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Signed and Unsigned Numbers

Csci136 Computer Architecture II Lab#5 Arithmetic Review ALU Design Ripple Carry Adder & Carry lookahead HW #4: Due on Feb 22, before class Feb.16, 2005. Signed and Unsigned Numbers. Conversion to Decimal Num’s d n ,d n-1 ,…,d 0  d n *base n +d n-1 *base n-1 +…+d 0 *base 0 Example:

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Signed and Unsigned Numbers

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  1. Csci136 Computer Architecture IILab#5 Arithmetic Review ALU Design Ripple Carry Adder & Carry lookahead HW #4: Due on Feb 22, before classFeb.16, 2005

  2. Signed and Unsigned Numbers • Conversion to Decimal Num’s dn,dn-1,…,d0  dn*basen+dn-1*basen-1+…+d0*base0 • Example: • 0xABCD  10*163+11*162+12*161+13*160 = 43981 • 1101  1*23+1*22+0*21+1*10 = 13

  3. Signed and Unsigned Numbers • Signed versus unsigned comparison • Assume 8-bit words $s0: 1111 0001two $s1: 0000 0001two slt $t0, $s0, $s1 #signed comparison sltu $t0, $s0, $s1 #unsigned comparison

  4. Signed and Unsigned Numbers • Sign Extension • 16-bit to 32-bit: • 12ten, -12ten • When? • when you shift a register right … the sign bit is repeated to keep a negative number negative  Arithmetic Shift • If the sign bit isn’t extended.. (i.e. empty spots filled with 0s) then the shift is called “Logical”variety of shift

  5. Sign Extension • Assume 8 bit words… • Shift right logical -11 by 2? • srl • Shift right arithmetic -11 by 2? • sra • Shift left logical -11 by 2? • sll

  6. Useful Shifts • Considering an 8 bit word… how would we compute dividing -6 by 2? • Consider an 8 bit word… how would we compute multiplying -7 by 4?

  7. Useful Shifts • Shifts are fast. Compilers will often substitute shifts for multiplication and division of integers by powers of 2. • Fast multiplication: int fmult(int a,b) { if(b == 0) { return 0; } if (even(b)) return fmult(2*a,b/2); if (odd(b)) return (a+fmult(a, b-1)); }

  8. Addition and Subtraction • Sub: a – b = a + (-b) • -bten = invert(b) + 1 • To get negative number from a positive.. Invert all digits and add 1. • To get a positive number from a negative… Invert all digits and add 1.

  9. What is Overflow? • Consider unsigned 8 bit integers.. What happens when we add 0x73 and 0xA9 ? • Can we get overflow when we add a positive and a negative number? • Consider signed 8 bit (2’s complement) integers.. what happens when we add 0xBC and 0xB0?

  10. Overflow Conditions for Addition and Subtraction

  11. ALU Design • Truth Tables • One logic function that is used for a variety of purposes (including within adders and to compute parity) is exclusive OR. The output of a two-input exclusive OR function is true only if exactly one of the inputs is true. Show the truth table for a two-input exclusive OR function and implement this function using AND gates, OR gates, and inverters.

  12. ALU Design • Building Logic Gates • Prove that the NOR gate is universal by showing how to build the AND, OR, and NOT functions using a two-input NOR gate. • Prove that the NAND gate is universal by showing how to build the AND, OR, and NOT functions using a two-input NAND gate.

  13. Ripple Carry Adder CarryOut = b ∙ CarryIn + a ∙ CarryIn + a ∙ b

  14. Faster Addition: Carry Lookahead • Objective • Speed up the computation • Simplify the hardware • 1st-level of Abstraction • Motivation: ci = f (a, b, ci-1)  ci = f (a, b, c0) c1 = (a0+b0)∙c0 + a0∙b0 c2 = (a1+b1)∙c1 + a1∙b1 = (a1+b1)∙((a0+b0)∙c0 + a0∙b0 )+ a1∙b1 c3 = (a2+b2)∙c2 + a2∙b2 = (a2+b2)∙((a1+b1)∙((a0+b0)∙c0 + a0∙b0 )+ a1∙b1)+ a2∙b2 ∙∙∙∙∙ • Simplify: gi = ai∙bi , pi = ai + bi ci = gi-1 + gi-2∙pi-1 + … + g0∙pi-1∙pi-2∙…∙p1 + c0∙pi-1∙…∙p0

  15. Faster Addition: Carry Lookahead • 2nd-level of Abstraction • Motivation: 1st-level abstraction isstill expensive! c8 = g7 + g6∙p7 + g5∙p7∙p6 + g4∙p7∙p6∙p5 + g3∙p7∙p6∙p5∙p4 + g2∙p7∙p6∙p5∙p4∙p3 + g1∙p7∙p6∙p5∙p4∙p3∙p2∙p1 + c0∙p7∙p6∙p5∙p4∙p3∙p2∙p1∙p0 • How: • Divide the bits into groups(sizeof n), each group using the carry-lookahead logic • Connect them in ripple carry way

  16. Faster Addition: Carry Lookahead c8 = G7,4+ G3,0∙P7,4+ c0∙P7,4∙P3,1

  17. HW#4 • Give MIPS code for abs $t2, $t3 If a>=0 then abs(a)=a else abs(a)=-a end

  18. HW#4 • Load 32-bit address lui $t0, A_upper_adjusted lw $s0, A_lower($t0) • A_lower will be sign-extended to compute the address! e.g. A=0x0001A001 ? + 0xFFFFA001 = 0x0001A001

  19. HW#4 • Carry out bit? • Overflow conditions for addition and subtraction

  20. HW#4 • slt $t0, $s0, $s1 • slt: R[rd] = (R[rs]<R[rt])? 1:0 • Compare? bne, beq

  21. HW#4 • Relative Performance of Adders • Equal time for AND, OR operation • Ripple carry: • c1  c2  c3  c4 • Time: add them together! • Carry lookahead: • {p1, p2, p3, p4, g1, g2, g3, g4} •  {c1, c2, c3, c4} • Time?

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