Digit Serial Methods with Applications to Division and Square Root

We present a generic digit serial method (DSM) to compute the digits of a real number <inline-formula> <tex-math notation="LaTeX">$V$</tex-math><alternatives><inline-graphic xlink:href="ferguson-ieq1-2759764.gif"/> </alternatives></inline-formula>. Bounds on these digits, and on the errors in the associated estimates of <inline-formula><tex-math notation="LaTeX">$V$</tex-math><alternatives> <inline-graphic xlink:href="ferguson-ieq2-2759764.gif"/></alternatives></inline-formula> formed from these digits, are derived. To illustrate our results, we derive such bounds for a parameterized family of high-radix algorithms for division and square root. These bounds enable a DSM designer to determine, for example, whether a given choice of parameters allows rapid formation and rounding of its approximation to <inline-formula><tex-math notation="LaTeX">$V$ </tex-math><alternatives><inline-graphic xlink:href="ferguson-ieq3-2759764.gif"/></alternatives></inline-formula>.

[1]  John Harrison,et al.  HOL Light: A Tutorial Introduction , 1996, FMCAD.

[2]  David W. Matula,et al.  Further Reducing the Redundancy of a Notation Over a Minimally Redundant Digit Set , 2003, J. VLSI Signal Process..

[3]  Tomás Lang,et al.  On recoding in arithmetic algorithms , 1996, J. VLSI Signal Process..

[4]  Tomás Lang,et al.  Very-high radix combined division and square root with prescaling and selection by rounding , 1995, Proceedings of the 12th Symposium on Computer Arithmetic.

[5]  David W. Matula,et al.  Recoders for Partial Compression and Rounding , 1997 .

[6]  John Harrison,et al.  HOL Light Tutorial , 2015 .

[7]  R. Petit.  A Tutorial Introduction , 1980 .

[8]  Tomás Lang,et al.  Very-High Radix Division with Prescaling and Selection by Rounding , 1994, IEEE Trans. Computers.

[9]  Constantin P. Niculescu,et al.  Convex Functions and Their Applications: A Contemporary Approach , 2005 .

[10]  Clarence Zener,et al.  Geometric Programming : Theory and Application , 1967 .

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[12]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[13]  Nicholas Pippenger.  On-the-Fly Algorithms and Sequential Machines , 2011, IEEE Transactions on Computers.

[14]  Tomás Lang,et al.  On-the-Fly Conversion of Redundant into Conventional Representations , 1987, IEEE Transactions on Computers.

[15]  John Harrison,et al.  Formal Verification of IA-64 Division Algorithms , 2000, TPHOLs.

[16]  Milos D. Ercegovac,et al.  Variable radix real and complex digit-recurrence division , 2005, 2005 IEEE International Conference on Application-Specific Systems, Architecture Processors (ASAP'05).

[17]  Tomás Lang,et al.  On-the-fly rounding for division and square root , 1989, Proceedings of 9th Symposium on Computer Arithmetic.

[18]  M. Ercegovac,et al.  Division and Square Root: Digit-Recurrence Algorithms and Implementations , 1994 .

[19]  David W. Matula,et al.  A 17 /spl times/ 69 bit multiply and add unit with redundant binary feedback and single cycle latency , 1993, Proceedings of IEEE 11th Symposium on Computer Arithmetic.