ISO/IEC 10967-2:2001

Information technology — Language independent arithmetic — Part 2: Elementary numerical functions

This part of ISO/IEC 10967 de_nes the properties of numerical approximations for many of the real elementary numerical functions available in standard libraries for a variety of programming languages in common use for mathematical and numerical applications. An implementor may choose any combination of hardware and software support to meet the speci_cations of this part. It is the computing environment, as seen by the programmer/user, that does or does not conform to the speci_cations. The term implementation (of this part) denotes the total computing environment pertinent to this part, including hardware, language processors, subroutine libraries, exception handling facilities, other software, and documentation.

Technologies de l'information — Arithmétique de langage indépendant — Partie 2: Fonctions numériques élémentaires

General Information

• Status: Published
• Publication Date: 15-Aug-2001
• ICS: 35.060 - Languages used in information technology
• Technical Committee: ISO/IEC JTC 1/SC 22 - Programming languages, their environments and system software interfaces
• Drafting Committee: ISO/IEC JTC 1/SC 22 - Programming languages, their environments and system software interfaces
• Current Stage: 9093 - International Standard confirmed
• Completion Date: 05-Sep-2023

Standards Content (Sample)

INTERNATIONAL ISO/IEC
STANDARD 10967-2
First edition
2001-08-15
Information technology — Language
independent arithmetic —
Part 2:
Elementary numerical functions
Technologies de l'information — Arithmétique de langage indépendant —
Partie 2: Fonctions numériques élémentaires
Reference number
ISO/IEC 10967-2:2001(E)
©
ISO/IEC 2001

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ISO/IEC 10967-2:2001(E)
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ISO/IEC 2001 { All rights reserved ISO/IEC 10967-2:2001(E)
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Scope 1
1.1 Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Conformity 2
3 Normative references 3
4 Symbols and denitions 4
4.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.1 Sets and intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.2 Operators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.3 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1.4 Exceptional values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1.5 Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 Denitions of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 Specications for integer and
oating point operations 10
5.1 Basic integer operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1.1 The integer result and wrap helper functions . . . . . . . . . . . . . . . . . 10
5.1.2 Integer maximum and minimum . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.3 Integer diminish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.4 Integer power and arithmetic shift . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.5 Integer square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.6 Divisibility tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.7 Integer division (with
oor, round, or ceiling) and remainder . . . . . . . . 13
5.1.8 Greatest common divisor and least common positive multiple . . . . . . . . 13
5.1.9 Support operations for extended integer range . . . . . . . . . . . . . . . . . 14
5.2 Basic
oating point operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2.1 The rounding and
oating point result helper functions . . . . . . . . . . . 15
5.2.2 Floating point maximum and minimum . . . . . . . . . . . . . . . . . . . . 17
5.2.3 Floating point diminish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2.4 Floor, round, and ceiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2.5 Remainder after division with round to integer . . . . . . . . . . . . . . . . 20
5.2.6 Square root and reciprocal square root . . . . . . . . . . . . . . . . . . . . . 20
5.2.7 Multiplication to higher precision
oating point datatype . . . . . . . . . . 20
5.2.8 Support operations for extended
oating point precision . . . . . . . . . . . 21
5.3 Elementary transcendental
oating point operations . . . . . . . . . . . . . . . . . 22
5.3.1 Maximum error requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3.2 Sign requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3.3 Monotonicity requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3.4 The result helper function . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3.5 Hypotenuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3.6 Operations for exponentiations and logarithms . . . . . . . . . . . . . . . . 24
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5.3.6.1 Integer power of argument base . . . . . . . . . . . . . . . . . . . 24
5.3.6.2 Natural exponentiation . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3.6.3 Natural exponentiation, minus one . . . . . . . . . . . . . . . . . . 26
5.3.6.4 Exponentiation of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3.6.5 Exponentiation of 10 . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3.6.6 Exponentiation of argument base . . . . . . . . . . . . . . . . . . . 28
5.3.6.7 Exponentiation of one plus the argument base, minus one . . . . . 29
5.3.6.8 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.6.9 Natural logarithm of one plus the argument . . . . . . . . . . . . . 30
5.3.6.10 2-logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3.6.11 10-logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3.6.12 Argument base logarithm . . . . . . . . . . . . . . . . . . . . . . . 31
5.3.6.13 Argument base logarithm of one plus each argument . . . . . . . . 32
5.3.7 Introduction to operations for trigonometric elementary functions . . . . . . 32
5.3.8 Operations for radian trigonometric elementary functions . . . . . . . . . . 33
5.3.8.1 Radian angle normalisation . . . . . . . . . . . . . . . . . . . . . . 34
5.3.8.2 Radian sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.8.3 Radian cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.8.4 Radian tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.8.5 Radian cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.8.6 Radian secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.8.7 Radian cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.8.8 Radian cosine with sine . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.8.9 Radian arc sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.8.10 Radian arc cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.8.11 Radian arc tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.8.12 Radian arc cotangent . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3.8.13 Radian arc secant . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.8.14 Radian arc cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.8.15 Radian angle from Cartesian co-ordinates . . . . . . . . . . . . . . 42
5.3.9 Operations for trigonometrics with given angular unit . . . . . . . . . . . . 43
5.3.9.1 Argument angular-unit angle normalisation . . . . . . . . . . . . . 43
5.3.9.2 Argument angular-unit sine . . . . . . . . . . . . . . . . . . . . . . 44
5.3.9.3 Argument angular-unit cosine . . . . . . . . . . . . . . . . . . . . 45
5.3.9.4 Argument angular-unit tangent . . . . . . . . . . . . . . . . . . . . 45
5.3.9.5 Argument angular-unit cotangent . . . . . . . . . . . . . . . . . . 46
5.3.9.6 Argument angular-unit secant . . . . . . . . . . . . . . . . . . . . 47
5.3.9.7 Argument angular-unit cosecant . . . . . . . . . . . . . . . . . . . 47
5.3.9.8 Argument angular-unit cosine with sine . . . . . . . . . . . . . . . 48
5.3.9.9 Argument angular-unit arc sine . . . . . . . . . . . . . . . . . . . 48
5.3.9.10 Argument angular-unit arc cosine . . . . . . . . . . . . . . . . . . 48
5.3.9.11 Argument angular-unit arc tangent . . . . . . . . . . . . . . . . . 49
5.3.9.12 Argument angular-unit arc cotangent . . . . . . . . . . . . . . . . 50
5.3.9.13 Argument angular-unit arc secant . . . . . . . . . . . . . . . . . . 51
5.3.9.14 Argument angular-unit arc cosecant . . . . . . . . . . . . . . . . . 51
5.3.9.15 Argument angular-unit angle from Cartesian co-ordinates . . . . . 52
5.3.10 Operations for angular-unit conversions . . . . . . . . . . . . . . . . . . . . 53
5.3.10.1 Converting radian angle to argument angular-unit angle . . . . . . 53
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5.3.10.2 Converting argument angular-unit angle to radian angle . . . . . . 54
5.3.10.3 Converting argument angular-unit angle to (another) argument
angular-unit angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.11 Operations for hyperbolic elementary functions . . . . . . . . . . . . . . . . 56
5.3.11.1 Hyperbolic sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.11.2 Hyperbolic cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.11.3 Hyperbolic tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.11.4 Hyperbolic cotangent . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.11.5 Hyperbolic secant . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.11.6 Hyperbolic cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.11.7 Inverse hyperbolic sine . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.11.8 Inverse hyperbolic cosine . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.11.9 Inverse hyperbolic tangent . . . . . . . . . . . . . . . . . . . . . . 60
5.3.11.10 Inverse hyperbolic cotangent . . . . . . . . . . . . . . . . . . . . . 60
5.3.11.11 Inverse hyperbolic secant . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.11.12 Inverse hyperbolic cosecant . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Operations for conversion between numeric datatypes . . . . . . . . . . . . . . . . 62
5.4.1 Integer to integer conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.2 Floating point to integer conversions . . . . . . . . . . . . . . . . . . . . . . 63
5.4.3 Integer to
oating point conversions . . . . . . . . . . . . . . . . . . . . . . 64
5.4.4 Floating point to
oating point conversions . . . . . . . . . . . . . . . . . . 64
5.4.5 Floating point to xed point conversions . . . . . . . . . . . . . . . . . . . . 65
5.4.6 Fixed point to
oating point conversions . . . . . . . . . . . . . . . . . . . . 66
5.5 Numerals as operations in a programming language . . . . . . . . . . . . . . . . . . 67
5.5.1 Numerals for integer datatypes . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.5.2 Numerals for
oating point datatypes . . . . . . . . . . . . . . . . . . . . . 68
6 Notication 68
6.1 Continuation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Relationship with language standards 69
8 Documentation requirements 70
Annex A (normative) Partial conformity 73
A.1 Maximum error relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.2 Extra accuracy requirements relaxation . . . . . . . . . . . . . . . . . . . . . . . . 74
A.3 Relationships to other operations relaxation . . . . . . . . . . . . . . . . . . . . . . 74
A.4 Very-close-to-axis angular normalisation relaxation . . . . . . . . . . . . . . . . . . 74
A.5 Part 1 requirements relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Annex B (informative) Rationale 77
B.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B.1.1 Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B.1.2 Exclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
B.2 Conformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
B.2.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.3 Normative references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.4 Symbols and denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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B.4.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.4.1.1 Sets and intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.4.1.2 Operators and relations . . . . . . . . . . . . . . . . . . . . . . . . 80
B.4.1.3 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . 80
B.4.1.4 Exceptional values . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
B.4.1.5 Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.4.2 Denitions of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.5 Specications for the numerical functions . . . . . . . . . . . . . . . . . . . . . . . 81
B.5.1 Basic integer operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B.5.1.1 The integer result and wrap helper functions . . . . . . . . . . . . 82
B.5.1.2 Integer maximum and minimum . . . . . . . . . . . . . . . . . . . 82
B.5.1.3 Integer diminish . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B.5.1.4 Integer power and arithmetic shift . . . . . . . . . . . . . . . . . . 83
B.5.1.5 Integer square root . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.5.1.6 Divisibility tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.5.1.7 Integer division (with
oor, round, or ceiling) and remainder . . . 83
B.5.1.8 Greatest common divisor and least common positive multiple . . . 84
B.5.1.9 Support operations for extended integer range . . . . . . . . . . . 84
B.5.2 Basic
oating point operations . . . . . . . . . . . . . . . . . . . . . . . . . 84
B.5.2.1 The rounding and
oating point result helper functions . . . . . . 86
B.5.2.2 Floating point maximum and minimum . . . . . . . . . . . . . . . 86
B.5.2.3 Floating point diminish . . . . . . . . . . . . . . . . . . . . . . . . 86
B.5.2.4 Floor, round, and ceiling . . . . . . . . . . . . . . . . . . . . . . . 86
B.5.2.5 Remainder after division and round to integer . . . . . . . . . . . 87
B.5.2.6 Square root and reciprocal square root . . . . . . . . . . . . . . . 87
B.5.2.7 Multiplication to higher precision
oating point datatype . . . . . 88
B.5.2.8 Support operations for extended
oating point precision . . . . . . 88
B.5.3 Elementary transcendental
oating point operations . . . . . . . . . . . . . 89
B.5.3.1 Maximum error requirements . . . . . . . . . . . . . . . . . . . . . 89
B.5.3.2 Sign requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.5.3.3 Monotonicity requirements . . . . . . . . . . . . . . . . . . . . . . 90

B.5.3.4 The result helper function . . . . . . . . . . . . . . . . . . . . . . 90
B.5.3.5 Hypotenuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.5.3.6 Operations for exponentiations and logarithms . . . . . . . . . . . 91
B.5.3.7 Introduction to operations for trigonometric elementary functions 93
B.5.3.8 Operations for radian trigonometric elementary functions . . . . . 94
B.5.3.9 Operations for trigonometrics with given angular unit . . . . . . . 96
B.5.3.10 Operations for angular-unit conversions . . . . . . . . . . . . . . . 97
B.5.3.11 Operations for hyperbolic elementary functions . . . . . . . . . . . 98
B.5.4 Operations for conversion between numeric datatypes . . . . . . . . . . . . 98
B.5.5 Numerals as operations in a programming language . . . . . . . . . . . . . 99
B.5.5.1 Numerals for integer datatypes . . . . . . . . . . . . . . . . . . . . 99
B.5.5.2 Numerals for
oating point datatypes . . . . . . . . . . . . . . . . 99
B.6 Notication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.6.1 Continuation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.7 Relationship with language standards . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.8 Documentation requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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Annex C (informative) Example bindings for specic languages 103
C.1 Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.2 BASIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.4 C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.5 Fortran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.6 Haskell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
C.7 Java . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.8 Common Lisp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.9 ISLisp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.10 Modula-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.11 Pascal and Extended Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.12 PL/I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C.13 SML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Annex D (informative) Bibliography 173
Annex E (informative) Possible changes to part 1 177
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ISO/IEC 2001 { All rights reserved
Foreword
ISO (the International Organization for Standardization) and IEC (the International Electrotech-
nical Commission) form the specialized system for worldwide standardization. National bodies
that are members of ISO or IEC participate in the development of International Standards through
technical committees established by the respective organization to deal with particular elds of
technical activity. ISO and IEC technical committees collaborate in elds of mutual interest.
Other international organizations, governmental and non-governmental, in liaison with ISO and
IEC, also take part in the work.
International Standards are drafted in accordance with the rules given in the ISO/IEC Direc-
tives, Part 3.
In the eld of information technology, ISO and IEC have established a joint technical committee,
ISO/IEC JTC 1. Draft International Standards adopted by the joint technical committee are
circulated to national bodies for voting. Publication as an International Standard requires approval
by at least 75 % of the national bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO/IEC 10967
may be the subject of patent rights. ISO and IEC shall not be held responsible for identifying
any or all such patent rights.
International Standard ISO/IEC 10967-2 was prepared by Joint Technical Committee ISO/IEC
JTC 1, Information technology, Subcommittee SC 22, Programming languages, their environments
and system software interfaces.
ISO/IEC 10967 consists of the following parts, under the general title Information technology
| Language independent arithmetic:
{ Part 1: Integer and
oating point arithmetic
{ Part 2: Elementary numerical functions
{ Part 3: Complex integer and
oating point arithmetic and complex elementary numerical
functions
Additional parts will specify other arithmetic datatypes or arithmetic operations.
Annex A forms a normative part of this part of ISO/IEC 10967. Annexes B to E are for
information only.
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ISO/IEC 2001 { All rights reserved ISO/IEC 10967-2:2001(E)
Introduction
The aims
Portability is a key issue for scientic and numerical software in today's heterogeneous computing
environment. Such software may be required to run on systems ranging from personal computers
to high performance pipelined vector processors and massively parallel systems, and the source
code may be ported between several programming languages. Part 1 of ISO/IEC 10967 species
the basic properties of integer and
oating point types that can be relied upon in writing portable
software.
Programmers writing programs that perform a signicant amount of numeric processing have
often not been certain how a program will perform when run under a given language processor.
Programming language standards have traditionally been somewhat weak in the area of numeric
processing, seldom providing an adequate specication of the properties of arithmetic datatypes,
particularly
oating point numbers. Often they do not even require much in the way of documen-
tation of the actual arithmetic operations by a conforming language processor.
It is the intent of this part to help to redress these shortcomings, by setting out precise deni-
tions of elementary numerical functions, and requirements for documentation.
It is not claimed that this part will ensure complete certainty of arithmetic behaviour in all
circumstances; the complexity of numeric software and the diculties of analysing and proving
algorithms are too great for that to be attempted. Rather, this International Standard will provide
a rmer basis than hitherto for attempting such analysis.
The aims for this part, part 2 of ISO/IEC 10967, are extensions of the aims for part 1: to ensure
adequate accuracy for numerical computation, predictability, notication on the production of
exceptional results, and compatibility with programming language standards.
The content
The content of this part is based on part 1, and extends part 1's specications to specications for
operations approximating real elementary functions, operations often required (usually without
a detailed specication) by the standards for programming languages widely used for scientic
software. This part also provides specications for conversions between the \internal" values of
an arithmetic datatype, and a very close approximation in, e.g., the decimal radix. It does not
cover the further transformation to decimal string format, which is usually provided by language
standards. This part also includes specications for a number of other functions deemed useful,
even though they may not be stipulated by programming language standards.
The numerical functions covered by this part are computer approximations to mathematical
functions of one or more real arguments. Accuracy versus performance requirements often vary
with the application at hand. This is recognised by recommending that implementors support more
than one library of these numerical functions. Various documentation and (program available)
parameters requirements are specied to assist programmers in the selection of the library best
suited to the application at hand.
The benets
Adoption and proper use of this part can lead to the following benets.
Language standards will be able to dene their arithmetic semantics more precisely without
preventing the ecient implementation of their language on a wide range of machine architectures.
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ISO/IEC 10967-2:2001(E)
ISO/IEC 2001 { All rights reserved
Programmers of numeric software will be able to assess the portability of their programs in
advance. Programmers will be able to trade o program design requirements for portability in
the resulting program.
Programs will be able to determine (at run time) the crucial numeric properties of the imple-
mentation. They will be able to reject unsuitable implementations, and (possibly) to correctly
characterize the accuracy of their own results. Programs will be able to detect (and possibly
correct for) exceptions in arithmetic processing.
End users will nd it easier to determine whether a (properly documented) application program
is likely to execute satisfactorily on their platform. This can be done by comparing the documented
requirements of the program against the documented properties of the platform.
Finally, end users of numeric application packages will be able to rely on the correct execution
of those packages. That is, for correctly programmed algorithms, the results are reliable if and
only if there is no notication.
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ISO/IEC 2001
INTERNATIONAL STANDARD ISO/IEC 10967-2:2001(E)
Information technology |
Language independent arithmetic |
Part 2: Elementary numerical functions
1 Scope
This part of ISO/IEC 10967 denes the properties of numerical approximations for many of the
real elementary numerical functions available in standard libraries for a variety of programming
languages in common use for mathematical and numerical applications.
An implementor may choose any combination of hardware and software support to meet the
specications of this part. It is the computing environment, as seen by the programmer/user, that
does or does not conform to the specications.
The term implementation (of this part) denotes the total computing environment pertinent
to this part, including hardware, language processors, subroutine libraries, exception handling
facilities, other software, and documentation.
1.1 Inclusions
The specications of part 1 are included by reference in this part.
This part provides specications for numerical functions for which all operand values are of
integer or
oating point datatypes satisfying the requirements of part 1. Boundaries for the oc-
currence of exceptions and the maximum error allowed are prescribed for each specied operation.
Also the result produced by giving a special value operand, such as an innity, or a NaN, is
prescribed for each specied
oating point operation.
This part covers most numerical functions required by the ISO/IEC standards for Ada [11],
Basic [16], C [17], C++ [18], Fortran [22], ISLisp [24], Pascal [27], and PL/I [29]. In particular,
specications are provided for:
a) Some additional integer operations.
b) Some additional non-transcendental
oating point operations, including maximum and min-
imum operations.
c) Exponentiations, logarithms, and hyperbolics.
d) Trigonometrics, both in radians and for argument-given angular unit with degrees as a
special case.
This part also provides specications for:
1. Scope 1

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ISO/IEC 10967-2:2001(E)
ISO/IEC 2001 { All rights reserved
e) Conversions between integer and
oating point datatypes (possibly with dierent radices)
conforming to the requirements of part 1, and the conversion operations used, for example,
in text input and output of integer and
oating point numbers.
f) The results produced by an included
oating point operation when one or more argument
values are IEC 60559 special values.
g) Program-visible parameters that characterise certain aspects of the operations.
1.2 Exclusions
This part provides no specications for
a) Numerical functions whose operands are of more than one datatype (with one exception).
This part neither requires nor excludes the presence of such \mixed operand" operations.
b) An interval datatype, or the operations on such data. This part neither requires nor excludes
such data or operations.
c) A xed point datatype, or the operations on such data. This part neither requires nor
excludes such data or operations.
d) A rational datatype, or the operations on such data. This part neither requires nor excludes
such data or operations.
e) Complex, matrix, statistical, or symbolic operations. This part neither requires nor excludes
such data or operations.
f) The properties of arithmetic datatypes that are not related to the numerical process, such
as the representation of values on physical media.
g) The properties of integer and
oating point datatypes that properly belong in programming
language standards or other specications. Examples include
1) the syntax of numerals and expressions in the programming language,
2) the syntax used for parsed (input) or generated (output) character string forms for
numerals by any specic programming language or library,
3) the precedence of operators in the programming language,
4) the presence or absence of automatic datatype coercions,
5) the rules for assignment, parameter passing, and returning value,
6) the consequences of applying an operation to values of improper datatype, or to unini-
tialised data.
Furthermore, this part does not provide specications for how the operations shoul
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