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Burroughs
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BURROUGHS SCIENTIFIC PROCESSOR
FLOATING POINT ARITHMETIC
BSP ~~~-~- BURROU(;HS SCIENTIFIC PROCESSOR
CONTENTS
Section Page
1 INTRODUCTION 1
2 DATA REPRESENTATION IN MEMORY 3
Single Precision Floating Point Word Format 3
Integer Word Format 4
Double Precision Real Floating Point Word Format 5
3 DATA REPRESENTATION IN THE ARITHIVIETIC ELEl\1ENTS 7
Basic Data Representation 7
Representation of Zero 8
Complex Number (Single Precision) 8
4 HARDWARE ERROR CHECKING 9
5 ARITHMETIC ALGORITHMS 11
Implementation of Reciprocation and Square Root 11
Division 13
Square Root 13
6 ROUNDING AND NORMALIZATION 15
Rounding - Single Precision 15
Rounding - Double Precision 17
Normaliza tion 17
Appendix A - Arithmetic Operations 19
Appendix B - Error Estimates for Arithmetic Operations 27
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~~p ~~~~~~~~~~~~~~~~~~~~~BURROUGHSSCIENTIFICPROCESSOR
BSP
1. INTRODUCTION
One of the most important features of any computer is its arithmetic. This docu-
ment discusses the implementation of floating point arithmetic in the Burroughs
Scientific Processor (BSP). Data representation in both the BSP memory and
arithmetic element is described, as are the arithmetic algorithms used in the
BSP. Of particular interest are the techniques used for error checking in the
arithmetic element and for rounding in both the scalar processor and the parallel
processor. The BSP arithmetic operations, including instructions and cycle
operations, are described in detail in Appendix A, and the accuracy of arithmetic
operations is discussed in Appendix B.
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~~p ~~~~~~~~~~~~~~~~~~~~~BURROUGHSSCIENTIFICPROCESSOR
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1 i
BSP ; ! ~ F
2. DATA REPRESENTATION IN MEMORY
The representation of data in the memory of the Burroughs Scientific Processor
(BSP) is as follows:
1. Single precision floating point word format,
2. Integer word format,
3. Double precision real floating point word format.
SINGLE PRECISION FLOATING POINT WORD FORMAT
A single precision floating point number, X, is represented by an ordered pair of
numbers, E (exponent) and m (mantissa)" such that:
E
X = 2 *m
where E is an integer and m satisfies the condition:
-1/2 .::. m <- 1 or 1/2'::: m < 1 or m = O.
In order to meet this condition, that is, 1/2 ':::Iml < 1, the last step in every floating
point operation is the normalization of the mantissa (removal of leading zeroes).
The layout of the floating point word in memory is indicated below. The bits are
numbered from right to left. The least significant bit is numbered 0; the most
significant bit of the mantissa is bit 35; the least significant of the exponent is bit
36. The most significant bit of the exponent is bit 45. Bit 46 is the sign bit of the
mantissa; bit 47 is the sign bit of the exponent. Every group of consecutive bits is
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~~p ~~~~~~~~~~~~~~~~~~~~~BURROUGHSSCIENTIFICPROCESSOR
called a field. and is denoted by W [x:y]; W is the name of the data unit x is the
address of the left-most bit. and y is the length of the field. Thus. a data word
of the data unit. A. is defined as A [ 47:48]. and its null indicator is defined as
A[48:1].
Using field notation X [leading bit:numbers of bits]. the mantissa is represented
by X [35:36J. the exponent by X [45:10J. the sign bit of the mantissa by X[46:1].
47 46 45 36 35 o
Exponent Mantissa
Sign of Exponent
The range of representable numbers in single precision is as follows:
1. For positive X:
1023
2- * 1/2 2. X 2.21023 * (1_2- 36 >. where
2- 1023 * 1/2 ;' 10- 308 . 25
2. For negative X: ( \
I
_ 2-
1023
* 1/2 ~ X ~ _2 1023 * (1_2- 36 >. where
INTEGER WORD FORMAT
An inte~er. I. is defined by its absolute value m (1) and by its sign bit S (1). Field
I [35:36J contains the magnitude, and the sign bit is in field [46:1]. The unused bits
of the data word are set to O. The range of integer values is symmetric about zero.
36
_2 + 1 < 1< 2 36 _1
46 35 0
~ ~ Integer I
Sign of Integer
4
BSP----------~~ ----------~------------- BU R ROUGHS SCI ENTI F! C PROCESSO R
DOUBLE PRECISION REAL FLOATING POINT WORD FORMAT
A double precision floating pOint number X is represented by two single precision
numbers FIRST (X) and SECOND (X). Both of these numbers are normalized. The
mantissa sign bits must be the same. Due to normalization, the relationship be-
tween exponents is:
EXPONENT (SECOND (X