Skip to main content
SpringerLink
Log in

CORDIC Processor for Variable-Precision Interval Arithmetic

  • Published:
Journal of VLSI signal processing systems for signal, image and video technology Aims and scope Submit manuscript

Abstract

In this paper we present a specific CORDIC processor for variable-precision coordinates. This system allows us to specify the precision to perform the CORDIC operation, and control the accuracy of the result, in such a way that re-computation of inaccurate results can be carried out with higher precision. It permits a reliable and accurate evaluation of a wide range of elementary functions. The specific architecture designed greatly improves the computational time of previous solutions based on classic polynomial approximation. For controlling error in numerical computation (where intervals are normally narrow) the proposed design performs an interval operation in a time close to that of a point operation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Knofel, “Hardware Kernel for Scientific/Engineering Computations,” in Scientific Computing with Automatic Result Verification, E. Adams and U. Kulisch (Eds.), 1993, pp. 549-570.

  2. D. Goldberg, “What Every Computer Scientist Should Know About Floating-Point Arithmetic,” ACM Computing Surveys, vol. 23, 1991, pp. 5-48.

    Article  Google Scholar 

  3. R.B. Brent, “A FORTRAN Multiprecision Arithmetic Package,” ACM Trans. Mathematical Software, vol. 4, 1978, pp. 57-70.

    Article  Google Scholar 

  4. O. Knuppel, “PROFIL/BIAS—A Fast Interval Library,” Computing, vol. 53, 1994, pp. 277-288.

    Article  MathSciNet  Google Scholar 

  5. R. Klatte, U. Kulisch, A. Wiethoff, C. Lawo, and M. Rauch, CXSC: A C++ Class Library for Extended Scientific Computing. Springer-Verlag, 1993.

  6. J.S. Ely, “The VPI Software Package for Variable Precision Interval Arithmetic,” Interval Computation, vol. 2, 1993, pp. 135-153.

    Google Scholar 

  7. R.E. Moore, Interval Analysis. Prentice Hall, 1966.

  8. R.E. Moore, Reliability in Computing: The Role of Interval Methods in Scientific Computations. Academic Press, 1988.

  9. R.E. Moore, “Computing to Arbitrary Accuracy,” in Computational and Applied Mathematics, I: Algorithms and Theory, C. Brezinski and U. Kulisch (Eds.), North-Holland, 1992, pp. 327-336.

  10. M.J. Schulte and E.E. Swartzlander, Jr. “A Family of Variable-Precision, Interval Arithmetic Processors,” IEEE Transactions on Computers, vol. 49, no. 5, 2000, pp. 387-397.

    Article  Google Scholar 

  11. M.J. Schulte and E.E. Swartzlander, Jr., “Hardware Design and Arithmetic Algorithms for a Variable-Precision, Interval Arithmetic Coprocessor,” in Proc. 12th Symposium on Computer Arithmetic, 1995, pp. 222-229.

  12. J.S. Ely and G.R. Baker, “High-Precision Calculations of Vortex Sheet Motion,” Journal of Computational Physics, vol. 111, no. 2, 1994, pp. 275-281.

    Article  MathSciNet  MATH  Google Scholar 

  13. R.B. Kearfott and V. Kreinovich, “Applications of Interval Computations: An Introduction,” in Applications of Interval Computations, R.B. Kearfott and V. Kreinovich (Eds.), Kluwer Academic Publishers, 1996, pp. 1-21.

  14. M.J. Schulte, A Variable-Precision, Interval Arithmetic Processor. PhD thesis, The University of Texas at Austin, May 1996.

  15. A. Knofel, “Fast Hardware Units for the Computation of Accurate Dot Products,” in Kornerup and Matula [42], pp. 70-75.

  16. M. Müller, C. Rüb, and W. Rülling, “Exact Accumulation of Floating-Point Numbers,” in Kornerup and Matula [42], pp. 64-69.

  17. T.M. Carter, “Cascade: Hardware for High/Variable Precision Arithmetic,” in Proceedings of the 9th Symposium on Computer Arithmetic, 1989, pp. 184-191.

  18. D.M. Chiarulli, W.G. Rudd, and D.A. Buell, “DRAFT: A Dynamically Reconfigurable Processor for Integer Arithmetic,” in Proceedings of the 7th Symposium on Computer Arithmetic, 1985, pp. 309-318.

  19. M.S. Cohen, T.E. Hull, and V.C. Hamacher. “CADAC: A Controlled-Precision Decimal Arithmetic Unit,” IEEE Transactions on Computers, vol. C-32, 1983, pp. 370-377.

    Article  Google Scholar 

  20. J.E. Volder, “The CORDIC Trigonometric Computing Technique,” IRE Trans. Electronic Computers, vol. EC, no. 8, 1959, pp. 330-334.

    Article  Google Scholar 

  21. J.S. Walther, “A Unified Algorithm for Elementary Functions,” in Proc. Spring. Joint Comput. Conf., 1971, pp. 379-385.

  22. Y.H. Hu, “The Quantization Effects of the CORDIC Algorithm,” IEEE Trans. on Signal Processing, vol. 40, no. 4, 1992, pp. 834-844.

    Article  MATH  Google Scholar 

  23. G.L. Haviland and A.A Tuszynski, “A CORDIC Arithmetic Processor Chip,” IEEE Trans. on Computers, vol. C-29, no. 2, 1980, pp. 68-79.

    Article  Google Scholar 

  24. D. Timmermann, H. Hahn, B.J. Hosticka, and B. Rix, “A New Addition Scheme and Fast Scaling Factor Compensation Methods for CORDIC Algorithms,” The VLSI Journal of Integration, no. 11, 1991, pp. 85-100.

  25. J. Villalba, T. Lang, and E.L. Zapata, “Parallel Compensation of Scale Factor for the CORDIC Algorithm,” Journal of VLSI of Signal Processing, vol. 19, no. 3, 1998, pp. 227-242.

    Article  Google Scholar 

  26. Y.H. Hu, “CORDIC-based VLSI Architectures for Digital Signal Processing,” IEEE Signal Processing Magazine, no. 7, 1992, pp. 16-35.

  27. J.M. Muller, Elementary Functions: Algorithms and Implementation. Birkhäuser Boston, 1997.

    Book  MATH  Google Scholar 

  28. J. Hormigo, J. Villalba, and E.L. Zapata, “A Hardware Approximation to Interval Arithmetic for Sine and Cosine Functions,” Volume of Extended Abstracts SCAN-98, Budapest (Hungary), Sept. 1998, pp. 58-59.

  29. J. Hormigo, J. Villalba, and E.L. Zapata, “Interval Sine and Cosine Functions Computation Based on Variable-Precision CORDIC Algorithm,” in Proc. 14th IEEE Symposium on Computer Arithmetic (ARITH14), 1999, pp. 186-193.

  30. J. Hormigo, J. Villalba, and E.L. Zapata, “Arithmetic Unit for the Computation of Interval Elementary Functions,” in 25th Euromicro Conference Workshop on Digital System Design, 1999, Sept. Milan, Italy.

  31. K. Kota and J.R. Cavallaro, “Numerical Accuracy and Hardware Tradeoffs for CORDIC Arithmetic for Special Purpose Processors,” IEEE Trans. Computers, vol 42, no. 7, 1993, pp. 769-779.

    Article  Google Scholar 

  32. K. Braune, “Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy,” in, Scientific Computing with Automatic Result Verification, U. Kulisch and H.J. Stetter (Eds.) Springer-Verlag, 1988, pp. 159-184.

  33. D.M. Priest, “Fast Table-Driven Algorithms for Interval Elementary Functions,” in Proc. 13th Symposium on Computer Arithmetic, 1997, pp. 168-174.

  34. J. Villalba, T. Lang, and E.L. Zapata, “Radix-4 Vectoring CORDIC Algorithm and Architectures,” Journal of VLSI of Signal Processing Systems, vol. 19, no. 2, 1998, pp. 127-148.

    Article  Google Scholar 

  35. S. Wang, V. Piuri, and E.E. Swartzlander, Jr., “Hybrid Cordic Algorithms,” IEEE Trans. Computers, vol. 46, no. 11, 1997, pp. 1202-1207.

    Article  Google Scholar 

  36. J. Villalba, J.A. Hidalgo, E. Antelo, J.D. Bruguera, and E.L. Zapata, “CORDIC Architecture with Parallel Compensation of the Scale Factor,” in Proc. Int. Conf. on Application Specific Array Processors (ASAP'95), 1995, pp. 258-269.

  37. M.D. Ercegovac and T. Lang, “Redundant and On-line CORDIC: Application to Matrix Triangularization and SVD,” IEEE Trans. on Computers, vol. 39, no. 6, 1990, pp. 725-740.

    Article  Google Scholar 

  38. D. Timmermann, B. Rix, H. Hahn, and B.J. Hosticka, “A CMOS Floating-Point Vector-Arithmetic Unit,” IEEE Journal of Solid-State Circuits, vol. 29, no. 5, 1994, pp. 634-639.

    Article  Google Scholar 

  39. J. Villalba, “Diseño de Arquitecturas CORDIC Multidimensionales,” Ph.D Universidad de Malaga, Nov. 1995.

  40. J. Hormigo, J. Villalba, and E.L. Zapata, “CORDIC Algorithm with Digit Skipping,” Proc. 32th. Asilomar Conference on Signals, Systems, and Computers, Nov. 1998, pp. 194-196.

  41. J. Hormigo, J. Villalba, and E.L. Zapata, “CORDIC and Variable-Precision,” Internal Report UMA-DAC-97/18. University of Malaga, Spain, 1997.

    Google Scholar 

  42. P. Kornerup and D.W. Matula (Eds.), in Proc. 10th Symposium on Computer Arithmetic, 1991.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Architecture, University of Málaga, Málaga (, Spain

    Javier Hormigo, Julio Villalba & Emilio L. Zapata

Authors
  1. Javier Hormigo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julio Villalba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilio L. Zapata
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hormigo, J., Villalba, J. & Zapata, E.L. CORDIC Processor for Variable-Precision Interval Arithmetic. The Journal of VLSI Signal Processing-Systems for Signal, Image, and Video Technology 37, 21–39 (2004). https://doi.org/10.1023/B:VLSI.0000017001.88149.f4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:VLSI.0000017001.88149.f4

Search

Navigation