Class Notes of CSCI 370 Computer Architecture (Spring 2025)
===========================================================

Slide 16.11.1
-------------

     1 / (0.66 * 10**(-9) sec/cycle)
   = 1.515 * 10**9 cycles/sec 

   AMAT = Time for a hit + Miss rate × Miss penalty
        = 0.66 ns + 0.053 * 50 ns
	= 3.31 ns

   Total CPI
        = 1 + (0.053 * 0.42 * 50 ns) / ( 0.66 ns/cycle )
	= 2.69 cycles/inst

   AMAT = Time for a hit + Miss rate × Miss penalty
        = 0.66 + 0.053*9.32 ns + 0.053*0.63*50 ns
	= 2.82 ns

   Total CPI
        = 1 + (0.053 * 0.42 * 9.32 ns) / ( 0.66 ns/cycle )
            + (0.053 * 0.42 * 0.63 * 50 ns) / ( 0.66 ns/cycle )
        = 1 + 0.314 +  1.062
	= 2.376 cycles/inst


Slide 16.10
-----------

     Execution timeY / Execution timeX
   = T (1) / T (2)
   = 9 cycles/inst / 3.4 cycles/inst 
   = 2.6

     100*10**(-9) sec * 4*10**9 cycles/sec
   = 400 cycles

   T(1) = 1 cycles/inst + 0.02*400 cycles/inst
        = 9 cycles/inst 

     5*10**(-9) sec * 4*10**9 cycles/sec
   = 20 cycles

   T(2) = 1 cycles/inst + 0.02 * 20 + 0.005 * 400 
        = 3.4


Slide 16.8
----------

Assuming a cache of 4096 blocks, a 4-word block size, and a 32-bit address, find

(i) the total number of sets and
(ii) the total number of tag bits:

Slide 16.8.4
------------

Fully associative cache:

    tag  index  offset
    ---  -----  ------
    28     0      4

    sets = 1 = 2**0
    tags = 32 - 0 - 4 = 28 bits
    tag bits = 1 * 4096 * 28 = 114688 bits


Slide 16.8.3
------------

4-way set associative cache:

    tag  index  offset
    ---  -----  ------
    18    10      4

    sets = 4096 / 4 = 1024 = 2**10
    tags = 32 - 10 - 4 = 18 bits
    tag bits = 1024 * 4 * 18 = 73728 bits


Slide 16.8.2
------------

2-way set associative cache:

    tag  index  offset
    ---  -----  ------
    17      11      4
		  
    sets = 4096 / 2 = 2048 = 2**11
    tags = 32 - 11 - 4 = 17 bits
    tag bits = 2048 * 2 * 17 = 69632 bits


Slide 16.8.1
------------

Direct-mapped cache:

    tag  index  offset
    ---  -----  ------
    16    12       4

    4096 = 2**12 sets

    offset = 2*4 = 16 bytes
    tag  = 32 - 12 - 4 = 16

    4096 x 16 = 2**12 x 2**4 = 2**16 tag bits
    

Slide: 16.3
-----------

   CPU times = Instruction count × CPI × Clock cycle time
   T = I x CPI x (cycle time)

   Execution timeY / Execution timeX
 = T (with stall) / T (without stall)
 = I x CPI (with) x (cycle time) / I x CPI (out) x (cycle time)
 = CPI (with) / CPI (out)
 =    ?       / 1.5
 = (1.5 + 2 + 1) /  1.5
 = 3


   I x 0.02 x 100 = 2 x I cycles => 2 cycles / inst

   I x 0.2 x 0.05 x 100 = I cycles => 1 cycles / inst






Slide: 16.2
-----------

     AMAT = Time for a hit + Miss rate × Miss penalty
          = 2 ns + 0.05 x 20 cycles / (2 ns / cycle)
	  = 2 ns + 2 ns
	  = 4 ns


Slide: 15.11
------------

   address (32 bits): tag | index | offset
                       22 |   6   |   4

   2**4 = 16 bytes / (4 bytes/word) = 4 words

   2**6 = 64 blocks


Slide: 15.10.3
--------------
Assume a direct-mapped cache has the features:

  (i) 64 KiB cache data size,
  (ii) 1-word cache block size, and
  (iii) a 32-bit byte-addressed cache reference.

   2**n * (2**m*32+31-n-m)
 = 2**14 * (1*32+31-14-0)
 = 2**4 * 49 Kibibits
 = 784 Kibibits

   n = 14
   m = 0 because 1 = 2**0

   64 * 2**10 bytes / ( 4 bytes / block)
 = 16 * 2**10 blocks
 = 2**14 blocks


Slide: 15.10.1
--------------

  address (32 bits): tag | index | offset
                      28 |   4   |   0


Slide: 15.7.2
-------------

   1600 / 16 = 100; R = 0
    100 / 16 =   6; R = 4
      6 / 16 =   0; R = 6 

   1600(10) = 640(16)
            = 0000 0000 0000 0000 0000 01 | 10 010 | 0 0000


Slide: 15.7.1
-------------

    01FFF8AC
  = 0000 0001 1111 1111 1111 1000 1010 1100

    offset = 4 bits since 2**4 = 16 bytes
    index  = 8 bits since 2**8 = 256 blocks
    tag    = 20 bits since 20 = 32 - 8 - 4

    offset = 1100 = 12
    index  = 1000 1010 = 8A
    tag    = 0000 0001 1111 1111 1111
           = 01FFF


Slide: 15.6
-----------

How many total bits are required for a direct-mapped
cache with 4 KiB of data and 2-word blocks, assuming
a 32-bit byte address?

   2**n * (2**m*32+31-n-m)
 = 2**9 * (2**1*32+31-9-1)
 = 2**9 * 85
 = 42.5 * 2**10
 = 42.5 Kibibits

   m = 1
   n = 9

   4 * 2**10 = 4 KiB
   # of blocks = 4 KiB / 2*4 bytes
               = (4*2**10)/(2*4)
	       = 2**9


Slide: 14.19
------------

    00 -- 01
    11 -- 10

    # taken (1-bit) = 0
    # taken (2-bit) = 00

    for ( i = 0; i < 100; i++ ) {
      # taken (1-bit) = 1
      # taken (2-bit) = 00
      $4 = $5 - $6
      ....
    }
    # taken (1-bit) = 0
    # taken (2-bit) = 10
    $1 = $2 + $3
    
    beq  $t1, $t2, offset    #  if ( $t1 == $t2 ) goto offset
    $6 = $7 * $8

    offset: $9 = $10 / $11

HW #3
-----

   A => 13 (10 + 3) bits
   Normalization
   Rounding => 13 bits

   B => 13 (10 + 3) bits
   Normalization
   Rounding => 13 bits

   C => 13 (10 + 3) bits
   Normalization
   Rounding => 13 bits


   A x B => 26 (13 + 13) bits => 13 (10 + 3) bits
   Normalization
   Rounding => 13 bits

   A x B - C  => 13 (10 + 3) bits
   Normalization
   Rounding => 10 bits
   No overflow or underflow


   beq  $t1, $t2, offset
   # branch if $t1 == $t2
   # branch if  rs ==  rt

   lw  $t1, offset($t2)
   # $t1=mem[[$t2]+offset]
   #  rt=mem[ [rs]+offset]

   1111 1111 1111 1111 = -1
   1111 1111 1111 1111 1111 1111 1111 1111 = -1 (sign-extended)

   1111 1111 1111 1111 = -1
   0000 0000 0000 0000 1111 1111 1111 1111 > 0  (zero-extended)
   
   0000 0000 0000 0001 = 1
   0000 0000 0000 0000 0000 0000 0000 0001 = 1

   add $t1, $t2, $t3
   # $t1 = $t2 + $t3
   #  rd =  rs +  rt


Tuesday, April 01, 2025
-----------------------

     0001 00 =  4
     0010 00 =  8
     0011 00 = 12
     0100 00 = 16
     ....
     

Thursday, March 27, 2025
------------------------

Slide: 10.18
------------

   1  5  10
   S  E  F

   2**5 = 32 - 2 = 30 / 2 = 15

    1 - 15 = -14
   30 - 15 =  15
   

Slide: 10.14
------------

Step 1. Calculating the exponent of the product

    (-1) + (-2) = -3

Step 2. Multiplying the significands

          1.010
        x 1.110
   ------------
           0000
          1010
         1010
      + 1010
   ------------
       10001100


=> -10.001100 x 2**(-3)

Step 3. Normalizing the product

     -10.001100 x  2**(-3)
  => -1.0001100 x  2**(-2)

Step 4. Rounding the significand

     -1.0001100 x  2**(-2)
  => -1.001     x  2**(-2)

Step 5. Checking for overflow or underflow

  No overflow or underflow


Wednesday, March 26, 2025
-------------------------

Slide: 10.11
------------

Step 1. Making exponents of two operands equal

   1.000 x 2**(-3) = 0.00001 x 2**2

Step 2. Adding or subtracting the significands

     01.000   x 2**2
     10.111   x 2**2
   +      1   
   ------------------
     11.000   x 2**2
   + 00.00001 x 2**2
   ------------------
     11.00001 x 2**2

Step 3. Normalizing the result

     11.00001 x 2**2
     00.11110 x 2**2
   +        1
   -----------------
    -00.11111 x 2**2

    -00.11111 x 2**2 = -1.1111 x 2**1

Step 4. Rounding the significand

     -1.1111 x 2**1 => -10.000 x 2**1

Step 3. Normalizing the result

     -10.000 x 2**1 = -1.000 x 2**2

Step 4. Rounding the significand

     -1.000 x 2**2
     
Step 5. Checking for overflow or underflow

     No overflow or underflow


Slide: 10.10
------------

Step 1. Making exponents of two operands equal

   1.011 x 2**(-3) = 0.01011 x 2**(-1)

Step 2. Adding or subtracting the significands

      1.111   x 2**(-1)
   +  0.01011 x 2**(-1)
   --------------------
     10.00111 x 2**(-1)

Step 3. Normalizing the result

     10.00111 x 2**(-1) = 1.000111 x 2**0

Step 4. Rounding the significand

     1.000111 x 2**0 = 1.001 x 2**0

Step 5. Checking for overflow or underflow

     No overflow or underflow
     

Tuesday, March 25, 2025
-----------------------

Slide: 10.5.6.2
---------------

    -5 < -3
    254 - 127 = 127


Slide: 10.5.6.1
---------------

   0 11111110 1111111111111111111 ...

   (-1)**0 × (1.111111111...) × 2**(254-127)
 ~ 2 x 2**127
 = 2**128

   0 1111111110 1111111111111111111111111...

   (-1)**0 × (1.111111111...) × 2**(2046-1023)
 ~ 2 x 2**1023
 = 2**1024


Slide: 10.5.2
-------------

   0.8125 x 2 = 1.6250 => 1
   0.6250 x 2 = 1.2500 => 1
   0.2500 x 2 = 0.5000 => 0
   0.5000 x 2 = 1.0000 => 1

   6 => 110

   6.8125 => 110.1101

  -6.8125 = -1.101101 x 2**2

  exponent = 2 + 127 (bias) = 129 = 128 + 1
           = 2**7 + 1 => 10000001
  1 10000001 101101 00000000000...

  exponent = 2 + 1023 (bias) = 1025 = 1024 + 1
           = 2**10 + 1 => 10000000001
  1 10000000001 101101 00000000000...


Monday, March 24, 2025
----------------------

Slide: 10.5.1
-------------

   Sign = 1 < 0
   Exponent = 01111111000 = 8 + 16 + 32 + 64 + 128 + 256 +512 = 1016
   val(E) = 1016 - 1023 = -7
   Significand = 1.100 = 1 + 2**(-1) = 1.5

   (-1)**S × (1.F)_2 × 2**val(E)
 = (-1)**1 x (1.1) x 2**(6)
 = -1.5 x 2**(-7)
 = -0.01171875


Slide: 10.4.3
-------------

   Sign = 0 > 0
   Exponent = 10000000101 = 5 + 2**10 = 1029
   val(E) = 1029 - 1023 = 6
   Significand = 1.0010101 = 1 + 2**(-3) + 2**(-5) + 2**(-7)
               = 1+(1/8)+(1/32)+(1/128)
               = 1.164

   (-1)**S × (1.F)_2 × 2**val(E)
 = (-1)**0 x (1.0010101) x 2**(6)
 = 1.164 x 64
 = 74.50


Slide: 10.4.2
-------------

   Sign = 0 > 0
   Exponent = 10000010 = 2 + 128 (2**7) = 130
   val(E) = 130 - 127 = 3
   Significand = 1.010011 = 1 + 2**(-2) + 2**(-5) + 2**(-6)
               = 1+(1/4)+(1/32)+(1/64)
	       = 1 + (16+2+1) / 64 = (64+16+2+1)/64
	       = 1.2969

   (-1)**S × (1.F)_2 × 2**val(E)
 = (-1)**0 x (1.010011) x 2**(3)
 = 1.2969 x 8
 = 10.375


Slide: 10.4.1
-------------

   Sign = 1 < 0
   Exponent = 01111100 = 4 + 8 + 16 + 32 + 64 = 124
   val(E) = 124 - 127 = -3
   Significand = 1.01 = 1 + 2**(-2) = 1+1/4 = 1.25

   (-1)**S × (1.F)_2 × 2**val(E)
 = (-1)**1 x (1.01) x 2**(-3)
 = -1 x 1.25 x 1/8
 = -0.15625
 

Wednesday, March 19, 2025
-------------------------

Slide: 8.11
-----------

        1110 ÷ 0011
      = 14   /   3

     => Q: 4 => 0100
        R: 2 => 0010

        0010

      - 0011
        1100
      +    1
      ------
        1101
      + 0010
      ------
        1111
	

        0001

      - 0011
        1100
      +    1
      ------
        1101
      + 0001
      ------
        1110


        0011

      - 0011
        1100
      +    1
      ------
        1101
      + 0011
      ------
      1 0000


        0001

      - 0011
        1100
      +    1
      ------
        1101
      + 0001
      ------
        1110


Slide: 8.10
-----------

        1110 ÷ 0011
      = 14   /    3

     => Q: 4 => 0100
        R: 2 => 0000 0010

        00000010

      - 00000011
        11111100
      +        1
      ----------
        11111101
      + 00000010
      ----------
        11111111
	

        00000010

      - 00000110
        11111001
      +        1
      ----------
        11111010
      + 00000010	
      ----------
        11111100


        00001110

      - 00001100
        11110011
      +        1
      ----------
        11110100
      + 00001110
      ----------
      1 00000010


        00001110

      - 00011000
        11100111
      + 00000001
      ----------
        11101000
      + 00001110
      ----------
        11110110
        

Tuesday, March 18, 2025
-----------------------

Slide: 8.5/6
------------

     1100
  +  0111
  -------
    10011


     1100 × 1101 = 10011100
  => 12   x 13   = 156

     10011100 = 4 + 8 + 16 + 128 = 156

     01100000
   + 00111100
   ----------
     10011100

     00110000
   + 00001100
   ----------
     00111100


Slide: 8.2.2
------------

    $a0: 7FFFFFFF => 7: 0111 > 0
  +             1 > 0
  ---------------
         80000000 => -2**31 = -2147483648


Slide: 8.2.1
------------

    $a0: 7FFFFFFF => 7: 0111 > 0
  +             1 > 0
  ---------------
         80000000 < 0


Monday, March 17, 2025
----------------------

Slide: 7.5
----------

       123450 => 543210
 $t1:       ^
 $t0:       0
 
    $t1: 0

     -
    |0|
    |1|
    |2|
    |3|
    |4|
    |5|


  A => B => C => D  


Slide: 7.3.2
------------

   5 + 4 + 3 + 2 + 1 = 15

     n: 5
   $t0: 5
   $t1: 0
   ans: 15


Thursday, March 06, 2025
------------------------

Slide: 7.3.1
------------

   5 + 4 + 3 + 2 + 1 = 15

     n: 5
   $t0: 0
   $t1: 15
   ans: 15


Slide: 6.13.3
-------------

     0 1 1 2 3 5 8
     0 1 1 2 3 
     
     $t0: 0
     $a0: 5
     $t1: 8
     $t2: 0

 L1:   $t2 = $t0 > 0
       if ( $t2 == 0 ) goto L2
       put $a0
       $t2 = $t1 + $a0
       $a0 = $t1
       $t1 = $t2
       $t0 = $t0 - 1
       goto L1
 L2:
 

Slide: 6.13.2
-------------

     $t0: -1 => FFFF FFFF 

     if ( $t0 > 0 )
       $t1 = 1;
     else
       $t1 = 0;

     if ( $t1 <> 0 )
       put 2
     else
       put 1


Slide: 6.13.1
-------------

     $t0: -1

     if ( $t0 > 0 )
       $t1 = 1;
     else
       $t1 = 0;

     if ( $t1 <> 0 )
       put 2
     else
       put 1


Wednesday, March 05, 2025
-------------------------

Slide: 6.12.3
-------------

    1 + 2 + 3 + 4 + 5 = 15
    5 + 4 + 3 + 2 + 1 = 15

    $v0: 5
    $a0: 0

L1: for ( $v0 = 5; $t0==0; --$v0 ) {
      if ( $v0 < 1)
        $t0 = 1
      else
        $t0 = 0
      $a0 := $a0 + $v0;
    }

L2:


Slide: 6.12.2
-------------

     lb   $t0, 0xFF

     $t0: -1


     if ( $t0 <> -1 )
       $t0 = 1
     else
       $t0 = 0

     if ( $t0 == 0 )
       put 2
     else
       put 1


Slide: 6.12.1
-------------

     li   $t0, 0xFF

     $t0: 255
     $t1:  -1


     if ( $t0 == $t1 )
       $t0 = 1
     else
       $t0 = 0

     if ( $t0 == 0 )
       put 2
     else
       put 1



Slide: 6.10.3
-------------

     lb   $t0, FF

     $t0: 0000 0000
       => 0000 00FF
       => 0000 0000 0000 0000 0000 0000 1111 1111
       => 1111 1111 1111 1111 1111 1111 1111 1111
          0000 0000 0000 0000 0000 0000 0000 0000
        +                                       1
	-----------------------------------------
          0000 0000 0000 0000 0000 0000 0000 0001 = -1

     $t0: 0

     if ( $t0 == 0 )
       put 2
     else
       put 1



Slide: 6.10.2
-------------

     li   $t0, -1

     $t0: FFFF FFFF = -1
          FFFF FFFE = -2

          0000 0002 =  2
	  FFFF FFFD
	+         1
	----------------
	  FFFF FFFE = -2


     $t0: FFFF FFFF = -1 => 2**32 - 1
          FFFF FFFE = -2 => 2**32 - 2

        1 0000 0000 = 2**32
      -           1 =
      ---------------------
          FFFF FFFF = 2**32 - 1

     if ( $t0 >= (2**32 - 2) )
       put 2
     else
       put 1


     0xFF => 1111 1111 = 16**2 - 1 = 256 - 1 = 255

           1 0000 0000 = 16**2
         -           1
         ---------------------
             1111 1111 = 256 - 1 = 255

     $t0: 0000 0000 => 0000 00FF = 255


Slide: 6.10.1
-------------

     li   $t0, 0xFF

     $t0: 255
     $t1:   0


     if ( $t0 >= $t1 )
       put 2
     else
       put 1


     0xFF => 1111 1111 = 16**2 - 1 = 256 - 1 = 255

           1 0000 0000 = 16**2
         -           1
         ---------------------
             1111 1111 = 256 - 1 = 255

     $t0: 0000 0000 => 0000 00FF = 255



Tuesday, March 04, 2025
-----------------------

Slide: 6.9.3
------------

    1 + 2 + 3 + 4 + 5 = 15
    5 + 4 + 3 + 2 + 1 = 15
    
    $v0: 5
    $a0: 0
    
L1: for ( $v0 = 5; $v0 == 0; --$v0 ) {
      $a0 := $a0 + $v0;
    }

L2:


Slide: 6.9.2
------------

     li   $t0, 0xFF

     $t0: 255
     $t1:  -1

     if ( $t0 <> $t1 )
       put 1
     else
       put 2
       

     0xFF => 1111 1111 = 16**2 - 1 = 256 - 1 = 255

           1 0000 0000 = 16**2
	 -           1
	 ---------------------
	     1111 1111 = 256 - 1 = 255

     $t0: 0000 0000 => 0000 00FF = 255



Slide: 6.7.3
------------

     $a0: 14
  /  $t0: -4 => positive
  -----------
           0

     14 / +nn = 0 (quotient)
     14 - (+nn*0) = 14 (remainder)


Slide: 6.7.1/2
--------------

     $a0: -14
  /  $t0:   4
  -----------
           -3

     -14 / 4 = -3 (quotient)
     -14 - (-3*4) = -14 + 12 = -2 (remainder)
     

Slide: 6.7.6
------------

             $t0:  8000 0000 =  -2**31
           x $t1:          2
   -----------------------------------
         0000 0001 0000 0000 =>  2**32
         FFFF FFFE FFFF FFFF
       +                   1
   -----------------------------------
         FFFF FFFF 0000 0000 => -2**32

         FFFF FFFF (negative)
	 0000 0000
       +         1
       -----------
         0000 0001 => -1


0000 00

=> FFFF FFFF 0000 0000 =  -2**32


Monday, March 03, 2025
----------------------

Slide: 6.7.4/5
--------------

     $t0:  8000 0000 = -2**31
   x $t1:          2
   --------------------------
         1 0000 0000 =  2**32


Slide: 6.7.3
------------

     $t0: -11
     $t1:  12
     $a0 = $t0 * 12 = -132


Slide: 6.7.1
------------

     $t0: -11
     $a0 = $t0 * 12 = -132
     

Slide: 6.5.3
------------

     X:  8000 0000 = -2**31

     $t0:   1000 0000 0000 0000 0000 0000 0000 0000 = -2**31
        +   1111 1111 1111 1111 1111 1111 1111 1111 = -1
     --------------------------------------------------
     $a0: 1 0111 1111 1111 1111 1111 1111 1111 1111 =  2**31-1


Slide: 6.5.1
------------

     X:  FF = -1

     $a0: 0000 0000 0000 0000
        = 0000 0000 0000 0000 0000 0000 0000 0000 
       => 0000 0000 0000 0000 0000 0000 1111 1111
       => 1111 1111 1111 1111 1111 1111 1111 1111 = -1

     F = 1111


Slide: 6.4.3
------------

     X:  1
   X+1:  0
   X+2:  0
   X+3:  0
     Y: 8000 0000 (- negative)

          7 => 0111 (+ positive)
          8 => 1000 (- negative)

   + $t1: 8000 0000 (- negative)
   - $t0:         1 (+ positive)
   ----------------

     0000 0000 0000 0000 0000 0000 0000 0001 =  1
     1111 1111 1111 1111 1111 1111 1111 1110
   +                                       1
   -----------------------------------------
     1111 1111 1111 1111 1111 1111 1111 1111 = -1

     $t1:   8000 0000 (- negative)
   + $t0:   FFFF FFFF (- negative)
   -------------------------------
     $a0: 1 7FFF FFFF (+ positive)


          7 => 0111 (+ positive)
          8 => 1000 (- negative)

              1000 0000 0000 0000 0000 0000 0000 0000 = 2**31
            -                                       1
            -------------------------------------------------
              0111 1111 1111 1111 1111 1111 1111 1111 = 2**31-1
           => 2147483648 - 1 = 2147483647
	   

Slide: 6.4.2
------------

     X:  1
   X+1:  0
   X+2:  0
   X+3:  0
     Y: 7FFF FFFF (+ positive)

     $t0:         1 (+ positive)
   + $t1: 7FFF FFFF (+ positive)
   ----------------
     $a0: 8000 0000 = -2**31 

          7 => 0111 (+ positive)
          8 => 1000 (- negative)

  8000 0000 = 1000 0000 0000 0000 0000 0000 0000 0000 = -2**31
              0111 1111 1111 1111 1111 1111 1111 1111
	    +                                       1
	    -----------------------------------------
	      1000 0000 0000 0000 0000 0000 0000 0000 = 2**31
	   => -2**31


Slide: 6.4.1
------------

     X:  1
   X+1:  0
   X+2:  0
   X+3:  0
     Y: 7FFF FFFF

     $t0:         1 (+ positive)
   + $t1: 7FFF FFFF (+ positive)
   ----------------
          8000 0000 (- negative)

          7 => 0111 (+ positive)
	  8 => 1000 (- negative)


Slide: 6.3.3
------------

   $t0: -20
   $t1:  20 <= -20
 - $t0: -20
 ------------
   $a0:  40


Slide: 6.3.2
------------

     c: '7' => 55

    $t0: 7 = 55 ('7') - 48 ('0')    
    $a0: 7


Slide: 6.3.1
------------

     X: 10
   X+1:  0
   X+2:  0
   X+3:  0
     Y: -20: X+4

    $t1: -20
  - $t0:  10
  ----------
    $a0: -30


Slide: 6.2.3
------------

     i: 7
     s: 55 => '7'
   s+1: 0

    $t0: 7 => '7' => 55 = '0' (48) + 7
    

Slide: 6.2.2
------------

     X: 10

    $t0:  10
  +      -20
  ----------
    $a0: -10
   

Slide: 6.2.1
------------

     X: 10
   X+1:  0
   X+2:  0
   X+3:  0
     Y: -20: X+4

    $t0:  10
  + $t1: -20
  ----------
    $a0: -10


Slide: 5.19.3
-------------

  $a0: 00 01 00 00 => 1x2**16 = 65536
                   => 1x16**4 = 65536
                    

Slide: 5.19.1
-------------

     msg: C
   msg+1: l 
   msg+2: a
   msg+3: s
   msg+4: s
   msg+5: ,
   msg+6: 
   msg+7: G
   msg+8: r
   msg+9: ...
   msg+10:
   msg+11:




Slide: 5.18.3
-------------

     X: 01 => 02
   X+1: 02 => 01
   X+2: 00
   X+3: 00

    $t0 = 00 00 00 01
    $t1 = 00 00 00 02

    
   0102: 0000 0001 0000 0010
       = 1x2**8 + 2
       = 256 + 2
       = 258

   a = 10 => 10
   b = 20 => 20

   t = a
   a = b
   b = t
   

Slide: 5.18.2
-------------

     X: 02
   X+1: 01 => 00
   X+2: 00
   X+3: 00


Thursday, February 20, 2025
---------------------------

Slide: 5.18.1
-------------

       X: 04
     X+1: 03
     X+2: 02
     X+3: 01

      $t0: X
      $a0: 3


Slide: 5.17.3
-------------

      s: a
    s+1: b
    s+2: d
    s+3: 0

     $t0: 00 00 00 d
     

Slide: 5.17.2
-------------

     X: 1
   X+1:
     Y: 1: X+2
   Y+1:  : X+3

    $t0: 00 00 00 01
    $a0: 00 00 00 01


     1111 1111 1111 1111 = -1
     0000 0000 0000 0000
   +                   1
   ---------------------
     0000 0000 0000 0001 = 1

     1111 1111 1111 1111 1111 1111 1111 1111 = -1
     0000 0000 0000 0000 0000 0000 0000 0000
   +                                       1
   -----------------------------------------
     0000 0000 0000 0000 0000 0000 0000 0001 = 1

                         0000 0000 0000 0001 = 1
     0000 0000 0000 0000 0000 0000 0000 0001 = 1


Slide: 5.17.1
-------------

     X: 1
   X+1:
   X+2:
   X+3:
   X+4: 1 
   X+5:
   X+6:
   X+7:
     Y: 3: X+8
   Y+1:    X+9
   Y+2:    X+10
   Y+3:    X+12

    $t0: 1
    $a0: 3


Slide: 5.17.6
-------------

    val: 01
  val+1:
  val+2: 02
  val+3: 03
  val+4: 04
  val+5: 05
  val+6: 06
  val+7: 07

    $t0: val+1
    $a0: 3


Slide: 5.17.5
-------------

    s: a
  s+1: b
  s+2: c
  s+3: 0

  $t0: 2
  $a0: s+2


Slide: 5.17.4
-------------

    s: a
  s+1: b
  s+2: c
  s+3: 0

  $t0: s+1
  $a0: b


Slide: 5.17.3
-------------

    s: a
  s+1: b
  s+2: c
  s+3: 0

  $t0: s+2
  $a0: s+2


Slide: 5.17.2
-------------

   val-1: 1
     val: 2
   val+1: 3

   $a0: 00 00 00 01


Slide: 5.17.1
-------------

     val: 00
   val+1: 01
   val+2: 10
   val+3: 00

   $a0: 00 00 00 10 = 16


Slide: 5.16.3
-------------

     val: 1
   val+1:
   val+2: 2
   val+3:
   val+4: 3
   val+5:
   val+6:
   val+7:

   $a0: 0
   

Tuesday, February 18, 2025
--------------------------

Slide: 5.16.2
-------------

     val: 1
   val+1:
   val+2: 2
   val+3:
   val+4: 3
   
   $a0: 


Slide: 5.16.1
-------------

    s: a
  s+1: b
  s+2: c
  s+3: 0

  $a0: s+1


Thursday, February 13, 2025
---------------------------

Slide: 5.15.3
-------------

     val: -10 => F6

     $t0 =  0000 0000
         => 0000 00F6

     $a0 =  0000 00F6 = 15x16 + 6 = 246
     

Slide: 5.15.2
-------------

    val0: 0 = 00
    val1: 1 = 01
    val2: 2 = 02

     $a0 =  0000 0000
         => 0000 0100 = 16**2 = 256

   
Slide: 5.14.4
-------------

     val: -10 => F6

     $a0 =  0000 0000
         => FFFF FFF6 = -10


Slide: 5.14.3
-------------

     val: -10 => F6
   val+1: 0
   val+2: 0 
   val+3: 0

     $a0 =  0000 0000
         => 0000 00F6 = 15x16+6 = 246

      0000 1010 = 10
      1111 0101
    +         1
    -----------
      1111 0110 = F6 => -10


Slide: 5.14.2
-------------

       val: F6
     val+1: FF

      0000 0000 0000 1010 = 10
      1111 1111 1111 0101
    +                   1
    ---------------------
      1111 1111 1111 0110 = FFF6 => -10


       $a0 =  0000 0000
           => 0000 FFF6
	   => FFFF FFF6

             FFFF FFF6
	     0000 0009
	   +         1
	   -----------
	     0000 000A => -10


Slide: 5.12.6
-------------

    X: 1
  X+1: 0
    Y: 2 : X+2
  Y+1: 0 : X+3
  Y+2: 0 : X+4
  Y+3: 0 : X+5
  ...

   $a0 = 0020


Wednesday, February 12, 2025
----------------------------

Slide: 5.12.5
-------------

   s: 100
 $a0: 100

Slide: 5.12.4
-------------

   X: 1
 X+1: 0 
   Y: 2 : X+2
 Y+1: 0 : X+3
 ....

Slide: 5.12.3
-------------

   s: x
 s+1: y
 s+2: z
 s+3: 0

Slide: 5.12.2
-------------

   s: x
 s+1: y
 s+2: z
 s+3: 0

Slide: 5.12.1
-------------

   s: x
 s+1: y
 s+2: z
 s+3: 0


Tuesday, February 11, 2025
--------------------------

Slide: 5.8
----------

   10000000 = 2**7 = 128 
 - 01111111 =       -127
 -----------------------
   00000001 =          1

   11111111 = -1
   00000000
 +        1
 ----------
 - 00000001

   10000000 = -128
   01111111
 +        1
 ----------
   10000000 = -2**7 = -128 


Slide: 5.8
----------

8 = 1000

A: 10
B: 11
C: 12
D: 13
E: 14
F: 15

75E0 = 7x16**3 + 5x16**2 + 14x16
     = 28672 + 1280 + 224
     = -30176
     

Slide: 5.8
----------

1462 / 16 = 91;  R: 6
  91 / 16 =  5;  R: B
   5 / 16 =  0;  R: 5


Slide: 5.7
----------

38 = 32 + 4 + 2
   = 00100110   


Slide: 5.7
----------

-15268 =

     15268 / 16 = 954;  R: 4
     954   / 16 =  59;  R: A
     59    / 16 =   3;  R: B
     3     / 16 =   0;  R: 3
     

Monday, February 10, 2025
-------------------------

Slide: 5.4
----------

    100000000 = 1x2**8 = 256
  -         1 =            1
  --------------------------
    011111111 =          255

    2*16 = 65536
  -    1 =     1
  --------------
         = 65535    


Thursday, February 6, 2025
--------------------------

Slide: 4.14.2
-------------

    s:   a
    s+1: b
    s+2: c
    s+3: 0
    s+4:
    ...

$a0: s
$a1: 4


Wednesday, February 5, 2025
---------------------------

for ( i = 0; i < 1000; i++ ) {

   ....

}


Tuesday, February 4, 2025
-------------------------

Slide 4.14.1
------------

s1: "abc";
s2: "xyz";

abcxyz0

print s1;


"abc" => abc0

A => ($ra=A) B => ($ra=B) C => ($ra=C) D ...

Runtime stak:

   A
   B
   C
   

Monday, February 3, 2025
------------------------

Slide 3.13
----------
     
   Average clock cycles per instruction (CPI)
 = CPU time / ( Instruction count × Cycle time )
 = ( 400 sec / 1750x10**9 I ) / ( 0.25 x 10**(-9) sec / cycle )
 = (0.23 x 10**(-9))/(0.25 x 10**(-9)) cycles / I
 = 0.914 cycles / I

SPECratio = T(reference) / T(Computer)
          = 8660 / 400 = 21.65

  ( Π SPECratioi )**1/n,
     where Π ai means the product a1×a2×...×an,  i = 1 .. n
= ( 21.65 x 8.5 )**(1/2)


Thursday, January 30, 2025
--------------------------

Slide 3.10
----------

 Power ∝ ½ × Capacitive load × Voltage**2 × Frequency switched

 0.8 = P2 / P1
     = ( ½ × Capacitive_2 × Voltage_2**2 × Frequency_2 ) /
       ( ½ × Capacitive_1 × Voltage_1**2 × Frequency_1 )
     = ( ½ × Capacitive_2 × Voltage_2**2 × Rate_2 ) /
       ( ½ × Capacitive_1 × Voltage_1**2 × Rate_1 )
     = ( Capacitive_2 x 3x3 x 10**9 ) /
       ( Capacitive_1 x 6x6 x 0.5x10**9 )
     = Capacitive_2 / (2xCapacitive_1)
 1.6 = Capacitive_2 / Capacitive_1


Slide 3.8
---------

Time = Seconds / Program = ( Instructions / Program ) ×
    ( Clock cycles / Instruction ) × ( Seconds / Clock cycle )

CPU times = ( Instruction count × CPI ) / Clock rate

  T(old) = I x CPI / R
         = 20 sec

  T(new) = 0.5 x I x 1.2 x CPI / R 
         = 0.6 x I x CPI / R
	 = 0.6 x 20
	 = 12 sec

Slide 3.6
---------

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time
 = CPU clock cycles for a program / Clock rate

   CPU clock cycles = Instructions for a program ×
        Average clock cycles per instruction (CPI)

   C1
 = 10000x0.3x1 + 10000x0.3x2 + 10000x0.4x3
 =  3000       +  6000       + 12000
 = 21000

   C2
 = 10000x0.3x2 + 10000x0.3x2 + 10000x0.4x2
 =  6000       +  6000       +  8000
 = 20000

   T1
 = 21000 cycles / (2x10**9 cycles/sec)
 = 10500x10**(-9) sec

   T2
 = 20000 cycles / (1.5x10**9 cycles/sec)
 = 13333x10**(-9) sec



Wednesday, January 29, 2025
---------------------------

Slide 3.5
---------

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time
 = CPU clock cycles for a program / Clock rate

   CPU clock cycles = Instructions for a program × 
        Average clock cycles per instruction (CPI)

    PerformanceX / PerformanceY = n

    PerformanceX / PerformanceY 
 =  Execution timeY / Execution timeX  = n

    T1
  = CPU clock cycles for P1 / Clock rate
  = Instructions for p1 × CPI(1) / R1
  = I x 2 cycles / (2.5 x 10**9 cycles/sec)
  = 0.8x10**(-9) x I sec
  = 800x10**(-12) x I sec

    T2
  = CPU clock cycles for P1 / Clock rate
  = Instructions for p1 × CPI(1) / R1
  = I x 2.5 cycles / (3.0 x 10**9 cycles/sec)
  = 0.833x10**(-9) x I sec
  = 833x10**(-12)  x I sec

    T2 / T1
  = [ 833x10**(-12) x I ] / [800 x10**(-12) x I ]
  = 1.04


Slide 3.4.2
-----------

    PerformanceX / PerformanceY = n

    PerformanceX / PerformanceY 
 =  Execution timeY / Execution timeX  = n

    5
 =  30 / T(c)
 => T(C) = 30 / 5 = 6


Tuesday, January 28, 2025
=========================

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time
 = CPU clock cycles for a program / Clock rate


Slide 3.4.1
-----------

   T(A) = 12 sec
   R(A) = 4 x 10**9 cycles/sec

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time
 = CPU clock cycles for a program / Clock rate
 = 12 sec
 = C(A) cycles / (4x10**9) cycles / sec

   C(A) cycles
 = 12 sec x (4x10**9) cycles / sec
 = 48x10**9 cycles

   T(B) = 8 sec
   R(B) = 2 x 10**9 cycles/sec

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time
 = CPU clock cycles for a program / Clock rate
 = 8 sec
 = C(B) cycles / (2x10**9) cycles / sec

   C(B) cycles
 = 8 sec x (2x10**9) cycles / sec
 = 16x10**9 cycles

   CPU execution time for a program
 = CPU clock cycles for a program × Clock cycle time


    PerformanceX = 1 / Execution timeX

    PerformanceX > PerformanceY
 ⇒  ( 1 / Execution timeX ) > ( 1 / Execution timeY )
 ⇒  Execution timeY > Execution timeX

    PerformanceX / PerformanceY = n

    PerformanceX / PerformanceY 
 =  Execution timeY / Execution timeX  = n

Slide 3.1.1
-----------

    PerformanceX / PerformanceY
 =  Execution timeY / Execution timeX  = n
 =  Execution timeB / Execution timeA 
 =  25 sec / 10 sec
 =  2.5
 
Slide 3.1.2
-----------

    PerformanceX / PerformanceY
 =  Execution timeY / Execution timeX  = n
 =  Execution timeB / Execution timeA  = 3
 =>  Execution timeB / 30 = 3
     Execution timeB = 90

Slide 2.21
----------

  Cost per die = Cost per wafer / ( Dies per wafer × Yield )
  Dies per wafer ≈ Wafer area / Die area
  Yield = 1 / ( 1 + Defects per area × Die area )n

  Yield = 1 / ( 1 + 0.02 x 3.14 x 10 x 10 / 120 )**2
        = 1 / ( 1 + 0.0523)**2
	= 1 / 1.1074
	= 0.903


word  b, c, a;          // 1 word = 4 bytes
lw  $t1, 8($t6)         // $t1 = a

relocatable address
$t6 = 1008 (1000 + 8)  // base address = 1000
$t6 = 2008 (2000 + 8)  // base address = 2000

4th generation:
SQL: FIND ALL RECORDS WHERE NAME IS "SMITH"

3rd generation:
for ( i = 0; i < 1000; i++ ) {
   if ( name[i] == "SMITH" )
     print record[i]
}
 
CPU (with +, but no x)
1 + 2 = 3

Microarchitecture (without x)
2 x 3 = 6 = 2 + 2 + 2 (microcode or firmware)

SIMD: Single Instruction/Multiple Data
RISC: Reduced Instruction Set Computer
CISC: Complex Instruction Set Computer

MIPS: https://mips.com/
ARM:  https://www.arm.com/products/silicon-ip-cpu

Load-store computers
Stored-program computers