sections in this module  City
College of San Francisco  CS270 Computer Architecture Module: MIPSI 
module list 
Bit
Instructions and Instruction Encoding
Bit instructions are used to manipulate data at the bit level. Although not common in highlevel code, their use is quite common in instructions generated.
The shift instructions
Consider a number 2^N where 31 > N > 0. This number is
represented in binary on our machine by a word with a single bit
set, bit #N. Thus 2^5 has bit 5 set (counting from 0) or 100000 in
binary. In decimal, of course, this is 32.
If we move this bit to the left one position, it becomes 2^6, or 64. Moving left one bit position multiplies the number by 2. This is a leftshift operation. A leftshift by P positions multiplies the value by 2^P.
Similarly, moving the bit one position to the right, divides it by 2. Thus, 2^6 (decimal 64) becomes 2^5 (decimal 32) when rightshifted one position.
Of course, integer division is not exact. Any bits lost (shifted off the right side) mean the result is inexact. We know this from integer division in general: 3/2 is 1 in integer division. This is because the binary number 011 (decimal 3), when rightshifted one position becomes 001, and the previous 2^0 bit is lost.
Rightshifting has another problem: what do we do when we rightshift a number that has its mostsignificant bit set? If we shift zero bits in from the left, the sign bit is no longer set! This would be correct if the original number was unsigned. Let's look at an example using a fourbit numbers:
1100 in a fourbit unsigned number is decimal 12. If we rightshift this one position and set the leftmost position to 0, the result is 0110, or 6 base 10. This is the correct answer. However, if the original number was interpreted as signed, its original value would have been 4. When we rightshift 4 by 1 position it shouldn't become 6! Instead we want to set the [shiftedin] mostsignificant bit to 1. This would produce 1110, which, when interpreted as a signed fourbit number is 2.
These two types of right shifts are called logical (when we treat the number as unsigned, and shift 0 bits in) and arithmetic (when we treat the number as signed, and replicate the sign bit in the bits shifted in). Since there is only one type of leftshift and it shifts 0 bits in, it is also called logical.
Example:
Implement the following operation in MIPS code:
Choose the correct code sequence from the following:
lw $t0,index sll $t1,$t0,1 sw $t1,index 
lw $t0,index srl $t1,$t0,1 sw $t1,index 
lw $t0,index sra $t1,$t0,1 sw $t1,index 
AND and OR operators
I hope we don't need to discuss what AND and OR mean for a pair of bits. Give two words of some size, then, x AND y (written xy. there could be a dot between them, but the dot wont display correctly here) is simply the AND of each corresponding bit. Similarly, the OR of x and y (written x + y) is the OR of each corresponding bit. Resorting to our fourbit numbers, if x is 0110 and y is 1100
That's how easy AND and OR are. The amazing thing is that these three primitives (and, or, and shift) form the basis of all arithmetic operations!
Example:
Given two 16bit unsigned nonzero numbers in $t0 and $t1, set $t2 so that $t2 has the number in $t1 as its most significant 16 bits and the number in $t0 as its least significant 16 bits.
Which code sequence is correct:
and $t2,$t0,$t1 
sll $t2,$t1,16 or $t2,$t2,$t0 
sll $t2,$t1,16 and $t2,$t2,$t0 
sll $t2,$t0,16 sra $t2,$t2,16 sll $t1,$t1,16 or $t2,$t1,$t2 
(One of the answers is correct all of the time and one is correct some of the time.)
Other bitwise operators
Besides AND and OR, there are two operators that are useful: NOR and XOR. For those who are unfamiliar with these operators, here are the basics:
Given two singlebit numbers X and Y, X NOR Y is 1 only if BOTH X
and Y are 0.
Similarly X XOR Y is 1 only if exactly one of X or Y is 1.
Here is a truthtable:
X  Y  X AND Y 
X OR Y 
X NOR Y 
X XOR Y 
0 
0 
0 
0 
1 
0 
0 
1 
0 
1 
0 
1 
1 
0 
0 
1 
0 
1 
1 
1 
1 
1 
0 
0 
MIPS Instructions
All of and, or, xor and nor have Rtype MIPS instructions where three registers are used:
op rd, rs, rt # rd = rs op rt for op=and,or,xor,nor
All of these except nor also have immediate counterparts where the 16bit immediate value is treated as unsigned (not signextended) when the operation is performed. These are useful for creating 16bit ANDs, ORs and XORs.
Extracting values stored in a bitfield
A common operation in lowlevel computer software requires the
extraction of a particular sequence of bits and their
interpretation as a value. A simple example of this are the basic
permissions on Unix.
Without getting too much into Unixland, the first 16bits of the master data structure stored for every data object on Unix, called an inode, comprise the object's mode. The mode includes both the filetype and the basic permissions. Within these 16 bits, bit #8 (counting from 0) indicates whether the owner of the file can read it. Thus, our data looks like
B
where B is the bit we want, and  indicates each bit that is 'in the way'.
Problem: if a file's mode is in $t0, set $t1 to 1 if the file's owner can read it, 0 otherwise.
There are several ways to solve this problem. Let's look at this graphically again. Here is the operation we are interested in:
B > 000000000000000B
You may already see a simple solution to this problem, but we are going to take the long way around and discuss the generallyuseful idea of a mask.
A mask is a special bit
pattern that is constructed so that an AND or OR operation can be
applied to a selected sequence of bits to isolate it. In our case,
we want to construct a mask so that we can extract our single bit,
i.e., so that the operation
B
> 0000000B00000000
can be performed. Of course, if we use an AND operator, the bit pattern is all zeros except in the position of B, where we have a 1:
B AND 0000000100000000 > 0000000B00000000
Once this operation is performed, we can simply shift B to the correct position:
0000000B00000000
> 000000000000000B
In our case, where the file's mode is in $t0, the following sequence would be used
We indicated earlier that this may not be the easiest solution, though it is the most general. A simpler solution would probably be
The concept of a mask
is very useful and can be used to isolate and extract any data
value. We will see it again later. As one further example, suppose
we wanted to isolate all
the permissions bits in the word in $t0, placing the isolated bits
in $t2. Since the permissions take up a total of 12 bits, we would
use
andi
$t2,$t0,0x0fff
Encoding MIPS instructions
As you might expect, instruction encoding for MIPS is significantly more complicated that it was for the Simple Machine. Consider two standard (but different) Rtype instructions:
As discussed in an earlier section (and as you can see from the two examples above), Rtype instructions must have room in the instruction encoding for the following parts:
Since there are 32 instructions, the register fields must have 5
bits. Similarly, since the maximum shift amount is 31, the shift
amount field must have 5 bits. This leaves 12 bits for the opcode.
To make the encoding for different instruction types more
compatible, the opcode field was broken into two 6bit fields,
called opcode and function. For Rtype instructions, the function
(funct) field
indicates the instruction and the opcode (op) field (which is 0 or
1 for an Rtype instruction) indicates to look in the funct field for the
operation code. This allows the fields to be laid out so the
instruction is symmetrical around the midpoint:
Rtype
Instruction Encoding 

op 
rs 
rt 
rd 
shamt 
funct 

31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12  11 
10 
9  8  7  6  5  4  3  2  1  0 
(The last row in the table above is for encoding the result in hexadecimal.)
Let's try encoding our example instructions above
We
need the numeric values. The values for the instructions are easiest to
see in section A.10 of your text, as are the instruction formats. The
register values, however, must be obtained from the Green Sheet in the
front of your textbook (front cover). The Green Sheet is also available
from a PDF link on my website's Syllabus for the class.)
The register positions in the add instruction is add rd,rs,rt. Now we can encode the first instruction: add $t0, $t1, $t2
op (0) 
rs (9) 
rt (10) 
rd (8) 
shamt (0) 
funct (32) 

0  0  0  0  0  0  0

1  0  0  1  0  1  0  1  0  0  1  0  0  0  0  0  0  0  0  1  0  0

0  0  0 
0 
1 
2 
A 
4 
0 
2 
0 
Checking on MARS, 0x012A4020 is indeed correct.
Let's try the second instruction: sll $t0, $t1, 4(The register positions in the shift instructions are sll
rd, rt, shamt ).
op (0) 
rs (0) 
rt (9) 
rd (8) 
shamt (4) 
funct (0) 

0  0  0  0  0  0  0

0  0  0  0  0  1  0  0  1  0  1  0  0  0  0  0  1  0  0  0  0  0

0  0  0 
0 
0 
0 
9 
4 
1 
0 
0 
Again, MARS agrees with 0x00094100 for this shift instruction.
That's all there is to encoding Rtype instructions. Let's move on to Itype:
The Itype instructions keep the same fields in the mostsignificant 16 bits of the instruction word, but merge the leastsignificant 16bits into the 16bit signed immediate (imm) field:
op 
rs 
rt 
imm 

31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12  11 
10 
9  8  7  6  5  4  3  2  1  0 
The normal encoding for Itype instructions is op rt, address, where address is imm(rs). Let's try a couple of Itype instructions:
lw $s1, 4($sp)
lui $at, 0x0400
Here are the values we need:
lw: opc=35, $s1=17, $sp=29
For lw $s1, 4($sp), here is the encoding
op (35) 
rs (29)  rt (17)  imm (4) 

1  0  0  0  1  1  1

1  1  0  1  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0

1  0  0 
8 
F 
B 
1 
0 
0 
0 
4 
lui: opc=15, $at=1
For lui $at, 0x0400
op (15) 
rs (0) 
rt (1) 
imm (0x400) 

0  0  1  1  1  1  0

0  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  0  0

0  0  0 
3 
C 
0 
1 
0 
4 
0 
0 
Both of these check out using MARS
Encoding Instructions using Instructions
Using our example instruction of add $t0, $t1, $t2 we will use our bitwise instructions to accomplish two tasks:
Both of these tasks will allow us to practice with masks and our bitwise operators.
Problem: Encode the add $t0, $t1, $t2 instruction, placing the result in $t0
If you remember from our earlier discussion, we had values for
these constants:
Let's assume we have the following register assignments already.
(We are doing the general solution here rather than optimizing for
0valued fields.)
Here is the easiest way.
The alternate way is to OR in each piece. Let's do this for
practice.
Interestingly, this took one less instruction.
Problem: Take an existing Rtype instruction and modify the rd field, setting it to $t4. Leave the remainder of the fields intact.
This involves ORing in the new value of rd. But first, the current instruction's rd field must be zeroed. This is a common use of masks. Here are the steps involved:
Assuming our existing instruction is in $t0 and the value of the new rd field is in $t7, here is the code
Note: We create the complement of the mask, then complement it to
get our correct mask (this is the nor instruction). We can't use a
load immediate instruction, as this will signextend the result.
Instead we must OR the value into a word of zeros. $zero comes in
very handy here.
(The code for these examples in in the file
online/mipsI/bitexample.s in the public work area on hills)
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