IEEE 754R标准：计算机编程环境的浮点数运算与交换规范

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IEEE 754标准，正式名称为"IEEE Standard for Floating-Point Arithmetic"（IEEE Std 754），是一项关键的计算机编程规范，于1985年首次发布，并在2008年进行了修订，即IEEE Std 754R。该标准主要关注二进制和十进制浮点数在计算机环境中的表示、交换以及算术运算。它定义了如何在硬件和软件混合实现的系统中处理数值计算，确保结果的一致性和精度。这个标准的核心内容包括： 1. **浮点数格式**：它规定了浮点数的二进制和十进制表示方式，包括阶码（用于表示指数部分）和尾数（表示小数部分）。这涵盖了不同规格的浮点数类型，如单精度（float）、双精度（double）等，每种类型有其特定的存储结构和操作规则。 2. **算术运算**：标准详细描述了浮点数之间的基本算术运算，如加法、减法、乘法、除法等，以及如何处理溢出、下溢、无穷大、NaN（Not-a-Number）等异常情况。这些运算的执行过程需保证正确性和可预测性。 3. **数据交换与表示**：规定了浮点数在不同程序之间、硬件与软件之间以及不同的处理器架构之间交换时的标准格式，确保数据的一致性。 4. **异常处理**：明确指出当遇到不适合的输入或运算条件时，应抛出何种异常，以及默认的处理方法。这有助于程序员理解和控制程序的行为。 5. **灵活性与兼容性**：尽管是标准化，但允许实施者根据具体需求在软件或硬件层面实现，提供了适应不同应用场景的可能性。 6. **历史更新**：IEEE 754R是对1985年版本的修订，反映了技术进步和需求变化，可能引入了新的特性或改进了现有规则。 IEEE 754标准是现代计算机科学和工程领域不可或缺的一部分，对于保证计算机浮点运算的精确性、一致性以及跨平台兼容性具有重要意义。了解并遵循此标准，程序员能够编写出更高效、健壮的代码，而硬件开发者则能设计出性能卓越的处理器架构。

9. Recommended operations ..........................................................................................................................41

9.1 Conforming language- and implementation-defined functions ........................................................41

9.2 Recommended correctly rounded functions .....................................................................................42

9.3 Operations on dynamic modes for attributes ....................................................................................46

9.4 Reduction operations ........................................................................................................................46

10. Expression evaluation ..............................................................................................................................48

10.1 Expression evaluation rules ............................................................................................................48

10.2 Assignments, parameters, and function values ...............................................................................48

10.3 preferredWidth attributes for expression evaluation ......................................................................49

10.4 Literal meaning and value-changing optimizations ........................................................................50

11. Reproducible floating-point results .........................................................................................................51

Annex A (informative) Bibliography .............................................................................................................53

Annex B (informative) Program debugging support .....................................................................................55

Index of operations ........................................................................................................................................57

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IEEE Standard for Floating-Point

Arithmetic

IMPORTANT NOTICE: This standard is not intended to assure safety, security, health, or environmental

protection in all circumstances. Implementers of the standard are responsible for determining appropriate safety,

security, environmental, and health practices or regulatory requirements.

This IEEE document is made available for use subject to important notices and legal disclaimers. These notices

and disclaimers appear in all publications containing this document and may be found under the heading

“Important Notice” or “Important Notices and Disclaimers Concerning IEEE Documents”. They can also be

obtained on request from IEEE or viewed at http://standards.ieee.org/IPR/disclaimers.html.

1. Overview

1.0

1.1 Scope

1.1.0

This standard specifies formats and methods for floating-point arithmetic in computer systems

—

standard

and extended functions with single, double, extended, and extendable precision

—

and recommends formats

for data interchange. Exception conditions are defined and standard handling of these conditions is

specified.

1.2 Purpose

1.2.0

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 Inclusions

1.3.0

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).

Authorized licensed use limited to: NORTHWESTERN POLYTECHNIC UNIVERSITY. Downloaded on April 13,2012 at 09:08:03 UTC from IEEE Xplore. Restrictions apply.

IEEE Std 754-2008

IEEE Standard for Floating-Point Arithmetic

1.4 Exclusions

1.4.0

This standard does not specify:

― Formats of integers.

― Interpretation of the sign and significand fields of NaNs.

1.5 Programming environment considerations

1.5.0

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 Word usage

1.6.0

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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