MIPS Architecture Arithmetic/Logic Instructions ALU Design

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MIPS Architecture Arithmetic/Logic Instructions
ALU Design Chapter 4

• By N. Guydosh
• 2/17/04

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Integer Representation

• 32 bit representation.. every bit is contributes
  to the magnitude of the number
• Range for unsigned integers is 0 through 232 -1
  or 0 through 4,294,967,294 0xFFFFFFFF
• But what about negative numbers?- The
  universal standard is signed twos
  Complement.- Automatically results in the most
  significant bit (bit 31, little endian notation)
  being a 1 for negative numbers and a 0 for
  positive numbers ... thus we have a sign bit.
  - Results in simple ALU design

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Twos Complement - 1

• Twos Complement Representation
• 0 (zero) is 0x00000000
• Positive numbers (231 - 1 of them) are
  0x00000001, 0x00000002, ... 0x7FFFFFFF 231
  -1 2,147,483,647
• Negative numbers (231 of them) are -1
  0xFFFFFFFF, -2 0xFFFFFFFE ,. . .
  -2,147,483,647 0x80000001 (2's complement)
  -(231 - 1) -2,147,483,648 0x80000000 (2's
  complement) -231 , see note below p. 213
  text.
• There are more negative numbers than positive
  but well live with it
• NOTE the 2s complement of 0x80000000 is also
  0x80000000 because of a net carry-out when
  forming the 2s complement an anomaly. This
  number has no positive counterpart in a 32 bit
  architecture. We will see later that when we use
  2s complement for the exponent of a floating
  point number, we may treat values such as 10000
  as -0 and 0000 as 0

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Twos Complement - 2

• Three formulas for doing 2's complement
• 1. Complement all the bits (1's complement) and
  add 1
• 2. Starting with the least significant bit,
  retain all low order zeros, retain the first
  nonzero digit, complement the remaining digits.
• 3. A real slick methodIf the digits in the 32
  bit number are B31B30B29 . . .B1B0 Then the
  sign/magnitude decimal equivalent is B31-231
  B30230 B29229 . . . B121 B020
  Note that the sign bit position is multiplied
  by -231 and not 231 !!! A generalization of
  converting from binary to decimal. which also
  works for 2's complement. ... see P. 213-214

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Uses of 2s complement

• How 2's complement is used
• For algebraic addition of signed numbers
  represent negative numbers by their 2's
  complement of the magnitude
• For the operation of subtraction add the 2's
  complement of the minuend to the subtrahend ...
  Take the 2's complement of the minuend even if it
  is a negative number. ... We converted the
  operation of subtraction to addition - All we
  need is an adder.
• Note there we distinguished between the sign of a
  number and the operation (add or subtract) being
  done. In either case we take the 2's
  complement.

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2s Complement Sign Extension

• To convert an integer shorter than 32 bits to a
  32 bit integer we simply extend the sign bit (0
  or 1) to the left until we have 32 bits - for
  example we convert a 16 bit version of 2 and -2
  (base 10) as follows 2 0x0002 gt
  0x00000002 (sign bit is 0) -2 0xFFFE gt
  0xFFFFFFFE (sign bit is 1)

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2s Complement and the Instruction Set

• How this affects the MIPS instruction set . . .
  see p. 215
• The less than test gets more complicated ...
  the slt and slti instructions interpret integers
  as signed -2d 0xfffffffe (2's comp) is less
  than 5280d 0x000014a0
• unsigned version of these instructions sltu and
  sltiu assume numbers are unsigned 0xfffffffe
  is now greater than 0x000014a0

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Data Types And Signs

• Fundamental Principle
• Unlike c language which intrinsically types data,
  data represented at the assembly language level
  is untyped - data types are interpreted by the
  operation being done on them
• Example in Cint num /
  represents a signed 32 bit integer /
  / num carries a label saying I am signed
  / unsigned int num / represents an unsigned
  32 bit integer / / num carries a
  label saying I am unsigned/
• In assembly language the operation or instruction
  type determines whether num is signed or
  unsigned.
• The unsigned counterparts of slt and slti are
  Assuming that 16 0xFFFFFFFE 17
  0x000014A0 Then slt 8, 16, 17 Signed
  comparison gt 8 1
• sltu 9 16, 17 Signed comparison gt 9
  0
• As an unsigned number 0xFFFFFFFE is a very large
  positive integer (4,294,967,294), but as a signed
  number it is -2.

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Addition, Subtraction And Overflow

• The most straight forward method of doing binary
  addition is the way we do it on paper - starting
  from low order we add corresponding pairs of bits
  and the carry-in generating a sum and carry-out.
  the carry-out is then propagated to the next
  stage.
• Carry-ripple approach
• Do subtraction by adding the 2's complement of
  the minuend to the subtrahend.
• What if the answer cannot be represented in 32
  bits?

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Overflow

• Example consider a 5 bit word for convenience.
  Assume the high (most significant) bit is the
  sign bit for 2s complement-9 10111-5
  11011-14 1 10010 gt there is a net
  carryout, but answer is correct because sign bit
  is correct
• 10010 gt2s compliment -01110 -14 )d
• -9 10111-7 11001-16 1 10000
  gt there is a net carryout, but answer is still
  correct because sign bit is correct 10000 gt
  2s complement -10000 -16 )d (the unique
  number that has no positive counterpart in a 5
  bit scheme)
• - 9 10111-10 10110
• -19 1 01101 gt overflow because sign
  bit wrong 01101 13)d

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Detecting overflow

• Rule (see p. 222, fig 4.4)
• Addition
• Overflow can occur only if the signs are the same
  and
• the sign of the answer is different from the
  sign of the operands.
• Subtraction
• Overflow (ie., underflow) results only if the
  operands have opposite signs
• and the sign of the answer is not the same as the
  sign of the 1st operand
• (or the sign of the answer is the same as the
  sign of the 2nd operand).
• Conditions for overflow (fig. 4.4)Operation Opera
  nd A Operand B Result A B ? 0
  ? 0 lt 0 A B lt 0
  lt 0 ? 0 A B
  ? 0 lt 0 lt 0
  A - B lt 0 ? 0 ? 0
• Simple hardware test for overflow in 2s
  complement Add/Subtract
• Overflow occurs if and only if the carry into
  the sign bit is not the same as the carry out
  from the sign bit (thus explaining why net
  carry-outs are not always overflows). The
  condition being tested for is necessary
  sufficient for overflow. See Exercise 4.42.
  Now review previous examples.

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Exception Handling And Overflows

• MIPS will generate an overflow exception
  (interrupt) only for signed arithmetic.
• The address of the offending instruction is saved
  in the exception program counter (EPC) and a
  branch is made to an error/interrupt handler for
  corrective action.
• MIPS instruction mfc0 is used to move the EPC to
  MIPS general purpose register. On returning from
  the exception handler, the software now has the
  option of returning back to the offending
  instruction via a jr instruction.

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UNSIGNED ARITHMETIC

• Data interpreted as unsigned ( by means of
  unsigned instructions) addresses and other data
  for which negative has no meaning.
• The unsigned counterparts of add, addi, and sub
  are addu, addiu, and subu with syntax the same
  as the signed counterparts.
• The main differences are the unsigned version do
  not cause exceptions in case of overflow

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Logical Operations ... See P. 225

• Shift Logical Left And Shift Logical Right sll
  rd, rt, shamt reg rd reg 16 ltlt shamt (in
  bits) srl rd, rt, shamt reg rd reg 16 gtgt
  shamt (in bits)
• Good for efficient multiplication and division by
  powers (shamt) of two
• and and or instructions ... see P. 226-227
  Good for bit manipulation - testing and setting
  bits in a word field
• Type R instructions and rd, rs, rt rd
  rs rt
• or rd, rs, rt rd rs rt , Note
  is theor operator

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Logical Operations (cont.)

• Type I instructions andi rt, rs, imm rt
  rs imm
• ori rt, rs, imm rt rs immimm
  treated as unsigned ... expanded to 32 bits by
  padding with zeros rather than extending the sign
  bit Application we can create a 32 bit
  constant using lui and ori in much the same way
  we did using lui and addi.

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Designing An ALU

• We now look at hardware design which will support
  the arithmetic instructions just described
• This is the number cruncher of the machine
• Start by building a 1 bit ALU and replicating it
  32 times
• Function of the ALU
• add, subtract, and, or, test for less than,
  overflow detection
• Simple example of an single stage adder design

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Single Stage One Bit Adder
From Mano, Digital Design 2nd ed. p. 121
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One Bit ALU
Each bit has a full adder, and, and or
function.
Unit for bits 0 - 30
Unit for bit 31. Has overflow detect andslt
capability using sign bitwhich will fed to
less inputof bit 0 For slt s1, s2, s3do
s2 - s3 and put sign bitin bit 0 zero out
remaining bits.
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Final 32 Bit ALU
For slt, zero out bits 1-32 via Less input and
feed sign bit into Less input of bit 0. Test
for zero if all bits 0 use or gate.
If Bnegate 1, and Operation 3, for set on
less than (slt), then subtraction is done
under the covers, but not transmitted outside.
Set is the sign bit of the subtract result.
lt Note wrap-around connection
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Carry Look Ahead

• Avoids the slow carry-ripple affect of the above
  circuit
• Carry ripple is very intuitive, but inefficient
  must sequence through all low order 31 bit
  positions before getting the carry-in for bit
  position 32 (bit 31).
• A less intuitive, but more efficient method
  calculates the carries in advance carry lookahead

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Carry Look Ahead (cont)
From Mano, Digital Design 2nd ed. p. 158
Reading the diagram directly Pi is carry
propagate for bit position i propagates
through bit position i even if position i does
not generate its own carry. Gi is carry
generate for bit position i generated directly
by this bit position independently of any carry
in Pi ai ? bi, Gi aibi Si Pi ? Ci, Ci1
Gi PiCi a recursive formula
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Carry Look Ahead (cont)

• By reading the full adder circuit diagram
  directly, we see that Pi Pixor ai ? bi.
  The book uses Pi Pior ai bi for this
  function a much cheaper and simpler circuit.
  How is this done? See below.
• If the latter (OR) is used, it must be used
  only for the carry out function. The sum
  function still requires the XOR (former version).
  Since the XOR version is already needed, why not
  also use it for the carry out? - maybe there are
  fan-out considerations.
• Thus we also may use Si Pixor ? Ci, where
  Pixor ai ? biCi1 Gi Pior Ci where Pior
  ai bi , and Gi aibiremember Pixor will
  also work for Ci is also but is a more
  complicated circuit.
• Proof of the equivalence of Pixor and Pior for
  carry out calculationsFrom the circuit diagram
  on previous slide we have the Boolean equations
  Ci1 Gi Pixor Ci aibi (ai ?
  bi)Ci aibi (aibi aibi) Ci prime means
  complement aibi aibi aibi Ci aibi
  Ci ai(bibi Ci ) bi(aiai Ci )
  ai(bi Ci ) bi(ai Ci ) aibi aibi aiCi
  biCi aibi (ai bi)Ci Gi
  Pior Ci Note we used boolean
  identities x xx, and xxy xy

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Carry Look Ahead (cont)

• Formulas for carries showing increasing
  fan-in.Recursively expand out the carry-out
  formulas

From Mano, Digital Design 2nd ed. p. 159
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Carry Look Ahead (cont)
Note that fan-in increases as the bit position
increases.
From Mano, Digital Design 2nd ed. p. 159
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Carry Look Ahead (cont)
4 bit carry look-ahead adder
From Mano, Digital Design 2nd ed. p. 160
See also fig. 4.24, p. 426 P H
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Carry Look-Ahead Reality sets in!

• Note that the fan-in of the and gates and the
  or gate for the carry-out any bit position
  increases with the bit position. At Some point
  it becomes impossible to implement due to
  fan-in/fan-out constraints of the technology.
• We can decompose a 32 bit adder to 4 groups of 8
  bit carry look-ahead adders, we can limit the
  fan-in of the carry look-ahead circuitry. The 4
  8 bit adders, in turn, can also utilize
  carry-look-ahead from group to group. This is
  an example of using a second level of abstraction
  of the carry-look-ahead concept.
• See the 16 bit example on pp. 242-248 for an
  illustration.