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Document:	Common Math Library Reference Manual Sep83
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Pages:	292
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TOPS-10/TOPS-20
Common Math Library
Reference Manual
Order No. AA-M400A-TK

September 1983
Abstract
This manual describes the mathematical routines that constitute the
TOPS-10ITOPS-20 Math Library.

OPERATING SYSTEM:

TOPS-20 Version 5.0 and 5.1
TOPS-10 Version 7.01A

SOFTWARE:

FORTRAN-10/20 Version 7
Pascal-10/20 Version 1

Software and manuals should be ordered by title and order number. In the United States. send orders
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First Printing, September 1983

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Contents
Page

Chapter 1 Introduction

1.1
1.2
1.3

1.4

1.5

The Math Library .
Math Symbols and Names used in Equations.
Data Types and Their Precision
1.3.1 Integer.
1.3.2 Single-Precision, Floating-Point .
1.3.3 Double-Precision, D-Floating-Point
1.3.4 Double-Precision, G-Floating-Point
1.3.5 Complex.
1.3.6 Complex, Double-Precision
Information About the Routines
1.4.1 Calling Sequence .
1.4.2 Entry Points .
1.4.3 Return Location
1.4.4 Register Usage.
Accuracy Tests

. 1-:~
. 1-9
1-10
1-10
I--H)

1-11
1--11
1-12
1-12
1-12
1-1~~

1-13
1-13
1-13
1-14

Chapter 2 Square Root Routines
SQRT . .
DSQRT.
GSQRT.
CSQRT.
CDSQRT
CGSQRT

· 2-3

· 2-5
· 2--7

.2-9
2-11
2-13

Chapter 3 Logarithm Routines
ALOG . .
ALOG10.
DLOG .
DLOG10
GLOG .
GLOG10
CLOG . .
CDLOG.
CGLOG.

· 3-:3

· 3-5
.3-7
.3-9
3-11
3-13
3-15
3-17
3-19

iii

Chapter 4 Exponential and Exponentiation Routines
EXP .
DEXP .
GEXP .
CEXP . .
CDEXP.
CGEXP.
EXPl. .
EXP2 . .
DEXP2 ..
GEXP2 ..
CEXP2 ..
EXP3 . .
DEXP3 ..
GEXP3 ..
CEXP3 ..

.4-3
.4-5
.4-7
.4-9
4-11
4-13
4-15
4-16
4-18
4-20
4-22
4-25
4-28
4-31
4-34

Chapter 5 Trigonometric Routines
SIN ..
SIND.
COS.
COSD.
DSIN.
DCOS.
GSIN.
GCOS.
CSIN.
CCOS.
CDSIN
CDCOS.
CGSIN .
CGCOS.
TAN . .
COTAN.
DTAN .
DCOTAN.
GTAN . .
GCOTAN.

.5-3
.5-5
.5-7
.5-9
5-11
5-13
5-15
5-17
5-19
5-21
5-23
5-25
5-27
5-29
5-31
5-33
5-35
5-37
5-39
5-41

Chapter 6 Inverse Trigonometric Routines
ASIN . .
ACOS . .
DASIN .
DACOS.
GASIN .
GACOS.
ATAN .
ATAN2 .
DATAN.
DATAN2
GATAN.
GATAN2
Iv

.6-3
.6-4
.6-5
.6-7
.6-9
6-11
6-13
6-15
6-17
6-19
6-21
6-23

Chapter 7 Hyperbolic Routines
SINH . .
COSH . .
DSINH.
DCOSH.
GSINH .
GCOSH . .
TANH.
DTANH.
GTANH.

· 7-3
.7-4
· 7-5

· 7-7
.7-8
7-10
7-11
7-12

7-13

Chapter 8 Random Number Generating Routines
RAN . . .
RANS . . .
SETRAN .
SAVRAN .

.8-3
.8-5
.8-6
.8-7

Chapter 9 Absolute Value Routines
lABS.
ABS .
DABS.
GABS.
CABS.
CDABS.
CGABS.

.9-3
.9-4
.9-5
.9-6
.9-7
.9-8
.9-9

Chapter 10 Data Type Conversion Routines
IFIX .
INT ..
IDINT
GFX.n.
REAL.
FLOAT.
SNGL . .
GSN.n .
DFLOAT ..
DBLE . .
GTOD .
GTODA.
GFL.n .
GDB.n .
DTOG .
DTOGA.
CMPL.I.
CMPLX.
CMPL.D
CMPL.G
CMPL.C

10-3
10-4

10-5
10-6

10-7
10-8

10-9
· 10-10
· 10-11
· 10--12

· 10-13
· 10-14
· 10-15
· 10-16
· 10-17
· 10-18
· 10-19
· 10-20
· 10-21
· 10-22

· 10-23
v

Chapter 11

Rounding and Truncation Routines
NINT . .
IDNINT.
IGNIN ..
ANINT.
DNINT.
GNINT ..
AINT.
DINT . .
GINT. .

11-3
11-4
11-5

11-6
11-7
11-8
11-9

· 11-10
· 11-11

Chapter 12 Product, Remainder, and Positive Difference Routines
DPROD.
GPROD.
MOD.
AMOD
DMOD
GMOD
IDIM.
DIM.
DDIM.
GDIM.

12-3
12-4
12-5
12-6
12-7
12-8
12-9
· 12-10
· 12-11
· 12-12

Chapter 13 Transfer of Sign Routines
ISIGN . .
SIGN . .
DSIGN .
GSIGN .

13-3
13-4
13-5
13-6

Chapter 14 Maximum/Minimum Routines
MAXO .
MAXI.
AMAXO.
AMAXI.
DMAXI.
GMAXI.
MINO . .
MINI . .
AMINO.
AMINI.
DMINI .
GMINI .

14-3
14-4
14-5
14-6
14-7
14-8
14-9
· 14-10
· 14-11
· 14-12
· 14-13
· 14-14

Chapter 15 Miscellaneous Complex Routines
REAL.C.
AIMAG.
CONJ.
CFM.
CFDV.

15-3
15-4
15-5

15-6
15-7

Appendix A ELEFUNT Test Results
Appendix B Using the Common Math Library with MACRO Programs

Tables

1-1
1-2

Math Library Routines. . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Comparison of Single-Precision, D-Floating-Point, and G-Floating-Point 1-11

vii

Preface
This manual describes the TOPS-I0/TOPS-20 Common Math Library. At
present, the library is included as part of each object-time system of each
language that uses it. In the future, the library will be a separate entity as
described in this manual. Chapter 1 introduces the library routines and gives
information on how they are described. A table of the routines, arranged in
alphabetical order, is included for easy reference. Chapters 2 through 15 contain the descriptions of the routines, grouped logically such that all like
routines are together (e.g., all the square root routines are in Chapter 2).
Appendix A gives the results of the ELEFUNT tests and Appendix B describes error handling for MACRO programs.

Chapter 1
Introduction

1.1 The Math Library
The TOPS-I0/TOPS-20 Common Math Library contains a set of routines
that perform the following mathematical functions for several types of data.
• square root
• natural and base-l0 logarithm
• exponential and exponentiation
• trigonometric
• inverse trigonometric
• hyperbolic
• random number generation
• absolute value
• data type conversion
• rounding and truncation
• product
• remainder
• positive difference
• transfer of sign
• maximum or minimum of a series
• complex conjugate
• complex multiplication or division
Most of the routines are functions; but some, notably the complex doubleprecision, are subroutines. The difference between the types of routines is the
way in which they are called from a program. Consult the applicable language
manual for more information.
The routines are listed alphabetically in Table 1-1 with a short description of
each and a page reference.

Introduction 1-3

Table 1-1: Math Library Routines
Routine Name

Page

Purpose

ABS

9-4

absolute value

ACOS

6-4

arc cosine

AIMAG

15-4

imaginary part of complex number

AINT

11-9

truncation to integer

ALOG

3-3

natural logarithm

ALOGlO

3-5

base-lO logarithm

AMAXO

14-5

largest of a series

AMAXI

14-6

largest of a series

AMINO

14-11

smallest of a series

AMINI

14-12

smallest of a series

AMOn

12-6

remainder

ANINT

11-6

nearest whole number

ASIN

6-3

arc sine

ATAN

6-13

arc tangent

ATAN2

6-15

polar angle of a point in the x-y plane

CABS

9-7

complex absolute value

CCOS

5-21

complex cosine

CDABS

9-8

complex, double-precision, D-floating-point absolute value

CDCOS

5-25

complex, double-precision, D-floating-point cosine

CDEXP

4-11

complex, double-precision, D-floating-point exponential

CDLOG

3-17

complex, double-precision, D-floating-point natural
logarithm

CDSIN

5-23

complex, double-precision, D-floating-point sine

CDSQRT

2-11

complex, double-precision, D-floating-point square root

CEXP

4-9

complex exponential

CEXP2.

4-22

exponentiation of a complex number to the power of an
integer

CEXP3.

4-34

exponentiation of a comple?, number to the power of
another complex number

CFDV

15-7

complex division

CFM

15-6

complex multiplication

CGABS

9-9

complex, double-precision, G-floating-point absolute value

CGCOS

5-29

complex, double-precision, G-floating-point cosine

(continued on next page)

1-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

Table Table 1-1 (Cont.):

Math Library Routines

Routine Name

Page

Purpose

CGEXP

4-13

complex, double-precision, G-floating-point exponential

CGLOG

3-19

complex, double-precision, G-floating-point natural
logarithm

CGSIN

5-27

complex, double-precision, G-floating-point sin

CGSQRT

2-13

complex, double-precision, G-floating-point square root

CLOG

3-15

complex natural logarithm

CMPL.C

10-23

conversion of two complex numbers to one complex number

CMPL.D

10-21

conversion of two double-precision, D-floating-point
numbers to complex format

CMPL.G

10-22

conversion of two double-precision, G-floating-point numbers to complex format

CMPL.I

10-19

conversion of two integers to complex format

CMPLX

10-20

conversion of two single-precision numbers to complex
format

CONJ

15-5

complex conjugate

COS

5-7

cosine (angle in radians)

COSD

5-9

cosine (angle in degrees)

COSH

7-4

hyperbolic cosine

COTAN

5-33

cotangent

CSIN

5-19

complex sine

CSQRT

2-9

complex square root

DABS

9-5

double-precision, D-floating-point absolute value

DACOS

6-7

double-precision, D-floating-point arc cosine

DASIN

6-5

double-precision, D-floating-point arc sine

DATAN

6-17

double-precision, D-floating-point arc tangent

DATAN2

6-19

double-precision, D-floating-point polar angle of a point in
the x-y plane

DBLE

10-12

conversion from single-precision to dou bIe-precision,
D-floating-point format

DCOS

5-13

double-precision, D-floating-point cosine

DCOSH

7-7

double-precision, D-floating-point hyperbolic cosine

DCOTAN

5-37

double-precision, D-floating-point cotangent

DDIM

12-11

double-precision, D-floating-point positive difference

DEXP

4-5

double-precision, D-float.ing-point exponent.ial

(continued on next page)

Introduction

1-5

Table 1-1 (cont.):

Math Library Routines

Routine Name

Page

Purpose

DEXP
DEXP2.

4-5
4-18

double-precision, D-floating-point exponential

DEXP3.

4-28

exponentiation of a double-precision, D-floating-point
number to the power of another double-precision,
D-floating-point number

DFLOAT

10-11

conversion of an integer to double-precision,
D-floating-point format

DIM

12-10

positive difference

DINT

11-10

double-precision, D-floating,point truncation

DLOG

3-7

double-precision, D-floating-point natural logarithm

DLOG10

double-precision, D-floating-point base-lO logarithm

DMAX1

3-9
14-7

double-precision, D-floating-point largest in a series

DMIN1

14-13

double-precision, D-floating-point smallest in a series

DMOD

12-7

double-precision, D-floating-point remainder

DNINT
DPROD
DSIGN
DSIN

11-7
12-3

double-precision, D-floating-point nearest whole number
double-precision, D-floating-point product

13-5

double-precision, D-floating-point transfer of sign

5-11

double-precision, D-floating-point sine

DSINH

7-5

double-precision, D-floating-point hyperbolic sine

DSQRT

2-5

double-precision, D-floating-point square root

DTAN

5-35

double-precision, D-floating-point tangent

DTANH

7-12

double-precision, D-floating-point hyperbolic tangent

DTOG

10-17

conversion of a double-precision, D-floating-point number
to double-precision, G-floating-point format

DTOGA

10-18

conversion of an array of double-precision, D-floating-point
numbers to double-precision, G-floating-point format

EXP
EXP1.

4-3
4-15

exponential

EXP2.

4-16

exponentiation of a single-precision number to the power of
an integer

EXP3.

4-25

exponentiation of a single-precision number to the power of
another single-precision number

FLOAT

10-8

conversion of an integer to single-precision format

GABS

9-6

double-precision, G-floating-point absolute value

GACOS

6-11

double-precision, G-·floating-point arc cosine

GASIN

6-9

double-precision, G-floating-point arc sine

GATAN

6-21

double-precision, G-floating-point arc tangent

exponentiation of a double-precision, D-floating-point
number to the power of an integer

exponentiation of an integer to the power of another integer

(continued on next page)

1-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

Table 1-1 (cont.):

Math Library Routines

Routine Name

Page

Purpose

GATAN2

6-23

double-precision, G-floating-point polar angle of a point in
the x-y plane

GCOS
GCOSH
GCOTAN
GDB.n

5-17

double-precision, G-floating-point cosine

7-10
5-41

double-precision, G-floating-point hyperbolic cosine

10-16

conversion of a single-precision number to
double-precision, G-floating-point format

GDIM

12-12

double-precision, G-floating-point positive difference

GEXP
GEXP2.

4-7

double-precision, G-floating-point exponential

4-20

exponentiation of a double-precision, G-floating-point
number to the power of an integer

GEXP3.

4-31

exponentiation of a double-precision, G-floating-point
number to the power of another double-precision, G-floating-point number

GFL.n

10-15

conversion of an integer to double-precision,
G-floating-point format

GFX.n

10-6

conversion of a double-precision, G-floating-point number
to integer format

GINT.
GLOG
GLOGI0

11-11

double-precision, G-floating-point truncation

3-11
3-13

double-precision, G-floating-point natural logarithm

GMAXI

14-8
14-14

double-precision, G-floating-point largest of a series

GMOD
GNINT.

12-8

double-precision, G-floating-point remainder

11-8

dquble-precision, G-floating-point nearest whole number

GPROD.
GSIGN

double-precision, G-floating-point product

GSIN
GSINH
GSN.n

12-4
13-6
5-15
7-8
10-10

GSQRT
GTAN
GTANH

2-7
5-39
7-13

GTOD

10-13

conversion of a double-precision, G-floating-point number
to double-precision, D-floating-point format

GTODA

10-14

copversion of an array of double-precision, G--floating-point
nqmbers to double-precision, D-floating-point format

GMIN1

double-precision, G-floating-point cotangent

double-precision, G-floating-point base-lO logarithm

double-precision, G-floating-point smallest of a series

dciuble-precision, G-floating-point transfer of sign
double-precision, G-floating-point sine
double-precision, G-floating-point hyperbolic sine
conversion of a double-precision, G-floating-point number
to single-precision format
double-precision, G-floating-point square root
double-precision, G-floating- point tangent
double-precision, G-floating-point hyperbolic tangent

(continued on next page)

Introduction 1-7

Table 1-1 (cont.): Math Library Routines
Routine Name

Page

Purpose

lABS

9-3

integer absolute value

IDIM

12--9

integer positive difference

IDINT

10--5

conversion of a double-precision, D-floating-point number
to integer format

IDNINT

11-4

integer nearest whole number for a double-precision,
D-floating-point number

IFIX

10-3

conversion of a single-precision number to integer format

IGNIN.

11-5

integer nearest whole number for a double-precision,
G-floating-point number

INT

10-4

conversion of a single-precision number to integer format

ISIGN

13-3

integer transfer of sign

MAXO

14-3

largest of a series

MAXI

14-4

largest of a series

MINO

14-9

smallest of a series

MINI

14-10

smallest of a series

MOD

12-5

integer remainder

NINT

11-3

integer nearest whole number for a single-precision number

RAN

8-3

random number generator

RANS

8-5

random number generator with shuffling

REAL

10-7

conversion of an integer to single-precision format

REAL.C

15-3

real part of a complex number

SAVRAN

8-7

save the seed for the last random number generated

SETRAN

8-6

set the seed value for the random number generator

SIGN

13-4

transfer of sign

SIN

5-3

sine (angle in radians)

SIND

5-5

sine (angle in degrees)

SINH

7-3

hyperbolic sine

SNGL

10-9

conversion of a double-precision, D-floating-point number
to single-precision format

SQRT

2-3

square root

TAN

5-31

tangent

TANH

7-11

hyperbolic tangent

The routines in this library are available to most of the languages available
with TOPS-10 and TOPS-20. Consult the applicable language manual for
specific information on how to use the Math Library_ Although all of the
routines listed in Table 1-1 exist in the library, not all of them can be called
from all languages. That is, some languages or compilers have restrictions
that disallow calling of a particular routine from a user program. For example,
1-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

the complex data type does not exist in PASCAL, so the routines that perform
complex mathematics are never called by a PASCAL program. However, a
compiler may itself call a routine because a user program has a statement that
necessitates use of a Math Library routine. For example, a FORTRAN program cannot call any of the routines whose names contain a period (.). However, the compiler recognizes when a statement within a program requires use
of one of those routines, and the compiler calls the appropriate routine. Similarly, a statement in an APL program may require a mathematical function,
so the APL interpreter translates that statement into a call to the appropriate
Math Library routine.

1.2 Math Symbols and Names Used In Equations
Throughout this manual, certain mathematical symbols and names are used
to indicate values, quantities, actions, or states. These symbols and their
meanings are listed below.
+

x
/
>
~

<
~

±
[]

eX
sin
cos
tan
cot
sin- 1
cos- 1
tan- 1
sinh
cosh
tanh
sgn
conj

equal to
plus
minus
multiplied by (used in equations)
multiplied by (used in numbers)
divided by
greater than
greater than or equal to
less than
less than or equal to
not equal to
square root
Pi (3.14159265358979323846264950338327)
plus or minus
greatest integer in
absolute value
equals approximately
subscript
superscript or raised to the power
natural logarithm
base-10 logarithm
imaginary number (yCi)
exponential
sine of an angle
cosine of an angle
tangent of an angle
cotangent of an angle
arc sine
arc cosine
arc tangent
hyperbolic sine
hyperbolic cosine
hyperbolic tangent
sign of
complex conjugate
Introduction 1-9

In addition, some equations use the names of routines to indicate a state or
action. These routines and their meanings are as follows.
FLOAT

convert and round from an integer to a single-precison, floatingpoint number

INT

convert and truncate from a single-precision, floating-point number to an integer

MAX

largest of a series

MIN

smallest of a series

MOD

remainder

Each of these routines is described in detail in this manual.
Also, machine infinity (or infinity) is a term used to indicate the largest or
smallest number representable in the machine.
+ machine infinity = 3777777777778 for single-precision
377777777777, 3777777777778 for dou ble-precision
-machine infinity = 4000000000008 for single precision
400000000000, 0000000000018 for double-precision

1.3 Data Types and Their Precision
The Common Math Library routines can handle several data types - integer;
single-precision, floating-point (also called real); double-precision, D-floatingpoint; double-precision, G-floating-point; complex; complex, double-precision, D-floating-point; and complex, double-precision, G-floating-point. Each
data type is described in detail in one of the following sections.

1.3.1 Integer
An integer value is a string of one to eleven digits that represents a whole
decimal number (a number without a fractional part). Integer values must be
within the range of _2 3f:i to +235 _1 (-34359738368 to +34359738367).

1.3.2 Single-Precision, Floating-Point
Single-precision, floating-point values may be of any size; however, each will
be rounded to fit the precision of 27 bits (7 to 9 dedmal digits).
Precision for single-precision, floating-point values is maintained to approximately eight significant digits; the absolute precision depends upon the numbers involved.
The range of magnitude permitted a single-precision, floating-point value is
from approximately 1.47x10-39 to 1.70x10+38 •

1-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

1.3.3 Double-Precision, D-Floating-Point
Double-precision, D-floating-point values are similar to single-precision,
floating-point values; the differences between these two values are:
• Double-precision, D-floating-point values, depending on their magnitude,
have precision of 62 bits, rather than the 27-bit precision obtained for single-precision, floating-point values.
• Each double-precision, D-floating-point value occupies two storage locations.
The range of magnitude permitted a double-precision, D-floating-point value
is from approximately 1.47x10-39 to 1.70x10-f:~8.

1.3.4 Double-Precision G-Floating-Point 1
Double-precision, G-floating-point values are similar to double-precision,
D-floating-point values. They differ in:
• the number of bits of exponent
• the number of bits of mantissa
• the range of numbers they can represent
• the digits of precision
Table 1-2 summarizes the differences among single-precision and the two
forms of double-precision.
Table 1-2: Comparison of Single-Precision, D-Floatlng-Polnt, and
G-Floatlng-Polnt
Digits of
Precision

Bits of
Exponent

Bits of
Mantissa

Range

single-precision

1. 47x10- 39 .
to 1. 70xlO+ 38

8.1

D-floating-point

1. 47x 10-::19
to 1. 70x10+:~8

18.7

G-floating-point

fi9

2.78xlO--:m9
to 8.99xlO+:l<17

17.8

1 Double-precision, G-floating-point data type is available only with TOPS-20 Version 5 (or

later) on the DECSYSTEM-20 KLlO model B.

Introduction 1-11

1.3.5 Complex
A complex value contains two numbers; it is assumed that the first (leftmost)
value of the pair represents the real part of the number and that the second
value represents the imaginary part of the number. The values that represent
the real and imaginary parts of a complex value occupy two consecutive
storage locations.

1.3.6 Complex, Double-Precision
You can use two types of complex, double-precision values - D-floating-point
and G-floating-point. Both are assumed to be double-precision arrays with
two elements. The first element is the real part, and the second element is the
imaginary part.

1.4 Information About the Routines
Each routine described in this manual has the following information provided.
• A short description
• The names of other routines called by the routine
• The data type and range of the argument(s)
• The data type and range of the result
• The accuracy of the result
• The algorithm used to calculate the result
• A reference to any text used for information about the algorithm (where
applicable)
• Any error conditions and the messages that result
Some additional information about the routines not included in each write-up
1S:

• Calling sequence
• Entry points
• Return location(s)
• Register usage
This information is described below. It is not included for each routine because it is identical for most routines and is relevant only for MACRO and
BLISS users.

1-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

1.4.1 Calling Sequence
Most routines are called by an identical calling sequence. This calling sequence is:
XMOVEI

PUSHJ

L,ARG
P, routine-name

ARG is the address of the argument block. L is the pointer to the argument
list for the routine; it is ACI6. P is the stack pointer; it is ACI7. Note that the
contents of L (ACI6) are not preserved.
For example, the SQRT routine is called by:
XMOVEI

PUSHJ

16,ARG
17,SQRT

Those routines called by a different calling sequence contain the calling sequence in their descriptions.

1.4.2 Entry Points
In most cases each routine has at least two entry points - its name and its
name followed by a period. For example, SQRT and SQRT. are entry points
for the SQRT routine. The name with the period is the one used by the
FORTRAN compiler. Some routines have additional entry points because
they perform more than one function. Thus, one routine calculates both sine
and cosine, so SIN, SIN., COS, and COS. are all entry points into that
routine. If you are calling a routine from a MACRO or BLISS program, you
can use the name of the routine as the entry point; it will always work.

1.4.3 Return Location
The result of the calculation of most routines is returned to one or two registers. For integer and single-precision results, the return location is register O.
For double-precision and complex (single-precision) results, the return locations are registers 0 and 1. For complex, double-precision results, the return
location must be specified as the second argument included in the call to the
routine. The requirements for the arguments included in the call are included
with each write-up of the complex, double-precision routines.

1.4.4 Register Usage
All the routines have similar register usage. Some may use more registers than
others, however. As stated above, registers 0 and 1 are used for the return
locations; therefore the original contents of one or both are lost on return from
a routine. These registers are also occasionally used to store the argument
initially. Registers 2 through 15 are saved, used, and restored. The number of
such registers used depends on the routine.

Introduction

1-13

1.5 Accuracy Tests
Each routine contains a section headed "Accuracy of Result." The accuracy
figures were obtained from the tests described below. These tests were run
with typical values for arguments. There may be unusual arguments that
could cause larger errors; for example, if you get too close to a threshold that
could cause overflow or underflow, larger errors can occur. The format of the
accuracy section is as follows. Note that the elements are explained with the
descri ptions of the tests.
Accuracy of Result

test interval:

0.00000 through 1.0000

MRE:

1.55x10-8 (25.9 bits)

RMS:

3.76x10-9 (28.0 bits)

LSB error distribution:

-2

-1

83%

+2
0%

To test a routine, several representative intervals for each routine were chosen. Sample values were then chosen randomly from each interval, approximately 200,000 for single-precision and 20,000 for double-precision. Each routine was then called using these values. The relative error of each result was
then obtained by the following equation.

For example:

result - result of rO\ldtinz I
,~ctual exact
actual
exact result
I

sinix) -~, SIN (x)
sin(x)

The test computed the maximum relative error (MRE) and the average relative error, called the root mean square (RMS). To interpret the MRE and
RMS, consider an "exact" routine, one that always returns an exact result
rounded to machine precision. Such a routine would show a maximum relative error of 2- 27 for single-precision; 2- 62 for double-precision, D-floatingpoint; and 2-59 for double-precision, G-floating-point. To make the MRE and
RMS more understandable in terms of bits of accuracy, the tests also give the
number of bits of accuracy by finding the negative base-2 logarithm of the
MRE and RMS. For the "exact" routine, the negative base-2 logarithm of the
MRE would be 27 for single-precision; 62 for double-precision, D-floatingpoint; and 59 for double-precision, G-floating-point. The negative base-210garithm of the RMS error from an "exact" routine would be about 28.3, 63.3,
and 60.3, respectively. These numbers are slightly larger than those for the

1-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

MRE because they reflect the RMS average of the "worst case" of exactness
(only 27 or 62 or 59 bits correct) and the "best case" (infinite bits correct).
Therefore, the closer the number of bits of accuracy of a routine approaches
that of an "exact" routine, the more accurate the routine. The accuracy
figures for "exact" routines for the three levels of precision are as follows.
Single-Precision

test interval:

0.00000 through 8192.0

MRE:

7.44x10-9 (27.0 bits)

RMS:

3.11xlO-9 (28.3 bits)

LSB error distribution:

-2

-1

100% 0%

Double-precision, D-floatlng-polnt

test interval:

-infinity to +infinity

MRE:

2.17x10- 19 (62.0 bits)

RMS:

8.81x10- 20 (63.3 bits)

LSB error distribution:

-2

-1

100% 0%

Double-precision, G-floatlng-polnt

test interval:

-infinity to +infinity

MRE:

1.73x10- 18 (59.0 bits)

RMS:

7.05x10- 19 (60.3 bits)

LSB error distribution:

-2

-1

100% 0%

A second test compared the result of the routines with the exact result
rounded to single- or double-precision. It counted the number of times the
routine's result agreed exactly with the rounded exact result, the number of
times they differed by ±1 bit, ±2 bits, and so on. The result of these comparisons is expressed as a percent of error distribution for the least significant bit
(LSB).
Appendix A shows accuracy results derived from the ELEFUNT tests of W. J.
Cody, Argonne National Laboratory. These tests show accuracy derived by
testing carefully-chosen identities for each function. This appendix is provided for your information, not for comparison with the test results described
above. Such a comparison would not be meaningful.

Introduction 1-15

Chapter 2
Square Root Routines

SQRT
Description
The SQRT routine calculates the single-precision, floating-point square root
of its single-precision, floating-point argument. That is:
.!.

SQRT(x) = Vx = X 2

Routines Called
SQRT calls the MTHERR routine.
Type of Argument
The argument must be a single-precision, floating-point value greater than or
equal to 0.0.
Type of Result
The result returned is a single-precision, floating-point value greater than or
equal to 0.0.
Accuracy of Result

test interval:

0.00000 through 8192.0

MRE:

8.09x10-9 (26.9 bits)

RMS:

3.21x1O-9 (28.3 bits)

LSB error distribution:

-2
0%

-1
0%

98%

Algorithm Used
SQRT(x) is calculated as follows.

First the routine does a linear, single-precision approximation on the argument to provide an initial guess for$. The routine then does two iterations
of the Newton-Raphson method, which results in an answer that is correct to,
but not always including, the last bit.
If x < 0.0
SQRT(x) = SQRT(lxl)
If x = 0.0
SQRT(x) = 0.0
If x > 0.0
Let x = 22be f where .25 ~ f < 1.0
then Vi = 2b e v'f
and Zo = 2b e (af-b)

a = .82812500 if .25 ~ f < .5
= .58593750 if .5 ~ f < 1.0
b = .29722518 if .25 ~ f < .5
= .42060167 if .5 ~ f < 1.0

Square Root Routines

2-3

The Newton-Raphson method, as applied to the SQRT function, yields the
following iterative approximation.
Zk+l =

1/2· (Zk+X/Zk)

Zk+l =

the next iteration

= the current iteration

= the initial approximation calculated by the linear approximation

the number whose square root is being calculated

Error Conditions
If the argument is negative, the following message is issued and the absolute
value of the argument is used.

SORT: Negative arg; result = SORT(ABS(arg))

2-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DSQRT
Description

The DSQRT routine calculates the double-precision, D-floating-point square
root of its double-precision, D-floating-point argument. That is:
DSQRT(x) = .JX = xt
Routines Called

DSQRT calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value greater than
or equal to 0.0.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 0.0.
Accuracy of Result

test interval:

0.00000 through 8192.0

MRE:

3.25x10- 19 (61.4 bits)

RMS:

1.23x10- 19 (62.8 bits)

LSB error distribution:

-2

-1

75%

25%

+2
0%

Algorithm Used

DSQRT(x) is calculated as follows.
First the routine does a linear, single-precision approximation on the highorder word. Then the routine does two single-precision iterations of the Newton-Raphson method, followed by two double-precision iterations of the Newton-Raphson method using a value derived from the linear approximation.
The linear approximation is as follows.
If x < 0.0
DSQRT(x) = DSQRT(lxl)
If x = 0.0
DSQRT(x) = 0.0
If x > 0.0
Let x = 22be f where .25 ~ f < 1.0
then Vx = 2b e v'f
and Zo = 2b • (af-b)

a = .82812500 if .25 ~ f < .5
= .58593750 if .5 ~ f < 1.0
b = .29722518 if .25 ~ f < .5
= .42060167 if .5 ~ f < 1.0

Square Root Routines 2-5

The Newton-Raphson method yields the following iterative approximation.
Zk+l = 1/2·(Zk+ X/ Zk)
Zk+l = the next iteration
Zk

the current iteration

the number whose square root is being calculated

Zo =

the initial approximation calculated by the linear approximation

For the single-precision approximations, x is truncated to single-precision and
all calculations are done in single-precision. For the double-precision iterations, the full double-precision value of x is used, the current value of Z2 is
zero-extended to double-precision, and all remaining calculations are done in
dou ble-precision.
Error Conditions
If the argument is negative, the following message is issued and the absolute

value of the argument is used.
DSQRT: Negative arg; result = DSQRT(ABS(arg))

2-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

GSQRT

Description
The GSQRT routine calculates the double-precision, G-floating-point square
root of its double-precision, G-floating-point argument. That is:

GSQRT(x) = .JX = xt
Routines Called
GSQRT calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, G-floating-point value greater than
or equal to 0.0.
Type of Result
The result returned is a double-precision, G-floating-point value greater than
or equal to 0.0.
Accuracy of Result
test interval:

0.00000 through 8192.0

MRE:

2.60x1O-- 18 (58.4 bits)

RMS:

9.87x10- 19 (59.8 bits)

LSB error distribution:

-2

-1

75%

25%

Algorithm Used
GSQRT(x) is calculated as follows.

First the routine does a linear, single-precision approximation on the highorder word. Then the routine does two single-precision iterations of the Newton-Raphson method, followed by two double-precision iterations of the Newton-Raphson method using a value derived from the linear approximation.
The linear approximation is as follows.
If x < 0.0
GSQRT(x) = GSQRT(lxl)
If x = 0.0
GSQRT(x) = 0.0
If x> 0.0
Let x = 22bo f where .25 ~ f < 1.0
then .JX = 2b e.Jf
and Zo = 2b e (af-b)
a = .82812500 if .25 ~ f < .5
a = .58593750 if .5 ~ f < 1.0
b = .29722518 if .25 ~ f < .5
b = .42060167 if .5 ~ f < 1.0

Square Root Routines 2-7

The Newton-Raphson method yields the following iterative approximation.
Zk + 1 =

1/2· ( Zk + xlZk)

Zk+l =

the next iteration

the current iteration

the number whose square root is being calculated

Zo =

the initial approximation calculated by the linear approximation

For the single-precision approximations, x is truncated to single-precision and
all calculations are done in single-precision. For the double-precision iterations, the full double-precision value of x is used, the current value of Z2 is
zero-extended to double-precision, and all remaining calculations are done in
dou ble- precision.
Error Conditions
If the argument is negative, the following message is issued and the absolute
value of the argument is used.
GSQRT: Negative arg; result = GSQRT(ABS(arg))

2-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

CSQRT
Description

The CSQRT routine calculates the complex, single-precision square root of its
complex, single-precision argument. That is:
CSQRT(z) = .JZ = zt
Routines Called

CSQRT calls the SQRT and MTHERR routines.
Type of Argument

The argument must be a complex, single-precision, floating-point value; it
can be any such value.
Type of Result

The result returned is a complex, single-precision, floating-point value, the
real part of which is greater than or equal to 0.0.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
-1000.0 through 1000.0 imaginary

MRE:

3.07x1o-8 (25.0 bits) real
3.05x1o-8 (25.0 bits) imaginary

RMS:

7.05x10-9 (27.1 bits) real
7.33x1o- 9 (27.0 bits) imaginary

LSB error distribution:

-2
2%
2%

-1
16%
19%

o
59%
55%

+1
20%
20%

+2
2% real
3 % imaginary

Algorithm Used

CSQRT(z) is calculated as follows.
Let z = x+i·y
then CSQRT(z) = u+i ·v, which is defined as follows.
If x~O.O
u = v"-(-I
x-I+-1z-I)-/2-.0-

v = y/(2.0·u)
If x < 0.0 and y~O.O
u = y/(2.0·v)
v = .J (lxl+lzl)/2.0

If x and yare both < 0.0
u = y/(2.0·v)
v = -.J (lxl+lzl);2.0

The result is in the right half plane; that is, the polar angle of the result lies in
the closed interval (-1r/2,+1r/2]. That is, the real part of the result is greater
than or equal to 0.0.

Square Root Routines 2-9

Error Conditions
If the imaginary part of the input value is too small, underflow can occur on
y/(2.0·u) or y/(2.0·v). If such underflow occurs, one of the following messages
is issued and the relevant part of the result is set to 0.0.
CSQRT: Real part underflow
CSQRT: Imaginary part underflow

2-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

CDSQRT

Description
The CDSQRT subroutine calculates the complex, double-precision, D-floating-point square root of its complex, double-precision, D-floating-point argument. That is:
1

CDSQRT(z,r) = viz = z"2
Z = location of input value
r = location of result
Routines Called
CDSQRT calls the DSQRT and MTHERR routines.
Type of Arguments
CDSQRT is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, D-floating-point value; it can be any such value.
Type of Result
The result returned is a complex, double-precision, D-floating-point value,
the real part of which is greater than or equal to 0.0. It is returned in the
second vector (r) supplied in the call. The real part of the result is returned in
the first element of r; the imaginary part is returned in the second element
of r.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
-1000.0 through 1000.0 imaginary

MRE:

1.10x10- 1R (59.7 bits) real
1.04x10- 18 (59.7 bits) imaginary

RMS:

2.69x10- 19 (61.7 bits) real
2.75x10- 19 (61.7 bits) imaginary

LSB error distribution:

-2
4%
5%

-1
17%
24%

0
43%
41%

32%
25%

5% real
5% imaginary

Square Root Routines 2-11

Algorithm Used
CDSQRT is calculated as follows.

Let z = x+i·y
then CDSQRT(z) = u+i ·v, which is defined as follows.
If x ~ 0.0
u = V--(I-x-I+-1z-I)/-2-.0-

v = y/(2.0·u)
If x < 0.0 and y ~ 0.0
u = y/(2.0·v)
v = V (lxl+lzl)/2.0
If x and yare both < 0.0
u = y/(2.0·v)
v = -v (lil+lzl)/2.0

The result is in the right half plane; that is, the polar angle of the result lies in
the closed interval [-11'/2, +71"/2]. That is, the real part of the result is greater
than or equal to 0.0.
Error Conditions
If the imaginary part of the input value is too small, underflow can occur on
y/(2.0·u) or y/(2.0·v). If such underflow occurs, one of the following messages
is issued and the relevant part of the result is set to 0.0.
CDSQRT: Real part underflow
CDSQRT: Imaginary part underflow

2-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGSQRT
Description
The CGSQRT subroutine calculates the complex, double-precision, G-floating-point square root of its complex, double-precision, G-floating-point argument. That is:
1

CGSQRT(z,r) = .Ji = ZT
Z = location of input value
r = location of result
Routines Called
CGSQRT calls the GSQRT and MTHERR routines.
Type of Argument
CGSQRT is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, G-floating-point value; it can be any such value.
Type of Result
The result returned is a complex, double-precision, G-floating-point value; it
may be any such value. It is returned in the second vector (r) supplied in the
call. The real part of the result is returned in the first element of r; the
imaginary part is returned in the second element of r.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
-1000.0 through 1000.0 imaginary

MRE:

8.61x10- 18 (56.7 bits) real
8.78x10- 18 (56.7 bits) imaginary

RMS:

2.16x10- 18 (58.7 bits) real
2.21x10- 18 (58.7 bits) imaginary

-2
LSB error distribution:

5%
5%

-1
16%
25%

41%
40%

32%
25%

5% real
5% imaginary

Square Root Routines

2"':13

Algorithm Used

CGSQRT(z) is calculated as follows.
Let z = x+i·y
then CGSQRT(z) = u+i·v is defined as follows.
Ifx~O.O
u = V'--(I-x-I+-lz-I)/-2-.0

v = y/(2.0·u)
If x < 0.0 and y ~ 0.0
u = y/(2.0·v)
v = J (lxl+lzl)/2.0

If x and yare both < 0.0
u = y/(2.0·v)
v =

-v (lxl+lzl)/2.0

The result is in the right half plane; that is, the polar angle of the result lies in
the closed interval (-1r/2, +11"/2].
Error Conditions

If the imaginary part of the argument is too small, underflow can occur on
y/(2.0·u) or y/(2.0·v). If this occurs, one of the following messages is issued
and the relevant part of the result is set to 0.0.
CGSQRT: Real part underflow
CGSQRT: Imaginary part underflow

2-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 3
Logarithm Routines

ALOG
Description
The ALOG routine calculates the single-precision, floating-point naturallogarithm of its argument. That is:

ALOG(x) = loge (x)
Routines Called

ALOG calls the MTHERR routine.
Type of Argument
The argument must be a single-precision, floating-point value greater than
0.0.
Type of Result
The result returned is a single-precision, floating-point value in the range
-89.415 to 88.029.
Accuracy of Result
test interval:

MRE:
RMS:

LSB error distribution:

1.46937x10-39 through 256.00
1.84x10-8 (25.7 bits)
5.21x10-9 (27.5 bits)
-2
0%

-1

81%

+1
18%

+2
0%

Algorithm Used

ALOG(x) is calculated as follows.
If x = 0.0

ALOG(x) = -machine infinity
If x < 0.0

ALOG(x) = ALOG(lxl)
If x is close to 1.0
ALOG(x) = L3·z 7 +L4ez5+L5ez3+L6ez
Z = (x-l)/(x+l)
L3 = .301003281
L4 = .39965794919
L5 = .666669484507
L6 = 2.0
If x is not close to 1.0
ALOG(x) = (k-.5) eloge(2)+loge(f·v'2)
x = 2k ·f
loge(f e v2) = L3 ez7 +L4ez5+L5ez3+L6ez
Z =

(f-~ )/(f+..[5 )

Logarithm Routines 3-3

Reference

Hart et. aI., Computer Approximations, (New York, N.Y.: John Wiley and
Sons, 1968).
The algorithm used is #2662, the coefficients are listed on page 193, and the
range of validity is on page 111.
Error Conditions

1. If the argument is equal to 0.0, the following message is issued and the
result is set to -machine infinity.
ALOG: Arg is zero; result = -infinity.

2. If the argument is less than 0.0, the following message is issued and the
absolute value of the argument is used.
ALOG: Negative arg, result = ALOG(ABS(arg»

3-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

ALOG10

Description

The ALOG 10 routine calculates the single-precision, floating-point base-IO
logarithm of its single-precision, floating-point argument. That is:
ALOG 10(x) = loglO(X)
Routines Called

ALOGI0 calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value greater than
0.0.
Type of Result

The result returned is a single-precision, floating-point value in the range
-38.832 to 38.230.
Accuracy of Result

test interval:
MRE:
RMS:
LSB error distribution:

1.46937xl0-39 through 256.00
2.52xI0-8 (25.2 bits)
5.99xI0- 9 (27.3 bits)
2
1%

I
19%

0
64%

+1
15C!(1

+2
0%

Algorithm Used

ALOG 10(x) is calculated as follows.
If x = 0.0

ALOG 10(x) = -machine infinity
If x < 0.0

ALOGIO(x) = ALOG10(lxl)
If x is close to 1. 0

ALOGI0(x) = loge(x) eloglO (e)
loge(x) = L3 ez 7+L4 ez5+L5 ez3+L6 ez
Z = (x-l)/(x+ 1)
L3 = .301003281
L4 = .39965794919
L5 = .666669484507
L6 = 2.0
If x is not close to 1.0

ALOG10(x) = loge(x)eloglO (e)
x = 2ke f
loge(x) = (k-.5) eloge(2)+loge(f e..£)
loge(f ev'2) = L3ez7+L4ez5+L5·z3+L6·z
z = (f-v.5 )/(f+v.5 )

Logarithm Routines 3-5

Reference

Hart et. aI, Computer Approximations, (New York, N.Y.: John Wiley and
Sons, 1968). The algorithm used is #2662, the coefficients are listed on page
193, and the range of validity is on page Ill.
Error Conditions

1. If the argument is 0.0, the following message is issued and the result is set
to -machine infinity.
ALOG 10: Arg is zero; result = -infinity
2. If the argument is less than 0.0, the following message is issued and the

absolute value of the argument is used.
ALOG 10: Negative arg; result = ALOG 1O(ABS(arg))

3-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

OLOO

Description
The DLOG routine calculates the double-precision, D-floating-point natural
logarithm of its double-precision, D-floating-point argument. That is:

DLOG(x) = loge(x)
Routines Called
DLOG calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, D-floating-point value greater than
0.0.
Type of Result
The result returned is a double-precision, D-floating-point value in the range
-89.415 to 88.029.
Accuracy of Result
test interval:

1.46937x10-39 through 256.00

MRE:

9.78x10- 19 (59.8 bits)

RMS:

3.03x10- 19 (61.5 bits)

LSB error distribution:

-2

-1

12%

51%

+1
23%

+2
13%

Algorithm Used
DLOG(x) is calculated as follows.
If x = 0.0
DLOG(x) = -machine infinity
If x < 0.0
DLOG(x) = DLOG(lxl)
If x> 0.0
x = 2k ·f where .5 < f < 1.0
and g and n are defined so that
f = 2- n • g where 1/v'2s g < .J2
Then DLOG(x) = (k-n) ·loge(2) +loge(g)
loge(g) is evaluated by defining
s = (g -l)/(g+ 1) and
z = 2·s
and then calculating
loge(g) = loge«1+z/2)/(1 --z/2)) using a minimax
rational approximation.

Logarithm Routines 3-7

Error Conditions

1. If the argument is equal to 0.0, the following message is issued and the
result is set to -machine infinity.
DLOG: Arg is zero; result = -infinity
2. If the argument is less than 0.0, the following message is issued and the

absolute value of the argument is used.
DLOG: Negative arg; result = DLOG(ABS(arg))

3-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

DLOG10

Description

The DLOG 10 routine calculates the double-precision, D-floating-point base10 logarithm of its double-precision D-floating-point argument. That is:
DLOG 10(x) = loglO(x)
Routines Called

DLOG 10 calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value greater than
0.0.
Type of Result

The result returned is a double-precision, D-floating-point value in the range
-38.832 to 38.320.
Accuracy of Result

test interval:
MRE:
RMS:
LSB error distribution:

1.46937xl0-:~9 through 256.00

1.20xl0- 18 (59.5 bits)
3.65xlo-- 19 (61.2 bits)

-2
30;()

-1
17%

0
38%

+1
26%

+2
14%

+3
2%

Algorithm Used

DLOG 10(x) is calculated as follows.
If x = 0.0
DLOG 10(x) = -machine infinity
If x < 0.0
DLOGIO(x) = DLOGIO(lxl)
If x > 0.0
x ~ 2k -f where .5 < f < 1.0
and g and n are defined so that
f = 2- n -g where l/Vi :5 g < V2
Then DLOG 10(x) = 10glO(e) -loge(x) = loge(x)/loge(lO)
loge(g) is evaluated by defining
s = (g -l)/(g+l) and
z = 2-s
and then calculating
loge(g) = loge«l +z/2)/(l -z/2» using a minimax
rational approximation.

Logarithm Routines 3-9

Error Conditions
1. If the argument is equal to 0.0, the following message is issued and the

result is set to -machine infinity.
DLOG 10: Arg is zero; result = -infinity

2. If the argument is less than 0.0, the following message is issued and the

absolute value of the argument is used.
DLOG10: Negative arg; result = DLOG10(ABS(arg»

3-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

GLOG

Description

The GLOG routine calculates the double-precision, G-floating-point natural
logarithm of its double-precision, G-floating-point argument. That is:
GLOG(x) = loge(x)
Routines Called

GLOG calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value greater than
0.0.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
-710.475 to 709.089.
Accuracy of Result

test interval:

0.00000 through 256.00

MRE:

5.13x10-18 (57.4 bits)

RMS:

1.26x10- 18 (59.5 bits)

LSB error distribution:

-2
0%

-1
10%

o
74%

+1
16%

+2
0%

Algorithm Used

GLOG(x) is calculated as follows.
If x = 0.0
GLOG(x) = machine infinity
If x < 0.0
GLOG(x) = GLOG(lxl)

If x> 0.0
x = 2k -f where .5 < f < 1.0
and g and n are defined so that
f = 2- n - g where 1/v'2::; g < v2

Then GLOG(x) = (k-n) -loge(2) +loge(g)
loge(g) is evaluated by defining
s = (g-1)/(g+ 1) and
z = 2-s
and then calculating
loge(g) = loge((1+z/2)/(l-z/2))
using a minimax rational approximation.

Logarithm Routines 3-11

Error Conditions
1. If the argument is equal to 0.0, the following message is issued and the

result is set to -machine infinity.
GLOG: Arg is zero; result = -infinity

2. If the argument is negative, the following message is issued and the absolute value of the argument is used.
GLOG: Negative arg; result = GLOG(ABS(arg»

3-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

GLOG10

Description

The GLOG 10 routine calculates the double-precision, G-floating-point base10 logarithm of its double-precision, G-floating-point argument. That is:
GLOG 10(x) = 10glO(x)
Routines Called

GLOGI0 calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value greater than
0.0.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
-308.555 to 307.953.
Accuracy of Result

test interval:

2.78134xl0--:109 through 256.00

MRE:

6.05xlO"IR (57.2 bits)

RMS:

1.42xl0- 1R (59.3 bits)

LSB error distribution:

o
62%

+1
18%

+2
0%

Algorithm Used

GLOG 10(x) is calculated as follows.
If x = 0.0
GLOG 10(x) = -machine infinity
If x < 0.0
GLOGI0(x) = GLOG10(lxl)
If x > 0.0
x =" 2k ·f where .5 < f < 1.0
and g and n are defined so that
f = 2- n - g where 1/V2 ~ g < .J2

Then GLOG 10(x) = 10glO( e) -loge(x) = loge(x)/loge(10)
loge(g) is evaluated by defining
s = (g-l)/g+l) and
z = 2-s
and then calculating
loge(g) = loge( (1 +z/2)/(1-z/2»
using a minimax rational approximation.

Logarithm Routines 3-13

Error Conditions

1. If the argument is equal to 0.0, the following message is issued and the

result is set to -machine infinity.
GLOG10: Arg is zero; result = -infinity

2. If the argument is negative, the following message is issued and the absolute value of the argument is used.
GLOG10: Negative arg; result = GLOG10(ABS(arg))

3-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

CLOG

Description

The CLOG routine calculates the complex, single-precision, floating-point
natural logarithm of its complex, single-precision, floating-point argument.
That is:
CLOG(z) = loge(z)
Routines Called

CLOG calls the ALOG, ATAN, ATAN2, and MTHERR routines.
Type of Argument

The argument must be a complex, single-precision, floating-point value, both
parts of which cannot be equal to 0.0, although either can be equal to 0.0.
Type of Result

The result returned is a complex, single-precision, floating-point value. The
real part of the result is in the range -89.415 to 88.029; the imaginary part is in
the range -7r to 7r.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
-100.00 through 100.00 imaginary

MRE:

5.30x10-5 (14.2 bits) real
1.49x1O-8 (26.0 bits) imaginary

RMS:

1.06x10-7 (23.2 bits) real
3.44x1O-9 (28.1 bits) imaginary
-3 -2 -1
0 +1 +2
1% 1% 1C]'o 6% 82% 7% 1% real
0% 0% 0% 3% 94% 3% 0% imaginary
~4+

LSB error distribution:
Algorithm Used

CLOG(z) is calculated as follows.
Let z = x+i-y
If x = 0.0 and y = 0.0
CLOG(z) = (+infinity, 0.0)

If x = 0.0 and y 0.0
CLOG(z) = loge(lyl)+i-sgn(Y)-7r/2

Logarithm Routines 3-15

If x =1= 0.0 and y = 0.0
If x > 0.0
CLOG(z) = loge(x)+i -0.0
If x < 0.0
CLOG(z) = loge(lxl) +i-1I"
If x =1= 0.0 and y =1= 0.0
CLOG(z) = u+i-v
u = .5 -loge(x 2+y2)
v = tan-1(y/x)
Scaled values are calculated on occurences of overflow/underflow
for (X 2,y2) or (X2+y2) and propagated to give a valid in-range result
for u.
Error Conditions

1. If both parts of the argument equal 0.0, the following message is issued
and the result is set to (+infinity, 0.0).
CLOG; Arg is zero; result = (+infinity, zero)
2. If either part of the result underflows, one or both of the following mes-

sages are issued and the relevant part of the result is set to 0.0.
CLOG: Real part underflow
CLOG: Imaginary part underflow

3-16 TOPS-10/TOPS-20 Common Math Library Reference Manual

CDLOG

Description

The CDLOG subroutine calculates the complex, double-precision, D-floatingpoint natural logarithm of its complex, double-precision, D-floating-point argument. That is:
CDLOG(z,r) = loge(z)
z = location of input value
r = location of result
Routines Called

CDLOG calls the DLOG, DATAN, DATAN2, and MTHERR routines.
Type of Argument

CDLOG is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, D-floating-point value, both parts of which cannot be
equal to 0.0, although either can be equal to 0.0.
Type of Result

The result returned is a complex, double-precision, D-floating-point value.
The real part of the result is in the range -89.415 to 88.376; the imaginary part
is in the range -7r to 7r. The result is returned in the second vector (r) supplied
in the call. The real part of the result is returned in the first element of r; the
imaginary part is returned in the second element of r.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
--100.00 through 100.00 inlaginary

MRE:

9.07x10- 16 (50.0 bits) real
5.09x1o-- 19 (60.8 bits) imaginary

RMS:

1.59x10- 18 (59.1 bits) real
1.04xlo- 19 (63.1 bits) imaginary

LSB error distribution:

-4+ -3 -2 -1
0 + 1 +2
1% 1% 1% 5% 84% 6% 10;(-) real
0% 0% 0% 4% 92% 4% 0% imaginary

Logarithm Routines 3-17

Algorithm Used

CDLOG is calculated as follows.
Let z = x+i·y
If x = 0.0 and y = 0.0
CDLOG(z) = (+infinity, 0.0)
If x = 0.0 and y 7'= 0.0
CDLOG(z) = loge(lyl)+i ·sgn(y) ·7r/2
If x 7'= 0.0 and y = 0.0
If x> 0.0
CDLOG(z) = loge(x)+i ·0.0
If x < 0.0
CDLOG(z) = loge(lxl) +i "7r
If x 7'= 0.0 and y 7'= 0.0
CDLOG(z) = u+i·v
u = .5 ·loge(x 2+y2)
v = tan- 1 (y,x)
Scaled values are calculated on occurrences of overflow/
underflow for (x 2, y2) or (X2+y2) and progagated to give a valid inrange result for u.
Error Conditions

1. If both parts of the argument equal 0.0, the following message is issued
and the result is set to (+infinity, 0.0).
CDLOG: Arg is zero; result = (+infinity, zero)

2. If either part of the result underflows, one or both of the following messages are issued and the relevant part of the result is set to 0.0.
CDLOG: Imaginary part underflow
CDLOG: Real part underflow

3-18 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGLOG

Description

The CGLOG subroutine calculates the complex, double-precision, G-f1oatingpoint natural logarithm of its complex, double-precision, G-floating-point argument. That is:
CGLOG(z,r) = loge(z)
z = location of input value
r = location of result
Routines Called

CGLOG calls the GLOG, GATAN, GATAN2, and MTHERR routines.
Type of Argument

CGLOG is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, G-floating-point value, both parts of which cannot be
equal to 0.0, although either can be equal to 0.0.
Type of Result

The result returned is a complex, double-precision, G-floating-point value.
The real part of the result is in the range -710.475 to 709.436; the imaginary
part is in the range -7r to 7r. The result is returned in the second vector (r)
supplied in the call. The real part of the result is returned in the first element
of r; the imaginary part is returned in the second element of r.
Accuracy of Result

test interval:

-1000.0 through 1000.0 real
-100.00 through 100.00 imaginary

MRE:

7.15x10-- 11i (47.0 bits) real
3.54x10- 18 (58.0 bits) imaginary

RMS:

1.77x10-- 17 (55.7 bits) real
8.19x10- 19 (60.1 bits) imaginary

LSB error distribution:

-4+ -3 -2 -1
0 +1 +2
1% 0% 1% 5% 86~)'i) 6% 1% real
0% 0% 0% 4% 92% 4(Yc, 0% imaginary

Logarithm Routines

3-19

Algorithm Used
CGLOG(z) is calculated as follows.

Let z = x+i-y
If x = 0.0 and y = 0.0
CGLOG(z) = +machine infinity
If x = 0.0 and y =1= 0.0
CGLOG(g) = loge(lyl)+i -sgn(y) -11'"/2
If x =1= 0.0 and y = 0.0
If x> 0.0
CGLOG(z) = loge(x) +i -0.0
If x < 0.0
CGLOG(z) = loge(lxl)+i-1I'"
If x =1= 0.0 and y =1= 0.0
CGLOG(z) = u+i-v
u = .5 -loge(x 2+ y2)
V = tan- 1(y/x)
Scaled values are calculated on occurrence of overflow/underflow
for (x 2, y2) or (X2+y2) and propagated to give a valid in-range result
for u.
Error Conditions
1. If both parts of the argument equal 0.0, the following message is issued
and the result is set to (+machine infinity, 0.0).
CGLOG: Arg is zero; result = (+infinlty, zero)

2. If either part of the result underflows, one or both of the following messages are issued and the relevant part of the result is set to 0.0.
CGLOG: Real part underflow
CGLOG: Imaginary part underflow

3-20 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 4
Exponential and Exponentiation Routines

EXP

Description

The EXP routine calculates· the single-precision, floating-point exponential
function of its single-precision, floating-point argument. That is:
EXP(x) = eX
Routines Called

EXP calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value in the range
-89.4159863 to 88.0296919.
Type of Result

The result returned is a single-precision, floating-point value greater than
zero.
Accuracy of Result

test interval:

-89.000 through 88.000

MRE:

1.74x10-8 (25.8 bits)

RMS:

3.98x10-9 (27.9 bits)

LSB error distribution:

-2
0%

-1
2%

o
86%

+1
12%

+2
0%

Algorithm Used

EXP(x) is calculated as follows.
If x < -89.4159863
EXP(x) = 0.0
If x > 88.0296919
EXP(x) = +machine infinity

Otherwise, the argument is reduced as follows:
Let n = the nearest integer to x/loge(2)
The reduced argument is:
g = x-n -loge(2)
The calculation is:
EXP(x) = R(g) _2(n+1)
R(g) = .5+g·p/(q-g-p)
P = p1-g 2 +.25
q = q1-g 2 +.5
pI = .00416028863
q1 = .0499871789

Exponential and Exponentiation Routines

4-3

Error Conditions

1. If the argument is less than -89.4159863, the following message is issued
and the result is set to 0.0.
EXP: Result underflow

2. If the argument is greater than 88.0296919, the following message is issued
and the result is set to +machine infinity.
EXP: Result overflow

4-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DEXP
Description
The DEXP routine calculates the double-precision, D-floating-point exponential function of its double-precision, D-floating-point argument. That is:

DEXP(x) = eX
Routines Called
DEXP calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, D-floating-point value in the range
-89.415986292232944914 to 88.029691931113054295.
Type of Result
The result returned is a double-precision, D-floating-point value greater than
zero.
Accuracy of Result
test interval:

-89.000 through 88.000

MRE:

4.89x10- 19 (60.8 bits)

RMS:

1.17x10-- 19 (62.9 bits)

LSB error distribution:

-2
0%

-1
2%

o
86%

+1
12%

+2
0%

Algorithm Used
DEXP(x) is calculated as follows.

If x < -89.415986292232944914
DEXP(x) = 0.0
If x > 88.029691931113054295

DEXP(x) = +rnachine infinity
Otherwise, the argument is reduced as follows:
Let xl = [x], the greatest integer in x
x2 = x-xl
n = the nearest integer to x/loge (2)
The reduced argument is:
g = x1-n·c1+x2+n·c2
cl = .543 R
c2 = loge(2)-.543 R

Exponential and Exponentiation Routines 4-5

The calculation is:
DEXP(x) = R(g) e 2(n+1)
R(g) = .5+g ep/(q_gep)
p = (((p2 eg2+p1) eg2)+pO) eg2
q = ((((q3 eg2+q2) -g2)+q1) eg2)+qO
pO = .250
p1 = .757531801594227767x10-2
p2 = .315551927656846464x10-4
qO =.5
q1 = .568173026985512218x10- 1
q2 = .631218943743985036x10-3
q3 = .751040283998700461x10-6
Error Conditions

1. If the argument is less than -89.415986292232944914, the following message is issued and the result is set to 0.0.
OEXP: Result underflow

2. If the argument is greater than 88.029691931113054295, the following message is issued and the result is set to +machine infinity.
OEXP: Result overflow

4-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

GEXP

Description

The GEXP routine calculates the double-precision, G-floating-point exponential function of its double-precision, G-floating-point argument. That is:
GEXP(x) = eX
Routines Called

GEXP calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value in the range
-710.475860073943942 to 709.08956571282405l.
Type of Result

The result returned is a double-precision, G-floating-point value greater than
or equal to zero.
Accuracy of Result

test interval:

-89.000 through 88.000

MRE:

3.99x10- 18 (57.8 bits)

RMS:

9.40x10- 19 (59.9 bits)

-2
0%

LSB error distribution:

-1
2%

o
85%

+1
13%

+2
0%

Algorithm Used

GEXP(x) is calculated as follows.
If x :s; -710.475860073943942
GEXP(x) = 0.0
If x > 709.089565712824051
GEXP(x) = +machine infinity

Otherwise, the argument is reduced as follows:
Let xl = [x], the greatest integer in x
x2 = x-xl
n = the nearest integer to x/loge(2)
The reduced argument is:
g = x1-n e c1+x2+n e c2
c1 = .5438
c2 = loge(2)-.543s

Exponential and Exponentiation Routines 4-7

The calculation is:
GEXP(x) = R(g) ·2(n+l)
R(g) = .5+g ep/(q_gep)
p = «(p2eg2+pl) eg2)+pO) eg2
q = ««q3 eg2+q2)·g2)+ql)·g2)+qO
pO = .250
pi = .757531801594227767xl0- 2
p2 = .315551927656846464xlO-4
qO =.5
ql = .568173026985512218xl0- 1
q2 = .631218943743985036xlO- 3
q3 = .751040283998700461xl0- 6
Error Conditions

1. If the argument is less than or equal to -710.475860073943942, the following message is issued and the result is set to 0.0.
GEXP: Result underflow

2. If the argument is greater than 709.089565712824051, the following Inessage is issued and the result is set to +machine infinity.
GEXP: Result overflow

4-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

CEXP

Description
The CEXP routine calculates the complex, single-precision, floating-point
exponential function of its complex, single-precision, floating-point argument.
That is:

CEXP(z) = eZ
Routines Called
CEXP calls the EXP, COS, SIN, and MTHERR routines.
Type of Argument
The argument must be a complex, single-precision, floating-point value in the
range -89.4159863 to 176.0593838 for the real part and less than 823549.66 for
the imaginary part.
Type of Result
The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

test interval:

-40.000 through 12.000 real
-10.000 through 157.08 imaginary

MRE:

2.77x10-8 (25.1 bits) real
2.88x10-8 (25.0 bits) imaginary

RMS:

6.51x10-9 (27.2 bits) real
6.38x10-9 (27.2 bits) imaginary

LSB error distribution:

-2
1%
1%

-1
19%
17%

58%
59%

21%
23%

1% real
1% imaginary

Algorithm Used
CEXP(z) is calculated as follows.

Letz=x+i·y

If Iyl > 823549.66
CEXP(z) = (0.0,0.0)
If x < -89.4159863
CEXP(z) = (0.0,0.0)
If x > 88.0296919 and y = 0.0
CEXP(z) = (+infinity, 0.0)
If 88.0296919 < x < 176.0593838
and a component of the result is out of range,
that component is set to +infinity.
If x > 176.0593838 and y =1= 0.0
CEXP(z) = (± infinity, ± infinity)
Otherwise
CEXP(z) = eXe(cos(y)+iesin(y»
Exponential and Exponentiation Routi nes 4-9

Error Conditions

The following table gives the possible error conditions and the resulting error
messages.
Error Conditions for CEXP

Real Part
of Argument

Imaginary Part
of Argument

Result

Error Message(s)

Any Value

> 823549.66

(0.0,0.0)

Not 0.0 and
-s; 823549.66

(0.0,0.0)

#2 and #3

Not 0.0 and
~ 823549.66

Underflow may
occur on neither,
either, or both
parts

None or #2
or #3 or
#2 and #3

(+infinity, 0.0)

< -89.4159863

Between
-89.41598663
and 88.0296919

> 88.0296919

0.0

> 176.0593838

Not 0.0 and
~ 823549.66

(± infinity,
± infinity)

#4 and #5

Between
88.0296919 and
176.0593838

Not 0.0 and
~ 823549.66

Overflow may occur on neither, either, or both
parts

None or #4
or #5 or
#4 and #5

Error Messages:
1. CEXP:ABS(IMAG(arg» too large; result = zero
2. CEXP: Real part underflow
3. CEXP: Imaginary part underflow
4. CEXP: Real part overflow
5. CEXP: Imaginary part overflow

4-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

CDEXP

Description

The CDEXP subroutine calculates the complex, double-precision, D-floatingpoint exponential function of its complex, double-precision, D-floating-point
argument. That is:
CDEXP(z,r)

eZ
Z = location of input value
r = location of result
=

Routines Called

CDEXP calls the DEXP, DSIN, DCOS, and MTHERR routines.
Type of Argument

CDEXP is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex
double-precision, D-floating-point value in the range -89.415986292232944914
to 176.059383862226109 for the real part and less than 6746518850.429 for the
imaginary part.
Type of Result

The result returned is a complex, double-precision, D-floating-point value. It
is returned in the second vector (r) supplied in the call. The real part of the
result is returned in the first element of r; the imaginary part is returned in
the second element of r.
Accuracy of Result

test interval:

-40.000 through 12.000 real
-10.000 through 157.08 imaginary

MRE:

8.78x10- 19 (60.0 bits) real
9.49x10- 19 (59.9 bits) imaginary

RMS:

1.90x10- 19 (62.2 bits) real
1.87x1o- 19 (62.2 bits) imaginary

LSB error distribution:

-2

-1

1%
1%

23%
20%

57%
59%

18%
19%

1% real
1% imaginary

Exponential and Exponentiation Routines

4--11

Algorithm Used

CDEXP is calculated as follows.
Letz=x+i·y
If Iyl > 6746518850.429

CDEXP(z) = (0.0,0.0)
If x < -89.415986292232944914

CDEXP(z) = (0.0,0.0)
If x > 88.029691931113054295 and y = 0.0

CDEXP(z) = (+infinity, 0.0)
If 88.029691931113054295 < x < 176.059383862226109

and a component of the result is out of range,
that component is set to +infinity.
If x > 176.059383862226109 and y :#= 0.0
CDEXP(z) = (± infinity, ± infinity).

Otherwise
CDEXP(z) = eXe(cos(y)+i esin(y))
Error Conditions

The following table gives the possible error conditions and the resulting error
messages.
Error Conditions for CDEXP
Real Part
of Argument

Imaginary Part
of Argument

Result

Error Message(s)

Any Value

> 6746518850.429

(0.0,0.0)

< -89.415986292232944914

0.0

(0.0,0.0)

Not 0.0 and

(0.0,0.0)

#2 and #3

Underflow may
occur on neither,
either, or both
parts

None or #2
or #3 or
#2 and #3

6746518850.429

Between

Not 0.0 and

-89.415986292232944914
and 88.02969193113054295

> 88.02969193113054295

0.0

(+infinity, 0.0)

> 176.059383862226109

Not 0.0 and

(± infinity,
± infinity)

#4 and #5

Overflow may occur on neither, either, or both
parts

None or #4
or #5 or
#4 and #5

6746518850.429

$ 6746518850.429

Between

Not 0.0 and

88.02969193113054295 and
176.059383862226109

6746518850.429

Error Messages:
1. CDEXP:ABS(IMAG(arg» too large; result = zero
2. CDEXP: Real part underflow
3. CDEXP: Imaginary part underflow
4. CDEXP: REAL(arg) too large; REAL(result) = +infinity
5. CDEXP: REAL(arg) too large; IMAG(result) = +infinity

4-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGEXP

Description

The CGEXP subroutine calculates the complex, double-precision, G-floatingpoint exponential function of its complex, double-precision, G-floating-point
argument. That is:
CGEXP(z,r)

eZ
Z = location of input value
r = location of result
=

Routines Called

CGEXP calls the GEXP, GSIN, GCOS, and the MTHERR routines.
Type of Argument

CGEXP is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, G-floating-point value in the range
-710.475860073943942 to 1418.179131425648102 for the real part and less than
1686629713.065 for the imaginary part.
Type of Result

The result returned is a complex, double-precision, G-floating-point value. It
is returned in the second vector (r) supplied in the call. The real part of the
result is returned in the first element of r; the imaginary part is returned in
. the second element of r.
Accuracy of Result

test interval:

-40.000 through 12.000 real
-10.000 through 157.08 imaginary

MRE:

6.50xl0- 18 (57.1 bits) real
6.67xl0- 18 (57.1 bits) imaginary

RMS:

·1.53xl0- 18 (59.2 bits) real
1.44xl0-18 (59.3 bits) imaginary

LSB error distribution:

-2
1%
0%

-1
19%
16%

57%
60%

22%
22%

1% real
1% imaginary

Exponential and Exponentiation Routi nes 4-13

Algorithm Used
CGEXP(z) is calculated as follows.

Let z = x+iey
If Iyl > 1686629713.065
CGEXP(z) = (0.0,0.0)
If x < -710.475860073943942
CGEXP(z) = (0.0,0.0)
If x > 709.089565 and y = 0.0
CGEXP(z) = (+infinity, 0.0)
If 709.089565 < x < 1418.179131425648102
and a component of the result is out of range,
that component is set to +infinity.
If x > 1418.179131425648102 and y =1= 0.0
CGEXP(z) = (±infinity, ±infinity)

Otherwise
CGEXP(z) = eXe(cos(y)+i esin(y»
Error Conditions
The table below shows the possible values of the argument that could cause
error conditions.
Error Conditions for CGEXP
Real Part
of Argument

Imaginary Part
of Argument

Result

Error Messages

Any value

> 1686629713.065

(0.0,0.0)

< -710.475860073943942

0.0

(0.0,0.0)

Not 0.0 and

(0.0,0.0)

#2 and #3

Underflow may
occur on neither,
either, or both
parts

None or #2 or #3
or #2 and #3

=:; 1686629713.065

Between

Not 0.0 and

-710.475860073943942
and 709.089565

=:; 1686629713.065

> 709.089565

0.0

(infinity, 0.0)

> 1418.179131425648102

Not 0.0 and

( ± infinity,
± infinity)

#4 and #5

Overflow may occur on neither, either, or both
parts

None or #4 or #5
or #4 and #5

=:; 1686629713.065

Between

Not 0.0 and

709.089565 and
1418.179131425648102

=:; 1686629713.065

Error Messages:
1. CGEXP: ABS(lMAG(arg» too large; result = zero
2. CGEXP: Real part underflow
3. CGEXP: Imaginary part underflow
4. CGEXP: REAL(arg) too large; REAL(result) = +infinity
5. CGEXP: REAL(arg) too large; IMAG(result) = +infinity

4-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

EXP1.

Description
The EXPl. routine raises one integer to the power of another integer. That is:

EXPl.(m,n) = mn
Routines Called
EXPl. calls the MTHERR routine.
Type of Arguments
The two arguments must be integer values; they can be any such values.
Type of Result
The result returned is an integer value; it may be any such value.
Accuracy of Result
The result is exact.
Algorithm Used
EXPl.(m,n) is calculated as shown in the following table.
Calculations for EXP1.
Value of m

Value of n

Result

+infinity

any value

-1

even

-1

odd

-1

*±1

Error Conditions

1. If the exponent is too large a number, the following message is issued and
the result is set to ± infinity.
EXP1.: Result overflow

2. If both the base and the exponent are 0, the following message is issued
and the result is set to O.
EXP1.: Zero**zero is indeterminate, result = zero

Exponential and Exponentiation Routines 4-15

EXP2.
Description

The EXP2. routine raises a single-precision, floating-point number to the
power of an integer. That is:
EXP2.(x,n) = xn
Routines Called

EXP2. calls the MTHERR routine.
Types of Arguments

There are two arguments. The base must be a single-precision, floating-point
value, and the exponent must be an integer value. They can be any such
values.
Type of Result

The result returned is a single-precision, floating-point value; it may be any
such value.
Accuracy of Result
MRE

test Interval
x

RMS

n
9

3.48xlO-9 (28.1 bits)

3.07xlO- (25.0 bits)

8.88xl0-9 (26.7 bits)

5.53xlO-B (24.1 bits)

1.61xl0- B (25.9 bits)

.50000 through 1.0000

-12

7.91xlO- (23.6 bits)

.50000 through 1.0000

9.08xlO- (23.4 bits)

.50000 through 1.0000

-20

1.27xlO- (22.9 bits)

.50000 through 1.0000

7.45xlO- (27.0 bits)

.50000 through 1.0000

-·5

.50000 through 1.0000

total

2.37xl0- (25.3 bits)

2.70x10- B (25.1 bits)

3.95xlO- (24.6 bits)

2.65xlO- (21.8 bits)

7.87x10- B (23.6 bits)

2.65xlO- 7 (21.8 bits)

3.67xlO-8 (24.7 bits)

LSB error distribution according to the value of n
2

-410%

-3
0%

-2
0%

-1
0%

0
100%

+1
0%

+2
0%

+3
0%

+4+
0%

n= -5

24%

41%

25%

13%

21%

23%

21%

13%

n = -12

13%

15%

12%

n= 15

12%

13%

12%

n = -20

20%

n= 40

34%

total

10%

12%

29%

12%

10%

4-16 TOPS-10/TOPS-20 Common Math Library Reference Manual

Algorithm Used
EXP2.(x,n) is calculated as shown in the following table.
Calculations for EXP2.
Value of x

Value of n

Result

*0.0

1.0

0.0

+infinity

> 0.0

Error Conditions

1. If the exponent has sufficiently large magnitude, overflow occurs in one of
the following ways:
Base

Exponent

Result

> 1.0

positive

+infinity

< -1.0

positive, even
positive, odd

+infinity
-infinity

0.0 to 1.0

negative

+infinity

-1.0 to 0.0

negative, even
negative, odd

+infinity
-infinity

and the following message is issued.
EXP2.: Result overflow

2. If the exponent has sufficiently large magnitude, underflow occurs in one
of the following ways:
Magnitude of Base

Exponent

Result

> 1.0

negative

0.0

< 1.0

positive

0.0

and the following message is issued.
EXP2.: Result underflow

3. If both the exponent and the base are zero, the following message is issued
and a result of zero is returned.
EXP2.: Zero··zero is indeterminate, result = zero

Exponential and Exponentiation Routines 4-17

DEXP2.
Description

The DEXP2. routine raises a double-precision, D-floating-point number to
the power of an integer. That is:
DEXP2.(x,n) = xn
Routines Called

DEXP2. calls the MTHERR routine.
Type of Arguments

There are two arguments. The base must be a double-precision, D-floatingpoint value, and the exponent must be an integer value. They can be any such
values.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result
test Interval
x

.50000 through 1.0000

MRE

RMS

2.16xlO- 19 (62.0 bits)

1.01x10- 19 (63.1 bits)

.50000 through 1.0000

-9

1.62xlO- 18 (59.1 bits)

4.72x10- 19 (60.9 bits)

.50000 through 1.0000

2.27x10- 18 (58.6 bits)

6.79x10- 19 (60.4 bits)

.50000 through 1.0000

2.73xlO- 18 (58:3 bits)

7.89x10- 19 (60.1 bits)

.50000 through 1.0000

-40

7.50x10- 18 (56.9 bits)

2.31xlO- 18 (58.6 bits)

7.50xlO- 18 (56.9 bits)

1.15x10- 18 (59.6 bits)

total

LSB error distribution according to the value of n
2

-4+
0%

-3
0%

-2
0%

-1
0%

0
100%

+1
0%

+2
0%

+3
0%

+4+
0%

n= -9

12%

20%

23%

20%

12%

n= 12

12%

15%

16%

15%

13%

n= 15

12%

13%

12%

n = -40

34%

total

10%

11%

31%

11%

10%

4-18 TOPS-10/TOPS-20 Common Math Library Reference Manual

Algorithm Used

DEXP2.(x,n) is calculated as shown in the following table.
Calculations for DEXP2.
Value of x

Value of n

Result

*0.0

1.0

0.0

+infinity

> 0.0

n
X

Error Conditions

1. If the exponent has sufficiently large magnitude, overflow occurs in one of
the following ways:
Base

Exponent

Result

> 1.0

positive

+infinity

<-1.0

positive, even
positive, odd

+infinity
-infinity

0.0 to 1.0

negative

+infinity

-1.0 to 0.0

negative, even
negative, odd

+infinity
-infinity

and the following error message is issued.
DEXP2.: Result overflow

2. If the exponent has sufficiently large magnitude, underflow occurs in one
of the following ways:
Magnitude of Base

Exponent

Result

> 1.0

negative

0.0

< 1.0

positive

0.0

and the following message is issued.
DEXP2.: Result underflow

3. If both the exponent and the base are zero, the following message is issued
and the result is set to zero.
DEXP2.: Zero··zero is indeterminate, result = zero

Exponential and Exponentiation Routines 4-19

GEXP2.

Description

The GEXP2. routine raise a double-precision, G-floating-point number to the
power of an integer. That is:
GEXP2.(x,n) = xn
Routines Called

GEXP2. calls the MTHERR routine.
Type of Arguments

There are two arguments. The base must be a double-precision, G-floatingpoint value; it can be any such value. The exponent must be an integer value;
it can be any such value.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result
test Interval

MRE

RMS

1.72xlO- 18 (59.0 bits)
1.26xlO- 17 (56.1 bits)

8.11xl(f19 (60.1 bits)
3.79xl0- 18 (57.9 bits)

1.69xlO- 17 (55.7 bits)
2.13xlO- 17 (55.4 bits)

5.45xl0- 18 (57.3 bits)
6.27xl0- 18 (57.1 bits)

-40

5.64xlo- 17 (54.0 bits)

1.85xHr 17 (55.6 bits)

5.64xlo- 17 (54.0 bits)

9.25xlo- 18 (56.6 bits)

.50000 through 1.0000

-9

.50000 through 1.0000

.50000 through 1.0000
.50000 through 1.0000

total

LSB error distribution according to the value of n
2

-4+
0%

-3
0%

-2
0%

-1
0%

0
100%

+1
0%

+2
0%

+3
0%

+4+
0%

n= -9

12%

21%

23%

20%

12%

n= 12

13%

16%

15%

13%

n= 15

12%

13%

14%

13%

12%

n = -40

34%

total

10%

11%

31%

10%

4-20 TOPS-10/TOPS-20 Common Math Library Reference Manual

Algorithm Used

GEXP2.(x,n) is calculated as shown in the following table.
Calculations for GEXP2.
Value of x

Value of n

Result

*0.0

1.0

0.0

+infinity

> 0.0

Error Conditions

1. If the exponent has sufficiently large magnitude, overflow occurs in one of
the following ways:
Base

Exponent

Result

> 1.0

positive

+infinity

<-1.0

positive, even
positive, odd

+infinity
-infinity

0.0 to 1.0

negative

+infinity

-1.0 to 0.0

negative, even
negative, odd

+infinity
-infinity

and the following error message is issued:
G EXP2.: Result overflow

2. If the exponent has sufficiently large magnitude, underflow occurs in one
of the following ways:
Magnitude of Base

Exponent

Result

> 1.0

negative

0.0

< 1.0

positive

0.0

and the following message is issued:
GEXP2.: Result underflow

3. If both the exponent and the base are zero, the following message is issued
and the result is set to zero.
GEXP2.: Zero**zero is indeterminate, result = zero

Exponential and Exponentiation Routines 4-21

CEXP2.
Description

The CEXP2. routine raises a complex, single-precision, floating-point number
to the power of an integer. That is:
CEXP2.(z,n) = zn
Routines Called

CEXP2. calls the CDLOG, DLOG, DSIN, DCOS, DEXP, and MTHERR
routines.
Type of Arguments

There are two arguments. The base must be a complex, single-precision,
floating-point value, and the exponent must be an integer. They can be any
such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

test interval:

.50000 through 1.0000 for Z (real)
.50000 through 1.0000 for z (imaginary)
-10 through 20 for n

MRE:

7,45x10-9 (27.0 bits) real
7,45x1o-9 (27.0 bits) imaginary

RMS:

3.17x10- 9 (28.2 bits) real
3.16x1o-9 (28.2 bits) imaginary

LSB error distribution:

-2
0%
0%

-1
0%
0%

o +1
100% 0%
100% 0%

+2
0% real
0% imaginary

When the ratio of the imaginary part of the base to the real part is less than
-1010, one part of the result is less accurate. Which part is less accurate
depends on the exponent. For example:
test interval:

LSB error distribution:

test interval:

LSB error distribution:

-1.00000x10- 10 thro\Jgh -1.00000x10- 15 for z (real)
-2.0000 through -1.0000 for z (imaginary)
-1 for n
-2
0%
0%

-1
6%
0%

0
+1
65% 28%
100% 0%

+2
2% real
0% imaginary

-1.00000x1O- lO through -1.00000x1o- 15 for z (real)
-2.0000 through -1.0000 for z (imaginary)
2 for n
-2
0%
6%

-1
0%
27%

0
+1
100% 0%
60% 8%

4-22 TOPS-10/TOPS-20 Common Math Library Reference Manual

+2
0% real
0% imaginary

Algorithm Used

CEXP2.(z,n) is calculated as follows.
Let Z = x+i·y
First the routine checks for the special cases shown in the following table.
Special Cases for CEXP2.
Value of x

Value of y

any value

0.0

(0.0,0.0)

0.0

(O.O,Q.O)

(1.0,0.0)

not both 0.0

Value of n

Result

x+i .y
(+infinity, +infinity)

If none of the special cases applies, the routine continues calculations as
follows.
The CEXP2. function is evaluated as the complex exponential of
n • (LNRHO + i -THETA).
LNRHO is the real part of:
loge(x+i -y)
THETA is the imaginary part of:
loge(x+i -y)
The real part of n-(LNRHO+i·THETA) is:
ALPHA = n-LNRHO
and the imaginary part is:
PHI = n-THETA
Since it is ultimately ei·PHI that is needed, it would appear that sin(PHI)
and cos(PHI) are needed. However, these functions will be multiplied by
eALPHA, and the handling of exception boundaries on the product will be
expedited by use of 10ge(sin(PHI)) and 10ge(cos(PHI)), which will be added
to ALPHA before the call to the DEXP function. The absolute values of
sin(PHI) and cos(PHI) are used as arguments of the CDLOG function; the
signs of sin(PHI) and cos(PHI) are stored for use in determining the signs
for the real and imaginary parts of the complex exponential, CEXP.
The real part of the final result is:
sgn(cos(PHI)) _eALPHA+loge(1 cos(PHI)I)
The imaginary part of the final result is:
sgn(sin(PHI)) -eALPHA+loge(1 sin(PHI)I)

Exponential and Exponentiation Routines 4-23

Error Conditions

The following error messages are returned for error conditions detected during
the check for the special cases shown above. Other errors detected will result
in error messages relating to the CEXP3. routine because CEXP2. is part of
the CEXP3. routine.
1. If both the real and iraaginary parts of the argument are zero and the

exponent is also zero, the following message is issued and the result is set
to (0.0,0.0).
CEXP2.: Zero··zero is indeterminate, result = zero

2. If both the real and imaginary parts of the argument are zero and the
exponent is negative, the following message is issued and the result is set
to (infinity, infinity).
CEXP2.: Zero·· negative exponent, result = infinity

3. If PHI ~ 6746518852, argument reduction for sin/cos is impossible so the
following message is issued and the result is set to (+infinity, +infinity).
CEXP2.: Both parts indeterminate

4. If the base and/or the exponent are such that one or both parts of the
result overflow, one of the following messages is issued and the corresponding result is set to ± infinity.
CEXP2.: Real part overflow
CEXP2.: Imaginary part overflow
CEXP2.: Both parts ove~flow

5. If the base and/or the exponent are such that one or both parts of the
result underflows, one of the following messages is issued and the corresponding result is set to 0.0.
CEXP2.: Real part underflow
CEXP2.: Imaginary part underflow
CEXP2.: Both parts underflow

4-24 TOPS-10/TOPS-20 Common Math Library Reference Manual

EXP3.

Description

The EXP3. routine raises a single-precision, floating-point number to the
power of another single-precision, floating-point number. That is:
EXP3.(x,y) = xY
Routines Called

EXP3. calls the MTHERR routine.
Type of Arguments

There are two arguments; both must be single-precision, floating-point values. The base must not be less than zero unless the exponent is an integer.
The base must not be equal to zero unless the exponent is greater than zero.
Type of Result

The result returned is a single-precision, floating-point value in the range
2- 129 to 2127.
Accuracy of Result
test Interval

MRE

RMS

5.1

1.52xlO-B (26.0 bits)

4.70x1o-9 (27.7 bits)

.50000 through 1.0000

-10.1

1.86xHrB (25.7 bits)

4.92x1o- 9 (27.6 bits)

.50000 through 1.0000

15.1

2.27x1O-B (25.4 bits)

5.42x1o-9 (27.5 bits)

.50000 through 1.0000

-20.1

3.14x1o-B (24.9 bits)

6.05x1o-9 (27.3 bits)

.50000 through 1.0000

30.1

3.90x1o--8 (24.6 bits)

7.32x1O-9 (27.0 bits)

.50000 through 1.0000

-50.1

6.18x1o--8 (23.9 bits)

1.07x1o-8 (26.5 bits)

.50000 through 1.0000

80.1

9.04x1o-- 8 (23.4 bits)

1.60x1o-8 (25.9 bits)

9.04x1o-B (23.4 bits)

8.74xlo-9 (26.8 bits)

.50000 through 1.0000

total

LSB error distribution according to the value of Y
5.1

-4+
0%

-3
0%

-2
0%

-1
12%

0
74%

+1
14%

+2
0%

+3
0%

+4+
0%

Y = -10.1

11%

70%

19%

15.1

18%

66%

16%

Y = -20.1

1411()

61%

24%

Y= 30.1

21%

56%

18%

Y = -50.1

17%

46%

23%

Y= 80.1

19%

36%

19%

total

16%

58%

19%

Exponential and Exponentiation Routi nes 4-25

Algorithm Used

EXP3. (x,y) is calculated as follows.
First the routine checks for the special cases shown in the following table.
Special Cases for EXP3.
Value of x

Value of y

Result

0.0

> 0.0

0.0

<0.0

infinity

*0.0

0.0

1.0

<0.0

odd integer

<0.0

even integer

> 0.0

<0.0

not integer

(-x)Y

Otherwise
xY = 2W

w = y ·log2(x)
log2(x) is calculated as follows:
x = 2m ·f where .5 $ f < 1.0
Let p be an odd integer < 16 and
let a = 2- p/ 16
Then select p to minimize la-fl
now x = 2m ·a·(f/a)
Then log2(x) = m+log2(a)+log2(f/a) or
log2(x) = m-p/16+log2(f/a)
Let ul = m-p/16 and
u2 = log2(f/a) = log2( (1 +s)/(1-s))
Then log2(x) = ul+u2 and
s = (f-a)/(f+a)
A rational approximation is used to evaluate u2; ul and u2 are then
used to determine wI and w2.
w = y·log2(x) = wl+w2 and
wI = FLOAT(INT(w·16.0))/16.0 = ml+pl/16
ml and pI are integers with 0 $ pI $ 15
Finally
If -129 $ w < 127
EXP3.(x,y) = xY = 2W is reconstructed as:
EXP3.(x,y) = 2w1 ·2w2
2w1 is evaluated by table lookup and 2w2 is evaluated from another rational approximation.

4-26 TOPS-10/TOPS-20 Common Math Library Reference Manual

Error Conditions
1. If the base is a negative value and the exponent is not an integer, the
following message is issued and the calculation proceeds using the absolute value of the base.
EXP3.: Negative base**non-integer; ABS(base) used

2. If the base is 0.0 and the exponent is negative, the following message is
issued and the result is set to infinity.
EXP3.: Zero**negative exponent; result = infinity

3. If both the base and the exponent are 0.0, the following message is issued
and the result is set to 0.0.
EXP3.: Zero**zero is indeterminate; result = zero

4. If y -log2(x) ;;::: 127, the result overflows. Then the following message is issued and the result is set to -infinity if x is less than 0.0 and y is an odd
integer. Otherwise, the result is set to +infinity.
EXP3.: Result overflow

5. If y·log2(x) < -129, the result underflows. Then the following message is
issued and the result is set to 0.0.
EXP3.: Result underflow

Exponential and Exponentiation Routines 4-27

DEXP3.
Description

The DEXP3. routine raises a double-precision, D-floating-point number to
the power of another double-precision, D-floating-point number. That is:
DEXP3.(x,Y) = xY
Routines Called

DEXP3. calls the MTHERR routine.
Type of Argument

There are two arguments; both must be double-precision, D-floating-point
values. The base must not be less than zero unless the exponent is an integer.
The base must not be equal to zero unless the exponent is greater than zero.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 2- 129 and less than or equal to 2127.
Accuracy of Result
test Interval
x

.50000 through 1.0000

MRE

RMS

5.1

5.23xlO- 19 (60.7 bits)

1.45x1(f19 (62.6 bits)

.50000 through 1.0000

-10.1

5.50xHr 19 (60.7 bits)

1.46xlo- 19 (62.6 bits)

.50000 through 1.0000

20.1

9.07x1o- 19 (59.9 bits)

1.B4xHt- 19 (62.2 bits)

.50000 through 1.0000

-50.1

1.97x1o- 18 (5B.B bits)

3.27xHt-19 (61.4 bits)

.50000 through 1.0000

BO.1

3.02x1o- 18 (5B.2 bits)

5.10xlo- 19 (60.B bits)

3.02x1o-18 (5B.2 bits)

2.9Bxlo-19 (61.5 bits)

total

LSB error distribution according to the value of y
-4+
0%

-3

5.1

-2
0%

-1
7%

0
73%

+1
20%

+2
0%

+3
0%

+4+
0%

y = -10.1

13%

70%

17%

Y= 20.1

11%

63%

25%

Y = -50.1

19%

46%

21%

Y = -BO.1

16%

35%

22%

10%

total

13%

57%

21%

Algorithm Used

DEXP3.(x,y) is calculated as follows.
First the routine checks for the special cases shown in the following table.

4-28 TOPS-10/TOPS-20 Common Math Library Reference Manual

Special Cases for DEXP3.
Value of x

Value of y

Result

0.0

> 0.0

0.0

<0.0

infinity

=1=0.0

0.0

1.0

< 0.0

odd integer

<0.0

even integer

> 0.0

<0.0

not integer

(-x)Y

Otherwise
xY = 2W
w = y ·log2(x)
log2(x) is calculated as follows:
x = 2m ·f where .5 ~ f < 1.0
Let p be an odd integer < 16 and
let a = 2- p/ 16
Then select p to minimize la-fl
now x = 2me a·(f/a)
Then log2(x) = m+log2(a)+log2(f/a) or
log2(x) = m-p/16+log 2(f/a)
Let u1 = m-p/16 and
u2 = log2(f/a) = log2( (1 +s)/(1-s»
Then log2(x) = u1 +u2 and
s = (f-a)/(f+a)
A rational approximation is used to evaluate u2; u1 and u2 are then
used to determine wI and w2.
w = y·log2(x) = w1+ w2 and
wI = FLOAT(INT(w·16.0»/16.0 = ml+p1/16
m1 and pI are integers with 0:::; pI ~ 15
Finally
If -129 ~ w < 127
DEXP3.(x,y) = xY = 2 is reconstructed as:
DEXP3.(x,y) = 2w1 ·2w2
2w1 is evaluated by table lookup and 2w2 is evaluated from another rational approximation.
W

Exponential and Exponentiation Routines

4-29

Error Conditions

1. If the base is a negative value and the exponent is not an integer, the

following message is issued and the calculation proceeds using the absolute value of the base.
DEXP3.: Negative base**non-integer; ABS(base) used

2. If the base is 0.0 and the exponent is negative, the following message is

issued and the result is set to infinity.
DEXP3.: Zero**negative exponent; result = infinity

3. If both the base and the exponent are 0.0, the following message is issued
and the result is set to 0.0.
DEXP3.: Zero**zero is indeterminate; result = zero

4. If y ·log2(x) ~ 127, the result overflows. Then the following message is issued and the result is set to -infinity if x is less than 0.0 and y is an odd
integer. Otherwise, the result is set to +infinity.
DEXP3.: Result overflow

5. If y ·log2(x) < -129, the result underflows. Then the following message is
issued and the result is set to 0.0.
DEXP3.: Result underflow

4-30 TOPS-10/TOPS-20 Common Math Library Reference Manual

GEXP3.
Description

The GEXP3. routine raises a double-precision, G-floating-point number to
the power of another double-precision, G-floating-point number. That is:
GEXP3.(x,y) = xY
Routines Called

GEXP3. calls the MTHERR routine.
Type of Arguments

There are two arguments; both must be double-precision, G-floating-point
values. The base must not be less than zero unless the exponent is an integer.
The base must not be equal to zero unless the exponent is greater than zero.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
2- 1025 to 21023 .
Accuracy of Result
test Interval
x

.50000 through 1.0000

MRE

RMS

5.10

3.69xlO- 18 (57.9 bits)

1.18xHr 18 (59.6 bits)

.50000 through 1.0000

-10.10

4.91x1o- 18 (57.5 bits)

1.22x1Q-18 (59.5 bits)

.50000 through 1.0000

20.10

7.92x1o- 18 (56.8 bits)

1.49x1Q-18 (59.2 bits)

.50000 through 1.0000

-50.10

1.46x1o- 17 (55.9 bits)

2.70x1Q-18 (58.4 bits)

.50000 through 1.0000

80.10

2.17x1o-17 (55.4 bits)

4.13x1o- 18 (57.7 bits)

2.17x1o-17 (55.4 bits)

2.43x1Q-18 (58.5 bits)

total

LSB error distribution according to the value of Y
5.10

-4+
0%

-3
0%

-2
0%

-1
14%

0
70%

+1
16%

+2
0%

+3
0%

+4+
0%

Y = --10.10

12%

68%

20%

Y= 20.10

19%

60%

19%

Y = -50.10

17%

43%

24%

Y= 80.10

18%

34%

19%

total

16%

55%

20%

Algorithm Used

GEXP3.(x,y) is calculated as follows.
First the routine checks for the special cases shown in the following table.

Exponential and Exponentiation Routines 4-31

Special Cases for GEXP3.
Value of x

Value of y

Result

0.0

> 0.0

0.0

<0.0

infinity

*0.0

0.0

1.0

<0.0

odd integer

<0.0

< 0.0

even integer

> 0.0

<0.0

not integer

(-x)Y

Otherwise
x Y = 2W

w = y -log2(x)
log2(x) is calculated as follows:
x = 2m -f where .5:5; f < 1.0
Let p be an odd integer < 16 and
let a = 2- p/ 16
Then select p to minimize la-fl
now x = 2m -a-(f/a)
Then log2(x) = m+log2(a)+log2(f/a) or
log2(x) = m-p/16+log 2(f/a)
Let ul = m-p/16 and
u2 = log2(f/a) = log2«(1+s)/(I-s))
Then log2(x) = ul+u2 and
s = (f-a)/(f+a)
A rational approximation is used to evaluate u2; ul and u2 are then
used to determine wI and w2.
w = y-Iog2(x) = wl+w2 and
wI = FLOAT(INT(w-16.0))/16.0 = ml+pl/16
ml and pI are integers with 0:5; pI :5; 15
Finally
If -1025 :5; w < 1023

GEXP3.(x,y) = xY = 2W is reconstructed as:
GEXP3.(x,y) = 2wl _2w2
2w1 is evaluated by table lookup and 2w2 is evaluated from another rational approximation.

4-32 TOPS-10/TOPS-20 Common Math Library Reference Manual

Error Conditions

1. If the base is a negative value and the exponent is not an integer, the
following message is issued and the calculation proceeds using the absolute value of the base.
GEXP3.: Negative base**non-integer; ABS(base) used

2. If the base is 0.0 and the exponent is negative, the following message is
issued and the result is set to infinity.
GEXP3.: Zero**negative exponent; result = infinity

3. If both the base and the exponent are 0.0, the following message is issued
and the result is set to 0.0.
GEXP3.: Zero**zero is indeterminate, result = zero

4. If y·log2(x) ~ 1023, the result overflows, the following message is issued,
and the result is set to -infinity if x less than 0.0 and y is an odd integer.
Otherwise, the result is set to +infinity.
GEXP3.: Result overflow

5. If y·log2(x) < -1025, the result underflows, the following message is issued,
and the result is set to 0.0.
GEXP3.: Result underflow

Exponential and Exponentiation Routines 4-33

CEXP3.
Description

The CEXP3. routine raises a complex, single-precision, floating-point number
to the power of another complex, single-precision, floating-point number.
That is:
CEXP3.(z,g) = zg
Routines Called

CEXP3. calls the CDLOG, DLOG, DSIN, DCOS, DEXP, and MTHERR
routines.
Type of Arguments

There are two arguments; both must be complex, single-precision, floatingpoint values. They can be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value. It may
be any such value.
Accuracy of Result

test interval:

.50000 through 1.0000 for z (real)
.50000 through 1.0000 for z (imaginary)
-100.00 through 207.00 for g (real)
-163.00 through 7.00 for g (imaginary)

MRE:

7.45xlO-9 (27.0 bits) real
7.45x10-9 (27.0 bits) imaginary

RMS:

:3.17xlO-9 (28.2 bits) real
3.17x10-9 (28.2 bits) imaginary

LSB error distribution:

-2
0%
0%

-1
0%
0%

o
+1
100% 0%
100% 0%

+2
0% real
0% imaginary

LSB error distribution:

test interval:

LSB error distribution:

-1.00000x10- 10 through -1.00000x10- 15 for z (real)
-2.0000 through -1.0000 for z (imaginary)
(-1,0) for g
-2
0%
0%

-1
6%
0%

0
+1
65% 28%
100% 0%

+2
2% real
0% imaginary

-1.00000x10- 10 through -1.00000x10- 15 for z (real)
-2.0000 through -1.0000 for z (imaginary)
(2,0) for g
-2
0%
6%

-1
0%
27%

0
+1
100% 0%
60% 8%

4-34 TOPS-10/TOPS-20 Common Math Library Reference Manual

+2
0% real
0% imaginary

Algorithm Used
CEXP3. (z,g) is calculated as follows.

Let z = x+i·y
g = a+i·b
First the routine checks for the special cases shown in the following table.
Special Cases for CEXP3.
Value of x

0.0
0.0
0.0

Value of y

0.0
0.0
0.0

Value of a

Result

> 0.0

(0.0,0.0)

~O.O

0.0

(+infinity, +infinity)

(0.0,0.0)

If none of the special cases applies, the routine continues calculation as
follows.
If x and y#:O
x+i·y is rewritten as
e1oge(X+i'Y)

The CEXP3. function is evaluated as the complex exponential of
(a+i· b) ·(LNRHO+i·THETA).
LNRHO is the real part of:
loge(x+i .y)
THETA is the imaginary part of:
loge(x+i -y)
The real part of (a+i-b)-(LNRHO+i-THETA) is:
ALPHA = a-LNRHO-b-THETA
and the imaginary part is:
PHI = a-THETA+b-LNRHO
Since it is ultimately ei · PHI that is needed, it would appear that sin(PHI)
and cos(PHI) are needed. However, these functions will be multiplied by
eALPHA, and the handling of exception boundaries on the product will be
expedited by use of loge(sin(PHI) and loge(cos(PHI), which will be added
to ALPHA before the call to the DEXP function. The absolute values of
sin (PHI) and cos(PHI) are used as arguments of the CDLOG function; the
signs of sin(PHI) and cos(PHI) are stored for use in determining the signs
for the real and imaginary parts of the complex exponential, CEXP.
The real part of the final result is:
sgn( cos(PHI) -eALPHA+loge(l cos(PHI)I)
The imaginary part of the final result is:
sgn(sin(PHI» -eALPHA+loge(i sin(PHI)I)

Exponential and Exponentiation Routines

4-35

Error Conditions
1. If both the real and imaginary parts of both arguments are 0.0, the follow-

ing message is issued and the result is set to (0.0,0.0).
CEXP3.: Zero**zero is indeterminate; result = zero

2. If both the real and imaginary parts of the base are zero and the real part
of the exponent is negative, the following message is issued and the result
is set to (+infinity,+infinity).
CEXP3.: Zero**(negative,non-zero) is indeterminate,
result = (infinity,infinity)

3. If PHI ~ 6746518852, argument reduction for sin/cos is impossible so the
following message is issued and the result is set to (+infinity,+infinity).
CEXP3.: Both parts indeterminate

4. If the base and/or the exponent are such that one or both parts of the

result overflow, one of the following messages is issued and the corresponding result is set to ± infinity.
CEXP3.: Real part overflow
CEXP3.: Imaginary part overflow
CEXP3.: Both parts overflow

5. If the base and/or the exponent are such that one or both parts of the
result underflows, one of the following messages is issued and the corresponding result is set to (0.0).
CEXP3.: Real part underflow
CEXP3.: Imaginary part underflow
CEXP3.: Real and imaginary parts underflow

4-36 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 5
Trigonometric Routines

SIN
Description

The SIN routine calculates the single-precision, floating-point sine of the
single-precision, floating-point angle given' in radians as the argument. That
is:
SIN (x) = sin(x)
Routines Called

SIN calls the MTHERR routine.
Type of Argument

The argument nlust be a single-precision, floating-point value less than or
equal to 210828714.
Type of Result

The result returned is a single-precision, floating-point value in the range -1.0
to 1.0.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

1.95x10-8 (25.6 bits)

RMS:

3.87x10-9 (27.9 bits)

LSB error distribution:

-2
0%

-1
12%

0
78%

+1
10%

+2
0%

Algorithm Used

SIN(x) is calculated as follows. Note that SIN(x) = -SIN(-x).
Let Ixl = 7r en+f
If I < 7r/2
The argument reduction is as follows.
n = the nearest integer to Ixl/1r
Then the reduced argument is:
f = Ixl-7r en
If If I < 863167530x10-4
sin(f) = f

Otherwise
sin(f) = f+feR(g)
g=(2
R(g) = ((((r5 eg+r4) eg+r3) eg+r2) -g+r1)-g
r1 = -.166666666
r2 = .833333072x10-2
r3 = -.198408328x10-3
r4 = .275239711x10-5
r5 = -.238683464x10- 7
Finally
SIN(x) = sgn(x) -(-l)n esin(f)

Trigonometric Routines 5-3

Error Conditions
If the absolute value of the argument is greater than 210828714, the following

message is issued and the result is set to 0.0.
SIN: ABS(arg) too large; result = zero

5-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

SIND
Description

The SIND routine calculates the single-precision, floating-point sine of the
single-precision, floating-point angle given in degrees as the argument. That
IS:

SIND(x) = sin (x)
Routines Called

SIND calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value less than or
equal to 47185919.
Type of Result

The result returned is a single-precision, floating-point value in the range -1.0
to 1.0.
Accuracy of Result

test interval:

-1000.0 through 3600.0

MRE:

1.95x10-8 (25.6 bits)

RMS:

4.11x10- 9 (27.9 bits)

LSB error distribution:

-2

-1

13%

73%

+1
14%

+2
0%

Algorithm Used

SIND(x) is calculated as follows. Note that SIND(x) = -SIND( -x).
Let Ixl = 180-n+f
If I ::; 90
The argument reduction is as follows.
n = the nearest integer to Ixl/180
Then the reduced argument, converted to radians is:
f = (Ix 1-180· n)· (71"/180)
If IfI < 863167530x10-- 4
sin(f) = f

Otherwise
sin(f) = f+f·R(g)
g = f2
R(g) = (( ((r5· g+r4)· g+r3)· g+r2)· g+r1)·g
r1 = -.166666666
r2 = .833333072x10- 2
r3 = -.198408328x10-3
r4 = .275239711x1Q-5
r5 = -.238683464x10-7
Finally
SIND(x) = sgn(x)· (-l)n· s in(f)
Trigonometric Routines

5-5

Error Conditions

If the absolute value of the argument is greater than 47185919, the following
message is issued and the result is set to 0.0.
SIND: ABS(arg) too large; result = zero

5-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

cos
Description

The COS routine calculates the single-precision, floating-point cosine of the
single-precision, floating-point angle given in radians as the argument. That
is:
COS(x) = cos(x)
Routines Called

COS calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value less than
210828714.
Type of Result

The result returned is a single-precision, floating-point value in the range -1.0
to 1.0.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

1.86x10-8 (25.7 bits)

RMS:

4.26x10-9 (27.8 bits)

LSB error distribution:

-2
0%

-1
12%

0
70%

+1
17%

+2
0%

Algorithm Used

COS(x) is calculated as follows. Note that COS(x) = COS(-x).
Let Ixl = 1I" en+f
If I < 11"/2
The argument reduction is as follows.
n = .5 + the nearest integer to Ixl/1I"
Then the reduced argument is:
f = Ixl-1I" en
If If I < .863167530x10-4

sin(f) = f
Otherwise
sin(f) = f +f eR(g)
g = f2
R(g) = ««r5 eg+r4) eg+r3) eg+r2) eg+r1) eg
r1 = -.166666666
r2 = .833333072x10- 2
r3 = -.198408328x10-3
r4 = .275239711x10-5
r5 = -.238683464x10- 7
Finally
COS (x) = (-1)n+ 1e sin(f)
Trigonometric Routines • 5-7

Error Conditions
If the absolute value of the argument is greater than or equal to 210828714, the

following message is issued and the result is set to 0.0.
COS: ABS(arg) too large; result = zero

5-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

coso
Description

The COSD routine calculates the single-precision, floating-point cosine of the
single-precision, floating-point angle given in degrees as the argument. That
IS:

COSD(x) = cos(x)
Routines Called

COSD calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value less than
47185919.
Type of Result

The result returned is a single-precision, floating-point value in the range -1.0
to 1.0.
Accuracy of Result

test interval:

-1000.0 through 3600.0

MRE:

1. 75x10-8 (25.8 bits)

RMS:

4.20x10- 9 (27.8 bits)

LSB error distribution:

-2

-1

12%

72%

+1
16%

+2
0%

Algorithm Used

COSD(x) is calculated as follows. Note that COSD(x) = COSD(-x).
Let Ixl = 180-n+f
If I ~ 90
The argument reduction is:
n = .5+ the nearest integer to Ixl/180
Then the reduced argument, converted to radians, is:
f = (lxl-180-n) -(11"/180)
If If I < .863167530x10-4

sin(f) = f
Otherwise
sin(f) = f +f -R(g)
g = f2
R(g) = ««r5-g+r4)-g+r3)-g+r2)-g+r1)-g
r1 = -.166666666
r2 = . 833333072x 10-2
r3 = -.198408328x10-3
r4 = .275239711x10-5
r5 = -.238683464x10-7
Finally
COSD(x) = (-1)n+ 1e sin(f)
Trigonometric Routi nes 5-9

Error Conditions

If the absolute value of the argument is greater than or equal to 47185919, the
following message is issued and the result is set to 0.0.
COSO: ABS(arg) too large; result = zero

5-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

DSIN

Description
The DSIN routine calculates the double-precision, D-floating-point sine of the
double-precision, D-floating-point angle given in radians as the argument.
That is:
DSIN(x) = sin (x)

Routines Called
DSIN calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, D-floating-point value less than or
equal to 6746518852 (or 231e 1l").
Type of Result
The result returned is a double-precision, D-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result
test interval:

-10.000 through 201.06

MRE:

6.06x10- 19 (60.5 bits)

RMS:

1.35x10- 19 (62.7 bits)

LSB error distribution:

-2
0%

-1

22%

68%

+1
10%

+2
0%

Algorithm Used
DSIN(x) is calculated as follows. Note that DSIN(x) = -DSIN(-x).
Let Ixl = 1I" e n+f
If I < 11"/2
The argument reduction is as follows.
f = ((lxl-n e c1)-n e c2)-n e c3
c1 = high-order 34 bits of 11"
c2 = next 31 bits of1l"
c3 = next 62 bits of 11"

If If I < 2- 31
sin(f) = f

Trigonometric Routines

5-11

Otherwise
sin(f) = f +f -R(g)
g=(2

R(g) = (g-XNUM/XDEN+rpl)eg
XNUM = «rp5 eg+rp4)eg+rp3)eg+rp2
XDEN = «g·q2)eg+ql)eg+qO
rpl = -.166666666666666667
rp2 = .451456904704461990x1Of
rp3 = -.489487151969463797x1Gr
rp4 = .428183075897778265x10
rp5 = -.121560740596710190x101
qO = .541748285645351853xl07
q1 = .702492288221842518xlO'>
q2 = .394924723520450141x1Gr
Finally
DSIN(x) = sgn(x)·(-l)n es in(f)
Error Conditions
If the absolute value of the argument is greater than 6746518850, the following

message is issued and the result is set to 0.0.
DSIN: ABS(arg) too large; result = zero

5-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

DCOS
Description

The DCOS routine calculates the double-precision, D-floating-point cosine of
the double-precision, D-floating-point angle given in radians as the argument.
That is:
DCOS(x} = cos(x}
Routines Called

DCOS calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value less than
6746518852 (or 231 e 7r).
Type of Result

The result returned is a double-precision, D-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

4.96x10- 19 (60.8 bits)

RMS:

1.41x10- 19 (62.6 bits)

LSB error distribution:

-2
0%

-1

16%

66%

+1
18%

+2
0%

Algorithm Used

DCOS(x} is calculated as follows. Note that DCOS(x} = DCOS(-x}.
Let Ixl = 7r e n+f
IfI< 7r/2
The argument reduction is as follows.
f = (lxl-n e c1}-n e c2}-n a c3
c1 = high-order 34 bits of 7r
c2 = next 31 bits of 7r
c3 = next 62 bits of 7r
If IfI < 2-31
sin(f) = f

Trigonometric Routines

5-13

Otherwise
sin(f) = f+feU(g)
g= f 2
R(g) = (g-XNUM/XDEN+rp1)·g
XNUM = «rp5-g+rp4) -g+rp3) eg+rp2
XDEN = «g·q2)eg+q1)-g+qO
rp1 = .166666666666666667
rp2 = .451456904704461990x1Gr
rp3 = -.489487151969463797x103
rp4 = .428183075897778265x10
rp5 = -.121560740596710190x1~1
qO = .541748285645351853x107
q1 = .702492288221842518x101)
q2 = .394924723520450141x103
Finally
DCOS(x) = (_1)n+l_ sin(f)
Error Conditions
If the absolute value of the argument is greater than or equal to 6746518852,
the following Inessage is issued and the result is set to 0.0.

DCOS: ABS(arg) too large; result = zero

5-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

GSIN

Description
The GSIN routine calculates the double-precision, G-floating-point sine of the
double-precision, G-floating-point angle given in radians as the argument.
That is,

GSIN (x) = sin (x)
Routines Called
GSIN calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, G-floating-point value less than or
equal to 1686629713 (or 229 _1r).
Type of Result
The result returned is a double-precision, G-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result
test interval:

-10.000 through 201.06

MRE:

3.30x10- 18 (58.1 bits)

RMS:

8.85x10- 19 (60.0 bits)

LSB error distribution:

-2
0%

-1
13%

o
78%

+1
9%

+2
0%

Algorithm Used
GSIN(x) is calculated as follows. Note that GSIN(x) = -GSIN(-x).

Let Ixl = 1r-n+f
IfI < 1r/2
The argument reduction is as follows.
f = ((lxl-n e c1)-n-c2)-n e c3
c1 = high-order 30 bits of 1r
c2 = next 28 bits of 1r
c3 = next 62 bits of 1r
If Ifl < 2-30
sin(f) = f

Trigonometric Routines

5-15

Otherwise
sin(f) = f+fe R(g)
g = f2
R(g) = (g-XNUM/XDEN+rp1)eg
XNUM = «rp5 eg+rp4) eg+rp3) eg+rp2
XDEN = «geq2)eg+q1)eg-qO
rp 1 = - .166666666666666667
rp2 = .451456904704461990x1of
rp3 = -.489487151969463797x103
rp4 = .428183075897778265x101
rp5 = -.121560740596710190x10- 1
qO = .541748285645351853x107
ql = .702492288221842518x105
q2 = .394924723520450141x1Gr
Finally
GSIN(x) = sgn(x)e(-1) ne sin(f)
Error Conditions
If the absolute value of the argument is greater than 1686629713, the following
message is issued and the result is set to 0.0.

GSIN: ABS(arg) too large; result = zero

5-16 TOPS-10/TOPS-20 Common Math Library Reference Manual

Geos
Description

The GCOS routine calculates the double-precision, G-floating-point cosine of
the double-precision, G-floating-point angle given in radians as the argument.
That is:
GCOS(x) = cos(x)
Routine Called

GCOS calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value less than
1686629713 (or 22g e7r ).
Type of Result

The result returned is a double-precision, G-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

3.44x10- 18 (58.0 bits)

RMS:

9.84x10- 19 (59.8 bits)

LSB error distribution:

-2
0%

-1
14%

o
72%

+1
15%

+2
0%

Algorithm Used

GCOS(x) is calculated as follows. Note that GCOS(x) = GCOS(-x).
Let Ixl = 7r-n+f
Ifl < 7r/2
The argument reduction is as follows.
f = «lxl-n ec1)-n ec2)-n ec3
c1 = high-order 30 bits of 7r
c2 = next 28 bits of 7r
c3 = next 62 bits of 7r
If If I < 2- 30

sin(f) = f

Trigonometric Routines

5-17

Otherwise
sin(f) = f+f- R(g)
g = f2
R(g) = (g-XNUM/XDEN+rp1)-g
XNUM = «rp5-g+rp4)-g+rp3)-g+rp2
XDEN = «g-q2)-g+q1)-g+qO
rp1 = -.166666666666666667
rp2 = .451456904704461990x1Gr
rp3 = -.489487151969463797x1OS
rp4 = .428183075897778265x101
rp5 = -.121560740596710190x1~1
qO = .541748285645351853x107
q1 = .702492288221842518x1Gr
q2 = .394924723520450141x1OS
Finally
GCOS(x) = (_1)n+1- sin(f)

Error Conditions
If the absolute value of the argument is greater than or equal to 1686629713,
the following message is issued and the result is set to 0.0.

GCOS: ABS(arg) too large; result = zero

5-18 TOPS-10/TOPS-20 Common Math Library Reference Manual

CSIN

Description

The CSIN routine calculates the complex, single-precision, floating-point sine
of the complex, single-precision, floating-point angle given in radians as the
argument. That is:
CSIN(z) = sin(z)
Routines Called

CSIN calls the SIN, COS, EXP, ALOG, and MTHERR routines.
Type of Argument

The argument must be a complex, single-precision, floating-point value, the
real part of which must be less than 210828714 (or 226e 7r).
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 i.maginary

MRE:

3.30x10-8 (24.9 bits) real
3.44x10-8 (24.8 bits) imaginary

RMS:

7.68x10- 9 (27.0 bits) real
6.75x10-9 (27.1 bits) imaginary

LSB error distribution:

-2
2%
1%

-1
23%
19%

0
51%
57%

+1
22%
22%

+2
2% real
1% imaginary

Algorithm Used

CSIN(z) is calculated as follows.
Let z = x+i·y
If Ixl > 210828714

CSIN(z) = (0.0,0.0)
If Iyl > 88.029692, calculation proceeds as follows.

For the real part of the result:
Let t = Isin(x)1
If t = 0.0

x = 0.0
If loge(t)+lyl > 88.722839

x = ±machine infinity
(88.722839 = 88.029692+1oge(2))

Trigonometric Routines 5-19

For the imaginary part of the result:
Let t = Icos(x)I*O
If loge(t)+lyl < 88.722839
y = ± infinity

Otherwise
CSIN(z) = sin(x) ·cosh(y)+i ·cos(x) ·sinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
210828714, the following message is issued and the result is set to (0.0,0.0).
CSIN: ABS(REAL(arg)) too large; result = zero

2. If Iy I+loge{lsin(x) I) > 88.722839, the real part overflows. If
lyl+loge(lcos(x» > 88.722839, the imaginary part overflows. If either part
overflows, one of the following Inessages is issued and the relevant part of
the result is set to ± machine infinity.
CSIN: Imaginary part overflow
CSiN: Real part overflow

3. If the imaginary part of the result is too small a number, the following
message is Issued and the imaginary part of the result is set to 0.0.
CSIN: Imaginary part underflow

5-20 TOPS-10/TOPS-20 Common Math Library Reference Manual

ccos
Description

The CCOS routine calculates the complex, single-precision, floating-point
cosine of the complex, single-precision, floating-point angle given in radians
as the argument. That is:
CCOS(z) = cos(z)
Routines Called

CCOS calls the SIN, COS, EXP, ALOG, and MTHERR routines.
Type of Argument

The argument must be a complex, single-precision, floating-point value, the
real part of which must be less than 210828714 (or 226e 7r).
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 imaginary

MRE:

3.35x10-8 (24.8 bits) real
3.57x10-8 (24.7 bits) imaginary

RMS:

7.76x10-9 (26.9 bits) real
6.68x10-9 (27.2 bits) imaginary

LSB error distribution:

-2
2%
1%

-1
20%
20CJCl

50%

25Cj()

57(H,

20%

+2
3% real
1% imaginary

Algorithm Used

CCOS(z) is calculated as follows.
Let z = x+iey
If Ixl >210828714

CCOS(z) = (0.0,0.0)
If Iyl > 88.029692 calculation proceeds as follows.

For the real part of the result:
Let t = Icos(x)I:;i:O
If loge (t) + Iyl > 88.722839

x = ± machine infinity
(88.722839 = 88.029692+loge(2))

Trigonometric Routines

5-21

For the imaginary part of the result:
Let t = Isin(x)1
If t = 0.0
y = 0.0
If loge(t)+lyl > 88.722839
y = ± machine infinity

Otherwise
CCOS(z) = cos(x) ·cosh(y)-i ·sin(x) ·sinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
210828714, the following message is issued and the result is set to (0.0,0.0).

eeos: ABS(REAL(arg)) too large: result = zero
2. If Iy I+loge(lcos(x) I) > 88.722839, the real part overflows. If
lyl+loge(lsin(x)l) > 88.722839, the imaginary part overflows. If either part
overflows, one of the following messages is issued and the relevant part of
the result is set to ± machine infinity.

eeos: Imaginary part overflow
eeos: Real part overflow
3. If the imaginary part of the result is too small a number, the following
message is issued and the imaginary part of the result is set to 0.0.

eeos: Imaginary part underflow

5-22 TOPS-10/TOPS-20 Common Math Library Reference Manual

eOSIN

Description

The CDSIN subroutine calculates the complex, double-precision, D-floatingpoint sine of the complex, double-precision, D-floating-point angle given in
radians as the argument. That is:
CDSIN(z,r) = sin(z)
z = location of input value
r = location of result
Routines Called

CDSIN calls the DSIN, DCOS, DEXP, DLOG, and MTHERR routines.
Type of Argument

CDSIN is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, D-floating-point value, the real part of which must be
less than 231e 7r -7r/2.
Type of Result

The result returned is a complex, double-precision, D-floating-point value; it
may be any such value. It is returned in the second vector (r) supplied in the
call. The real part of the result is returned in the first element of r; the
imaginary part is returned in the second element of r.
Accuracy of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 imaginary

1.09x10- 18 (59.7 bits) real
9.86x10- 19 (59.8 bits) imaginary
2.22x10- 19 (62.0 bits) real
RMS:2.08x10-19 (62.1 bits) imaginary

MRE:

LSB error distribution:

-2

-1

2%
2%

22%
26%

51 %
54%

23%
17%

2% real
1% imaginary

Trigonometric Routines 5-23

Algorithm Used
CDSIN(z) is calculated as follows.
Let z = x+i-y

If Ixl > 231e 7r - 7r/2
CDSIN(z) = (0.0,0.0)
If Iyl > 88.029692, calculations proceed as follows.
For the real part of the result:
Let t = Isin(x)1

If t = 0.0
x = 0.0
If loge(t)+lyl > 88.722839
x = ± infinity
(88.722839 = 88.029692 + loge (2) )
For the imaginary part of the result:
Let t = Icos(x)1 *-

If loge(t)+lyl > 88.722839
y = ± infinity
Otherwise
CDSIN(z) = sin(x) ecosh(y)+i ·cos(x) ·sinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
231 .7r - 7r/2, the following message is issued and the result is set to (0.0,0.0).
COSIN: ABS(REAL(arg)) too large; result = zero
2. If Iy I+loge{lsin(x) I) > 88.722839, the real part overflows. If
lyl+loge(lcos(x)1) > 88.722839, the imaginary part overflows. If either part
overflows, one of the following messages is issued and the relevant part of
the result is set to ± machine infinity.

COSIN: ABS(lMAG(arg)) too large; REAL(result) = infinity
COSIN: ABS(IMAG(arg)) too large; IMAG(result) = Infinity
3. If the imaginary part of the result is too small a number, the following
message is issued and the imaginary part of the result is set to 0.0.

COSIN: Imaginary part underflow

5-24 TOPS-10/TOPS-20 Common Math Library Reference Manual

cocos
Description

The CDCOS subroutine calculates the complex, double-precision, D-floatingpoint cosine of the complex, double-precision, D-floating-point angle given in
radians as the argument. That is:
CDCOS(z) = cos(z)
z = location of input value
r = location of result
Routines Called

CDCOS calls the DSIN, DCOS, DEXP, DLOG, and MTHERR routines.
Type of Argument

CDCOS is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, D-floating-point value, the real part of which must be
less than 231 .7r - 7r/2.
Type of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 imaginary

MRE:

9.89x10- 19 (59.8 bits) real
9.98x10- 19 (59.8 bits) imaginary

RMS:

2.25x10- 19 (61.9 bits) real
2.03x10- 19 (62.1 bits) imaginary

LSB error distribution:

-2

-1

3%
1%

24%
21%

50%
55%

21%
21%

2% real
1% imaginary

Trigonometric Routines

5-25

Algorithm Used

CDCOS(z) is calculated as follows.
Let z = x+iey
If Ixl > 231e 1r -- 1r/2
CDCOS(z) = (0.0,0.0)
If Iyl > 88.029692, calculation proceeds as follows.

For the real part of the result:
Let t = Icos(x) I =1=

If loge(t)+lyl > 88.722839
x = ± infinity

(88.722839 = 88.029692+loge(2))
For the imaginary part of the result:
Let t = Isin(x)1
If t = 0.0
y = 0.0
If loge(t)+lyl > 88.722839
y = ± infinity

Otherwise
CDCOS(z) = cos(x) ·cosh(y)-i ·sin(x) esinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
231e 1r-1r/2, the following message is issued and the result is set to (0.0,0.0).

cocos: ABS(REAL(arg)) too large; result = zero
2. If lyl+loge(lcos(x)l) > 88.722839, the real part overflows. If
lyl+loge(lsin(x)l) > 88.722839, the imaginary part overflows. If either part
overflows, one of the following messages is issued and the relevant part of
the result is set to ± machine infinity.

cocos: ABS(IMAG(arg)) too large; REAL(result) = infinity
COCOS: ABS(IMAG(arg)) too large; IMAG(result) = infinity

3. If the imaginary part of the result is too small a number, the following
message is issued and the imaginary part of the result is set to 0.0

cocos: Imaginary part underflow

5-26 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGSIN

Description

The CGSIN subroutine calculates the complex, double-precision, G-floatingpoint sine of the complex, double-precision, G-floating-point angle given in
radians as the argument. That is,
CGSIN(z,r) = sin(z)
z = location of input value
r = location of result
Routines Called

CGSIN calls the GSIN, GCOS, GEXP, GLOG, and MTHERR routines.
Type of Argument

CGSIN is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, G-floating-point value, the real part of which must be
less than 229 • 7r-7r/2.
Type of Result

The result returned is a complex, double-precision, G-floating-point value; it
may be any such value. It is returned in the second vector (r) supplied in the
call. The real part of the result is returned in the first element of r; the
imaginary part is returned in the second element of r.
Accuracy of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 imaginary

MRE:

7.35xlO- 18 (56.9 bits) real
7.01xlO- 18 (57.0 bits) imaginary

RMS:

1.76x10- 18 (59.0 bits) real
.1.61xlO- 18 (59.1 bits) imaginary

LSB error distribution:

-2

-1

2%
1%

22%
20%

51%
55%

+1
23%
22%

2% real
2% imaginary

Trigonometric Routi nes

5-27

Algorithm Used

CGSIN(z) is calculated as follows.
Let z = x+iey
If Ixl > 229 ·7r-7r/2
CGSIN(z) = (0.0,0.0)
If Iyl > 709.089565712824, calculation proceeds as follows.

For the real part of the result:
Let t = Isin(x)1
If t = 0.0

x = 0.0
If loge(t)+lyl > 709.782712893384

x = ±machine infinity
(709.782712893384 = 709.089565712824+1o~(2))
For the imaginary part of the result:
Let t = Icos(x)1 0.0

If loge(t)+lyl > 709.782712893384

y = ±machine infinity
Otherwise
CGSIN(z) = sin(x) ·cosh(x)+i ecos(x) esinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
22ge 7r-7r/2, the following message is issued and the result is set to (0.0,0.0).
CGSIN: ABS(REAL(arg)) too large; result = zero

2. If lyl+loge(lsin(x)l) > 709.782712893384, the real part of the result will overflow. If lyl+loge(lcos(x)l) > 709.782712893384, the imaginary part of the
result will overflow. Any overflowed result is set to ±machine infinity and
one of the following messages is issued.
CGSIN: ABS(IMAG(arg)) too large; REAL(result) = infinity
CGSIN: AGS(IMAG(arg)) too large; IMAG(result) = infinity

3. If the imaginary part of the result underflows, the following message is
issued and the imaginary part of the result is set to 0.0.
CGSIN: Imaginary part underflow

5-28 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGCOS
Description

The CGCOS subroutine calculates the complex, double-precision, G-floatingpoint cosine of the complex, double-precision, G-floating-point angle given in
radians as the argument. That is:
CGCOS(z,r) = cos(z)
z = location of input value
r = location of result
Routines Called

CGCOS calls the GSIN, GeOS, GEXP, GLOG, and MTHERR routines.
Type of Argument

CGCOS is a subroutine that is called with two arguments. Both arguments
must be two-element, double-precision vectors. The first vector (z) contains
the input value; the second vector (r) will contain the result. The real part of
the input value must be stored in the first element of z; the imaginary part
must be stored in the second element of z. The input value must be a complex, double-precision, G-floating-point value, the real part of which must be
less than 22ge 1r--7r/2.
Type of Result

test interval:

-200.00 through 200.00 real
-10.000 through 10.000 imaginary

MRE:

8.31x10- 18 (56.7 bits) real
7.00x10- 18 (57.0 bits) inlaginary

RMS:

1.83x10- 18 (58.9 bits) real
1.53x10- 18 (59.2 bits) imaginary

LSB error distribution:

--2
2%
2%

-1

20%
20%

50%
58%

25%
20%

3% real
1% imaginary

Trigonometric Routines

5-29

Algorithm Used
CGCOS(z) is calculated as follows.

Let z = x+i-y
If Ixl>2 29 _1I"-1I"/2
CGCOS(z) = (0.0,0.0)
If Iyl > 709.089565712824, calculation proceeds as follows.

For the real part of the result:
Let t = Icos(x)1 =I=- 0.0
If loge(t)+lyl > 709.782712893384
x = ±machine infinity
(709.782712893384 = 709.089565712824+loge(2))

For the imaginary part of the result:
Let t = Isin(x)1
If t = 0.0
y = 0.0
If loge(t)+lyl > 709.782712893384
y = ±machine infinity
Otherwise
CGCOS(z) = cos(x) ecosh(y)-i esin(x) esinh(y)
Error Conditions

1. If the absolute value of the real part of the argument is greater than
22ge 1l"-1I"/2, the following message is issued and the result is set to (0.0,0.0).
CGCOS: ABS(REAL(arg)) too large; result = zero

2. If lyl+loge(lcos(x)l) > 709.782712893384, the real part of the result will
overflow. If lyl+loge(lsin(x)l) > 709.782712893384, the imaginary part of the
result will overflow. Any overflowed result is set to ±machine infinity and
one of the following messages is issued.
CGCOS: ABS(IMAG(arg)) too large; REAL(result) = infinity
CGCOS: ABS(IMAG(arg)) too large; IMAG(result) = Infinity

3. If the imaginary part of the result underflows, the following message is
issued and the imaginary part is set to 0.0.
CGCOS: Imaginary part underflow

5-30 TOPS-10/TOPS-20 Common Math Library Reference Manual

TAN
Description
The TAN routine calculates the single-precision, floating-point tangent of the
single-precision, floating-point angle given in radians as the argument. That
is:

TAN(x) = tan(x)
Routines Called
TAN calls the MTHERR routine.
Type of Argument
The argument must be a single-precision, floating-point value less than or
equal to 226 ·rr/2.
Type of Result
The result returned is a single-precision, floating-point value; it may be any
such value.
Accuracy of Result
test interval:

-10.000 through 201.06

MRE:

2.35x10-8 (25.3 bits)

RMS:

5.28x10-9 (27.5 bits)

LSB error distribution:

-2

-1

13%

70%

+1
16%

+2
0%

Algorithm Used
TAN(x) is calculated as follows.
If Ixl > 226 • rr/2
TAN(x) = 0.0

Otherwise, the identities:
tan( rr/2.0-g) = 1.0/tan(g)
tan(n· rr+h) = tan(h) where -rr/2.0 < h ~ rr/2.0
tan(-x) = -tan(x)
are used to reduce TAN(x) to a problem with
-rr/2.0 < x ~ rr/2.0
Then nand f are defined so that:
x = n ·rr/4.0+f where 0.0 ~ f s rr/4.0
If f < 2-- 14
tan(f) = f

Trigonometric Routines 5-31

Otherwise
tan(f) = fe R(f2)
R(f2) = (pO+f2e(p1+f2ep2))/(qO+f2e(q1+f2))
pO = 62.604
p1 = -6.9716
p2 = 6.7309
qO = pO
q1 = -27.839
Then, TAN(x) can be derived if L is an integer and n has the values shown
in the following table.
Deriving TAN(x)
Value of n

Low-order two
bits of n

TAN(x)

sgn(x) etan(f)

4L+1

sgn(x) e (l/tan(f»

4L+2

sgn(x) e (-l/tan(f»

4L+3

sgn(x) e_tan(f)

Reference
Coefficients are derived flom those given in Cody and Waite, Software Manual for Elementary Functions (Englewood Cliffs, N.J.: Prentice Hall, 1980) for
machines with 25-32 bit precision.
Error Conditions
If the absolute value of the argument is greater than 226 e1r/2, the following
message is issued and the result is set to 0.0.
TAN: ABS(arg) too large; result = zero

5-32 TOPS-10/TOPS-20 Common Math Library Reference Manual

COTAN

Description

The COTAN routine calculates the single-precision, floating-point cotangent
of the single-precision, floating-point angle given in radians as the argument.
That is:
COTAN(x) = cot(x)
Routines Called

COTAN calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value less than or
equal to 226 • rr/2 and greater than 2- 126 • (1/2+2- 27 ).
Type of Result

The result returned is a single-precision, floating-point value; it may be any
such value.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

2.42xlO-8 (25.3 bits)

RMS:

5.29x10-9 (27.5 bits)

LSB error distribution:

-2

-1

18%

66%

+1
16%

+2
0%

Algorithm Used

COTAN (x) is calculated as follows.
If Ixl > 226 • 7r/2
COTAN(x) = 0.0
If Ixl < 2- 126 • (1/2+2- 27 )
COTAN(x) = +machine infinity

Otherwise, the identities:
tan( 7r/2.0-g) = 1.0/tan(g)
tan(n ·7r+h) = tan(h) where -7r/2.0 < h ~ 7r/2.0
tan( -x) = -tan(x)
cot(x) = 1.0/tan(x)
cot( -x) = -cot(x)
are used to reduce COTAN(x) to a problem with
-rr/2.0 < x ~ 7r/2.0

Trigonometric Routines 5-33

Then nand f are defined so that:
x = n ·1r/4.0+f where 0.0 ~ f ~ 1r/4.0
If f < 2- 14
tan(f) = f

Otherwise
tan(f) = f· R(f2)
R(f2) = (pO+f2. (pI +f 2• p2) )/(qO+f2• (ql +f2»
pO = 62.604
pI = -6.9716
p2 = 6.7309
qO = pO
ql = -27.839
Then COTAN(x) can be derived if L is an integer and n has the value
shown in the following table.
Deriving COTAN(x)
Value of n

Low-order two
bits of n

COTAN(x)

sgn(x) • (l/tan(f))

4L+1

sgn(x) ·tan(f)

4L+2

sgn(x) • -tan(f)

4L+3

sgn(x) • -(l/tan(f))

Reference

Coefficients are derived from those given in Cody and Waite, Software Manual for Elementary Functions (Englewood Cliffs, N.J.: Prentice Hall, 1980) for
machines with 25-32 bit precision.
Error Conditions

1. If the absolute value of the argument is less than 2- 126 • (l/2+Z-27), the following message is issued and the result is set to +machine infinity.
COT AN: result overflow

2. If the absolute value of the argument is greater than 226 ·1r/2, the following
message is issued and the result is set to 0.0.
COTAN: ABS(arg) too large; result = zero

5-34 TOPS-10/TOPS-20 Common Math Library Reference Manual

DTAN
Description

The DTAN routine calculates the double-precision, D-floating-point tangent
of the double-precision, D-floating-point angle given in radians as the argument. That is:
DTAN(x) = tan(x)
Routines Called

DTAN calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value less than or
equal to 231 ·7r/2.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

9.60x1o-- 19 (59.9 bits)

RMS:

2.08x10- 19 (62.1 bits)

LSB error distribution:

-2
1%

-1

18%

55%

+1
22%

+2
3%

Algorithm Used

DTAN(x) is calculated as follows.
If Ixl > 231 ·7r/2
DTAN(x) = 0.0

Otherwise, the identities:
tan( 7r/2.0-g) = 1.0/tan(g)
tan(n ·7r+h) = tan(h) where -7r/2.0 < h S 7r/2.0
tan( -x) = -tan(x)
are used to reduce DTAN(x) to a problem with
-7r/2.0 < x S 7r/2.0
Then nand f are defined so that:
x = n e7r/2.0+f where -7r/4.0 S f S 7r/4.0
If f < 2-31
tan(f) = f

Trigonometric Routines

5-35

Otherwise
tan(f) = R(f)
R(f) = ««(xp4-g+xp3) -g+xp2)-g+xp1) -g) -f+f)/
« «q4 -g+q3) -g+q2) -g+q1) -g+l.O)
g = f-f
xp1 = -.1372889460941120802
xp2 = .3925934686364577602 -10- 2
xp3 = -.2882482747560198194 -10- 4
xp4 = .2927308283322907641-10- 7
q1 = -.4706222794274454135
q2 = .2746669449551304872-10- 1
q3 = - .4030063705745304384 -10--:3
q 4 = .1312960309685759549 -10- 5
If n is even
DTAN(x) = tan(f)
If n is odd
DTAN(x) = -1/tan(f)

Reference
Coefficients are derived from those given in Cody and Waite, Software Manual for Elementary Functions, (Englewood Cliffs, N .~J.: Prentice Hall, 1980)
for machines with 25-32 bit precision.
Error Conditions
If the absolute value of the argument is greater than ~1 - 7r/2, the following
message is issued and the result is set to 0.0.
DTAN: ABS(arg) too large; result = zero

5-36 TOPS-10/TOPS-20 Common Math Library Reference Manual

DCOTAN

Description

The DCOTAN routine calculates the double-precision, D-floating-point cotangent of the double-precision, D-floating-point angle given in radians as the
argument. That is:
DCOTAN(x) = cot(x)
Routines Called

DCOTAN calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value less than or
equal to 2'f3 1 -7r/2 and greater than 2- 127 -(1+2-61 ).
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

9.09x10- 19 (59.9 bits)

RMS:

2.08x10- 19 (62.1 bits)

LSB error distribution:

-2
2%

-1

23%

55%

+1
19%

+2
1%

Algorithm Used

DCOTAN(x) is calculated as follows.
If Ixl > 231 _7r/2

DCOTAN(x) = 0.0
If Ixl < 2- 127 - (1 +2- 61 )

DCOTAN(x) = +machine infinity
Otherwise, the identities:
tan( 7r/2.0-g) = 1.0/tan(g)
tan(n -7r+h) = tan(h) where -7r/2.0 < h $; 7r/2.0
tan( -x) = -tan(x)
cot(x) = 1.0/tan(x)
cot( -x) = -cot(x)
are used to reduce DCOTAN(x) to a problem with
-7r/2.0 < x $; 7r/2.0
Then nand f are defined so that:
x = n -7r/2.0+f where -7r/4.0 $; f $; 7r/4.0
If f < 2- 31

tan(f) = f

Trigonometric Routines 5-37

Otherwise
tan(f) = R(f)
R(f) = « (xp4 -g+xp3) -g+xp2) -g+x'p1) -g) -f+f)/
««q4-g+q3) -g+q2) -g+q1) -g+1.0)
g = f-f
xp1 = -.1372889460941120802
xp2 = .3925934686364577602 -10-- 2
xp3 = -.2882482747560198194 -10--4
xp4 = .2927308283322907641-10-7
q1 = -.4706222794274454135
q2 = .2746669449551304872-10--1
q3 = -.4030063705745304384-10-3
q4 = .1312960309685759549-10-5

If n is even
DCOTAN(x) = l/tan(f)

If n is odd
DCOTAN(x) = -tan(f)
References
Coefficients are derived from those given in Cody and Waite, Software Manual for Elementary Functions, (Englewood Cliffs, N.J.: Prentice Hall, 1980)
for machines with 25-32 bit precision.
Error Conditions

1. If the absolute value of the argument is greater than 231 _1r/2, the following
message is issued and the result is set to 0.0.
DCOT AN: ABS(arg) too large; result = zero

2. If the absolute value of the argument is less than 2- 127 - (1 + (Z-61 », the
following message is issued and the result is set to +machine infinity.
DCOT AN: Result overflow

5-38 TOPS-10/TOPS-20 Common Math Library Reference Manual

GTAN
Description

The GTAN routine calculates the double-precision, G-floating-point tangent
of the double-precision, G-floating-point angle given in radians as the argument. That is:
GTAN(x) = tan(x)
Routines Called

GTAN calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value less than or
equal to 22ge 7r/2.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result

test interval:

-10.000 through 201.06

MRE:

5.95x10- 18 (57.2 bits)

RMS:

1.43x10-18 (59.3 bits)

LSB error distribution:

-2
1%

-1

20%

60%

+1
18%

+2
0%

Algorithm Used

GTAN(x) is calculated as follows.
If Ixl > 22ge 7r/2
GTAN(x) = 0.0
Otherwise, the identities:
tan( 7r/2.0-g) = 1.0/tan(g)
tan(n e7r+h) = tan(h) where -7r/2.0 < h ::; 7r/2.0
tan( -x) = -tan(x)
are used to reduce GTAN(x) to a problem with
-7r/2.0 < x ::; 7r/2.0
Then nand f are defined so that:
x = n e7r/2.0+f where -7r/4.0 ~ f ~ 7r/4.0
If f < 2-30
tan(f) = f

Trigonometric Routi nes 5-39

Otherwise
tan(f) = R(f)
R(f) = ««(xp4-g+xp3) -g+xp2)-g+xp1) -g) -f+f)/
««q4-g+q3) -g+q2) -g+q1) -g+1.0)
g = f-f
xp1 = -.1372889460941120802
xp2 = .3925934686364577602 -10- 2
xp3 = -.2882482747560198194-10-4
xp4 = .2927308283322907641-10-7
q1 = -.4706222794274454135
q2 = .2746669449551304872-10- 1
q3 = -.4030063705745304384-10-3
q4 = .1312960309685759549-10-5
If n is even

GTAN(x) = tan(f)
If n is odd

GTAN(x) = -l/tan(f)
Reference

Coefficients are derived from those given in Cody and Waite, Software Manual for the Elementary Functions, (Englewood, N.J.: Prentice Hall, 1980) for
machines with 25-32 bit precision.
Error Conditions
If the absolute value of the argument is greater than 229 _1r/2, the following

message is issued and the result is set to 0.0.
GT AN: ABS(arg) too large; result = zero

5-40 TOPS-10/TOPS-20 Common Math Library Reference Manual

GCOTAN

Description

The GCOTAN routine calculates the double-precision, G-floating-point cotangent of the double-precision, G-floating-point angle given in radians as the
argument. That is:
GCOTAN(x) = cot(x)
Routines Called

GCOTAN calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, G-floating-point value less than or
equal to 229 _1(/2 and greater than 2-1023_(1+2-58).
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result

test interval:
MRE:

-10.000 through 201.06
6.46x10- 18 (57.1 bits)

RMS:

1.43x10-18 (59.3 bits)

LSB error distribution:

-2

-1

18%

60%

+1
20%

+2
1%

Algorithm Used

GCOTAN (x) is calculated as follows.
If Ixl > 229 -1(/2
GCOTAN(x) = 0.0
If Ix I < 2- 1023 - (1 + 2-58 )
GCOTAN (x) = + machine infinity
Otherwise, the identities
tan( 1(/2.0-g) = 1.0/tan(g)
tan(n· 1(+h) = tan(h) where -1(/2.0 < h:s; 1(/2.0
tan( -x) = -tan(x)
cot(x) = 1.0/tan(x)
cot( -x) = -cot(x)
are used to reduce GCOTAN(x) to a problem with
-1(/2.0 < x :s; 1(/2.0
Then nand f are defined so that:
x = n -1(/2.0+f where -1(/4.0 :s; f:s; 1(/4.0
If f <2- 30
tan(f) = f

Trigonometric Routines 5-41

Otherwise
tan(f) = R(f)
R(f) = (((((xp4-g+xp3)-g+xp2)-g+xp1)-g)-f+f)/
(( ((q4-g+q3)" g+q2) -g+q1) -g+1.0)
g = f-f
xp1 = -.1372889460941120802
xp2 = .3925934686364577602 -10- 2
xp3 = -.2882482747560198194-10- 4
xp4 = .2927308283322907641-10-7
q1 = -.4706222794274454135
q2 = .2746669449551304872-10-- 1
q3 = -.4030063705745304384-10-3
q4 = .1312960309685759549-10-5
If n is even
GCOTAN (x) = l/tan(f)
If n is odd
GCOTAN(x) = -tan(f)

Reference

Coefficients are derived from those given in Cody and Waite, Software Manual for Elementary Functions, (Englewood Cliffs, N.J.: Prentice Hall, 1980)
for machines with 25-32 bit precision.
Error Conditions

1. If the absolute value of the argument is greater than 229 ·1r/2, the following
message is issued and the result is set to 0.0.

Gcor AN:ABS(arg) to large; result = zero
2. If the absolute value of the argument is less than 2- 1023 • (1+2- 58 ), the following message is issued and the result is set to + machine infinity.

Gcor AN: Result overflow

5-42 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 6
Inverse Trigonometric Routines

ASIN
Description

The ASIN routine calculates, in radians, the single-precision, floating-point
arc sine of its single-precision, floating-point argument. That is:
ASIN(x) = sin- 1 (x)
Routines Called

ASIN calls the SQRT and MTHERR routines.
Type of Argument

The argument must be a single-precision, floating-point value in the range
-1.0 to 1.0.
Type of Result

The result returned is a single-precision, floating-point value in the range
-7r/2 to 7r/2.
Accuracy of Result

test interval:

0.00000 through 1.0000

MRE:

2.56xlO-8 (25.2 bits)

RMS:

5.34xlO-9 (27.5 bits)

LSB error distribution:

-2

-1
10%

83%

+1
7%

+2
0%

Algorithm Used

ASIN(x) is calculated as follows.
Let R(z) = ze(pO+ze(pl+zep2))/(qO+ze(ql+z))
pO = .564915737
pI = -.409490163
p2 = 1.93496723xl0-2
qO = 3.38949412
ql = -3.98220081
Let s = y+y eR(z)
Then, the following table gives the value of ASIN(x) depending on the
values of x, z, and y.
range of x

ASIN(x)

-1.0 to -.5

(1+x)/2

-2$

-(1r/2+s)

-.5 to 0.0

-x

-8

0.0 to .5

.5 to 1.0

(1-x)/2

-2$

1r/2+s

Error Conditions

If the absolute value of the argument is greater than 1.0, the following message is issued and the result is set to +machine infinity.
ASIN: ABS(arg) greater than 1.0; result = +infinity
Inverse Trigonometric Routines 6-3

ACOS

Description

The ACOS routine calculates, in radians, the single-precision, floating-point
arc cosine of its single-precision, floating-point argument. That is:
ACOS(x) = cos- 1 (x)
Routines Called

ACOS calls the SQRT and MTHERR routines.
Type of Argument

The argument must be a single-precision, floating-point value in the range
-1.0 to 1.0.
Type of Result

The result returned is a single-precision, floating-point value in the range
0.0 to 7r.
Accuracy of Result

test interval:

0.00000 through 1.0000

MRE:

1.5fixl0-8 (25.9 bits)

RMS:

3.76xl0-9 (28.0 hits)

LSB error distribution:

-2

-1

83%

+1
9%

+2
0%

Algorithm Used

ACOS(x) is calculated as follows.
Let R(z) = z-(pO+z-(pl+z-p2))/(qO+z-(ql+z))
pO = .564915737
pI = -.409490163
p2 = .93496723xl0-2
qO = 3.38949412
ql = -3.98220081
Let s = y+y·R(z)
Then, the following table gives the values of ACOS(x) depending on the
values of x, z, and y.
range of x

ACOS(x)

-1.0 to-.5

(1+x)/2

-2vz

1\"+8

-.5 to 0.0

-x

1\"/2+8

1\"/2 -8

-2$'

-8

0.0 to .5

.5 to 1.0

(l-x)/2

Error Conditions
If the absolute value of the argument is greater than 1.0, the following mes-

sage is issued and the result is set to +machine infinity.
ACOS: ABS(arg) greater than 1.0; result = +infinity

6-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DASIN
Description
The DASIN routine calculates, in radians, the double-precision, D-floatingpoint arc sine of its double-precision, D-floating-point argument. That is:

DASIN(x) = sin- 1(x)
Routines Called
DASIN calls the DSQRT and MTHERR routines.
Type of Argument
The argument must be a double-precision, D-floating-point value in the range
-1.0 to 1.0.
Type of Result
The result returned is a double-precision, D-floating-point value in the range
-7r/2 to 7r/2.
Accuracy of Result
test interval:

0.00000 through 1.0000

MRE:

8.96x10- 19 (60.0 bits)

RMS:

1.88x10-19 (62.2 bits)

LSB error distribution:

-2
1%

-1
25%

0
69%

+1
5%

+2
0%

Algorithm Used
DASIN(x) is calculated as follows.

Let R(g) = (g e(rp1+g e(rp2+g e(rp3+g e(rp4+g-rpfi)))))/
(qO+g e(q1+ge(q2+ge (q3+g e(q4+g)))))
rp1 = -.27368494524164255994x10 2
rp2 = .57208227877891731407x10 2
rp3 = -.39688862997504877339x10 2
rp4 = .10152522233806463645x102
rp5 = -.69674573447350646411
qO = -.16421096714498560795xl03
ql = .41714430248260412556xl03
q2 = -.38186303361750149284xl03
q3 = .15095270841030604719xl0 3
q4 = -.23823859153670238830xl0 2
Let s = y+yeR(g)
Then, the following table gives the values of DASIN(x) depending on the
values of x, Z, and y.

Inverse Trigonometric Routines 6-5

range of x

DASIN(x)

-1.0 to -.5

(l+x)/2

-2$

-(1r/2+8)

-.5 to 0.0

-x

-8

0.0 to .5

.5 to 1.0

(l-x)/2

-2VZ

1r/2+8

Error Conditions
If the absolute value of the argument is greater than 1.0, the following message is issued and the result is set to +machine infinity.
DASIN: ABS(arg) greater than 1.0; result = +infinlty

6-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

DACOS
Description

The DACOS routine calculates, in radians, the double-precision, D-floatingpoint arc cosine of its double-precision, D-floating-point argument. That is:
DACOS(x) = cos- 1(x)
Routines Called

DACOS calls the DSQRT and MTHERR routines.
Type of Argument

The argument must be a double-precision, D-floating-point value in the range
-1.0 to 1.0.
Type of Result

The result returned is a double-precision, D-floating-pointvalue in the range
0.0 to 11".
Accuracy of Result

test interval:

0.00000 through 1.0000

MRE:

4.48x10- 19 (61.0 bits)

RMS:

1.25x10- 19 (62.8 bits)

LSB error distribution:

-2
0%

-1
19%

0
75%

+1
6%

+2
0%

Algorithm Used

DACOS(x) is calculated as follows.
Let R(g) = (ge(rp1+ge(rp2+ge(rp3+g e(rp4+gerp5)))))/
(qO+ge (q1+ge (q2+ge (q3+g e(q4+g)))))
rp1 = -.27368494524164255994x102
rp2 = .57208227877891731407x102
rp3 = -.39688862997504877339x102
rp4 = .10152f)22233806463645x102
rp5 = -.69674573447350646411
qO = -.16421096714498f)6079f)x103
q1 = .41714430248260412556x103
q2 = -.381863033617f)0149284x103
q3 = .1f)09f)270841030604719x103
q4 = -.238238f)91f)3670238830x102
Let s = y+y e R(g)
Then, the following table gives the values of DACOS(x) depending on the
values of x, Z, and y.

Inverse Trigonometric Routines 6-7

range of x

ACOS(x)

-1.0 to -.5

(1+x)/2

-2yz

7r+8

-.5 to 0.0

-x

1r/2+8

0.0 to .5

1r/2-8

.5 to 1.0

(1-x)/2

-2yz

-8

Error Conditions
If the absolute value of the argument is greater than 1.0, the following mes-

sage is issued and the result is set to +machine infinity.
DACOS: ABS(arg) greater than 1.0; result = +infinity

6-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

GASIN

Description
The GASIN routine calculates, in radians, the double-precision, G-floatingpoint arc sine of its double-precision, G-floating-point argument. That is:

GASIN (x) = sin- 1(x)
Routines Called
GASIN calls the GSQRT and MTHERR routines.
Type of Argument
The argument must be a double-precision, G-floating-point value in the range
-1.0 to 1.0.
Type of Result
The result returned is a double-precision, G-floating-point value in the range
-7r/2 to 7r/2.
Accuracy of Result
test interval:

0.00000 through 1.0000

MRE:

6.69x10- 18 (57.1 bits)

RMS:

1.54x10- 18 (59.2 bits

LSB error distribution:

-2
1%

-1
26%

0
72%

+1
2%

+2
0%

Algorithm Used
GASIN(x) is calculated as follows.

Let R(g) = (ge(rp1+ge(rp2+ge(rp3+g e(rp4+gerp5)))))/
(qO+ge (q1+ge (q2+g e(q3+g e(q4+g)))))
rp1 = -.27368494f)2416425f)994x102
rp2 = .57208227877891731407x102
rp3 = -.39688862997f)04877339x102
rp4 = .101f)2f)22233806463645x102
rp5 = ~.69674573447350646411
qO = -.16421096714498f)60795x103
q1 = ,417144302482604125f)6x103
q2 = -.381863033617fi0149284x103
q3 = .1f)09f)270841030604719x103
q4 = -.238238f)91f)3670238830x102
Let s = y+y eR(g)
Then, the following table gives the value of GASIN(x) depending on the
values of x, z, and y.

Inverse Trigonometric Routines 8-9

range of x

GASIN(x)

~1.0 to -.5

(1+x)/2

-2VZ

-( 11"/2+8)

-.5 to 0.0

2
X

-x

-8

0.0 to .5

.5 to 1.0

(1- x)/2

-2VZ

11"/2+8

Error Conditions

If the absolute value of the argument is greater than 1.0, the following message is issued and the result is set to +machine infinity.
GASIN: ABS(arg) greater than 1.0; result = +infinity

6-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

GACOS
Description

The GACOS routine calculates, in radians, the double-precision, G-floatingpoint arc cosine of its double-precision, G-floating-point argument. That is:
GACOS(x) = cos- 1 (x)
Routines Called

GACOS calls the GSQRT and MTHERR routines.
Type of Argument

The argument must be a double-precision, G-floating-point value in the range
-1.0 to 1.0.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
0.0 to 1T'.
Accuracy of Result

test interval:

0.00000 through 1.0000

MRE:

4.18x10- 18 (57.7 bits)

RMS:

1.03x10- 18 (59.8 bits)

LSB error distribution:

-2
0%

1
14%

0
72%

1
+
15%

2
+
0%

Algorithm Used

GACOS(x) is calculated as follows.
Let R(g) = (ge (rp1 +ge (rp2+ge (rp3+g e(rp4+gerp5»»)/
(qO+g· (q1+ge (q2+ge (q3+g e(q4+g»»)
rp1 = -.27368494524164255994x1(f
rp2 = .57208227877891731407x1(f
rp3 = -.39688862997504877339x1(f
rp4 = .10152522233806463645x1(f
rp5 = -.69674573447350646411
qO = -.16421096714498560795x1Q3
q1 = .41714430248260412556x1Q3
q2 = -.38186303361750149284x1Q3
q3 = .15095270841030604719x1(i3
q4 = -.23823859153670238830x1(f
Let s = y+yeR(g)
Then the following table gives the value of GACOS(x) depending on the
values of x, z, and y.

Inverse Trigonometric Routines 6-11

GACOS(x)

(l+x)/2

-2$

1r+S

-x

1r/2+s

0.0 to .5

1r/2-s

.5 to 1.0

(l-x)/2

-2$

-s

range of x

-1.0 to -.5
-.5 to 0.0

Error Conditions
If the absolute value of the argument is greater than 1.0, the following mes-

sage is issued and the result is set to machine infinity.
GACOS: ABS(arg) greater than 1.0; result = +infinity

6-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

ATAN
Description

The ATAN routine calculates, in radians, the single-precision, floating-point
arc tangent of its single-precision, floating-point argument. That is:
ATAN(x) = tan- 1(x)
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is a single-precision, floating-point value in the range
-7r/2 to 7r/2.
Accuracy of Result

test interval:

-80.000 through 80.000

MRE:

8.07x10-9 (26.9 bits)

RMS:

2.99x10-9 (28.3 bits)

LSB error distribution:

-2
0%

-1

98%

+1
1%

+2
0%

Algorithm Used

ATAN(x) is calculated as follows.
If x < 0.0
ATAN(x) = -ATAN(lxl)
If x > 0.0
ATAN(x) = tan-1(XHI)+tan-1(z)
z = (x-XHI)/(1+x·XHI)
XHI is chosen so that
Izi ~ tan( 7r/32)

tan- 1(XHI) is found by table lookup. It is stored as ATANHI and
ATANLO to provide guard bits for improved accuracy.
tan- 1(z) is evaluated by means of a polynomial approximation (see
"Reference" below).
If x < tan( 7r/32)
z=x
ATAN(x) = tan-1(z)
If x > l/tan( 7r/32)
z = l/x
ATAN(x) = 7r/2-tan- 1 (z)
If tan(7r/32) < x < l/tan( 7r/32)
an appropriate XHI is obtained from a table. The table contains values for XHI for various ranges of x.
Inverse Trigonometric Routines 6-13

Reference

The polynomial approximation used in the algorithm is formula #4901 from
Hart et aI., Computer Approximations, (New York, N.Y.: John Wiley and
Sons, 1968).
Error Conditions

None

8-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

ATAN2
Description

The ATAN2 routine calculates, in radians, the single-precision, floating-point
polar angle for the two single-precision, floating-point coordinates of a point
in the x-y plane that are included as the arguments. That is:
ATAN2(y,x) = tan- 1(y/x)
Routines Called

ATAN2 calls the ATAN and MTHERR routines.
Type of Arguments

The arguments must be single-precision, floating-point values; they can be
any such values provided both arguments are not zero.
Type of Result

The result returned is a single-precision, floating-point value in the range
-1r to 1r.
Accuracy of Result

test interval:

-80.000 through 1.0000 for x
-80.000 through 1.0000 for y

MRE:

1.46x10-8 (26.0 bits)

RMS:

3.08x10-9 (28.3 bits)

LSB error distribution:

-2
0%

-1
1%

o
98%

+1
1%

+2
0%

Algorithm Used

ATAN2 (y,x) is calculated as follows.
Let u = Iyl and
v = Ixl and compute tan-1(u,v)
Then find ATAN2(y,x) based on the signs of y and x as follows.
x

ATAN2(y,x)

tan-1(u,v)
-tan-1(u, v)

+
+

-(tan-l( u, V)-1I")
tan-l (U,V)-1I"

Inverse Trigonometric Routines 6-15

The reduced argument for ATAN2 is:
z = (u/v-XHI)/(1+u/v· XHI)
This is rewritten as:
z = (u-v·XHI)/(v+u·XHl)
The numerator is calculated to be:
u-v·XHI = u-VHI·XHI-VLO·XHI
v = VHI+VLO
VHI has, at most, 27 significant bits
VLO has, at most, 35 significant bits
XHI is tabulated with, at most, 13 significant bits
This guarantees that the numerator of z is calculated exactly.
Error Conditions

1. If both arguments are 0.0, the following message is issued and the result is
set to 0.0.
AT AN2: Both arguments are zero, result = zero
2. If y/x underflows and x is greater than 0.0, the following message is issued

and the result is set to 0.0.
ATAN2: Result underflow

6-16 TOPS-10/TOPS-20 Common Math Library Reference Manual

DATAN
Description

The DATAN routine calculates, in radians, the double-precision D-floatingpoint arc tangent of its double-precision, D-floating-point argument. That is:
DATAN(x) = tan- 1 (x)
Routines CaUed

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point value in the range
-7r/2 to 7r/2.
Accuracy of Result

test interval:

-80.000 through 80.000

MRE:

3.40x10- 19 (61.3 bits)

RMS:

9.37x10- 20 (63.2 bits)

LSB error distribution:

-2
0%

-1
1%

94%

Algorithm Used

DATAN(x) is calculated as follows.
If x < 0.0
DATAN(x) = -DATAN(lxl)
If x> 0.0
DATAN(x) = tan- 1(XHI)+tan- 1 (z)
z = (x-XHI)/(1+x·XHI)
XHI is chosen so that
Izl $; tan( 7r/32)

tan- 1 (XHI) is found by table lookup. It is stored as ATANHI and
ATANLO to provide guard bits for improved accuracy.
tan- 1(z) is evaluated by means of a polynomial approximation
(see"Reference" below).
If x < tan( 7r/32)
z= x
DATAN(x) = tan- 1 (z)
If x > 1/tan( 7r/32)
z = 1/x
DATAN(x) = 7r/2-tan- 1 (z)
If tan( 7r/32) < x < 1/tan( 7r/32)
an appropriate XHI is obtained from a table. The table contains values for XHI for various ranges of x.
Inverse Trigonometric Routines 6-17

Reference

The polynomial approximation used in the algorithm is formula #4904 from
Hart et aI., Computer Approximation~, (New York, N.Y.: John Wiley and
Sons, 1968).
Error Conditions

None

6-18 TOPS-10/TOPS-20 Common Math Library Reference Manual

DATAN2

Description

The DATAN2 routine calculates, in radians, the double-precision, D-floatingpoint polar angle for the two double-precision, D-floating-point coordinates of
a point in the x-y plane that are included as the arguments. That is:
DATAN2(y,x) = tan- 1(y/x)
Routines Called

DATAN2 calls the DATAN and M1'HERR routines.
Type of Arguments

The arguments must be double-precision, D-floating-point values; they can
be any such values provided both arguments are not zero.
Type of Result

The result returned is a double-precision, D-floating-point value in the range
-1r to 1r.
Accuracy of Result

test interval:

-80.000 through 1.0000 for x
-80.000 through 1.0000 for y

MRE:

5.27x10- 19 (60.7 bits)

RMS:

9.09x10-9 (63.3 bits)

LSB error distribution:

-2
0%

-1

97%

+2
0%

Algorithm Used

DATAN2(y,x) is calculated as follows.
Let u = Iyl and
v = Ixl and compute tan-I (u/v)
Then find DATAN2(y,x) based on the signs of y and x as follows.
x

DATAN2(y,x)

tan-l (u/v)
-tan-l (u/v)

+
+

-(tan-I (u/v)-1I")
tan -I (u/v )-11"

Inverse Trigonometric Routines 6-19

The reduced argument for DATAN2 is:
z = (u/v-XHI)/(1+u/v· XHI)
This is rewritten as:
z = (u-v·XHI)/(v+u·XHI)
The numerator is calculated to be:
u-v·XHI = u-VHI·XHI-VLO·XHI
v = VHI+VLO
VHI has, at most, 27 significant bits
VLO has, at rnost, 35 significant bits
XHI is tabulated with, at most, 13 significant bits
This guarantees that the numerator of z is calculated exactly.
Error Conditions

1. If both arguments are 0.0, the following message is issued and the result is
set to 0.0.
DA T AN2: Both arguments are zero, result = zero

2. If y/x underflows and x is greater than 0.0, the following message is issued
and the result is set to 0.0.
OAT AN2: Result underflow

6-20 TOPS-10/TOPS-20 Common Math Library Reference Manual

GATAN
Description

The GATAN routine calculates, in radians, the double-precision, G-floatingpoint arc tangent of its double-precision, G-floating-point argument. That is:
GATAN(x) = tan-1(x)
Routines Called

None
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
- 1r/2 to 1r/2.
Accuracy of Result

test interval:

-80.000 through 80.000

MRE:

2.04x10- 18 (58.8 bits)

RMS:

7.03x10- 19 (60.3 bits)

LSB error distribution:

-2

-1

97%

Algorithm Used

GATAN(x) is calculated as follows.
If x < 0.0

GATAN(x) = -GATAN(lxl)
If x> 0.0

GATAN(x) = tan- 1(XHI)+tan- 1 (z)
z = (x-XHI)/(1+x·XHI)
XHI is chosen so that
Izi ~ tan( 1r/32)
tan- 1(XHI) is found by table lookup. It is stored as ATANHI and
ATANLO to provide guard bits for improved accuracy.
tan- 1 (z) is evaluated by means of a polynomial approximation (see
"Reference" below).
If x < tan( 1r/32)
z=x

GATAN(x) = tan- 1 (z)
If x > tan( 1r/32)

z = l/x
GATAN(x) = 1r/2-tan- 1 (z)
If tan( 1r/32) < x < l/tan( 1r/32)

an appropriate XiiI is obtained from a table. The table contains values for XHI for various ranges of x.
Inverse Trigonometric Routines 6-21

Reference

The polynomial approxinlation used in the algorithm is formula 4904 from
Hart et ai., Computer Approximations, (New York, N.Y.: John Wiley and
Sons, 1968).
Error Conditions

None

6-22 TOPS-10/TOPS-20 Common Math Library Reference Manual

GATAN2

Description

The GATAN2 routine calculates, in radians, the double-precision, G-floatingpoint polar angle for the two double-precision, G-floating-point coordinates of
a point in the x-y plane that are included as the arguments. That is:
GATAN2(y,x) = tan-l(y/x)
Routines Called

GATAN2 calls the GATAN and MTHERR routines.
Type of Arguments

The arguments must be double-precision, G-floating-point values; they can
be any such values provided both arguments are not zero.
Type of Result·

The result returned is a double-precision, G-floating-point value in the range
to 7r.

- 7r

Accuracy of Result

test interval:

-80.000 through 1.0000 for x
-80.000 through 1.0000 for y

MRE:

3.28x10-- 18 (58.1 bits)

RMS:

7.15xlO- 19 (60.3 bits)

LSB error distribution:

-2

-1

98%

+1
2%

+2
0%

Algorithm Used

GATAN2(y,x) is calculated as follows.
Let u = Iyl and
v = Ixl and compute tan-leu/v)
Then find GATAN2(y,x) based on the signs of y and x as follows.
x

GATAN2(y,x)

tan-- l ( u/v )
-tan-l (u/v)

+
+

-( tan-I (u/v )-7r)
tan-l (U/V)-7r

Inverse Trigonometric Routines

6-23

The reduced argument for GATAN2 is:
z = (u/v-XHI)/(I+u/v·XHI)
This is rewritten as:
z = (u-v·XHI)/(v+u·XHI)
The numerator is calculated to be:
u-v·XHI = u-VHI·XHI-VLO·XHI
v = VHI+VLO
VHI has, at most, 27 significant bits
VLO has, at most, 35 significant bits
XHI is tabulated with, at most, 13 significant bits
This guarantees that the numerator of z is calculated exactly.
Error Conditions
1. If both arguments are 0.0, the following message is issued and the result is
set to 0.0.
GAT AN2: Both arguments are zero, result = zero

2. If y/x underflows and x is greater than 0.0, the following message is issued
and the result is set to 0.0.
GAT AN2: Result underflow

6-24 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 7
Hyperbolic Routines

SINH
Description

The SINH routine calculates the single-precision, floating-point hyperbolic
sine of its single-precision, floating-point argument. That is:
SINH(x) = sinh(x)
Routines Called

SINH calls the EXP and MTHERR routines.
Type of Argument

The argument must be a single-precision, floating-point value in the range
-88.722 to 88.722.
Type of Result

The result returned is a single-precision, floating-point value; it may be any
such value.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

2.61x1o--8 (25.2 bits)

RMS:

4.24x10-9 (27.8 bits)

LSB error distribution:

-2
0%

-1
4%

+1
11%

85%

+2
0%

Algorithm Used

SINH(x) is calculated as follows.
The table below gives the value of SINH(x) depending upon the range of
values for Ixl.
range of Ixl

SINH(x)

0.0 to 2- 13

2- 13 to 1.0

x ·p4(x2 )

1.0 to 9.7 = 14·1oge(2)

(ex -e- X )/2· sgn(x)

9.7 to 88.03 = 127·1oge(2)

eX/2·sgn(x)

88.03 to 88.722 = 128 ·loge(2)

ex-loge(2) • sgn(x)

88.722 to infinity

infinity ·sgn(x)

If z = x2
p4(z) = 1+z·(cl+z·(c2+z·(c3+c4·z)))
cl = 1.666666643xlO-1
c2 = 8.333352593xlO- 3
c3 = 1.983581245xlO-4
c4 = 2.818523951xlO-6
Error Conditions
If the absolute value of the argument is greater than 88.722, the following
message is issued and the result is set to ± machine infinity using the sign of
the argurnent.

SINH: Result overflOw
, Hyperbolic Routines 7-3

COSH
Description

The COSH routine calculates the single-precision, floating-point hyperbolic
cosine of its single-precision, floating-point argument. That is:
COSH(x) = cosh(x)
Routines Called

COSH calls the EXP and MTHERR routines.
Type of Argument

The argument must be a single-precision, floating-point value in the range
-88.722 to 88.722.
Type of Result

The result returned is a single-precision, floating-point value greater than or
equal to 1.0.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

2.12x10-8 (25.5 bits)

RMS:

4.49x10-9 (27.7 bits)

LSB error distribution:

-2

-1

82%

+1
14%

+2
0%

Algorithm Used

COSH(x) is calculated as follows.
The table below gives the value of COSH(x) depending upon the range of
.
values for Ixl.
range of Ixl

COSH(x)

0.0 to 2- 14

1.0

to 9.7 = 14-1oge(2)

(ex +e- x )/2

9.7 to 88.03 = 127 -loge(2)
88.03 to 88.722 = 128 -loge(2)

eX/2
ex-!oge(2)

88.722 to infinity

infinity

Error Conditions

If the absolute value of the argument is greater than 88.722, the following
message is issued and the result is set to ± machine infinity using the sign of
the argument.
COSH: Result overflow

7-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DSINH

Description

The DSINH routine calculates the double-precision, D-floating-point hyperbolic sine of its double-precision, D-floating-point argument. That is:
DSINH(x) = sinh(x)
Routines Called

DSINH calls the DEXP and MTHERR routines.
Type of Argument

The argument must be a double-precision, D-floating-point value in the range
-88.722 to 88.722.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

6.82x10-8 (60.3 bits)

RMS:

1.27x10-9 (62.8 bits)

LSB error distribution:

-2
0%

-1
6%

83%

11%

+2
0%

Algorithm Used

DSINH(x) is calculated as follows.
The table below gives the value of DSINH(x) depending upon the range of
values for Ixl.
range of Ixl

DSINH(x)

0.0 to 2-a1

to 1.0

1.0 to 22.0 = 32 ·loge(2)

x+x·R(x 2 )
(ex -e-X)/2· sgn(x)

22.0 to 88.03 = 127 ·loge(2)

eX /2 • sgn(x)

88.03 to 88.722 =;: 128 ·loge(2)

ex-!oge(2) • sgn(x)

88.722 to infinity

infinity • sgn(x)

Hyperbolic Routines 7-5

If z = x2
R(z) = (rpO+z· (rp1+z· (rp2+z·rp3) »/(qO+z· (q1+z· (q2+z»)
rpO =.35181283430177117881x106
rp1 = .11563521196851768270x105
rp2 = .16375798202630751372x103
rp3 = .78966127417357099479
qO = -.21108770058106271242x107
q1 = .36162723109421836460x105
q2 = -.27773523119650701667x103
Error Conditions
If the absolute value of the argument is greater than 88.722, the following
message is issued and the result is set to ± machine infinity using the sign of
the argument.

DSINH: Result overflow

7-6

TOPS~10/TOPS-20 Common Math Library Reference Manual

DCOSH

Description

The DCOSH routine calculates the double-precision, D-floating-point hyperbolic cosine of its double-precision, D-floating-point argument. That is:
DCOSH(x) = cosh(x)
Routines Called

DCOSH calls the DEXP and MTHERR routines.
Type of Argument

The argument must be a double-precision, D-floating-point value in the range
-88.722 to 88.722.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 1.0.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

5.90x10- 19 (60.6 bits)

RMS:

1.34x10- 19 (62.7 bits)

LSB error distribution:

-2

-1

81%

14%

+2
0%

Algorithm Used

DCOSH(x) is calculated as follows.
The table below gives the value of DCOSH(x) depending upon the range
of values for Ixl.
range of Ixl

DCOSH(x)

0.0 to 2- 32

1.0

to 22.0 =·32 -loge(2)

(ex +e- x )/2

22.0 to 88.03 = 127 -loge(2)
88.03 to 88.722 = 128 -loge(2)

eX/2
ex.-}oge(2)

88.722 to infinity

infinity

Error Conditions

If the absolute value of the argument is greater than 88.722, the following
message is issued and the result is set to ± machine infinity using the sign of
the argument.
DCOSH: Result overflow

Hyperbolic Routines 7-7

GSINH
Description

The GSINH routine calculates the double-precision, G-floating-point hyperbolic sine of its double-precision, G-floating-point argument. That is:
GSINH(x) = sinh(x)
Routines Called

GSINH calls the GEXP and MTHERR routines.
Type of Argument

The argument must be a double-precision, G-floating-point value in the range
-709.782713 to 709.782713.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

6.40x1o- 18 (57.1 bits)

RMS:

9.44x1o- 19 (59.9 bits)

LSB error distribution:

-2

-1

87%

+1
10%

+2
0%

Algorithm Used

GSINH(x) is calculated as follows.
The table below gives the value of GSINH(x) depending upon the range of
values for Ixl.
range of Ixl

GSINH(x)

0.0 to 2-30

to 1.0

1.0 to 22.0 = 32 -loge(2)

x+x -R(x2)
(ex -e-X)/2 - sgn(x)

22.0 to 709.089565

eX /2 - sgn(~)

709.089565 to 709.782713

ex-loge(2) -sgn(x)

709.782713 to infinity

infinity -sgn(x)

7-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

If z = X2
R(z) = (rpO+z -(rp1+z -(rp2+z -rp3)) )/(qO+z- (q1+z -(q2+z)))
rpO = .35181283430177117881.106
rp1 = .11563521196851768270-1cr
rp2 = .16375798202630751372-10.1
rp3 = .78966127417357099479
qO = -.21108770058106271242-107
q1 = .36162723109421836460-1cr
q2 = -.27773523119650701667-1fr1
Error Conditions
If the absolute value of the argument is greater than 709.782713, the following
message is issued and the result is set to ± machine infinity, using the sign of
the argument.

GSINH: Result overflow

Hyperbolic Routines 7-9

GCOSH

Description

The GCOSH routine calculates the double-precision, G-floating-point hyperbolic cosine of its double-precision, G-floating-point argument. That is:
GCOSH(x) = cosh(x)
Routines Called

GCOSH calls the GEXP and MTHERR routines.
Type of Argument

The argument must be a double-precision, G-floating-point value in the range
-709.782713 to 709.782713.
Type of Result

The result returned is a double-precision, G-floating-point value greater than
or equal to 1.0.
Accuracy of Result

test interval:

0.00000 through 88.721

MRE:

4.84x10- 18 (57.5 bits)

RMS:

1.00x10-18 (59.8 bits)

LSB error distribution:

-2

-1

84%

+1
13%

Algorithm Used

GCOSH(x) is calculated as follows.
The table below gives the value of GCOSH(x) depending upon the range
of values for Ixl.
range of Ixl

GCOSH(x)

0.0 to 2- 30

1.0

to 22.0 = 32 -loge(2)

(ex +e- X )/2

22.0 to 709.089565

eX/2

709.089565 to 709.782713

ex-loge(2)

709.782713 to infinity

infinity

Error Conditions
If the absolute value of the argument is greater than 709.782713, the following

message is issued and the result is set to ± machine infinity, using the sign of
the argument.
GCOSH: Result overflow

7-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

TANH
Description

The TANH routine calculates the single-precision, floating-point hyperbolic
tangent of its single-precision, floating-point argument. That is:
TANH(x) = tanh(x)
Routines Called

TANH calls the EXP routine.
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is a single-precision, floating-point value in the range -1.0
to 1.0.
Accuracy of Result

test interval:

0.00000 through 90.000

MRE:

2.69x10-8 (25.1 bits)

RMS:

5.53x10-9 (27.4 bits)

LSB error distribution:

-2

-1

79%

+1
21%

+2
0%

Algorithm Used

TANH(x) is calculated as follows.
The table below gives the value of TANH(x) depending upon the range of
values for Ixl.
range of Ixl

TANH(x)

0.0 to 2- 15

to loge(3)/2

x+x - R(x 2 )

loge(3)/2 to 9.8479016

(1-2/( e2 -Ixl + 1» - sgn(x)

9.8479016 to infinity

1.0-sgn(x)

If g = x 2

R(g) = g-(a+b-g)/(c+g)
a = -.823772813
b = -.383101067x10- 2
C = 2.47131965
Error Conditions

None

Hyperbolic Routines 7-11

DTANH

Description

The DTANH routine calculates the double-precision, D-floating-point hyperbolic tangent of its double-precision, D-floating-point argument. That is:
DTANH(x) = tanh(x)
Routines Called

DTANH calls the EXP routine.
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result

test interval:
MRE:
RMS:
LSB error distribution:

0.00000 through 90.000
7.17x1019 (60.3 bits)
1.75x1019 (62.3 bits)
-2
0%

-1

70%

+1
30%

+2
0%

Algorithm Used

DTANH(x) is calculated as follows.
The table below gives the value of DTANH(x) depending upon the range
of values for Ixl.
range of Ixl

DTANH(x)

0.0 to 2- 32 -~.

2- -y3' to loge(3)/2

x+x-R(x2 )

loge(3)/2 to 22.1807100

(l-2/(e2 -Ixl +1» -sgn(x)

22.1807100 to infinity

1.0 - sgn(x)

If g = x 2
R(g) = g- (rpO+g- (rp1+rp2- g) )/(qO+g- (q1+g- (q2+g»)
rpO = -.161341190239962281x1Q4
rp1 = -.992259296722360833x1()2
rp2 = -.964374927772254698
qO = .484023570719886887x1Q4
q1 = .22337720718962312926x1Q4
q2 = .112744743805349493x1Q3
Error Conditions

None
7-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

GTANH
Description

The GTANH routine calculates the double-precision, G-floating-point hyperbolic tangent of its double-precision, G-floating-point argument. That is:
GTANH(x) = tanh(x)
Routines Called

GTANH calls the GEXP routine.
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
-1.0 to 1.0.
Accuracy of Result

test interval:

0.00000 through 90.000

MRE:

6.44x10- 18 (57.1 bits)

RMS:

1.33x10-18 (59.4 bits)

LSB error distribution:

-2

-1

80%

+1
20%

+2
0%

Algorithm Used

GTANH(x) is calculated as follows.
The table below gives the value of GTANH(x) depending upon the range
of values for Ixl.
range of Ixl

GTANH(X)

0.0 to 2- 32 - v3

2- _y3" tologe(3)/2

x+x -R(x2 )

loge(3)/2 to 22.1807100

(l-2/(e2 -Ixl +1)) -sgn(x)

22.1807100 to infinity

1.0-sgn(x)

If g = x 2
R(g) = g-(rpO+g-(rpl+rp2·g»/(qO+g-(q1+g·(q2+g»)
rpO = -.161341190239962281xlcr
rpl = -.992259296722360833xl02
rp2 = -.964374927772254698
qO = .484023570719886887xlcr
q1 = .22337720718962312926xl04
q2 = .112744743805349493xl()3
Error Conditions

None
Hyperbolic Routines 7-13

Chapter 8
Random Number Generating Routines

RAN
Description
The RAN routine returns pseudo random numbers between 0.0 and 1.0, but
not including 0.0 or 1.0. The period of the sequence is 2147483647; that is, the
numbers repeat every 2147483647 calls.

RAN uses a pure multiplicative congruential random number generator with
prime modulus. The seed value can be supplied by the system or supplied by
a call to the SETRAN subroutine. (See SETRAN, p. 8-6).
Routines Called
RAN does not call any routines; but you can call the SETRAN subroutine to
provide a seed value and the SA VRAN subroutine (see SA VRAN, p. 8-7) to
determine the last seed, used by RAN.
Type of Argument
The argument is a dummy value that is not used.
Type of Result
The result returned is a single-precision, floating-point value that is greater
than 0.0 and less than 1.0.
Accuracy of Result
The independence of successive random numbers generated by multiplicative
congruential methods can be measured by the spectral test. For this generator, with seed 630360016 and modulus 2147483647, the spectral test yields the
following results.

mu{n)

bits

2.446
.4766
3.715
4.944
.8183

3
4
5
6

8
6
5

mu(n)

measures how densely n-tuples of random numbers cover an
n-dimensional square.

bits

is the number of independent bits in successive n-tuples of numbers returned by RAN.

For example, successive pairs of random numbers can be considered to be
independent in their first 15 bits. The remaining 12 bits are not independent.

Random Number Generating Routines 8-3

Algorithm Used

RAN(n) is calculated as follows.
Using a seed value supplied from a call to the SETRAN subroutine or the
default seed value 524287(=2 19_1), the seed value is calculated by:
RAN(n) = seed/231 , truncated
On subsequent calls to RAN, a new seed is calculated from the previous
seed value by:
seed = seed -630360016 mod (231 _1)
and the random number is then generated.
References

A full description of the spectral test is given in R.R. Coveyan and R.D.
MacPherson, Journal of the ACM 14 (1967), pp. 100-119 and in D.E. Knuth,
Seminumerical Algorithms (Reading, Mass.: Addison-Wesley, 1981), Section
3.3.4.
Error Conditions

None

8-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

RANS
Description
The RANS routine returns pseudo random numbers between 0.0 and 1.0, but
not including 0.0 or 1.0. The period of the sequence 2484877906816; that is, the
numbers repeat every 2484877906816 calls.

RANS is based on the same multiplicative random number generator as RAN
(p. 8-3). In addition, it shuffles the numbers using a 128-word table.
Routines Called
RAN8 calls the RAN and SA VRAN routines.
Type of Argument
The argurnent is a dumm.y value that is not used.
Type of Result
The result returned is a single-precision, floating··point value that. is greater
than 0.0 and less than 1.0.
Accuracy of Result
Not applicable
Algorithm Used
RANS(n) is calculated as follows.
On the initial reference to RAN8, RAN is called 128 times to generate 8 1 ,
8 2, ... ,8 12R (uniform random deviates in (0,1») and a new seed Xo. Xo is
obtained from a call to the 8AVRAN subroutine (see 8AVRAN, p.8-7)
after 8 128 has been generated. Then:

Xi; 1= 630360016·X j mod(2 31 _1)
j = (xi~l mod(128)+1
s·J = X.I." 1/231
t = s·J
RANS(n) = t
Error Conditions
None

Random Number Generating Routines

8-5

SETRAN

Description

The SETRAN subroutine provides the internal integer seed value for the RAN
routine.
SETRAN is used to reset RAN to return the same sequence of random numbers again, or to set RAN to an arbitrary value (such as the time of day) so
that it will return an entirely new sequence.
Routines Called

SETRAN does not call any routines; but you can call the SAVRAN subroutine to save and return the last seed value used by RAN.
Type of Argument

The argument must be an integer value in the range 0 to 231. If the argument
is 0, the default seed value for RAN is used.
Type of Result

Not applicable
Accuracy of Result

Not applicable
Algorithm Used

SETRAN(n) is calculated as follows.
Using the value supplied, SETRAN computes:
seed = Iseedl mod (2147483647)
Error Conditions

None

8-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

SAVRAN
Description

The SAVRAN subroutine saves and returns the last seed used by the RAN
routine.
Routines Called

None
Type of Argument

The argument must be an integer variable in which the seed value will be
stored.
Type of Result

The result returned is an integer value between 1 and 2147483647.
Accuracy of Result

Not applicable .
Algorithm Used

Not applicable
Error Conditions

None

Random Number Generating Routines 8-7

Chapter 9
Absolute Value Routines

lABS
Description

The lABS routine returns the integer absolute value of its integer argument.
That is:
IABS(n) = Inl
Routines Called

None
Type of Argument

The argument must be an integer value; it can be any such value.
Type of Result

The result returned is an integer value greater than or equal to O.
Accuracy of Result

The result is exact.
Algorithm Used

IABS(n) is calculated as follows.
If n ~ 0
ABS(n) = n
If n < 0

ABS(n) = -n
Error Conditions
If the argument is the "most negative integer" (4000000000008 ), overflow oc-

curs and the result is set to machine infinity.

Absolute Value Routines

9-3

ASS
Description

The ABS routine returns the single-precision, floating-point absolute value of
its single-precision, floating-point argument. That is:
ABS(x) = Ixl
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is a single-precision, floating-point value greater than or
equal to 0.0.
Accuracy of Result

The result is exact.
Algorithm Used

ABS(x) is calculated as follows.
If x ~ 0.0
ABS(x) = x
If x < 0.0
ABS(x) = -x
Error Conditions

None

9-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DABS
Description

The DABS routine returns the double-precision, D-floating-point absolute
value of its double-precision, D-floating-point argument. That is:
DABS(x) = Ixl
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 0.0.
Accuracy of Result

The result is exact.
Algorithm Used

DABS(x) is calculated as follows.
If x ~ 0.0

DABS(x) = x
If x < 0.0

DABS(x) =-x
Error Conditions

None

Absolute Value Routines 9-5

GABS
Description

The GABS routine returns the double-precision, G-floating-point absolute
value of its double-pl'ecision, G-floating-point argument. That is:
GABS(x) = Ixl
Routines Called

None
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point value greater than
or equal to 0.0.
Accuracy of Result

The result is exact.
Algorithm Used

GABS(x) is calculated as follows.
If x ~ 0.0

GABS(x) = x
If x < 0.0

GABS(x) = -x
Error Conditions

None

9-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

CABS

Description
The CABS routine returns the single-precision, floating-point absolute value
of its complex, single-precision; floating-point argument. That is:

CABS(z) = Izl
Routines Called
CABS calls the SQRT and MTHERR routines.
Type of Argument
The argument must be a complex, single-precision, f10ating-point value; it
can be any such value.
Type of Result
The result returned is a single-precision, floating-point value greater than or
equal to 0.0.
Accuracy of Result

test interval:

-1.00000xl0 18 through 1.00000xl018 real
-1.00000xl018 through 1.00000xl018 imaginary

MRE:

1.84xlO-8 (25.7 bits)

RMS:

5.36xIQ-9 (27.5 bits)

LSB error distribution:

-2

-1

14%

65%

+1
21%

+2
0%

Algorithm Used
CABS(z) is calculated as follows.

Let z = x+i·y
v = MAX(lxl,lyl)
w = MIN(lxl,lyl)
Then CABS(z) =

v·v 1.0+ (w/v)2

Error Conditions
If the argument is so large that it causes an overflow, the following message is
issued and the result is set to +machine infinity.
CABS: Result overflow

Absolute Value Routines

9-7

CDABS

Description

The CDABS routine calculates the double-precision, D-floating-point absolute value of its complex, double-precision, D-floating-point argument. That
IS:

CDABS(z) = Izl
z = location of input value
Routines Called

CDABS calls the DSQRT and MTHERR routines.
Type of Argument

The argument must be a two-element, double-precision vector that contains
the input value, (z). Z must be a complex, double-precision, D-floating-point
value; it can be any such value.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 0.0.
Accuracy of Result

test interval:

-1.00000x1018 through 1.00000x1()18 real
-1.00000x1018 through 1.00000x1()18 imaginary

MRE:

6.32x1o- 19 (60.5 bits)

RMS:

1.89x1o-19 (62.2 bits)

LSB error distribution:

-2

-1

56%

+1
38%

+2
2%

Algorithm Used

CDABS(z) is calculated as follows.
Let z = x+i·y
v = MAX(lxl,lyl)
w = MIN(lxl,lyl)
Then CDABS(z) = v

·v 1.0+ (w/v)2

Error Conditions
If the argument is so large that overflow occurs, the' following message is

issued and the result is set to +machine infinity.
CDABS: Result overflow

9-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

CGABS
Description

The CGABS routine calculates the double-precision, G-floating-point absolute value of its complex, double-precision, G-floating argument. That is:
CGABS(z) = Izl
z = location of input value
Routines Called

CGABS calls the GSQRT and MTHERR routines.
Type of Argument

The argument must be a two-element, double-precision vector that contains
the input value (z). Z must be a complex, double-precision, G-floating-point
value; it can be any such value.
Type of Result

The result returned is a double-precision, G~floating-point value greater than
or equal to 0.0.
Accuracy of Result

test interval:

-1.00000x10 18 through 1.00000x1018 real
-1.00000x1018 through 1.00000x1018 imaginary

MRE:

4.88x10- 18 (57.5 bits)

RMS:

1.51x10- 18 (59.2 bits)

LSB error distribution:

-2

-1

56%

38%

+2
2%

Algorithm Used

CGABS(z) is calculated as follows.
Let z = x+i·y
v = MAX(lxl,lyl)
w = MIN(lxl,lyl)
Then CGABS(z) = v • .J 1.0+ (w/v)2
Error Conditions

If the argument is so large that overflow occurs, the following message is
issued and the result is set to +machine infinity.
CGABS: Result overflow

Absolute Value Routines

9-9

Chapter 10
Data Type Conversion Routines

IFIX
Description

The IFIX routine converts and truncates its single-precision, floating-point
argument to an integer value.
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value less than 235.
Type of Result

The result returned is an integer value; it may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

IFIX(x) is calculated by means of the FIX machine instruction. This instruction converts and truncates the argument to an integer.
Error Conditions

If the argument is greater than 235 , an overflow occurs and the result is set to
machine infinity.

Data Type Conversion Routines 10-3

INT
Description

The INT routine converts and truncates its single-precision, floating-point
argument to an integer value.
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value less than 235 •
Type of Result

The result returned is an integer value; it may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

INT(x) is calculated by means of the FIX machine instruction. This instruction converts and truncates the argument to an integer.
Error Conditions
If the argument is greater than 235 , an overflow occurs and the result is set to

machine infinity.

10-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

IOINT

Description

The IDINT routine converts and truncates its double-precision, D-floatingpoint argument to an integer value.
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is an integer value; it may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

IDINT(x) is calculated as follows.
The routine, working on the magnitude of the argument, copies the exponent field to a scratch register. It then clears the exponent field of the
magnitude of the argument, and uses the copy of the exponent to control a
shift to leave the integer in the location of the r~sult. If necessary, the
routine negates the result.
Error Conditions
If the shift results in a loss of significant bits on the left, an overflow occurs
and the result is set to machine infinity.

Data Type Conversion Routines 10-5

GFX.n
Description

The GFX.n routine converts and truncates its double-precision, G-floatingpoint argument to an integer value. n is an even octal number from 0
through 14 that designates a register (AC).
Routines Called

None
Calling Sequence

GFX.n is not called like most of the other routines in the library (see Section
1.4.1). It is called by:
EXTEND n, GFX.n
Type of Argument

The argument must be a double-precision, G-floating-point value less than
235. It must be stored in the AC specified in the routine name.
Type of Result

The result returned is an integer value; it may be any such value. It is returned in the AC specified in the routine name.
Accuracy of Result

The result is exact.
Algorithm Used

GFX.n(x) is calculated by means of the GFIX machine instruction. This
instruction converts and truncates the argument to an integer.
Error Conditions

If the argument is greater than 235 , an overflow occurs and the result is set to
machine infinity.

10-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

REAL
Description

The REAL routine converts and rounds its integer argument into a singleprecision, floating-point value.
Routines Called

None
Type of Argument

The argument must be an integer value; it can be any such value.
Type of Result

The result returned is a single-precision, floating-point value less than 235 •
Accuracy of Result

The result is rounded with an error bound of half a least significant bit.
Algorithm Used

REAL(n) is calculated by means of the FLTR machine instruction. This
instruction converts and rounds the argument to a single-precision, floatingpoint value.
Error Conditions

None

Data Type Conversion Routines 10-7

FLOAT
Description
The FLOAT routine converts and rounds its integer argument to a singleprecision, floating-point value.
Routines Called
None
Type of Argument
The argument must be an integer value; it can be any such value.
Type of Result
The result returned is a single-precision, floating-point value less than 2::15.
Accuracy of Result
The result is rounded with an error bound of half a least significant bit.
Algorithm Used
FLOAT(n) is calculated by means of the FLTR machine instruction. This
instruction converts and rounds the argument to a single-precision floatingpoint value.
Error Conditions
None

10-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

SNGL

Description

The SNGL routine converts and rounds its double-precision, D-floating-point
argument to a single-precision, floating-point value.
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a single-precision, floating-point value; it may be any
such value.
Accuracy of Result

The result is accurate to half a least significant bit because of rounding.
Algorithm Used

SNGL(x) is calculated as follows.
The routine tests the most significant bit of the low word of the magnitude of
the argument.
If it is 0, the high word is returned.
If it is 1, the low bit of the high word of the magnitude is tested.
If it is 0, it is made 1 and negated if necessary.
If it is 1, the high word of the magnitude is incremented and negated if

necessary.
Error Conditions
If overflow occurs, the result is set to machine infinity.

Data Type Conversion Routines 10-9

GSN.n

Description

The GSN.n routine converts and rounds its double-precision, G-floating-point
argument to a single-precision, floating-point value. n is an even octal number from through 14 that designates a register (AC).

Routines Called

None
Calling Sequence

GSN.n is not called like most of the other routines in the library (see Section
1.4.1). It is called by:
EXTEND n GSN.n
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value. It must be etored in the AC specified in the routine name.
Type of Result

The result returned is a single-precision, floating-point value; it may be any
such value. It is returned in the AC specified in the routine name.
Accuracy of Result

The result is exact to half a least significant bit because of rounding.
Algorithm Used

GSN .n(x) is calculated as follows.
The routine tests the most significant bit of the low word of the magnitude of
the argument.
If it is 0, the high word is returned.
If it is 1, the low bit of the high word of the magnitude is tested.
If it is 0, it is made 1 and negated if necessary.
If it is 1, the high word of the magnitude is incremented and ~egated if
necessary.
Error Conditions

1. If overflow occurs, the result is set to machine inf~nity.
2. If underflow occurs, the result is set to 0.0.

10-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

DFLOAT

Description

The DFLOAT routine converts its integer argument to a double-precision,
D-floating-point value.
Routines Called

None
Type of Argument

The argument must be an integer value; it can be any such value.
Type of Result

The result returned is a double-precision, D-floating-point value less than 2:35 •
Accuracy of Result

The result is exact.
Algorithm Used

DFLOAT(n) is calculated by moving the value of the argument to the locations used by a double~precision result. See Chapter 1 for a discussion of the
location of the result.
Error Conditions

None

Data Type Conversion Routines 10-11

DBlE
Description

The DBLE routine converts its single-precision floating-point argument to a
double-precision, D-floating-point value.
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

The result is exact.
Algorithm Used

DBLE(x) is calculated by moving the value of the argument to the locations
used by a double-precision result. (See Chapter 1 for a discussion of the
location of the result.) The low order word is set to O.
Error Conditions

None

10-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

GTOD
Description

The GTOD routine converts its double-precision, G-floating point argument
to a double-precision, D-floating-point value.
Routines Called

GTOD calls the MTHERR routine.
Type of Argument

The argument must be a double-precision G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

The result is exact.
Algorithm Used

GTOD(x) is calculated by converting the double-precision, G-floating-point
value to double-precision, D-floating point and setting the low-order three bits
to O.
Error Conditions

1. If the resulting exponent is too small to be represented as a doubleprecision, D-floating-point number, the following message is issued and
the result is set to 0.0.
GTOD: Result underflow
2. If the resulting exponent is too large to be represented as a double-

precision, D-floating-point number, the following message is issued and
the result is set to + machine infinity.
GTOD: Result overflow

Data Type Conversion Routines 10-13

GTODA

Description

The GTODA subroutine converts an array of double-precision, G-floatingpoint values to an array of double-precision, D-floating-point values. It is
called as:
GTODA (x,y,i)
x = input array
y = array used for result
i = number of elem.ents to convert
Routines Called

GTODA calls the MTHERR routine.
Type of Arguments

GTODA is a subroutine that is called with three arguments. The first and
second arguments must be double-precision arrays. The third argument must
be an integer value representing the number of elements to be converted. The
first array (x) contains the input values; the second array (y) will contain the
results. The input values must be double-precision, G-floating-point values;
they can be any such values.
Type of Result

The result returned is an array of double-precision, D-floating-point values;
they may be any such values. They are returned in the second array (y)
supplied in the call.
Accuracy of Result

The result is exact for each value converted.
Algorithm Used

GTODA(x) is calculated as follows.
Using the number specified in the third argument, GTODA converts each
double-precision, G-floating-point value to a double-precision, D-floatingpoint value and sets the low-order three bits to O. Each converted value is
stored in the second array.
Error Conditions

1. For each resulting exponent that is too small to be represented as a

double-precision, D-floating-point number, the following message is issued and the result is set to 0.0.
GTODA: Result underflow

2. For each resultIng exponent that is too large to be represented as a double-

precision, D-floating-point number, the following message is issued and
the result is set to +machine infinity.
GTODA: Result overflow

10-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

GFL.n

Description
The GFL.n· routine converts its integer argument to a double-precision,
G-floating-point value. n is an even octal number from 0 through 14 that
designates a register (AC).
Routines Called
None
Calling Sequence
GFL.n is not called like most of the routines in the library (see Section 1.4.1).
It is called by:

EXTEND n, GFL.n
Type of Argument
The argument must be an integer value; it can be any such value. It must be
stored in the AC specified in the routine name.
Type of Result
The result returned is a double-precision, G-floating-point value less than 2:ll).
It is returned in the AC specified in the routine name.
Accur:::y of Result
The result is exact.
Algorithm Used
GFL.n(n) is calculated by moving the value of the argument to the locations
used by a double-precision result (see Chapter 1).
Error Conditions
None

Data Type Conversion Routines 10-15

GOB.n·
Description

The GDB.n routine converts its single-precision, floating-point argument to a
double-precision, G-floating-point value. n is an even octal number from 0
through 14 that designates a register (AC).
Routines Called

None
Calling Sequence

GDB.n is not called like most of the routines in the library (see Section 1.4.1).
It is called by:

EXTEND n, GDB.n
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value. It must be stored in the AC specified in the routine name.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value. It is returned in the AC specified in the routine name.
Accuracy of Result

The result is exact.
Algorithm Used

GDB.n(x) is calculated as follows.
The routine uses the GDBLE machine instruction to convert the argument
and move it to the locations used for double-precision results.
Error Conditions

None

10-16 TOPS-10/TOPS-20 Common Math Library Reference Manual

DTOG

Description

The DTOG routine converts its double-precision, D-floating-point argument
to a double-precision, G-floating-point value.
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result

The result is rounded with an error bound of half a least significant bit.
Algorithm Used

DTOG(x) is calculated by converting the double-precision, D-floating-point
value to a double-precision, G-floating-point value and rounding the converted value.
Error Conditions

None

Data Type Conversion Routines 10-17

DTOGA

Description

The DTOGA subroutine converts an array of double-precision, D-floatingpoint values to an array of double-precision, G-floating-point values. It is
called as:
DTOGA(x,y,i)
x = input array
y = array used for result
i = number of elements to convert
Routines Called

None
Type of Arguments

DTOGA is a subroutine that is called with three arguments. The first and
second arguments must be double-precision arrays. The third argument must
be an integer value representing the number of elements to be converted. The
first array (x) contains the input values; the second array (y) will contain the
result. The input values must be double-precision, D.floating-point values;
they can be any such values.
Type of Result

The result returned is an array of double-precision, G-floating-point values;
they may be any such values. They are returned in the second array (y)
supplied in the call.
Accuracy of Result

Each element of the result is rounded with an error bound of half a least
significant bit.
Algorithm Used

DTOGA(x) is calculated as follows.
Using the number specified in the third argument, DTOGA converts each
double-precision, D-floating-point value to a double-precision, G-floatingpoint value and rounds the converted value. Each converted value is
stored in the second array.
Error Conditions

None

10-18 TOPS-10/TOPS-20 Common Math Library Reference Manual

CMPL.I

Description

The CMPL.I routine converts its two integer arguments into a complex,
single-precision, floating-point value.
Routines Called

None
Type of Arguments

Both arguments must be integer values; they can be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

The result is rounded with an error bound of half a least significant bit for
each part (real and imaginary).
Algorithm Used

CMPL.I(n,m) is calculated as follows.
The two arguments are converted to single-precision, floating-point values
using the FLTR machine instructions. These values are then moved to the
locations where the result is stored as a complex value (see Chapter 1).
The first argument is used as the real part of the complex number and the
second argument as the imaginary part.
Error Conditions

None

Data Type Conversion Routines

10-19

CMPLX

Description

The CMPLX routine converts two single-precision arguments into one complex single-precision, floating-point value.
Routines Called

None
Type of Arguments

Both arguments must be single-precision, floating-point values; they can be
any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

CMPLX(x,y) is calculated by moving the arguments to the locations used for
a complex result (see Chapter 1). The first argument is used as the real part of
the complex number and the second argument as the imaginary part.
Error Conditions

None

10-20 TOPS-10/TOPS-20 Common Math Library Reference Manual

CMPL.D

Description

The CMPL.D routine converts its two double-precision, D-floating-point arguments into a complex, single-precision, floating-point value.
Routines Called

None
Type of Arguments

The arguments must be double-precision, D-floating-point values; they can
be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

The result is accurate to half a least significant bit for each part because of
rounding.
Algorithm Used

CMPL.D(x,y) is calculated by converting the arguments to single-precision
and then moving them to the locations used for the real and imaginary parts
of the complex result (see Chapter 1). The first argument is used as the real
part of the complex number and the second argument as the imaginary part.
Error Conditions
If overflow occurs on the conversions, the result is set to machine infinity for

either or both of the parts of the result.

Data Type Conversion Routines

10-21

CMPL.G

Description

The CMPL.G routine converts its two double-precision, G-floating-point arguments into a complex, single-precision, floating-point value.
Routines Called

None
Type of Arguments

The arguments must be double-precision, G-floating-point values; they can
be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

The result is accurate to half a least significant bit for each part because of
rounding.
Algorithm Used

CMPL.G(x,y) is calculated by converting the arguments to single-precision
and then moving them to the locations used for the real and imaginary parts
of the complex result (see Chapter 1). The first argument is used as the real
part of the complex number and the second argument as the imaginary part.
Error Conditions
1. If overflow occurs on the conversions, the result is set to machine infinity

for either or both of the parts of the result.
2. If underflow occurs on the conversions, the result is set to 0.0 for either or
both parts of the result.

10-22 TOPS- 10/TOPS-20 Common Math Library Reference Manual

CMPL.C

Description

The CMPL.C routine creates a complex, single-precision, floating-point value
from the real parts of two complex, single-precision, floating-point values.
Routines Called

None
Type of Arguments

The arguments must be complex, single-precision, floating-point values; they
can be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

CMPL.C(z,g) is calculated by moving the arguments to the locations used for
a complex result (see Chapter 1). The first argument is used as the real part of
the complex number and the second argument as the imaginary part.
Error Conditions

None

Data Type Conversion Routi nes

10-23

Chapter 11
Rounding and Truncation Routines

NINT
Description

The NINT routine rounds its single-precision, floating-point argument to the
nearest integer.
Routines Called

NINT calls the MTHERR routine.
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is an integer value; it may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

NINT(x) is calculated as follows.
Let j = INT(lxl+.5)
If j < 235 and
If x 2: 0.0
NINT(x) = j
If x < 0.0
NINT(x) = -j
If j = 235 and
If x < 0.0
NINT(x) = -j

Otherwise, overflow occurs and
If x > 0.0
NINT(x) = 235 -1
If x < 0.0
NINT(x) = _2 35
Error Conditions
If x is greater than or equal to 235 or less than -2 35 , the result overflows. When

overflow occurs, the following message is issued and the result is set to +machine infinity if x is greater than 0.0 or to -machine infinity if x is less than
0.0.
NINT: Result overflow

Rounding and Truncation Routines 11-3

IDNINT

Description

The IDNINT routine rounds its double-predsion, D-floating-point argument
to the nearest integer.
Routines Called

IDNINT calls the MTHERR routine.
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is an integer value; it may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

IDNINT(x) is calculated as follows.
Let j = INT(lxl+.5)
If j < 235 and

If x ;?: 0.0
IDNINT(x) = j
If x < 0.0
IDNINT(x) = -j
If j = 235 and

If x < 0.0
IDNINT(x) = -j
Otherwise, overflow occurs and
If x > 0.0
IDNINT(x) = 235 _1
If x < 0.0
IDNINT(x) = _2 35
Error Conditions
If x is greater than or equal to 235 or less than _2 35 , the result overflows. When

overflow occurs, the following message is issued and the result is set to +machine infinity if x is greater than 0.0 or to -machine infinity if x is less than
0.0.
IONINT: Result overflow

11-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

IGNIN.
Description
The IGNIN. routine rounds its double-precision, G-floating-point argument
to the nearest integer.
Routines Called
IGNIN. calls the MTHERR routine.
Type of Argument
The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result
The result returned is an integer value; it may be any such value.
Accuracy of Result
The result is exact.
Algorithm Used
IGNIN.(x} is calculated as follows.

Let j = INT(lxl+.5)

If j < 235 and
If x ~ 0.0
IGNIN.(x) = j
If x < 0.0
IGNIN.(x) = -j
If j = 23fi and
If x < 0.0
IGNIN.(x) = -j
Otherwise, overflow occurs and
If x > 0.0
IGNIN.(x) = 2:15_1
If x < 0.0
IGNIN .(x) = _2 35
Error Conditions
If x is greater than or equal to 23fi or less than _2 35 , the result overflows. When
overflow occurs, the following message is issued and the result is set to +machine infinity if x is greater than 0.0 or - machine infinity if x is less than 0.0.
IGNIN.: Result overflow

Rounding and Truncation Routines 11-5

ANINT

Description
The ANINT routine rounds its single-precision, floating-point argument to
the nearest single-precision, floating-point whole number.
Routines Called
None
Type of Argument
The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Return
The result returned is a single-precision, floating-point whole value; it may be
any such value.
Accuracy of Result
The result is exact.
Algorithm Used
ANINT(x) is calculated as follows.
If Ixl :;::: 226

ANINT(x) = x because x is an integer
If Ixl < 226
If X > 0.0

ANINT(x) = ((lxl+2 26 )rounded)-2 26
If x < 0.0

ANINT(x) = -(((lxl+2 26 )rounded)-2 26 )
Error Conditions
None

11-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

DNINT

Description

The DNINT routine rounds its double-precision, D-floating-point argument
to the nearest double-precision, D-floating-point whole number.
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point whole value; it
may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

DNINT is calculated as follows.
If Ixl2 261
DNINT(x) = x because x is an integer

If Ixl < 261
If x> 0.0
DNINT(x) = «lxl+2 61 )rounded)-261
If x < 0.0
DNINT(x) = -«(lxl+2 61 )rounded)-261 )
Error Conditions

None

Rounding and Truncation Routines 11-7

GNINT.

Description

The GNINT. routine rounds its double-precision, G-floating-point argument
to the nearest double-precision, G-floating-point whole number.
Routines Called

None
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point whole value; it
may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

GNINT.(x) is calculated as follows.
If Ixl ~ 258

GNINT.(x) = x because x is an integer
If Ixl < 258
If X > 0.0

GNINT.(x) = «lxl+2 58 )rounded)-258
If x < 0.0

GNINT.(x) = -«(lxl+2 58 )rounded)-2 58 )
Error Conditions

None

11-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

AINT

Description

The AINT routine truncates its single-precision, floating-point argument to a
single-precision, floating-point whole number.
Routines Called

None
Type of Argument

The argument must be a single-precision, floating-point value; it can be any
such value.
Type of Result

The result returned is a single-precision, floating-point whole value; it may be
any such value.
Accuracy of Result

The result is exact.
Algorithm Used

AINT(x) is calculated as follows.
If Ixl ;? 226
AINT(x) = x because x is an integer
If Ixl < 226
If x> 0.0
AINT(x) = ((Ixl +2 26 )truncated)-2 26
If x < 0.0
AINT(x) = -( ((lxl+2 26 )truncated)-2 26 )
Error Conditions

None

Rounding and Truncation Routines 11-9

DINT

Description

The DINT routine truncates its double-precision, D-floating-point argument
to a double-precision, D-floating-point whole number.
Routines Called

None
Type of Argument

The argument must be a double-precision, D-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, D-floating-point whole value; it
may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

DINT(x) is calculated as follows.
If Ixl ~ 261

DINT(x) = x because x is an integer
If Ixl < 1.0

DINT(x) = 0.0
Otherwise
DINT(x) = sgn(x) -(Ixl with fraction bits replaced by zeroes)
Error Conditions

None

11-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

GINT.

Description

The GINT. routine truncates its double-precision, G-floating-point argument
to a double-precision, G-floating-point whole number.
Routines Called

None
Type of Argument

The argument must be a double-precision, G-floating-point value; it can be
any such value.
Type of Result

The result returned is a double-precision, G-floating-point whole value; it
may be any such value.
Accuracy of Result

The result is exact.
Algorithm Used

GINT.(x) is calculated as follows.
If Ixl ;::: 258
GINT.(x) = x because x is an integer
If Ixl < 1.0
GINT.(x) = 0.0

Otherwise
GINT.(x) = sgn(x) -(Ixl with fraction bits replaced by zeroes)
Error Conditions

None

Rounding and Truncation Routines 11-11

Chapter 12
Product, Remainder, and Positive Difference
Routines

DPROD
Description

The DPROD routine multiplies two single-precision, floating-point numbers
and returns a double-precision, D-floating-point product. That is:
DPROD(x,y) = x·y
Routines Called

OPROD calls the MTHERR routine.
Type of Arguments

Both arguments must be single-precision, floating-point values; they can be
any such values.
Type of Result

The result returned is a double-precision, D-floating-point value; it may be
any such value.
Accuracy of Result

The result is exact.
Algorithm Used

DPROD(x,y) is calculated as follows.
Let x = DBLE(x)
y = DBLE(y)
DPROD(x,y) = x·y
Error Conditions
1. If overflow occurs, the following message is issued and the result is set to

±machine infinity.
DPROD: Result overflow

2. If underflow occurs, the following message is issued and the result is set to
0.0.
DPROD: Result underflow

Product, Remainder, and Positive Difference Routines 12-3

GPROD.

Description

The GPROD. routine multiplies two single-precision, floating-point numbers
and returns a double-precision, G-floating-point product. That is:
GPROD.(x,y) = x·y
Routines Called

GPROD. calls the MTHERR routine.
Type of Arguments

Both arguments must be single-precision, floating-point values; they can be
any such values.
Type of Result

The result returned is a double-precision, G-floating-point value; it may be
any such value.
Accuracy of Result

The result is exact.
Algorithm Used

GPROD.(x,y) is calculated as follows.
Let x = GDB.O(x)
y = GDB.O(y)
GPROD.(x,y) = x·y
Error Conditions

None

12-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

MOD
Description

The MOD routine returns the integer remainder of the quotient of its integer
arguments. That is:
MOD(i,j) = i-[i/j] ej
Routines Called

None
Type of Arguments

Both arguments must be integer; the second argument cannot equal zero. If
the first argument is negative, the result is negative.
Type of Result

The result returned is an integer value in the range -Ijl to Ijl.
Accuracy of Result

The result is exact.
Algorithm Used

MOD(i,j) is calculated as follows.
MOD(i,j) = (Iil-[I il/j] ej). sgn(i)
[Iil/j] = the greatest integer in lil/j
Error Conditions

None

Product, Remainder, and Positive Difference Routines 12-5

AMOD
Description

The AMOD routine returns the single-precision, floating-point remainder of
the quotient of its single-precision, floating-point arguments. That is:
AMOD(x,y) = x-[x/y]·y
Routines Called

AMOD calls the MTHERR routine.
Type of Arguments

Both arguments must be single-precision, floating-point values; the second
argument cannot equal zero. If the first argument is negative, the result will
be negative.
Type of Result

The result returned is a single-precision, floating-point value in the range - Iyl
to Iyl.
Accuracy of Result

The result is exact.
Algorithm Used

AMOD(x,y) is calculated as follows.
AMOD(x,y) = (Ixl-[lxl/y] .y) ·sgn(x)
[I x I/y] = largest integer in Ixl/y
Error Conditions

Underflow may occur if y is too small a number. If underflow occurs, the
following message is issued and the result is set to 0.0.
AMOD: Result underflow

12-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

DMOD
Description

The DMOD routIne returns the double-precision, D-floating-point remainder
of the quotient of its double-precision, D-floating-point arguments. That is:
DMOD(x,y) = x-[x/y]·y
Routines Called

DMOD calls the MTHERR routine.
Type of Arguments

Both arguments must be double-precision, D-floating-point val~es; the second argument cannot equal zero. If the first argument is negative, the result
will be negative.
Type of Result

The result returned is a double-precision, D-floating-point value in the range
- Iyl to Iyl.
Accuracy of Result

The, result is exact.
Algorithm Used

DMOD(x,y) is calculated as follows.
DMOD(x,y) = (Ixl-[Ixl/y] .y) ·sgn(x)
[Ixl/y] = largest integer in Ixl/y
Error Conditions

Underflow may occur if y is too small a number. If underflow occurs, the
following message is issued and the result is set to 0.0.
DMOD: Result underflow

Product, Remainder, and Positive Difference Routines 12-7

GMOD
Description

The GMOD routine returns the double-precision, G-floating-point remainder
of the quotient of its double-precision, G-floating-point arguments. That is:
GMOD(x,y) = x-[x/y]·y
Routines Called

GMOD calls the MTHERR routine.
Type of Arguments

Both arguments must be double-precision, G-floating-point values; the second argument cannot equal zero. If the first argument is negative, the result
will be negative.
Type of Result

The result returned is a double-precision, G-floating-point value in the range
- Iyl to Iyl.
Accuracy of Result

The result is exact.
Algorithm Used

GMOD(x,y) is calculated as follows.
GMOD(x,y) = (Ixl-[Ixl/y] .y). sgn(x)
[lxl/y] = largest integer in Ixl/y
Error Conditions

Underflow nlay occur if y is too small a number. If underflow occurs, the
following message is issued and the result is set to 0.0.
GMOD: Result underflow

12-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

101M

Description
The IDIM routine returns the integer difference between its integer argU>l
ments, provided that the difference is positive. If the difference is negative,
IDIM returns zero. That is:

IDIM(i,j) = i-j
Routines Called
IDIM calls the MTHERR routine.
Type of Arguments
Both arguments JIlust be integer values; they can be any such values.
Type of Result
The result returned is an integer value greater than or equal to O.
Accuracy of Result
The result is exact.
Algorithm Used
IDIM is calculated as follows.
If i ~ j

IDIM(i,j) = 0
If i > j

IDIM(i,j) = i-j
Error Conditions
If overflow occurs during subtraction, the following message is issued and the

result is set to machine infinity.
101M: Result overflow

Product, Remainder, and Positive Difference Routines 12-9

DIM
Description

The DIM routine returns the single-precision, floating-point difference between its single-precision, floating-point arguments, provided that the difference is positive. If the difference is negative, DIM returns zero. That is:
DIM(x,y) = x-y
Routines Called

DIM calls the MTHERR routine.
Type of Arguments

Both arguments must be single-precision, floating-point values; they can be
any such values.
Type of Result

The result returned is a single-precision, floating-point value greater than or
equal to 0.0.
Accuracy of Result

The result is rounded with an error bound of half a least significant hit.
Algorithm Used

DIM(x,y) is calculated as follows.
IfxsY

DIM(x,y) = 0.0
If x > y

DIM(x,y) = x-y
Error Conditions

1. If overflow occurs during subtraction, the following message is issued and

the result is set to machine infinity.
01 M: Result overflow
2. If underflow occurs during subtraction, the following message is issued

and the result is set to 0.0.
DIM: Result underflow

12-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

DDIM
Description

The DDIM routine returns the double-precision, D-floating-point difference
between its double-precision, D-floating-point arguments, provided that the
difference is positive. If the difference is negative, DDIM returns zero. That is:
DDIM(x,y) = x-y
Routines Called

DDIM calls the MTHERR routine.
Type of Arguments

Both arguments must be double-precision, D-floating-point values; they can
be any such values.
Type of Result

The result returned is a double-precision, D-floating-point value greater than
or equal to 0.0.
Accuracy of Result

The result is rounded with an error bound of half a least significant bit.
Algorithm Used

DDIM(x,y) is calculated as follows.
Ifx:5Y
DDIM(x,y) = 0.0
If x > y
DDIM(x,y) = x-y
Error Conditions
1. If overflow occurs during subtraction, the following message is issued and

the result is set to machine infinity.
DDIM: Result overflow

2. If underflow occurs during subtraction, the following message is issued
and the result is set to 0.0.
DDIM: Result underflow

Product, Remainder, and Positive Difference Routines 12-11

GDIM
Description

The GDIM routine returns the double-precision, G-floating-point difference
between its double-precision, G-floating-point arguments, provided that the
difference is positive. If the difference is negative, GDIM returns zero. That is:
GDIM(x,y) = x-y
Routines Called

GDIM calls the MTHERR routine.
Type of Arguments

Both arguments must be double-precision, G-floating-point values; they can
be any such values.
Type of Result

The result returned is a double-precision, G-floating-point value greater than
or equal to 0.0.
Accuracy of Result

The result is rounded with an error bound of half a least significant bit.
Algorithm Used

GDIM(x,y) is calculated as follows.
If x::::; y

GDIM(x,y) = 0.0
If x > y

GDIM(x,y) = x-y
Error Conditions

1. If overflow occurs during subtraction, the following message is issued and
the result is set to machine infinity.
GDIM: Result overflow
2.

If underflow occurs during subtraction, the following message is issued
and the result is set to 0.0.
GDIM: Result underflow

12-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 13
Transfer of Sign Routines

ISIGN
Description

The ISIGN routine transfers the sign of its integer second argument to its
integer first argument, ignoring the sign of the first argument. That is:
ISIGN (i,j) = lilesgn(j)
Routines Called

ISIGN calls the MTHERR routine.
Type of Arguments

Both arguments must be integer values; they can be any such values.
Type of Result

The result returned is an integer value; it has the same magnitude as the first
argument.
Accuracy of Result

The result is exact.
Algorithm Used

ISIGN(i,j) is calculated as follows.
ISIGN(i,j) = lilesgn(j)
If j 2:: 0
ISIGN(i,j) = Iii
If j < 0
ISIGN(i,j) = -Iii
Error Conditions
If i = _2 35 and j > 0, overflow occurs. If overflow occurs, the following message
is issued and the result is set to machine infinity.

ISIGN: Result overflow

Transfer of Sign Routines 13-3

SIGN

Description

The SIGN routine transfers the sign of its single-precision, floating-point
second argument to its single-precision, floating-point first argument, ignoring the sign of the first argument. That is:
SIGN (x,y) = Ixlesgn(y)
Routines Called

None
Type of Arguments

Both arguments must be single-precision, floating-point values; they can be
any such values.
Type of Result

The result returned is a single-precision, floating-point value; it has the same
magnitude as the first argument.
Accuracy of Result

The result is exact.
Algorithm Used

SIGN(x,y) is calculated as follows.
SIGN(x,y) = Ixlesgn(y)
If y;? 0.0

SIGN(x,y) = Ixl
If y < 0.0

SIGN(x,y) = -Ixl
Error Conditions

None

13-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DSIGN
Description
The DSIGN routine transfers the sign of its double-precision, D-floating-point
second argument to its double-precision, D-floating-point first argument, ignoring the sign of the first argument. That is:

DSIGN(x,y) = Ixlesgn(y)
Routines Called
None
Type of Arguments
Both arguments must be double-precision, D-floating-point values; they can
be any such values.
Type of Result
The result returned is a double-precision, D-floating-point value; it has the
same magnitude as the first argument.
Accuracy of Result
The result is exact.
Algorithm Used
DSIGN(x,y) is calculated as follows.

DSIGN(x,y) = Ixlesgn(y)
If y ~ 0.0
DSIGN(x,y) = Ixl
If y < 0.0
DSIGN(x,y) = -Ixl
Error Conditions
None

Transfer of Sign Routines 13-5

GSIGN

Description

The GSIGN routine transfers the sign of its double-precision, G-floating-point
second argument to its double-precision, G-floating-point first argument, ignoring the sign of the first argument. That is:
GSIGN(x,y) = Ixl-sgn(y)
Routines Called

None
Type of Arguments

Both arguments must be double-precision, G-floating-point values; they can
be any such values.
Type of Result

The result returned is a double-precision, G-floating-point value; it has the
same magnitude as the first argument.
Accuracy of Result

The result is exact.
Algorithm Used

GSIGN(x,y) is calculated as follows.
GSIGN(x,y) = Ixl-sgn(y)
If y ~ 0.0
GSIGN(x,y) = Ixl
If y < 0.0

GSIGN(x,y) = -Ixl
Error Conditions

None

13-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 14
Maximum/Minimum Routines

MAXO
Description

The MAXO routine finds the integer maximum of a series of integer arguments.
Routines Called

None
Type of Arguments

All the arguments must be integer values; they can be any such values. There
can be as many arguments as desired.
Type of Result

The result returned is an integer value; it is the largest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

MAXO(i, ... j) is calculated as follows.
The MAXO routine compares each argument in succession with the current
largest argument, which is held in a register. Each time an argument exceeds
the current largest argument, the register is updated. This loop continues
until the final argument is processed. The contents of the register are then
returned as the result.
Error Conditions

None

Maximum/Minimum Routines 14-3

MAX1

Description

The MAXI routine finds the integer maximum of a series of single-precision,
floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be single-precision, floating-point values; they can be
any such values. There can be as many arguments as desired.
Type of Result

The result returned is the largest value in the series converted to integer
format.
Accuracy of Result

The result is exact except for possible overflow during the conversion to integer.
Algorithm Used

MAXI(x, ... y) is calculated as follows.
The MAXI routine compares each argument in succession with the current
largest argument, which is held in a register. Each time an argument exceeds
the current largest argument, the register is updated. This loop continues
until the final argument is processed. The contents of the register are then
converted to integer format and returned as the result.
Error Conditions

Overflow can occur during conversion to integer. If overflow occurs, the result
is set to ± machine infinity.

14-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

AMAXO

Des~rlptlon

The AMAXO routine finds the single-precision, floating-point maximum of a
series of integer arguments.
Routines Called

None
Type of Arguments

All the arguments must be integer; they can be any such values. There can be
as many arguments as desired.
Type of Result

The result returned is the largest value in the series converted to singleprecision, floating-point format.
Accuracy of Result

The result is exact unless a rounding error occurs during conversion, in which
case the error could be half a least significant bit.
Algorithm Used

AMAXO(i, ... j) is calculated as follows.
The AMAXO routine compares each argument in succession with the current
largest argument, which is held in a register. Each time an argument exceeds
the current largest argument, the register is updated. This loop continues
until the final argument is processed. The contents of the register are then
converted to single-precision, floating-point format and returned as the result.
Error Conditions

None

Maximum/Minimum Routines 14-5

AMAX1

Description

The AMAXI routine finds the single-precision, floating-point maximum of a
series of single-precision, floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be single-precision, floating-point values; they can be
any such values. There can be as many arguments as desired.
Type of Result

The result returned is a single-precision, floating-point value; it is the largest
value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

AMAXl(x, ... y) is calculated as follows.
The AMAXI routine compares each argument in succession with the current
largest argument, which is held in a register. Each time an argument exceeds
the current largest argument, the register is updated. This loop continues
until the final argument is processed. The contents of the register are then
returned as the result.
Error Conditions

None

14-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

DMAX1

Description

The DMAXI routine finds the double-precision, D-floating-point maximum
of a series of double-precision, D-floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be double-precision, D-floating-point values; they can
be any such values. There can be as many arguments as desired.
Type of Result

The result returned is a double-precision, D-floating-point value; it is the
largest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

DMAXl(x, ... y) is calculated as follows.
The DMAXI routine compares each argument in succession with the current
largest argument, which is held in two registers. Each time an argument
exceeds the current largest argument, the registers are updated. This loop
continues until the final argument is processed. The contents of the registers
are then returned as the result.
Error Conditions

None

Maximum/Minimum Routines 14-7

GMAX1

Description

The GMAXI routine finds the double-precision, G··floating-point maximurn
of a series of double-precision, G-floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be double-precision, G-floating-point values; they can
be any such values. There can be as many arguments as desired.
Type of Result

The result returned is a double-precision, G-floating-point value; it is the
largest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

GMAXl(x, ... y) is calculated as follows.
The GMAXI routine compares each argument in succession with the current
largest argument, which is held in two registers. Each time an argulnent
exceeds the current largest argument, the registers are updated. This loop
continues until the final argument is processed. The contents of the registers
are then returned as the result.
Error Conditions

None

14-8 TOPS-10/TOPS-20 Common Math Library Reference Manual

MINO

Description

The MINO routine finds the integer minimum of a series of integer arguments.
Routines Called

None
Type of Arguments

All the arguments must be integer values; they can be any such values. There
can be as many arguments as desired.
Type of Result

The result returned is an integer value; it is the smallest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

MINO(i, ... j) is calculated as follows.
The MINO routine compares each argument in succession to the current
smallest argument, which is held in a register. Each time an argument is less
than the current smallest argument, the register is updated. This loop continues until the final argument is processed. The contents of the register are then
returned as the result.
Error Conditions

None

Maximum/Minimum Routines 14-9

MIN1

Description

The MINI routine finds the integer minimum of a series of single-precision,
floating- point arguments.
Routines Called

None
Type of Arguments

All the arguments must be single-precision, floating-point values; they can be
any such values. There can be as many arguments as desired.
Type of Result

The result returned is the smallest value in the series converted to integer
format.
Accuracy of Result

The result is exact except for possible overflow during the conversion to integer.
Algorithm Used

MINl(x, ... y) is calculated as follows.
The MINI routine compares each argument in succession with the current
smallest argument, which is held in a register. Each time an argulnent is
smaller than the current smallest argument, the register is updated. This loop
continues until the final argument is processed. The contents of the register
are then converted to integer and returned as the result.
Error Conditions

Overflow can occur during conversion to integer. If overflow occurs, the result
is set to ± machine infinity.

14-10 TOPS-10/TOPS-20 Common Math Library Reference Manual

AMINO

Description

The AMINO routine finds the single-precision, floating-point minimum of a
series of integer arguments.
Routines Called

None
Type of Arguments

All the arguments must be integer; they can be any such values. There can be
as many arguments as desired.
Type of Result

The result returned is the smallest value in the series converted to singleprecision, floating-point format.
Accuracy of Result

The result is exact unless a rounding error occurs during conversion, in which
case the error could be half a least significant bit.
Algorithm Used

AMINO(i, ... j) is calculated as follows.
The AMINO routine compares each argument in succession with the current
smallest argument, which is held in a register. Each time an argument is
smaller than the current smallest argument, the register is updated. This loop
continues until the final argument is processed. The contents of the register
are then converted to single-precision, floating-point format and returned as
the result.
Error Conditions

None

Maximum/Minimum Routines 14-11

AMIN1

Description

The AMINI routine finds the single-precision, floating-point minimum of a
series of single-precision, floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be single-precision, floating-point values; they can be
any such values. There can be as many arguments as desired.
Type of Result

The result returned is a single-precision, floating-point value; it is the smalJest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

AMINI(x, ... y) is calculated as follows.
The AMINI routine compares each argument in succession with the current
smallest argument, which is held in a register. Each time an argument is
smaller than the current smallest argunlent, the register is updated. This loop
continues until the final argument is processed. The contents of the register
are then returned as the result.
Error Conditions

None

14-12 TOPS-10/TOPS-20 Common Math Library Reference Manual

DMIN1

Description

The DMINI routine finds the double-precision, D-floating-point minimum of
a series of double-precision, D-floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be double-precision, D-floating-point values; they can
be any such values. There can be as many arguments as desired.
Type of Result

The result returned is a double-precision, D-floating-point value; it is the
smallest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

DMINl(x, ... y) is calculated as follows.
The DMINI routine compares each argument in succession with the current
smallest argument, which is held in two registers. Each time an argument is
less than the current smallest argument, the registers are updated. This loop
continues until the final argument is processed. The contents of the registers
are then returned as the result.
Error Conditions

None

Maximum/Minimum Routines

14-13

GMIN1

Description

The GMINI routine finds the double-precision, G-floating-point minimum of
a series of double-precision, G-floating-point arguments.
Routines Called

None
Type of Arguments

All the arguments must be double-precision, G-floating-point values; they can
be any such values. There can be as many arguments as desired.
Type of Result

The result returned is a double-precision, G-floating-point value; it is the
smallest value in the series.
Accuracy of Result

The result is exact.
Algorithm Used

GMINl(x, ... y) is calculated as follows.
The GMINI routine compares each argument in succession with the current
smallest argument, which is held in two registers. Each time an argument is
less than the current smallest argument, the registers are updated. This loop
continues until the final argument is processed. The contents of the registers
are then returned as the result.
Error Conditions

None

14-14 TOPS-10/TOPS-20 Common Math Library Reference Manual

Chapter 15
Miscellaneous Complex Routines

REAL.C

Descriptio n

The REAL.C routine returns the real part of a complex number. That is:
REAL.C(z) = REAL.C(x+i .y) = x
Routines Called

None
Type of Argument

The argument must be a complex value; it can be any such value.
Type of Result

The result returned is a single-precision, floating-point value.
Accuracy of Result

The result is exact.
Algorithm Used

REAL.C(z) is calculated by copying the real part of the argument to the
return location.
Error Conditions

None

Miscellaneous Complex Routines 15-3

AIMAG

Description

The AIMAG routine returns the imaginary part of a complex number. That is:
AIMAG(z) = AIMAG(x+i .y) = y
Routine'S Called

None
Type of Argument

The argument must be a complex value; it can be any such value.
Type of Result

The result returned is a single-precision, floating-point value; it is the imaginary part of the number.
Accuracy of Result

The result is exact.
Algorithm Used

AIMAG(z) is calculated by copying the imaginary part of the argument to the
return location.
Error Conditions

None

15-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

CONJ

Description

The CONJ routine finds the conjugate of a complex number. That is:
CONJ(z) = conj(x+i .y) = x-i·y
Routines Called

None
Type of Argument

The argument must be a complex value; it can be any such value.
Type of Result

The result returned is a complex value; it is the conjugate of the argument
value.
Accuracy of Result

The result is exact.
Algorithm Used

CONJ(z) is calculated as follows.
Let z = x+i·y
conj(x+i .y) = x+( -i .y)
CONJ(z) = x-i·y
Error Conditions

None

Miscellaneous Complex Routines 15-5

CFM

Descr1ptlon

The CFM subroutine finds the complex, single-precision, floating-point product of two complex, single-precision, floating-point values. That is:
CFM(z,g) = zeg
Routines Called

CFM calls the MTHERR routine.
Type of Arguments

CFM is a subroutine with two arguments; both must be complex, singleprecision, floating-point values. They can be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value.
Accuracy of Result

test interval:

-10000. through 10000. for z (real)
-10000. through 10000. for z (imaginary)
-10000. through 10000. for g (real)
-10000. through 10000. for g (imaginary)

MRE:

1.20x10-5 (16.4 bits) real
1.47x10-6 (19.4 bits) imaginary

RMS:

2.64x10- 7 (21.9 bits) real
5.81x10-8 (24.0 bits) imaginary

LSB error distribution:

-4+ -3 -2 -1 a
+1 +2 +3 +4+
2% 1% 1% 14% 64% 15% 1% 1% 2% real
1% 1% 1% 15% 64% 14% 1% 1% 2% imaginary

Algorithm Used

CFM(z,g) is calculated as follows.
Let z = a+i· b
Let g = c+i·d
If CFM(z,g) = (a+i· b)· (c+i· d)

CFM(z,g) = (a·c-b-d)+i-(b-c+a·d)
Error Conditions

1. If either part of the result overflows, the following message is issued and
that part of the result is set to machine infinity.
CMATH: Complex overflow

2. If either part of the result underflows, the following message is issued and
that part of the result is set to 0.0.
CMATH: Complex underflow

15-6 TOPS-10/TOPS-20 Common Math Library Reference Manual

CFDV

Description

The CFDV subroutine finds the complex, single-precision, floating-point quotient of two complex, single-precision, floating-point values. That is:
CFDV(z,g) = zig
Routines Called

CFDV calls the MTHERR routine.
Type of Arguments

CFDV is a subroutine with two arguments; both must be complex, singleprecision, floating-point values. They can be any such values.
Type of Result

The result returned is a complex, single-precision, floating-point value; it may
be any such value.
Accuracy of Result

test interval:

-10000. through 10000. for z (real)
-10000. through 10000. for z (imaginary)
-10000. through 10000. for g (real)
-10000. through 10000. for g (imaginary)

MRE:

2.87x10-7 (21.7 bits) real
7.60x10- 7 (20.3 bits) imaginary

RMS:

1.33x10-8 (26.2 bits) real
2.30x10-8 (25.4 bits) imaginary

LSB error distribution:

-4+ -3 -2 -1 0
+1 +2 +3 +4+
1% 1% 3% 22% 49% 21% 2% 0% 1% real
1% 1% 3% 21% 50% 20% 3% 1% 1% imaginary

Algorithm Used

CFDV(z,g) is calculated as follows.
Let z = a+i-b
Letg=c+i-d
If CFDV(z,g) = (a+i-b)/(c+i-d)

CFDV(z,g) = ((a-c+b-d)+i-(b-c-a-d))/(c 2 +d 2 )
Error Conditions

1. If either part of the result underflows, the following message is issued and
that part of the result is set to 0.0.
CMATH: Complex underflow

2. If either part of the result overflows, that part of the result is set to
machine infinity.

Miscellaneous Complex Routines 15-7

Appendix A
ELEFUNT Test Results
This appendix contains the results of the ELEFUNT tests of W. J. Cody,
Argonne National Laboratory. For each test, the test interval, maximum relative error (MRE), and root mean square (RMS) relative error are given. Note
that it is not meaningful to compare these test results with the test results
given for each routine under the heading "Accuracy of Result."
ACOS(x) vs Taylor Series
test interval: -1.0000 through -0.7500
MRE: 0.1231x10-7
(26.3 bits)
RMS: 0.2868x10-8
(28.4 bits)
ACOS(x) vs Taylor Series
test interval: 0.7500 through 1.0000
MRE: 0.1488x10-7
(26.0 bits)
RMS: 0.1330x10-8
(29.5 bits)
ACOS(x) vs Taylor Series
test interval: -0.1250 through 0.1250
MRE: 0.1030x10-7
(26.5 bits)
8
RMS: 0.2647x10(28.5 bits)
ALOG(x·x) vs 2·loge x
test interval: 0.1600x102 through 0.2400x103
MRE: 0.1466xlO- 7
(26.0 bits)
RMS: 0.2292x10- 8
(28.7 bits)
ALOG(x) vs Taylor Series expansion of ALOG(1+y)
test interval: 1-0.1953x10-2 through 1+0.1953x1o-2
MRE: 0.2466x10-7
(25.3 bits)
8
RMS: 0.6614x10(27.2 bits)
ALOG(x) vs ALOG(17x/16)-ALOG(17/16)
test interval: 0.7071 through 0.9375
(25.4 bits)
MRE: 0.2264x10-7
R1\1S: 0.6426x10-8
(27.2 bits)

A-1

ALOG10(x) vs ALOG10(11x/10)-ALOG10(11/10)
test interval: 0.3162 through 0.9000
MRE: 0.3863x10-7
(24.6 bits)
7
RMS: 0.1122x10(26.4 bits)
ASIN (x) vs Taylor Series
test interval: 0.7500 through 1.0000
MRE: 0.1478x10-7
(26.0 bits)
RMS: 0.3245x10-8
(28.2 bits)
ASIN (x) vs Taylor Series
test interval: -0.1250 through 0.1250
MRE: 0.1190x10-7
(26.3 bits)
RMS: 0.6733x10-9
(30.5 bits)
ATAN (x) vs truncated Taylor Series
test interval: -0.6250x10- 1 through 0.6250xl0- 1
MRE: 0.8032x10-8
(26.9 bits)
RMS: 0.1796x10-9
(32.4 bits)
ATAN(x) vs ATAN(1/16)+ATAN«x-1/16)/(1+x/16»
test interval: 0.6250.10- 1 through 0.2679
MRE: 0.1488x10-7
(26.0 bits)
RMS: 0.6219x10-8
(27.3 bits)
2·ATAN(x) vs ATAN(2x/(1-x·x»
test interval: 0.2679 through 0.4142
MRE: 0.1423x10-7
(26.1 bits)
RMS: 0.6597x10-8
(27.2 bits)
2·ATAN(x) vs ATAN(2x/(1-x·x»
test interval: 0.4142 through 1.0000
MRE: 0.1484xlO-7
(26.0 bits)
8
RMS: 0.3894x10(27.9 bits)
COS (x) vs 4·COS(x/3)3_3·COS(x/3)
test interval: 0.2199x102 through 0.2356xl02
MRE: 0.2070x10-7
(25.5 bits)
RMS: 0.6463x10-8
(27.2 bits)
COSH(x) vs C·(COSH(x+1)+COSH(x-1»
test interval: 3.0000 through 0.8803x1p2
MRE: 0.2219x10-7
(25.4 bits)
RMS: 0.7007x1o-8
(27.1 bits)
COSH(x) vs Taylor Series expansion of COSH(x)
test interval: 0.0000 through 0.5000
MRE: 0.1490x10-7
(26.0 bits)
(27.4 bits)
RMS: 0.5491x10-8
COT(x) vs (COT(x/2)2_1)/(2·COT(x/2»
test interval: 0.1885x102 through 0.1963xl02
MRE: 0.2975x10-7
(25.0 bits)
RMS: 0.8629x10-8
(26.8 bits)

A-2 TOPS-10/TOPS-20 Common Math Library Reference Manual

DACOS(x) vs Taylor Series
test interval: -1.0000 through -0.7500
MRE: 0.3582x10- 18
(61.3 bits)
RMS: 0.1211x10- 18
(62.8 bits)
DACOS(x) vs Taylor Series
test interval: -0.1250 through -0.1250
MRE: 0.3000x10- 18
(61.5 bits)
RMS: 0.1224x10- 18
(62.8 bits)
DACOS(x) vs Taylor Series
test interval: 0.7500 through 1.0000
MRE: 0.4337x10- 18
(61.0 bits)
RMS: 0.1682x10- 18
(62.4 bits)
DASIN (x) vs Taylor Series
test interval: -0.1250 through 0.1250
MRE: 0.4334x10- 18
(61.0 bits)
RMS: 0.1715x10- 18
(62.3 bits)
DASIN(x) vs Taylor Series
test interval: 0.7500 through 1.0000
MRE: 0.4326x10- 18
(61.0 bits)
RMS: 0.1168x1o-- 18
(62.9 bits)
DATAN(x) vs truncated Taylor Series
test interval: -0.6250x10- 1 through -0.6250x10- 1
MRE: 0.4326x10- 18
(61.0 bits)
RMS: 0.1370x10- 18
(62.7 bits)
DATAN(x) vs DATAN(1/16)+DATAN«x-1/16)/(1+x/16))
test interval: 0.6250x10- 1 through 0.2679
MRE: 0.4333x10- 18
(61.0 bits)
RMS: 0.1755x10- 18
(62.3 bits)
2 -DATAN(x) vs DATAN(2x/(l-x-x))
test interval: 0.2679 through 0.4142
MRE: O.6610x10- 18
(60.4 bits)
18
RMS: 0.1987x10(62.1 bits)
2-DATAN(x) vs DATAN(2x/(1-x-x))
test interval: 0.4142 through 1.0000
MRE: 0.4319x10- 18
(61.0 bits)
RMS: 0.1167x10- 18
(62.9 bits)
DCOS(x) vs 4-DCOS(x/3)3_3-DCOS(x/3)
test interval: 0.2199x10 2 through 0.2356x10 2
MRE: 0.6523x10- 18
(60.4 bits)
RMS: 0.1960x10- 18
(62.2 bits)
DCOSH(x) vs Taylor Series expansion of DCOSH(x)
test interval: 0.0000 through 0.5000
MRE: 0.4337x10- 18
(61.0 bits)
RMS: 0.1550x10- 18
(62.5 bits)

ELEFUNT Test Results

A-3

DCOSH(x) vs C·(DCOSH(x+l)+DCOSH(x-·l)
test interval: 3.0000 through 0.8803xl0 2
MRE: 0.8440xlO- 18
(60.0 bits)
RMS: 0.2805x10- 18
(61.6 bits)
DCOT(x) vs (DCOT(x/2)2-1)/(2·DCOT(x/2»
test interval: 0.1885xl0 2 through 0.1963xl0 2
MRE: 0.9064xlO- 18
(59.9 bits)
RMS: 0.2632xlO'-.18
(61. 7 bits)
DEXP(x-0.0625) vs DEXP(x)/DEXP(0.0625)
test interval: -0.2841 through 0.3466
MRE: 0.4336xlO-- 18
(61.0 bits)
18
RMS: 0.1689xlO(62.4 bits)
DEXP(x-2.8125) vs DEXP(x)/DEXP(2.8125)
test interval: -3.4660 through -0.4505xl0 2
MRE: 0.6394xlO- 18
(60.4 bits)
RMS: 0.1670x10- 18
(62.4 bits)
DEXP(x-2.8125) vs DEXP(x)/DEXP(2.8125)
test interval: -6.9310 through 0.8792xl0 2
MRE: 0.6350x10- 18
(6004 bits)
RMS: 0.1808xlO- 18
(62.3 bits)
DEXP3. (x 1.0 vs x)
test interval: 0.5000 through 1.0000
The result is exact.
DEXP3. (XSQ1.5 vs XSQ ·x)
test interval: 0.5000 through 1.0000
MRE: 0.4336xlO- 18
(61.0 bits)
RMS: O.1585xl0 18
(62.4 bits)
DEXP3. (XSQ1.5 vs XSQ ·x)
test interval: 1.0000 through 0.5541x10 13
MRE: 0.4330xlO- 18
(61.0 bits)
R1\1S: 0.1678xlO- 18
(62.4 bits)
DEXP3. (x Y vs XSQy/2)
test interval: 0.1000xlO- 1 through 0.1000xl0 2 for x
-O.1942xl02 through 0.1942xl0 2 for y
]\tIRE: 0.5499xIO- 18
(60.7 bits)'
RMS: 0.1196xlO-- 18
(62.9 hits)
DLOG(x) vs Taylor Series expansion of DLOG(1+y)
test interval: 1-9537xlO-6 through 1+9537xlO-6
l\1RE: O.5605xlO- 18
(60of) bits)
Rl\tlS: 0.1922xl0 18
(62.2 bits)
DLOG(x) vs DLOG(17x/16)--DLOG(17/16)
test interval: 0.7071 through 0.9375
MHE: O.9228xlO- 18
(59.9 bits)
RMS: O.3347xlO- 18
(61.4 bits)

A-4 TOPS-10/TOPS-20 Common Math Library Reference Manual

DLOG(x·x) vs 2·DLOG(x)
test interval: 0.1600x10 2 through 0.2400xl03
MRE: 0.4306xlO- 18
(61.0 bits)
RMS: 0.7895x10- 19
(63.5 bits)
DLOG10(x) vs DLOG10(11x/10)-DLOG10(11/10)
test interval: 0.3162 through 0.9000
MRE: 0.1476x10- 17
(59.2 bits)
RMS: 0.3747x10- 18
(61.2 bits)
DSIN(x) vs 3·DSIN(x/3)-4·nSIN(x/3)3
test interval: 0.0000 through 1.5710
MRE: 0.5378x10- 18
(60.7 bits)
18
RMS: 0.1802xlO(62.3 bits)
DSIN(x) vs 3·DSIN(x/3)-4·DSIN(x/3)3
test interval: 0.1885x10 2 through 0.2042x10 2
MRE: 0.6115x10- 18
(60.5 bits)
18
RMS: 0.1960x10(62.2 bits)
DSINH(x) vs Taylor Series expansion of DSINH(x)
test interval: 0.0000 through 0.5000
MRE: 0.4336x10- 18
(61.0 bits)
RMS: 0.8776x10- 19
(63.3 bits)
DSINH(x) vs C·(DSINH(x+1)+DSINH(x-1))
test interval: 3.0000 through 0.8803x10 2
MRE: 0.8643x10- 18 ' (60.0 bits)
RMS: 0.2736x10- 18
(61. 7 bits)
DSQRT(x -x)-x
test interval: 0.7071 through 1.0000
MRE: 0.3064x10- 18
(61.5 bits)
19
RMS: 0.7383x10(63.6 bits)
DSQRT(x ·x)-x
test interval: 1.0000 through 1.4140
The result is exact.
DTAN(x) vs 2·TAN(x/2)/(l-DTAN(x/2)2)
test interval: 0.1885x10 2 through 0.1963x10 2
MRE: 0.1262x10- 17
(59.5 bits)
RMS: 0.3402x10- 18
(61.4 bits)
DTAN(x) vs 2-DTAN(x/2)/(l-DTAN(x/2)2)
test interval: 2.7490 through 3.5340
MRE: O.1216xlO- 17
(59.5 bits)
RMS: O.2492xlO- 18
(61.8 bits)
DTAN(x) vs 2-DTAN(x/2)/(l-DTAN(x/2)2)
test interval: 0.0000 through 0.7854
MRE: 0.1094xlO- 17
(59.7 bits)
RMS: 0.3331xlO- 18
(61.4 bits)

ELEFUNT Test Results

A-S

DTANH(x) vs (DTANH(x-1/B) +DTANI-I(1/8) )/( 1+DTANH(x-1/8)DTANH(1/8»
test interval: 0.1250 through 0.5493
MRE: 0.8436x10- 18
·(60.0 bits)
RMS: 0.2150x10· 18
(62.0 bits)
DTANH(x) vs (DTANH(x-1/B) +DTANH(1/8) )/(1 +DTANH(x-1/8)DTANH(1/B»
test interval: 0.6743 through 0.2253x10 2
MRE: 0.4952x10·· 18
(60.B bits)
RMS: 0.1966x10- 18
(62.1 bits)
EXP(x-0.0625) vs EXP(x)/EXP(0.0625)
test interval: -0.2841 through 0.3466
MRE: O.1489xlO-7
(26.0 bits)
RMS: 0.5BOlx10- 8
(27.4 bits)
EXP(x-2.8125) vs EXP(x)/EXP(2.8125)
test interval: -3.4660 through -0.6931xl02
MRE: 0.1489xlO- 7
(26.0 bits)
RMS: 0.5879xl0--8
(27.3 bits)
EXP(x-2.8125) vs EXP(x)/EXP(2.8125)
test interval: 6.9310 through 0.8792xl0 2
MRE: O.2108xlO-7
(25.5 bits)
RMS: 0.576BxlO-8
(27.4 bits)
EXP3. (x LO vs x)
test interval: 0.5000 through 1.0000
The result is exact.
EXP3. (XSQ1.5 vs XSQ ·x)
test interval: 0.5000 through 1.0000
MRE: 0.1487xlO- 7
(26.0 bits)
RMS: 0.5433xl0 8
(27.5 bits)
L~XP3. (XSQ1.5 vs XSQ ·x)

test interval: 1.0000 through 0.5541xl0 13
MHE: O.1461xlO- 7
(26.0 bits)
RMS: 0.5347x10- 8
(27.5 bits)
EXP3. (x Y vs XSQy/2)
test interval: 0.1.000xlO- 1 through 0.1000xl0 2 for x
-0.1942xl02 through 0.1942xl0 2 for y
MRE: 0.2065xl0 7
(25.5 bits)
RMS: 0.3572xlO-8
(28.0 bits)
GACOS(x) vs Taylor Series
test interval: -1.0000 through -0.7500
MRE: 0.2869xlO-· 17
(58.3 hits)
H1\1S: 0.1515xlO- 17
(59.2 bits)
GACOS(x) vs Taylor Series

test interval: 0.7500 through 1.0000
MRE: O.3443xlO- 17
(58.0 hits)
18
HMS: 0.4924xlO·
(60.B bits)

A-6

TOPS- -, OlTOPS-20 Common Math Library Reference Manual

GACOS(x) vs Taylor Series
test interval: -0.1250 through 0.1250
MRE: 0.2399x10- 17
(58.5 bits)
RMS: O.1297x10- 17
(59.4 bits)
GASIN(x) vs Taylor Series
test interval: 0.7500 through 1.0000
MRE: 0.3457x10- 17
(58.0 bits)
RMS: 0.1452x10- 17
(59.3 bits)
GASIN(x) vs Taylor Series
test interval: -0.1250 through 0.1250
MRE: 0.3462x10- 17
(58.0 bits)
RMS: 0.4997x10- 18
(60.8 bits)
GATAN(x) vs truncated Taylor Series
test interval: -O.6250x10- 1 through 0.6250x10- 1
MRE: 0.3389x10- 17
(58.0 bits)
18
RMS: 0.3674x10(61.2 bits)
GATAN(x) vs GATAN(1/16)+GATAN«x-1/16)/(1+x/16))
test interval: 0.6250x10- 1 through 0.2679
MRE: 0.3899x10- 17
(57.8 bits)
RMS: 0.1436x10- 17
(59.3 bits)
2 ·GATAN(x) vs GATAN(2x/(1-x·x))
test interval: 0.2679 through 0.4142
MRE: 0.3308x10- 17
(58.1 bits)
RMS: 0.1601x10- 17
(59.1 bits)
2 ·GATAN(x) vs GATAN(2x/(1-x·x))
test interval: 0.4142 through 1.0000
MRE: 0.4360x10- 17
(57.7 bits)
18
RMS: 0.9839x10(59.8 bits)
GCOS(x) vs 4·GCOS(x/3)3_3·GCOS(x/3)
test interval: 0.2199x10 2 through 0.2356xl0 2
MRE: 0.4779x10- 17
(57.5 bits)
RMS: 0.1515x10- 17
(59.2 bits)
GCOSH(x) vs C·(GCOSH(x+1)+GCOSH(x-1))
test interval: 3.0000 through 0.7091x103
MRE: 0.4770x10- 17
(57.5 bits)
RMS: 0.1712x10- 17
(59.0 bits)
GCOSH(x) vs Taylor Series expansion of GCOSH(x)
test interval: 0.0000 through 0.5000
MRE: O.3469x10- 17
(58.0 bits)
RMS: 0.1234x10- 17
(59.5 bits)
GCOT(x) vs (GCOT(x/2)2_1)/(2·GCOT(x/2))
test interval: 0.1885xl0 2 through 0.1963x10 2
MRE: 0.7609x10- 17
(56.9 bits)
RMS: 0.2096x10- 17
(58.7 bits)

ELEFUNT Test Results A-7

GEXP(x-2.8125) vs GEXP(x)/GEXP(2.8125)
test interval: 6.9310 through 0.7090x103
MRE: 0.4706x10-- 17
(57.6 bits)
RMS: 0.1391x10- 17
(59.3 bits)
GEXP(x-2.8125) vs GEXP(x)/GEXP(2.8125)
test interval: -3.4660 through -0.6682x103
MRE: 0.4690x10- 17
(57.6 bits)
RMS: 0.1395x10- 17
(59.3 bits)
GEXP(x-0.0625) vs GEXP(x)/GEXP(0.0625)
test interval: -0.2841 through 0.3466
MRE: 0.3469x10-- 17
(58.0 bits)
17
RMS: 0.1384x10(59.3 bits)
GEXP3. (xl. O vs x)
test interval: 0.5000 through 1.0000
The result is exact.
GEXP3. (XSQ1.5 vs XSQ ·x)
test interval: 0.5000 through 1.0000
MRE: 0.3464x10- 17
(58.0 bits)
RMS: 0.1334x10-- 17
(59.4 bits)
GEXP3. (XSQ1.5 vs XSQ ·x)
test interval: 1.0000 through 0.4479x10 103
l\1RE: 0.3464x10- 17
(58.0 bits)
17
RMS: 0.1347x10(59.4 bits)
GEXP3. (xY vs XSQy/2)
test interval: 1.0000 through 0.1000x10 2 for x
-O.1543x103 through 0.1543x103 for y
MRE: 0.3371xlO·· 16
(54.7 bits)
RMS: 0.4759x10- 17
(57.5 bits)
GLOG(x) vs Taylor Series expansion of GLOG(1+y)
test interval: 1-0.1907xlO-5 through 1+0.1907x1o-5
MRE: 0.5771x10- 17
(57.3 bits)
RMS: 0 ..1557xlO-· 17
(59,,2 bits)
GLOG(x) vs GLOG(17x/16)-GLOG(17/16)
test interval: 0.7071 through 0.9375
MRE: O.3501xlO- 17
(58.0 bits)
RMS: 0.1488xlo-- 17
(59.2 bits)
GLOG(x·x) vs 2·GLOG(x)
test interval: 0.1600xl02 through 0.2400x103
MRE: O.:~393xl017
(58.0 bits)
RMS: 0.4781xl0- 18
(60.9 bits)
GLOGIO(x) vs GLOGI0(11x/lO)-GLOG10(11/10)
test interval: 0.3162 through 0.9000
MRE: O.9112x10- 17
(56.6 bits)
RMS: 0.2560xlO- 17
(58.4 bits)

A-~8 TOPS-10/TOPS-20 Common Math Library Reference Manual

GSIN(x) vs 3·GSIN(x/3)-4-GSIN(x/3P
test interval: 0.0000 through 1.5710
MRE: 0.3794xlO- 17
(57.9 bits)
RMS: 0.1394xlO- 17
(59.3 bits)
GSIN(x) vs 3·GSIN(x/3)-4·GSIN(x/3)3
test interval: 0.1885x10 2 through 0.2042x10 2
MRE: 0.5320x10- 17
(57.4 bits)
RMS: 0.1719x10- 17
(59.0 bits)
GSINH(x) vs C·(GSINH(x+1)+GSINH(x-1»
test interval: 3.0000 through 0.7091x103
MRE: 0.5035x10- 17
(57.5 bits)
17
RMS: 0.1730x10(59.0 bits)
GSINH(x) vs Taylor Series expansion of GSINH(x)
test interval: 0.0000 through 0.5000
MRE: 0.3459x10- 17
(58.0 bits)
RMS: 0.2973x10- 18
(61.5 bits)
GSQRT(x ·x)-x
test interval: 0.7071 through 1.0000
MRE: 0.2450x10- 17
(58.5 bits)
18
RMS: 0.6269x10(60.5 bits)
GSQRT(x -x)-x
test interval: 1.0000 through 1.4140
The result is exact.
GTAN(x) vs 2·GTAN(x/2)/(1-GTAN(x/2)2)
test interval: 2.7490 through 3.5340
MRE: 0.6827x10- 17
(57.0 bits)
RMS: O.2028x10- 17
(58.8 bits)
GTAN(x) vs 2-GTAN(x/2)/(l-GTAN(x/2)2)
test interval: 0.1885x10 2 through 0.1963x102
MRE: 0.9834x10- 17
(56.5 bits)
RMS: 0.2760x10- 17
(58.3 bits)
GTAN(x) vs 2-GTAN(x/2)/(1-GTAN(x/2)2)
test interval: 0.0000 through 0.7854
MRE: 0.9663x10- 17
(56.5 bits)
RMS: O.2678x10- 17
(58.4 bits)
GTANH(x) vs (GTANH(x-1/8)+GTANH(1/8»/(1+GTANH(x-l/8)GTANH(1/8»
test interval: 0.1250 through 0.5493
(57.6 bits)
MRE: 0.4684xlO- 17
17
RMS: O.1608xlO(59.1 bits)
GTANH(x) vs (GTANH(x-1/8) +GTANH(1/8) )/(l+GTANH(x-1/8)GTANH(1/8»
test interval: 0.6743 through 2149x10 2
MRE: O.3750xlO-- t7
(57.9 bits)
17
RMS: 0.1621x10(59.1 bits)

ELEFUNT Test Results

A-9

SIN(x) vs 3 SIN(x/3)-4-SIN(x/~))3
test interval: 0.0000 through 1.5710
MRE: 0.1934x10- 7
(25.6 bits)
RMS: 0.5980x1o--s
(27.3 bits)
SIN(x) vs 3-SIN(x/3)-4-SIN(x/3)3
test interval: 0.1885x10 2 through 0.2042x10 2
MRE: 0.2736x10- 7
(25.1 bits)
RMS: O.6923xlO-8
(27.1 bits)
SINH(x) vs C -(SINH(x+1)+SINH(x-1»
test interval: 3.0000 through 0.8803x10 2
MRE: O.3020x10- 7
(25.0 bits)
RMS: 0.7083x10-8
(27.1 bits)
SINH(x) vs Taylor Series expansion of SINH(x)
test interval: 0.0000 through 0.5000
MRE: 0.1479x10-- 7
(26.0 bits)
RMS: 0.1143xlO-s
(29.7 bits)
SQRT(x -x)-x
test interval: 0.7071 through 1.0000
The result is exact.
SQRT(x ·x)-x
test interval: 1.0000 through 1.4140
The result is exact.
TAN(x) vs 2·TAN(x/2)/(l-TAN(x/2)2)
test interval: O.1885xl0 2 through 0.1963xl0 2
MRE: O,3059xlO- 7
(25.0 bits)
RMS: 0.1039xlO-7
(26,5 bits)
TAN (x) vs 2-TAN(x/2)/(1-TAN(x/2)2)
test interval: 2.7490 through 3.5340
MRE: O.2940x10-7
(25.0 bits)
RMS: 0.7439x10-- s
(27.0 hits)
TAN(x) vs 2-TAN(x/2)/(l--TAN(x/2)2)
test interval: 0.0000 through 0.7854
MRE: O.2994xlo--7
(25.0 bits)
7
Rl\IS: 0.1074x10-(26.5 bits)
TANH(x) vs (TANH(x--l/8)+TANH(1/8»/(1+TANH(x-1/8)TANH(1/B»
test interval: 0.1250 through 0.5493
MRE: 0.2020x10-7
(25.6 bits)
RMS: O.6944x10-8
(27.1 bits)
TANH(x) vs (TANH(x-l/8)+TANH(1/8»/(1+TANH(x-1/B)TANH(1/B»
test interval: 0.6743 through O.1040xl0 2
IV1RE: O.2156xlO- 7
(25.5 bits)
HMS: O.6360xlo-- s
(27.2 bits)

Appendix B
Using the Common Math Library with MACRO
Programs
The Math Library was designed to be used mainly by compiler-level languages. The object-time systems of such languages have facilities to handle
error conditions that may occur when a routine from the Math Library is
executed. MACRO programmers must include such facilities in their programs.
There are two facilities necessary for use of the Math Library: a trap handler
and an error handler. The trap handler is needed, since under certain circumstances the Math Library executes floating-point instructions which may
overflow or underflow. In these cases, the library routines expect that the
result will be set to the largest possible number for floating overflow, or set to
zero for underflow. The central processor does not set the results - the overflows and underflows must be detected by the APR trapping system and
interpreted by the trap handler. If the overflow/underflow settings are not
done properly, the math routine in question will very likely return mathematically incorrect results.
The error handler is a general error printout routine. It is called by the Math
Library when the arguments passed to a Math Library routine are out of range
or otherwise incorrect.
Provided with the Math Library are modules for handling APR traps and
properly setting the results (MTHTRP) and for providing error handling and
reporting (MTHDUM). A MACRO program must initialize these modules
before using any other components of the Math Library, as follows:
PUSHJ
PUSHJ

P,%TRPIN##
P,%ERINI##

;INITIALIZE TRAP HANDLER
;INITIALIZE ERROR HANDLER

8-1

Index
A
ABS routine, 9-4
Absolute value
complex, 9-7
double-precision D-floating-point, 9-8
. double-precision G-floating-point, 9-9
double-precision,
D-floating-point, 9-5
G-floating-point, 9-6
integer, 9-3
single-precision, 9-4
Accuracy tests, 1-14
ACOS routine, 6-4
AIMAG routine, 15-4
AINT routine, 11-9
ALOG routine, 3-3
ALOG10 routine, 3-5
AMAXO routine, 14-5
AMAX1 routine, 14-6
AMINO routine, 14-11
AMINI routine, 14-12
AMOD routine, 12-6
ANINT routine, 11-6
Arc cosine
double-precision,
D-floating-point,6-7
G-floating-point, 6-11
single-precision, 6-4
Arc sine
double-precision,
D-floating-point, 6-5
G-floating-point, 6-9
single-precision, 6-3
Arc tangent
double-precision,
D-floating-point, 6-17
G-floating-point, 6-21
single-precision, 6-13

ASIN routine, 6-3
ATAN routine, 6-13
ATAN2 routine, 6-15
Average relative error, 1-14
B

Base-10 logarithm,
double-precision,
D-floating-point, 3-9
G-floating-point, 3-13
single-precision, 3-5

c
CABS routine, 9-7
Calling sequence, 1-13
CCOS routine, 5-21
CDABS routine, 9-8
CDCOS routine, 5-25
CDEXP routine, 4-11
CDLOG routine, 3-17
CDSIN routine, 5-23
CDSQRT routine, 2-11
CEXP routine, 4-9
CEXP2. routine, 4-22
CEXP3. routine, 4-34
CFDV routine, 15-7
CFM routine, 15-6
CGABS routine, 9-9
CGCOS routine, 5-29
CGEXP routine, 4-13
CGLOG routine, 3-19
CGSIN routine, 5-27
CGSQRT routine, 2-13
CLOG routine, 3-15
CMPL.C routine, 10--23
CMPL.D routine, 10--21
CMPL.G routine, 10--22

Index-1

CMPL.I routine, 10-19
CMPLX routine, 10-20
Cody, W. J., 1-15, A-I
Cody and Waite, Software Manual for
Elementary Functions, 5-32, 5-34,
5-36,5-38,5-40
Complex,
absolute value, 9-7
conjugate, 15-5
conversion,
complex to complex, 10-23
cosine, 5-21
data types, 1-12
division, 15-7
double-precision D-floating-point, 1-12
absolute value, 9-8
cosine, 5-25
exponential, 4-11
natural logarithm, 3-17
sine, 5-23
square root, 2-11
double-precision G-floating-point, 1-12
absolute value, 9-9
cosine, 5-29
exponential, 4-13
natural logarithm, 3-19
sine, 5-27
square root, 2-13
exponential, 4-9
exponentiation,
complex to complex, 4-34
complex to integer, 4-22
multiplication, 15-6
natural logarithm, 3-15
number,
imaginary part, 15-4
real part, 15-3
product, 15-6
quotient, 15-7
sine, 5-19
square root, 2-9
Computer Approximations,
Hart et.al., 3-4, 3-6, 6-14, 6-18, 6-22
CONJ routine, 15-5
Conjugate
complex, 15-5
Conversion
complex to complex, 10-23
double-precision,
D-floating-point to complex, 10-20
D-floating-point to G-floating-point,
10-17, 10-18
D-floating-point to integer, 10-5

2-lndex

Conversion (Cont.)
D-floating-point to single-precision, 10-9
G~floating-point to complex, 10-22
G-floating-point to D-floating-point,
10-13, 10-14
G-floating-point to integer, 10-6
G-floating-point to single-precision,
10-10
integer,
to complex, 10-19,
to double-precision D-floating-point,
10-11
to double-precision G-floating-point,
10-15
to single-precision, 10-7, 10-8
single-precision,
to complex, 10-20
to double-precision D-floating-point,
10-12
to double-precision G-floating-point,
10-16
to integer, 10-3, 10-4
COS routine, 5-7
COSD routine, 5-9
COSH routine, 7-4
Cosine,
complex, 5-21
double-precision D-floating-point, 5-25
double-precision G-floating-point, 5-29
double-precision,
D-floating-point, 5-13
G-floating-point, 5-17
single-precision, 5-7, 5-9
COTAN routine, 5-33
Cotangent,
double-precision,
D-floating-point, 5-37
G-floating-point, 5--41
single-precision, 5-33
Coveyan, R. R. and MacPherson,
R. D., Journal of the ACM, #14, 8-4
CSIN routine, 5-19
CSQRT routine, 2-9

DABS routine, 9-5
DACOS routine, 6-7
DASIN routine, 6-5
DATAN routine, 6-17
DATAN2 routine, 6-19

Data types, 1-10
complex, 1-12
double-precision,
D-floating-point, 1-11
G-floating-point, 1-11
integer, 1-10
single-precision, 1-10
DBLE routine, 10-12
DCOS routine, 5-13
DCOSH routine, 7-7
DCOTAN routine, 5-37
DDIM routine, 12-11
DEXP routine, 4-5
DEXP2. routine, 4-18
DEXP3. routine, 4-28
DFLOAT routine, 10-11
D-floating-point,
absolute value, 9-5
arc cosine, 6-7
arc sine, 6-5
arc tangent, 6-17
base-l0 logarithm, 3-9
conversion,
to complex, 10-21
to G-floating-point, 10-17, 10-18
to integer, 10-5
to single-precision, 10-9
cosine, 5-13
cotangent, 5-37
data type, 1-11
exponential, 4-5
exponentiation,
to D-floating-point, 4-28
to integer, 4-18
hyperbolic cosine, 7-7
hyperbolic sine, 7-5
hyperbolic tangent, 7-12
maximum of a series, 14-7
minimum of a series, 14-13
natural logarithm, 3-7
polar angle of two points, 6-19
positive difference, 12-11
product, 12-3
remainder, 12-7
rounding,
to D-floating-point, 11-7
to integer, 11-4
sine, 5-11
square root, 2-5
tangent, 5-35
transfer of sign, 13-5
truncation, 11-10
DIM routine, 12-10

DINT routine, 11-10
Division, complex, 15-7
DLOG routine, 3-7
DLOGI0 routine, 3-9
DMAXI routine, 14-7
DMINI routine, 14-13
DMOD routine, 12-7
DNINT routine, 11-7
Double precision,
data types, 1-11
D-floating-point, 1-11
absolute value, 9-5
arc cosine, 6-7
arc sine, 6-5
arc tangent, 6-17
base-l0 logarithm, 3-9
conversion,
to complex, 10-21
to G-floating-point, 10-17, 10-18
to integer, 10-5
to single-precision, 10-9
cosine, 5-13
cotangent, 5-37
exponential, 4-5
exponentiation,
to D-floating-point, 4-28
to integer, 4-18
hyperbolic cosine, 7-7
hyperbolic sine, 7-5
hyperbolic tangent, 7-12
maximum of a series, 14-7
minimum of a series, 14-13
natural logarithm, 3-7
polar angle of two points, 6-19
positive difference, 12-11
product, 12-3
remainder, 12-7
rounding,
to D-floating-point, 11-7
to integer, 11-4
sine, 5-11
square root, 2-5
tangent, 5-35
transfer of sign, 13-5
truncation, 11-10
G-floating-point, 1-11
absolute value, 9-6
arc cosine, 6-11
arc sine, 6-9
arc tangent, 6-21
base-l0 logarithm, 3-13
conversion,
to complex, 10-22

Index-3

Double Precision (Cont.)
to D-floating-point, 10-13, 10-14
to integer, 10-6
to single-precision, 10-10
cosine, 5-17
cotangent, 5-41
exponential, 4-7
exponentiation,
to G-floating-point, 4-31
to integer, 4-20
hyperholic cosine, 7-10
hyperbolic sine, 7-8
hyperbolic tangent, 7-13
maximum of a series, 14-8
minimum of a series, 14-14
natural logarithm, 3-11
polar angle of two points, 6-23
positive difference, 12-12
product, 12-4
remainder, 12-8
rounding,
to G-floating-point, 11-8
to integer, 11-5
sine, 5-15
square root, 2-7
tangent, 5-39
transfer of sign, 13-6
truncation, 11-11
DPROD routine, 12-3
DSIGN routine, 13-5
DSIN routine, 5-11
DSINH routine, 7-5
DSQRT routine, 2-5
DTAN routine, 5-35
DTANH routine, 7-12
DTOG routine, 10-17
DTOGA routine, 10-18
E

ELEFUNT tests, 1-15, A-I
Entry points, 1-13
Error,
maximum relative (MRE), 1-14
average relative (RMS), 1-14
EXP routine, 4-3
EXPI. routine, 4-15
EXP2. routine, 4-16
EXP3. routine, 4-25
Exponential,
complex, 4-9
double-precision D-floating-point, 4-11
double-precision G-floating-point, 4-13

4--lndex

Exponential (Cont.)
double-precision,
D-floating-point, 4-5
G-floating-point, 4-7
single-precision, 4-3
Exponentiation,
complex to complex, 4-34
complex to integer, 4-22
D-floating-point to D-floating-point, 4-28
D-floating-point to integer, 4-18
G-floating-point to G-floating-point, 4-31
G-floating-point to integer, 4-20
integer to integer, 4-15
single-precision to integer, 4-16
single-precision to single-precision, 4-25
F

FLOAT routine, 10-8
Functions,
math library, 1-3
G

GABS routine, 9-6
GACOS routine, 6-11
GASIN routine, 6-9
GATAN routine, 6-21
GATAN2 routine, 6-23
GCOS routine, 5-17
GCOSH routine, 7-10
GCOTAN routine, 5-41
GDB.n routine, 10-16
GDIM routine, 12-12
GEXP routine, 4-7
GEXP2. routine, 4-20
GEXP3. routine, 4-31
GFL.n routine, 10-15
G-floating-point,
absolute value, 9-6
arc cosine, 6-11
arc sine, 6-9
arc tangent, 6-21
base-10 logarithm, 3-13
conversion,
to complex, 10-22
to D-floating-point, 10-13, 10-14
to integer, 10-6
to single-precision, 10-10
cosine, 5-17
cotangent, 5-41
data type, 1-11
exponential, 4-7

G-floating-point (Cont.)
exponentiation,
to G-floating-point, 4-31
to integer, 4-20
hyperbolic cosine, 7-10
hyperbolic sine, 7-8
hyperbolic tangent, 7-13
maximum of a series, 14-8
minimum of a series, 14-14
natural logarithm, 3-11
polar angle of two points, 6-23
positive difference, 12-12
product, 12-4
remainder, 12-8
rounding, 11-8
to G-floating-point, 11-8
to integer, 11-5
sine, 5-15
square root, 2-7
tangent, 5-39
transfer of sign, 13-6
truncation, 11-11
GFX.n routine, 10-6
GINT. routine, 11-11
GLOG routine, 3-11
GLOG 10 routine, 3-13
GMAX1 routine, 14-8
GMIN1 routine, 14-14
GMOD routine, 12-8
GNINT. routine, 11-8
GPROD. routine, 12-4
GSIGN routine, 13-6
GSIN routine, 5-15
GSINH routine, 7-8
GSN.n routine, 10-10
GSQRT routine, 2-7
GTAN routine, 5-39
GTANH routine, 7-13
GTOD routine, 10-13
GTODA routine, 10-14
H

Hart et.al., Computer Approximations,
3-4,3-6,6-14,6-18,6-22
Hyperbolic cosine,
double-precision,
D-floating-point, 7-7
G-floating-point, 7-10
single-precision, 7-4
Hyperbolic sine,
double-precision,
D-floating-point, 7-5
G-floating-point,7-8

Hyperbolic sine (Cont.)
single-precision, 7-3
Hyperbolic tangent,
double-precision,
D-floating-point, 7-12
G-floating-point, 7-13
single-precision, 7-11
I

lABS routine, 9-3
IDIM routine, 12-9
IDINT routine, 10-5
IDNINT routine, 11-4
IFIX routine, 10-3
IGNIN. routine, 11-5
Imaginary part of a complex number, 15-4
INT routine, 10-4
Integer,
absolute value, 9-3
conversion,
to complex, 10-19
to D-floating-point, 10-11
to G-floating-point, 10-15
to single-precision, 10-7, 10-8
data type, 1-10
exponentiation, 4-15
maximum, 14-3, 14-4
minimum, 14-9, 14-10
positive difference, 12-9
remainder, 12-5
transfer of sign, 13-3
ISIGN routine, 13-3
J

Journal of the ACM, #14,
Coveyan, R. R. and MacPherson, R. D., 8-4
K

Knuth, D. E., Seminumerical Algorithms, 8-4
L

Logarithm, see natural logarithm,
base-10 logarithm
LSB (least significant bit) error distribution,
1-15
M

MACRO programs, using the math
library with, B-1

Index-5

Math library,
functions, 1-3
restrictions, 1-8
with MACRO programs, B-1
Mathematical names, 1-9
Mathematical symbols, 1-9
MAXO routine, 14-3
MAXI routine, 14-4
Maximum of a series,
double-precision,
D-floating-point, 14-7
G-floating-point, 14-8
integer, 14-3, 14-4
single-precision, 14-5, 14-6
Maximum relative error, 1-14
MINO routine, 14-9
MINI routine, 14-10
Minimum of a series,
double-precision,
D-floating-point, 14-13
G-floating-point, 14-14
integer, 14-9, 14-10
single-precision, 14-11, 14-12
MOD routine, 12-5
MRE (maximum relative error), 1-14
Multiplication, complex, 15-6
N

Names, mathematical, 1-9
Natural logarithm
complex, 3-15
double-precision D-floating-point, 3-17
double-precision G-floating-point, 3-19
double-precision,
D-floating-point, 3-7
G-floating-point, 3-11
single-precision, 3-3
Newton-Raphson method, 2-4, 2-6, 2-8
NINT routine, 11-3
p

Polar angle of two points,
double-precision,
D-floating-point, 6-19
G-floating-point, 6-23
single-precision, 6-15
Positi ve difference,
double-precision,
D-floating-point, 12--11
G-floating-point, 12-12
integer, 12-9
single-precision, 12-10

6-lndex

Precision, 1-10
Product,
complex, 15-6
double-precision,
D-floating-point, 12-3
G-floating-point, 12-4

Q
Quotient, complex, 15-7

R
RAN routine, 8-3
Random number generator, 8-3
spectral test with, 8-3
with shuffiing, 8-5
Random number seed,
saving, 8-7
setting, 8-6
RANS routine, 8-5
REAL routine, 10-7
REAL.C routine, 15-3
Real part of a complex number, 15-3
Register usage, 1-13
Relative error
average (RMS), 1-1.4
maximum (MRE), 1-14
Remainder,
double-precision,
D-floating-point, 12-7
G-floating-point, 12-8
integer, 12-5
single-precision, 12-6
Restrictions, math library, 1-8
Return location, 1-13
RMS (root mean square), 1-14
Root mean square (RMS), 1-14
Rounding,
double-precision,
D-floating-point,
to D-floating-point, 11-7
to integer, 11-4
G-floating-point,
to G-floating-point, 11-8
to1nteger, 11-5
single-precision,
to integer, 11-3
to single-precision, 11-6

s
Saving random number seed, 8-7
SAVRAN routine, 8-7

Seminumerical algorithms,
Knuth, D. E., 8-4
SETRAN routine, S--6
Setting random number seed, S--6
SIGN routine, 13-4
Sign, transfer,
double-precision,
D-floating-point, 13-5
G-floating-point, 13-6
integer, 13-3
single-precision, 13-4
SIN routine, 5--3
SIND routine, 5--5
Sine,
complex, 5--19
double-precision D-floating-point, 5-23
double-precision G-floating-point, 5-27
double-precision,
D-floating-point, 5-11
G-floating-point, 5-15
single-precision, 5-3, 5-5
Single-precision,
absolute value, 9--4
arc cosine, 6-4
arc sine, 6-3
arc tangent, 6-13
base-10 logarithm, &-5
conversion,
to complex, 10--20
to D-floating-point, 10--12
to G-floating-point, 10--16
to integer, 10--3, 10-4
cosine, 5-7, 5-9
cotangent, 5--33
data type, 1-10
exponential, 4-3
exponentiation,
to integer, 4-16
to single-precision, 4-25
hyperbolic cosine, 7-4
hyperbolic sine, 7-3
hyperbolic tangent, 7-11
maximum of a series, 14-5, 14-6
minimum of a series, 14-11, 14-12
natural logarithm, 3-3
polar angle of two points, 6-15
positive difference, 12-10
remainder, 12-6

Single-precision (Cont.)
rounding,
to integer, 11-3
to single-precision, 11-6
sine, 5-3, 5-5
square root, 2-3
tangent, 5-31
transfer of sign, 13-4
truncation, 11-9
SINH routine, 7-3
SNGL routine, 10--9
Software Manual for Elementary Functions,
Cody and Waite, 5--32, 5-34, 5-36, 5-38,
5-40
Spectral test with random number generator,
8-3

SQRT routine, 2-3
Square root,
complex, 2-9
double-precision D-floating-point, 2-11
double-precision G-floating-point, 2-13
double-precision,
D-floating-point, 2-5
G-floating-point, 2-7
single-precision, 2-3
Symbols, mathematical, 1-9
T

TAN routine, 5-31
Tangent,
double-precision,
D-floating-point, 5-35
G-floating-point, 5-39
single-precision, 5-31
TANH routine, 7-11
Test interval, 1-14
Tests, accuracy, 1-14
Transfer of sign,
double-precision,
D-floating-point, 13-5
G-floating:.point, 13-6
integer, 13-3
single-precision, 13-4
Truncation,
double-precision,
D-floating-point, 11-10
G-floating-point, 11-11
single-precision, 11-9

Index-7

TOPS-10/TOPS-20
Common Math Library
Reference Manual
AA-M400A-TK
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