The Power Architecture and Power.org word marks and the Power and Power.org logos and related marks are trademarks and service marks licensed by Power.org. Stamatis Vassiliadis Symposium Future Research in Computer Arithmetic September 28, 2007 Eric Schwarz, IBM
Future Research in Computer Arithmetic. September 28, 2007 Eric Schwarz, IBM. Topics. Binary Multiplication Proofs of Overlapped ScanningFoundations 7:3 Counter DesignFuture Division Direct Division Remainder Avoidanceactive Decimal Floating-Point extremely active - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
The Power Architecture and Power.org word marks and the Power and Power.org logos and related marks are trademarks and service marks licensed by Power.org.
»Recode multiplier and separate into digits»Create multiples of the multiplicand»Multiplex the multiples»Sum all partial products in counter tree»reduce final 2 partial products in CLA
Stamatis VassiliadisSymposium
History of Overlapped Scanning
» A.D. Booth in 1951 showed overlapped scanning» L. Rubinfield in 1975 proved radix-4 Booth» S. Vassiliadis in 1989 proved Booth for any radix
Partial products are formed from two multiples to reduce area
ii
p
ii NMNMP 100)](10)[( 1
2/
0
ip
iiNMP 10)(
0
Multiple A B
1X 0 1X
2X 2X 0
3X 2X 1X
4X 2X 2X
5X 5X 0
6X 5X 1X
7X 5X 2X
8X 10X -2X
9X 10X -1X
34 digit multiplication on 36 digit dataflow 1 digit every 2 cycles:
16 digit multiplication on dual 18 digit dataflows 1 digit every cycles:
Stamatis VassiliadisSymposium
Performance of Arithemetic Operations
Cycles Required for Execution
Doubleword Operands Quadword Operands
Case 1 Add/Sub 9 to 13 11 to 15
Case 2 Add/Sub 11 to 15 13 to 17
Case 3 Add/Sub 13 to 17 15 to 19
Multiplication * 19 + N 21 + 2N
Division 82 154
* N is the number of digits in the first operand excluding leading zeros
Stamatis VassiliadisSymposium
Future
» Pipelined Adder with Rounding Injection • Lia-Kai Wang
» Decimal Multiplication • Mark Erle and Michael Schulte – 3:2 Counter• Tomas Lang and A. Nannarelli - 4:2 Counter• Alvaro Vazquez et. al. 4221• Luigi Dadda - counters
» Decimal Multiply-Add Pipeline with Rounding Injection
» Divide • Tomas Lang and A. Nannarelli – Base 2 and Base 5
» Intel Format
Stamatis VassiliadisSymposium
Future of Computer Arithmetic
» Is based on clear proofs and expositions of the fundamental concepts.
• the easier to understand, the easier to build on
»Arithmetic is very active• IEEE 754R Standard currently in ballot• Decimal Floating-Point pipelined designs• new adder designs, new multiplier designs• vector processing, image processing, video game