Chapter 12: Handling Binary Integer Data

This chapter covers binary data, which refers to integer data that are stored in the form of two’s–complement numbers of either 2 bytes (16 bits) or 4 bytes (32 bits).  Later versions of the IBM mainframe, certainly the zSeries, also include 8 byte (64 bit) integers.

While it is true that all data in a stored–program computer are stored in binary form, it is the interpretation of those data by the CPU that determines the format to be used.  Consider the following ambiguous declaration.

DATA    DC X‘81 6C’

If this field is processed as a character string, say using MVC, it will be interpreted as the two printable characters “a%”.  If the field is processed as a packed decimal, say using ZAP, it will be interpreted as the three–digit positive number with value equal to +816.

This field contains four hexadecimal digits, or 16–bits.  It can be viewed as a 16–bit signed integer in two’s–complement format.  A bit of reflection will show that, interpreted in this format, the field represents a negative number.  We now convert it to the decimal value.

The value itself is X‘81 6C’ or binary        1000 0001 0110 1100.

Take the one’s complement to get                  0111 1110 1001 0011.

Add one to get                                                            0111 1110 1001 0100.

Convert this back to hexadecimal                   X‘7E 94’.

The decimal value for the last is 32,404.  The data field, interpreted as an 8–bit integer
stored in two’s–complement form is an integer with the negative value –32,404.

The two standard binary formats are as follows.

F       The fullword format is a 32–bit integer, requiring four bytes of storage.

H      The halfword format is a 16–bit integer, requiring two bytes of storage.

The ranges are what would be expected for standard two’s–complement arithmetic.

Type          Bits       Minimum       Maximum          Minimum             Maximum

Half–word      16           –(215)               (215) – 1                   –32,768                   32,767

Full–word       32           –(231)               (231) – 1       –2,147,483,648       2,147,483,647

Those of us trained on computers other than IBM mainframes will unconsciously equate integer data with one of the standard two’s–complement formats.  The 16–bit and 32–bit forms were rather popular when the System/360 was first designed.  These two formats
were continued into the System/370 and later models.  As noted above, newer models
include a 64–bit integer format.

Those programmers trained primarily on IBM mainframes might consider the packed decimal format as an equally good way to handle integers.  Recall that the packed format can handle integers of lengths up to 31 digits, as opposed to the 11 digit maximum on the 32–bit two’s–complement format.  In this view, binary arithmetic is done only in the registers and usually is applied only for address computations.  Your author’s opinion is that each integer representation has its strengths; pay your money and take your choice.

Declaring Binary Storage

There are many ways to declare binary storage.  The four most useful are

1.      B         Ordinary binary,

2.      F          Full–word (32–bit binary two’s–complement integer),

3.      H         Half–word (16–bit) binary two’s–complement integer), and

Each of the B and X declarations may declare a storage area with length from 1 through 256 bytes.  The lengths of the F and H declarations are fixed at 4 and 2 bytes respectively.
Apparently, it is possible to assign a length in bytes to either type, but this is strange.

Note that the two declarations below have an identical effect.  Each defines a 32–bit binary integer with value equal to 14,336 in decimal.

F1       DC F‘14336’       DEFAULT SIZE IS FOUR BYTES.

X1       DC XL4‘00003800’  SIZE SPECIFIED AS FOUR BYTES.

While the second declaration is unusual for a full–word, it makes some examples easier.

More On DC (Define Constant)

The general format of the DC statement is as follows.

 Name DC dTLn ‘constant’

The name is an optional entry, but required if the program is to refer to the field by name.  The standard column positions apply here.

The declarative, DC, comes next in its standard position.

The entry “dTLn” is read as follows.

d       is the optional duplication factor.  If not specified, it defaults to 1.

T       is the required type specification.  The types for binary are B, F, H, and X.
Note that the data actually stored at the location does not need to be
of this type, but it is a good idea to restrict it to that type.

L       is an optional length of the data field in bytes.

The ‘constant’ entry is required and is used to specify a value.  If the length attribute is omitted, the length is specified implicitly by this entry.  Again, it is rarely desirable
to specify a length for the F and H data types.

Alignment and Value Ranges

Remember that the System/360 is a byte–addressable machine.  The type F declares a full–word, which is a four–byte field aligned on a full–word boundary; i.e., its address is a multiple of four.  The type H declares a half–word, which is a two–byte field aligned on a
half–word boundary; i.e., its address is a multiple of two.

If the value declared in either a type F or type H constant is greater than that
allowed by the data type, the assembler merely truncates the leftmost digits.

Consider the following example

This is truncated to a value of 8,192, which is X‘2000’.  The leading 1 is dropped
from the hexadecimal representation, because only the last four digits fit into the
half–word storage allocation; 4 hexadecimal digits = 2 bytes = 1 half–word.

Sequential Memory

Consider the following two declarations which are sequential.  Each is a half–word,
which is declared using the hexadecimal construct to make the example clear.

H1       DC   XL2‘0102’   DECIMAL 258

H2       DC   XL2‘0304’   DECIMAL 772  At address H1+2

The half–word value stored at address H1 is hexadecimal 0102 or decimal 258.

The full–word value stored at address H1 is hexadecimal 01020304, or
16, 909, 060 in decimal.  This fact can present problems for the incautious coder.

To load the value of the half–word at address H1 into a register, one uses the Load
Half–word instruction; e.g., LH R4,H1.  Register R4 gets 258.  But if I accidentally write a full–word load instruction, as in L R4,H1, then register R4 will get the decimal value
16, 909, 060.  This is due to the fact that the four bytes beginning at address H1 have the value X‘0102 0304’.  The fact that H1 and H2 are defined separately matters not at all.

Similarly, suppose I declare a full–word as follows.

F1       DC XL4 ‘11121314’   DECIMAL 17,899,828

If the code says LH R4,F1, then F1 gets hexadecimal X‘1112’ or decimal 4370.

Binary Constants and Hexadecimal Constants

The type B declaration uses binary numbers (0 or 1) to define a string of bits.  The type X declaration uses hexadecimal digits to define what is also just a string of bits.

Consider the following pairs of declarations.

B1       DC B‘10101110’

X1       DC XL1‘AE’             READ AS 1010 1110

B2       DC B‘0001001000010011’

X2       DC XL2‘1213’           READ AS 0001 0010 0001 0011

B1 and X1 each declare the same bit pattern.

B2 and X2 each declare the same bit pattern.

Personally, I find the hexadecimal constants much easier to read, and would suggest not using the B declaration.  The most common use for the binary declaration would be to set bit patterns to be sent to registers that control Input/Output devices.  In standard programming, we do not have access to those registers on a System/360 or later mainframe..

Input and Output of Binary Data

All data are input originally as EBCDIC characters.

All data printed must be output as EBCDIC characters.

The standard input process for binary data is a two–step one, in which the character
data are first packed to form decimal data and then converted to binary.

The standard process to output binary data from a register is also a two–step one.
First convert the binary to decimal data and then use unpack or the edit instruction
to produce the printable EBCDIC characters.

Conversion between Packed Decimal and Binary

These two conversion instructions are each a type RX instruction.

CVB (Convert to Binary) converts packed decimal data from storage into binary form in a general–purpose register.  This is a type RX instruction with opcode X‘4F’.

CVD (Convert to Decimal) converts binary data in a general–purpose register into packed decimal form in storage.  This is a type RX instruction with opcode X‘4E’.

The format of each is OP R1,D2(X2,B2).

Template for the instructions:             CVB Register,Storage_Location

CVD Register,Storage_Location

For the CVB instruction, the Storage Location contains the packed decimal value that
is to be converted to binary and placed in the register.

For the CVD instruction, the Storage Location is the field that will receive the packed decimal value resulting from the conversion of the value in the register.

It is standard practice to use the floating point data type D (double word) to
declare the storage location.

Why A Floating Point Type Here?

The D data type declares a double precision floating point value, which occupies eight bytes (64 bits) and is automatically aligned on a double–word boundary.  In other words, its address is a multiple of 8.  The true requirement for the operand is that it be exactly eight bytes long and begin on a double–word boundary.  The D declaration fills the bill.

Consider the following code, which is rather standard.

CVB  R6,D1

D1   DS D      DOUBLE WORD OR 8 BYTES

One might also write the following, if one is careful.

CVB  R6,D2

D2   DS PL8    EIGHT BYTES FOR UP TO 15 DIGITS

The difficulty here is insuring that D2 is properly aligned on a double–word boundary.
While this can be done, it is less error–prone to use the D type and have the assembler
automatically do the alignment for you.

How many digits do I really need?  The biggest value storable as a 32–bit binary number is 2,147,483,647.  This number has 10 digits, which will be converted to 11 digits for storage in
Packed Decimal format.  A 4–byte full–word will store only seven digits.  It takes a six–byte packed decimal field to store 11 digits.  There is no data size that automatically takes 6 bytes and no provision for aligning an address on a multiple of six.  The obvious choice for the packed decimal intermediary form is storage as a double–word.

Input example

ZAP  D1,AMTPACK  TRANSFER TO THE DOUBLE WORD

CVB  R5,D1       CONVERT TO BINARY

D1       DS   D           THIS RESERVES EIGHT BYTES

Output example

CVD R5,D2        PLACE INTO A DOUBLE WORD

ZAP AMTPACK,D2   TRANSFER TO THE PACKED WORD

D2       DS  D            THIS ALSO RESERVES EIGHT BYTES

Each of these examples assumes that a field, AMTPACK in each, has been properly declared with the proper length.  Recall that each example is a part of a larger process.

The input process has several steps:
1.      Read in the sequence of digits as EBCDIC characters.
2.      Use the PACK command to place the result in the field AMTPACK.
3.      Use the above sequence to convert the number to binary form in the register.

The output process has several steps:
1.      Use the above sequence to convert the binary number in the register to
a packed form in the field AMTPACK.
2.      Use UNPK or ED, preferably the latter, to generate the EBCDIC characters
that form the printable output.

RX (Register–Indexed Storage): Explicit Base Register Usage

This is a four–byte instruction of the form OP R1,D2(X2,B2).

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) OP R1 X2 B2 D2 D2D2

The first byte contains the 8–bit instruction code.

The second byte contains two 4–bit fields, each of which encodes a register number.  The first hexadecimal digit, denoted R1, identifies the register to be used as either the source or destination for the data.  The second hexadecimal digit, denoted X2, identifies the register to be used as the index.  If the value is 0, indexed addressing is not used.

The third and fourth bytes contain a standard address in base/displacement format.

As an examples of this type, we consider the two following instructions:
L          Load Fullword                        Opcode is X‘58’
A          Add Fullword             Opcode is X‘5A’

We consider a number of examples based on the following data declarations.  Note that the data are defined in consecutive fullwords in memory, so that fixed offset addressing can be employed.  Each fullword has a length of four bytes.

DAT1      DC F‘1111’

DAT2      DC F‘2222’     AT ADDRESS (DAT1 + 4)

DAT3      DC F‘3333’     AT ADDRESS (DAT2 + 4) OR (DAT1 + 8)

A standard code block might appear as follows.

L R5,DAT1

A R5,DAT2

A R5,DAT3      NOW HAVE THE SUM.

One variant of this code might be the following.  See page 92 of R_17.

LA R3,DAT1      GET ADDRESS INTO R3

L  R5,0(,3)     LOAD DAT1 INTO R5
A  R5,4(,3)     ADD DAT2, AT ADDRESS DAT1+4.

A  R5,8(,3)     ADD DAT3, AT ADDRESS DAT1+8.

Note the leading comma in the construct (,3), which is of the form (Index, Base).  This indicates that no index register is being used, but that R3 is being used as a base register.  It is equivalent to the construct (0,3), which might be preferred.

Here is another variant of the above code.

LA R3,DAT1      GET ADDRESS INTO R3

LA R8,4         VALUE 4 INTO REGISTER 8

LA R9,8         VALUE 8 INTO REGISTER 9

L  R5,0(0,3)    LOAD DAT1 INTO R5
A  R5,0(8,3)    ADD DAT2, AT ADDRESS DAT1+4.

A  R5,0(9,3)    ADD DAT3, AT ADDRESS DAT1+8.

Loading Values: L, LH, LR, and LCR

The general–purpose registers are designed to store and manipulate binary data that are stored in the form of 32–bit two’s–complement integers.  As an aside, remember two facts about such numbers.

1.      The IBM standard is to number the bits from left to right as 0 through 31.
The sign bit is called “Bit 0” and the units bit on the right “Bit 31”.

2.      IBM will often call this “31 bit data”, as the value has a 31–bit magnitude
(stored in bits 1 – 31) and a sign bit.

We first discuss three of the standard instructions used to load values into a register.

L                Load a full–word value into the register.

LH             Load a half–word value into the register.
The 16–bit value is sign extended into 32–bits for the register.

LR             Copy a value from one register to another register.

LCR          Load the first register with the two’s–complement of the value in the second.

Note:            None of these instructions will set a condition code.

Do not load a register and expect a condition code to reflect the value loaded.

L (Load 32–bit Full–word)

The instruction is a type RX, with format L R1,D2(X2,B2).  The opcode is X‘58’.  The object code format is as follows.

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) X‘58’ R1 X2 B2 D2 D2D2

The first operand specifies any general–purpose register.  This is indicated by the first
hexadecimal digit in the second byte of the object code.

The second operand references a full–word in storage, usually aligned on a full–word
boundary.  If the second operand is a literal, the assembler will align it properly.  The address of this second word is computed from the standard base/displacement form (B2 D2 D2 D2 in bytes 3 and 4) with an index register (the second hexadecimal digit in byte 2).

Here is a template for the instruction:  L Reg,Full_Word

Here are some examples of common usage.  Other examples will be discussed later.

L1       L R2,=F‘4000’  R2 GETS DECIMAL 4000

L2       L R3,F1        R3 ALSO GETS DECIMAL 4000

L3       L R4,H1        THIS IS PROBABLY A MISTAKE.

L4       L R5,=A(H1)    LOAD THE ADDRESS INTO R5.

F1       DC F‘4000’

H1       DC H‘2000’     Stored as X‘07 D0’

H2       DC H‘3000’     Stored as X‘0B B8’

Note again, it is usually a mistake to attempt to use a full–word load to place a half–word value into a register.  What will happen when the instruction at address L3 is executed is that register R4 will be loaded with the value X‘07 D0 0B B8’, or decimal 131, 075, 000.

The execution of the instruction at address L4 causes the address of the halfword H1, not its value, to be loaded into register R5.  For the System/370, the address is a 24–bit unsigned integer that is extended to a 32–bit value for storage in the register.

LH (Load 16–bit Half–word)

The instruction is a type RX, with format LH R1,D2(X2,B2).  The opcode is X‘48’.
The object code format is as follows.

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) X‘48’ R1 X2 B2 D2 D2D2

The first operand specifies any general–purpose register.  This is indicated by the first
hexadecimal digit in the second byte of the object code.

The second operand references a full–word in storage, usually aligned on a half–word
boundary.  If the second operand is a literal, the assembler will align it properly.  The address of this second word is computed from the standard base/displacement form (B2 D2 D2 D2 in bytes 3 and 4) with an index register (the second hexadecimal digit in byte 2).

The assembler loads the half–word into the rightmost 16 bits of the register (16 – 31)
and then propagates the half–word’s sign bit through the left 16 bits of the register.

Here is a template for the instruction:  LH Reg,Half_Word

Here are some examples of common usage.  Other examples will be discussed later.

L1       LH R2,=H‘4000’  R2 GETS DECIMAL 4000

L2       LH R3,H1        R3 GETS DECIMAL 2000

L3       LH R4,F1        THIS IS PROBABLY A MISTAKE.

F1       DC F‘4000’      Stored as X‘00 00 0F A0’

H1       DC H‘2000’

The difficulty with the instruction at address L3 is that it will access the two bytes at the addresses F1 and F1+1.  The halfword stored there has value X‘00 00’, or just 0.

Sign Extension for LH

Consider two 16–bit integers that are stored as half–words in two’s–complement form.

The positive number + 100 is stored as 0000 0000 0110 0100, or X‘0064’.

The negative number –100 is stored as 1111 1111 1001 1100 or X‘FF9C’

The LH sign extends the halfword data into fullword data with the proper sign.  This it does by copying bits 0 through 15 of the halfword into bits 16 through 31 of the register and then copying the sign bit (now in register bit 16) into bits 0 through 15 of the register.

Consider the code fragment below.

LH R7,=H‘100’

After this, register R7 contains the full–word value +100, as shown below.

 Left half–word Right half–word 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 0000 0000 0000 0000 0000 0000 0110 0100

Now consider the code fragment.

LH R8,=H‘-100’

After this, register R8 contains the full–word value –100, as shown below.

 Left half–word Right half–word 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 1111 1111 1111 1111 1111 1111 1001 1100

LR (Load Register) and LCR (Load Complement Register)

Each instruction is a type RR, with format LR R1,R2.  The opcode for LR is X‘18’.

The opcode for LCR is X‘13’.  The object code format for each is as follows.

 Type Bytes Operands RR 2 R1,R2 OP R1 R2

Each operand specifies any general–purpose register.  The contents of the register specified as the second operand are copied into the register specified as the first operand.

Consider the code fragment below.

L  R9,=H‘200’  REGISTER 9 GETS DECIMAL 200

LR R7,R9       REGISTER 7 ALSO GETS 200
THIS TIME IT IS COPIED FROM R9

LCR R8,R9      REGISTER 8 GETS DECIMAL -200, STORED
IN PROPER 2’S-COMPLEMENT FORMAT.

LM (Load Multiple Registers)

The LM instruction loads data from main storage into more than one register.

The instruction is a type RS with format LM R1,R3,D2(B2).  The opcode is X‘98’.

This is a four–byte instruction with object code format as follows:

 Type Bytes Operands 1 2 3 4 RS 4 R1,R3,D2(B2) X‘98’ R1 R3 B2 D2 D2D2

The first byte contains the 8–bit instruction code.

The second byte contains two 4–bit fields, each of which encodes a register number.  These two bytes specify the range of registers to be loaded.

The third and fourth bytes together contain a 4–bit register number and 12–bit displacement used to specify the memory address of the operand in storage.  This operand is considered as the first fullword a block of fullwords; the size of the block is determined by the number of registers specified in byte 2.  This is a type RS instruction; indexed addressing is not used.

Recall that each label in the assembly language program references an address,
which must be expressed in the form of a base register with displacement.

Any address in the format of base register and displacement will appear in the form.

 B D1 D2 D3

B is the hexadecimal digit representing the base register.

The register numbers “wrap around”, so that 15,1 specifies the three registers 15, 0, 1.

Example code:

LM R6,R8,F1    LOAD R6, R7, R8 FROM F1, F2, F3

LM R15,R2,F1   LOAD R15, R0, R1, R2 FROM F1 TO F4

F1       DC F‘1111’

F2       DC F‘2222’

F3       DC F‘3333’

F4       DC F‘4444’

LM and the Standard Closing Code

Look again at part of the standard closing code for our programs.

*******************  END LOGIC    **************************

L     R13,SAVEAREA+4          POINT AT OLD SAVE AREA

LM    R14,R12,12(R13)         RESTORE THE REGISTERS

LA    R15,0                   RETURN CODE = 0

The label SAVEAREA references a sequence of full words used to save information
used when returning to the operating system.

The second full–word in this area, at address SAVEAREA+4, holds the address of
the block of memory used to save the register information.

The instruction LM R14,R12,12(R13) loads the 15 registers R14 through R12, omitting only R13, with the 15 full–word values beginning at the specified address.  More specifically, the old register values are saved in a block beginning with the fourth full–word (at offset 12) in the block with address now in R13.  The address 12(R13) is specified in base/displacement format and references the start address of the 60–byte part of the save area that is used to store the values of the registers.

The instruction LA R15,0 is a use of a Load Address instruction that we shall discuss
very shortly.  I would prefer something like LH R15,=H‘0’, which appears to be equivalent, but can lead to addressability issues.  The LA format is safer.

Up to now, we have discussed “value loaders”, such as the following example.

L R3,FW1

This finds the full–word at address FW1 and loads its value into register R3.

At times, we shall need not the value stored at an address but the address itself.
One possibility would be to store a return address to be used by a subroutine.

There are two common ways to access the address and store it into a register.

1.      Use the L (Load full–word) instruction and use an address literal

2.      Use the LA (Load Address) instruction and use the label.

The following two statements are equivalent.  Each loads R1 with the address FW1.

L  R1,=A(FW1)

LA R1,FW1

In the System/360 and System/370 the address is treated as a 24–bit unsigned integer,
which can be represented by six hexadecimal digits.

If the address of FW1 is X‘112233’, register R1 gets X‘00112233’.

The instruction is a type RX, with format LA R1,D2(X2,B2).  The opcode is X‘41’.

The object code format is as follows.

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) X‘41’ R1 X2 B2 D2 D2D2

Here is a template for the instruction:  LA Reg,Address

The first operand specifies any general–purpose register.  This is indicated in the
object code by the first hexadecimal digit in the second byte.

The second operand references a storage address in the form D2(X2,B2).  The index register is specified by the second hexadecimal digit in the second byte.  Bytes 3 and 4 together contain an address in base/displacement form, to which the index value is added.

Consider the following fragment of code, which indicates one use of the instruction.

LA R9,A10

A10      DC F‘100’

Suppose that label A10 is subject to base register 3 containing value X‘9800’
with a displacement of X‘260’.  The object code for the LA instruction is as follows.

41 90 32 60

The code for the LA instruction is X‘41’.  The second byte “90” is of the form R1X2,
where R1 is the target register and X2 is the index register.  As is standard, a value of 0 indicates that indexing is not used in this address; it is pure base/displacement form.

The LA instruction causes register R9 to be get value X‘9800’ + X‘260’ = X‘9A60’.

LA: A Second Look

The instruction is a type RX, with format LA R1,D2(X2,B2).

Consider the example above, coded as LA R9,X‘260’(0,3).

Again, the object code for this is 41 90 32 60.

Let’s analyze this object code.  What it says is the following:

1)      Take the contents of register 3                                                   X‘9800’

2)      Add the value of the offset                                                          X‘260’

3)      Add the contents of the index                                                      X‘000’
(here no index register is used)

4)      Get the value                                                                              X‘9A60’

5)      Place that value into register R9, which now contains X‘0000 9A60’.

But note:              While we call this an address, it is really just an unsigned binary number.

This gives rise to a common use of the LA instruction to load a constant
value into a general–purpose register.

LA: Load Register with Explicit Value

Consider the instruction LA R8,4(0,0).

The object code for this is 41 80 00 04.

The code is executed assuming no base register and no index register.
The number 4 is computed and loaded into register 8.

The following instruction is considered identical: LA R8,4.

Note that the second operand in this form of the instruction is a non–negative
integer that is treated by the assembler as a displacement.

This implies that the value must be in a form that can be represented as a 12–bit unsigned integer, specifically that it must be a non–negative integer not larger than 4,095 (decimal).

Consider now the line from the standard ending code of our programs.

LA    R15,0                   RETURN CODE = 0

This places the value 0 into the destination register.

Instructions: Surface Meaning and Uses

In the previous example, we see a trick that is commonly used by assembly language programmers: find what the instruction really does and exploit it.  The surface meaning of the LA instruction is simple: load the address of a label or symbolic address into a given register.

The usage to load a register with a small non–negative constant value is an immediate and logical result of the way the object code is executed.  The goals of such tricks seem to be:

1)      To gain coding efficiency, and

2)      To avoid addressing problems that sometimes arise in the use of literals.

Storing Register Values: ST, STH, and STM

ST (Store Full Word) is a type RX instruction, with format ST R1,D2(X2,B2).

STH (Store Half Word) is a type RX instruction, with format STH R1,D2(X2,B2).

STM (Store Multiple) is a type RS instruction, with format STM R1,R3,D2(B2).

The ST instruction stores the full–word contents of the register, specified in the
first operand, into the full word at the address specified by the second operand.

The STH instruction stores the rightmost 16 bits of the register specified by the
first operand into the half word at the address specified by the second operand.

For STM (Store Multiple Registers), the first two operands specify a range of
registers to be stored.  Remember that the register numbers “wrap around”

STM R7,R10,X2    STORE THE FOUR REGISTERS R7,R8,R9,AND R10
INTO FOUR FULL-WORDS BEGINNING AT X2

STM R10,R7,X4    STORE THE 14 REGISTERS R10 THROUGH R7
(ALL BUT R8 AND R9) INTO 14 FULL-WORDS

While each of these instructions is quite similar to its load register partner, we shall
spend a bit of time discussing the instructions.  After all, this is a textbook.

ST: Store Fullword

The ST (Store Full Word) is a type RX instruction, with format ST R1,D2(X2,B2)
and opcode X‘50’.  The object code format is as follows:

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) X‘50’ R1 X2 B2 D2 D2D2

The first operand specifies any general–purpose register.  This is indicated by the first
hexadecimal digit in the second byte of the object code.

The second operand references a full–word in storage, usually aligned on a full–word
boundary.  The address of this second word is computed from the standard base/displacement form (B2 D2 D2 D2 in bytes 3 and 4) with an index register (the second hexadecimal digit in byte 2).  Here is a template for the instruction:  ST Reg,Full_Word

Here are some examples of common usage.  Other examples will be discussed later.
Suppose that R3 contains the decimal value 163840, which is X‘0002 8000’.

ST1      ST R3,F1

ST2      ST R3,H1         NOTE THE TYPE MISMATCH.

F1       DC X‘0000 0000’

H1       DC X‘0000’

H2       DC X‘0000’

The instruction at address ST1 works as advertised, storing the register value into the fullword at the given address.  The instruction at address ST2 is almost certainly a mistake.
The register value is stored into the four bytes beginning at address H1.  Halfword H1 is set to the value X‘0002’ and halfword H2 is set to the value X‘8000’.

STH: Store Halfword

The STH (Store Half Word) is a type RX instruction, with format ST R1,D2(X2,B2)
and opcode X‘40’.  This instruction stores the rightmost 16 bits (bits 16 – 31) of the source register into the halfword at the given address.  The object code format is as follows:

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) X‘40’ R1 X2 B2 D2 D2D2

The first operand specifies any general–purpose register.  This is indicated by the first
hexadecimal digit in the second byte of the object code.

The second operand references a half–word in storage, usually aligned on a half–word
boundary.  The address of this second word is computed from the standard base/displacement form (B2 D2 D2 D2 in bytes 3 and 4) with an index register (the second hexadecimal digit in byte 2).  Here is a template for the instruction:  ST Reg,Half_Word

Here are some examples of common usage.  Other examples will be discussed later.
Suppose that R3 contains the decimal value 163840, which is X‘0002 8000’.

ST1      STH R3,F1        NOTE THE TYPE MISMATCH.

ST2      STH R3,H1

F1       DC  X‘0000 0000’

H1       DC  X‘0000’

H2       DC  X‘0000’

The instruction at address ST2 works as advertised, though perhaps not as intended.  The rightmost 16 bits of register R3 contain a value represented in hexadecimal as X‘8000’.  This value is copied into the halfword at address H1, correctly setting its value.

The instruction at address ST1 is almost certainly a mistake.  It loads the halfword at address F1 with the hexadecimal value X‘8000’.  Note that it does not matter that the assembly listing defines F1 as a fullword.  The halfword at address F1 comprises the two bytes, the first at address F1 and the second at address F1+1.  After this instruction is executed, F1 contains the value X‘8000 0000’; the rightmost 16 bits have been copied into the two leftmost bytes associated with the address F1.

STM: Store Multiple Registers

The STM instruction stores data from one or more registers into main memory.

The instruction is a type RS with format STM R1,R3,D2(B2).  The opcode is X‘98’.

This is a four–byte instruction with object code format as follows:

 Type Bytes Operands 1 2 3 4 RS 4 R1,R3,D2(B2) X‘98’ R1 R3 B2 D2 D2D2

The first byte contains the 8–bit instruction code.

The second byte contains two 4–bit fields, each of which encodes a register number.  These two bytes specify the range of registers to be loaded.

The third and fourth bytes together contain a 4–bit register number and 12–bit displacement used to specify the memory address of the operand in storage.  This operand is considered as the first fullword a block of fullwords; the size of the block is determined by the number of registers specified in byte 2.  This is a type RS instruction; indexed addressing is not used.

Since this is a type RS instruction, indexed addressing is not used.

Recall that each label in the assembly language program references an address,
which must be expressed in the form of a base register with displacement.

Any address in the format of base register and displacement will appear in the form.

 B D1 D2 D3

B is the hexadecimal digit representing the base register.

The register numbers “wrap around”, so that 15,1 specifies the three registers 15, 0, 1.

Example code:

STM R6,R8,F1       STORE R6, R7, R8 INTO F1, F2, F3

STM R15,R2,F1      STORE R15, R0, R1, R2 INTO
F1, F2, F3, F4

F1       DC F‘1111’

F2       DC F‘2222’

F3       DC F‘3333’

F4       DC F‘4444’

Standard Boilerplate Code

Once again, we examine some of the standard code used in all of our programs.

The standard startup code includes the following fragment.

SAVE  (14,12)     SAVE THE CALLER’S REGISTERS

This macro generates the following code.

STM 14,12,12(13)  STORE REGISTERS 14 THROUGH 12
(15 IN ALL) INTO THE ADDRESS
12 OFFSET FROM BASE REGISTER 13.

We might have concluded our code with the macro

RETURN (14,12)

This expands into the code we actually use in our programs.

LM 14,12,12(13)

LA    R15,0     RETURN CODE = 0

Binary Arithmetic: Addition and Subtraction

There are six instructions for addition and subtraction.

Mnemonic                        Description                       Type               Format

A               Add full–word to register                   RX         A R1,D2(X2,B2)

S                Subtract full–word from register        RX         S R1,D2(X2,B2)

AH            Add half–word to register                  RX         AH R1,D2(X2,B2)

SH             Subtract half–word from register       RX         SH R1,D2(X2,B2)

AR            Add register to register                       RR          AR R1,R2

SR             Subtract register from register            RR          SR R1,R2

In each of these, the first operand is a register.  It is this register that has its
value changed by the addition or subtraction.

For the half–word instructions (AH and SH), the second operand references a half–word storage location.  The 16–bit contents of this location are sign extended to a full 32–bit word before the arithmetic is performed.

Binary Arithmetic: Half–word arithmetic

Examples of the instructions

L    R7,FW1     LOAD REGISTER FROM FW1

A    R7,FW2     ADD FW2 TO REGISTER 7

S    R7,=F‘2’   SUBTRACT 2 FROM R7

ST   R7,FW3     STORE VALUE IN R7 INTO FW3

AR   R7,R8      ADD CONTENTS OF R8 TO R7

SR   R7,R9      SUBTRACT R9 FROM R7

SR   R8,R8      SET R8 TO ZERO

FW1      DC   F‘2’

FW2      DC   F‘4’

FW3      DC   F‘0’

As noted indirectly above, one has two options for operating on one register.

AR   R7,R7      DOUBLE THE CONTENTS OF R7
(ADD R7 TO ITSELF)

SR   R9,R9      SET R9 TO ZERO.

Comparing Binary Data: C, CH, and CR

There are three instructions for binary comparison with the value in a register.

Mnemonic                     Description                       Type               Format

C                  Compare full–word                       RX            C R1,D2(X2,B2)

CH               Compare half–word                      RX            CH R1,D2(X2,B2)

CR               Compare register to register          RR             CR R1,R2

Each comparison sets the expected condition code.

Condition                      Condition Code            Branch Taken

Operand 1 = Operand 2             0 (Equal/Zero)                BE, BZ

Operand 1 < Operand 2             1 (Low/Minus)               BL, BM

Operand 1 > Operand 2             2 (High/Plus)                  BH, BP

Don’t forget that literal arguments can be used with either C or CH, as in this example.

C  R9,=F‘0’   COMPARE THE REGISTER TO ZERO

BH ISPOS      IT IS POSITIVE

BL ISNEG      NO, IT IS NEGATIVE.

If this line is reached, R9 contains the value 0.

An Extended Example

This example takes the value in HW1, makes it non–negative, and then sums
backwards N + (N – 1) + … + 2 + 1 + 0.

SR  R6,R6    SET R6 TO ZERO

LH  R5,HW1   GET THE VALUE INTO R5

SR  R6,R5    SUBTRACT TO CHANGE THE SIGN

C   R6,=F‘0’ IS R6 POSITIVE? (IF SO R5 IS NEGATIVE)

BH  POS      YES R6 IS POSITIVE.

LR  R6,R5    R5 IS NOT NEGATIVE. COPY R5 INTO R6

*    NOW R6 CONTAINS THE ABSOLUTE VALUE OF THE HALF-WORD

POS  SR  R5,R5    R5 WILL HOLD THE TOTAL.  SET TO ZERO.

LOOP AR  R5,R6    ADD R6 TO R5

S   R6,=F‘1’ DECREMENT R5 BY 1

C   R6,=F‘0’ IS THE VALUE STILL POSITIVE?

BH  LOOP     YES, GO BACK AND ADD AGAIN.

*    THE SUM IS FOUND IN R5.

Register Shift Operations

We now discuss a number of shift operations performed on registers.

Mnemonic                     Description                       Type                     Format

SLA             Shift left algebraic                     RS                SLA R1,D2(B2)

SRA             Shift right algebraic                   RS                SRA R1,D2(B2)

SLL             Shift left logical                         RS                SLL R1,D2(B2)

SRL             Shift right logical                       RS                SRL R1,D2(B2)

SLDA          Shift left double algebraic         RS                SLDA R1,D2(B2)

SRDA          Shift left double algebraic         RS                SRDA R1,D2(B2)

SLDL          Shift left double logical             RS                SLDL R1,D2(B2)

SRDL          Shift right double logical           RS                SRDL R1,D2(B2)

The algebraic shifts preserve the sign bit in a register, and thus are useful for arithmetic.

The logical shifts do not preserve the sign bit.

The shift operations set the standard condition codes, for use by BC and BCR.

The register numbers for the double shift instructions must be an even number,
referencing the first of an even–odd register pair (see below).

Shift Instructions: Object Code Format

All shift instructions are four–byte instructions of the form OP R1,R3,D2(B2).

 Type Bytes 1 2 3 4 RS 4 R1,R3,D2(B2) OP R1 R3 B2 D2 D2D2

The first byte contains the 8–bit instruction code.

The second byte contains two 4–bit fields, each of which encodes a register number.
The first register number (R1) is the register to be shifted.  The second register number
(R3) is not used and conventionally set to 0.

The third and fourth bytes contain a 4–bit register number and 12–bit value.  In many
type RS instructions, these would indicate a base register and a displacement to be
used to specify the memory address for the operand in storage.

For the shift instructions, this field is considered as a value to indicate the shift count.
The value is in the form below.  B is the number of the register to be used as a base
for the value.  The next three hexadecimal digits are added to the value in that register.

 B D1 D2 D3

The sum is used as a shift count, not as an address.  The two conventional uses are to specify a constant shift count and to use a register to contain the shift count.  Consider the following two examples, each of which uses the SRA instruction with opcode X‘8A’.

Object Code    Source Code

8A 90 00 0A    SRA 9,10    BASE REGISTER = 0, DISPLACEMENT
= 10; THE SHIFT COUNT IS 10.

8A 90 B0 00    SRA 9,0(11) HERE REGISTER 11 (X‘B’) CONTAINS
THE SHIFT COUNT.

Example Object Code Analysis: SLL

Shift Left Logical       Operation code = X‘89’

This is also a type RS instruction, though the appearance of a typical use seems to deny
this.  Consider the following instruction which shifts R6 left by 12 bits.

SLL R6, 12    Again, I assume we have set R6 EQU 6

The above would be assembled as 89 60 00 0C, as decimal 12 is X‘C’.

The deceptive part concerns the value 12, used for the shift count.  Where is that stored?

The answer is that it is not stored, but is used as a value for the shift count.

The object code 00 0C literally indicates the computation of a value that is an sum of
decimal 12 from the value in base register 0.  But “0” indicates that no base register is
used, hence the value for the shift is decimal 12.

Here are three lines from a working program I wrote on 2/23/2009.

000014 5840 C302     00308    47          L     R4,=F’1’

000018 8940 0001     00001    48          SLL   R4,1

00001C 8940 0002     00002    49          SLL   R4,2

Note that the load instruction makes use of a literal.  The assembler will create an entry in the literal pool and populate it with the value 1.  Here, my code calls for register 12 (X‘C’) to serve as the base register.  The literal is stored at offset X‘302’ from the address stored
in that base register.

While it might seem plausible that the SLL instructions similarly generate literals, this is not the case.  In each, as noted above, the value is stored as a count in the base/displacement format, which is here pressed into duty to store a value and not an address.

Single Shifts: Algebraic and Logical

Here are some diagrams describing shifts in a single register.  These examples will
assume an 8–bit register with the IBM bit numbering scheme; 32 bits are hard to draw.

This figure illustrates logical shifts by 1 for these imaginary 8–bit registers.

This figure illustrates algebraic shifts by 1 for these imaginary 8–bit registers.

The actual IBM assembler shift instructions operate on 32–bit registers and can shift by
any number of bit positions.  For single register shifts, the  shift count should be a
non–negative integer less than 32.  For double register shifts, the upper limit is 63.

Double Register Shifts

Each of these four instructions operates on an even–odd register pair.

The algebraic shifts preserve the sign bit of the even register; the logical shifts do not.

Here is a diagram illustrating a double algebraic right shift.

If the above example were a logical double right shift, a 0 would have been
inserted into the leftmost bit of the even register.

Remember to consider the shifts in register pairs, preferably even–odd pairs.

Consider the following code:             SR R9,R9    This clears R9

SRDL R8,32

The double–register right shift moves the contents of R8 into R9 and clears R8,
as it is a logical shift.

Single Register Left Shifts: Another View

First consider the left shifts.  There are two single–register variants: SLL and SLA.

For an N–bit logical left shift, bits 0 through (N – 1) are shifted out of the register
and discarded.  Bits 31 through (32 – N) are filled with 0.

Bit 0 is not considered as a sign bit in a logical shift; it may change values.

For an N–bit arithmetic left shift, bits 1 through N are shifted out of the register
and discarded.  Bits 31 through (32 – N) are filled with 0.  Bit 0 (the sign bit)
is not changed.

The overflow bit can be set by an arithmetic left shift.  This will occur if
the bit shifted out does not match the sign bit that is retained in bit 0.

We shall see later that setting the overflow bit indicates that the result of the
shift cannot be viewed as a valid result of an arithmetic operation.

Single Register Right Shifts: Another View

Now consider the left shifts.  There are two single–register variants: SRL and SRA.

For either of these shift types, a shift by N bit will cause the N least significant bits
to be shifted out of the register and discarded.

For an N–bit logical right shift, the value 0 is shifted into the N most significant bits,
bits 0 through (N – 1) of the register.  Bit 0 is not considered a sign bit and is
shifted into bit N of the register.  The sign of the number may change.

For an N–bit arithmetic right shift, bit 0 is considered as a sign bit.  Bit 0 is not changed,
but is shifted into bits 1 through N of the register.  At the end, the (N + 1) most
significant bits of the register contain what used to be bit 0 (the sign bit).

For an arithmetic right shift, the sign of the shifted result is the same as that of the
original.  If the sign bit originally is 0, the SRL and SRA produce identical results.

Double Register Shifts: Another View

The double register shifts are just generalizations of the single register shifts.

In these double register shifts, a pair of registers is viewed as a single 64–bit value.

The IBM coding convention (and possibly the CPU hardware) calls for this pair to be
what is called an even–odd pair, in which the odd number is one more than the even.

Examples of even–odd register pairs are: 4 and 5, 6 and 7, 8 and 9, 10 and 11.

Consider the two registers R5 and R6.  While it is true that 5 is an odd number
and 6 is an even number; these two registers do not form an even–odd pair.

Each of these is a member of a distinct even–odd pair.

Shift Examples

Here are some typical shift examples, with comments.

SRA R9,2      Algebraic right shift by 2 bit positions, equivalent to division
by 4.  SRA by N bit positions is equivalent to division by 2N.

SLA R8,3    Algebraic left shift by 3 bit positions, equivalent to multiplication
by 8.  SLA by N bit positions is equivalent to multiply by 2N.

NOTE:   Multiplication using the M, MH, or MR instructions is rather slow, as is
division with either D or DR.  It is almost universal practice to use
arithmetic left shifts to replace multiplication by a power of 2 and
arithmetic right shifts to replace division by a power of 2.

Example:               Consider the following three lines of code.

L R5,AVAL    ASSUME AVAL IS THE LABEL FOR A FULL-WORD

LR R6,R5     COPY VALUE INTO R6

SRA R6,3     SAME AS MULTIPLY BY 8

AR R6,R5     R6 NOW HAS 9 TIMES THE VALUE IN R5.

More on Shifting and Arithmetic

The association of arithmetic left shifting with multiplication, and arithmetic right
shifting with division is useful.  However, there are limits to this interpretation.

To illustrate this for multiplication, I select an integer that is a simple power of 2,
4096 = 212.  As a 16–bit integer, this would be stored in memory as follows.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Taking the two’s complement of the above, we find that –4096 is stored as follows.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

We shall use each of these two integer values to illustrate the limits of the arithmetic
left shift.  We shall then consider the following pair as subject to an arithmetic right shift.

+32 = 25 is stored as follows.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

–32 is stored as follows.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

Arithmetic Left Shifts as Multiplication

We first consider some left shifts that can validly be interpreted as multiplication.
For each of these integers, consider a SLA 2 (Arithmetic Left Shift by 2 bit positions).

According to our interpretation, a SLA 2 should be equivalent to multiplication by 22 = 4.

The 4096 = 212 becomes 16384 = 214.  This is as it should be.

4096·4 = 16384 and 212·22 = 214.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The –4096 = –(212) becomes –16384 = –(214).  This is as it should be.

(–4096)·4 = –16384 and –(212)·22 = –(214).

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Overflow on Shifting Left (Multiplication)

Consider again 4096 = 212, stored as a 16–bit integer.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Consider the result of SLA 3 (Arithmetic Left Shift by 3 bit positions).

According to our interpretation, a SLA 3 should be equivalent to multiplication by 23 = 8.
We note that 4096·8 = 32768 and 212·23 = 215 = 32768.

But, the 4096 = 212 becomes –32768 = –(215).  The sign has “gone bad”, as a result of
arithmetic overflow.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

But consider the same operation on –4096 = –(212).

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

After the shift, we have the proper result; –4096·8 = –32768.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

More on Overflow While Shifting Left

In this illustration we continue to focus on 16–bit two’s complement integers.
A 32–bit representation would show the same problem, only at larger values.

Suppose we have the valid integer –32,768 = –(215).  This is stored as follows.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Suppose we attempt a SLA (Shift Left Arithmetic) by any positive bit count.
The result will remain the same.  The sign bit is always preserved in an arithmetic shift.

In attempting a SLA as a substitute for multiplication by a power of two, we find that.

(–32,768)·2 = –32,768.

(–32,768)·4 = –32,768.

(–32,768)·8 = –32,768.

In other words, once overflow has been hit, SLA ceases to serve as multiplication.

Arithmetic Right Sifting as Division

Here the results are a bit less strange.  First consider our positive number, +32.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

A SRA 4 (Arithmetic Right Shift by 4) should yield 32/16 = 2.  It does.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

Further shifting this result by 1 bit position will give the value 1 (as expected).

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

However, any more SRA (Arithmetic Right Shifts) will give the value 0.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

This is as expected for integer division, and is not surprising.

More on Arithmetic Right Sifting as Division

Here the results are a bit less strange.  Now consider our negative number, –32.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

A SRA 3 (Arithmetic Right Shift by 3) should yield (–32)/8 = (–4).  It does.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0

A SRA 2 (Arithmetic Right Shift by 2) should yield (–4)/4 = (–1).  It does.

 Sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But note that further Arithmetic Right Shifts continue to produce the result –1.

What we are saying is that (–1) / 2 = –1.  If the above is acceptable, then the SRA works well as a substitution for division by a power of two.

Register Pairs: Multiplication and Division

We now discuss two instructions that, in their full–word variants, demand the use of a
64–bit “double word”.  Rather than use the type, we use a pair of registers.

The assembly language definition calls for “even–odd register pairs”.

Each pair of registers is referenced by its (lower numbered) even register.

The standard pairs from the general–purpose registers that are not reserved for other use are shown in the following list.

R4 and R5                                        R8 and R9

R6 and R7                                        R10 and R11

When such a pair is referenced by a multiply or divide instruction, it is treated as
a 64–bit two’s–complement integer with the sign in bit 0 of the even register.

Remember that the bits of a register are numbered left to right, so that bit 0 is
the sign bit and bit 31 is the rightmost (least significant) bit.

Examples:              M R4,F2  MULTIPLY VALUE IN R5 BY VALUE IN
FULL-WORD F2. RESULTS IN (R4, R5)

D R6,F3  DIVIDE 64-BIT NUMBER IN (R6, R7) BY F3

Full–Word Multiplication

This slide will cover the two multiplication instructions based on full words.
The half–word multiplication instruction will be discussed later.

The two instructions of interest here are:

Mnemonic                     Description                    Type                           Format

M                 Multiply full–word                 RX                  M R1,D2(X2,B2)

MR              Multiply register                      RR                   MR R1,R2

For each of these, one uses a selected even–odd pair to hold the 64–bit product.
Here is the status of the registers in the selected pair; think (4, 5) or (8, 9), etc.

 Even Register Odd Register Before multiplication Not used: contents are ignored Multiplicand After multiplication Product: high–order 32 bits Product: low–order 32 bits

If the product can be represented as a 32–bit number, the even register will contain
the extended sign bit, so that the 64–bit number in the register pair has the right sign.

Note that the multiplication overwrites the value of the multiplicand in the odd register.

Full–Word Multiplication: Examples

In the first fragment, the starting value in R4 is irrelevant, as it is ignored.

Each example assumes two full–words: MULTCAND and MULTPLER.

L R5,MULTCAND     LOAD THE MULTIPLICAND INTO R5.

SR R4,R4          CLEAR R4. THIS IS REALY USELESS.

M  R4,MULTPLER    MULTIPLY BY A FULLWORD

*    R4 NOW HAS BITS  0 – 31 OF THE 64-BIT PRODUCT

*    R5 NOW HAS BITS 32 – 63 OF THE 64-BIT PRODUCT

Another code fragment:

L R9,MULTCAND     LOAD THE MULTIPLICAND INTO R9.

L R5,MULTPLER     LOAD MULTIPLIER INTO R5

MR R8,R5          MULTIPLY BY FULL-WORD VALUE IN R5

*    R8 NOW HAS BITS  0 – 31 OF THE 64-BIT PRODUCT

*    R9 NOW HAS BITS 32 – 63 OF THE 64-BIT PRODUCT

Half–Word Multiplication

Mnemonic                     Description                       Type                           Format

MH              Multiply half–word                 RX                     MH R1,D2(X2,B2)

This instruction requires only one register.  It is loaded with the multiplicand before the
multiplication, and receives the product.

Note that this is the product of a 32–bit number (in the register) and a 16–bit number
in the half–word in memory.  This will result in a 48–bit product.

Of bits 0 – 47 of the product, only bits 16 – 47 are retained and kept in the 32–bit
register as the product.  If the absolute value of the product is greater than 231, the sign
bit of the result (as found in the register) might not be the actual sign of the product.

Here is an example of a proper use of the instruction, which will give correct results.

LH  R3,MULTCAND   Each of these two arguments is a half–word

MH  R3,MULTPLER   with value in the range: –215 Ł N Ł (215 – 1).

MULTCAND DC H‘222’

MULTPLER DC H‘44’

The magnitude of the product will not exceed (215)·(215) = 230, an easy fit for a register.

Full–Word Division

This slide will cover the two division instructions based on full words.
The half–word division instruction will be discussed later.

The two instructions of interest here are:

Mnemonic                           Description                 Type            Format

D                  Divide full–word                    RX               D R1,D2(X2,B2)

DR               Divide register                        RR                DR R1,R2

For each of these, one uses a selected even–odd pair to hold the 64–bit dividend.

Here is the status of the registers in the selected pair; think (4, 5) or (8, 9), etc.

 Even Register Odd Register Before division Dividend: high–order 32 bits Dividend: low–order 32 bits After division Remainder from division Quotient from division

In each of the full–word division operations, it is important to initialize the even register
of the pair correctly.  There are two cases to consider.

1.      The dividend is a full 64–bit number, possibly loaded with a LM instruction.

2.      The dividend is a 32–bit number.  In that case, we need to initialize both registers.

Full–Word Division: Example 1

In this example, I am assuming a full 64–bit dividend that is stored in two adjacent
full words in memory.  I use this memory structure to avoid adding anything new.

LM R10,R11, DIVHI   LOAD TWO FULLWORDS

D  R10,DIVSR        NOW DIVIDE

*       R10 CONTAINS THE REMAINDER

*       R11 CONTAINS THE QUOTIENT

DIVHI   DC F‘1111’    ARBITRARY NUMBER THAT IS NOT TOO BIG

DIVLO   DC F‘0003’    ANOTHER ARBITRARY NUMBER

DIVSR   DC F‘19’      THE DIVISOR

Important Note:    This process of assembling a 64–bit dividend from two full words
might run into problems if DIVLO is seen as negative.

Here, I choose to ignore that point.

Full–Word Division: Example 2

In this example, I am assuming a 32–bit dividend and using a more standard
approach.  Please note that it works only for positive dividends.

SR  R10,R10         SET R10 TO 0

L   R11,DIVIDEND    LOAD FULL–WORD DIVIDEND

D   R10,DIVISOR     DO THE DIVIDING

*       R10 CONTAINS THE REMAINDER

*       R11 CONTAINS THE QUOTIENT

DIVIDEND DC F‘812303    JUST SOME NUMBER.

DIVISOR  DC F‘16384’    A POWER OF TWO, SEE NOTE BELOW

NOTES:    1.         This works only for a positive dividend.  The reason is that, by clearing
the even register of the even–odd pair, I have declared the 64–bit dividend
to be a positive number, even if R11 is loaded with a negative number.

2.         There is a much faster way to divide any number by a power of two.
This method, using a shift instruction, will be discussed later.

Full–Word Division: Example 3

In this example, I am assuming a 32–bit dividend and using the standard approach
that will work correctly for all dividends.  The dividend is first loaded into the even
register of the even–odd pair and then shifted into the odd register.

This shifting causes the sign bit of the 64–bit dividend to be set correctly.

L   R10,DIVIDEND    LOAD INTO THE EVEN REGISTER

SRDA R10,32         SHIFTING BY 32 BITS PLACES

*                            THE DIVIDEND INTO R11.

*                            R10 RETAINS THE SIGN BIT D   R10,DIVISOR     DO THE DIVIDING

*       R10 CONTAINS THE REMAINDER

*       R11 CONTAINS THE QUOTIENT

DIVIDEND DC F‘812303    JUST SOME NUMBER.

DIVISOR  DC F‘16384’    A POWER OF TWO, SEE NOTE BELOW

We shall discuss this a bit more after we have discussed the shift operations.

Full–Word Division: Example 4

Here is a more realistic example of the use of a full 64–bit dividend.

Code fragment 1: Create the 64–bit product and store in adjacent full words.

L   R5,MCAND        LOAD THE MULTIPLICAND INTO R5.

M   R4,MPLER        MULTIPLY BY A FULLWORD

*       R4 NOW HAS BITS  0 – 31 OF THE 64-BIT PRODUCT

*       R5 NOW HAS BITS 32 – 63 OF THE 64-BIT PRODUCT STM R4,R5,PRODHI    STORE THE 64-BIT PRODUCT

Code fragment 2: Some time later use the 64–bit product as a dividend for division.

LM  R10,R11,PRODHI  LOAD TWO FULLWORDS

D   R10,DIVSR       NOW DIVIDE

*       R10 CONTAINS THE REMAINDER

*       R11 CONTAINS THE QUOTIENT

PRODHI  DC F‘0’        TWO FULL WORDS SET ASIDE

PRODLO  DC F‘0’        64 BITS (8 BYTES) OF STORAGE.

Diversion: Shifting the Dividend into Place

Consider two possible dividends: + 100 and – 100.

Consider the code fragment below.

LH R6,=H‘100’

SRDA R6,32

After the first instruction is executed, register R6 contains the full–word value +100, as shown below.

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 0000 0000 0000 0000 0000 0000 0110 0100

After the shift in the second instruction, the contents of R6 have been shifted to R7,
leaving only the sign bit in R6.

R6

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 0000 0000 0000 0000 0000 0000 0000 0000

R7

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 0000 0000 0000 0000 0000 0000 0110 0100

Shifting the Dividend Into Place (Part 2)

Now consider the code fragment.

LH R8,=H‘-100’

SRDA R8,32

After the first instruction is executed, register R8 contains the full–word value –100,
as shown below.

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 1111 1111 1111 1111 1111 1111 1001 1100

After the shift in the second instruction, the contents of R8 have been shifted to R9,
leaving only the sign bit in R8.

R8

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 1111 1111 1111 1111 1111 1111 1111 1111

R9

 0 – 3 4 – 7 8 – 11 12 – 15 16 – 19 20 – 23 24 – 27 28 – 31 1111 1111 1111 1111 1111 1111 1001 1100

Boolean Operators: AND, OR, XOR

We now conclude our investigation of binary integer data by examining the Boolean operators, which treat binary data one bit at a time.  We shall repeat the basic definitions, discuss the implementation by IBM, and close by repeating a natural application.  The three Boolean operators directly supported by IBM are the logical AND, OR, and NOT.

Each of these operates on binary data, one bit at a time according to the following tables.

AND   0·0 = 0            OR      0+0 = 0            XOR   0Ĺ0 = 0
0·1 = 0                        0+1 = 1                        0Ĺ1 = 1
1·0 = 0                        1+0 = 1                        1Ĺ0 = 1
1·1 = 1                        1+1 = 1                        1Ĺ1 = 0

To show the bitwise nature of these operations, we consider a few examples as
applied to four–bit integers.

1010   1010      0101      0101      0101

· 0111  · 1101    + 0000    + 1111    Ĺ 1111
0010   1010      0101      1111      1010

Note that the XOR function can be used to generate the Boolean not function.  The Boolean NOT, denoted by  and defined by .  As seen above, this can be extended bitwise, so that the rightmost example takes the logical NOT of the digits 0101.

One of the more natural uses of the Boolean operators is to do bitwise operations on data
represented in 8–bit bytes and denoted by two 4–bit hexadecimal digits.  There are three operations that will commonly be seen in assembly language programs.

1.      Select a bit position in a byte and force that bit to have the value 1.

2.      Select a bit position in a byte and force that bit to have the value 0.

3.      Select a bit position in a byte and flip the value of that bit.

We shall examine the use of these operations on 4–bit fields, as longer data structures
can be analyzed one hexadecimal digit at a time.  We use the IBM bit numbering.

 Bit number 0 1 2 3 Bit value 8 4 2 1

Here are the basic masking operations that can be performed on a 4–bit hexadecimal digit.

 Bits Affected To set the bit, use OR with To clear the bit, use AND with None 0000 X‘0’ 1111 X‘F’ 0 1000 X‘8’ 0111 X‘7’ 1 0100 X‘4’ 1011 X‘B’ 2 0010 X‘2’ 1101 X‘D’ 3 0001 X‘1’ 1110 X‘E’ 0 and 1 1100 X‘C’ 0011 X‘3’ 0 and 2 1010 X‘A’ 0101 X‘5’ 0 and 3 1001 X‘9’ 0110 X‘6’ 1 and 2 0110 X‘6’ 1001 X‘9’ 1 and 3 0101 X‘5’ 1010 X‘A’ 2 and 3 0011 X‘3’ 1100 X‘C’ 0, 1, and 2 1110 X‘E’ 0001 X‘1’ 0, 1, and 3 1101 X‘D’ 0010 X‘2’ 0, 2, and 3 1011 X‘B’ 0100 X‘4’ 1, 2, and 3 0111 X‘7’ 1000 X‘8’ ALL 1111 X‘F’ 0000 X‘0’

System/370 architecture supports three Boolean functions, each in four formats.

 Instruction Format Operands Logical AND Logical OR Logical XOR NR OR XR RR Two registers N O X RX Register and storage NI OI XI SI Register and immediate NC OC XC SS Two storage locations

Each of these twelve instructions sets the condition codes used by the conditional branch instructions in the same way.  If every bit in the result is 0, the result is 0 and condition code 0 is set.  If any bit in the result is 1, the result is not negative and condition code 1 is set, as if the result were negative.  Here are two equivalent ways to test results.

To determine                     Yes                        No
All target bits are 0           Use BZ                  Use BNZ
Any target bit is 1             Use BM                 Use BNM

Here are the logical instructions, grouped by type.

Type RR

This is a two–byte instruction of the form OP R1,R2.

 Type Bytes Operands RR 2 R1,R2 OP R1 R2

The first byte contains the 8–bit instruction code.  The second byte contains two 4–bit fields, each of which encodes a register number.  This instruction format is used to process data between registers.

Here are the three Boolean instructions of this type.

NR   Logical AND                       Opcode is X‘14’

OR   Logical OR                          Opcode is X‘16’

XR   Logical Exclusive OR          Opcode is X‘17’

Type RX

This is a four–byte instruction of the form OP R1,D2(X2,B2).

 Type Bytes Operands 1 2 3 4 RX 4 R1,D2(X2,B2) OP R1 X2 B2 D2 D2D2

The first byte contains the 8–bit instruction code.  The second byte contains two 4–bit fields, each of which encodes a register number.  The first operand, encoded as R1, is the target register for the instruction.  The second register number, encoded as X2, is the optional index register.  Bytes 3 and 4 together contain the address of the second operand in base and displacement form, which may be modified by indexing if the index register field is not zero.

Here are the three Boolean instructions of this type.

N      Logical AND                       Opcode is X‘54’

O      Logical OR                          Opcode is X‘56’

X      Logical Exclusive OR          Opcode is X‘57’

Type SI

This is a four–byte instruction of the form OP D1(B1),I2.

 Type Bytes Operands 1 2 3 4 SI 4 D1(B1), I2 OP I2 B1 D1 D1D1

The first byte contains the 8–bit instruction code.  The second byte contains the 8–bit value of the second operand, which is treated as an immediate operand.  The instruction contains the value of the operand, not its address.  The first operand is an address, specified in standard base register and displacement form.  Note that this first operand must reference the address of a single byte, as this is a byte–oriented operation.

Here are the three Boolean instructions of this type.

NI     Logical AND                       Opcode is X‘94’

OI     Logical OR                          Opcode is X‘96’

XI     Logical Exclusive OR          Opcode is X‘97’

Type SS

These are of the form OP D1(L,B1),D2(B2), which provide a length for only operand 1.  The length is specified as an 8–bit byte.

 Type Bytes Operands 1 2 3 4 5 6 SS(1) 6 D1(L,B1),D2(B2) OP L B1 D1 D1D1 B2 D2 D2D2

The first byte contains the operation code.  The second byte contains a value storing one less than the length of the first operand, which is the destination for the operation.

Bytes 3 and 4 specify the address of the first operand, using the standard base register and displacement format.  Bytes 5 and 6 specify the address of the second operand, using the standard base register and displacement format.

Here are the three Boolean instructions of this type.

NC   Logical AND                       Opcode is X‘D4’

OC   Logical OR                          Opcode is X‘D6’

XC   Logical Exclusive OR          Opcode is X‘D7’

Another Look at Case Conversion

In order to investigate the difference between upper case and lower case letters, we here present a slightly different version of the EBCDIC table.  Admittedly, we have covered this in a previous chapter, but cover it again within the context of the Boolean operators.

 Zone 8 C 9 D A E Numeric 1 “a” “A” “j” “J” 2 “b” “B” “k” “K” “s” “S” 3 “c” “C” “l” “L” “t” “T” 4 “d” “D” “m” “M” “u” “U” 5 “e” “E” “n” “N” “v” “V” 6 “f” “F” “o” “O” “w” “W” 7 “g” “G” “p” “P” “x” “X” 8 “h” “H” “q” “Q” “y” “Y” 9 “i” “I” “r” “R” “z” “Z”

The structure implicit in the above table will become more obvious when we compare
the binary forms of the hexadecimal digits used for the zone part of the code.

Upper Case     C = 1100   D = 1101   E = 1110
Lower Case     8 = 1000   9 = 1001   A = 1010

Note that it is only one bit in the zone that differentiates upper case from lower case.
In binary, this would be noted as 0100 or X‘4’.  As this will operate on the zone field of a character field, we extend this to the two hexadecimal digits X‘40’.  The student should verify that the one’s–complement of this value is X‘BF’.  Consider the following operations.

UPPER CASE

‘A’             X’1100 0001’                                          X’1100 0001’
OR      X ‘40’        X‘0100 0000’  AND    X ‘BF’       X‘1011 1111’
X’1100 0001’                 X’1000 0001’

Converted to                  ‘A’                                                                      ‘a’

Lower case

‘a’              X’1000 0001’                                          X’1000 0001’
OR      X ‘40’        X‘0100 0000’  AND    X ‘BF’       X‘1011 1111’
X’1100 0001’                 X’1000 0001’

Converted to                  ‘A’                                                                      ‘a’

We now have a general method for changing the case of a character, if need be.
Assume that the character is in a one byte field at address LETTER.

Convert a character to upper case.      OI,LETTER,=X‘40’

This leaves upper case characters unchanged.

Convert a character to lower case.      NI,LETTER,=X‘BF’

This leaves lower case characters unchanged.

Change the case of the character.       XI,LETTER,=X‘40’
This changes upper case to lower case and lower case to upper case.