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COMPUTER ARITHMETIC

COMPUTER ARITHMETIC

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COMPUTER ARITHMETIC

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  1. COMPUTER ARITHMETIC • Arithmetic with Signed-2's Complement Numbers • Multiplication and Division • Floating-Point Arithmetic Operations • Decimal Arithmetic Unit • Decimal Arithmetic Operations

  2. Addition and Subtraction Subtract Magnitude Add Operation Magnitude When A>B When A<B When A=B - (B - A) +(B - A) - (B - A) +(B - A) (+A) + (+B) (+A) + (- B) (- A) + (+B) (- A) + (- B) (+A) - (+B) (+A) - (- B) (- A) - (+B) (- A) - (- B) +(A + B) - (A + B) +(A + B) - (A + B) +(A - B) - (A - B) +(A - B) - (A - B) +(A - B) +(A - B) +(A - B) +(A - B) SIGNED MAGNITUDEADDITION AND SUBTRACTION Addition: A + B ; A: Augend; B: Addend Subtraction: A - B: A: Minuend; B: Subtrahend Hardware Implementation Bs B Register M(Mode Control) AVF Complementer Input Carry Output Carry E Parallel Adder S Load Sum As A Register

  3. Addition and Subtraction SIGNED 2’S COMPLEMENT ADDITION AND SUBTRACTION Hardware B Register Complementer and Parallel Adder V Overflow AC Algorithm Subtract Add Minuend in AC Subtrahend in B Augend in AC Addend in B AC  AC + B V  overflow AC  AC + B’+ 1 V  overflow END END

  4. Multiplication MULTIPLICATION Multiplication: B * A; B: Multiplicand; A: Multiplier; P: Partial Product Multiplication of Unsigned Positive Numbers A = An-1An-2 ... A0 B = Bn-1Bn-2 ... B0 P = B * A n-1 2i * Ai ) = B * ( i=0 = An-1 * (B2n-1) + An-2 * (B2n-2) + ... + A0 * (B20) B shifted left B shifted left B shifted left n-1 bits n-2 bits 0 bits = A Or B shifted (n-1) bits to the left P = An-1*(B2n-1 * 20) + An-2*(B2n-1 *2-1) + ... + A0*(B2n-1 * 2-(n-1)) B2n-1 B2n-1 shifted right B2n-1 shifted right 1 bit (n-1) bits

  5. Multiplication EXAMPLE Multiplicand B=10111 E A Q SC Multiplier in Q Q0 = 1; add B First partial product Shift right EAQ Q0 = 1; add B Second Partial Product Shift right EAQ Q0 = 0; shift right EAQ Q0 = 0; shift right EAQ Q0 = 1; add B Fifth partial product Shift right EAQ Final Product in AQ = 0110110101 0 00000 10011 101 10111 0 10111 0 01011 11001 100 10111 1 00010 0 10001 01100 011 0 01000 10110 010 0 00100 01011 001 10111 0 11011 0 01101 10101 000

  6. Multiplication SIGNED MAGNITUDE MULTIPLICATION Bs Hardware B Register Sequence Counter Complementer and Parallel Adder Qn As Qs 0 E AC Q Register EAQ B <- Multiplicand B Q <- MultiplierA Algorithm As,Qs <- Qs Bs A <- 0, E <- 0 SC <- n-1 = 0 =1 Q0 EA <- A + B shr EAQ SC <- SC+1 = 0 END Product in AQ =0 SC

  7. Multiplication BOOTH MULTIPLICATION ALGORITHM FOR SIGNED 2’S COMPLEMENT Multiplier Strings of 0’s: No addition; Simply shifts Strings of 1’s: String of 1’s from mp to mq: 2p+1 - 2q Example 001110 (14) -> p = 3, q = 1 001110 = 23+1 - 21 M * 14 = M24 - M21 Algorithm [1] Subtract multiplicand for the first least significant 1 in a string of 1’s in the multiplier [2] Add multiplicand for the first 0 after the string of 1’s in the multiplier [3] Partial Product does not change when the multiplier bit is identical to the previous bit 110010 = -24 + 22 - 21 = -16 + 4 - 2 = -14 subtract subtract 24 21 Add 22

  8. Multiplication BOOTH ALGORITHM FOR SIGNED 2’S COMPLEMENT B <- Multiplicand B Q <- Multiplier A AC <- 0 Q-1 <- 0 SC <- n Q0Q-1 ? 10 01 Q-1 : shifted out bit on shr of Q 11 00 AC<-AC+B’+1 AC <- AC + B ashr(AC&Q) SC <- SC + 1  0 SC ? =0 END

  9. Multiplication EXAMPLE OF BOOTH MULTIPLIER B = 10111 Q0Q-1 B’+1=01001 AC Q Q-1 SC 10 11 01 00 10 Initial Subtract B ashr ashr Add B ashr ashr Subtract B ashr 00000 10011 0 101 01001 01001 00100 11001 1 100 00010 01100 1 011 10111 11001 11100 10110 0 010 11110 01011 0 001 01001 00111 00011 10101 1 000

  10. Multiplication ARRAY MULTIPLIER A = a1a0: Multiplier B = b1b0: Multiplicand C = B * A = c3c2c1c0 b1 b0 a1 a0 a0b1 a0b0 a1b1 a1b0 c3 c2 c1 c0 b1 b0 a0 b1 b0 a1 HA HA C S C S c3 c2 c1 c0

  11. Multiplication ARRAY MULTIPLIER 4-BIT X 3-BIT a0 a1 b3 b2 b1 b0 b3 b2 b1 b0 0 Addend Augend 4-bit Adder Sum and Carry Outputs a2 b3 b2 b1 b0 Addend Augend 4-bit Adder Sum and Carry Outputs c6 c5 c4 c3 c2 c1 c0

  12. Division DIVISION A / B = Q + R A: Dividend; B: Divisor; Q: Quotient; R: Remainder Divisor B = 10001, B’+ 1 = 01111 E A Q SC 01110 00000 5 0 11100 00000 01111 1 01011 1 01011 00001 4 0 10110 00010 01111 1 00101 1 00101 00011 3 0 01010 00110 01111 0 11001 00110 10001 1 01010 2 0 10100 01100 01111 1 00011 1 00011 01101 1 0 00110 11010 01111 0 10101 11010 10001 1 00110 11010 0 00110 11010 Dividend: shl EAQ add B’+1 E=1 Set Q0=1 shl EAQ Add B’+1 E=1 Set Q0=1 shl EAQ add B’+1 E=0; Q0=0 add B restore remainder shl EAQ add B’+1 E=1 Set Q0=1 shl EAQ add B’+1 E=0; Q0=0 add B restore remainder neglect E remainder in A quotient in Q

  13. Division FLOWCHART OF DIVIDE OPERATION Dividend in AQ Divisor in B Qs As Bs SC<- n - 1 shl EAQ E EA  A + B’+1 EA  A+B’+1 A  A+B’+1 1 0 E 1 A B A<B E 0(A<B) A B Q0 1 EA  A+B EA  A+B EA  A+B DVF  1 DVF  0 SC  SC-1  0 0 SC END (Divide overflow) END (Quotient in Q Remainder in R)

  14. Floating Point Arithmetic FLOATING POINT ARITHMETIC OPERATIONS F = m x re where m: Mantissa r: Radix e: Exponent Registers for Floating Point Arithmetic Bs B b BR Parallel Adder and Comparator Parallel Adder E As A1 A a AC Qs Q q QR

  15. Floating Point Arithmetic FLOATING POINT ADD AND AUBTRACT  0  0 =0 AC BR CHECK FOR 0 =0 a<b a>b a:b Align Mantissa AC  BR shr A a  a+1 shr B b  b+1 add op sub sub add op As  A’s 1 1 As Bs As Bs + or - of mantissa 0 0 EA  A+B EA<-A+B’+1 A  A’+1 As  A’s 0 E 1  0 =0 A Normalization E A1 AC  0 shl A a  a+1 shr A A1  E a  a+1 END

  16. Floating Point Arithmetic FLOATING POINT MULTIPLICATION BR  Multiplicand QR  Multiplier =0 BR  0 =0 QR  0 AC  0 a  q a  a + b a  a - bias Multiply mantissa (finxed point multiplication) shl AQ a  a - 1 0 A1 1 END (Product is in AC)

  17. Floating Point Arithmetic FLOATING POINT DIVISION BR  Divisor AC  Dividend =0 BR  0 =0 AC  0 QR  0 Qs  As + Bs Q  0 SC  n-1 divide by 0 EA  A+B’+1 1 0 E A<B A>=B A  A+B shr A a  a+1 A  A+B a  a+b’+1 a  a+bias q  a Divide Magnitude of mantissa as in fixed point numbers

  18. BCD Arithmetic BCD ADD BCD digit < 10 BCD digit + BCD digit + carry =< 19 Binary Sum BCD Sum K Z8 Z4 Z2 Z1 C S8 S4 S2 S1 Decimal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 2 0 0 0 1 1 0 0 0 1 1 3 0 0 1 0 0 0 0 1 0 0 4 0 0 1 0 1 0 0 1 0 1 5 0 0 1 1 0 0 0 1 1 0 6 0 0 1 1 1 0 0 1 1 1 7 0 1 0 0 0 0 1 0 0 0 8 0 1 0 0 1 0 1 0 0 1 9 0 1 0 1 0 1 0 0 0 0 10 0 1 0 1 1 1 0 0 0 1 11 0 1 1 0 0 1 0 0 1 0 12 0 1 1 0 1 1 0 0 1 1 13 0 1 1 1 0 1 0 1 0 0 14 0 1 1 1 1 1 0 1 0 1 15 1 0 0 0 0 1 0 1 1 0 16 1 0 0 0 1 1 0 1 1 1 17 1 0 0 1 0 1 1 0 0 0 18 1 0 0 1 1 1 1 0 0 1 19

  19. BCD Arithmetic BCD ADDER If we can convert Binary Sums to BCD Sum , we can use a binary adder to add two BCD numbers SUM =< 9 BCD Sum = Binary Sum BCD Carry = Binary Carry 19 >= SUM > 9 BCD Sum = Binary Sum + 0110 BCD Carry = Carry(Binary Sum + 0110) 4-bit Binary Add 1 0 K Take next higher digit 0 Z8 1 1 Z4 done ? 0 END 1 0 Z2 BCD Sum = Sum BCD C  Carry(Sum) BCD Sum  Sum + 0110 BCD C  Carry(BCD Sum)

  20. BCD Arithmetic BCD ADDER HARDWARE Addend Augend Carry Out 4-bit Binary Addr Carry In K Z8 Z4 Z2 Z1 BCD Carry 0 0 1 1 0 4-bit Binary Adder S8 S4 S2 S1

  21. Decimal Arithmetic DECIMAL ARITHMETIC OPERATIONS Addition - Identical to the BCD addition - 9’s complement and 10’s complement are identical to 1’s complement and 10’s complement, respectively