Digital PDFs

AA-LA72A-TE

December 1988

196 pages

Original

5.3MB

Document:	VMS RTL Mathematics {MTH$) Manual
Order Number:	AA-LA72A-TE
Revision:
Pages:	196
Original Filename:	http://bitsavers.org/pdf/dec/vax/vms/5.0/AA-LA72A-TE_VMS_5.0_RTL_Mathematics_MTH$_Manual_198804.pdf

OCR Text

VMS RTL Mathematics
(MTH$) Manual
Order Number: AA-LA 7 2A-TE

April 1988
This manual documents the mathematics routines contained in the MTH$
facility of the VMS Run-Time Library.

Revision/Update Information:

This document supersedes Section 4
and the MTH$ portion of Part II of the
VAX/VMS Run- Time Library Routines
Reference Manual, Version 4.4.

Software Version:

VMS Version 5.0

digital equipment corporation
maynard, massachusetts

April 1988
The information in this document is subject to change without notice and should
not be construed as a commitment by Digital Equipment Corporation. Digital
Equipment Corporation assumes no responsibility for any errors that may appear
in this document.
The software described in this document is furnished under a license and may be
used or copied only in accordance with the terms of such license.
No responsibility is assumed for the use or reliability of software on equipment
that is not supplied by Digital Equipment Corporation or its affiliated companies.
Copyright © 1988 by Digital Equipment Corporation
All Rights Reserved.
Printed in U.S.A.
The postpaid READER'S COMMENTS form on the last page of this document
requests the user's critical evaluation to assist in preparing future documentation.
The following are trademarks of Digital Equipment Corporation:
DEC
DEC/CMS
DEC/MMS
DECnet
DECsystem-10
DECSYSTEM-20
DECUS
DECwriter

DIBOL
EduSystem
IAS
MASSBUS
PDP
PDT
RSTS
RSX

UNIBUS
VAX
VAXcluster
VMS
VT

~U~UIJ~D™
ZK4610

HOW TO ORDER ADDITIONAL DOCUMENTATION
DIRECT MAIL ORDERS
USA & PUERTO RICO*

CANADA

INTERNATIONAL

Digital Equipment Corporation
P.O. Box CS2008
Nashua, New Hampshire
03061

Digital Equipment
of Canada Ltd.
100 Herzberg Road
Kanata, Ontario K2K 2A6
Attn: Direct Order Desk

Digital Equipment Corporation
PSG Business Manager
c/o Digital§ local subsidiary
or approved distributor

In Continental USA and Puerto Rico call 800-258-1710.
In New Hampshire, Alaska, and Hawaii call 603-884-6660.
In Canada call 800-267-6215.
*Any prepaid order from Puerto Rico must be placed with the local Digital subsidiary (809-754-7575).
Internal orders should be placed through the Software Distribution Center (SOC), Digital Equipment
Corporation, Westminster, Massachusetts 01473.

Production Note
This book was produced with the VAX DOCUMENT electronic publishing
system, a software tool developed and sold by DIGITAL. In this system,
writers use an ASCII text editor to create source files containing text and
English-like code; this code labels the structural elements of the document,
such as chapters, paragraphs, and tables. The VAX DOCUMENT software,
which runs on the VMS operating system, interprets the code to format the
text, generate a table of contents and index, and paginate the entire document.
Writers can print the document on the terminal or line printer, or they can use
DIGITAL-supported devices, such as the LN03 laser printer and PostScript®
printers (PrintServer 40 or LN03R ScriptPrinter), to produce a typeset-quality
copy containing integrated graphics.

PostScript is a trademark of Adobe Systems, Inc.

Contents
ix

PREFACE

CHAPTER 1 INTRODUCTION TO MTH$

1-1

1.1

ENTRY POINT NAMES

1-1

1.2

CALLING CONVENTIONS

1-2

1.3

ALGORITHMS

1-2

1.4

CONDITION HANDLING

1-3

1.5

COMPLEX NUMBERS

1-3

1.6

ROUTINES NOT DOCUMENTED IN THE MTH$ REFERENCE
SECTION

1-4

EXAMPLES OF CALLS TO RUN-TIME LIBRARY MATHEMATICS
ROUTINES
BASIC Example
COBOL Example
FORTRAN Examples
MACRO Examples
PASCAL Examples
PL/I Examples

1-9
1-9
1-9
1-10
1-12
1-14
1-15

1.7
1.7.1
1.7.2
1.7.3
1.7.4
1.7.5
1.7.6

MTH$ REFERENCE SECTION
MTH$XACOS
MTH$XACOSD
MTH$XASIN
MTH$XASIND
MTH$XATAN
MTH$XATAND
MTH$XATAN2

MTH-3
MTH-6
MTH-9
MTH-11
MTH-13
MTH-15
MTH-17
v

Contents

MTH$XATAND2
MTH$XATANH
MTH$CXABS
MTH$CCOS
MTH$CXCOS
MTH$CEXP
MTH$CXEXP
MTH$CLOG
MTH$CXLOG
MTH$CMPLX
MTH$XCMPLX
MTH$CONJG
MTH$XCONJG
MTH$XCOS
MTH$XCOSD
MTH$XCOSH
MTH$CSIN
MTH$CXSIN
MTH$CSQRT
MTH$CXSQRT
MTH$CVT_x_x
MTH$CVT_)(A_)(A
MTH$XEXP
MTH$HACOS
MTH$HACOSD
MTH$HASIN
MTH$HASIND
MTH$HATAN
MTH$HATAND
MTH$HATAN2
MTH$HATAND2
MTH$HATANH
MTH$HCOS
MTH$HCOSD
MTH$HCOSH
MTH$HEXP
MTH$HLOG
MTH$HLOG2
MTH$HLOG10
MTH$HSIN
MTH$HSIND
MTH$HSINH
vi

MTH-19
MTH-21
MTH-23
MTH-26
MTH-28
MTH-31
MTH-33
MTH-36
MTH-38
MTH-40
MTH-42
MTH-44
MTH-45
MTH-47
MTH-49
MTH-51
MTH-53
MTH-54
MTH-57
MTH-59
MTH-62
MTH-64
MTH-66
MTH-69
MTH-71
MTH-73
MTH-75
MTH-77
MTH-79
MTH-81
MTH-83
MTH-85
MTH-87
MTH-88
MTH-89
MTH-91
MTH-93
MTH-95
MTH-97
MTH-99
MTH-100
MTH-102

Contents

MTH$HSQRT
MTH$HTAN
MTH$HTAND
MTH$HTANH
MTH$XIMAG
MTH$XLOG
MTH$XLOG2
MTH$XLOG10
MTH$RANDOM
MTH$XREAL
MTH$XSIN
MTH$XSINCOS
MTH$XSINCOSD
MTH$XSIND
MTH$XSINH
MTH$XSQRT
MTH$XTAN
MTH$XTAND
MTH$XTANH
MTH$UMAX
MTH$UMIN

APPENDIX A UNDOCUMENTED MTH$ ROUTINES

MTH-104
MTH-106
MTH-108
MTH-110
MTH-112
MTH-114
MTH-116
MTH-118
MTH-120
MTH-122
MTH-124
MTH-126
MTH-129
MTH-132
MTH-134
MTH-136
MTH-138
· MTH-140
MTH-142
MTH-144
MTH-145

A-1

INDEX

TABLES
1-1

Additional Mathematics Routines

1-4

A-1

Undocumented MTH$ Routines

A-1

vii

Preface
This manual provides users of the VMS operating system with detailed usage
and reference information on mathematics routines supplied in the MTH$
facility of the Run-Time Library.
Run-Time Library routines can only be used in programs written in languages
that produce native code for the VAX hardware. At present, these languages
include VAX MACRO and the following compiled high-level languages:
VAX Ada
VAX BASIC
VAX BLISS-32
VAXC
VAX COBOL
VAX COBOL-74
VAX CORAL
VAX DIBOL
VAX FORTRAN
VAX Pascal
VAX PL/I
VAX RPG
VAX SCAN
Interpreted languages which can also access Run-Time Library routines
include VAX DSM and DATATRIEVE.

Intended Audience
This manual is intended for system and application programmers who want
to call Run-Time Library routines.

Document Structure
This manual is organized into two parts as follows:
•

The introductory chapters provide guidelines on using the MTH$
mathematics routines.

•

The MTH$ Reference Section provides detailed reference information on
each mathematics routine contained in the MTH$ facility of the RunTime Library. This information is presented using the documentation
format described in the Introduction to the VMS Run-Time Library. Routine
descriptions appear in alphabetical order by routine name.

Preface

Associated Documents
The Run-Time Library routines are documented in a series of reference
manuals. A general overview of the Run-Time Library and a description
of how the Run-Time Library routines are accessed is presented in the
Introduction to the VMS Run-Time Library. Descriptions of the other RTL
facilities and their corresponding routines and usages are discussed in the
following books:
•

The VMS RTL DECtalk (DTK$) Manual

•

The VMS RTL Library (LIB$) Manual

•

The VMS RTL General Purpose (OTS$) Manual

•

The VMS RTL Parallel Processing (PPL$) Manual

•

The VMS RTL Screen Management (SMG$) Manual

•

The VMS RTL String Manipulation (STR$) Manual

The VAX Procedure Calling and Condition Handling Standard, which is
documented in the Introduction to System Routines, contains useful information
for anyone who wants to call Run-Time Library routines.
Applications programmers of any language may refer to the Guide to Creating
VMS Modular Procedures for the Modular Programming Standard and other
guidelines.
High-level language programmers will find additional information on calling
Run-Time Library routines in their language reference manual. Additional
information may also be found in the language user's guide provided with
your VAX language.
The Guide to Using VMS Command Procedures may also be useful.
For a complete list and description of the manuals in the VMS documentation
set, see the Overview of VMS Documentation.

Preface

Conventions
Convention

Meaning
In examples, a key name (usually abbreviated)
shown within a box indicates that you press
a key on the keyboard; in text, a key name is
not enclosed in a box. In this example, the key
is the RETURN key. (Note that the RETURN
key is not usually shown in syntax statements
or in all examples; however, assume that you
must press the RETURN key after entering a
command or responding to a prompt.)

CTRL/C

A key combination, shown in uppercase with a
slash separating two key names, indicates that
you hold down the first key while you press the
second key. For example, the key combination
CTRL/C indicates that you hold down the key
labeled CTRL while you press the key labeled C.
In examples, a key combination is enclosed in a
box.

$SHOW TIME
05-JUN-1988 11 :55:22

In 9-Xamples, system output (what the system
displays) is shown in black. User input (what
you enter) is shown in red.

$TYPE MYFILE.DAT

In examples, a vertical series of periods, or
ellipsis, means either that not all the data that
the system would display in response to a
command is shown or that not all the data a
user would enter is shown.

input-file, ...

In examples, a horizontal ellipsis indicates
that additional parameters, values, or other
information can be entered, that preceding
items can be repeated one or more times, or
that optional arguments in a statement have
been omitted.

[logical-name]

Brackets indicate that the enclosed item is
optional. (Brackets are not, however, optional
in the syntax of a directory name in a file
specification or in the syntax of a substring
specification in an assignment statement.)

quotation marks
apostrophes

The term quotation marks is used to refer
to double quotation marks (" ). The term
apostrophe ( ') is used to refer to a single
quotation mark.

Other conventions used in the documentation of Run-Time Library routines
are described in the Introduction to the VMS Run-Time Library.

Introduction to MTH$

The Run-Time Library mathematics routines may be called to perform a wide
variety of computations including the following:
•

Complex exponentiation

•

Complex function evaluation

•

Exponentiation

•

Floating-point trigonometric function evaluation

•

Miscellaneous function evaluation

The OTS$ facility provides additional language-independent arithmetic
support routines.
This introduction to Run-Time Library mathematics routines includes
examples of how to call mathematics routines from BASIC, COBOL,
FORTRAN, MACRO, PASCAL, and PL/I.

1 .1

Entry Point Names
The names of the mathematics routines are formed by adding the MTH$
prefix to the function names.
When function arguments and returned values are of the same data type, the
first letter of the name indicates this data type. When function arguments and
returned values are of different data types, the first letter indicates the data
type of the returned value, and the second letter indicates the data type of the
argument(s).
The letters used as data type prefixes are listed below.
Letter

Data Type

Word

Longword

O_floating

G_floating

H_floating

F_floating complex

O_floating complex

G_floating complex

Generally, F-floating data types have no letter designation. For example,
MTH$SIN returns an F-floating value of the sine of an F-floating argument
and MTH$DSIN returns a D-floating value of the sine of a D-floating
argument. However, in some of the miscellaneous functions, F-floating
data types are referenced by the letter designation A.

1-1

Introduction to MTH$
1.2 Calling Conventions

1.2

Calling Conventions
All calls to mathematics routines, as described in the FORMAT section of
each routine, accept arguments passed by reference. JSB entry points accept
arguments passed by value.
All mathematics routines return values in RO or RO /Rl except those routines
for which the values cannot fit in 64 bits. D-floating complex, G-floating
complex and H-floating values are data structures which are larger than 64
bits. Routines that return values which cannot fit in registers RO/Rl return
their function values into the first argument in the argument list.
The notation JSB MTH$NAME_Rn, where n is the highest register number
referenced, indicates that an equivalent JSB entry point is available. No
registers are saved; only registers RO:Rn are changed.
Routines with JSB entry points accept a single argument in RO:Rm, where m,
which is defined below, is dependent on the data type.
Data Type

F_floating

D_floating
G_floating

H_floating

A routine which returns one value returns it to registers RO:Rm.
When a routine returns two values, for example MTH$SINCOS, the
fi:rst value is returned in RO:Rm and the second value is returned in
(R <m+l> :R <2•m+l> ).
Note that for routines that return a single value, n> =m. For routines that
return two values, n> =2•m + 1.
All CALL entry points for mathematics routines do the following:
•

Disable floating-point underflow

•

Enable integer overflow

•

Cause no floating-point overflow or other arithmetic traps or faults

•

Preserve all other enabled operations across the CALL

JSB entry points execute in the context of the caller with the enable operations
as set by the caller. Since the routines do not cause arithmetic traps or faults,
their operation is not affected by the setting of the arithmetic trap enables,
except as noted.
For more detailed information on CALL and JSB entry points, refer to the
Introduction to the VMS Run-Time Library.

1 .3

Algorithms
For those mathematics routines that have corresponding algorithms, the
· complete algorithm can be found in the Description section of the routine
description appearing in the MTH$ Reference Section of this manual.

1-2

Introduction to MTH$
1 .4 Condition Handling

1 .4

Condition Handling
Error conditions are indicated by using the VAX signaling mechanism. The
VAX signaling mechanism signals all conditions in mathematics routines
as SEVERE by calling LIB$SIGNAL. When a SEVERE error is signaled, the
image is caused to exit after printing an error message. A user-established
condition handler can be written to cause execution to continue at the point
of the error by returning SS$_CQNTINUE. A mathematics routine returns to
its caller after the contents of RO/Rl have been restored from the mechanism
argument vector CHF$L _MCH_SAVRO /Rl. Thus, the user-established
handler should correct CHF$L_MCH_SAVRO/Rl to the desired function
value to be returned to the caller of the mathematics routine.
D-floating complex, G-floating complex, and H-floating values cannot be
corrected with a user-established condition handler, because R2/R3 are not
available in the mechanism argument vector.
Note that it is more reliable to correct RO and Rl to resemble RO and Rl of
a double-precision floating-point value. A double-precision floating-point
value correction works for both single- and double-precision values. If
the correction is not performed, the floating-point reserved operand -0.0
is returned. A floatingLpoint reserved operand is a floating-point datum
with a sign bit of 1 and a biased exponent of zero. Accessing the floatingpoint reserved operand will cause a reserved operand fault. See the VMS
RTL Library (LIB$) Manual for a complete description of how to write user
condition handlers for SEVERE errors.
A few mathematics routines signal floating underflow if the calling program
OSB or CALL) has enabled floating underflow faults or traps.
All mathematics routines access input arguments and the real and imaginary
parts of complex numbers using floating-point instructions. Therefore, a
reserved operand fault can occur in any mathematics routine.

1 .5

Complex Numbers
A complex number y is defined as an ordered pair of real numbers r and i,
where r is the real part and i is the imaginary part of the complex number.
y=(r,i)

VMS supports three floating-point complex types: F-floating complex, Dfloating complex, and G-floating complex. There is no H-floating complex
data type.
Run-Time Library mathematics routines that use complex arguments require
two x-floating values to be passed by reference for each argument. The
first x-floating value contains r, the real part of the complex number. The
second x-floating value contains i, the imaginary part of the complex number.
Similarly, Run-Time Library mathematics routines that return complex
function values return two x-floating values. Some Language Independent
Support (OTS$) routines also calculate complex functions.
Note that complex functions have no JSB entry points.

1-3

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

1.6

Routines Not Documented in the MTH$ Reference Section
The mathematics routines in Table 1-1 are not found in the reference section
of this manual. Instead, their entry points and argument information are
listed in Appendix A of this manual.
A reserved operand fault can occur for any floating-point input argument
in any mathematics routine. Other condition values signaled by each
mathematics routine are indicated in the footnotes.
Table 1-1
Entry Point

Additional Mathematics Routines
Function

Absolute Value Routines
MTH$ABS

F-floating absolute value

MTH$DABS

D-floating absolute value

MTH$GABS

G-floating absolute value

MTH$HABS

H-floating absolute value 1

MTH$11ABS

Word absolute value 2

MTH$JIABS

Longword absolute value 2

Bitwise AND Operator Routines
MTH$11AND

Bitwise AND of two word arguments

MTH$JIAND

Bitwise AND of two longword arguments

F-floating Conversion Routines
MTH$DBLE

Convert F-floating to D-floating (exact)

MTH$GDBLE

Convert F-floating to G-floating (exact)

MTH$11FIX

Convert F-floating to word (truncated) 2

MTH$JIFIX

Convert F-floating to longword (truncated) 2

1 Returns value to the first argument; value exceeds 64 bits.
2 1nteger overflow exceptions can occur.

1-4

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

Table 1-1 (Cont.)

Additional Mathematics Routines

Entry Point

Function

Floating-Point Positive Difference Routines
MTH$DIM

Positive difference of two F-floating arguments 3 •4

MTH$001M

Positive difference of two 0-floating arguments 3 •4

MTH$GDIM

Positive difference of two G-floating arguments3 •4

MTH$HOIM

Positive difference of two H-floating arguments 1,3 ,4

MTH$11DIM

Positive difference of two word arguments 2

MTH$JIOIM

Positive difference of two longword arguments 2

Bitwise Exclusive OR Operator Routines
MTH$11EOR

Bitwise exclusive OR of two word arguments

MTH$JIEOR

Bitwise exclusive OR of two longword arguments

Integer to Floating-point Conversion Routines
MTH$FLOATI

Convert word to F-floating (exact)

MTH$0FLOTI

Convert word to 0-floating (exact)

MTH$GFLOTI

Convert word to G-floating (exact)

MTH$FLOATJ

Convert longword to F-floating (exact)

MTH$0FLOTJ

Convert word to 0-floating (exact)

MTH$GFLOTJ

Convert longword to G-floating (exact)

Conversion to Greatest Floating-point Integer Routines
MTH$FLOOR

Convert F-floating to greatest F-floating integer

MTH$0FLOOR

Convert 0-floating to greatest 0-floating integer

MTH$GFLOOR

Convert G-floating to greatest G-floating integer

MTH$HFLOOR

Convert H-floating to greatest H-floating integer 1

1 Returns value to the first argument; value exceeds 64 bits.

2 1nteger overflow exceptions can occur.
3 Floating-point overflow exceptions can occur.
4 Floating-point underflow exceptions can occur.

1-5

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

Table 1-1 (Cont.)

Additional Mathematics Routines

Entry Point

Function

Floating-point Truncation Routines
MTH$AINT

Convert F-floating to truncated F-floating 3

MTH$DINT

Convert D-floating to truncated D-floating

MTH$11DINT

Convert D-floating to truncated word 2

MTH$JIDINT

Convert D-floating to truncated longword 2

MTH$GINT

Convert G-floating to truncated G-floating

MTH$11GINT

Convert G-floating to truncated word 2

MTH$JIGINT

Convert G-floating to truncated longword 2

MTH$HINT

Convert H-floating to truncated H-floating 1 •3

MTH$11HINT

Convert H-floating to truncated word 2

MTH$JIHINT

Convert H-floating to truncated longword 2

MTH$11NT

Convert F-floating to truncated word 2

MTH$JINT

Convert F-floating to truncated longword 2

Bitwise Inclusive OR Operator Routines
MTH$110R

Bitwise inclusive OR of two word arguments

MTH$JIOR

Bitwise inclusive OR of two longword arguments

Maximum Value Routines
MTH$AIMAXO

F-floating maximum of n word arguments

MTH$AJMAXO

F-floating maximum of n longword arguments

MTH$1MAXO

Word maximum of n word arguments

MTH$JMAXO

Longword maximum of n longword arguments

MTH$AMAX1

F-floating maximum of n F-floating arguments 2

MTH$DMAX1

D-floating maximum of n D-floating arguments

MTH$GMAX1

G-floating maximum of n G-floating arguments

MTH$HMAX1

H-floating maximum of n H-floating arguments 1

MTH$1MAX1

Word maximum of n F-floating arguments 2

MTH$JMAX1

Longword maximum of n F-floating arguments 2

Returns value to the first argument; value exceeds 64 bits.

2 1nteger overflow exceptions can occur.

3 Floating-point overflow exceptions can occur.

1-6

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

Table 1-1 (Cont.)

Additional Mathematics Routines

Entry Point

Function

Minimum Value Routines
MTH$AIMINO

F-floating minimum of n word arguments

MTH$AJMINO

F-floating minimum of n longword arguments

MTH$1MINO

Word minimum of n word arguments

MTH$JMINO

Longword minimum of n longword arguments

MTH$AMIN1

F-floating minimum of n F-floating arguments 2

MTH$0MIN1

0-floating minimum of n 0-floating arguments

MTH$GMIN1

G-floating minimum of n G-floating arguments

MTH$HMIN1

H-floating minimum of n H-floating arguments 1

MTH$1MIN1

Word minimum of n F-floating arguments 2

MTH$JMIN1

Longword minimum of n F-floating arguments 2

Remainder Routines
MTH$AMOO

Remainder of two F-floating arguments, arg 1/arg2 3

MTH$0MOO

Remainder of two 0-floating arguments, arg 1/arg2 3

MTH$GMOO

Remainder of two G-floating arguments, arg 1/arg2 3

MTH$HMOO

Remainder of two H-floating arguments, arg 1/arg2 1' 3

MTH$1MOO

Remainder of two word arguments, arg 1/arg2 5

MTH$JMOO

Remainder of two longword arguments, arg 1/arg2 5

Floating-point Conversion to Nearest Value Routines
MTH$ANINT

Convert F-floating to nearest F-floating integer

MTH$0NINT

Convert 0-floating to nearest 0-floating integer3

MTH$110NNT

Convert 0-floating to nearest word integer

MTH$JIONNT

Convert 0-floating to nearest longword integer

MTH$GNINT

Convert G-floating to nearest G-floating integer3

MTH$11GNNT

Convert G-floating to nearest word integer2

MTH$JIGNNT

Convert G-floating to nearest longword integer2

MTH$HNINT

Convert H-floating to nearest H-floating integer 1

1 Returns value to the first argument; value exceeds 64 bits.
2 1nteger overflow exceptions can occur.

3 Floating-point overflow exceptions can occur.
5 Divide-by-zero exceptions can occur.

1-7

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

Table 1-1 (Cont.)

Additional Mathematics Routines

Entry Point

Function

MTH$11HNNT

Convert H-floating to nearest word integer2

MTH$JIHNNT

Convert H-floating to nearest longword integer2

MTH$1NINT

Convert F-floating to nearest word integer2

MTH$JNINT

Convert F-floating to nearest longword integer3 •6

Bitwise Complement Operator Routines
MTH$1NOT

Bitwise complement of word argument

MTH$JNOT

Bitwise complement of longword argument

Floating-point Multiplication Routines
MTH$DPROD

D-floating product of two F-floating arguments3

MTH$GPROD

G-floating product of two F-floating arguments 3

Bitwise Shift Operator Routines
MTH$11SHFT

Bitwise shift of word

MTH$JISHFT

Bitwise shift of longword

Floating-point Sign Function Routines
MTH$SGN

F- or D-floating sign function

MTH$SIGN

F-floating transfer of sign of y to sign of x

MTH$DSIGN

D-floating transfer of sign of y to sign of x

MTH$GSIGN

G-floating transfer of sign of y to sign of x

MTH$HSIGN

H-floating transfer of sign of y to sign of x 1

1 Returns value to the first argument; value exceeds 64 bits.
2 Integer overflow exceptions can occur.

3 Floating-point overflow exceptions can occur.
6 Returns contents of RO if a negative argument is input.

1-8

Introduction to MTH$
1.6 Routines Not Documented in the MTH$ Reference Section

Table 1-1 (Cont.)

Additional Mathematics Routines

Entry Point

Function

MTH$11SIGN

Word transfer of sign of y to sign of x

MTH$JISIGN

Longword transfer of sign of y to sign of x

Conversion of Double to Single Floating-point Routines

MTH$SNGL

Convert D-floating to F-floating (rounded) 3

MTH$SNGLG

Convert G-floating to F-floating (rounded) 3 •4

3 Floating-point overflow exceptions can occur.
4 Floating-point underflow exceptions can occur.

1.7

Examples of Calls to Run-Time Library Mathematics Routines

1.7.1

BASIC Example
The following BASIC program uses the H-floating data type. BASIC also
supports the D-floating, F-floating and G-floating data types, but does not
support the complex data types.
10

!+
! Sample program to demonstrate a call to MTH$HEXP from BASIC.
!-

EXTERNAL SUB MTH$HEXP ( HFLOAT, HFLOAT )
DECLARE HFLOAT X,Y
! X and Y are H-floating
DIGITS$ = '###.#################################'
X = '1.2345678901234567891234567892'H
CALL MTH$HEXP (Y,X)
A$ = 'MTH$HEXP of ' + DIGITS$ + ' is ' + DIGITS$
PRINT USING A$, X, Y
END

The output from this program is as follows:
MTH$HEXP of 1.234567890123456789123456789200000
is 3.436893084346008004973301321342110

1.7.2

COBOL Example
The following COBOL program uses the F-floating and D-floating data types.
COBOL does not support the G-floating and H-floating data types or the
complex data types.
This COBOL program calls MTH$EXP and MTH$DEXP.

1-9

Introduction to MTH$
1 . 7 Examples of Calls to Run-Time Library Mathematics Routines

IDENTIFICATION DIVISION.
PROGRAM-ID.
FLOATING_POINT.

* Calls MTH$EXP using a Floating Point data type.
* Calls MTH$DEXP using a Double Floating Point data type.

ENVIRONMENT DIVISION.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 FLOAT_PT
COMP-1.
01 ANSWER_F
COMP-1.
01 DOUBLE_PT
COMP-2.
01 ANSWER_D
COMP-2.
PROCEDURE DIVISION.
PO.
MOVE 12.34 TO FLOAT_PT.
MOVE 3.456 TO DOUBLE_PT.
CALL "MTH$EXP" USING BY REFERENCE FLOAT_PT GIVING ANSWER_F.
DISPLAY" MTH$EXP of ". FLOAT_PT CONVERSION, " is",
ANSWER_F CONVERSION.
CALL "MTH$DEXP" USING BY REFERENCE DOUBLE_PT GIVING ANSWER_D.
DISPLAY" MTH$DEXP of ", DOUBLE_PT CONVERSION, " is ".
ANSWER_D CONVERSION .
STOP RUN.

The output from this example program is as follows:
MTH$EXP of 1.234000E+01 is 2.286620E+05
MTH$DEXP of 3.456000000000000E+OO is
3.168996280537917E+01

1.7.3

FORTRAN Examples
The first two FORTRAN programs below use the D-floating and H-floating
data types. The third FORTRAN program below uses the F-floating complex
data type. FORTRAN supports the four floating data types and the three
complex data types.

C+
C This FORTRAN program computes exp(x) in
C double precision by using the RTL routine "MTH$DEXP x "·

C Declare X,Y and MTH$DEXP as double precision values.
C MTH$DEXP(X) will return a double precision value to variable Y.

cREAL*8 X,Y,MTH$DEXP

x = 3.456

Y = MTH$DEXP(X)
WRITE(6,1) X,Y
FORMAT(' ','MTH$DEXP(',F20.15,') IS' ,F20.15)
END

The output generated by this FORTRAN example is as follows:
MTH$DEXP(3.456000000000000) IS
31.689962805379165

1-10

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

C+
C This FORTRAN program computes exp(x) using
C the RTL routine MTH$HEXP. MTH$HEXP is CALLed by
C MTH$HEXP(return_value , argument)

C Declare X,Y as H-floating point values.
C Given X MTH$HEXP will return the value of exp(X) in Y by the call
C CALL MTH$HEXP(Y,X).

REAL*16 X,Y

x = 1.2345678901234567891234567892
1

CALL MTH$HEXP(Y,X)
WRITE(6,1) X,Y
FORMAT(' , ,'MTH$HEXP of , ,E35.30,' is ',E35.30)
END

This FORTRAN program generates the following output:
MTH$HEXP of .123456789012345678912345678920E+01
is .343689308434600800497330132134E+01
~

C+
C This FORTRAN program computes the complex log
C of x using the RTL routine MTH$CLOG. This program also demonstrates
C two ways the user can create a complex number.

C Declare Z,Z_LOG,MTH$CMPLX, and MTH$CLOG as complex values and R and I
C as real values. MTH$CMPLX takes two real arguments and returns one
C complex number: Z = MTH$CMPLX(R,I) is a complex number with "real"
C part R and "imaginary" part I.

C Given a complex number Z, MTH$CLOG(Z) returns the complex natural
C logarithm of Z.

COMPLEX Z,Z_LOG,MTH$CMPLX,MTH$CLOG
REAL*4 R,I
R = 3.142563
I= 7.4367846
Z = MTH$CMPLX(R,I)
C+
C Z is a complex number with real part Rand imaginary part I.

TYPE *• , The complex number z is' ,z
C+
C Compute the natural logarithm of Z = (2,1).
C Directly define the complex number Z.

z = (2.0,1.0)
Z_LOG = MTH$CLOG(Z)
TYPE*·' The complex log of (2,1) is
END

',Z_LOG

The output generated by this program is as follows:
The complex number z is (3.142563,7.436785)
The complex log of (2,1) is
(0.8047190,0.4636476)

1-11

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

1.7.4

MACRO Examples
MACRO and BLISS support JSB entry points as well as CALLS and CALLG
entry points. Both MACRO and BLISS support the four floating data types
and the three complex data types.
The MACRO programs below illustrate the use of the CALLS and CALLG
instructions, as well as JSB entry points .
. TITLE EXAMPLE_JSB
;+

; This example calls MTH$DEXP by using a Macro JSB command.
; The JSB command expects RO/R1 to contain the quadword input value X.
; The result of the JSB will be located in RO/R1 .

. EXTRN MTH$DEXP_R6
;MTH$DEXP is an external routine .
. PSECT DATA, PIC, EXE, NOWRT
.DOUBLE 2.0
; X is 2.0
.ENTRY EXAMPLE_JSB, AM<>
MOVQ
X, RO
; X is in registers RO and R1
JSB
GAMTH$DEXP_R6
; The result is returned in RO/R1.
RET
.END
EXAMPLE_JSB

This MACRO program generates the following output:
RO <-- 732541EC
R1 <-- ED6EC6A6
That is, MTH$DEXP(2) is 7.3890560989306502
.TITLE EXAMPLE_CALLG
;+

This example calls MTH$HEXP by using a Macro CALLG command.
The CALLG command expects that the address of the return value
Y, the address of the input value X, and the·argument count 2 be
stored in memory; this program stores this information in ARGUMENTS;
The result of the CALLG will be located in RO/R1 .
. EXTRN MTH$HEXP
; MTH$HEXP is an external routine .
. PSECT DATA, PIC, EXE, WRT
ARGUMENTS:
.LONG 2
The CALLG will use two arguments .
. ADDRESS Y, X
The first argument must be the address
receiving the computed value, while
the second argument is used to
compute exp(X) .
X:
. H_FLOATING 2
x = 2.0
.H_FLOATING 0
Y:
Y is the result, initially set to 0 .
. ENTRY EXAMPLE_G, AM<>
CALLG ARGUMENTS, GAMTH$HEXP ; CALLG returns the value to Y.
RET
.END
EXAMPLE_G

1-12

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

The output generated by this MACRO program is as follows:
address of Y <-- D8E64003
<-- 4DDA4B8D
<-- 3A3BDCC3
<-- B68BA206
That is, MTH$HEXP of 2.0 returns
7.38905609893065022723042746057501
.TITLE EXAMPLE_CALLS
;+

This example calls MTH$HEXP by using the Macro CALLS command.
The CALLS command expects the SP to contain the H-floating address of
the return value, the address of the input argument X and the argument
count 2. The result of the CALLS will be located in registers RO-R3 .
. EXTRN MTH$HEXP
; MTH$HEXP is an external routine .
. PSECT DATA, PIC, EXE, WRT
.H_FLOATING 0
Y is the result, initially set to 0.
.H_FLOATING 2
X=2
.ENTRY EXAMPLE_S, AM<>
MOVAL X, -(SP)
The address of X is in the SP.
MOVAL Y, -(SP)
The address of Y is in the SP
CALLS Y, GAMTH$HEXP
The value is returned to the address of Y.
RET
.END
EXAMPLE_S

Y:
X:

The output generated by this program is as follows:
address of Y <-- D8E64003
<-- 4DDA4B8D
<-- 3A3BDCC3
<-- B68BA206
That is, MTH$HEXP of 2.0 returns
7.38905609893065022723042746057501
.TITLE COMPLEX_EX1
;+

This example calls MTH$CLOG by using a MACRO CALLG command.
To compute the complex natural logarithm of Z = (2.0,1.0) register
RO is loaded with 2.0, the real part of Z, and register R1 is loaded
with 1.0, the imaginary part of Z. The CALLG to MTH$CLOG
returns the value of the natural logarithm of Z in
registers RO and R1. RO gets the real part of Zand R1
gets the imaginary part .

ARGS:

REAL:
!MAG:

. EXTRN MTH$CLOG
.PSECT DATA, PIC, EXE, NOWRT
.LONG
1
The CALLG will use one argument .
. ADDRESS REAL
The one argument that the CALLG
uses is the address of the argument
of MTH$CLOG.
.FLOAT 2
real part of Z is 2.0
.FLOAT 1
imaginary part Z is 1.0
.ENTRY COMPLEX_EX1, AM<>
CALLG ARGS, GAMTH$CLOG; MTH$CLOG return the real part of the
complex natural logarithm in RO and
the imaginary part in R1.
RET
.END
COMPLEX_EX1

1-13

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

This program generates the following output:
RO <--R1 <---

0210404E
63383FED

That is, MTH$CLOG(2.0,1.0) is
(0.8047190,0.4636476)
.TITLE COMPLEX_EX2
;+

This example calls MTH$CLOG by using a MACRO CALLS command.
To compute the complex natural logarithm of Z = (2.0,1.0) register
RO is loaded with 2.0. the real part of z. and register R1 is loaded
with 1.0. the imaginary part of Z. The CALLS to MTH$CLOG
returns the value of the natural logarithm of Z in registers RO
and R1. RO gets the real part of Zand R1 gets the imaginary
part.

REAL:
!MAG:

.EXTRN MTH$CLOG
.PSECT DATA. PIC, EXE. NOWRT
; real part of Z is 2.0
.FLOAT 2
; imaginary part Z is 1.0
.FLOAT 1
.ENTRY COMPLEX_EX2, ~M<>
SP <-- address of Z. Real part of Z is
MOVAL REAL. -(SP)
in ©(SP) and imaginary part is in
©(SP)+4.
CALLS
MTH$CLOG return the real part of the
complex natural logarithm in RO and
the imaginary part in R1.
RET
.END
COMPLEX_EX2

This MACRO example program generates the following output:
RO <--R1 <---

0210404E
63383FED

That is, MTH$CLOG(2.0,1.0) is
(0.8047190.0.4636476)

1.7.5

PASCAL Examples
The following PASCAL programs use the D-floating and H-floating data
types. PASCAL also supports the F-floating and G-floating data types.
PASCAL does not support the complex data types, however.

{+}

{ Sample program to demonstrate a call to MTH$DEXP from PASCAL.
{-}

PROGRAM CALL_MTH$DEXP (OUTPUT);
{+}

{ Declare variables used by this program.
{-}

VAR
X
Y

1-14

DOUBLE := 3.456;
DOUBLE;

{ X,Y are D-floating unless overridden}
{ with /DOUBLE qualifier on compilation }

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

{+}

{ Declare the RTL routine used by this program.
{-}

[EXTERNAL.ASYNCHRONOUS] FUNCTION MTH$DEXP (VAR value
BEGIN
Y := MTH$DEXP (x);
WRITELN ('MTH$DEXP of '
END.

DOUBLE)

DOUBLE; EXTERN;

X:5:3, ' is ', Y:20:16);

The output generated by this PASCAL program is as follows:
MTH$DEXP of 3.456 is· 31.6899656462382318
~

{+}

{ Sample program to demonstrate a call to MTH$HEXP from PASCAL.
{-}

PROGRAM CALL_MTH$HEXP (OUTPUT);
{+}

{ Declare variables used by this program.
{-}

VAR
X
Y

QUADRUPLE := 1.2345678901234567891234567892; { X is H-floating }
QUADRUPLE;
{ Y is H-floating }

{+}

{ Declare the RTL routine used by this program.
{-}

[EXTERNAL.ASYNCHRONOUS] PROCEDURE MTH$HEXP (VAR h_exp
value : QUADRUPLE); EXTERN;
BEGIN
MTH$HEXP (Y,X);
WRITELN ('MTH$HEXP of '
END.

QUADRUPLE;

X:30:28, ' is ', Y:35:33);

This PASCAL program generates the following output:
MTH$DEXP of 3.456 is

1.7.6

31.6899656462382318

PL/I Examples
The following PL/I programs use the D-floating and H-floating data types to
test entry points. PL/I also supports the F-floating and G-floating data types.
PL/I does not support the complex data types, however.

1-15

Introduction to MTH$
1 .7 Examples of Calls to Run-Time Library Mathematics Routines

*
*
*
*/
TEST:

*
*
*

This program tests a MTH$D entry point
PROC OPTIONS (MAIN) ;
DCL (MTH$DEXP)
ENTRY (FLOAT(53)) RETURNS (FLOAT(53));
DCL OPERAND FLOAT(53);
DCL RESULT FLOAT(53);

/*** Begin test ***/
OPERAND = 3.456;
RESULT= MTH$DEXP(OPERAND);
PUT EDIT ('MTH$DEXP of ',OPERAND, ' is ', RESULT)(A(12),F(5,3),A(4),F(20,15));
END TEST;

The output generated by this PL/I program is as follows:
MTH$DEXP of 3.456 is

31.689962805379165

*
*
*
*

This program tests a MTH$H entry point.
Note that in the PL/I statement below, the /G-float switch
is needed to compile both G- and H-floating point MTH$ routines.

TEST:

PROC OPTIONS (MAIN) ;

*
*
*

DCL (MTH$HEXP)
ENTRY (FLOAT (113), FLOAT (113))
DCL OPERAND FLOAT (113);
DCL RESULT FLOAT (113);

/*** Begin test ***/
OPERAND= 1.234578901234567891234567892;
CALL MTH$HEXP(RESULT,OPERAND);
PUT EDIT ('MTH$HEXP of ',OPERAND, 'is ', RESULT) (A(12),F(29,27),A(4),F(29,27));
end test;

To run this program, you must use the following DCL commands:
$ PLI/G_FLOAT EXAMPLE
$ LINK EXAMPLE
$ RUN EXAMPLE

This program generates the following output:
MTH$HEXP of 1.234578901234567891234567892 is
3.436930928565989790506225633

1-16

MTH$ Reference Section
This section provides detailed descriptions of the routines provided by the
VMS RTL Mathematics (MTH$) Facility.

MTH$xACOS

Arc Cosine of Angle Expressed in
Radians

Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Radians routine returns that angle (in radians).

FORMAT

MTH$ACOS cosine
MTH$DACOS cosine
MTH$GACOS cosine
Each of the above three formats accepts as input one of the floating-point
types.

jsb entries

MTH$ACOS_R4
MTH$DACQS_R7
MTH$GACQS_R7
Each of the above three JSB entries accepts as input one of the floating-point
types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
F_floating, D_floating, G_floating
write only
by value

Angle in radians. The angle returned will have a value in the range
0 ~angle~ 7r

MTH$ACOS returns an F-floating number. MTH$DACOS returns. a Dfloating number. MTH$GACOS returns a G-floating number.

ARGUMENTS

cosine
VMS usage:
type:
access:
mechanism:

floating_point
F_floating, D_floating, G_floating
read only
by reference

The cosine of the angle whose value (in radians) is to be returned. The cosine
argument is the address of a floating-point number that is this cosine. The
absolute value of cosine must be less than or equal to 1. For MTH$ACOS,
cosine specifies an F-floating number. For MTH$DACOS, cosine specifies a
D-floating number. For MTH$GACOS, cosine specifies a G-floating number.

MTH-3

MTH$xACOS

DESCRIPTION

The angle in radians whose cosine is X is computed as:
Value of
Cosine

Value Returned

7r/2

,.-l

O<X< 1

zAT AN(zSQRT(l - X 2 )/ X), where zATAN and zSORT are the
Math Library arc tangent and square root routines, respectively,
of the appropriate data type

-1<x<0
1 < IXI

zAT AN(zSQRT(l - X 2 )/ X) + 7r
The error MTH$_1NV ARGMA T is signaled

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HACOS.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xACOS routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of
one and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. The absolute value of cosine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L _MCH _SA VRO /R 1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

EXAMPLES

100

! This BASIC program demonstrates the use of
! MTH$ACOS.
!-

300
400
500
32767

MTH-4

EXTERNAL REAL FUNCTION MTH$ACOS
DECLARE REAL COS_VALUE, ANGLE
INPUT "Cosine value between -1 and +1 "; COS_VALUE
IF (COS_VALUE < -1) OR (COS_VALUE > 1)
THEN PRINT "Invalid cosine value"
GOTO 300
ANGLE = MTH$ACOS( COS_VALUE )
PRINT "The angle with that cosine is 11 • ANGLE; "radians"
END

MTH$xACOS

This BASIC program prompts for a cosine value and determines the angle
that has that cosine. The output generated by this program is as follows:
$ RUN ACOS
Cosine value betwen -1 and +1 ? .5
The angle with that cosine is 1.0472 radians
~

PROGRAM GETANGLE(INPUT, OUTPUT);
{+}

{ This PASCAL program uses MTH$ACOS to determine
{ the angle which has the cosine given as input.
{-}

VAR
COS : REAL;
FUNCTION MTH$ACOS(COS
EXTERN;

REAL)

REAL;

BEGIN
WRITE('Cosine value between -1 and +1: ');
READ (COS);
WRITELN('The angle with that cosine is', MTH$ACOS(COS),
'radians');
END.

This PASCAL program prompts for a cosine value and determines the angle
that has that cosine. The output generated by this program is as follows:
$ RUN ACOS
Cosine value between -1 and +1: .5
The angle with that cosine is 1.04720E+OO radians

MTH-5

MTH$xACOSD

Arc Cosine of Angle Expressed in
Degrees

Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Degrees routine returns that angle (in degrees).

FORMAT

MTH$ACOSD cosine
MTH$DACOSD cosine
MTH$GACOSD cosine
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ACOSD_R4
MTH$DACOSD_R7
MTH$GACOSD_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Angle in degrees. The angle returned will have a value in the range
0 ~ angle ~ 180

MTH$ACOSD returns an F-floating number. MTH$DACOSD returns a
D-floating number. MTH$GACOSD returns a G-floating number.

ARGUMENTS

cosine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, G_floating, D_floating
read only
by reference

Cosine of the angle whose value (in degrees) is to be returned. The cosine
argument is the address of a floating-point number that is this cosine. The
absolute value of cosine must be less than or. equal to 1. For MTH$ACOSD,
cosine specifies an F-floating number. For MTH$DACOSD, cosine specifies
a D-floating number. For MTH$GACOSD, cosine specifies a G-floating
number.

MTH-6

MTH$xACOSD

DESCRIPTION

The angle in degrees whose cosine is X is computed as:
Value of
Cosine

Angle Returned

-1

180

O<X< 1

zAT AN D(zSQRT(l - X 2 )/ X), where zAT AND and zSQRT
are the Math Library arc tangent and square root routines,
respectively, of the appropriate data type

-1<x<0

zAT AN D(zSQRT(l - X 2 )/ X) + 180

1 < IXI

The error MTH$_1NV ARGMA T is signaled

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HACOSD.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xACOSD routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of
one and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. The absolute value of cosine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

EXAMPLE
PROGRAM ACOSD(INPUT,OUTPUT);
{+}

{ This PASCAL program demonstrates the use of
{ MTH$ACOSD.
{-}

FUNCTION MTH$ACOSD(COS

REAL): REAL; EXTERN;

VAR
COSINE : REAL;
RET_STATUS : REAL;
BEGIN
COSINE := 0.5;
RET_STATUS := MTH$ACOSD(COSINE);
WRITELN('The angle, in degrees, is: '
END.

RET_STATUS);

MTH-7

MTH$xACOSD

The output generated by this PASCAL example program is as follows:
The angle, expressed in degrees, is:

MTH-8

6.00000E+01

MTH$xASIN

Arc Sine in Radians

Given the sine of an angle, the Arc Sine in Radians routine returns that
angle (in radians).

FORMAT

MTH$ASI N sine
MTH$DASIN sine
MTH$GASIN sine
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ASIN_R4
MTH$DASIN_R7
MTH$GASIN_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Angle in radians. The angle returned will have a value in the range
-7r

/2 ~ angle ~ 7r /2

MTH$ASIN returns an F-floating number. MTH$DASIN returns a D-floating
number. MTH$GASIN returns a G-floating number.

ARGUMENTS

sine
VMS usage:
type:
access:
mechanism:

floating_point
F_floating, D_floating, G_floating
read only
by reference

The sine of the angle whose value (in radians) is to be returned. The sine
argument is the address of a floating-point number that is this sine. The
absolute value of sine must be less than or equal to 1. For MTH$ASIN, sine
specifies an F-floating number. For MTH$DASIN, sine specifies a D-floating
number. For MTH$GASIN, sine specifies a G-floating number.

MTH-9

MTH$xASIN

DESCRIPTION

The angle in radians whose sine is X is computed as:
Value of Sine

Angle Returned

7r/2

-1

-Tr/2

0<IXI<1

zATAN(X/zSQRT(l - X 2 )), where zATAN and zSQRT
are the Math Library arc tangent and square root routines,
respectively, of the appropriate data type

1 < IXI

The error MTH$_1NV ARGMA T is signaled

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HASIN.

CONDITION
VALUES
SIGNALED

MTH-10

SS$_ROPRAND

Reserved operand. The MTH$xASIN routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. The absolute value of sine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

MTH$xASIND

MTH$xASIND Arc Sine in Degrees
Given the sine of an angle, the Arc Sine in Degrees routine returns that
angle (in degrees).

FORMAT

MTH$ASIND sine
MTH$DASIND sine
MTH$GASIND sine
Each of the above formats accepts as input one of the floating-point types.
r

jsb entries

MTH$ASI N D_R4
MTH$DASIND_R7
MTH$GASIND_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Angle in degrees. The angle returned will have a value in the range
-90 ::; angle ::; 90

MTH$ASIND returns an F-floating number. MTH$DASIND returns a Dfloating number. MTH$GASIND returns a G-floating number.

ARGUMENTS

sine
VMS usage: floating_point
type:
· f _floating, O_floating, G_floating
access:
read only
mechanism: by reference

Sine of the angle whose value (in degrees) is to be returned. The sine
argument is the address of a floating-point number that is this sine. The
absolute value of sine must be less than or equal to 1. For MTH$ASIND,
sine specifies an F-floating number. For MTH$DASIND, sine specifies a
D-floating number. For MTH$GASIND, sine specifies a G-floating number.

MTH-11

MTH$xASIND

DESCRIPTION

The angle in degrees whose sine is X is computed as:
Value of Sine

Value Returned

-1

-90

0 < IXI < 1

zATAND(X/zSQRT(l - X 2 )), where zATAND and zSORT
are the Math Library arc tangent and square root routines,
respectively, of the appropriate data type

1 < IXI

The error MTH$_1NV ARGMA T is signaled

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HASIND.

CONDITION
VALUES
SIGNALED

MTH-12

SS$_RQPRAND

Reserved operand. The MTH$xASIND routine
encountered a floating point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of
one and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. The absolute value of sine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R 1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L _MCH_SA VRO /R 1.

MTH$xATAN

Arc Tangent in Radians

Given the tangent of an angle, the Arc Tangent in Radians routine returns
that angle (in radians).

FORMAT

MTH$ATAN tangent
MTH$DATAN tangent
MTH$GATAN tangent
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ATAN_R4
MTH$DATAN_R7
MTH$GATAN_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Angle in radians. The angle returned will have a value in the range
-Jr

/2 ~ angle ~ 7r /2

MTH$ATAN returns an F-floating number. MTH$DATAN returns a Dfloating number. MTH$GATAN returns a G-floating number.

ARGUMENTS

tangent
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The tangent of the angle whose value (in radians) is to be returned. The
tangent argument is the address of a floating-point number that is this
tangent. For MTH$ATAN, tangent specifies an F-floating number. For
MTH$DATAN, tangent specifies a D-floating number. For MTH$GATAN,
tangent specifies a G-floating number.

MTH-13

MTH$xATAN

DESCRIPTION

In radians, the computation of the arc tangent function is based on the
following identities:
arctan(X) = X - X 3 /3 + X 5 /5 - X 7 /7 + ...
arctan(X) = X + X * Q(X2 ),
where Q(Y) = -Y/3 + Y 2 /5- Y3 /7 + ...
arctan(X) = X * P(X 2 ),
where P(Y) = 1- Y/3 +·Y 2 /5 - Y 3 /7 + ...
arctan(X) = 7r/2 - arctan(l/X)
arctan(X) = arctan(A) + arctan((X - A)/(1 +A* X))
for any real A

The angle in radians whose tangent is X is computed as:
Value of X

Angle Returned

o:::;X::-;3/32

X+X*Q(X 2 )

3/32 < x:::;11

ATAN(A) + V * (P(V 2 )), where A and ATAN(A) are
chosen by table lookup and V = (X -A)/(1 +A* X)

11<X

7r/2 - W * (P(W 2 )) where W = 1/X

X<O

-zAT AN(IXI)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HATAN.

CONDITION
VALUE
SIGNALED

MTH-14

SS$_ROPRAND

Reserved operand. The MTH$xA TAN routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xATAND

MTH$xATAND Arc Tangent in Degrees
Given the tangent of an angle, the Arc Tangent in Degrees routine returns
that angle (in degrees).

FORMAT

MTH$ATAND tangent
MTH$DATAND tangent
MTH$GATAND tangent
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ATAND_R4
MTH$DATAND_R7
MTH$GATAND_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Angle in degrees. The angle returned will have a value in the range
-90 ~ angle ~ 90

MTH$ATAND returns an F-floating number. MTH$DATAND returns a
D-floating number. MTH$GATAND returns a G-floating number.

ARGUMENTS

tangent
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The tangent of the angle whose value (in degrees) is to be returned. The
tangent argument is the address of a floating-point number that is this
tangent. For MTH$ATAND, tangent specifies an F-floating number. For
MTH$DATAND, tangent specifies a D-floating number. For MTH$GATAND,
tangent specifies a G-floating number.

MTH-15

MTH$xATAND

DESCRIPTION

The computation of the arc tangent function is based on the following
identities:
arctan(X) = (180/'rr) * (X - X 3 /3 + X 5 /5 - X 7 /7 + ... )
arctan(X) = 64 * X + X * Q(X 2 ),
4
where Q(Y) = 180/7r * [(1 - 64 * 7r /180)] - Y /3 + Y 2 /5 - Y 3 /7 + Y /9
arctan(X) = X * P(X 2 ),
where P(Y) = 180/7r * [1 - Y /3 + Y 2 /5 - Y 3 /7 + Y 4 /9 ... ]
arctan(X) = 90 - arctan(l/ X)
arctan(X) = arctan(A) + arctan((X - A)/(1 +A* X))

The angle in degrees whose tangent is Xis computed as:
Tangent

Angle Returned

X9/32

64*X+X*Q(X 2 )

3/32 < X:Sll

ATAN D(A) + V * P(V 2 ), where A and ATAND(A) are
chosen by table lookup and V = (X - A)/(1 +A* X)

ll<X

90- W * (P(W 2 )), where W = 1/X

X<O

-zATAND(IXI)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HATAND.

CONDITION
VALUE
SIGNALED

MTH-16

SS$_ROPRAND

Reserved operand. The MTH$xA T AND routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xATAN2

Arc Tangent in Radians with Two
Arguments

Given sine and cosine, the Arc Tangent in Radians with Two Arguments
routine returns the angle (in radians) whose tangent is given by the
quotient of sine and cosine, (sine/cosine).

FORMAT

MTH$ATAN2 sine ,cosine
MTH$DATAN2 sine ,cosine
MTH$GATAN2 sine ,cosine
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Angle in radians. MTH$ATAN2 returns an F-floating number.
MTH$DATAN2 returns a D-floating number. MTH$GATAN2 returns a
G-floating number.

ARGUMENTS

sine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
read only
by reference

Dividend. The sine argument is the address of a floating.;point number that
is this dividend. For MTH$ATAN2, sine specifies an F-floating number. For
MTH$DATAN2, sine specifies a D-floating number. For MTH$GATAN2, sine
specifies a G-floating number.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Divisor. The cosine argument is the address of a floating-point number that
is this divisor. For MTH$ATAN2, cosine specifies an F-floating number. For
MTH$DATAN2, cosine specifies a D-floating number. For MTH$GATAN2,
cosine specifies a G-floating number.

MTH-17

MTH$xATAN2

DESCRIPTION

The angle in radians whose tangent is Y /Xis computed as follows, where f is
defined in the description of MTH$zCOSH.
Value of Input Arguments

X = 0 or Y/X > 211+ 11
X > 0 and Y/X$. 21!+ 11
X < 0 and Y/X$. 21J+ 1l

Angle Returned
7r

/2 * (signY)

zATAN(Y/X)
1r

* (signY) + zATAN(Y/X)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HATAN2.

CONDITION
VALUES
SIGNALED

MTH-18

SS$_RQPRAND

Reserved operand. The MTH$xA T AN2 routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. Both cosine and sine are zero.
LIB$SIGNAL copies the floating-point reserved
operand to the mechanism argument vector
CHF$L _MCH _SA VRO /R 1 . The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH_SAVRO/R 1.

MTH$xATAN 02

MTH$xATAND2

Arc Tangent in Degrees with
Two Arguments

Given sine and cosine, the Arc Tangent in Degrees with Two Arguments
routine returns the angle (in degrees) whose tangent is given by the
quotient of sine and cosine, (sine/cosine).

FORMAT

MTH$ATAND2 sine ,cosine
MTH$DATAND2 sine ,cosine
MTH$GATAND2 sine ,cosine
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Angle (in degrees). MTH$ATAND2 returns an F-floating number.
MTH$DATAND2 returns a D-floating number. MTH$GATAND2 returns
a G-floating number.

ARGUMENTS

sine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
read only
by reference

Dividend. The sine argument is the address of a floating-point number that
is this dividend. For MTH$ATAND2, sine specifies an F-floating number. For
MTH$DATAND2, sine specifies a D-floating number. For MTH$GATAND2,
sine specifies a G-floating number.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
F_floating, O_floating, G_floating
read only
by reference

Divisor. The cosine argument is the address of a floating-point number
that is this divisor. For MTH$ATAND2, cosine specifies an F-floating
number. For MTH$DATAND2, cosine specifies a D-floating number. For
MTH$GATAND2, cosine specifies a G-floating number.

MTH-19

MTH$xATAND2

DESCRIPTION

The angle in degrees whose tangent is Y/X is computed below and where f is
defined in the description of MTH$zCOSH.
Value of Input Arguments

Angle Returned

X = 0 or Y/X > 2U+ll

90 * (signY)

X > 0 and Y/X:::; 2U+ll

zAT AN D(Y / X)

X < 0 and Y/X:::; 2lf+l)

180 * (signY) + zAT AN D(Y / X)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HATAND2.

CONDITION
VALUES
SIGNALED

MTH-20

SS$_ROPRAND

Reserved operand. The MTH$xA T AND2 routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. Both cosine and sine are zero.
LIB$SIGNAL copies the floating-point reserved
operand to the mechanism argument vector
CHF$L _MCH _SA VRO /R 1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH_SAVRO/R1.

MTH$xATANH

Hyperbolic Arc Tangent

Given the hyperbolic tangent of an angle, the Hyperbolic Arc Tangent
routine returns the hyperbolic arc tangent of that angle.

FORMAT

MTH$ATANH hyperbolic-tangent
MTH$DATANH hyperbolic-tangent
MTH$GATANH hyperbolic-tangent
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The hyperbolic arc tangent of hyperbolic-tangent. MTH$ATANH returns
an F-floating number. MTH$DATANH returns a D-floating number.
MTH$GATANH returns a G-floating number.

ARGUMENTS

hyperbolic-tangent
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Hyperbolic tangent of an angle. The hyperbolic-tangent argument is the
address of a floating-point number that is this hyperbolic tangent. For
MTH$ATANH, hyperbolic-tangent specifies an F-floating number. For
MTH$DATANH, hyperbolic-tangent specifies a D-floating number. For
MTH$GATANH, hyperbolic-tangent specifies a G-floating number.

DESCRIPTION

The hyperbolic arc tangent function is computed as follows:
Value of x

Value Returned

IXI < 1

zAT AN H(X) = zLOG((X + 1)/(X - 1))/2

IXl2::1

An invalid argument is signaled

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HATANH.

MTH-21

MTH$xATANH

CONDITION
VALUES
SIGNALED

MTH-22

SS$_ROPRAND

Reserved operand. The MTH$xAT ANH routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument: IXl~l. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

MTH$CxABS

Complex Absolute Value

The Complex Absolute Value routine returns the absolute value of a
complex number (r,i).

FORMAT

MTH$CABS complex-number
MTH$CDABS complex-number
MTH$CGABS complex-number
Each of the above three formats accepts as input one of the three floatingpoint complex types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The absolute value of a complex number. MTH$CABS returns an F-floating
number. MTH$CDABS returns a D-flo'1ting number. MTH$CGABS returns a
G-floating number.

ARGUMENT

complex-number
VMS usage: complex_number
type:
f _floating complex, D_floating complex, G_floating
complex
access:
read only
mechanism: by reference

A complex number (r,i), where r and i are both floating-point complex values.
The complex-number argument is the address of this complex number. For
MTH$CABS, complex-number specifies an F-floating complex number. For
MTH$CDABS, complex-number specifies a D-floating complex number. For
MTH$CGABS, complex-number specifies a G-floating complex number.

DESCRIPTION

The complex absolute value is computed as follows, where MAX is the larger
of lrl and Iii, and MIN is the smaller of lrl and Iii.
result= MAX* SQRT((MIN/MAX) 2 +1)

MTH-23

MTH$CxABS

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

MTH$_FLOOVEMA T

Reserved operand. The MTH$CxABS routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.
Floating-point overflow in Math Library when both
r and i are large.

EXAMPLES

C+

C
C
C

This FORTRAN example forms the absolute value of an
F-f loating complex number using MTH$CABS and the
FORTRAN random number generator RAN.

C
C

Declare Z as a complex value and MTH$CABS as a REAL*4 value.
MTH$CABS will return the absolute value of Z:
Z_NEW = MTH$CABS(Z).

COMPLEX Z
COMPLEX CMPLX
REAL*4 Z_NEW,MTH$CABS
INTEGER M
M = 1234567
C+

Generate a random complex number with the FORTRAN generic CMPLX.
Z = CMPLX(RAN(M),RAN(M))

C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE *· ' The complex number z is' ,z
TYPE*· ' It has real part' ,REAL(Z), 'and imaginary part' ,AIMAG(Z)
TYPE *·

C+

Compute the complex absolute value of Z.
Z_NEW = MTH$CABS(Z)
TYPE*· 'The complex absolute value of',z,' is',Z_NEW
END

This example uses an F-floating complex number for complex-number. The
output of this FORTRAN example is as follows:
The complex number z is (0.8535407,0.2043402)
It has real part 0.8535407
and imaginary part

0.2043402

The complex absolute value of (0.8535407,0.2043402) is

MTH-24

0.8776597

MTH$CxABS

C+

C
C
C
C

This FORTRAN example forms the absolute
value of a G-floating complex number using
MTH$CGABS and the FORTRAN random number
generator RAN.

C
C
C

Declare Z as a complex value and MTH$CGABS as a
REAL*8 value. MTH$CGABS will return the absolute
value of Z: Z_NEW = MTH$CGABS(Z).

COMPLEX*16 Z
REAL*8 Z_NEW,MTH$CGABS
C+

C
C

Generate a random complex number with the FORTRAN
generic CMPLX.

z = (12.34567890123,45.536376385345)
TYPE*· 'The complex number z is',z
TYPE *, ' '
C+

Compute the complex absolute value of Z.

cZ_NEW = MTH$CGABS(Z)
TYPE*· 'The complex absolute value of',z,' is' ,Z_NEW
END

This FORTRAN example uses a G-floating complex number for complexnumber. Because this example uses a G-floating number, it must be compiled
as follows:
$ FORTRAN/G MTHEX.FOR

Notice the difference in the precision of the output generated:
The complex number z is (12.3456789012300,45.5363763853450)
The complex absolute value of (12.3456789012300,45.5363763853450) is
47.1802645376230

MTH-25

MTH$CCOS

Cosine of a Complex Number
(F-floating Value)
The Cosine of a Complex Number (F-floating Value) routine returns the
cosine of a complex number as an F-floating value.

FORMAT

MTH$CCOS

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
F_floating complex
write only
by value

The complex cosine of the complex input number. MTH$CCOS returns an
F-floating complex number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
read only
by reference

A complex number (r,i) where r and i are floating-point numbers. The
complex-number argument is the address of this complex number. For
MTH$CCOS, complex-number specifies an F-floating complex number.

DESCRIPTION

The complex cosine is calculated as follows:
result= (COS(r) * COSH(i), -SIN(r) * SINH(i))

The routine descriptions for the D- and G-floating point versions of this
routine are listed alphabetically under MTH$CxCOS.

CONDITION
VALUES
SIGNALED

MTH-26

SS$_ROPRAND

Reserved operand. The MTH$CCOS routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of i is greater than about 88.029
for F-floating values.

MTH$CCOS

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the complex
cosine of an F-f loating complex number using
MTH$CCOS and the FORTRAN random number
generator RAN.

C
C
C

Declare Z and MTH$CCOS as complex values.
MTH$CCOS will return the cosine value of
Z:
Z_NEW = MTH$CCOS(Z)

COMPLEX Z,Z_NEW,MTH$CCOS
COMPLEX CMPLX
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic CMPLX.
Z = CMPLX(RAN(M),RAN(M))

C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE *• ' The complex number z is' ,z
TYPE *· ' It has real part' ,REAL(Z), 'and imaginary part' ,AIMAG(Z)
TYPE *,

C+

Compute the complex cosine value of Z.
Z_NEW = MTH$CCOS(Z)
TYPE *· ' The complex cosine value of' ,z,' is' ,Z_NEW
END

This FORTRAN example demonstrates the use of MTH$CCOS, using the
MTH$CCOS entry point. The output of this program is as follows:
The complex number z is (0.8535407,0.2043402)
It has real part 0.8535407
and imaginary part 0.2043402
The complex cosine value of (0.8535407,0.2043402) is (0.6710899,-0.1550672)

MTH-27

MTH$CxCOS

Cosine of a Complex Number

The Cosine of a Complex Number routine returns the cosine of a complex
number.

FORMAT

MTH$CDCOS
MTH$CGCOS

complex-cosine ,complex-number
complex-cosine ,complex-number

Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

complex-cosine
VMS usage:
type:
access:
mechanism:

complex_number
O_floating complex, G_floating complex
write only
by reference

Complex cosine of the complex-number. The complex cosine routines that
have D-floating and G-floating complex input values write the address of
the complex cosine into the complex-cosine argument. For MTH$CDCOS,
the complex-cosine argument specifies a D-floating complex number. For
MTH$CGCOS, the complex-number argument specifies a G-floating complex
number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
O_floating complex, G_floating complex
read only
by reference

A complex number (r,i) where r and i are floating-point numbers. The
complex-number argument is the address of this complex number. For
MTH$CDCOS, complex-number specifies a D-floating complex number. For
MTH$CGCOS, complex-number specifies a G-floating complex number.

DESCRIPTION

The complex cosine is calculated as follows:
result= (COS(r) * COSH(i), -SIN(r) * SINH(i))

MTH-28

MTH$CxCOS

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CxCOS routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of i is greater than about 88.029
for F-floating and D-floating values or greater than
709.089 for G-floating values.

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the complex
co.sine of a D-floating complex number using
MTH$CDCOS and the FORTRAN random number
generator RAN.

C
C
C

Declare Z and MTH$CDCOS as complex values.
MTH$CDCOS will return the cosine value of
Z:
Z_NEW = MTH$CDCOS(Z)

COMPLEX*16 Z,Z_NEW,MTH$CDCOS
COMPLEX*16 DCMPLX
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic DCMPLX.
Z = DCMPLX(RAN(M) ,RAN(M))

C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE*· 'The complex number z is',z
TYPE *·

C+

Compute the complex cosine value of Z.
Z_NEW = MTH$CDCOS(Z)
TYPE *· ' The complex cosine value of' ,z,' is' ,Z_NEW
END

MTH-29

MTH$CxCOS

This FORTRAN example program demonstrates the use of MTH$CxCOS,
using the MTH$CDCOS entry point. Notice the high precision of the output
generated:
The complex number z is (0.8535407185554504,0.2043401598930359)
The complex cosine value of (0.8535407185554504,0.2043401598930359) is
(0.6710899028500762,-0.1550672019621661)

MTH-30

MTH$CEXP

Complex Exponential (F-floating
Value)
The Complex Exponential (F-floating Value) routine returns the complex
exponential of a complex number as an F-floating value.

FORMAT

MTH$CEXP

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
F_floating complex
write only
by value

Complex exponential of the complex input number. MTH$CEXP returns an
F-floating complex number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
f _floating complex
read only
by reference

Complex number whose complex exponential is to be returned. This complex
number has the form (r,i), where r is the real part and i is the imaginary part.
The complex-number argument is the address of this complex number. For
MTH$CEXP, complex-number specifies an F-floating number.

DESCRIPTION

The complex exponential is computed as follows:

complex - exponent= (EXP(r) * COS(i),EXP(r) * SIN(i))
The routine descriptions for the D- and G-floating point versions of this
routine are listed alphabetically under MTH$CxEXP.

MTH-31

MTH$CEXP

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CEXP routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of r is greater than about 88.029
for F-floating values.

EXAMPLE
C+

C
C
C

This FORTRAN example forms the complex exponential
of an F-f loating complex number using MTH$CEXP
and the FORTRAN random number generator RAN.

Declare Z and MTH$CEXP as complex values. MTH$CEXP
will return the exponential value of Z: Z_NEW = MTH$CEXP(Z)

C
c-

COMPLEX Z,Z_NEW,MTH$CEXP
COMPLEX CMPLX
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic CMPLX.

cZ = CMPLX(RAN(M),RAN(M))
C+

C
C

Z is a complex number (r,i) with real part "r"
and imaginary part "i".
TYPE *· ' The complex number z is' ,z
TYPE *· ' It has real part' ,REAL(Z), 'and imaginary part' ,AIMAG(Z)
TYPE *·

C+

Compute the complex exponential value of Z.
Z_NEW = MTH$CEXP(Z)
TYPE*· ' The complex exponential value of' ,z,' is' ,Z_NEW
END

This FORTRAN program demonstrates the use of MTH$CEXP as a function
call. The output generated by this example is as follows:
The complex number z is (0.8535407,0.2043402)
It has real part 0.8535407
and imaginary part 0.2043402
The complex exponential value of (0.8535407,0.2043402) is
(2.299097,0.4764476)

MTH-32

MTH$CxEXP

Complex Exponential

The Complex Exponential routine returns the complex exponential of a
complex number.

FORMAT

MTH$CDEXP
MTH$CGEXP

complex-exponent ,complex-number
complex-exponent, complex-number

Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

complex-exponent
VMS usage:
type:
access:
mechanism:

complex_number
O_floating complex, G_floating complex
write only
by reference

Complex exponential of complex-number. The complex exponential routines
that have D-floating complex and G-floating complex input values write the
complex-exponent into this argument. For MTH$CDEXP, complex-exponent
argument specifies a D-floating complex number. For MTH$CGEXP,
complex-exponent specifies a G-floating complex number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
O_floating complex, G_floating complex
read only
by reference

Complex number whose complex exponential is to be returned. This complex
number has the form (r,i), where r is the real part and i is the imaginary
part. The complex-number argument is the address of this complex number.
For MTH$CDEXP, complex-number specifies a D-floating number. For
MTH$CGEXP, complex-number specifies a G-floating number.

DESCRIPTION

The complex exponential is computed as follows:

complex - exponent= (EXP(r) * COS(i),EXP(r) * SIN(i))

MTH-33

MTH$CxEXP

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CxEXP routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of r is greater than about 88.029
for 0-floating values or greater than about 709 .089
for G-floating values.

EXAMPLE
C+

C
C
C

This FORTRAN example forms the complex exponential
of a G-f loating complex number using MTH$CGEXP
and the FORTRAN random number generator RAN.

C
C
C

Declare Z and MTH$CGEXP as complex values.
MTH$CGEXP will return the exponential value
of Z:
CALL MTH$CGEXP(Z_NEW,Z)

COMPLEX*16 Z,Z_NEW
COMPLEX*16 MTH$GCMPLX
REAL*8 R,I
INTEGER M
M = 1234567
C+

C
C-

Generate a random complex number with the FORTRAN
generic CMPLX.
R = RAN(M)
I = RAN(M)
Z = MTH$GCMPLX(R,I)
TYPE *· ' The complex number z is' ,z
TYPE *· ' '

C+

Compute the complex exponential value of Z.
CALL MTH$CGEXP(Z_NEW,Z)
TYPE*· 'The complex exponential value of' ,z,' is',Z_NEW
END

This FORTRAN example demonstrates how to access MTH$CGEXP as a
procedure call. Because G-floating numbers are used, this program must be
compiled using the command "FORTRAN/G filename".

MTH-34

MTH$CxEXP

Notice the high precision of the output generated:
The complex number z is (0.853540718555450,0.204340159893036)
The complex exponential value of (0.853540718555450,0.204340159893036) is
(2.29909677719458,0.476447678044977)

MTH-35

MTH$CLOG

Complex Natural Logarithm
(F-floating Value)
The Complex Natural Logarithm (F-floating Value) routine returns the
complex natural logarithm of a complex number as an F-floating value.

FORMAT

MTH$CLOG

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
F_floating complex
write only
by value

The complex natural logarithm of a complex number. MTH$CLOG returns
an F-floating complex number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
read only
by reference

Complex number whose complex natural logarithm is to be returned. This
complex number has the form (r,i}, where r is the real part and i is the
imaginary part. The complex-number argument is the address of this
complex number. For MTH$CLOG, complex-number specifies an F-floating
number.

DESCRIPTION

The complex natural logarithm is computed as follows:

CLOG(x) = (LOG(CABS(x)), ATAN2(i, r))
The routine descriptions for the D- and G-floating point versions of this
routine are listed alphabetically under MTH$CxLOG.

CONDITION
VALUE
SIGNALED

MTH-36

SS$_ROPRAND

Reserved operand. The MTH$CLOG routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$CLOG

EXAMPLE
Examples of using MTH$CLOG from VAX MACRO (using both the CALLS
and the CALLG instructions) appear in the introductory section of this
manual.

MTH-37

MTH$CxLOG

Complex Natural Logarithm

The Complex Natural Logarithm routine returns the complex natural
logarithm of a complex number.

FORMAT

MTH$CDLOG
MTH$CGLOG

complex-natural-log , complex-number
complex-natural-log ,complex-number

Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

complex-natural-log
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
write only
by reference

Natural logarithm of the complex number specified by complex-number.
The complex natural logarithm routines that have D-floating complex
and G-floating complex input values write the address of the complex
natural logarithm into complex-natural-log. For MTH$CDLOG, the
complex-natural-log argument specifies a D-floating complex number.
For MTH$CGLOG, the complex-natural-log argument specifies a G-floating
complex number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
read only
by reference

Complex number whose complex natural logarithm is to be returned. This
complex number has the form (r,i), where r is the real part and i is the
imaginary part. The complex-number argument is the address of this
complex number. For MTH$CDLOG, complex-number specifies a D-floating
number. For MTH$CGLOG, complex-number specifies a G-floating number.

DESCRIPTION

The complex natural logarithm is computed as follows:

CLOG(x) = (LOG(CABS(x)), ATAN2(i, r))

MTH-38

MTH$CxLOG

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CxLOG routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

EXAMPLE
C+

C
C
C

This FORTRAN example forms the complex logarithm
of a D-f loating complex number by using MTH$CDLOG
and the FORTRAN random number generator RAN.

C
c

Declare Z and MTH$CDLOG as complex values. Then MTH$CDLOG
will return the logarithm of Z: CALL MTH$CDLOG(Z_NEW,Z).

C
C
C

Declare Z,Z_LOG, and MTH$DCMPLX as complex values,
and R and I as real values. MTH$DCMPLX takes two real
arguments and returns one complex number.

C
C

Given a complex number Z, MTH$CDLOG(Z) returns the
complex natural logarithm of Z.

c
c

cCOMPLEX*16 Z,Z_NEW,MTH$DCMPLX
REAL*8 R,I
R = 3.1425637846746565
I = 7.43678469887
Z = MTH$DCMPLX(R,I)
C+

C
C

Z is a complex number (r,i) with real part "r" and imaginary
part "i".
TYPE*· ' The complex number z is' ,z
TYPE *,
CALL MTH$CDLOG(Z_NEW,Z)
TYPE*,' The complex logarithm of',z,' is' ,Z_NEW
END

This FORTRAN example program uses MTH$CDLOG by calling it as a
procedure. The output generated by this program is as follows:
The complex number z is (3.142563784674657,7.436784698870000)
The complex logarithm of (3.142563784674657,7.436784698870000) is
(2.088587642177504,1.170985519274141)

MTH-39

MTH$CMPLX

MTH$CMPLX Complex Number Made from
F-floating-Point
The Complex Number Made from F-floating-Point routine returns a
complex number from two floating-point input values.

FORMAT

MTH$CMPLX real-part ,imaginary-part

RETURNS

VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
write only
by value

A complex number. MTH$CMPLX returns an F-floating complex number.

ARGUMENTS

real-part
VMS usage:
type:
access:
mechanism:

floating_point
f _floating
read only
by reference

Real part of a complex number. The real-part argument is the address
of a floating-point number that contains this real part, r, of (r,i). For
MTH$CMPLX, real-part specifies an F-floating number.

imaginary-part
VMS usage:
type:
access:
mechanism:

floating_point
f _floating
read only
by reference

Imaginary part of a complex number. The imag-parg argument is the address
of a floating-point number that contains this imaginary part, i, of (r,i). For
MTH$CMPLX, imaginary-part specifies an F-floating number.

DESCRIPTION

MTH-40

The MTH$CMPLX routines return a complex number from two F-floating
input values. The routine descriptions for the D- and G-floating point
versions of this routine are listed alphabetically under MTH$xCMPLX.

MTH$CMPUC

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CMPLX routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

EXAMPLE
C+

C
C
C

This FORTRAN example forms two F-f loating
point complex numbers using MTH$CMPLX
and the FORTRAN random number generator RAN.

C
C
C

Declare Z and MTH$CMPLX as complex values, and R
and I as real values. MTH$CMPLX takes two real
F-floating point values and returns one COMPLEX*8 number.

C
C
C

Note, since CMPLX is a generic name in FORTRAN, it would be
sufficient to use CMPLX.
CMPLX must be declare to be of type COMPLEX*8.

Z = CMPLX(R,I)

COMPLEX Z,MTH$CMPLX,CMPLX
REAL*4 R,I
INTEGER M
M = 1234567
R = RAN(M)
I = RAN(M)
Z = MTH$CMPLX(R,I)
C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE *· ' The two input values are:' ,R,I
TYPE*· ' The complex number z is' ,z
z = CMPLX(RAN(M),RAN(M))
TYPE *·
TYPE*· 'Using the FORTRAN generic CMPLX with random Rand I:'
TYPE*· 'The complex number z is',z
END

This FORTRAN example program demonstrates the use of MTH$CMPLX.
The output generated by this program is as follows:
The two input values are: 0.8535407
0.2043402
The complex number z is (0.8535407,0.2043402)
Using the FORTRAN generic CMPLX with random Rand I:
The complex number z is (0.5722565,0.1857677)

MTH-41

MTH$xCMPUC

MTH$xCMPLX

Complex Number Made from Dor G-floating-Point

The Complex Number Made from D- or G-floating-Point routine returns a
complex number from two D- or G-floating input values.

FORMAT

MTH$DCMPLX complx ,real-part ,imaginary-part
MTH$GCMPLX complx ,real-part ,imaginary-part
Each of the above formats accepts as input one of floating-point complex
types.

RETURNS
ARGUMENTS

None.

compix
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
write only
by reference

The floating-point complex value of a complex number. The complex
exponential functions that have D-floating complex and G-floating complex
input values write the address of this floating-point complex value into
complx. For MTH$DCMPLX, complx specifies a D-floating complex number.
For MTH$GCMPLX, complx specifies a G-floating complex number. For
MTH$CMPLX, complx is not used.

real-part
VMS usage:
type:
access:
mechanism:

floating_point
D_floating, G_floating
read only
by reference

Real part of a complex number. The real-part argument is the address
of a floating-point number that contains this real part, r, of (r,i).
For MTH$DCMPLX, real-part specifies a D-floating number. For
MTH$GCMPLX, real-part specifies a G-floating number.

imaginary-part
VMS usage:
type:
access:
mechanism:

floating_point
D_floating, G_floating
read only
by reference

Imaginary part of a complex number. The imag-parg argument is the
address of a floating-point number that contains this imaginary part, i, of
(r,i). For MTH$DCMPLX, imaginary-part specifies a D-floating number. For
MTH$GCMPLX, imaginary-part specifies a G-floating number.

MTH-42

MTH$xCMPLX

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xCMPLX routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

EXAMPLE
C+

C
C
C

This FORTRAN example forms two D-f loating
point complex numbers using MTH$CMPLX
and the FORTRAN random number generator RAN.

C
C
C
C

Declare Z and MTH$DCMPLX as complex values, and R
and I as real values. MTH$DCMPLX takes two real
D-f loating point values and returns one
COMPLEX*16 number.

c
c-

COMPLEX*16 Z
REAL*8 R,I
INTEGER M
M = 1234567
R = RAN(M)
I = RAN(M)
CALL MTH$DCMPLX(Z,R,I)
C+

C
C

Z is a complex number (r,i) with real part "r" and imaginary
part "i".
TYPE*• 'The two input values are:' ,R,I
TYPE *· ' The complex number z is' ,Z
END

This FORTRAN example demonstrates how to make a procedure call
to MTH$DCMPLX. Notice the difference in the precision of the output
generated.
The two input values are: 0.8535407185554504
0.2043401598930359
The complex number z is (0.8535407185554504,0.2043401598930359)

MTH-43

MTH$CONJG

Conjugate of a Complex Number
(F-floating Value)

The Conjugate of a Complex Number (F-floating Value) routine returns the
complex conjugate (r,-i) of a complex number (r,i) as an F-floating value.

FORMAT

MTH$CONJG

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
f _floating complex
write only
by value

Complex conjugate of a complex number. MTH$CONJG returns an F-floating
complex number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
read only
by reference

A complex number (r,i), where r and i are floating-point numbers. The
complex-number argument is the address of this floating-point complex
number. For MTH$CONJG, complex-number specifies an F-floating number.

DESCRIPTION

CONDITION
VALUE
SIGNALED

MTH-44

The MTH$CONJG routine return the complex conjugate (r,-i) of a complex
number (r,i) as an F-floating value. The routine descriptions for the Dand G-floating point versions of this routine are listed alphabetically under
MTH$xCONJG.

SS$_ROPRAND

Reserved operand. The MTH$CONJG routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xCONJG

Conjugate of a Complex Number

The Conjugate of a Complex Number routine returns the complex
conjugate (r,-i) of a complex number (r,i).

FORMAT

MTH$DCONJG
MTH$GCONJG

complex-conjugate ,complex-number
complex-conjugate ,complex-number

Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

comp/ex-conjugate
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
write only
by reference

The complex conjugate (r,-i) of the complex number specified by complexnumber. MTH$DCONJG and MTH$GCONJG write the address of this
complex conjugate into complex-conjugate. For MTH$DCONJG, the
complex-conjugate argument specifies the address of a D-floating complex
number. For MTH$GCONJG, the complex-conjugate argument specifies the
address of a G-floating complex number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
read only
by reference

A complex number (r,i), where r and i are floating-point numbers. The
complex-number argument is the address of this floating-point complex
number. For MTH$DCONJG, complex-number specifies a D-floating
number. For MTH$GCONJG, complex-number specifies a G-floating
number.

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xCONJG routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-45

MTH$xCONJG

EXAMPLE
C+

C
C
C

This FORTRAN example forms the complex conjugate
of a G-f loating complex number using MTH$GCONJG
and the FORTRAN random number generator RAN.

C
C
C

Declare Z, Z_NEW, and MTH$GCONJG as a complex values.
MTH$GCONJG will return the complex conjugate
value of Z:
Z_NEW = MTH$GCONJG(Z).

COMPLEX*16 Z,Z_NEW,MTH$GCONJG
COMPLEX*16 MTH$GCMPLX
REAL*8 R,I,MTH$GREAL,MTH$GIMAG
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic CMPLX.

R = RAN(M)
I = RAN(M)
Z = MTH$GCMPLX(R,I)
TYPE*· 'The complex number z is',z
TYPE 1,MTH$GREAL(Z) ,MTH$GIMAG(Z)
FORMAT(' with real part ',F20.16,' and imaginary part',F20.16)
TYPE *, ' '

C+

Compute the complex absolute value of Z.
Z_NEW = MTH$GCONJG(Z)
TYPE*· 'The complex conjugate value of' ,z,' is',Z_NEW
TYPE 1,MTH$GREAL(Z_NEW),MTH$GIMAG(Z_NEW)
END

This FORTRAN example demonstrates how to make a function call to
MTH$GCONJG. Because G-floating numbers are used, the examples must be
compiled with the statement "FORTRAN/G filename".
The output generated by this program is as follows:
The complex number z is (0.853540718555450,0.204340159893036)
with real part
0.8535407185554504
and imaginary part 0.2043401598930359
The complex conjugate value of
(0.853540718555450,0.204340159893036) is
(0.853540718555450,-0.204340159893036)
with real part
0.8535407185554504
and imaginary part -0.2043401598930359

MTH-46

MTH$xCOS

Cosine of Angle Expressed in
Radians
The Cosine of Angle Expressed in Radians routine returns the cosine of a
given angle (in radians).

FORMAT

MTH$COS angle-in-radians
MTH$DCOS angle-in-radians
MTH$GCOS angle-in-radians
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$COS_R4
MTH$DCOS_R7
MTH$GCOS_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Cosine of the angle. MTH$COS returns an F-floating number. MTH$DCOS
returns a D-floating number. MTH$GCOS returns a G-floating number.

ARGUMENTS

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The angle in radians. The angle-in-radians argument is the address of a
floating-point number. For MTH$COS, angle-in-radians is an F-floating
number. For MTH$DCOS, angle-in-radians specifies a D-floating number.
For MTH$GCOS, angle-in-radians specifies a G-floating number.

DESCRIPTION

See the MTH$xSINCOS routine for the algorithm used to compute the cosine.
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HCOS.

MTH-47

MTH$xCOS

CONDITION
VALUE
SIGNALED

MTH-48

SS$_ROPRAND

Reserved operand. The MTH$xCOS procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xCOSD

Cosine of Angle Expressed in
Degrees

The Cosine of Angle Expressed in Degrees routine returns the cosine of a
given angle (in degrees).

FORMAT

MTH$COSD angle-in-degrees
MTH$DCOSD angle-in-degrees
MTH$GCOSD angle-in-degrees
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$COSD_R4
MTH$DCOSD_R7
MTH$GCOSD_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Cosine of the angle. MTH$COSD returns an F-floating number.
MTH$DCOSD returns a D-floating number. MTH$GCOSD returns a Gfloating number.

ARGUMENTS

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Angle (in degrees). The angle-in-degrees argument is the address of a
floating-point number. For MTH$COSD, angle-in-degrees specifies an Ffloating number. For MTH$DCOSD, angle-in-degrees specifies a D-floating
number. For MTH$GCOSD, angle-in-degrees specifies a G-floating number.

DESCRIPTION

See the MTH$SINCOSD routine for the algorithm used to compute the
cosine.
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HCOSD.

MTH-49

MTH$xCOSD

CONDITION
VALUE
SIGNALED

MTH-50

SS$_ROPRAND

Reserved operand. The MTH$xCOSD procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xCOSH

Hyperbolic Cosine

The Hyperbolic Cosine routine returns the hyperbolic cosine of the input
value.

FORMAT

MTH$COSH floating-point-input-value
MTH$DCOSH floating-point-input-value
MTH$GCOSH floating-point-input-value
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The hyperbolic cosine of the input value floating-point-input-value.
MTH$COSH returns an F-floating number. MTH$DCOSH returns a Dfloating number. MTH$GCOSH returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
this input value. For MTH$COSH, floating-point-input-value specifies an
F-floating number. For MTH$DCOSH, floating-point-input-value specifies a
D-floating number. For MTH$GCOSH, floating-point-input-value specifies
a G-floating number.

DESCRIPTION

Computation of the hyperbolic cosine depends on the magnitude of the
input argument. The range of the function is partitioned using four datatype-dependent constants: a(z), b(z), and c(z). The subscript z indicates the
data type. The constants depend on the number of exponent bits (e) and the
number of fraction bits (f) associated with the data type (z).
The values of e and f are:
z

F
D

MTH-51

MTH$xCOSH

The values of the constants in terms of e and f are:
Variable

Value

a(z)

2(-J/2)

b(z)

CEILING[ (! + 1)/2 * ln(2)]

c(z)

(2e- 1 ) * ln(2)

Based on the above definitions, zCOSH(X) is computed as follows:
Value of X
IXI

Value Returned

< a(z)

s; IXI < .25
.25 s; IXI < b(z)
b(z) s; IXI < c(z)
c(z) s; lxl

a(z)

Computed using a power series expansion in IXl 2

(zEXP(IXI) + l/zEXP(IXl))/2
zEXP(IXl)/2
Overflow occurs

This routine description for the H-floating point value is listed alphabetically
under MTH$HCOSH.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xCOSH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of floating-point-input-value is
greater than about yyy; LIB$SIGNAL copies the
reserved operand to the signal mechanism vector.
The result is the reserved operand -0.0 unless a
condition handler changes the signal mechanism
vector.
The values of yyy are:
MTH$COSH-88. 722
MTH$DCOSH-88. 722
MTH$GCOSH-709. 782

MTH-52

MTH$CSIN

Sine of a Complex Number
(F-floating Value)
The Sine of a Complex Number (F-floating Value) routine returns the sine
of a complex number (r,i) as an F-floating value.

FORMAT

MTH$CSIN

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
f _floating complex
write only
by value

Complex sine of the complex number. MTH$CSIN returns an F-floating
complex number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
read only
by reference

A complex number (r,i), where r and i are floating-point numbers. The
complex-number argument is the address of this complex number. For
MTH$CSIN, complex-number specifies an F-floating complex number.

DESCRIPTION

The complex sine is computed as follows:
complex- sine= (SIN(r) * COSH(i),COS(r) * SINH(i))

The routine descriptions for the D- and G-floating point versions of this
routine are listed alphabetically under MTH$CxSIN.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CSIN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Fioating-point overflow in Math Library: the
absolute value of i is greater than about 88.029
for F-floating values.

MTH-53

MTH$CxSIN

Sine of a Complex Number
The Sine of a Complex Number routine returns the sine of a complex
number (r,i).

FORMAT

MTH$CDSIN
MTH$CGSIN

complex-sine ,complex-number
complex-sine , complex-number

Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

complex-sine
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
write only
by reference

Complex sine of the complex number. The complex sine routines with Dfloating complex and G-floating complex input values write the complex sine
into this complex-sine argument. For MTH$CDSIN, complex-sine specifies
a D-floating complex number. For MTH$CGSIN, complex-sine specifies a
G-floating complex number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
read only
by reference

A complex number (r,i), where r and i are floating-point numbers. The
complex-number argument is the address of this complex number. For
MTH$CDSIN, complex-number specifies a D-floating complex number. For
MTH$CGSIN, complex-number specifies a G-floating complex number.

DESCRIPTION

The complex sine is computed as follows:
complex - sine= (SIN(r) * COSH(i), COS(r) * SINH(i))

MTH-54

MTH$CxSIN

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CxSIN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of i is greater than about 88.029
for D-floating values or greater than about 709.089
for G-floating values.

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the complex
sine of a G-f loating complex number using
MTH$CGSIN and the FORTRAN random number
generator RAN.

C
C
C

Declare Z and MTH$CGSIN as complex values.
MTH$CGSIN will return the sine value
of Z:
CALL MTH$CGSIN(Z_NEW,Z)

cCOMPLEX*16 Z,Z_NEW
COMPLEX*16 DCMPLX
REAL*8 R,I
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic DCMPLX.
R = RAN(M)
I = RAN(M)
Z = DCMPLX(R,I)

C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE *· ' The complex number z is' ,z
TYPE *·

C+

Compute the complex sine value of Z.

cCALL MTH$CGSIN(Z_NEW,Z)
TYPE *· ' The complex sine value of' ,z,' is' ,Z_NEW
END

MTH-55

MTH$CxSIN

This FORTRAN example demonstrates a procedure call to MTH$CGSIN.
Because this program uses G-floating numbers, it must be compiled with the
statement "FORTRAN/G filename".
The output generated by this program is as follows:
The complex number z is (0.853540718555450,0.204340159893036)
The complex sine value of (0.853540718555450,0.204340159893036) is
(0.769400835484975,0.135253340912255)

MTH-56

MTH$CSQRT

Complex Square Root (F-floating
Value)

The Complex Square Root (F-floating Value) routine returns the complex
square root of a complex number (r,i).

FORMAT

MTH$CSQRT

RETURNS

VMS usage:
type:
access:
mechanism:

complex-number

complex_number
f _floating complex
write only
by value

The complex square root of complex-number. MTH$CSQRT returns an
F-floating number.

ARGUMENTS

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
F_floating complex
read only
by reference

Complex number (r,i). The complex-number argument contains the address
of this complex number. For MTH$CSQRT, complex-number specifies an
F-floating number.

DESCRIPTION

The complex square root is computed as follows.
First, calculate ROOT and Q using the following equations:

ROOT= SQRT((ABS(r) + (CABS(r, i))/2)

Q = i/(2 *ROOT)
Then, the complex result is given as follows:
CSQRT((r,i))

20
<O
<O

Any

(ROOT,Q)

20
<O

(Q,ROOT)
(-Q,-ROOT)

The routine descriptions for the D- and G-floating point versions of this
routine are listed alphabetically under MTH$CxSQRT.

MTH-57

MTH$CSQRT

CONDITION
VALUE
SIGNALED

MTH-58

SS$_RQPRAND

Reserved operand. The MTH$CSQRT procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$CxSQRT

Complex Square Root

The Complex Square Root routine returns the complex square root of a
complex number (r ,i).

FORMAT

MTH$CDSQRT complex-square-root ,complex-number
MTH$CGSQRT complex-square-root ,complex-number
Each of the above formats accepts as input one of the floating-point complex
types.

RETURNS
ARGUMENTS

None.

complex-square-root
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
write only
by reference

Complex square root of the complex number specified by complex-number.
The complex square root routines that have D-floating complex and Gfloating complex input values write the complex square root into complexsquare-root. For MTH$CDSQRT, complex-square-root specifies a D-floating
complex number. For MTH$CGSQRT, complex-square-root specifies a
G-floating complex number.

complex-number
VMS usage:
type:
access:
mechanism:

complex_number
D_floating complex, G_floating complex
read only
by reference

Complex number (r,i). The complex-number argument contains the address
of this complex number. For MTH$CDSQRT, complex-number specifies a Dfloating number. For MTH$CGSQRT, complex-number specifies a G-floating
number.

DESCRIPTION

The complex square root is computed as follows.
First, calculate ROOT and Q using the following equations:
ROOT= SQRT((ABS(r) + (CABS(r,i))/2)

Q = i/(2 *ROOT)

MTH-59

MTH$CxSQRT

Then, the complex result is given as follows:
CSQRT((r,i))

2::0
<O
<O

CONDITION
VALUE
SIGNALED

any

(ROOT,Q)

2::0
<O

(Q,ROOT)

SS$_ROPRAND

(-0,-ROOT)

Reserved operand. The MTH$CxSQRT procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the complex square
root of a D-f loating complex number using
MTH$CDSQRT and the FORTRAN random number
generator RAN.

C
C
C

Declare Z and Z_NEW as complex values. MTH$CDSQRT
will return the complex square root of
Z:
CALL MTH$CDSQRT(Z_NEW,Z).

COMPLEX*16 Z,Z_NEW
COMPLEX*16 DCMPLX
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic CMPLX.
Z = DCMPLX(RAN(M),RAN(M))

C+

C
C

Z is a complex number (r,i) with real part "r" and imaginary
part "i".
TYPE *· ' The complex number z is' ,z
TYPE*·

C+

Compute the complex complex square root of Z.
CALL MTH$CDSQRT(Z_NEW,Z)
TYPE*· 'The complex square root of',z,' is' ,Z_NEW
END

MTH-60

MTH$CxSQRT

This FORTRAN example program demonstrates a procedure call to
MTH$CDSQRT. The output generated by this program is as follows:
The complex number z is (0.8535407185554504,0.2043401598930359)
The complex square root of (0.8535407185554504,0.2043401598930359) is
(0.9303763973040062,0.1098158554350485)

MTH-61

MTH$CVT_x_x

MTH$CVT_x_x Convert One Double-Precision
Value
The Convert One Double-Precision Value routines convert one doubleprecision value to the destination data type and return the result as a
function value. MTH$CVT_D_G converts a D-floating value to G-floating
and MTH$CVT_G_D converts a G-floating value to a D-floating value.

FORMAT

MTH$CVT_D_G floating-point-input-val
MTH$CVT_G_D floating-point-input-val

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
G_floating, D_floating
write only
by value

The converted value. MTH$CVT_D_G returns a G-floating value.
MTH$CVT_G_D returns a D-floating value.

ARGUMENT

floating-point-input-val
VMS usage:
type:
access:
mechanism:

floating_point
D_floating, G_floating
read only
by reference

The input value to be converted. The floating-point-input-val argument is
the address of this input value. For MTH$CVT_D_G, the floating-pointinput-val argument specifies a D-floating number. For MTH$CVT_G_D, the
floating-point-input-val argument specifies a G-floating number.

DESCRIPTION

MTH-62

These procedures are designed to function as hardware conversion
instructions. They fault on reserved operands. If floating-point overflow
is detected, an error is signaled. If floating-point underflow is detected and
floating-point underflow is enabled, an error is signaled.

MTH$CVT_x_x

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CVT_x_x procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library.

MTH$_FLOUNDMA T

Floating-point underflow in Math Library.

MTH-63

MTH$CVT_xA_xA

MTH$CVT_xA_xA
Convert an Array of Double-Precision Values
The Convert an Array of Double-Precision Values routines convert a
contiguous array of double-precision values to the destination data type
and return the results as an array. MTH$CVT_OA_GA converts D-floating
values to G-floating and MTH$CVT_GA_DA converts G-floating values to
D-floating.

FORMAT

MTH$CVT_DA_GA

RETURNS

MTH$CVT_DA_GA and MTH$CVT_GA_DA return the address of the
output array to the floating-point-dest-array argument.

ARGUMENTS

floating-point-input-array
, floating-point-dest-array
{,array-size]
MTH$CVT_GA_DA floating-point-input-array
, floating-point-dest-array
{,array-size]

floating-point-input-array
VMS usage:
type:
access:
mechanism:

floating_point
D_floating, G_floating
read only
by reference, array reference

Input array of values to be converted. The floating-point-input-array
argument is the address of an array of floating-point numbers. For
MTH$CVT_DA_GA, floating-point-input-array specifies an array of Dfloating numbers. For MTH$CVT_GA_DA, floating-point-input-array
specifies an array of a G-floating numbers.

floating-point-dest-array
VMS usage:
type:
access:
mechanism:

floating_point
G_floating, D_floating
write only
by reference, array reference

Output array of converted values. The floating-point-dest-array argument
is the address of an array of floating-point numbers. For MTH$CVT_DA_
GA, floating-point-dest-array specifies an array of G-floating numbers.
For MTH$CVT_GA_DA, floating-point-dest-array specifies an array of
D-floating numbers.

MTH-64

MTH$CVT_xA_xA

array-size
VMS usage: longword_signed
longword (signed)
access:
read only
mechanism: by reference

type:

Number of array elements to be converted. The default value is 1. The
array-size argument is the address of a longword containing this number of
elements.

DESCRIPTION

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$CVT_xA _xA
procedure encountered a floating-point reserved
operand due to incorrect user input. A floatingpoint reserved operand is a floating-point datum
with a sign bit of 1 and a biased exponent of zero.
Floating-point reserved operands are reserved for
future use by DIGIT AL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library.

MTH$_FLOUNDMA T

Floating-point underflow in Math Library.

MTH-65

MTH$xEXP

Exponential
The Exponential routine returns the exponential of the input value.

FORMAT

MTH$EXP floating-point-input-value
MTH$DEXP floating-point-input-value
MTH$GEXP floating-point-input-value
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$EXP_R4
MTH$DEXP_R6
MTH$GEXP_R6
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The exponential of floating-point-input-value. MTH$EXP returns an Ffloating number. MTH$DEXP returns a D-floating number. MTH$GEXP
returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
a floating-point number. For MTH$EXP, floating-point-input-value specifies
an F-floating number. For MTH$DEXP, floating-point-input-value specifies
a D-floating number. For MTH$GEXP, floating-point-input-value specifies a
G-floating number.

MTH-66

MTH$xEXP

DESCRIPTION

The exponential of xis computed as:
Value of x

Value Returned

X > c(z)
x:::; - c(z)
IXI < ru+1)

Overflow occurs

Otherwise

where:
Y = INTEGER(x * ln2(E))
V = FRAC(x * ln2(E)) * 16
U = INTEGER(V)/16
W = FRAC(V)/16
2W =polynomial approximation of degree 4,8, or 8 for z = F, D, or G.

See also the section on the hyperbolic cosine for definitions off and c(z).
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HEXP.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xEXP routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: floatingpoint-input-value is greater than yyy; LIB$SIGNAL
copies the reserved operand to the signal
mechanism vector. The result is the reserved
operand -0.0 unless a condition handler changes
the signal mechanism vector.
The values of yyy are approximately:
MTH$EXP-88.029
MTH$DEXP-88.029
MTH$GEXP-709.089

MTH-67

MTH$xEXP

MTH$_FLOUNDMA T

Floating-point underflow in Math Library: floatingpoint-input-value is less than or equal to yyy
and the caller (CALL or JSB) has set hardware
floating-point underflow enable. The result is set
to 0.0. If the caller has not enabled floating-point
underflow (the default), a result of 0.0 is returned
but no error is signaled.
The values of yyy are approximately:
MTH$EXP- -88. 722
MTH$DEXP- -88. 722
MTH$GEXP- -709.774

EXAMPLE
IDENTIFICATION DIVISION.
PROGRAM-ID.
FLOATING_POINT.

** Calls MTH$EXP using a Floating Point data type.
* Calls MTH$DEXP using a Double Floating Point data type.
*
ENVIRONMENT DIVISION.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 FLOAT_PT
COMP-1.
01 ANSWER_F
COMP-1.
01 DOUBLE_PT
COMP-2.
01 ANSWER_D
COMP-2.
PROCEDURE DIVISION.
PO.
MOVE 12.34 TO FLOAT_PT.
MOVE 3.456 TO DOUBLE_PT.
CALL 11 MTH$EXP 11 USING BY REFERENCE FLOAT_PT GIVING ANSWER_F.
DISPLAY" MTH$EXP of 11 , FLOAT_PT CONVERSION, 11 is 11 ,
ANSWER_F CONVERSION.
CALL 11 MTH$DEXP 11 USING BY REFERENCE DOUBLE_PT GIVING ANSWER_D.
DISPLAY 11 MTH$DEXP of 11 , DOUBLE_PT CONVERSION, 11 is 11 ,
ANSWER_D CONVERSION .
STOP RUN.

This sample program demonstrates calls to MTH$EXP and MTH$DEXP from
COBOL.

The output generated by this program is as follows:
MTH$EXP of 1.234000E+01 is 2.286620E+05
MTH$DEXP of 3.456000000000000E+OO is
3.168996280537917E+01

MTH-68

MTH$HACOS

Arc Cosine of Angle Expressed in
Radians (H-floating Value)

Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Radians (H-floating Value) routine returns that angle (in radians) in Hfloating-point precision.

FORMAT

MTH$HACOS

jsb entries

MTH$HACOS_R8

RETURNS

None.

ARGUMENTS

h-radians ,cosine

h-radians
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Angle (in radians) whose cosine is specified by cosine. The h-radians
argument is the address of an H-floating number that is this angle. ·
MTH$HACOS writes the address of the angle into h-radians.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

MTH-69

MTH$HACOS

DESCRIPTION

The angle in radians whose cosine is Xis computed as:
Value of
Cosine

Value Returned

7r/2

-1

O<X< 1

zAT AN(zSQRT(l - X 2 )/ X), where zAT AN and zSQRT are the
Math Library arc tangent and square root routines, respectively,
of the appropriate data type

CONDITION
VALUES
SIGNALED

MTH-70

-1<x<0

zAT AN(zSQRT(l - X 2 )/ X) + 7r

1 < IXI

The error MTH$_1NV ARGMA T is signaled

SS$_ROPRAND

MTH$_1NV ARGMA T

Invalid argument. The absolute value of cosine
is greater than 1. LIB$SIGNAL copies the
floating-point reserv~d operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L _MCH_SAVRO/R 1.

MTH$HACOSD

Arc Cosine of Angle Expressed in
Degrees {H-Floating Value)

Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Degrees (H-Floating Value) routine returns that angle (in degrees) as an
H-floating value.

FORMAT

MTH$HACOSD

jsb entries

MTH$HACOSD_R8

RETURNS

None.

ARGUMENTS

h-degrees ,cosine

h-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Angle (in degrees) whose cosine is specified by cosine. The h-degrees
argument is the address of an H-floating number that is this angle.
MTH$HACOSD writes the address of the angle into h-degrees.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Cosine of the angle whose value (in degrees) is to be returned. The cosine
argument is the address of a floating-point number that is this cosine. The
absolute value of cosine must be less than or equal to 1. For MTH$HACOSD,
cosine specifies an H-floating number.

MTH-71

MTH$HACOSD

DESCRIPTION

The angle in degrees whose cosine is Xis computed as:
Value of
Cosine

Angle Returned

-1

180

0<X<1

zATAND(zSQRT(l - X 2 )/X), where zATAND and zSORT
are the Math Library arc tangent and square root routines,
respectively, of the appropriate data type

CONDITION
VALUES
SIGNALED

MTH-72

-1<x<0

zAT AN D(zSQRT(l - X 2 )/ X) + 180

1 < IXI

The error MTH$_1NV ARGMA T is signaled

SS$_ROPRAND

MTH$_1NV ARGMA T

Invalid argument. The absolute value of cosine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

MTH$HASIN

Arc Sine in Radians (H-floating
Value)

Given the sine of an angle, the Arc Sine in Radians (H-floating Value)
routine returns that angle (in radians) as an H-floating value.

FORMAT

MTH$HASIN

jsb entries

MTH$HASIN_R8

RETURNS

None.

ARGUMENTS

h-radians ,sine

h-radians
VMS usage: floating_point
type:
H_floating
access:
write only
mechanism: by reference

Angle (in radians) whose sine is specified by sine. The h-radians argument is
the address of an H-floating number that is this angle. MTH$HASIN writes
the address of the angle into h-radians.

sine
VMS usage: floating_point
type:
H_floating
access:
read only
mechanism: by reference

MTH-73

MTH$HASIN

DESCRIPTION

The angle in radians whose sine is X is computed as:
Value of
Sine

Angle Returned

7r/2

-1

-7r/2

0 < IXI < 1

zATAN(X/zSQRT(l - X 2 )), where zATAN and zSQRT are the
Math Library arc tangent and square root routines, respectively,
of the appropriate data type

1 < IXI

CONDITION
VALUES
SIGNALED

MTH-74

The error MTH$_INV ARGMA T is signaled

SS$_ROPRAND

MTH$_1NV ARGMA T

MTH$HASIND

Arc Sine in Degrees (H-Floating
Value)

Given the sine of an angle, the Arc Sine in Degrees (H-Floating Value)
routine returns that angle (in degrees) as an H-floating value.

FORMAT

MTH$HASIND

jsb entries

MTH$HASIND_R8

RETURNS

None.

ARGUMENTS

h-degrees ,sine

h-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H _floating
write only
by reference

Angle (in degrees) whose sine is specified by sine. The h-degrees argument
is the address of an H-floating number that is this angle. MTH$HASIND
writes the address of the angle into h-degrees.

sine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

MTH-75

MTH$HASIND

DESCRIPTION

The angle in degrees whose sine is Xis computed as:
Value of
Sine

Value Returned

-1

-90

0 < IXI < 1

zATAND(X/zSQRT(l - X 2 )), where zATAND and zSQRT
are the Math Library arc tangent and square root routines,
respectively, of the appropriate data type

1 < IXI

CONDITION
VALUES
SIGNALED

MTH-76

The error MTH$_1NV ARGMA T is signaled

SS$_ROPRAND

MTH$_1NV ARGMA T

Invalid argument. The absolute value of sine
is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R 1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.

MTH$HATAN

Arc Tangent in Radians (H-floating
Value)

Given the tangent of an angle, the Arc Tangent in Radians (H-floating
Value) routine returns that angle (in radians) as an H-floating value.

FORMAT

MTH$HATAN

jsb entries

MTH$HATAN_R8

RETURNS

None.

ARGUMENTS

h-radians ,tangent

h-radians
VMS usage:· floating_point
type:
H_floating
access:
write only
mechanism: by reference

Angle (in radians) whose tangent is specified by tangent. The h-radians
argument is the address of an H-floating number that is this angle.
MTH$HATAN writes the address of the angle into h-radians.

tangent
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The tangent of the angle whose value (in radians) is to be returned. The
tangent argument is the address of a floating-point number that is this
tangent. For MTH$HATAN, tangent specifies an H-floating number.

DESCRIPTION

In radians, the computation of the arc tangent function is based on the
following identities:
arctan(X) = X - X 3 /3 + X 5 /5 - X 7 /7 + ...
arctan(X) = X + X * Q(X2 ),

where Q(Y) = -Y/3 + Y 2 /5 - Y 3 /7 + ...

arctan(X) = X * P(X 2 ),

where P(Y) = 1 - Y /3 + Y 2 /5 - Y 3 /7 + ...

arctan(X) = 7r/2 - arctan(l/X)
arctan(X) = arctan(A) + arctan((X - A)/(1 +A* X))
for any real A

MTH-77

MTH$HATAN

The angle in radians whose tangent is X is computed as:

CONDITION
VALUE
SIGNALED

MTH-78

Value of X

Angle Returned

o::;X::;3/32

X+X*Q(X 2 )

3/32 < X::;ll

ATAN(A) + V * (P(V 2 )), where A and ATAN(A) are
chosen by table lookup and V = (X - A)/(1 +A* X)

ll<X

7r/2 - W * (P(W 2 )) where W = 1/X

X<O

-zAT AN(IXI)

SS$_ROPRAND

Reserved operand. The MTH$xAT AN routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$HATAND

Arc Tangent in Degrees
(H-floating Value)

Given the tangent of an angle, the Arc Tangent in Degrees (H-floating
Value) routine returns that angle (in degrees) as an H-floating point value.

FORMAT

MTH$HATAND

jsb entries

MTH$HATAND_R8

RETURNS

None.

ARGUMENTS

h-degrees ,tangent

h-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Angle (in degrees) whose tangent is specified by tangent. The h-degrees
argument is the address of an H-floating number that is this angle.
MTH$HATAND writes the address of the angle into h-degrees.

tangent
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The tangent of the angle whose value (in degrees) is to be returned. The
tangent argument is the address of a floating-point number that is this
tangent. For MTH$HATAND, tangent specifies an H-floating number.

DESCRIPTION

The computation of the arc tangent function is based on the following
identities:
arctan(X) = 180/11" * (X - X 3 /3 + X 5 /5 - X 7 /7 + ... )
arctan(X) = 64 * X + X * Q(X 2 ),
where Q(Y) = 180/rr * [(1-64 * 7r/180) - Y/3+
y2 /5- y3 /7 + y4 /9 ... ]
arctan(X) = X * P(X 2 ),
where P(Y) = 180/rr * [1 - Y /3 + Y 2 /5 - y 3 /7+
y4/9 ... ]
arctan(X) = 90 - arctan(l/ X)
arctan(X) = arctan(A) + arctan{{X - A)/{1 +A* X))

MTH-79

MTH$HATAND

The angle in degrees whose tangent is X is computed as:

CONDITION
VALUE
SIGNALED

MTH-80

Tangent

Angle Returned

X9/32

64 * X + X * Q(X )

3/32 < X::;ll

ATAND(A) + V * P(V 2 ), where A and ATAND(A) are
chosen by table lookup and V = (X - A)/(l +A* X)

ll<X

90- W * (P(W 2 )), where W = 1/X

X<O

-zAT AN D(IXI)

SS$_ROPRAND

Reserved operand. The MTH$xAT AND routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$HATAN2

Arc Tangent in Radians
(H-floating Value) with Two
Arguments

Given sine and cosine, the Arc Tangent in Radians (H-floating Value) with
Two Arguments routine returns the angle (in radians) as an H-floating value
whose tangent is given by the quotient of sine and cosine, (sine/cosine).

FORMAT

MTH$HATAN2

RETURNS

None.

ARGUMENTS

h-radians ,sine ,cosine

h-radians
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Angle (in radians) whose tangent is specified by (sine/cosine). The hradians argument is the address of an H-floating number that is this angle.
MTH$HATAN2 writes the address of the angle into h-radians.

sine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Dividend. The sine argument is the address of a floating-point number that
is this dividend. For MTH$HATAN2, sine specifies an H-floating number.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Divisor. The cosine argument is the address of a floating-point number that
is this divisor. For MTH$HATAN2, cosine specifies an H-floating number.

MTH-81

MTH$HATAN2

DESCRIPTION

CONDITION
VALUES
SIGNALED

MTH-82

The angle in radians whose tangent is Y /X is computed as follows, where f is
defined in the description of MTH$zCOSH.
Value of Input Arguments

Angle Returned

X = 0 or Y/X > 2U+ 1I

/2 * (signY)

X > 0 and Y/X~ 2U+ 11

zATAN(Y/X)

X < 0 and Y/X~ 2(/+ 11

* (signY) + zATAN(Y/X)

SS$_ROPRAND

Reserved operand. The MTH$HA T AN2 routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. Both cosine and sine are zero.
LIB$SIGNAL copies the floating-point reserved
operand to the mechanism argument vector
CHF$L_MCH_SAVRO/R1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH_SA VRO /R 1.

MTH$HATAND2

Arc Tangent in Degrees
(H-floating Value) with Two
Arguments

Given sine and cosine, MTH$xHT AND2 returns the angle (in degrees)
whose tangent is given by the quotient of sine and cosine, (sine/cosine).

FORMAT

MTH$HATAND2

RETURNS

None.

ARGUMENTS

h-degrees ,sine ,cosine

h-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Angle (in degrees) whose tangent is specified by (sine/cosine). The hdegrees argument is the address of an H-floating number that is this angle.
MTH$HATAND2 writes the address of the angle into h-degrees.

sine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Dividend. The sine argument is the address of a floating-point number that
is this dividend. For MTH$HATAND2, sine specifies an H-floating number.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Divisor. The cosine argument is the address of a floating-point number that
is this divisor. For MTH$HATAND2, cosine specifies an H-floating number.

MTH-83

MTH$HATAND2

DESCRIPTION

The angle in degrees whose tangent is Y/Xis computed below. The value of

f is defined in the description of MTH$zCOSH.
Value of Input Arguments

Angle Returned

X = 0 or Y/X > 2(/+ 1)

90 * (signY)

X > 0 and Y/X-5:. 2(J+ l

zAT AN D(Y/ X)

X < 0 and Y/X-5:. 2IJ+ l

180 * (signY) + zAT AN D(Y / X)

CONDITION
VALUES
SIGNALED

MTH-84

SS$_ROPRAND

Reserved operand. The MTH$HA T AND2 routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a .floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_1NV ARGMA T

Invalid argument. Both cosine and sine are zero.
LIB$SIGNAL copies the floating-point reserved
operand to the mechanism argument vector
CHF$L_MCH_SAVRO/R1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L _
MCH_SAVRO/R1.

MTH$HATANH

Hyperbolic Arc Tangent
(H-floating Value)

Given the hyperbolic tangent of an angle, the Hyperbolic Arc Tangent (Hfloating Value) routine returns the hyperbolic arc tangent (as an H..:floating
value) of that angle.

FORMAT

MTH$HATANH

RETURNS

None.

ARGUMENTS

h-atanh ,hyperbolic-tangent

h-atanh
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Hyperbolic arc tangent of the hyperbolic tangent specified by hyperbolictangent. The h-atanh argumentis the address of an H-floating nuthber that
is this hyperbolic arc tangent. MTH$HATANH writes the address of the
hyperbolic arc tangent into h-atanh.

hyperbolic-tangent
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Hyperbolic tangent of an angle. The hyperbolic-tangent argumerit is the
address of a floating-point number that is this hyperbolic tangent, ,For
MTH$HATANH, hyperbolic-tangent specifies an H-floating number.

DESCRIPTION

The hyperbolic arc tangent function is computed as follows:
Value of x

Value Returned

IXI < 1
1x1::::1

zATANH(X) = zLOG((X + 1)/(X -1))/2
An invalid argument is signaled

MTH-85

MTH$HATANH

CONDITION
VALUES
SIGNALED

MTH-86

SS$_ROPRAND

MTH$_1NV ARGMA T

Invalid argument: IX I 2'.: 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R 1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L _MCH _SA VRO /R 1.

MTH$HCOS

Cosine of Angle Expressed in
Radians (H-floating Value)
The Cosine of Angle Expressed in Radians (H-floating Value) routine
returns the cosine of a given angle (in radians) as an H-floating value.

FORMAT

MTH$HCOS

jsb entries

MTH$HCOS_R5

RETURNS

None.

ARGUMENTS

h-cosine ,angle-in-radians

h-cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Cosine of the angle specified by angle-in-radians. The h-cosine argument is
the address of an H-floating number that is this cosine. MTH$HCOS writes
the address of the cosine into h-cosine.

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The angle in radians. The angle-in-radians argument is the address of
a floating-point number. For MTH$HCOS, angle-in-radians specifies an
H-floating number.

DESCRIPTION
CONDITION
VALUE
SIGNALED

See the MTH$xSINCOS routine for the algorithm used to compute the cosine.

SS$_ROPRAND

Reserved operand. The MTH$HCOS procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-87

MTH$HCOSD

Cosine of Angle Expressed in
Degrees (H-floating 'Value)

The Cosine of Angle Expressed in Degrees (H-floating Value) routine
returns the cosine of a given angle (in degrees) as an H-floating value.

FORMAT

MTH$HCOSD

jsb entries

MTH$HCOSD_R5

RETURNS

None.

ARGUMENTS

h-cosine ,angle-in-degrees

h-cosine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Cosine of the angle specified by angle-in-degrees. The h-cosine argument is
the address of an H-floating number that is this cosine. MTH$HCOSD writes
this cosine into h-cosine.

angle-in-degrees
VMS usage:
type:
·
access:
mechanism:

floating_point
H _floating
read only
by reference

Angle (in degrees). The angle-in-degrees argument is the address of a
floating-point number. For MTH$HCOSD, angle-in-degrees specifies an
H-floating number.

DESCRIPTION

CONDITION
VALUE
SIGNALED

MTH-88

See the MTH$SINCOSD routine for the algorithm used to compute the
cosine.

SS$_ROPRAND

Reserved operand. The MTH$HCOSD procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$HCOSH

Hyperbolic Cosine (H-floating
Value)

The Hyperbolic Cosine routine returns the hyperbolic cosine of the input
value as an H-floating value.

FORMAT

MTH$HCOSH

RETURNS

None.

ARGUMENTS

h-cosh ,floating-point-input-value

h-cosh
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Hyperbolic cosine of the input value specified by floating-point-input-value.
The h-cosh argument is the address of an H-floating number that is this
hyperbolic cosine. MTH$HCOSH writes the address of the hyperbolic cosine
into h-cosh.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
this input value. For MTH$HCOSH, floating-point-input-value specifies an
H-floating number.

DESCRIPTION

f = 113

MTH-89

MTH$HCOSH

The values of the constants in terms of e and f are:
Variable

Value

a(z)

rfl2

b(z)

(! + 1)/2 * ln{2)
2e-l * ln{2)

c(z)

Based on the above definitions, zCOSH(X) is computed as follows:
Value of X

Value Returned

IXI < a(z)

CONDITION
VALUES
SIGNALED

MTH-90

a(z)~IXI < .25

Computed using a power series expansion in IXl 2

.25~IXI < b(z)

(zEXP(IXI) + 1/zEXP(IXl))/2

b{z)~IXI < c(z)

zEXP(IXl)/2

c(z)~IXI

Overflow occurs

SS$_ROPRAND

Reserved operand. The MTH$HCOSH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

MTH$HEXP

Exponential (H-floating Value)
The Exponential routine returns the exponential of the input value as an
H-floating value.

FORMAT

MTH$HEXP

jsb entries

MTH$HEXP_R6

RETURNS

None.

ARGUMENTS

h-exp ,floating-point-input-value

h-exp
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Exponential of the input value specified by floating-point-input-value.
The h-exp argument is the address of an H-floating number that is this
exponential. MTH$HEXP writes the address of the exponential into h-exp.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H _floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number. For MTH$HEXP, floating-point-input-value
specifies an H-floating number.

DESCRIPTION

The exponential of xis computed as:
Value of x

Value Returned

x > c(z)
x::; - c(z)

Overflow occurs

lxl < 2-(/+11

Otherwise

2Y * 2U * 2W

where:
Y = INTEGER(x * ln2(E))
V = FRAC(x * ln2(E)) * 16
U = INTEGER(V)/16

MTH-91

MTH$HEXP

W = FRAC(V)/16

2W =polynomial approximation of degree 14 for z = H.

See also the section on the hyperbolic cosine for definitions of f and c(z).

CONDITION
VALUES
SIGNALED

MTH-92

SS$_RQPRAND

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: floatingpoint-input-value is greater than yyy; LIB$SIGNAL
copies the reserved operand to the signal
mechanism vector. The result is the reserved
operand -0.0 unless a condition handler changes
the signal mechanism vector. The value. of yyy is
approximately 11355.830 for MTH$HEXP.

MTH$_FLOUNDMA T

Floating-point underflow in Math Library: floatingpoint-input-value is less than or equal to yyy and
the caller (CALL or JSB) has set hardware floatingpoint underflow enable. The result is set to 0.0. If
the caller has not enabled floating-point underflow
(the default), a result of 0.0 is returned but no
error is signaled. The value of yyy is approximately
-11356.523 for MTH$HEXP.

MTH$HLOG

Natural Logarithm (H-floating Value)

MTH$HLOG

The Natural Logarithm (H-floating Value) routine returns the natural (base
e) logarithm of the input argument as an H-floating value.

FORMAT

MTH$HLOG

jsb entries

MTH$HLOG_R8

RETURNS

None.

ARGUMENTS

h-natlog ,floating-point-input-value

h-natlog
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Natural logarithm of floating-point-input-value. The h-natlog argument
is the address of an H-floating number that is this natural logarithm.
MTH$HLOG writes the address of this natural logarithm into h-natlog.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
a floating-point number that is this value. For MTH$HLOG, floating-pointinput-value specifies an H-floating number.

DESCRIPTION

Computation of the natural logarithm routine is based on the following:
1

ln(X * Y) = ln(X) + ln(Y)

In(l + x) = x - x 2 /2 + x 3 /3 - x 4 /4 ...
for IXI < 1

ln(X) = in(A) + 2 * (V + V 3 + V 5 /5 + V 7 /7 ... )
where V = (X - A)/(X +A), A> 0,
and p(y) = 2 * (1 + y/3 + y2 /5 ... )

For x = 2n * f, where n is an integer and f is in the interval of 0.5 to 1,
define the following quantities:
If n~l, then N = n - 1 and F = 2f
If n::;O, then N = n and F = f

MTH-93

MTH$HLOG

From (1) above it follows that:

ln(X) = N * ln(2) + ln(F)

Based on the above relationships, zLOG is computed as follows:

CONDITION
VALUES
SIGNALED

MTH-94

If IF - ll < 2- 5 ,
zLOG(X) = N * zLOG(2) + W + W * p(W),
where W = F-1.

Otherwise,
zLOG(X) = N * zLOG(2) + zLOG(A) + V * p(V 2 ),
where V = (F - A)/(F +A) and A and zLOG(A)
are obtained by table look up.

SS$_ROPRAND

Reserved operand. The MTH$HLOG procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LOGZERNEG

Logarithm of zero or negative value. Argument
floating-point-input-value is less than or equal
to 0.0. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument
vector CHF$L _MCH_SA VRO /R 1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L _
MCH_SAVRO/R1.

MTH$HLOG2

Base 2 Logarithm (H-floating
Value)

The Base 2 Logarithm (H-floating Value) routine returns the base 2
logarithm of the input value specified by floating-point-input-value as
an H-floating value.

FORMAT

MTH$HLOG2

RETURNS

None.

ARGUMENTS

h-log2 ,floating-point-input-value

h-log2
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Base 2 logarithm of floating-point-input-value. The h-log2 argument is the
address of an H-floating number that is this base 2 logarithm. MTH$HLOG2
writes the address of this logarithm into h-log2.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number that is this input value. For MTH$HLOG2,
floating-point-input-value specifies an H-floating number.

DESCRIPTION

The base 2 logarithm function is computed as follows:

zLOG2(X) = zLOG2(E) * zLOG(X)

MTH-95

MTH$HLOG2

CONDITION
VALUES
SIGNALED

MTH-96

SS$_ROPRAND

Reserved operand. The MTH$HLOG2 procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LQGZERNEG

Logarithm of zero or negative value. Argument
floating-point-input-value is less than or equal
to 0.0. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument
vector CHF$L_MCH_SAVRO/R 1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH_SAVRO/R 1.

MTH$HLOG10

Common Logarithm (H-floating
Value)

The Common Logarithm (H-floating Value) routine returns the common
(base 10) logarithm of the input argument as an H-floating value.

FORMAT

MTH$H LOG 1 0

jsb entries

MTH$HLOG1 O_R8

RETURNS

None.

ARGUMENTS

h-log 10, floating-point-input-value

h-log10
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Common logarithm of the input value specified by floating-point-inputvalue. The h-log10 argument is the address of an H-floating number that is
this common logarithm. MTH$HLOG10 writes the address of the common
logarithm into h-loglO.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number. For MTH$HLOG10, floating-point-input-value
specifies an H-floating number.

DESCRIPTION

The common logarithm function is computed as follows:
zLOGIO(X) = zLOGIO(E) * zLOG(X)

MTH-97

MTH$HLOG10

CONDITION
VALUES
SIGNALED

MTH-98

SS$_ROPRAND

Reserved operand. The MTH$HLOG 10 procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LOGZERNEG

Logarithm of zero or negative value. Argument
floating-point-input-value is less than or equal
to 0.0. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument
vector CHF$L_MCH_SAVRO/R1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L _
MCH_SA VRO /R 1.

MTH$HSIN

Sine of Angle Expressed in Radians
(H-floating Value)
The Sine of Angle Expressed in Radians (H-floating Value) routine returns
the sine of a given angle (in radians) as an H-floating value.

FORMAT

MTH$HSIN

jsb entries

MTH$HSIN_R5

RETURNS

None.

ARGUMENTS

h-sine ,angle-in-radians

h-sine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

The sine of the angle specified by angle-in-radians. The h-sine argument is
the address of an H-floating number that is this sine. MTH$HSIN writes the
address of the sine into h-sine.

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Angle (in radians). The angle-in-radians argument is the address of a
floating-point number that is this angle. For MTH$HSIN, angle-in-radians
specifies an H-floating number.

DESCRIPTION
CONDITION
VALUE
SIGNALED

See the MTH$SINCOS routine for the algorithm used to compute this sine.

SS$_ROPRAND

Reserved operand. The MTH$HSIN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-99

MTH$HSIND

Sine of Angle Expressed in Degrees
(H-floating Value)

The Sine of Angle Expressed in Degrees (H-floating Value) routine returns
the sine of a given angle (in degrees) as an H-floating value.

FORMAT

MTH$HSIND h-sine ,angle-in-degrees

jsb entries

MTH$HSIND_R5

RETURNS

None.

ARGUMENTS

h-sine
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Sine of the angle specified by angle-in-degrees. The h-sine argument is the
address of an H-floating number that is this sine. MTH$HSIND writes the
address of the angle into h-sine.

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

Angle (in degrees). The angle-in-degrees argument is the address of a
floating-point number that is this angle. For MTH$HSIND, angle-in-degrees
specifies an H-floating number.

DESCRIPTION

MTH-100

See MTH$SINCOSD for the algorithm used to compute the sine.

MTH$HSIND

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The ivlTH$HSIND procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased ecponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOUNDMAT

Floating-point underflow in Math Library. The
absolute value of the input angle is less than
180/7r * 2-m (where m = 16,384 for H-floating).

MTH-101

MTH$HSINH

Hyperbolic Sine (H-floating Value)

The Hyperbolic Sine (H-floating Value) routine returns the hyperbolic sine
of the input value specified by floating-point-input-value as an H-floating
value.

FORMAT

MTH$HSINH

RETURNS

None.

ARGUMENTS

h-sinh ,floating-point-input-value

h-sinh
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Hyperbolic sine of the input value specified by floating-point-input-value.
The h-sinh argument is the address of an H-floating number that is this
hyperbolic sine. MTH$HSINH writes the address of the hyperbolic sine into
h-sinh.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
a floating-point number that is this value. For MTH$HSINH, floating-pointinput-value specifies an H-floating number.

DESCRIPTION

Computation of the hyperbolic sine function depends on the magnitude of
the input argument. The range of the function is partitioned using four data
type dependent constants: a(z), b(z), and c(z). The subscript z indicates the
data type. The constants depend on the number of exponent bits (e) and the
number of fraction bits (f) associated with the data type (z).
The values of e and f are as follows:

e = 15

f = 113

MTH-102

MTH$HSINH

The values of the constants in terms of e and f are:
Variable

Value

a(z)

2(-/ /2)

b(z)

(! + 1)/2 * ln(2)

c(z)

2e-i * ln(2)

Based on the above definitions, zSINH(X) is computed as follows:
Value of X

CONDITION
VALUES
SIGNALED

Value Returned

IXI < a(z)

a(z)::;IXI < LO

zSINH(X) is computed using a power series
expansion in IXl 2

i.o::;jXI < b(z)

(zEXP(X) - zEXP(-X))/2

b(z)::;jXj < c(z)

SIGN(X) * zEXP(IXl)/2

c(z)::;IXI

Overflow occurs

SS$_ROPRAND

Reserved operand. The MTH$HSINH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of floating-point-input-value
is greater than yyy. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R 1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R 1. The value of yyy is
approximately 11356.523.

MTH-103

MTH$HSQRT

MTH$HSQRT Square Root (H-floating Value)
The Square Root (H-floating Value) routine returns the square root of the
input value floating-point-input-value as an H-floating value.

FORMAT

MTH$HSQRT h-sqrt ,floating-point-input-value

jsb entries

MTH$HSQRT_R8

RETURNS

None.

ARGUMENTS

h-sqrt
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write. only
by reference

Square root of the input value specified by floating-point-input-value. The
h-sqrt argument is the address of an H-floating number that is this square
root. MTH$HSQRT writes the address of the square root into h-sqrt.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H _floating
read only
by reference

Input value. The floating-point-input-value argument is the address of
a floating-point number that contains this input value. For MTH$HSQRT,
floating-point-input-value specifies an H-floating number.

DESCRIPTION

The square root of Xis computed as follows:
If X < 0, an error is signaled.
Let X = 2K * F
where:
K is the exponential part of the floating-point data
F is the fractional part of the floating-point data
If K is even:

x = 212 •P) * F,
zSQRT(X) = 2P * zSQRT(F),
1/2$F < 1, where P = K/2

MTH-104

MTH$HSQRT

If K is odd:

x = 2(2*P+1) * F = 2(2*P+2) * (F /2),

zSQRT(X) = 2IP+ll * zSQRT(F /2),
1/45:.F/2 < 1/2, where p = (K-1)/2
Let F' = A * F + B, when K is even:
A = 0.95F6198 (hex)
B = 0.6BA5918 (hex)
Let F' =A* (F /2) + B, when K is odd:
A= O.D413CCC (hex)
B = 0.4C 1E248 (hex)
Let K' = P, when K is even
Let K' = P+1, when K is odd

Let Y[O] = 2K' * F' be a straight line approximation within the given interval
using coefficients A and B which minimize the absolute error at the midpoint
and endpoint.
Starting with Y[O], n Newton-Raphson iterations are performed:
Y[n + 1] = 1/2 * (Y[n] + X/Y[n])

where n = 5 for H-floating.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$HSORT procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_SQUROONEG

Square root of negative number. Argument
floating-point-input-value is less than 0.0.
LIB$SIGNAL copies the floating-point reserved
operand to the mechanism argument vector
CHF$L _MCH _SA VRO /R 1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH _SA VRO /R 1.

MTH-105

MTH$HTAN

Tangent of Angle Expressed in
Radians (H-floating Value)
The Tangent of Angle Expressed in Radians (H-floating Value) routine
returns the tangent of a given angle (in radians) as an H-floating value.

FORMAT

MTH$HTAN

jsb entries

MTH$HTAN_R5

RETURNS

None.

ARGUMENTS

h-tan ,angle-in-radians

h-tan
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Tangent of the angle specified by angle-in-radians. The h-tan argument is
the address of an H-floating number that is this tangent. MTH$HTAN writes
the address of the tangent into h-tan.

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
H _floating
read only
by reference

The input angle (in radians). The angle-in-radians argument is the address of
a floating-point number that is this angle. For MTH$HTAN, angle-in-radians
specifies an H-floating number.

DESCRIPTION

When the input argument is expressed in radians, the tangent function is
computed as follows:
1

If IXI < 2<-11 2>, then zTAN(X) = X (see the section on MTH$zCOSH
for the definition of f)

Otherwise, call MTH$zSINCOS to obtain zSIN(X) and zCOS(X); then
a. If zCOS(X) = 0, signal overflow
b. Otherwise, zT AN(X) = zSI N(X)/ zCOS(X)

MTH-106

MTH$HTAN

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$HTAN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMAT

Floating-point overflow in math library.

MTH-107

MTH$HTAND

Tangent of Angle Expressed in
Degrees (H-floating Value)

The Tangent of Angle Expressed in Degrees (H-floating Value) routine
returns the tangent of a given angle (in degrees) as an H-floating value.

FORMAT

MTH$HTAND h-tan ,angle-in-degrees

jsb entries

MTH$HTAND_R5

RETURNS

None.

ARGUMENTS

h-tan
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Tangent of the angle specified by angle-in-degrees. The h-tan argument
is the address of an H-floating number that is this tangent. MTH$HTAND
writes the address of the tangent into h-tan.

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read on'y
by reference

The input angle (in degrees). The angle-in-degrees argument is the address
of a floating-point number which is this angle. For MTH$HTAND, angle-indegrees specifies an H-floating number.

DESCRIPTION

When the input argument is expressed in degrees, the tangent function is
computed as follows:
1

If IXI < (180/7r)*2<- 2 /(e-l)) and underflow signaling is enabled, underflow
is signaled (see the section on MTH$zCOSH for the definition of e).

Otherwise, if IXI < (180/7r) *2(-f/ 2), then zTAND(X) = (7r/l80) *X. See
the description of MTH$zCOSH for the definition off.

Otherwise, call MTH$zSINCOSD to obtain zSIND(X) and zCOSD(X).
a. Then, if zCOSD(X) = 0, signal overflow
b. Else, zT AN D(X) = zSI N D(X)/ zCOSD(X)

MTH-108

MTH$HTAND

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$HT AND procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in math library.

MTH-109

MTH$HTANH

Compute the Hyperbolic Tangent
(H-floating Value)

The Compute the Hyperbolic Tangent (H-floating Value) routine returns the
hyperbolic tangent of the input value as an H-floating value.

FORMAT

MTH$HTANH

RETURNS

None.

ARGUMENTS

h-tanh ,floating-point-input-value

h-tanh
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
write only
by reference

Hyperbolic tangent of the value specified by floating-point-input-value. The
h-tanh argument is the address of a H-floating number that is this hyperbolic
tangent. MTH$HTANH writes the address of the hyperbolic tangent into
h-tanh.

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
H_floating
read only
by reference

The input value. The floating~point-input-value argument is the address of
a floating-point number that contains this input value. For MTH$HTANH,
floating-point-input-value specifies an H-floating number.

DESCRIPTION

MTH-110

For MTH$HTANH, the hyperbolic tangent of X is computed using a value of
56 for g and a value of 40 for h. The hyperbolic tangent of X is computed as
follows:
Value of x

Hyperbolic Tangent Returned

1x19-g

rg < 1x1:::;0.25

zSI N H(X)/ zCOSH(X)

0.25 < IXI < h

(zEXP(2 * X) - 1)/(zEXP(2 * X) + 1)

h:::;IXI

sign(X) * 1

MTH$HTANH

CONDITION
VALUE
SIGNALED

SS$_RQPRAND

Reserved operand. The MTH$HTANH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-111

MTH$xlMAG

MTH$x1MAG

Imaginary Part of a Complex
Number

The Imaginary Part of a Complex Number routine returns the imaginary
part of a complex number.

FORMAT

MTH$AIMAG
MTH$DIMAG
MTH$GIMAG

complex-number
complex-number
complex-number

Each of the above three formats corresponds to one of the three floating-point
complex types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Imaginary part of the input complex-number. MTH$AIMAG returns an Ffloating number. MTH$DIMAG returns a D-floating number. MTH$GIMAG
returns a G-floating number.

ARGUMENT

complex-number
VMS usage: complex_number
type:
F_floating complex, D_floating complex,
G_floating complex
access:
read only
mechanism: by reference

The input complex number. The complex-number argument is the address
of this floating-point complex number. For MTH$AIMAG, complex-number
specifies an F-floating number. For MTH$DIMAG, complex-number specifies
a D-floating number. For MTH$GIMAG, complex-number specifies a Gfloating number.

CONDITION
VALUE
SIGNALED

MTH-112

SS$_ROPRAND

Reserved operand. The MTH$xlMAG routine
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xlMAG

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the imaginary part of
a G-f loating complex number using MTH$GIMAG
and the FORTRAN random number generator
RAN.

C
C
C

Declare Z as a complex value and MTH$GIMAG as
a REAL*8 value. MTH$GIMAG will return the imaginary
part of Z:
Z_NEW = MTH$GIMAG(Z).

COMPLEX*16 Z
COMPLEX*16 DCMPLX
REAL*8 R,I,MTH$GIMAG
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the
FORTRAN generic CMPLX.
R = RAN(M)
I = RAN(M)
Z = DCMPLX(R,I)

C+

C
C

Z is a complex number (r,i) with real part "r" and
imaginary part "i".
TYPE*· 'The complex number z is',z
TYPE *· ' It has imaginary part' ,MTH$GIMAG(Z)
END

This FORTRAN example demonstrates a procedure call to MTH$GIMAG.
Because this example uses G-floating numbers, it must be compiled with the
statement "FORTRAN/G filename".
The output generated by this program is as follows:
The complex number z is (0.8535407185554504,0.2043401598930359)
It has imaginary part 0.2043401598930359

MTH-113

MTH$xLOG

Natural Logarithm
The Natural Logarithm routine returns the natural (base e) logarithm of the
input argument.

FORMAT

MTH$ALOG
MTH$DLOG
MTH$GLOG

floating-point-input-value
floating-point-input-value
floating-point-input-value

Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ALOG_R5
MTH$DLOG_R8
MTH$GLOG_R8
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The natural logarithm of floating-point-input-value. MTH$ALOG returns an
F-floating number. MTH$DLOG returns a D-floating number. MTH$GLOG
returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usag_e:
type:
access:
mechanism:

floating_point
F_floating, D_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number that is this value. For MTH$ALOG, floatingpoint-input-value specifies an F-floating number. For MTH$DLOG,
floating-point-input-value specifies a D-floating number. For MTH$GLOG,
floating-point-input-value specifies a G-floating number.

DESCRIPTION

MTH-114

Computation of the natural logarithm routine is based on the following:

ln(X * Y) = ln(X) + ln(Y)

In(l + X) = x - x 2 /2 + x 3 /3 - x 4 /4 ...
for IXI < 1

ln(X) = In(A) + 2 * (V + V 3 /3 + V 5 /5 + V 7 /7 ... )
= ln(A) + V * p(V 2 ), where V = (X - A)/(X +A),
A> 0, and p(y) = 2 * (1 + y/3 + y2/5 ... )

MTH$xLOG

For x = 2n * f, where n is an integer and f is in the interval of 0.5 to l,
define the following quantities:
If n~l, then N = n - 1 and F = 2f
If n~O, then N = n and F = f

From ( 1 ) above it follows that:
4

ln(X) = N * ln(2) + ln(F)

Based on the above relationships, zLOG is computed as follows:

If IF - ll < 2- 5 , zLOG(X) = N * zLOG(2) + W + W * p(W),
where W = F-1.

Otherwise, zLOG(X) = N * zLOG(2) + zLOG(A) + V * p(V 2 ),
where V = (F -A)/(F +A) and A and zLOG(A)
are obtained by table look up.

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HLOG.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xLOG procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LQGZERNEG

MTH-115

MTH$xLOG2

Base 2 Logarithm

The Base 2 Logarithm routine returns the base 2 logarithm of the input
value specified by floating-point-input-value.

FORMAT

MTH$ALOG2
MTH$DLOG2
MTH$GLOG2

floating-point-input-value
floating-point-input-value
floating-point-input-value

Each of the above formats accepts as input one of the floating-point types.

RETURNS ,

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

The base 2 logarithm of floating-point-input-value. MTH$ALOG2
returns an F-floating number. MTH$DLOG2 returns a D-floating number.
MTH$GLOG2 returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address of
a floating-point number that is this input value. For MTH$ALOG2, floatingpoint-input-value specifies an F-floating number. For MTH$DLOG2,
floating-point-input-value specifies a D-floating number. For MTH$GLOG2,
floating-point-input-value specifies a G-floating number.

DESCRIPTION

The base 2 logarithm function is computed as follows:

zLOG2(X) = zLOG2(E) * zLOG(X)
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HLOG2.

MTH-116

MTH$xLOG2

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xLOG2 procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LOGZERNEG

Logarithm of zero or negative value. Argument
floating-point-input-value is less than or equal
to 0.0. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument
vector CHF$L_MCH_SAVRO/A1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L_
MCH_SAVRO/R1.

IVITH~117

MTH$xLOG10

Common Logarithm

The Common Logarithm routine returns the common (base 10) logarithm
of the input argument.

FORMAT

MTH$ALOG10
MTH$DLOG10
MTH$GLOG10

floating-point-input-value
floating-point-input-value
floating-point-input-value

Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$ALOG1 O_R5
MTH$DLOG1 Q_R8
MTH$GLOG1 Q_R8
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The common logarithm of floating-point-input-value. MTH$ALOG 10
returns an F-floating number. MTH$DLOG10 returns a D-floating number.
MTH$GLOG10 returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number. For MTH$ALOG10, floating-point-input-value
specifies an F-floating number. For MTH$DLOG10, floating-point-inputvalue specifies a D-floating number. For MTH$GLOG10, floating-pointinput-value specifies a G-floating number.

DESCRIPTION

The common logarithm function is computed as follows:

zLOGlO(X) = zLOGlO(E) * zLOG(X)
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HLOG10.

MTH-118

MTH$xLOG10

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xLOG 10 procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_LQGZERNEG

Logarithm of zero or negative value. Argument
floating-point-input-value is less than or equal
to 0.0. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument
vector CHF$L_MCH_SAVRO/R 1. The result is the
floating-point reserved operand unless you have
written a condition handler to change CHF$L _
MCH_SAVRO/R1.

MTH-119

MTH$RANDOM

Random Number Generator,
Uniformly Distributed

The Random Number Generator, Uniformly Distributed routine is a general
random number generator.

FORMAT

MTH$RANDOM

RETURNS

VMS usage:
type:
access:
mechanism:

seed

floating_point
F_floating
write only
by value

MTH$RANDOM returns an F-floating random number.

ARGUMENT

seed
VMS usage:
type:
access:
mechanism:

longword_unsigned
longword (unsigned)
modify
by reference

The integer seed, a 32-bit number whose high-order 24 bits are converted by
MTH$RANDOM to an F-floating random number. The seed argument is the
address of an unsigned longword that contains this integer seed. The seed is
modified by each call to MTH$RANDOM.

DESCRIPTION

This routine must be called again to obtain the next pseudorandom number.
The seed is updated automatically.
The result is a floating-point number that is uniformly distributed between
0.0 inclusively and 1.0 exclusively.
There are no restrictions on the seed, although it should be initialized
to different values on separate runs in order to obtain different random
sequences. MTH$RANDOM uses the following method to update the seed
passed as the argument:
SEED= (69069 *SEED+ l)(modulo2 32 )

MTH-120

MTH$RANDOM

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$RANDOM procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

EXAMPLE
RAND:
PROCEDURE OPTIONS (MAIN);
DECLARE FOR$SECNDS ENTRY (FLOAT BINARY (24))
RETURNS (FLOAT BINARY (24));
DECLARE MTH$RANDOM ENTRY (FIXED BINARY (31))
RETURNS (FLOAT BINARY (24));
DECLARE TIME FLOAT BINARY (24);
DECLARE SEED FIXED BINARY (31);
DECLARE I FIXED BINARY (7);
DECLARE RESULT FIXED DECIMAL (2);
/* Get floating random time value
*/
TIME= FOR$SECNDS (OEO);
/* Convert to fixed
*/
SEED = TIME;
/* Generate 100 random numbers between 1 and 10 */
DO I = 1 TO 100;
RESULT= 1 +FIXED ( (10EO * MTH$RANDOM (SEED) ) ,31 );
PUT LIST (RESULT);
END;
END RAND;

This PL/I program demonstrates the use of MTH$RANDOM. The value
returned by FOR$SECNDS is used as the seed for the random-number
generator to insure a different sequence each time the program is run. The
random value returned is scaled so as to represent values between 1 and 10.
Because this program· generates random numbers, the output generated will
be different each time the program is executed. One example of the outut
generated by this program is as follows:
7
4
1
6
8
1
3

4
4
10
2
5
2
8

6
2
10
3
4
2
1

5
4
6
6
9
3
5

9
4
7
10
8
6

10
8
3
8
7
5

5
3
2

5
6
2

5
8
2
5
6
3

3
9
1
4
8
4

8
1
2
2
10
4

8
7
6
8
9
8

1
6
5
5
9

3
8
3
9
9
2

1
6
9
6
4
8

3
9
5
4
5
5

2
10
8
2
7
5

MTH-121

MTH$xREAL

Real Part of a Complex Number

The Real Part of a Complex Number routine returns the real part of a
complex number.

FORMAT

MTH$REAL complex-number
MTH$DREAL complex-number
MTH$GREAL complex-number
Each of the above three formats accepts as input one of the three floatingpoint complex types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Real part of the complex number. MTH$REAL returns an F-floating number.
MTH$DREAL returns a D-floating number. MTH$GREAL returns a Gfloating number.

ARGUMENT

complex-number
VMS usage: complex_number
type:
f _floating complex, D_floating complex, G_floating
complex
access:
read only
mechanism: by reference

The complex number whose real part is returned by MTH$REAL. The
complex-number argument is the address of this floating-point complex
number. For MTH$REAL, complex-number is an F-floating complex number.
For MTH$DREAL, complex-number is a D-floating complex number. For
MTH$GREAL, complex-number is a G-floating complex number.

CONDITION
VALUE
SIGNALED

MTH-122

SS$_ROPRAND

Reserved operand. The MTH$xREAL procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xREAL

EXAMPLE
C+

C
C
C
C

This FORTRAN example forms the real
part of an F-floating complex number using
MTH$REAL and the FORTRAN random number
generator RAN.

C
C

Declare Z as a complex value and MTH$REAL as a
REAL*4 value. MTH$REAL will return the real
part of Z:
Z_NEW = MTH$REAL(Z).

c
c

COMPLEX Z
COMPLEX CMPLX
REAL*4 MTH$REAL
INTEGER M
M = 1234567
C+

C
C

Generate a random complex number with the FORTRAN
generic CMPLX.
Z = CMPLX(RAN(M),RAN(M))

C+

C
C

Z is -a complex number (r,i) with real part "r" and imaginary
part "i".
TYPE *· ' The complex number z is' ,z
TYPE *· ' It has real part' ,MTH$REAL(Z)
END

This FORTRAN example demonstrates the use of MTH$REAL. The output of
this program is as follows:
The complex number z is (0.8535407,0.2043402)
It has real part 0.8535407

MTH-123

MTH$xSIN

Sine of Angle Expressed in Radians
The Sine of Angle Expressed in Radians routine returns the sine of a given
angle (in radians).

FORMAT

MTH$SIN angle-in-radians
MTH$DSIN angle-in-radians
MTH$GSIN angle-in-radians
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$SIN_R4
MTH$DSIN_R7
MTH$GSIN_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

Sine of the angle specified by angle-in-radians. MTH$SIN returns an Ffloating number. MTH$DSIN returns a D-floating number. MTH$GSIN
returns a G-floating number.

ARGUMENTS

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Angle (in radians). The angle-in-radians argument is the address of a
floating-point number that is this angle. For MTH$SIN, angle-in-radians
specifies an F-floating number. For MTH$DSIN, angle-in-radians specifies a
D-floating number. For MTH$GSIN, angle-in-radians specifies a G-floating
number.

DESCRIPTION

See the MTH$SINCOS routine for the algorithm used to compute this sine.
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HSIN.

MTH-124

MTH$xSIN

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xSIN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-125

MTH$xSI NCOS

MTH$xSINCOS

Sine and Cosine of Angle
Expressed in Radians

The Sine and Cosine of Angle Expressed in Radians routine returns the
sine and cosine of a given angle (in radians).

FORMAT

MTH$SINCOS angle-in-radians ,sine ,cosine
MTH$DSINCOS angle-in-radians ,sine ,cosine
MTH$GSINCOS angle-in-radians ,sine ,cosine
MTH$HSINCOS angle-in-radians ,sine ,cosine
Each of the above four formats accepts as input one of the four floating-point
types.

jsb entries

MTH$SI NCOS_R5
••TLI~"'~•l\1,...n~

o-,

IVI I n~Ll'11 l""'V~-n I

MTH$GSI NCQS_R7
MTH$HSINCOS_R7
Each of the above four JSB entries accepts as input one of the four floatingpoint types.

RETURNS

ARGUMENTS

MTH$SINCOS, MTH$DSINCOS, MTH$GSINCOS, and MTH$HSINCOS
return the sine and cosine of the input angle by reference in the sine and
cosine arguments.

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating, H_floating
read only
by reference

Angle (in radians) whose sine and cosine are to be returned. The anglein-radians argument is the address of a floating-point number that
is this angle. For MTH$SINCOS, angle-in-radians is an F-floating
number. For MTH$DSINCOS, angle-in-radians is a D-floating number.
For MTH$GSINCOS, angle-in-radians is a G-floating number. For
MTH$HSINCOS, angle-in-radians is an H-floating number.

sine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating, H_floating
write only
by reference

Sine of the angle specified by angle-in-radians. The sine argument is the
address of a floating-point number. MTH$SINCOS writes an F-floating
MTH-126

MTH$xSI NCOS

number into sine. MTH$DSINCOS writes a D-floating number into sine.
MTH$GSINCOS writes a G-floating number into sine. MTH$HSINCOS
writes an H-floating number into sine.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating, H_floating
write only
by reference

Cosine of the angle specified by angle-in-radians. The cosine argument is
the address of a floating-point number. MTH$SINCOS writes an F-floating
number into cosine. MTH$DSINCOS writes a D-floating number into cosine.
MTH$GSINCOS writes a G-floating number into cosine. MTH$HSINCOS
writes an H-floating number into cosine.

DESCRIPTION

All routines with JSB entry points accept a single argument in RO:Rm, where
m, which is defined below, is dependent on the data type.
Data Type

F_floating

D_floating
G_floating

H_floating

In general, Run-Time Library routines with JSB entry points return one value
in RO:Rm. The MTH$SINCOS routine returns two values, however. The sine
of angle-in-radians is returned in RO:Rm and the cosine of angle-in-radians
is returned in (R <m+l> :R <2*m+l> ).
In radians, the computation of zSIN(X) and zCOS(X) is based on the following
polynomial expansions:
sin(X) = X - X 3 /(3!) + X 5 /(5!) - X 7 /(7!) ...
= X + X * P(X 2 ), where
P(y) = y/(3!) + y2 /(5!) + y 3 /(7!) ...
cos(X) = 1-X2 /(2!) +x 4 /(4!)-X 6 /(6!) ...
= Q(X 2 ), where
Q(y) = (1- y/(2!) + y2/(4!) + y 3 /(6!) ... )

If IXI < 2<-f12>,
then zSIN(X) = X and zCOS(X) = 1
(see the section on MTH$zCOSH for
the definition of f)

If rt 1 2 ~1XI < 7r/4,
then zSIN(X) = X + P(X 2 )
and zCOS(X) = Q(X 2 )

3 If 7r/4~IXI and x > o,

a. Let J = INT(X/(7r/4))

and I = J modulo 8
MTH-127

MTH$xSINCOS

b. If J is even, let Y = X - J * (7r/4)
otherwise,
let Y = (J + 1) * (7r / 4) - x
With the above definitions, the following table relates zSIN(X)
and zCOS(X) to zSIN(Y) and zCOS(Y):
Value of I

zSIN(X)

zSIN(Y)

zCOS(Y)

zSIN(Y)

zCOS(Y)

-zSIN(Y)

zSIN(Y)

-zCOS(Y)

-zSIN(Y)

-zCOS(Y)

-zSIN(Y)

-zCOS(Y)

zSIN(Y)

-zSIN(Y)

zCOS(Y)

zCOS(X)

c. zSIN(Y) and zCOS(Y) are computed as follows:
,,,QTl\.T(V\
•\.I. j -

wV.1. ..L

V
.._

.;..L...
.1

p(y2\
,.._
j)

.._

and zCOS(Y) = Q(Y2)

CONDITION
VALUE
RETURNED

MTH-128

If 7r/4~IXI and x < o,
then zSIN(X) = -zSIN(IXI)
and zCOS(X) = zCOS(IXI)

SS$_ROPRAND

Reserved operand. The MTH$xSINCOS procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$xSINCOSD

Sine and Cosine of Angle
Expressed in Degrees

The Sine and Cosine of Angle Expressed in Degrees routine returns the
sine and cosine of a given angle (in degrees).

FORMAT

MTH$SINCOSD angle-in-degrees ,sine ,cosine
MTH$DSINCOSD angle-in-degrees ,sine ,cosine
MTH$GSINCOSD angle-in-degrees ,sine ,cosine
MTH$HSINCOSD angle-in-degrees ,sine ,cosine
Each of the above four formats accepts as input one of the four floating-point
types.

jsb entries

MTH$SI NCOSD_R5
MTH$DSI NCOSD_R7
MTH$GSINCOSD_R7
MTH$HSI NCOSD_R7
Each of the above four JSB entries accepts as input one of the four floatingpoint types.

RETURNS

ARGUMENTS

MTH$SINCOSD, MTH$DSINCOSD, MTH$GSINCOSD, and
MTH$HSINCOSD return the sine and cosine of the input angle by reference
in the sine and cosine arguments.

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
f_floating, O_floating, G_floating, H_floating
read only
by reference

Angle (in degrees) whose sine and cosine are returned by MTH$xSINCOSD.
The angle-in-degrees argument is the address of a floating-point number
that is this angle. For MTH$SINCOSD, angle-in-degrees is an F-floating
number. For MTH$DSINCOSD, angle-in-degrees is a D-floating number.
For MTH$GSINCOSD, angle-in-degrees is a G-floating number. For
MTH$HSINCOSD, angle-in-degrees is an H-floating number.

sine
VMS usage:
type:
access:
mechanism:

floating_point
f_floating, O_floating, G_floating, H_floating
write only
by reference

Sine of the angle specified by angle-in-degrees. The sine argument is the
address of a floating-point number. MTH$SINCOSD writes an F-floating

MTH-129

MTH$xSINCOSD

number into sine. MTH$DSINCOSD writes a D-floating number into sine.
MTH$GSINCOSD writes a G-floating number into sine. MTH$HSINCOSD
writes an H-floating number into sine.

cosine
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating, H_floating
write only
by reference

Cosine of the angle specified by angle-in-degrees. The cosine argument
is the address of a floating-point number. MTH$SINCOSD writes an Ffloating number into cosine. MTH$DSINCOSD writes a D-floating number
into cosine. MTH$GSINCOSD writes a G-floating number into cosine.
MTH$HSINCOSD writes an H-floating number into cosine.

DESCRIPTION

All routines with JSB entry points accept a single argument in RO:Rm, where
m, which is defined below, is dependent on the data type.
Data Type

F_floating

D_floating
G_floating

H_floating

In general, Run-Time Library routines with JSB entry points return one value
in RO:Rm. The MTH$SINCOSD mutine returns two values, however. The
sine of angle-in-degrees is returned in RO:Rm and the cosine of angle-indegrees is returned in (R <m+l> :R <2•m+l> ).
In degrees, the computation of zSIND(X) and zCOSD(X) is based on the
following polynomial expansions:

SIND(X) = (C * X) - (C * X) 3 /(3!)+
(C * X) 5 /(5!) - (C * x)7 /(7!) ...
= X/2 6 + X * P(X 2 ), where
P(y) = -y/(3!) + y 2 /(5!) - y3 /(7!) ...
COSD(X) = 1- (C * X) 2 /(2!)+
(C * X) 4 /(4!) - (C * X) 6 /(6!) ...
= Q(X2 ), where
Q(y) = 1 - y/(2!) + y 2 /(4!) - y 3 /(6!) ...
and C = 7r/180
1

If IXI < (180/7r) * 2- 2 e-i and underflow signaling is enabled,
underflow is signaled for zSIND(X) and zSINCOSD(X).
See MTH$zCOSH for the definition of e.
OTHERWISE:

MTH-130

If IXI < (180/7r) * 2<-f/ 2),
then zSIND(X) = (7r/180) * X and zCOSD(X) = 1.
(See MTH$zCOSH for the definition of f.)

MTH$xSINCOSD

If (180/7r) * 2<-f / 2) ~ IXI < 45
then zSI N D(X) = X/2 6 + P(X 2)
and zCOSD(X) = Q(X2)

If 45~IXI and x > o,
a. Let J = JNT(X/(45))and
I= J modulo 8
b. If J is even, let Y = X - J * 45;
otherwise, let Y = (J + 1) * 45 - X.
With the above definitions, the following table relates
zSIND(X) and zCOSD(X) to zSIND(Y) and zCOSD(Y):
Value of I

zSIND(X)

zCOSD(X)

zSIND(Y)

zCOSD(Y)

zSIND(Y)

zCOSD(Y)

-zSIND(Y)

zSIND(Y)

-zCOSD(Y)

-zSIND(Y)

-zCOSD(Y)

-zSIND(Y)

-zCOSD(Y)

zSIND(Y)

-zSIND(Y)

zCOSD(Y)

c. zSIND(Y) and zCOSD(Y) are computed as follows:
zSIND(Y) = Y/2 6 + P(Y 2)
zCOSD(Y) = Q(Y2)

d. If 45~ IXI and X < 0,

then zSIND(X) = -zSIND(IXI)
and zCOSD(X) = zCOSD(IXI)

CONDITION
VALUES
SIGNALED

SS$_RQPRAND

Reserved operand. The MTH$xSINCOSD
procedure encountered a floating-point reserved
operand due to incorrect user input. A floatingpoint reserved operand is a floating-point datum
with a sign bit of 1 and a biased exponent of zero.
Floating-point reserved operands are reserved for
future use by DIGIT AL.

MTH$_FLOUNDMA T

Floating-point underflow in math library. The
absolute value of the input angle is less than
180/7r * 2-m (where m = 128 for F-floating and
0-floating, 1,024 for G-floating, and 16,384 for
H-floating).

MTH-131

MTH$xSIND

Sine of Angle Expressed in Degrees

The Sine of Angle Expressed in Degrees routine returns the sine of a given
angle (in degrees).

FORMAT

MTH$SIND angle-in-degrees
MTH$DSI ND angle-in-degrees
MTH$GSIND angle-in-degrees
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$SIND_R4
MTH$DSIND_R7
MTH$GSIND_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D,;_floating, G_floating
write only
by value

The sine of the angle. MTH$SINO returns an F-floating number.
MTH$0SINO returns a 0-floating number. MTH$GSINO returns a G-floating
number.

ARGUMENTS

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Angle (in degrees). The angle-in-degrees argument is the address of a
floating-point number that is this angle. For MTH$SINO, angle-in-degrees
specifies an F-floating number. For MTH$0SINO, angle-in-degrees specifies
a 0-floating number. For MTH$GSINO, angle-in-degrees specifies a Gfloating number.

DESCRIPTION

See MTH$SINCOSO for the algorithm that is used to compute the sine.
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HSINO.

MTH-132

MTH$xSIND

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$SIND procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased ecponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOUNDMA T

Floating-point underflow in math library. The
absolute value of the input angle is less than
180/7r * rm (where m = 128 for F-floating and
0-floating, and 1,024 for G-floating).

MTH-133

MTH$xSINH

Hyperbolic Sine

The Hyperbolic Sine routine returns the hyperbolic sine of the input value
specified by floating-point-input-value.

FORMAT

MTH$SINH floating-point-input-value
MTH$DSINH floating-point-input-value
MTH$GSINH floating-point-input-value
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
write only
by value

The hyperbolic sine of f!oating-point-input-nlue, MTH$SINH returns an
F-floating number. MTH$DSINH returns a D-floating number. MTH$GSINH
returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, O_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the address
of a floating-point number that is this value. For MTH$SINH, floatingpoint-input-value specifies an F:-floating number. For MTH$DSINH,
floating-point-input-value specifies a D-floating number. For MTH$GSINH,
floating-point-input-value specifies a G-floating number.

DESCRIPTION

8
8

D
G

MTH-134

MTH$xSINH

The values of the constants in terms of e and fare:
Variable

Value

a(z)

2(-/ /2)

b(z)

CEILING[ (! + 1)/2 * ln{2)]
{2(e- 1) * ln{2))

c(z)

Based on the above definitions, zSINH(X) is computed as follows:
Value of X
IXI

Value Returned

< a(z)

a(z) ~ IXI

< 1.0

zSINH(X) is computed using a
power series expansion in IXl 2

1.0 ~ IXI

< b(z)
b(z) ~ IXI < c(z)

(zEXP(X) - zEXP(-X))/2
SIGN(X) * zEXP(IXl)/2

c(z) ~ IXI

Overflow occurs

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HSINH.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xSINH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library: the
absolute value of floating-point-input-value
is greater than yyy. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism
argument vector CHF$L_MCH_SAVRO/R1. The
result is the floating-point reserved operand unless
you have written a condition handler to change
CHF$L_MCH_SAVRO/R1.
The values of yyy are approximately:
MTH$SINH-88. 722
MTH$DSINH-88. 722
MTH$GSINH-709. 782

MTH-135

MTH$xSQRT

MTH$xSQRT Square Root
The Square Root routine returns the square root of the input value
floating-point-input-value.

FORMAT

MTH$SQRT floating-point-input-value
MTH$DSQRT floating-point-input-value
MTH$GSQRT floating-point-input-value
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$SQRT_R3
MTH$DSQRT_R5
MTH$GSQRT_R5
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f_floating, D_floating, G_floating
write only
by value

The square root of floating-point-input-value. MTH$SQRT returns an Ffloating number. MTH$DSQRT returns a D-floating number. MTH$GSQRT
returns a G-floating number.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

Input value. The floating-point-input-value argument is the address of
a .floating-point number that contains this input value. For MTH$SQRT,
floating-point-input-value specifies an F-floating number. For
MTH$DSQRT, floating-point-input-value specifies a D-floating number.
For MTH$GSQRT, floating-point-input-value specifies a G-floating number.

DESCRIPTION

MTH-136

MTH$xSQRT

If K is even:

x = 212 •P) * F,
zSQRT(X) = 2P * zSQRT(F),
1/2~F < 1, where P = K/2

If K is odd:

x = 2(2•P+1) * F = 2(2•P+2) * (F /2),
zSQRT(X) = 2IP+ll * zSQRT(F /2),
1/4~F/2 < 1/2, where p = (K-1)/2

Let F' = A * F + B, when K is even:
A= 0.95F6198 (hex)
B = 0.6BA5918 (hex)
Let F' =A* (F /2) + B, when K is odd:
A = O.D413CCC (hex)
B = 0.4C 1E248 (hex)
Let K' = P, when K is even
Let K' = P+1, when K is odd

Let Y[O) = 2K' * F' be a straight line approximation within the given interval
using coefficients A and B which minimize the absolu~e error at the midpoint
and endpoint.
Starting with Y[O], n Newton-Raphson iterations are performed:
Y[n + 1] = 1/2 * (Y[n] + X/Y[n])

where n = 2, 3, or 3 for z = F-floating, D-floating, or G-floating, respectively.
The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HSQRT.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xSORT procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_SQUROONEG

MTH-137

MTH$xTAN

Tangent of Angle Expressed in
Radians
The Tangent of Angle Expressed in Radians routine returns the tangent of
a given angle (in radians).

FORMAT

MTH$TAN angle-in-radians
MTH$DTAN angle-in-radians
MTH$GTAN angle-in-radians
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$TAN_R4
MTH$DTAN_R7
MTH$GTAN_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

The tangent of the angle specified by angle-in-radians. MTH$TAN
returns an F-floating number. MTH$DTAN returns a D-floating number.
MTH$GTAN returns a G-floating number.

ARGUMENTS

angle-in-radians
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input angle (in radians). The angle-in-radians argument is the address of
a floating-point number that is this angle. For MTH$TAN, angle-in-radians
specifies an F-floating number. For MTH$DTAN, angle-in-radians specifies a
D-floating number. For MTH$GTAN, angle-in-radians specifies a G-floating
number.

MTH-138

MTH$xTAN

DESCRIPTION

When the input argument is expressed in radians, the tangent function is
computed as follows:
1

If IXI < 2<-!1 2>, then zTAN(X) = X (see the section on MTH$zCOSH
for the definition of f)

Otherwise, call MTH$zSINCOS to obtain zSIN(X) and zCOS(X); then
a. If zCOS(X) = 0, signal overflow
b. Otherwise, zT AN(X) = zSI N(X)/ zCOS(X)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HTAN.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xT AN procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library.

MTH-139

MTH$xTAND

Tangent of Angle Expressed in
Degrees

The Tangent of Angle Expressed in Degrees routine returns the tangent of
a given angle (in degrees).

FORMAT

MTH$TAND angle-in-degrees
MTH$DTAND angle-in-degrees
MTH$GTAND angle-in-degrees
Each of the above formats accepts as input one of the floating-point types.

jsb entries

MTH$TAND_R4
MTH$DTAND_R7
MTH$GTAND_R7
Each of the above JSB entries accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
write only
by value

Tangent of the angle specified by angle-in-degrees. MTH$TAND returns
an F-floating number. MTH$DTAND returns a D-floating number.
MTH$GTAND returns a G-floating number.

ARGUMENTS

angle-in-degrees
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input angle (in degrees). The angle-in-degrees argument is the address
of a floating-point number which is this angle. For MTH$TAND, angle-indegrees specifies an F-floating number. For MTH$DTAND, angle-in-degrees
specifies a D-floating number. For MTH$GTAND, angle-in-degrees specifies
a G-floating number.

MTH-140

MTH$xTAND

DESCRIPTION

When the input argument is expressed in degrees, the tangent function is
computed as follows:
1

If IXI < (180/7r)*2<- 2/<e-l)) and underflow signaling is enabled, underflow
is signaled (see the section on MTH$zCOSH for the definition of e).

Otherwise, if IXI < (180/7r) * 2<-!! 2>, then zTAND(X) = (7r/l80) *X. See
the description of MTH$zCOSH for the definition of f.

Otherwise, call MTH$zSINCOSD to obtain zSIND(X) and zCOSD(X).
a. Then, if zCOSD(X) = 0, signal overflow
b. Else, zTAND(X) = zSIND(X)/zCOSD(X)

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HTAND.

CONDITION
VALUES
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xT AND procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH$_FLOOVEMA T

Floating-point overflow in Math Library.

MTH-141

MTH$xTANH

Compute the Hyperbolic Tangent

The Compute the Hyperbolic Tangent routine returns the hyperbolic
tangent of the input value. ·

FORMAT

MTH$TANH floating-point-input-value
MTH$DTANH floating-point-input-value
MTH $GTANH floating-point-input-value
Each of the above formats accepts as input one of the floating-point types.

RETURNS

VMS usage:
type:
access:
mechanism:
"'1"'1... -

floating_point
f _floating, D_floating, G_floating
write only
by value

1----- ---1--1! -

'--- - - - ' - -£ .tl--.t.!-- --!-.a. !--·-.t. -·-1·--

lilt: uypt:lUUU\,; li:lllot:lll Ui UUCl.U.lll:;;-J!U.U.U-.11.1.1:1u1-vcuu1C.

11. K'T'LJd'.''T'A 1'TLJ

1V.llll..pll"U'lll

returns an F-floating number. MTH$DTANH returns a D-floating number.
MTH$GTANH returns a G-floating number. Unlike the other three routines,
MTH$HTANH returns the hyperbolic tangent by reference in the h-tanh
argument.

ARGUMENTS

floating-point-input-value
VMS usage:
type:
access:
mechanism:

floating_point
f _floating, D_floating, G_floating
read only
by reference

The input value. The floating-point-input-value argument is the
address of a floating-point number that contains this input value. For
MTH$TANH, floating-point-input-value specifies an F-floating number.
For MTH$DTANH, floating-point-input-value specifies a D-floating number.
For MTH$GTANH, floating-point-input-value specifies a G-floating number.

MTH-142

MTH$xTANH

DESCRIPTION

In calculating the hyperbolic tangent of x, the values of g and h are:
z

For MTH$TANH, MTH$DTANH, and MTH$GTANH the hyperbolic tangent
of x is then computed as follows:
Value of x

Hyperbolic Tangent Returned

lxl9-g
2-g < IXI < 0.5

0.5~ IXI < 1.0

xT AN H(X) = X + X 3 * R(X2 ), where R(X 2 ) is a
rational function of X 2 •
xTANH(X) = xTANH(xHI)+xTANH(xLO)*C/B
where C = 1- xTANH(xHI) * xTANH(xHI),

B = 1 + xT AN H(xH I)* xT AN H(xLO),
xHI = 1/2+N/16+1/32 for N=O, 1, ... ,7,
and xLO = X-xHI.

1.0 < IXI < h

xTANH(X) = (xEXP(2*X)-1)/(xEXP(2*X) + 1)

h~IXI

xT AN H(X) = sign(X) * 1

The routine description for the H-floating point version of this routine is listed
alphabetically under MTH$HTANH.

CONDITION
VALUE
SIGNALED

SS$_ROPRAND

Reserved operand. The MTH$xT ANH procedure
encountered a floating-point reserved operand due
to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit
of 1 and a biased exponent of zero. Floating-point
reserved operands are reserved for future use by
DIGITAL.

MTH-143

MTH$UMAX

MTH$UMAX Compute Unsigned Maximum
The Compute Unsigned Maximum routine computes the unsigned
longword maximum of n unsigned longword arguments, where n is greater
than or equal to 1.

FORMAT

MTH$UMAX argument {argument, ... }

RETURNS

VMS usage:
type:
access:
mechanism:

longword_unsigned
longword {unsigned)
write only
by value

Maximum value returned by MTH$UMAX.

ARGUMENTS

argument
VMS usage:
type:
access:
mechanism:

iongword_unsigned
longword {unsigned)
read only
by reference

Argument whose maximum MTH$UMAX computes. Each argument
argument is an unsigned longword that contains one of these values.

argument
VMS usage:
type:
access:
mechanism:

longword_unsigned
longword {unsigned)
read only
by reference

Additional arguments whose maximum MTH$UMAX computes. Each
argument argument is an unsigned longword that contains one of these
values.

DESCRIPTION
CONDITION
VALUES
RETURNED

MTH-144

MTH$UMAX is the unsigned version of MTH$JMAXO.

None.

MTH$UMIN

Compute Unsigned Minimum
The Compute Unsigned Minimum routine computes the unsigned longword
minimum of n unsigned longword arguments, where n is greater than or
equal to 1.

FORMAT

MTH$UMIN

RETURNS

VMS usage:
type:
access:
mechanism:

argument [argument, ... ]

longword_unsigned
longword (unsigned)
write only
by value

Minimum value returned by MTH$UMIN.

ARGUMENTS

argument
VMS usage:
type:
access:
mechanism:

longword_unsigned
longword (unsigned)
read only
by reference

Argument whose minimum MTH$UMIN computes. Each argument
argument is an unsigned longword that contains one of these values.

argument
VMS usage:
type:
access:
mechanism:

longword_unsigned
longword (unsigned)
read only
by reference

Additional arguments whose minimum MTH$UMIN computes. Each
argument argument is an unsigned longword that contains one of these
values.

DESCRIPTION
CONDITION
VALUES
RETURNED

MTH$UMIN is the unsigned version of MTH$JMINO.

None.

MTH-145

Undocumented MTH$ Routines

This appendix lists all of the entry point and argument information for the
MTH$ routines not documented in the MTH$ Reference Section of this
manual.
Table A-1

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$ABS

F-f/oating Absolute Value Routine
Format:

MTH$ABS f-floating

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

Format:

MTH$DABS d-floating

Returns:

floating_point, D_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$DABS

D-f/oating Absolute Value Routine

G-f/oating Absolute Value Routine

MTH$GABS
Format:

MTHSGABS g-floating

Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference
H-f/oating Absolute Value Routine

MTH$HABS
Format:

MTHSABS h-abs-val, h-floating

Returns:

None

h-abs-val:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference

Format:

MTH$11ABS word

Returns:

word_signed, word (signed), write only, by value

word:

word_signed, word (signed), read only, by reference

MTH$11ABS

Word Absolute Value Routine

Longword Absolute Value Routine

MTH$JIABS
Format:

MTH$JIABS longword

Returns:

longword_signed, longword (signed), write only, by value

longword:

longword_signed, longword (signed), read only, by reference

A-1

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

Bitwise AND of Two Word Parameters Routine

MTH$11AND

Format:

MTHSllAND word 1, word2

Returns:

word_unsigned, word (unsigned), write only, by value

word1:

word_unsigned, word (unsigned), read only, by reference

word2:

word_unsigned, word (unsigned), read only, by reference

Format:

MTH$JIAN D longword 1, longword2

Returns:

longword_unsigned, longword (unsigned), write only, by value

longword1:

longword_unsigned, longword (unsigned), read only, by reference

longword2:

longword_unsigned, longword (unsigned), read only, by reference

Bitwise AND of Two Longword Parameters Routine

MTH$JIAND

Convert F-f/oating to D-f/oating (Exact) Routine

MTH$DBLE

Format:

MTH$DBLE f-floating

Returns:

floating_point, O_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$GDBLE

Convert F-f/oating to G-f/oating (Exact) Routine

Format:

MTH$GDBLE f-floating

Returns:

floating_point, G_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

Format:

MTH$DIM f-floating 1, f-floating2

Returns:

floating_point, F_floating, write only, by value

f-floating1:

floating_point, F_floating, read only, by reference

f-floating2:

floating_point, F_floating, read only, by reference

MTH$DIM

Positive Difference of Two F-floating Parameters Routine

Positive Difference of Two D-f/oating Parameters Ro~tine

MTH$DDIM

A-2

Format:

MTH$DDIM d-floating 1, d-floating2

Returns:

floating_point, O_floating, write only, by value

d-floating1:

floating_point, O_floating, read only, by reference

d-floating2:

floating_point, O_floating, read only, by reference

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$GDIM

Positive Difference of Two G-floating Parameters Routine
Format:

MTH$GDIM g-floating1, g-floating2

Returns:

floating_point, G_floating, write only, by value

g-floating1:

floating_point, G_floating, read only, by reference

g-floating2:

floating_point, G_floating, read only, by reference

MTH$HDIM

Positive Difference of Two H-floating Parameters Routine
Format:

MTH$HDIM h-floating, h-floating 1, h-floating2

Returns:

None

h-floating:

floating_point, H_floating, write only, by reference

h-floating1 :

floating_point, H_floating, read only, by reference

h-floating2:

floating_point, H_floating, read only, by reference

MTH$11DIM

Positive Difference of Two Word Parameters Routine
Format:

MTH$11DIM word1, word2

Returns:

word_signed, word (signed), write only, by value

word1:

word_signed, word (signed), read only, by reference

word2:

word_signed, word (signed), read only, by reference

MTH$JIDIM

Positive Difference of Two Longword Parameters Routine
Format:

MTH$JI DIM longword 1, longword2

Returns:

longword_signed, longword (signed), write only, by value

longword1:

longword_signed, longword (signed), read only, by reference

longword2:

longword_signed, longword (signed), read only, by reference

MTH$11EOR

Bitwise Exclusive OR of Two Word Parameters Routine
Format:

MTH$11EOR word1, word2

Returns:

word_unsigned, word (unsigned), write only, by value

word1:

word_unsigned, word (unsigned), read only, by reference

word2:

word_unsigned, word (unsigned), read only, by reference

MTH$JIEOR

Bitwise Exclusive OR of Two Longword Parameters Routine
Format:

MTH$JIEOR longword1, longword2

Returns:

longword_unsigned, longword (unsigned), write only, by value

longword1:

longword_unsigned, longword (unsigned), read only, by reference

longword2:

longword_unsigned, longword (unsigned), read only, by reference

A-3

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$11FIX

Convert F-f/oating to Word (Truncated) Routine
Format:

MTH$11FIX f-floating

Returns:

word_signed, word (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference
Convert F-f/oating to Longword (Truncated) Routine

MTH$JIFIX

Format:

MTH$JIFIX f-floating

Returns:

longword_signed, longword (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference
Convert Word to F-f/oating (Exact) Routine

MTH$FLOATI

Format:

MTH$FLOATI word

Returns:

floating_point, F_floating, write only, by value

word:

word_signed, word (signed), read only, by reference
Convert Word to D-f/oating (Exact) Routine

MTH$DFLOTI

Format:

MTH$DFLOTI word

Returns:

floating_point, D_floating, write only, by value

word:

word_signed, word (signed), read only, by reference

Format:

MTH$GFLOTI word

Returns:

floating_point, G_floating, write only, by value

word:

word_signed, word (signed), read only, by reference

Convert Word to G-f/oating (Exact) Routine

MTH$GFLOTI

Convert Longword to F-f/oating (Exact) Routine

MTH$FLOATJ

Format:

MTH$FLOATJ longword

Returns:

floating_point, F_floating, write only, by value

longword:

longword_signed, longword (signed), read only, by reference
Convert Longword to D-f/oating (Exact) Routine

MTH$DFLOTJ

A-4

Format:

MTH$DFLOTJ longword

Returns:

floating_point, D_floating, write only, by value

longword:

longword_signed, longword (signed), read only, by reference

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$GFLOTJ

Convert Longword to G-f/oating (Exact) Routine

Format:

MTH$GFLOTJ longword

Returns:

floating_point, G_floating, write only, by value

longword:

longword_signed, longword (signed), read only, by reference
Convert F-f/oating to Greatest F-floating Integer Routine

MTH$FLOOR

Format:

MTH$FLOOR f-floating

JSB:

MTH$FLOQR_R1 f-floating

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference
Convert D-f/oating to Greatest D-floating Integer Routine

MTH$DFLOOR

Format:

MTH$DFLOOR d-floating

JSB:

MTH$DFLOQR_R3 d-floating

Returns:

floating_point, O_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference
Convert G-floating to Greatest G-floating Integer Routine

MTH$GFLOOR

Format:

MTH$GFLOOR g-floating

JSB:

MTH$GFLOOR_R3 g-floating

Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference

Format:

MTH$HFLOOR max-h-float, h-floating

JSB:

MTH$HFLOQR_R7 h-floating

Convert H-f/oating to Greatest H-floating Integer Routine

MTH$HFLOOR

Returns:

None

max-h-float:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference
Convert F-floating to Truncated F-floating Routine

MTH$AINT

Format:

MTH$AI NT f-floating

JSB:

MTH$AINT_R2 f-floating

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

A-5

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$DINT

Convert 0-f/oating to Truncated 0-f/oating Routine
Format:

MTH$DINT d-floating

JSB:

MTH$DINT_R4 d-floating

Returns:

floating_point, D_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$11DINT

Convert 0-f/oating to Word (Truncated) Routine
Format:

MTH$11DINT d-floating

Returns:

word_signed, word (signed), write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$JIDINT

Convert 0-f/oating to Longword (Truncated) Routine
Format:

MTH$JIDINT d-floating

D ......+ ..........

1,....,,,..,
.. ,,.. .. ..1 .,.;,..,..,,n..I lnnn\A1nr..I /.,.innorH \Alri-to nnlu hu u!:lll110
IU11!:fVVUIU-'11!:f11'JU1 IVll~YVVl'-A \~l::JlllV'-Al1 vv111.v " " ' " J I ,_,,
""""'"'9""

d-floating:

floating_point, D_floati.ng, read only, by reference

1ui::;11.u111~.

MTH$GINT

Convert G-f/oating to G-f/oating (Truncated) Routine
Format:

MTH$GINT g-floating

JSB:

MTH$GINT_R4 g-floating

Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference

MTH$11GINT

Convert G-f/oating to Word (Truncated) Routine
Format:

MTH$11GINT g-floating

Returns:

word_signed, word (signed), write only, by value

g-floating:

floating_point, G_floating, read only, by reference

MTH$JIGINT

Convert G-floating to Longword (Truncated) Routine
Format:
Returns:

longword_signed, longword (signed), write only, by value

g-floating:

floating_point, G_floating, read only, by reference

Format:

MTH$HINT trunc-h-flt, h-floating

JSB:

MTH$HI NT_RS h-floating

Returns:

None

trunc-h-flt:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference

MTH$HINT

A-6

MTH$JIGINT g-floating

Convert H-f/oating to H-f/oating (Truncated) Routine

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$11HINT

Convert H-f/oating to Truncated Word Routine
Format:

MTH$11HINT h-floating

Returns:

word_signed, word (signed), write only, by value

h-floating:

floating_point, H_floating, read only, by reference
Convert H-f/oating to Truncated Longword Routine

MTH$JIHINT

Format:

MTH$JIHINT h-floating

Returns:

longword_signed, longword (signed), write only, by value

h-floating:

floating_point, H_floating, read only, by reference

Format:

MTH$11NT f-floating

Returns:

word_signed, word (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

Convert F-f/oating to Word (Truncated) Routine

MTH$11NT

Convert F-f/oating to Longword (Truncated) Routine

MTH$JINT

Format:

MTH$JINT f-floating

Returns:

longword_signed, longword (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference
Bitwise Inclusive OR of Two Word Parameters Routine

MTH$110R

Format:

MTH$110R word 1, word2

Returns:

word_unsigned, word (unsigned), write only, by value

word1:

word_unsigned, word (unsigned), read only, by reference

word2:

word_unsigned, word (unsigned), read only, by reference
Bitwise Inclusive OR of Two Longword Parameters Routine

MTH$JIOR

Format:

MTH$JIOR longword 1, longword2

Returns:

longword_unsigned, longword (unsigned), write only, by value

longword1:

longword_unsigned, longword (unsigned), read only, by reference

longword2:

longword_unsigned, longword (unsigned), read only, by reference
F-f/oating Maximum of N Word Parameters Routine

MTH$AIMAXO

Format:

MTH$AIMAXO word, ...

Returns:

floating_point, F_floating, write only, by value

word:

word_signed, word (signed), read only, by reference

A-7

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$AJMAXO

F-f/oating Maximum of N Longword Parameters Routine
Format:

MTH$AJMAXO longword, ...

Returns:

floating_point, F_floating, write only, by value

longword:

longword_signed, longword (signed), read only, by reference

Format:

MTH$1MAXO word, ...

Returns:

word_signed, word (signed), write only, by value

word:

word_signed, word (signed), read only, by reference

Word Maximum of N Word Parameters Routine

MTH$1MAXO

Longword Maximum of N Longword Parameters Routine

MTH$JMAXO
Format:

MTH$JMAXO longword, ...

Returns:

longword_signed, longword (signed), write only, by value

longword:

longword_signed, longword (signed), read only, by reference

MTH$AMAX1

F-f/oating Maximum of N F-floating Parameters Routine
Format:

MTH$AMAX1 f-floating, ...

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference
D-f/oating Maximum of N D-floating Parameters Routine

MTH$DMAX1
Format:

MTH$DMAX1 d-floating, ...

Returns:

floating~point, O_floating, write only, by value

d-floating:

floating_point, O_floating, read only, by reference
G-f/oating Maximum of N G-f/oating Parameters Routine

MTH$GMAX1
Format:

MTH$GMAX1 g-floating, ...

Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference
H-floating Maximum of N H-f/oating Parameters Routine

MTH$HMAX1

A-8

Format:

MTH$HMAX1 h-float-max, h-floating, ...

Returns:

None

h-float-max:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$1MAX1

Word Maximum of N F-f/oating Parameters Routine

Format:

MTH$1MAX1 f-floating, ...

Returns:

word_signed, word (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$JMAX1

Longword Maximum of N F-f/oating Parameters Routine

Format:

MTH$JMAX1 f-floating, ...

Returns:

longword_signed, longword (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$AIMINO

F-f/oating Minimum of N Word Parameters Routine

Format:

MTH$AIMINO word, ...

Returns:

floating_point, F_floating, write only, by value

word:

word_signed, word (signed), read only, by reference

Format:

MTH$AJMINO longword, ...

Returns:

floating_point, F_floating, write only, by value

longword:

longword_signed, longword (signed), read only, by reference

MTH$AJMINO

F-f/oating Minimum of N Longword Parameters Routine

MTH$1MINO

Word Minimum of N Word Parameters Routine

Format:

MTH$1MINO word, ...

Returns:

word_signed, word (signed), write only, by value

word:

word_signed, word (signed), read only, by reference

MTH$JMINO

Longword Minimum of N Longword Parameters Routine

Format:

MTH$JMINO longword, ...

Returns:

longword_signed, longword (signed), write only, by value

longword:

longword_signed, longword (signed), read only, by reference

MTH$AMIN1

F-f/oating Minimum of N F-f/oating Parameters Routine

Format:

MTH$AMIN1 f-floating, ...

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

A-9

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$DMIN1

D-f/oating Minimum of N D-f/oating Parameters Routine
Format:

MTH$DMI N 1 d-floating, ...

Returns:

floating_point, D_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$GMIN1

G-f/oating Minimum of N G-f/oating Parameters Routine
Format:
Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference
H-f/oating Minimum of N H-floating Parameters Routine

MTH$HMIN1
Format:

MTH$HMIN1 h-float-max, h-floating, ...

Returns:

None

h-float-max:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference
Word Minimum of N F-f/oating Parameters Routine

MTH$1MIN1
Format:

MTH$1MIN1 f-floating, ...

Returns:

word_signed, word (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference
Longword Minimum of N F-f/oating Parameters Routine

MTH$JMIN1
Format:

MTH$JMIN1 f-floating, ...

Returns:

longword_signed, longword (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

Format:

MTH$AMOD dividend, divisor

Returns:

floating_point, F_floating, write only, by value

dividend:

floating_point, F_floating, read only, by reference

divisor:

floating_point, F_floating, read only, by reference

Format:

MTH$DMOD dividend, divisor

Returns:

floating_point, D_floating, write only, by value

dividend:

floating_point, D_floating, read only, by reference

divisor:

floating_point, D_floating, read only, by reference

Remainder of Two F-f/oating Parameters Routine

MTH$AMOD

Remainder of Two D-f/oating Parameters Routine

MTH$DMOD

A-10

MTH$GMIN1 g-floating, ...

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines
Entry Point Information

Routine Name

MTH$GMOD

Remainder of Two G-f/oating Parameters Routine
Format:

MTH$GMOD dividend, divisor

Returns:

floating_point, G_floating, write only, by value

dividend:

floating_point, G_floating, read only, by reference

divisor:

floating_point, G_floating, read only, by reference

Format:

MTHSHMOD h-mod, dividend, divisor

Returns:

None

h-mod:

floating_point, H_floating, write only, by reference

Remainder of Two H-f/oating Parameters Routine

MTH$HMOD

dividend:

floating_point, H_floating, read only, by reference

divisor:

floating_point, H_floating, read only, by reference

MTH$1MOD

Remainder of Two Word Parameters Routine
Format:

MTH$1MOD dividend, divisor

Returns:

word_signed, word (signed), write only, by value

dividend:

word_signed, word (signed), read only, by reference

divisor:

word_signed, word (signed), read only, by reference

Format:

MTH$JMOD dividend, divisor

Returns:

longword_signed, longword (signed), write only, by value

dividend:

longword_signed, longword (signed), read only, by reference

divisor:

longword_signed, longword (signed), read only, by reference

Format:

MTH$ANINT f-floating

Returns:

floating_point, F_floating, write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$JMOD

Remainder of Two Longword Parameters Routine

MTH$ANINT

Convert F-f/oating to Nearest F-f/oating Integer Routine

MTH$DNINT

Convert D-floating to Nearest D-f/oating Integer Routine
Format:

MTH$DNINT d-floating

Returns:

floating_point, D_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference

A-11

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$11DNNT

Convert 0-floating to Word Integer Routine

Format:

MTH$11DNNT d-floating

Returns:

word_signed, word (signed), write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$JIDNNT

Convert D-floating to Nearest Longword Integer Routine

Format:

MTH$JIDNNT d-floating

Returns:

longword_signed, longword (signed), write only, by value

d-floating:

floating_point, D_floating, read only, by reference
Convert G-f/oating to Nearest G-f/oating Integer Routine

MTH$GNINT

Format:
Returns:

floating_point, G_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference

MTH$11GNNT

Convert G-f/oating to Nearest Word Integer Routine

Format:

MTH$11GNNT g-floating

Returns:

word_signed, word (signed), write only, by value

g-floating:

floating_point, G_floating, read only, by reference

MTH$JIGNNT

Convert G-floating to Nearest Longword Integer Routine

Format:

MTH$JIGNNT g-floating

Returns:

longword_signed, longword (signed), write only, by value

g-floating:

floating_point, G_floating, read only, by reference

MTH$HNINT

Convert H-f/oating to Nearest H-f/oating Integer Routine

Format:

MTH$HNINT nearst-h-flt, h-floating

Returns:

None

nearst-h-flt:

floating_point, H_floating, write only, by reference

h-floating:

floating_point, H_floating, read only, by reference

Format:

MTH$11HNNT h-floating

Returns:

word_signed, word (signed), write only, by value

h-floating:

floating_point, H_floating, read only, by reference

Convert H-floating to Nearest Word Integer Routine

MTH$11HNNT

A-12

MTH$GNINT g-floating

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$JIHNNT

Convert H-floating to Nearest Longword Integer Routine
Format:

MTH$JIHNNT h-floating

Returns:

longword_signed, longword (signed), write only, by value

h-floating:

floating_point, H_floating, read only, by reference

Format:

MTH$1NINT f-floating

Returns:

word_signed, word (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$1NINT

Convert F-floating to Nearest Word Integer Routine

MTH$JNINT

Convert F-floating to Nearest Longword Integer Routine
Format:

MTH$JNINT f-floating

Returns:

longword_signed, longword (signed), write only, by value

f-floating:

floating_point, F_floating, read only, by reference

MTH$1NOT

Bitwise Complement of Word Parameter Routine
Format:

MTH$1NOT word

Returns:

word_unsigned, word (unsigned), write only, by value

word:

word_unsigned, word (unsigned), read only, by reference

MTH$JNOT

Bitwise Complement of Longword Parameter Routine
Format:

MTH$JNOT longword

Returns:

longword_unsigned, longword (unsigned), write only, by value

longword:

longword_unsigned, longword (unsigned), read only, by reference

MTH$DPROD

D-floating Product of Two F-floating Parameters Routine
Format:

MTH$DPROD f-floating 1, f-floating2

Returns:

floating_point, D_floating, write only, by value

f-floating 1 :

floating_point, F_floating, read only, by reference

f-floating2:

floating_point, F_floating, read only, by reference

MTH$GPROD

G-f/oating Product of Two F-floating Parameters Routine
Format:

MTHSGPROD f-floating 1, f-floating2

Returns:

floating_point, G_floating, write only, by value

f-floating 1 :

floating_point, F_floating, read only, by reference

f-floating2:

floating_point, F_floating, read only, by reference

A-13

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

F-f/oating Sign Function

MTH$SGN

Format:

MTH$SGN f-floating

Returns:

longword_signed, longword (signed), write only, by reference

f-floating:

floating_point, F_floating, read only, by reference
D-f/oating Sign Function

MTH$SGN

Format:

MTH$SGN d-floating

Returns:

longword_signed, longword (signed), write only, by reference

d-floating:

floating_point, D_floating, read only, by reference
Bitwise Shift of Word Routine

MTH$11SHFT

Format:

MTH$11SHFT word, shift-cnt

Returns:

vvord_unsigned, word (unsigned), write only: by value

word:

word_unsigned, word (unsigned), read only, by reference

shift-cnt:

word_signed, word (signed), read only, by reference
Bitwise Shift of Longword Routine

MTH$JISHFT

Format:

MTH$JISHFT longword, shift-cnt

Returns:

longword_unsigned, longword (unsigned), write only, by value

longword:

longword_unsigned, longword (unsigned), read only, by reference

shift-cnt:

longword_signed, longword (signed), read only, by reference
F-floating Transfer of Sign of Y to Sign of X Routine

MTH$SIGN

Format:

MTH$SIGN f-float-x, f-float-y

Returns:

floating_point, F_floating, write only, by value

f-float-x:

floating_point, F_floating, read only, by reference

f-float-y:

floating_point, F_floating, read only, by reference

Format:

MTH$DSIGN d-float-x, d-float-y

Returns:

floating_point, D_floating, write only, by value

d-float-x:

floating_point, D_floating, read only, by reference

d-float-y:

floating_point, D_floating, read only, by reference

MTH$DSIGN

A-14

D-f/oating Transfer of Sign of Y to Sign of X Routine

Undocumented MTH$ Routines

Table A-1 (Cont.)

Undocumented MTH$ Routines

Routine Name

Entry Point Information

MTH$GSIGN

G-f/oating Transfer of Sign of Y to Sign of X Routine

Format:

MTH$GSIGN g-float-x, g-float-y

Returns:

floating_point, G_floating, write only, by value

g-float-x:

floating_point, G_floating, read only, by reference

g-float-y:

floating_point, G_floating, read only, by reference

MTH$HSIGN

H-f/oating Transfer of Sign of Y to Sign of X Routine

Format:

MTH$HSIGN h-result, h-float-x, h-float-y

Returns:

None

h-result:

floating_point, H_floating, write only, by reference

h-float-x:

floating_point, H_floating, read only, by reference

h-float-y:

floating_point, H_floating, read only, by reference

MTH$11SIGN

Word Transfer of Sign of Y to Sign of X Routine

Format:

MTH$11SIGN word-x, word-y

Returns:

word_signed, word (signed), write only, by value

word-x:

word_signed, word (signed), read only, by reference

word-y:

word_signed, word (signed), read only, by reference

MTH$JISIGN

Longword Transfer of Sign of Y to Sign of X Routine

Format:

MTH$JISIGN longwrd-x, longwrd-y

Returns:

longword_signed, longword (signed), write only, by reference

longwrd-x:

longword_signed, longword (signed), read only, by reference

longwrd-y:

longword_signed, longword (signed), read only, by reference

MTH$SNGL

Convert D-floating to F-f/oating (Rounded) Routine

Format:

MTH$SNGL d-floating

Returns:

floating_point, F_floating, write only, by value

d-floating:

floating_point, D_floating, read only, by reference

MTH$SNGLG

Convert G-f/oating to F-f/oating (Rounded) Routine

Format:

MTH$SNGLG g-floating

Returns:

floating_point, F_floating, write only, by value

g-floating:

floating_point, G_floating, read only, by reference

A-15

Index

Absolute value
See also Mathematics routine
of complex number• MTH-23
Arc cosine
in degrees• MTH-6, MTH-71
in radians• MTH-3, MTH-69
Arc sine
in degrees • MTH-11 , MTH-7 5
in radians•MTH-9, MTH-73
Arc tangent
hyperbolic• MTH-21, MTH-85
in degrees•MTH-15, MTH-19, MTH-79,
MTH-83
in radians• MTH-13, MTH-17, MTH-77,
MTH-81
Arrays
conversion of• MTH-64

Double-precision value
converting• MTH-62
converting an array of• MTH-64

c
Complex number• 1-3, MTH-5 7, MTH-59,
MTH-112, MTH-122
absolute value of• MTH-23
complex exponential of• MTH-31 , MTH-33
conjugate of• MTH-44, MTH-45
cosine of• MTH-26, MTH-28
made from floating-point• MTH-40, MTH-42
natural logarithm of• MTH-36, MTH-38
sine of• MTH-53, MTH-54
Conjugate of complex number• MTH-44,
MTH-45
Cosine
in radians• MTH-126
Cosine
hyperbolic• MTH-51 , MTH-89
in degrees• MTH-49, MTH-88, MTH-129
in radians• MTH-4 7, MTH-87
of complex number• MTH-26, MTH-28

E
Exponential• MTH-66, MTH-91
of complex number• MTH-31, MTH-33

H
Hyperbolic arc tangent• MTH-21, MTH-85
Hyperbolic cosine• MTH-51, MTH-89
Hyperbolic sine• MTH-102, MTH-134
Hyperbolic tangent• MTH-110, MTH-142

L
Logarithm
base 2 • MTH-95, MTH-116
common• MTH-97, MTH-118
natural• MTH-93, MTH-114
natural complex• MTH-36, MTH-38

M
Mathematics routine • 1-1
absolute value• 1-4
algorithm• 1-2
bitwise AND operator• 1-4
bitwise complement operator• 1-8
bitwise exclusive OR operator• 1-5
bitwise inclusive OR operator• 1-6
bitwise shift • 1-8
calling convention• 1-2
complex number• 1-3

lndex-1

Index

Mathematics routine (cont'd.)
condition handling • 1-3
conversion of double to single floating-point
value• 1-9
conversion to greatest floating-point integer•
1-5
entry point name • 1-1
F-floating conversion • 1-4
floating-point conversion to nearest value• 1-7
floating-point multiplication• 1-8
floating-point positive difference• 1-5
floating-point sign function• 1-8
integer to floating-point conversion• 1-5
JSB entry point• 1-2
maximum value• 1-6
minimum value• 1-7
remainder• 1-7
truncation of floating-point value• 1-6
undocumented routines• A-1 to A-15
list of• 1-4 to 1-9
MTH$ACOS • MTH-3
MTH$ACOSD • MTH-6
MTH$AIMAG • MTH-112
MTH$ALOG•MTH-114
MTH$ALOG 10 • MTH-118
MTH$ALOG2 • MTH-116
MTH$ASIN • MTH-9
MTH$ASIND • MTH-11
MTH$AT AN• MTH-13
MTH$ATAN2•MTH-17
MTH$AT AND• MTH-15
MTH$ATAND2•MTH-19
MTH$AT ANH• MTH-21
MTH$CABS • MTH-23
MTH$CCOS • MTH-26
MTH$CDABS•MTH-23
MTH$CDCOS i MTH-28
MTH$CDEXP•MTH-33
MTH$CDLOG • MTH-38
MTH$CDSIN • MTH-54
MTH$CDSORT•MTH-59
MTH$CEXP • MTH-31
MTH$CGABS • MTH-23
MTH$CGCOS • MTH-28
MTH$CGEXP•MTH-33
MTH$CGLOG • MTH-38
MTH$CGSIN • MTH-54
MTH$CGSORT•MTH-59
MTH$CLOG • MTH-36
MTH$CMPLX • MTH-40
MTH$CONJG • MTH-44
MTH$COS • MTH-4 7

lndex-2

!\llTH$COSD • MTH-49
MTH$COSH • MTH-51
MTH$CSIN • MTH-53
MTH$CSORT • MTH-5 7
MTH$CVT_DA_GA • MTH-64
MTH$CVT_D_G•MTH-62
MTH$CVT_GA_DA • MTH-64
MTH$CVT_G_D • MTH-62
MTH$DACOS • MTH-3
MTH$DACOSD•MTH-6
MTH$DASIN • MTH-9
MTH$DASIND • MTH-11

MTH$DATAN•MTH~13

MTH$DA T AN2 • MTH-17
MTH$DAT AND• MTH-15
MTH$DATAND2•MTH-19
MTH$DATANH•MTH-21
MTH$DCMPLX • MTH-42
MTH$DCONJG • MTH-45
MTH$DCOS. MTH-4 7
MTH$DCOSD. MTH-49
MTH$DCOSH • MTH-51
MTH$DEXP • MTH-66
MTH$DIMAG • MTH-112
MTH$DLOG • MTH-114
MTH$DLOG 10 • MTH-118
MTH$DLOG2 • MTH-116
MTH$DREAL•MTH-122
MTH$DSIN • MTH-124
MTH$DSINCOS • MTH-126
MTH$DSINCOSD • MTH-129
MTH$DSIND • MTH-132
MTH$DSINH • MTH-134
MTH$DSORT • MTH-136
MTH$DTAN•MTH-138
MTH$DT AND• MTH-140
MTH$DTANH•MTH-142
MTH$EXP • MTH-66
MTH$GACOS • MTH-3
MTH$GACOSD•MTH-6
MTH$GASIN • MTH-9
MTH$GASIND • MTH-11
MTH$GA TAN• MTH-13
MTH$GATAN2•MTH-17
MTH$GATAND•MTH-15
MTH$GATAND2•MTH-19
MTH$GA TANH• MTH-21
MTH$GCMPLX • MTH-42
MTH$GCONJG • MTH-45
MTH$GCOS • MTH-4 7
MTH$GCOSD • MTH-49

Index

MTH$GCOSH•MTH-51
MTH$GEXP • MTH-66
MTH$GIMAG • MTH-112
MTH$GLOG • MTH-114
MTH$GLOG 10 • MTH-118
MTH$GLOG2 • MTH-116
MTH$GREAL•MTH-122
MTH$GSIN • MTH-124
MTH$GSINCOS • MTH-126
MTH$GSINCOSD • MTH-129
MTH$GSIND • MTH-132
MTH$GSINH • MTH-134
MTH$GSQRT • MTH-136
MTH$GT AN• MTH-138
MTH$GT AND• MTH-140
MTH$GT ANH• MTH-142
MTH$HACOS•MTH-69
MTH$HACOSD•MTH-71
MTH$HASIN • MTH-73
MTH$HASIND • MTH-7 5
MTH$HATAN•MTH-77
MTH$HATAN2•MTH-81
MTH$HATAND•MTH-79
MTH$HATAND2•MTH-83
MTH$HATANH•MTH-85
MTH$HCOS • MTH-87
MTH$HCOSD•MTH-88
MTH$HCOSH • MTH-89
MTH$HEXP • MTH-91
MTH$HLOG • MTH-93
MTH$HLOG 10 • MTH-97
MTH$HLOG2•MTH-95
MTH$HSIN • MTH-99
MTH$HSINCOS • MTH-126
MTH$HSINCOSD • MTH-129
MTH$HSIND • MTH-100
MTH$HSINH • MTH-102
MTH$HSQRT•MTH-104
MTH$HT AN• MTH-106
MTH$HT AND• MTH-108
MTH$HT ANH • MTH-110
MTH$RANDOM • MTH-120
MTH$REAL • MTH-122
MTH$SIN • MTH-124
MTH$SINCOS • MTH-126
MTH$SINCOSD • MTH-129
MTH$SIND • MTH-132
MTH$SINH • MTH-134
MTH$SQRT • MTH-136
MTH$T AN• MTH-138
MTH$T AND• MTH-140

MTH$TANH•MTH-142
MTH$UMAX • MTH-144
MTH$UMIN • MTH-145

R
Random number generator• MTH-120

s
Sine
hyperbolic• MTH-102, MTH-134
in degrees• MTH-100, MTH-129, MTH-132
in radians•MTH-99, MTH-124, MTH-126
of complex number• MTH-53, MTH-54
Square root• MTH-104, MTH-136

T
Tangent• MTH-106, MTH-108, MTH-138,
MTH-140
hyperbolic• MTH-110, MTH-142

lndex-3

Reader's Comments

VMS RTL Mathematics
(MTH$) Manual
AA-LA 72A-TE

Please use this postage-paid form to comment on this manual. If you require a written reply to a software
problem and are eligible to receive one under Software Performance Report (SPR) service, submit your
comments on an SPR form.
Thank you for your assistance·.

I rate this manual's:
Accuracy (software works as manual says)
Completeness (enough information)
Clarity (easy to understand)
Organization (structure of subject matter)
Figures (useful)
Examples (useful)
Index (ability to find topic)
Page layout (easy to find information)

Excellent

Good

Fair

Poor

D
D
D
D
D
D
D
D

I would like to see more /less

What I like best about this manual is

What I like least about this manual is

I found the following errors in this manual:
Page
Description

Additional comments or suggestions to improve this manual:

I am using Version _ _ _ of the software this manual describes.

Name/Title

Dept.

Company

Date

Mailing Address
Phone

I
I
I
I
I
I

-- Do Not Tear - Fold Here and Tape

-------------------[lllr---------------1

mamaomn™

No Postage

~:~=j~=~y
in the

United States

BUSINESS REPLY MAIL
FIRST CLASS PERMIT NO. 33 MAYNARD MASS.
POST AGE WILL BE PAID BY ADDRESSEE

DIGIT AL EQUIPMENT CORPORATION
Corporate User Publications-Spit Brook
ZK01-3/J35 110 SPIT BROOK ROAD
NASHUA, NH 03062-9987

111 ••••• 11.11 •••• 11 •••• 1.11.1 •• 1.1 •• 1••.1.1 ••• 1.11 •• 1
·- Do Not Tear - Fold Here - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I
I
I