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019
– 10 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
IEEE Introduction
This introduction is not part of IEEE Std 754-2019, IEEE Standard for Floating-Point Arithmetic.
This standard is a product of the Floating-Point Working Group of, and sponsored by, the Microprocessor
Standards Committee of the IEEE Computer Society.
This standard provides a discipline for performing floating-point computation that yields results
independent of whether the processing is done in hardware, software, or a combination of the two. For
operations specified in the normative part of this standard, numerical results and exceptions are uniquely
determined by the values of the input data, the operation, and the destination, all under user control.
This standard defines a family of commercially feasible ways for systems to perform binary and decimal
floating-point arithmetic. Among the desiderata that guided the formulation of this standard were:
a) Facilitate movement of existing programs from diverse computers to those that adhere to this
standard as well as among those that adhere to this standard.
b) Enhance the capabilities and safety available to users and programmers who, although not
expert in numerical methods, might well be attempting to produce numerically sophisticated
programs.
c) Encourage experts to develop and distribute robust and efficient numerical programs that are
portable, by way of minor editing and recompilation, onto any computer that conforms to this
standard and possesses adequate capacity. Together with language controls it should be possible to
write programs that produce identical results on all conforming systems.
d) Provide direct support for
― execution-time diagnosis of anomalies
― smoother handling of exceptions
― interval arithmetic at a reasonable cost.
e) Provide for development of
― common elementary functions such as exp or cos
― high precision (multiword) arithmetic
― coupled numerical and symbolic algebraic computation.
f) Enable rather than preclude further refinements and extensions.
In programming environments, this standard is also intended to form the basis for a dialog between the
numerical community and programming language designers. It is hoped that language-defined methods for
the control of expression evaluation and exceptions might be defined in coming years, so that it will be
possible to write programs that produce identical results on all conforming systems. However, it is
recognized that utility and safety in languages are sometimes antagonists, as are efficiency and portability.
Therefore, it is hoped that language designers will look on the full set of operation, precision, and
exception controls described here as a guide to providing the programmer with the ability to portably
control expressions and exceptions. It is also hoped that designers will be guided by this standard to
provide extensions in a completely portable way.
Informative annexes provide additional information – Annex A lists bibliographical resources, Annex B
suggests programming environment features for debugging support, and Annex C lists all references to the
operations of the standard.
8
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019 – 11 –
I
Floating-Point Arithmetic
1.0
1. Overview
1.1.0
1.1 Scope
This standard specifies formats and operations for floating-point arithmetic in computer systems. Exception
conditions are defined and handling of these conditions is specified.
1.2.0
1.2 Purpose
This standard provides a method for computation with floating-point numbers that will yield the same
result whether the processing is done in hardware, software, or a combination of the two. The results of the
computation will be identical, independent of implementation, given the same input data. Errors, and error
conditions, in the mathematical processing will be reported in a consistent manner regardless of
implementation.
1.3.0
1.3 Inclusions
This standard specifies:
― Formats for binary and decimal floating-point data, for computation and data interchange.
― Addition, subtraction, multiplication, division, fused multiply add, square root, compare, and
other operations.
― Conversions between integer and floating-point formats.
― Conversions between different floating-point formats.
― Conversions between floating-point formats and external representations as character sequences.
― Floating-point exceptions and their handling, including data that are not numbers (NaNs).
1.4.0
1.4 Exclusions
This standard does not specify:
― Formats of integers.
― Interpretation of the sign and significand fields of NaNs.
11
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019
– 12 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
1.5.0
1.5 Programming environment considerations
This standard specifies floating-point arithmetic in two radices, 2 and 10. A programming environment
may conform to this standard in one radix or in both.
This standard does not define all aspects of a conforming programming environment. Such behavior should
be defined by a programming language definition supporting this standard, if available, and otherwise by a
particular implementation. Some programming language specifications might permit some behaviors to be
defined by the implementation.
Language-defined behavior should be defined by a programming language standard supporting this
standard. Then all implementations conforming both to this floating-point standard and to that language
standard behave identically with respect to such language-defined behaviors. Standards for languages
intended to reproduce results exactly on all platforms are expected to specify behavior more tightly than do
standards for languages intended to maximize performance on every platform.
Because this standard requires facilities that are not currently available in common programming
languages, the standards for such languages might not be able to fully conform to this standard if they are
no longer being revised. If the language can be extended by a function library or class or package to
provide a conforming environment, then that extension should define all the language-defined behaviors
that would normally be defined by a language standard.
Implementation-defined behavior is defined by a specific implementation of a specific programming
environment conforming to this standard. Implementations define behaviors not specified by this standard
nor by any relevant programming language standard or programming language extension.
Conformance to this standard is a property of a specific implementation of a specific programming
environment, rather than of a language specification.
However a language standard could also be said to conform to this standard if it were constructed so that
every conforming implementation of that language also conformed automatically to this standard.
1.6.0
1.6 Word usage
In this standard three words are used to differentiate between different levels of requirements and
optionality, as follows:
― may indicates a course of action permissible within the limits of the standard with no implied
preference (“may” means “is permitted to”)
― shall indicates mandatory requirements strictly to be followed in order to conform to the standard
and from which no deviation is permitted (“shall” means “is required to”)
― should indicates that among several possibilities, one is recommended as particularly suitable,
without mentioning or excluding others; or that a certain course of action is preferred but not
necessarily required; or that (in the negative form) a certain course of action is deprecated but not
prohibited (“should” means “is recommended to”).
Further:
― might indicates the possibility of a situation that could occur, with no implication of the
likelihood of that situation (“might” means “could possibly”)
― see followed by a number is a cross-reference to the clause or subclause of this standard identified
by that number
― NOTE introduces text that is informative (that is, is not a requirement of this standard).
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ISO/IEC 60559:2020(E)
IEEE Std 754-2019 – 13 –
IEEE Std 754-2019
IEEE Standard for Floating-Point Arithmetic
2.0
2. Definitions, abbreviations, and acronyms
2.1.0
2.1 Definitions
For the purposes of this standard, the following terms and definitions apply.
applicable attribute: The value of an attribute governing a particular instance of execution of a
computational
...
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