Our mutual friend Numeric Representation o Conversion to and from hex is not too Two's complement and friends BAD 16 = __________ 10 . o Octal is also useful* 0, 1, 2, 3, . . . o Why do machine language programmers confuse Halloween with Christmas? CS240 Computer Organization Department of Computer Science Wellesley College *And the same conversion trick works between binary and octal. Why? Numeric representation 4-2 Frankly, other conversions are no Arithmetic is the same everywhere where as nice ... o Given a number num , we find d n d n-1 ... d 1 d 0 so that Addition Subtraction num = d n r n + d n-1 r n-1 + ... + d 1 r 1 + d 0 r 0 10011 2 � CDB 16 � + � 00101 2 - � C3E 16 o The algorithm* is simple, but tedious. for i ← 0 to n � � � � do � d i ← num mod r � Multiplication Division � � � � num ← num div r � 134 8 � x � 27 8 11 2 100011 2 * num = d n r n-1 + ... + d 1 r 1 + d 0 r 0 = ( num div r ) * r + ( num mod r ). Repeat. Numeric representation 4-3 Numeric representation 4-4 1
Representing a range of numbers Signed magnitude representation o Using n bits, we could represent the natural numbers up o What range of numbers can be represented using an n-bit to 2 n - 1 . . . binary number. 0 1 2 3 4 5 . . . 2 n -1 � 0 1 1 0 0 1 1 1 � n-1 . . . 2 1 0 � 000...0 111...1 � o . . . or we could reserve one bit, usually the msb , to represent a number ’ s sign. -(2 n -1 -1) -2 -1 0 1 2 2 n-1 -1 � sign bit sign bit 111...1 011...1 � (positive) (negative) Numeric representation 4-5 Numeric representation 4-6 Some problems with signed magnitude Click and Clack ride again o Additional circuitry is needed to add positive numbers o An automobile odometer to negative numbers. 0 1 1 0 1 � represents +1101 � � provides an elementary solution, provided it runs � 1 1 1 0 1 � represents -1101 backwards as well as forwards � number line o And, how exactly does one represent zero?* � � -500 ... -3 -2 -1 0 1 2 3 ... 499 � � number line � � 500 ... 997 998 999 000 001 002 003 ... 499 � *Rule for taking the negative of a number: Subtract number from 999 *And why is this a problem? (to avoid borrowing) then add 1 Numeric representation 4-7 Numeric representation 4-8 2
Digital Tappet Brothers Addition works perfectly* o We build a binary odometer o We retain our handy-dandy 3-bit binary odometer -1 � 3 � o The number line works exactly the same as before + � -2 + � -1 number line � � � -4 -3 -2 -1 0 1 2 3 � � -1 � 2 � number line + � -4 + � -3 � � � 100 101 110 111 000 001 010 011 � *Rule for taking the negative of a number: Subtract number from 111 *And there is no need for a separate circuit to do subtraction. (to avoid borrowing) then add 1 But overflow remains a clear and present danger. Numeric representation 4-9 Numeric representation 4-10 Detecting overflow Logical operations o Some language, C for add add $s1,$s2,$s3 $s1=$s2 + $s3 Arithmetic subtract sub $s1,$s2,$s3 $s1=$s2 - $s3 example, ignore integer add immediate addi $s1,$s2,100 $s1=$s2 + 100 overflow. Others require that the program be and and $s1,$s2,$s3 $s1=$s2 & $ s3 notified. or or $s1,$s2,$s3 $s1=$s2 | $s3 o MIPS detects overflow nor nor $s1,$s2,$s3 $s1=~($s2 | $s3) with an exception. Logical and immediate andi $s1,$s2,100 $s1=$s2 & 100 o The exception program or immediate ori $s1,$s2,100 $s1=$s2 | 100 counter (EPC) contains the sll sll $s1,$s2,10 $s1=$s2 << 10 address of the offending srl srl $s1,$s2,10 $s1=$s2 >> 10 instruction. Data transfer load word lw $s1,100($s2) $s1=Mem[$s2+100] store word sw $s1, 100($s2) Mem[$s2+100]=$s1 Numeric representation 4-11 Numeric representation 4-12 3
andi $t2, $s0, 15 Alas, no NOT o AND applied can be used o In keeping with the two-operand format, the to mask or conceal certain designers of MIPS decided against a NOT bit patterns by forcing 0s. operation. o For example, suppose $s0 held the bit pattern o A NOR operation is offered in consolation.* 72 A8 8B AD? o Similarly, the OR nor $t0, $t1, $t2 # $t0 = ~($t1 | $t3) � operation may be used force 1s. *How does this help? Numeric representation 4-13 Numeric representation 4-14 4
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