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1 IKI10230 Pengantar Organisasi Komputer Bab 6: Aritmatika 7 & 14 Mei 2003 Bobby Nazief (nazief@cs.ui.ac.id) Qonita Shahab (niet@cs.ui.ac.id) bahan kuliah: http://www.cs.ui.ac.id/kuliah/iki10230/ Sumber: 1. Hamacher. Computer Organization, ed-5. 2. Materi kuliah CS61C/2000 & CS152/1997, UCB.

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2 Number Representation

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3 How to Represent Negative Numbers? °So far, unsigned numbers °Obvious solution: define leftmost bit to be sign! 0 => +, 1 => - Rest of bits can be numerical value of number °Representation called sign and magnitude

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4 Shortcomings of sign and magnitude? °Arithmetic circuit more complicated Special steps depending whether signs are the same or not °Also, Two zeros 0x00000000 = +0 ten 0x80000000 = -0 ten What would it mean for programming? °Sign and magnitude abandoned

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5 Another try: complement the bits °Example: 7 10 = 00111 2 -7 10 = 11000 2 °Called one’s Complement °Note: postive numbers have leading 0s, negative numbers have leadings 1s. 000000000101111... 11111 1111010000... °What is -00000 ? °How many positive numbers in N bits? °How many negative ones?

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6 Shortcomings of ones complement? °Arithmetic not too hard °Still two zeros 0x00000000 = +0 ten 0xFFFFFFFF = -0 ten What would it mean for programming? °One’s complement eventually abandoned because another solution was better

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7 Search for Negative Number Representation °Obvious solution didn’t work, find another °What is result for unsigned numbers if tried to subtract large number from a small one? Would try to borrow from string of leading 0s, so result would have a string of leading 1s With no obvious better alternative, pick representation that made the hardware simple: leading 0s positive, leading 1s negative 000000...xxx is >=0, 111111...xxx is < 0 °This representation called two’s complement

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8 Two’s Complement Number line °2 N-1 non-negatives °2 N-1 negatives °one zero °how many positives? °comparison? °overflow? 00000 00001 00010 11111 11110 10000 01111 10001 0 1 2 -2 -15 -16 15............

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9 Two’s Complement Numbers 0000... 0000 0000 0000 0000 two = 0 ten 0000... 0000 0000 0000 0001 two = 1 ten 0000... 0000 0000 0000 0010 two = 2 ten... 0111... 1111 1111 1111 1101 two = 2,147,483,645 ten 0111... 1111 1111 1111 1110 two = 2,147,483,646 ten 0111... 1111 1111 1111 1111 two = 2,147,483,647 ten 1000... 0000 0000 0000 0000 two = –2,147,483,648 ten 1000... 0000 0000 0000 0001 two = –2,147,483,647 ten 1000... 0000 0000 0000 0010 two = –2,147,483,646 ten... 1111... 1111 1111 1111 1101 two =–3 ten 1111... 1111 1111 1111 1110 two =–2 ten 1111... 1111 1111 1111 1111 two =–1 ten °One zero, 1st bit => >=0 or <0, called sign bit but one negative with no positive –2,147,483,648 ten

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10 Two’s Complement Formula °Can represent positive and negative numbers in terms of the bit value times a power of 2: d 31 x -2 31 + d 30 x 2 30 +... + d 2 x 2 2 + d 1 x 2 1 + d 0 x 2 0 °Example 1111 1111 1111 1111 1111 1111 1111 1100 two = 1x-2 31 +1x2 30 +1x2 29 +... +1x2 2 +0x2 1 +0x2 0 = -2 31 + 2 30 + 2 29 +... + 2 2 + 0 + 0 = -2,147,483,648 ten + 2,147,483,644 ten = -4 ten °Note: need to specify width: we use 32 bits

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11 Two’s complement shortcut: Negation °Invert every 0 to 1 and every 1 to 0, then add 1 to the result Sum of number and its one’s complement must be 111...111 two 111...111 two = -1 ten Let x’ mean the inverted representation of x Then x + x’ = -1 x + x’ + 1 = 0 x’ + 1 = -x °Example: -4 to +4 to -4 x : 1111 1111 1111 1111 1111 1111 1111 1100 two x’: 0000 0000 0000 0000 0000 0000 0000 0011 two +1: 0000 0000 0000 0000 0000 0000 0000 0100 two ()’: 1111 1111 1111 1111 1111 1111 1111 1011 two +1: 1111 1111 1111 1111 1111 1111 1111 1100 two

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12 Two’s comp. shortcut: Sign extension °Convert 2’s complement number using n bits to more than n bits °Simply replicate the most significant bit (sign bit) of smaller to fill new bits 2’s comp. positive number has infinite 0s 2’s comp. negative number has infinite 1s Bit representation hides leading bits; sign extension restores some of them 16-bit -4 ten to 32-bit: 1111 1111 1111 1100 two 1111 1111 1111 1111 1111 1111 1111 1100 two

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13 Distribusi Nilai UTS

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14 Addition of Positive Numbers

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15 One-Bit Full Adder (1/3) °Example Binary Addition: Carries °Thus for any bit of addition: The inputs are a i, b i, CarryIn i The outputs are Sum i, CarryOut i °Note: CarryIn i+1 = CarryOut i a: 0 0 1 1 b: 0 1 0 1 Sum:1 0 0 0

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16 One-Bit Full Adder (2/3) Definition Sum = ABC in + ABC in + ABC in + ABC in CarryOut = AB + AC in + BC in ¯ ¯ ¯ ¯ ¯ ¯

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17 One-Bit Full Adder (3/3) °To create one-bit full adder: implement gates for Sum implement gates for CarryOut connect all inputs with same name Sum A B CarryIn CarryOut +

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18 Ripple-Carry Adders: adding n-bits numbers °Critical Path of n-bit Rippled-carry adder is n*CP CP = 2 gate-delays (C out = AB + AC in + BC in ) A0 B0 1-bit FA Sum0 CarryIn0 CarryOut0 A1 B1 1-bit FA Sum1 CarryIn1 CarryOut1 A2 B2 1-bit FA Sum2 CarryIn2 CarryOut2 A3 B3 1-bit FA Sum3 CarryIn3 CarryOut3

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19 Fast Adders

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20 Carry Look Ahead: reducing Carry Propagation delay A B S G P A B S G P A B S G P C1 = G0 + C0 P0 = A0 B0 + C0 (A0+B0) C2 = G1 + G0 P1 + C0 P0 P1 C3 = G2 + G1 P2 + G0 P1 P2 + C0 P0 P1 P2 G = G3 + P3·G2 + P3·P2·G1 + P3·P2·P1·G0 C4 =... P = P3·P2·P1·P0 ABC-out 000“kill” 01C-in“propagate” 10C-in“propagate” 111“generate” A0 B0B0 S0S0 G P P = A + B G = A B Cin

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21 Carry Look Ahead: Delays °Expression for any carry: C i+1 = G i + P i G i-1 + … + P i P i-1 … P 0 C 0 °All carries can be obtained in 3 gate-delays: 1 needed to developed all P i and G i 2 needed in the AND-OR circuit °All sums can be obtained in 6 gate-delays: 3 needed to obtain carries 1 needed to invert carry 2 needed in the AND-OR circuit of Sum’s circuit °Independent of the number of bits (n) °4-bit Adder: CLA: 6 gate-delays RC: (3*2 + 3) gate-delays °16-bit Adder: CLA: 6 gate-delays RC: (15*2 + 3) gate-delays Sum = ABC in + ABC in + ABC in + ABC in ¯ ¯ ¯ ¯ ¯ ¯

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22 Cascaded CLA: overcoming Fan-in constraint CLACLA 4-bit Adder 4-bit Adder 4-bit Adder C1 = G0 + C0 P0 C2 = G1 + G0 P1 + C0 P0 P1 C3 = G2 + G1 P2 + G0 P1 P2 + C0 P0 P1 P2 G P G0 P0 C4 =... C0 Delay = 3 + 2 + 3 = 8 Delay RC = 15*2 + 3 = 33

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23 Signed Addition & Subtraction

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24 Addition & Subtraction Operations °Addition: Just add the two numbers Ignore the Carry-out from MSB Result will be correct, provided there’s no overflow 0 1 0 1(+5) +0 0 1 0(+2) 0 1 1 1(+7) 0 1 0 1(+5) +1 0 1 0(-6) 1 1 1 1(-1) 1 0 1 1(-5) +1 1 1 0(-2) 11 0 0 1(-7) 0 1 1 1(+7) +1 1 0 1(-3) 10 1 0 0(+4) 0 0 1 0(+2) 0 0 1 0 0 1 0 0(+4)+1 1 0 0(-4) 1 1 1 0 (-2) 1 1 1 0(-2) 1 1 1 0 1 0 1 1(-5)+0 1 0 1(+5) 10 0 1 1 (+3) °Subtraction: Form 2’s complement of the subtrahend Add the two numbers as in Addition

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25 Overflow °Examples: 7 + 3 = 10 but... ° - 4 5 = - 9 but... 2’s ComplementBinaryDecimal 00000 10001 20010 30011 0000 1111 1110 1101 Decimal 0 -2 -3 40100 50101 60110 70111 1100 1011 1010 1001 -4 -5 -6 -7 1000-8 0111 0011+ 1010 1 1100 1011+ 0111 110 7 3 1 – 6 – 4 – 5 7

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26 Overflow Detection °Overflow: the result is too large (or too small) to represent properly Example: - 8 < = 4-bit binary number <= 7 °When adding operands with different signs, overflow cannot occur! °Overflow occurs when adding: 2 positive numbers and the sum is negative 2 negative numbers and the sum is positive °On your own: Prove you can detect overflow by: Carry into MSB ° Carry out of MSB 0111 0011+ 1010 1 1100 1011+ 0111 110 7 3 1 – 6 –4 – 5 7 0

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27 Overflow Detection Logic °Carry into MSB ° Carry out of MSB For a N-bit Adder: Overflow = CarryIn[N - 1] XOR CarryOut[N - 1] A0 B0 1-bit FA Result0 CarryIn0 CarryOut0 A1 B1 1-bit FA Result1 CarryIn1 CarryOut1 A2 B2 1-bit FA Result2 CarryIn2 A3 B3 1-bit FA Result3 CarryIn3 CarryOut3 Overflow XYX XOR Y 000 011 101 110

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28 Arithmetic & Branching Conditions

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29 Condition Codes °CC Flags will be set/cleared by arithmetic operations: N (negative): 1 if result is negative (MSB = 1), otherwise 0 C (carry): 1 if carry-out(borrow) is generated, otherwise 0 V (overflow): 1 if overflow occurs, otherwise 0 Z (zero): 1 if result is zero, otherwise 0 0 1 0 1(+5) +1 0 1 0(-6) 1 1 1 1(-1) 0 1 1 1(+7) +1 1 0 1(-3) 10 1 0 0(+4) 0 1 0 1(+5) +0 1 0 0(+4) 1 0 0 1(-7?) 0 0 1 1(+3) +1 1 0 1(-3) 10 0 0 0(0)

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30 Multiplication of Positive Numbers

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31 Unsigned Multiplication °Paper and pencil example (unsigned): Multiplicand 1101(13) Multiplier 1011(11) 1101 1101 0000 1101 Product 10001111(143) °m bits x n bits = m+n bit product °Binary makes it easy: 0 => place 0 ( 0 x multiplicand) 1 => place a copy ( 1 x multiplicand)

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32 Unsigned Combinational Multiplier °Stage i accumulates A * 2 i if B i == 1 B0B0 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 0000

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33 How does it work? °at each stage shift A left ( x 2) °use next bit of B to determine whether to add in shifted multiplicand °accumulate 2n bit partial product at each stage B0B0 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 0000 000

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34 Multiplier Circuit Product (Multiplier) Multiplicand 4-bit FA Add/NoAdd Control 4 bits 8 bits Shift Right MUX 0 4 bits C Multiplicand 1101 C ProductMultiplier 0 00001011 0 1101 1011Add 0 01101101Shift 1 00111101Add 0 10011110Shift 0 10011110NoAdd 0 01001111Shift 1 00011111Add 0 10001111Shift

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35 Signed-Operand & Multiplication

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36 Signed Multiplication °Negative Multiplicand: Multiplicand 10011(-13) Multiplier 01011(+11) 1111110011 111110011 00000000 1110011 000000 Product 1101110001(-143) °Negative Multiplier: Form 2’s complement of both multiplier and multiplicand Proceed as above

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37 Motivation for Booth’s Algorithm °Works well for both Negative & Positive Multipliers °Example 2 x 6 = 0010 x 0110: 0010 x 0110 + 0000shift (0 in multiplier) + 0010 add (1 in multiplier) + 0100 add (1 in multiplier) +0000 shift (0 in multiplier) 00001100 °FA with add or subtract gets same result in more than one way: 6= – 2 + 8 0110 = – 00010 + 01000 = 11110 + 01000 °For example ° 0010 x 0110 00000000 shift (0 in multiplier) 1111110 sub (first 1 in multpl.) 000000 shift (mid string of 1s) +00010 add (prior step had last 1) 00001100

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38 Booth’s Algorithm Current BitBit to the RightExplanationExampleOp 1 0Begins run of 1s0001111000sub 1 1Middle of run of 1s0001111000none 0 1End of run of 1s0001111000add 0 0Middle of run of 0s0001111000none Originally for Speed (when shift was faster than add) Small number of additions needed when multiplier has a few large blocks of 1s

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39 Booths Example (2 x 7) 1a. P = P - m1110 +1110 1110 0111 0shift P (sign ext) 1b. 00101111 0011 111 -> nop, shift 2.00101111 1001 111 -> nop, shift 3.00101111 1100 101 -> add 4a.0010 +0010 0001 1100 1shift 4b.00100000 1110 0done OperationMultiplicandProductnext? 0. initial value00100000 0111 010 -> sub

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40 Booths Example (2 x -3) 1a. P = P - m1110 +1110 1110 1101 0shift P (sign ext) 1b. 00101111 0110 101 -> add + 0010 2a.0001 0110 1shift P 2b.00100000 1011 010 -> sub +1110 3a.00101110 1011 0shift 3b.0010 1111 0101 111 -> nop 4a1111 0101 1 shift 4b.00101111 1010 1 done OperationMultiplicandProductnext? 0. initial value00100000 1101 010 -> sub

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41 Fast Multiplication (read yourself!)

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42 Integer Division

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43 Divide: Paper & Pencil 1001 Quotient Divisor 1000 1001010 Dividend –1000 10 101 1010 –1000 10 Remainder (or Modulo result) °See how big a number can be subtracted, creating quotient bit on each step Binary => 1 * divisor or 0 * divisor °Dividend = Quotient x Divisor + Remainder => | Dividend | = | Quotient | + | Divisor |

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44 Division Circuit Remainder (Quotient) Divisor 33-bit FA Control 33 bits 65 bits Shift Left 33 bits Q Setting Sign-bit Checking

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45 RemainderQuotient Initially000001000 Shift00001000_ Sub(-11)11101 Set q 0 11110 Restore000010000 Shift00010000_ Sub(-11) 11101 Set q 0 11111 Restore000100000 Shift00100000_ Sub(-11) 11101 Set q 0 00001 000010001 Shift00010001_ Sub(-11) 11101001_ Set q 0 11111 Restore000100010 Restoring Division Algorithm 0010 Quotient Divisor 111000Dividend 11 10Remainder

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46 Floating-point Numbers & Operations

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47 Review of Numbers °Computers are made to deal with numbers °What can we represent in N bits? Unsigned integers: 0to2 N - 1 Signed Integers (Two’s Complement) -2 (N-1) to 2 (N-1) - 1

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48 Other Numbers °What about other numbers? Very large numbers? (seconds/century) 3,155,760,000 10 (3.15576 10 x 10 9 ) Very small numbers? (atomic diameter) 0.00000001 10 (1.0 10 x 10 -8 ) Rationals (repeating pattern) 2/3 (0.666666666...) Irrationals 2 1/2 (1.414213562373...) Transcendentals e (2.718...), (3.141...) °All represented in scientific notation

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49 Scientific Notation Review 6.02 x 10 23 radix (base) decimal pointmantissa exponent °Normalized form: no leadings 0s (exactly one digit to left of decimal point) °Alternatives to representing 1/1,000,000,000 Normalized: 1.0 x 10 -9 Not normalized: 0.1 x 10 -8,10.0 x 10 -10

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50 Scientific Notation for Binary Numbers 1.0 two x 2 -1 radix (base) “binary point”Mantissa exponent °Computer arithmetic that supports it called floating point, because it represents numbers where binary point is not fixed, as it is for integers Declare such variable in C as float

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51 Floating Point Representation (1/2) °Normal format: +1.xxxxxxxxxx two *2 yyyy two °Multiple of Word Size (32 bits) 0 31 SExponent 302322 Significand 1 bit8 bits23 bits °S represents Sign Exponent represents y’s Significand represents x’s °Represent numbers as small as 2.0 x 10 -38 to as large as 2.0 x 10 38

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52 Floating Point Representation (2/2) °What if result too large? (> 2.0x10 38 ) Overflow! Overflow => Exponent larger than represented in 8-bit Exponent field °What if result too small? (>0, < 2.0x10 -38 ) Underflow! Underflow => Negative exponent larger than represented in 8-bit Exponent field °How to reduce chances of overflow or underflow?

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53 Double Precision Fl. Pt. Representation °Next Multiple of Word Size (64 bits) °Double Precision (vs. Single Precision) C variable declared as double Represent numbers almost as small as 2.0 x 10 -308 to almost as large as 2.0 x 10 308 But primary advantage is greater accuracy due to larger significand 0 31 SExponent 302019 Significand 1 bit11 bits20 bits Significand (cont’d) 32 bits

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54 IEEE 754 Floating Point Standard (1/4) °Single Precision, DP similar °Sign bit:1 means negative 0 means positive °Significand: To pack more bits, leading 1 implicit for normalized numbers 1 + 23 bits single, 1 + 52 bits double always true: 0 < Significand < 1(for normalized numbers) °Note: 0 has no leading 1, so reserve exponent value 0 just for number 0

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55 IEEE 754 Floating Point Standard (2/4) °Kahan wanted FP numbers to be used even if no FP hardware; e.g., sort records with FP numbers using integer compares °Could break FP number into 3 parts: compare signs, then compare exponents, then compare significands °Wanted it to be faster, single compare if possible, especially if positive numbers °Then want order: Highest order bit is sign ( negative < positive) Exponent next, so big exponent => bigger # Significand last: exponents same => bigger #

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56 IEEE 754 Floating Point Standard (3/4) °Negative Exponent? 2’s comp? 1.0 x 2 -1 v. 1.0 x2 +1 (1/2 v. 2) 01111 000 0000 0000 0000 0000 0000 1/2 00000 0001000 0000 0000 0000 0000 0000 2 This notation using integer compare of 1/2 v. 2 makes 1/2 > 2! °Instead, pick notation 0000 0001 is most negative, and 1111 1111 is most positive 1.0 x 2 -1 v. 1.0 x2 +1 (1/2 v. 2) 1/2 00111 1110000 0000 0000 0000 0000 0000 01000 0000000 0000 0000 0000 0000 0000 2

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57 IEEE 754 Floating Point Standard (4/4) °Called Biased Notation, where bias is number subtract to get real number IEEE 754 uses bias of 127 for single prec. Subtract 127 from Exponent field to get actual value for exponent 1023 is bias for double precision °Summary (single precision): 0 31 SExponent 302322 Significand 1 bit8 bits23 bits °(-1) S x (1 + Significand) x 2 (Exponent-127) Double precision identical, except with exponent bias of 1023

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58 Special Numbers °What have we defined so far? (Single Precision) ExponentSignificandObject 000 0nonzero??? 1-254anything+/- fl. pt. # 2550+/- infinity 255nonzeroNaN

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59 Infinity and NaNs result of operation overflows, i.e., is larger than the largest number that can be represented overflow is not the same as divide by zero (raises a different exception) +/- infinity S 1... 1 0... 0 It may make sense to do further computations with infinity e.g., X/0 > Y may be a valid comparison Not a number, but not infinity (e.q. sqrt(-4)) invalid operation exception (unless operation is = or =) NaN S 1... 1 non-zero NaNs propagate: f(NaN) = NaN HW decides what goes here

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60 FP Addition °Much more difficult than with integers °Can’t just add significands °How do we do it? De-normalize to match exponents Add significands to get resulting one Keep the same exponent Normalize (possibly changing exponent) °Note: If signs differ, just perform a subtract instead.

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61 FP Subtraction °Similar to addition °How do we do it? De-normalize to match exponents Subtract significands Keep the same exponent Normalize (possibly changing exponent)

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62 Extra Bits for rounding "Floating Point numbers are like piles of sand; every time you move one you lose a little sand, but you pick up a little dirt." How many extra bits? IEEE: As if computed the result exactly and rounded. Addition: 1.xxxxx1.xxxxx1.xxxxx +1.xxxxx0.001xxxxx0.01xxxxx 1x.xxxxy 1.xxxxxyyy 1x.xxxxyyy post-normalization pre-normalization pre and post °Guard Digits: digits to the right of the first p digits of significand to guard against loss of digits – can later be shifted left into first P places during normalization. °Addition: carry-out shifted in °Subtraction: borrow digit and guard °Multiplication: carry and guard, Division requires guard

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63 Rounding Digits normalized result, but some non-zero digits to the right of the significand --> the number should be rounded E.g., B = 10, p = 3: 0 2 1.69 0 0 7.85 0 2 1.61 = 1.6900 * 10 = -.0785 * 10 = 1.6115 * 10 2-bias - one round digit must be carried to the right of the guard digit so that after a normalizing left shift, the result can be rounded, according to the value of the round digit IEEE Standard: four rounding modes: round to nearest (default) round towards plus infinity round towards minus infinity round towards 0 round to nearest: round digit < B/2 then truncate > B/2 then round up (add 1 to ULP: unit in last place) = B/2 then round to nearest even digit it can be shown that this strategy minimizes the mean error introduced by rounding

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64 Sticky Bit Additional bit to the right of the round digit to better fine tune rounding d0. d1 d2 d3... dp-1 0 0 0 0. 0 0 X... X X X S X X S + Sticky bit: set to 1 if any 1 bits fall off the end of the round digit d0. d1 d2 d3... dp-1 0 0 0 0. 0 0 X... X X X 0 X X 0 - d0. d1 d2 d3... dp-1 0 0 0 0. 0 0 X... X X X 1 - generates a borrow Rounding Summary: Radix 2 minimizes wobble in precision Normal operations in +,-,*,/ require one carry/borrow bit + one guard digit One round digit needed for correct rounding Sticky bit needed when round digit is B/2 for max accuracy Rounding to nearest has mean error = 0 if uniform distribution of digits are assumed

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65 Denormalized Numbers 02 2 2 -bias 1-bias 2-bias B = 2, p = 4 normal numbers with hidden bit --> denorm gap The gap between 0 and the next representable number is much larger than the gaps between nearby representable numbers. IEEE standard uses denormalized numbers to fill in the gap, making the distances between numbers near 0 more alike. 02 2 2 -bias 1-bias 2-bias p bits of precision p-1 bits of precision same spacing, half as many values! NOTE: PDP-11, VAX cannot represent subnormal numbers. These machines underflow to zero instead.

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