NBSIR 80-2138

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national bureau

07 STANDARDS

JUUBBART SEP 1 6 1981

MULTIPRECISION COMPUTING AT NBS:

YESTERDAY, TODAY, AND TOMORROW

no. $0-3/32

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Daniel W. Lozier

Mathematical Analysis Division

Center for Applied Mathematics

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

October 1 980

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology, and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

I. Introduction 1

II. The Four Basic Arithmetic Operations in Multiprecision • 5

III. Some Representative Examples of Software

for Multiprecision Computation 9

IV. Use of Brent’s MP Package with Crary' s Augment

Precompiler on the NBS Univac 1108 Interactive System.... 19

V. Summary and Conclusions........ 26

References 28

Appendix A

Appendix B

Appendix C
ABSTRACT

Multiprecision computing is a technique by which arithmetic operations may be

performed on a computer to precision levels that are higher than the directly

supported single and double precisions. The last ten years have seen the development of portable Fortran software of very high quality that essentially

duplicates all the capabilities of standard Fortran, so that an existing standard Fortran program can be re-executed to arbitrarily high precision. In

this paper some of the design techniques for such software, which have evolved

at NBS and elsewhere, will be discussed. Methods for using the software presently available at NBS will be described, and a complete example will be

given. Directions for further extensions and improvements will be indicated.

ii
I. INTRODUCTION

When setting out to perform a calculation on a computer, a person is

faced with a very limited set of possible arithmetic capabilities. - One has to

choose between single precision, which represents numbers on various computers

to a significance in the approximate range from 6 to 15 figures, or double

precision, which gives up to approximately 30 significant figures. Even on

computers with quadruple precision, about 35 significant figures is the upper

limit. The situation with regard to the range of numbers representable in a

computer is similar. It varies from approximately 75 orders of magnitude up

to several thousand orders of magnitude (nearly 10000 in the new VAX-11

architecture of the Digital Equipment Corporation). Much intellectual and

engineering effort has gone into optimizing, according to greatly different

criteria by different groups and organizations, the final choice of what

arithmetic capabilities will be implemented in hardware. Still, the result is

a limited set of possibilities that does not provide the means to do every

calculation that we would like to do. Perhaps the people who have a genuine

need for more extensive arithmetic capabilities should make a greater effort

to influence the computer designers.

If the hardware, or more correctly the computer environment presented to

the user, fails to provide for a computational need then our recourse is to

programs. This is true, in particular, if the arithmetic capabilities

provided by the computer environment are not sufficient for a particular computational task. The relative simplicity of algorithms for the four basic

multiprecision operations of addition, subtraction, multiplication and division has undoubtedly led to many undocumented programs as well as to programs

that were constructed for use in specific applications. Our interest here is

in software, which is distinguished for our purposes from the more general

1
term "programs" by the criteria of reliability, robustness, correctness,

portability, maintainability, availability of documentation, ease of use, completeness, and other desirable qualities. Completeness, in particular, is

important since a moment's reflection brings to mind the necessity of having

available all of the peripheral functions, such as input-output, type conversion, the mathematical functions, and so on. This leads to a software package

of considerable size, and not one that the casual programmer with only a

limited commitment is likely to produce.

Fortunately, software possessing most of the desirable properties listed

above has been developed during the past decade. A later section of this report will describe some representative contributions that have been made to

this development. The MP package of R. P. Brent [1] emerges as the most advanced product.

As in the case of other packages, every arithmetic or other operation

performed by Brent's package requires a subroutine call (the package is written in FORTRAN and is intended for use only in the FORTRAN environment).

Thus, the function

F(x) 3 /I x + cos x^ - ex

could be evaluated by the FORTRAN statement

F = SQRT(2.0)*X + C0S(X**2) - EXP(X)

whereas the code

CALL MPCRM(2. 0 , TWO) Convert 2 to MP form

CALL MPSQRT( TWO, TEMP) Compute n

CALL MPMUL(TEMP ,X,F) •• /l X

CALL MPMUL(X,X,TEMP) » X

2

CALL MP COS (TEMP, TEMP) » cos X

2

CALL MPADD(F,TEMP,F) •• /

x + COS X‘

2
CALL MPEXP (X ,TEMP ) Compute ex

CALL MP SUB ( F , TEMP , F ) " F(x)

or something similar would be required to use Brent's package. Here TWO,

TEMP, X and F are the names of FORTRAN arrays in which multiprecision numbers

are stored. It was to avoid such lists of operations, reminiscent of assembly

language code, that FORTRAN was developed in the early days of computers. But

FORTRAN was not designed to process MP variables , which are not accommodated

by any standard data type of FORTRAN.

The solution that has evolved for this problem is to extend FORTRAN in

such a way as to introduce a new data type for MP numbers. This involves

writing a FORTRAN-like compiler, an activity of a completely different kind

than writing the MP package itself. Two such compilers have emerged: the

FORPAK compiler of D. J. Orser [9] and the AUGMENT compiler of F. D. Crary

[4]. Both are written in FORTRAN and produce FORTRAN object code, which is

subsequently processed by an ordinary FORTRAN compiler. For this reason the

special compilers are known as precompilers. In each case the precompiler is

informed about all necessary details of the implementing package for the special computer arithmetic by means of an input "description deck." Thus, they

can be used to facilitate the use of any special arithmetic for which a valid

description deck can be written.

The precompilers are capable of "parsing" general FORTRAN-like expressions. For example, precompilation of the type declaration

MULTIPLE TWO ,X,F

and the statements

TWO = 2.0

F = SQRT(TW0)*X + C0S(X**2) - EXP(X)

*The representation of MP numbers will be discussed in the next section of

this report.

3
would result in automatic generation of the temporary variable and 8 lines of

code given above. One of the precompilers, AUGMENT, is being used widely with

Brent's MP package. An example of the use of Crary' s precompiler with Brent's

package on the NBS central computer will be given in this report. The use of

Orser's precompiler with another multiprecision package, the Super Precision

Package of Wyatt [9], will be discussed briefly.

4
II. THE FOUR BASIC ARITHMETIC OPERATIONS IN MULTIPRECISION

We have already remarked that the basic algorithms for addition, subtraction, multiplication and division are simple. They are straightforward implementations of the elementary arithmetic methods taught by the school system,

easily done in FORTRAN using integer arithmetic.

Many details on multiprecision arithmetic are given in Knuth [6] and references listed there. We give only a brief overview here. The most common

approach, and the one used in Brent's MP package, is the following. Some number of consecutive integer storage locations (or words), say n + 2, is decided

upon for the representation of MP numbers. A real number to be represented in

MP form is first expressed in a notation like scientific notation:

a = ± ( 0 .a^a2 a3 ...)be .

Here b > 0 is an integer (the base or radix) whose optimal value depends on

the computer wordlength, and the a^ are the (unique) digits of a to the base

b. That is, 0 _< a^ < b for each i. The representation is made unique by requiring the first digit to be nonzero so that we have 1 _< a.\ < b. The representation is then said to be normalized; 0.aia2a3 ... is called the (normalized) fraction of a and e is called the exponent of a. The MP form of a is

stored by placing the sign (equal to -1 or +1) in word 1, the exponent in word

2, and the first n digits of the normalized fraction in words 3 through n + 2.

The final digit may be modified according to some rounding rule. The representation of zero in MP form is nonunique; word 1 is set equal to zero and

words 2 through n + 2 are undetermined.

Now it becomes clear how the basic arithmetic operations can be coded

using FORTRAN integer arithmetic. After taking into account any necessary

shifting of the fractions, the sum of two fractions is obtained by ordinary

5
digit-by-digit "add and carry” operations. The difference of two fractions is

obtained by ordinary digit- by-digit "borrow and subtract" operations. These

operations are 0(n) in number and this order cannot be improved.

The full multiplication of two fractions requires each digit of one fraction to be multiplied by every digit of the other, together with appropriate

shifting and scaling operations, in order to produce the 2n-digit product.

Some practical savings can be obtained by not computing all of this, since

only n digits of the result are to be stored, but this procedure still requires O(n^) operations. Multiplication algorithms of lower order exist but

have not found their way into general purpose multiprecision packages, for the

reason that excessively large values of n are required before actual savings

in execution time can be achieved [1,9]. The "fast" algorithms require more

"overhead" which tends to dominate the multiplication process for moderate

values of n. As an indication of the execution times that can be expected,

for Brent's package on the UNIVAC 1108 the multiply time with n = 10 is

approximately 2 msec and with n = 14 it is approximately 2.8 msec. These

values of n correspond approximately to 43 and 62 decimal places of precision.

For comparison, the hardware double precision (18 decimals) multiply time is

5y sec.

However, one special multiplication algorithm is often implemented. It

is found, for example, in Brent's package. When one of the operands is a

FORTRAN integer then multiplication requires only 0(n) operations. But it is

interesting to note that this capability cannot be fully exploited by the precompiler. If the FORTRAN statement is

X = A * I

where X, A are multiprecision variables and I is a FORTRAN integer variable,

then the desired 0(n) multiplication is performed. Alternatively, suppose I

6
is a multiprecision variable with every digit of its fraction equal to zero

after the first digit. Then I is representable as a FORTRAN integer but the

precompiler cannot know to use the 0(n) multiplication. We conclude that, in

this case, writing FORTRAN code with mixed mode expressions is preferable to

the common programming practice of matching the types of all elements in a

FORTRAN expression.

Division is also an O(n^) operation, whether it is done by (i) reciprocating the denominator by use of Newton* s method followed by multiplication by

the numerator or (ii) the school, or divide-and-correct , method. Brent's

package uses method (i) but other programs, such as Wyatt's, use method (ii).

As in the case of multiplication, "fast" division methods have not been used

in general-purpose multiprecision software, except for the special case when

the divisor is typed as a FORTRAN integer. This special case has order n and

is exploitable by the precompiler only when the FORTRAN source statement mixes

multiprecision and integer types of variables.

We conclude this section with remarks on the optimal value of the base b

of representation of MP numbers. When two fractions are multiplied, each

digit of one is multiplied by every digit of the other. Since the largest

value of a digit is b-1, it is clear that ( b— 1

) ^ must not overflow a FORTRAN

integer storage location. Thus, there is an upper limit on the allowable bitlength of digits equal to approximately a half-wordlength. The actual upper

limit for b in Brent's software is specified by the condition that 8b2 -l may

not exceed the largest machine-representable integer.

For storage reasons we want b to be as large as possible. This suggests

that b should be a power of 2. However, if a great deal of input and output

is to be done, then a power of 10 might be preferable because of the cost of

conversion between binary and decimal representations. The following table,

7
taken from Brent's documentation [2] of the MP package, shows optimal powerof-2 and power-of-10 choices for b as a function of wordlength:

Wordlength Power-of-2 Choice Power-of-10 Choice

( 48

36

32

24

18

16

12

bits 4194304 1000000

bits 65536 100000

bits 16384 100000

bits 1024 1000

bits 128 100

bits 64 10

bits 16 10

8
III. SOME REPRESENTATIVE EXAMPLES OF SOFTWARE

FOR MULTIPRECISION COMPUTATION

Multiprecision computing can be viewed as having reached a fairly advanced level of development in Brent's package, particularly when used in combination with a precompiler such as Crary's. Other forms of special computer

arithmetic, less related to floating-point computation and to the FORTRAN language, have also evolved but will not concern us here. In this section we

will describe briefly several representative examples of earlier multiprecision work. Then we will describe Brent's package in somewhat more detail.

Lawson's Q-Precision Package

Development of this package was begun in 1964 or even earlier by C. L.

Lawson at Jet Propulsion Laboratories. Originally for the IBM 7094 and later

for the UNIVAC 1108, early versions were written mostly in assembly language.

These evolved into the present version in which FORTRAN is used extensively,

but the core of the program remains in assembly language. The package is limited to triple and quintuple precision with no extension of the exponent

range. It includes many elementary mathematical functions, since it was used

to test the accuracy of the FORTRAN function library, as well as a Gauss elimination subroutine for solving linear systems. Its limited design goals permitted the use of function approximations that have an intrinsic precision

limit associated with them (Chebyshev and minimax approximations). Such

approximations are favored for efficiency but are not suitable for a general

multiple-precision capability. The FORTRAN that was used contains nonstandard

expressions. The Q-precision package suffers badly from lack of portability

but was never viewed as general-purpose software. It was never published in

9
the open literature. Nevertheless, it serves as an excellent early example

because of its successful applications on two different computers.

Peavy* s Package

Like Lawson, B. A. Peavy of NBS began to develop a multiprecision package

in the 1960's. But Peavy' s package is coded entirely in FORTRAN. However, it

lacks portability because it uses nonstandard FORTRAN expressions and was designed with only the UNIVAC 1108 in mind. It includes addition, subtraction,

multiplication and division; square root, sine, cosine, arctangent, exponential and logarithm; double precision to multiple precision conversion and the

reverse conversion; comparison and printing capabilities. The package uses

from 3 to 20 words per multiprecision variable, the precise number being set

by the statement CALL ISTART(N). The package stores 10 decimal digits per

word, with the final 5 digits of the last word representing the exponent.

Thus precisions from 25 to 195 in steps of 10 are provided, with the exponent

(to the base 10) ranging from -4999 to 4999. Changing from one precision to

another is not facilitated by this choice of representation, which can be a

drawback for the use of the package. The package has been used by I. A.

Stegun and R. Zucker of the Center for Applied Mathematics of NBS to test subroutines for the sine, cosine, exponential, hyperbolic cosine integrals, and

to provide check values to 35 significant figures for these functions. Unfortunately, no publication describing Peavy' s package was ever prepared. An

informal write-up does exist, however.

Maximon's Package [8]

The motivation for development of this package, which appeared in 1971,

was somewhat different from Lawson's and Peavy' s. L. C. Maximon, an NBS

10
physicist, needed higher precision than was available on the UNIVAC 1108 in

order to compute "cross sections for the scattering of high energy electrons

and ... protons from nuclei." [8, p2]. The theoretical physicist/applied

mathematician's point of view is well stated by Maximon: "Although analytical

methods may be developed for a particular problem, the most direct and generally applicable technique is simply to perform the pertinent part of the

calculation (that part involving the cancellation with resultant loss of significant figures) with however many significant figures are needed in order

that the final result have the desired number of significant figures." [8,

pp2,3]. Maximon also recognized two other important applications of multiprecision arithmetic, namely the construction of multiprecision function routines

for the purpose of testing single, double or other fixed precision library

routines, and the computation of the special functions of mathematical physics

to arbitrary precision.

Maximon' s package is written entirely in FORTRAN with a view toward portability. It provides for addition, subtraction, multiplication, division, and

division by an integer; conversion from integer, single or double precision

variables to extended precision and the reverse conversions to double precision; and output printing. The working precision, number of base digits per

word, and the base (2,8,10 or 16) are specified by the user by calling an initialization routine, PRM(NDG,MDW,NBRT) . Parameters derived from these inputs

are communicated to the working subroutines via COMMON. Appropriate values

for NDG, NDW and NBRT depend on the wordlength of the computer and the declared DIMENSION lengths of the multiprecision number arrays. The ability to

change these values at any time during the execution of the user* s program,

within the operable limits, is a notable feature. It was absent in Lawson's

and Peavy' s packages but is present in the packages of Wyatt and Brent. The

11
change is made by calling the initialization subroutine with the desired new

parameter values.

Subroutines to evaluate selected mathematical functions were added later

to the package. These were used to test the accuracy of the UNIVAC 1108

Fortran Library [7],

A major design problem for any multiprecision package is how to handle

input and output. The problem does not stem from the requirement for base

conversion between incommensurate number bases, although accurate and fast

algorithms are not completely straightforward when numbers of all sizes are to

be accommodated. The problem is how to manage the physical arrangement of

numbers to be read in or printed out. In FORTRAN the FORMAT statement exists

for this purpose. FORMAT statements require interpretive processing at runtime, a task of greater magnitude in its full generality than any multiprecision package designer has yet undertaken, as far as this author is aware.

Consequently, simplified forms are found in all of the multiprecision packages

considered here.

Input is relatively simple; some form of free-form input format is the

usual solution. This, of course, requires character processing as well as numerical processing. Maximon does not provide for number input except by way

of conventional FORTRAN variables which can be converted (in their restricted

precision) to multiple-precision form. This was reasonable for the types of

applications he had in mind. The output of a single multiprecision number is

usually done by encoding a character string which then is printed using a

FORTRAN format of the form nAl. This may have to be continued on several

lines if the precision is very high. Maximon considered the output problem

with some care and came essentially to this solution. Annoyingly, the exponent, although printed in decimal form, refers to the internal number base

12
which could be other than 10. He provided for grouping the digits to improve

readability and for specifying the number of digits to print before the decimal point. Changing these specifications is cumbersome and may require recompilation of one subroutine. The packages of Wyatt and Brent are more flexible

in this regard.

Finally, we note that Maximon' s package includes no detection of underflow or overflow. The handling of errors in general is another difficult

design question which has received much attention from later multiprecision

software designers.

Wyatt* s Package [9]

W. T. Wyatt, Jr. of Harry Diamond Laboratories is a physicist. Like

Maximon, he needed higher precision than was (or is now) available on conventional computers. In an ambitious undertaking he produced portable FORTRAN

software for "Super Precison" computing that essentially duplicated all of the

capabilities of the FORTRAN language. An unpublished document describing his

package is dated March 21, 1973. In addition to the four basic arithmetic operations for Super Precision operands, and Super Precision mixed with integer

operands, he provided for complex arithmetic with Super Precision real and imaginary parts. Both the real and the complex mathematical functions found in

the 1966 FORTRAN standard language were included, as were the so-called "intrinsic functions" —MOD, ABS, INT, MIN, MAX, etc. Wyatt* s motivation was his

conviction that every existing FORTRAN program should be convertible for use

with Super Precision arithmetic. This inevitably led to the idea of a precompiler, for the reasons we have already discussed. Without a precompiler,

*See American Standard FORTRAN, X3. 9-1966, published by American Standards

Association, Inc. , now superseded.

13
every arithmetic expression would have to be "parsed" by a programmer and recoded, operation after operation. A fruitful collaboration with D. J. Orser

of the Applied Mathematics Division of NBS (now the Center for Applied Mathematics) resulted in Orser* s FORPAK precompiler [9],

Before discussing FORPAK, we describe some of the design features of the

Super Precision Package. By means of a call on the initializing subroutine,

ZSETUP(I,J,K) , the user selects the number of digits I for the representation

of multiprecision numbers; the number base J from the set 2, 3,..., 16; and

specifies the number of bits K in the computer integer word. The software determines the minimum number of words, NF, needed to meet or exceed the requested number of digits. The number of digits per word is determined so as to

minimize the amount of storage used. Accordingly, Wyatt's package incurs the

overhead of unpacking and repacking operations for every multiplication so as

to avoid the necessity of wasting more than half of every word. After the

call to ZSETUP, computation proceeds using approximately rounded arithmetic,

"approximately" meaning that something less than the full double length product of two multiprecision fractions is formed during multiplication.

Wyatt* s package permits the user to exercise additional control over its

operation by means of global variables in labelled COMMON blocks. For example, alternative rounding strategies can be selected. Another example is

"clip mode". In clip mode the excess digits, beyond digit I, are set to zero

after every arithmetic operation and rounding takes place in digit I instead

of in the final digit of word NF. This gives the user maximum control over

the precision. The desirability of this feature for error analysis has been

emphasized by T. E. Hull [5].

14
Wyatt's package includes an error-handling and traceback capability, in

contrast to the previous packages considered here. When an error is detected, such as an attempted division by zero or a negative argument for the

square root function, the subroutine that detected the error prints a message

and calls the traceback subroutine. The traceback subroutine prints the chain

of subroutine calls through the Super Precision Package that led to the error.

Execution is then terminated, but the user can easily modify the traceback

subroutine to perform a "standard fixup and return".

As previously mentioned, Orser's precompiler was developed with Wyatt's

package in mind. Nevertheless, it is, like Crary' s precompiler, a general

precompiler suitable for use with any new data type for FORTRAN which has a

suitable supporting software package. Orser and Wyatt had as a goal to minimize the number of changes necessary to transform an existing conventional

FORTRAN program into a program making valid use of the new SUPER PRECISION

data type. This goal implied that multiprecision variables appearing in

input/output statements should be treated automatically. Accordingly, a

method was developed to analyze the original FORMAT statements. Let us consider only the more interesting case of output formats. If no Super Precision

variable appeared in an output list, the corresponding FORMAT statement was

not altered and the output operations proceeded identically in both the original and transformed programs. On the other hand, suppose a Super Precision

variable does appear. Then the FORMAT statement is divided into substatements

by means of the numeric designators which correspond to Super Precision variables in the transformed program. These substatements are processed like ordinary FORMAT statements (they involve no nonstandard FORTRAN variables), with

a special fixed form of numerical output of the Super Precision variables in

between the output they generate. Thus, all output headings from the original

15
program are preserved. Each Super Precision number is printed starting on a

new line and continuing onto additional lines as necessary. In contrast to

the current arrangement with the Brent-Crary software, no programmer intervention is required.

Details of usage of the Wyatt-Orser software will not be given here. The

ease of its usage is comparable to the Brent-Crary software which is described

in detail in the next section of this report. Interested individuals are invited to contact either this author or D. J. Orser.

Brent 's Package [1]

The design of Brent’s package has already been discussed in some detail.

It includes most features of Wyatt' s package with some exceptions such as

"clip mode". It has quite a few more functions and fundamental constants.

From a computer programming point of view, it is more cleanly designed and implemented. This has important implications for the future maintenance and

modification of the package. Unlike Maximon and Wyatt, Brent is a numerical

analyst with strong interest in complexity theory and algorithm construction.

This has not resulted, in many cases, in very much different algorithms for

arithmetic and function procedures but Brent states his algorithms more explicitly. This also contributes to the maintainability of the MP package.

The MP User's Guide [2] provides complete documentation of Brent's MP

package, including its usage with Crary' s AUGMENT precompiler. AUGMENT and

MP, as well as the MP User's Guide itself, are resident in the NBS UNIVAC 1108

Interactive System, hereafter called System I. Current information for accessing this documentation and for using the software will be given in the next

section.

16
We have already introduced the (unpacked) representation of numbers and

the four basic arithmetic operations in MP. In addition to the unpacked representation, there exists a packed representation in which two digits, instead

of only one, are stored in each word. The use of the packed representation

saves storage at the expense of time.

The range of MP numbers is limited by the integer wordlength, since only

one word is used for the range. This is also true of Wyatt's and Maximon's

packages. On System I this translates into an exponent range of approximately

10*20686623784 gM with B = 2^, M = 2^2). Underflow in an arithmetic

operation results in the storing of zero; overflow causes termination with an

error message. For the arithmetic operations the user has the choice of truncation (the default choice), proper rounding, or directed rounding. The

directed roundings (round up and round down) are intended for an interval

arithmetic extension of MP that Brent is planning to implement.

Another useful extension of the package would be to allow a variable number of words to store the exponent instead of just one. This extension of

range is needed now in order to test Clenshaw and Olver's "unrestricted" exponential algorithm [3]. This algorithm produces values of ex guaranteed correct to any relative precision a, for any input argument x. The algorithm is

self-adaptive, i.e., it selects parameters so that an estimate of the running

time is minimized asymptotically for large values of x and/or small values of

a. Brent's package suffices to test the algorithm for small values of a but

not for sufficiently large values of x. Although Brent acknowledges the need

for a multiword range, he presently plans generalizations of the package

in other directions. Consequently, it appears to be necessary to undertake

the multiword range extension of MP here at NBS.

17
Appendix A is taken from the MP User's Guide [2]. It gives the names of

all MP subroutines that are likely to be called from a user's program. Most

of the names are suggestive of their functions. Every name has the prefix MP.

The suffix I indicates an arithmetic operation involving a multiprecision

operand with a FORTAN integer operand. The suffix Q indicates an arithmetic

operation involving a multiprecision operand with a rational operand, represented by two FORTRAN integers (the numerator and denominator). As an example

of usage typical of all MP subroutines, CALL MPADDQ (X,I,J,Y) adds the rational number I/J to the multiprecision number X, giving the multiprecision number

Y. Full details on the direct usage as well as the preferred usage via

AUGMENT of all MP subroutines may be found in the MP User's Guide.

18
IV. USE OF BRENT'S MP PACKAGE WITH CRARY’S AUGMENT PRECOMPILER

ON THE NBS UNIVAC 1108 INTERACTIVE SYSTEM

£ Files exist in the NBS Univac 1108 Interactive System, also referred to

as System I, which facilitate the use of Brent’s MP package with Crary'

AUGMENT precompiler. Although many possibilities exist, we restrict ourselves

here to consideration of only one type of usage with a choice of two different

precisions: 43 and 62 decimal places of precision. First we present the

suggested EXEC 8 runstream. Then we discuss typical aspects of preparing the

individual program units for precorapilation. The detailed user' s guide to the

use of Brent's package resides in AMDAI*MP.USERSGUIDE. This guide includes

pertinent information on the use of AUGMENT as well as the MP package.

The type of usage we consider is the conversion of a user-written FORTRAN

program to multiprecision from ordinary FORTRAN precision. The FORTRAN program consists of one or more program units (the main program plus FUNCTION and

SUBROUTINE subprograms), let us say MAIN, FUNl, FUN2, SUBl , SUB2. It may be

that not every program unit requires multiprecision, and therefore precompilation, but let us assume that they all do. If any do not then they would be

handled according to the usual rules of EXEC 8. The following runstream is

prepared:

1. @ASG,

AMDAI* AUGMENT.

2. @ASG,

AMDAI*MP.

3. @XQT AMDAI*AUGMENT . AUGMENT

4. (§ADD AMDAI*MP. DESCRIPTI0N/MP12

*It is necessary to assume some familiarity with certain system aspects of

UNIVAC 1100 Series computers. No attempt will be made here to define the

many terms, such as "files", which have precise technical definitions

peculiar to these computers. Reference: UNIVAC Series 1100 Executive System,

Vol. 2, EXEC Level 36R2 Programmer Reference, UP-4144.23, 1979, or successor,

published by Sperry Rand Corp.

19
5. *BEGIN

6. :F0R,IS MAIN

7. Source deck for MAIN

8. : FOR, IS FUN1

9. Source deck for FUNl

10. : FOR, IS FUN2

11. Source deck for FUN2

12. : FOR, IS SUBl

13. Source deck for SUBl

14. : FOR, IS SUB2

15. Source deck for SUB2

16. *END

17. @ADD 20.

18. @MAP, IS ,MAIN

19. IN MAIN, FUNl, FUN2, SUBl, SUB2

20. IN AMDAI*MP . MPINIT/ MP 1

2

21. LIB AMDAI*MP.

22. END

23. @XQT MAIN

24. Input data, if any

The result of this runstream will be the raultiprecision execution of the

user's FORTRAN program to the equivalent of 43 significant figures. If 62

significant figures are required, the number 12 in items 4 and 20 must be

changed to 16 (the numbers 12 and 16 indicate the number of FORTRAN integer

storage locations used to represent each MP number). We remark that (i) at

present only the FORTRAN V compiler can be used, which means :FTN cannot be

substituted for :F0R in items 6, 8, 10, 12, 14; (ii) the norr-EXEC 8 symbol

20
is changed to during processing by AUGMENT; and (iii) the output of AUGMENT

is a legitimate system runstream stored in File 20 and initiated by line 17 of

the runstream.

The best way of illustrating the typical aspects of preparing an individual program unit for pre-compilation is by means of an actual example. Figure

1 shows an ordinary EXEC 8 runstream of the most common type: a program is

read in, compiled and executed using every available system default. This is

an example of the runstream created when a naive user of System I submits a

card deck over the counter for processing.

Figure 2 shows the modified runstream ready for precompilation and execution. It is easily verified that all of the EXEC 8 and AUGMENT control

cards are in place according to the suggested runstream presented above. We

now focus attention on the actual FORTRAN code in Figure 1 and its transformation for multiprecision processing in Figure 2, referring to individual

lines of the codes by their line numbers. The transformation process , which

must be carried out by the programmer, is analogous to transforming a singleprecision program into double precision. Type declarations, constants, and

input/output statements must all be reviewed and modified appropriately.

Figure 1, Line 2 is replaced by Figure 2, Lines 8,9,10. This changes

every floating-point variable from double precision to multiple precision,

which in this example is the equivalent of 62 decimal places as specified by

the number 16 appearing in Figure 2, Lines 4 and 78.

Figure 1, Line 4 is replaced by Figure 2, Line 14. This replaces the

double-precision value of the constant tt by its 62-place multiprecision

*The procedures to be described here are suitable for use with the 1978 version of Brent's package. In 1980 an updated version appeared in which more

powerful input/ output subroutines support noticeably simpler procedures. For

further information, contact the author of this report.

21
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value. If this were not done, then errors in the 18th decimal place would

enter the multiprecision calculation, thereby invalidating it. The effect is

the same when a single-precision constant inadvertently enters a doubleprecision calculation.

Figure 1, Lines 5,6,7 and 8 read input values and print them out. The

values used in the multiprecision calculation should be identical. The transformed code appearing in Figure 2, Line 7, 15-20 accomplishes this. The technique simply introduces new double-precision variable names for the input

variables and establishes the multiprecision equivalents by means of the

assignment statements in Figure 2, Lines 16 and 17. The input and output conversion is done by System I identically in Figures 1 and 2. The establishment

of multiprecision equivalents by assignment is done exactly if the MP package

is working in a base that is a power of two (which it is if the techniques

presented here are used).

Output of numbers which will be represented in multiple precision is indicated in Figure 1, Lines 22, 31 and 48 according to the format of Line 53.

The transformed code appearing in Figure 2, Lines 12,34,35,44,45,62,63,68

accomplishes the corresponding multiprecision output. The technique introduces a new storage vector of length 64, equal to one storage location for

each of the 62 decimal digits plus one for the sign and one for the decimal

point. See Figure 2, Line 12. The MP subroutine MPOUTE encodes the fraction

part of a single multiprecision number by storing these 64 symbols into the

storage vector, and the exponent part of the multiprecision number into an integer storage location. Output of the resulting decimal-encoded number is

controlled by the transformed FORMAT statement appearing in Figure 2, Line 63.

Minor variations on this technique will permit some degree of flexibility,

such as the printing of two columns of numbers instead of one. It should be

24
noted that the role of the number 64 would be played by 45 if we were working

to 43 decimal places.

One final modification of the original code must always be done. It is

the insertion of the statement INITIALIZE MP as the first executable statement

of the transformed program; see Figure 2, Line 13.

The result of processing the transformed runstream of Figure 2 using

System I is shown in Appendix B. We see that the printed output of the

AUGMENT precompiler consists of the listing of the "description deck" for the

MP package (introduced by Figure 2, Line 4) followed by a commented version of

the original FORTRAN source. In our example the only comments are indications

of occurrences of mixed-mode operations. If there were any errors detected by

AUGMENT, they would be identified here by comments. As stated earlier, the

legitimate FORTRAN code produced by AUGMENT is stored as part of a system runstream in File 20. This runstream is initiated in Figure 2, Line 75, and it

results in the invocation of the FORTRAN V compiler. The compiler output

clearly shows the expansion of FORTRAN expressions into lists of subroutine

calls on the MP package. Following this is the output listing of the MAP processor and, finally, the multiprecision results of execution of the program.

Appendix C shows the computer-generated accounting information for the

original runstream of Figure 1 and the transformed runstream of Figure 2. We

observe factors of 7.66 in Total Time, 24.76 in CPU Time, 7.98 in Core Block

Seconds, and 2.74 in Total Run Cost.

25
V. SUMMARY AND CONCLUSIONS

Interest in multiprecision computing has existed since the earliest days

of modern computing. This interest has led to many implementations- of multiprecision programs. But only recently have truly general-purpose software

packages of high quality been developed. We have seen three representative

examples of earlier attempts to produce software of high quality, namely the

packages of Lawson, Peavy and Maximon. Then we considered the package of

Wyatt, important for its thoroughness and completeness as well as its stimulation of the precompiler of Orser. The close connection between compilation

processes and multiple-precision computation was joined in actual software.

Finally, we considered the package of Brent, the most advanced multiple-precision package known to this author. Not only is it built on a firm theoretical foundation, it also employs disciplined coding techniques that ease the

burden of making additions, corrections and modifications. Furthermore, it is

distributed with a user's guide, installation instructions, test programs, and

an interface which enables the use of the precompiler of Crary, all on the

same source magnetic tape. For these reasons and the fact that it is actively

supported by its author, the use of Brent's package with Crary' s precompiler

is to be recommended for most applications of multiple-precision arithmetic.

However, the use of Brent's package with Orser' s precompiler is an alternative

that should be investigated.

It should not be considered, however, that Brent's package leaves nothing

to be done. Additional mathematical functions could be added to the present

repertory, which includes in addition to the FORTRAN library functions a limited complement of special functions (Bessel's function of the first kind,

Dawson's integral, exponential integral, error function, complementary error

function, gamma function, logarithmic integral, log gamma function). Improved

26
algorithms for these functions could be implemented. A more fundamental modification of Brent's package would be to extend the MP number representation

so that an arbitrary number of words could be used for the exponent part as

well as the fraction part. Presently, only one word is used for this purpose.

Such an extension would make it possible to design algorithms in which underflow and overflow can always be avoided. Although exponents of one word are

satisfactory for many applications, current research in algorithms at NBS and

elsewhere is being hampered right now by this limitation. Finally, we note

that Brent' s own future development of the MP package seems to be centering on

the provision of an interval arithmetic capability.

Another important activity for the future would be to develop micro-coded

"instructions” for multiprecision operations on a micro-codable computer.

This would speed up multiprecision computations on that computer but the major

purpose would be to gain additional knowledge about arithmetic processes in

general. For example, one could attempt to build an efficient system for computing with base, precision, and range completely under the control of the

programmer. The system would be a prototype for future developments in the

design of arithmetic units far general-purpose computers. It also would have

an influence on future compiler requirements. The resulting freedom from unnatural restrictions on the precision and range of floating-point numbers

would be a real benefit for designers of numerical algorithms and subroutines.

27
REFERENCES

1. Brent, R. P. A FORTRAN Multiple Precision Package, ACM Transactions on Mathematical Software, Vol. 4, No. 1, March 1978, Pages 57-70.

2. Brent, R. P. MP User's Guide (Third Edition), Technical Report TR-CS-79-

08, Department of Computer Science, Australian National University, Box 4,

Camberra, ACT 2000, Australia (December 1979).

3. Clenshaw, C. W. and Olver, F. W. J. An Unrestricted Algorithm for the Exponential Function, SIAM J. Numer. Anal., Vol. 17, No. 2, April 1980.

4. Crary, F. D. A Versatile Precompiler for Nonstandard Arithmetics, ACM

Transactions on Mathematical Software, Vol. 5, No. 2, June 1979, Pages

204-217.

5. Hull, T. E. Desirable Floating-Point Arithmetic and Elementary Functions

for Numerical Computation, Proceedings of the 4th Symposium on Computer

Arithmetic, October 25-27, 1978, Santa Monica, Calif., sponsored by the

IEEE Computer Society in cooperation with the UCLA Computer Science

Department.

6. Knuth, D. E. The Art of Computer Programming, Vol. 2 /Semi numerical

Algorithms . Addisort-Wesley Publishing Co., Reading, Massachusetts, 1969.

7. Lozier, D. W.; Maximon, L. C.; and Sadowski, W. L. Peformance Testing of

a FORTRAN Library of Mathematical Function Routines—A Case Study in the

Application of Testing Techniques, J. Res. NBS, Vol. 77B, Nos. 3/4, JulyDee. 1973.

8. Maximon, L. C. Fortran Program for Arbitrary Precision Arithmetic, NBS

Report 10563, April 1, 1971.

9. Wyatt, W. T. , Jr.; Lozier, D. W.; and Orser, D. J. A Portable Extended

Precision Arithmetic Package and Library with Fortran Precompiler, ACM

Transactions on Mathematical Software, Vol. 2, No. 3, Sept. 1976, Pages

209-231.

28
APPENDIX A

Summary of useful

(Third Edition), R.

MP routines, taken from the MP User's Guide

P. Brent, The Australian National University.

Sec. 1.5 MP User* 3 Guide

1.5 Summary of useful MP routines

We mention here the names of those MP routines which are likely to be of

interest to someone using the MP package without the aid of the Augment

interface. Routines which are called by other MP routines but are

unlikely to be called directly by a user of the package are omitted.

For the user of Augment, the summary given in Section 5.2 will be more

useful. Details of all MP routines may be found in Section 6.

Basic arithmetic

MPADD, MPADDI , MPADDQ , MPDIV, MPDIVI, MPMUL, MPMULI, MPMULQ

,

MPREC, MPSUB

Powers and roots

MPPWR, MPPWR2, MPQPWR , MPROOT, MPSQRT

Elementary functions

MPASIN, MPATAN, MPATN2, MPCIS, MPCOS, MPCOSH, MPEXP, MPEXP1

,

MPLG10, MPLN, MPLNI, MPLNS, MPSIN, MPSINH, MPTAN, MPTANH

Special functions

MPBESJ, MPDAW, MPEI, MPERF, MPERFC, MPGAM, MPGAMQ , MPLI, MPLNGM

Constants

MPBERN, MPEPS, MPEUL, MPMAXR , MPMINR, MPPI, MPZETA

Input and output

MPFIN, MPFOUT, MPIN, MPOUT, MPUNFR, MPUNFW

Conversion

MPCAM, MPCDM, MPCIM, MPCMD, MPCMDE, MPCMI, MPCMR, MPCHRE, KPCQM,

MPCRM

Comparison

MPCMPA, MPCMPD, MPCMPI, MPCMPQ, MPCMPR, MPCOMP, MPEQ, MPC-E , MPGT

,

MPLE, MPLT, MPNE

General utility routines

MPABS, MPCEIL, MPCHEB, MPCHEV

,

MPCMF, MPCMIM

,

MPDIGS, MPDIM,

MPFLOR, MPGCDA, MPGCDB, MPINIT, MPMAX, MPMIN, MPMOD

,

MPNEG,

MPPACK, MPPARA, MPPARB, MPPOLY, MPSETR, MPSET2, MPSIGN, MPSTR

,

MPUNPK

Example and test programs

EXAMPLE, JACOBI, TEST, TEST2

A-l

APPENDIX B

The result of processing the runstream shown

in Figure 2 on System I

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B-15

APPENDIX C

Accounting information from actual execution of

shown in Figure 1 on System I, and from actual

the runstream shown in Figure 2 on System I.

the runstream

execution of

The printed

output of the latter execution appears in Appendix B
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C-2
NBS-114A (rev. 9-78)

2. Gov't. Accession No, SglU.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-2138

4. TITLE AND SUBTITLE

Multiprecision Computing at NBS: Yesterday, Today,

and Tomorrow

5. Publication Date October 1980

" Organization Code

- —

7. AUTHOR(S)

Daniel W. Lozier

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP) 13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS' Software Summary, is attached.

16. ABSTRACT (A 200-word or leas tactual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.) Multiprecision computing is a technique by which arithmetic operations may be performed on a computer to precision levels that are higher than the directly supported

single and double precisions. The last ten years have seen the development of portable Fortran software of very high quality that essentially duplicates all the

capabilities of standard Fortran, so that an existing standard Fortran program can be

re-executed to arbitrarily high precision. In this paper some of the design techniques for such software, which have evolved at NBS and elsewhere, will be discussed.

Methods for using the software presently available at NBS will be described, and a complete example will be given. Directions for further extensions and improvements

will be indicated.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

AUGMENT and MP; Fortran arithmetic extensions; multiple precision; multiprecision

Fortran software; multiprecision mathematical functions; precompilers for special

arithmetic.

18. AVAILABILITY j~x} Unlimited 19. SECURITY CLASS 21. NO. OF

(THIS REPORT) PRINTED PAGES

1 I For Official Distribution. Do Not Release to NTIS UNCLASSIFIED 52

1 5 Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

20402, SD Stock No. SN003-003-

20. SECURITY CLASS

(THIS PAGE)

22. Price

j?T~! Order From National Technical Information Service (NTIS), Springfield,

VA. 22161 UNCLASSIFIED $7.00

USCOMM-DC
,• 3