1. © Yohai Devir 2007 Fast Arithmetics

2. © Yohai Devir 2007 Full Adder Cout AiAi BiBi SiSi Cin

3. © Yohai Devir 2007 Carry Ripple Adder A0A0 B0B0 S0S0 A1A1 B1B1 S1S1 A2A2 B2B2 S2S2 A3A3 B3B3 S3S3 C 0 =0C1C1 C2C2 C3C3 C4C4 Note: MSB is rightmost

4. © Yohai Devir 2007 Reduction Changing the goal to: Calculate the Carry vector In one time unit we can calculate S out of A,B,C No effect on complexity

5. © Yohai Devir 2007 A0A0 B0B0 S0S0 C0C0 A1A1 B1B1 S1S1 C1C1 A2A2 B2B2 S2S2 C2C2 A3A3 B3B3 S3S3 C3C3 Note: MSB is rightmost

6. © Yohai Devir 2007 Current problem C 0 =0 A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 C1C1 C2C2 C3C3 C4C4 A4A4 B4B4 A5A5 B5B5 A6A6 B6B6 A7A7 B7B7 C5C5 C6C6 C7C7 Note: MSB is rightmost

7. © Yohai Devir 2007 Three Cases Generate: Cout = 1 11 ? Kill : Cout = 0 00 ? Propagate: Cout = Cin 10 ?

8. © Yohai Devir 2007 G/P gate AiAi BiBi GP G = A∙B P = A xor B

9. © Yohai Devir 2007 C 0 =0 A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 A4A4 B4B4 A5A5 B5B5 A6A6 B6B6 A7A7 B7B7 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 C1C1 C2C2 C3C3 C4C4 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 C5C5 C6C6 C7C7 1+O(N) ! C8C8 Note: MSB is rightmost

10. © Yohai Devir 2007 If we could have calculated every 2 nd Carry C 0 =0 C1C1 C3C3 C5C5 C7C7 C2C2 C4C4 C6C6 C8C8 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 Note: MSB is rightmost

11. © Yohai Devir 2007 Calculate every 2 nd GiGi PiPi G’P’ C i+2 =1 iff G i+1 or (G i and P i+1 ) C i+2 =0 iff G i+1 =0 and P i+1 =0 C i+2 =C i iff P i and P i+1 G i+1 P i+1 CiCi GiGi PiPi G i+1 P i+1 C i+1 C i+2 G’ = G i+1 + G i ∙P i+1 P’ = P i ∙P i+1 Note: MSB is rightmost

12. © Yohai Devir 2007 C 0 =0 A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 A4A4 B4B4 A5A5 B5B5 A6A6 B6B6 A7A7 B7B7 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 C1C1 C2C2 C3C3 C4C4 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 C5C5 C6C6 C7C7 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 C8C8 C2C2 C4C4 C3C3 C6C6 C8C8 2+O(N/2)+1 ! Note: MSB is rightmost

13. © Yohai Devir 2007 C 0 =0 A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 A4A4 B4B4 A5A5 B5B5 A6A6 B6B6 A7A7 B7B7 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 C1C1 C2C2 C3C3 C4C4 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 C5C5 C6C6 C7C7 C2C2 C4C4 C3C3 C6C6 G0G0 P0P0 G1G1 P1P1 G2G2 P2P2 G3G3 P3P3 G4G4 P4P4 G5G5 P5P5 G6G6 P6P6 G7G7 P7P7 C8C8 C4C4 C8C8 C8C8 log(n) +log(n) = 2log(n) = O(log(n)) Note: MSB is rightmost

14. © Yohai Devir 2007 Final stage A0A0 B0B0 S0S0 C0C0 A1A1 B1B1 S1S1 C1C1 A2A2 B2B2 S2S2 C2C2 A3A3 B3B3 S3S3 C3C3 Overall complexity: O(log(n))+1=O(log(n)) Note: MSB is rightmost

15. © Yohai Devir 2007 Comparator A==B, A>B A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 EQ 4 EQ 1 EQ 2 EQ 3 EQ 0 =1 Note: MSB is rightmost GR 4 GR 1 GR 2 GR 3 GR 0 =0

16. © Yohai Devir 2007 Comparator A==B, A>B A0A0 B0B0 A1A1 B1B1 ALL-EQ 1 Note: MSB is rightmost ALL-GR 1 ALL-EQ 1 iffA 0 A 1 ==B 0 B 1 ALL-EQ 1 = EQ 1 ∙EQ 0 ALL-GR 1 iffA 0 A 1 > B 0 B 1 ALL-GR 1 = GR 1 + GR 0 ∙EQ 1 EQ 0 GR 0

17. © Yohai Devir 2007 A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A3A3 B3B3 A4A4 B4B4 A5A5 B5B5 A6A6 B6B6 A7A7 B7B7 log(n) + 1 = O(log(n)) Note: MSB is rightmost