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Probabilistic Error Analysis of Approximate Adders and Multipliers

  • Sana Mazahir
  • Muhammad Kamran Ayub
  • Osman Hasan
  • Muhammad Shafique
Chapter

Abstract

Approximate adders and multipliers are widely being advocated to be used in error resilient applications. A very important performance metric in this regard is the probability of occurrence of error in these arithmetic circuits as this allows us to choose the most efficient configuration of an adder or multiplier for a given application. In this chapter, we present an analytical error analysis approach for approximate adders, which comprise of subadder units, and recursive approximate multipliers with approximate partial products. We also derive probability mass function (PMF) of error for both of the considered adder and multiplier models. The results show that the proposed analysis serves as an effective tool for predicting, evaluating, and comparing the accuracy of various approximate adders and multipliers. For illustration purposes, we also show that the comparative performance of different approximate adders and multipliers can be correctly predicted in practical applications of image processing.

References

  1. 1.
    Xu Q, Kim NS, Mytkowicz T (2016) Approximate computing: a survey. IEEE Des Test 33(1):8–22CrossRefGoogle Scholar
  2. 2.
    Shafique M, Hafiz R, Rehman S, El-Harouni W, Henkel J (2016) Cross-layer approximate computing: from logic to architectures. In: Proceedings of 53rd IEEE/ACM design automation conferenceGoogle Scholar
  3. 3.
    Zhang Q, Wang T, Tian Y, Yuan F, Xu Q (2015) ApproxANN: an approximate computing framework for artificial neural network. In: Proceedings of the 2015 design, automation & test in Europe conference & exhibition. EDA Consortium, San Jose, pp 701–706Google Scholar
  4. 4.
    Venkataramani S, Sabne A, Kozhikkottu V, Roy K, Raghunathan A (2012) SALSA: systematic logic synthesis of approximate circuits. In: Proceedings of 49th IEEE/ACM design automation conference, pp 796–801Google Scholar
  5. 5.
    Ranjan A, Raha A, Venkataramani S, Roy K, Raghunathan A (2014) ASLAN: synthesis of approximate sequential circuits. In: Proceedings of design, automation test Europe conference & exhibition, p 364Google Scholar
  6. 6.
    Shafique M, Ahmad W, Hafiz R, Henkel J (2015) A low latency generic accuracy configurable adder. In: Proceedings of 52nd annual design automation conference, p 86Google Scholar
  7. 7.
    Kahng AB, Kang S (2012) Accuracy-configurable adder for approximate arithmetic designs. In: Proceedings of 49th annual design automation conference, pp 820–825Google Scholar
  8. 8.
    Du K, Varman P, Mohanram K (2012) High performance reliable variable latency carry select addition. In: Proceedings of design, automation test Europe conference & exhibition, pp 1257–1262Google Scholar
  9. 9.
    Gupta V, Mohapatra D, Raghunathan A, Roy K (2013) Low-power digital signal processing using approximate adders. IEEE Trans Comput Aided Des Integr Circuits Syst 32(1):124–137CrossRefGoogle Scholar
  10. 10.
    Bhardwaj K, Mane PS (2013) ACMA: accuracy-configurable multiplier architecture for error-resilient system-on-chip. In: Proceedings of 8th international workshop on reconfigurable communication-centric systems-on-chip, pp 1–6Google Scholar
  11. 11.
    Kulkarni P, Gupta P, Ercegovac MD (2011) Trading accuracy for power in a multiplier architecture. J Low Power Electron 7(4):490–501CrossRefGoogle Scholar
  12. 12.
    Chen I-C, Hayes JP (2015) Low-area and high-speed approximate matrix-vector multiplier. In: IEEE 18th international symposium design diagnostics of electronic circuits & systems, pp 23–28Google Scholar
  13. 13.
    Momeni A, Han J, Montuschi P, Lombardi F (2015) Design and analysis of approximate compressors for multiplication. IEEE Trans Comput 64(4):984–994MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gupta V, Mohapatra D, Park SP, Raghunathan A, Roy K (2011) IMPACT: imprecise adders for low-power approximate computing. In: Proceedings of 17th IEEE/ACM international symposium low-power electronics and design, pp 409–414Google Scholar
  15. 15.
    Rehman S, El-Harouni W, Shafique M, Kumar A, Henkel J (2016) Architectural-space exploration of approximate multipliers. In: Proceedings of international conference on computer-aided design, pp 1–6Google Scholar
  16. 16.
    Ma J, Man K, Krilavicius T, Guan S, Jeong T (2011) Implementation of high performance multipliers based on approximate compressor design. In: Proceedings of international conference electrical and control technologyGoogle Scholar
  17. 17.
    Zhu N, Goh WL, Yeo KS (2009) An enhanced low-power high-speed adder for error-tolerant application. In: Proceedings of 12th international symposium on integrated circuits, pp 69–72Google Scholar
  18. 18.
    Ye R, Wang T, Yuan F, Kumar R, Xu Q (2013) On reconfiguration-oriented approximate adder design and its application. In: Proceedings of international conference computing-aided Design, pp 48–54Google Scholar
  19. 19.
    Hashemi S, Bahar R, Reda S (2015) Drum: a dynamic range unbiased multiplier for approximate applications. In: Proceedings of the IEEE/ACM international conference on computer-aided design. IEEE Press, Piscataway, pp 418–425Google Scholar
  20. 20.
    Nepal K, Li Y, Bahar R, Reda S (2014) Abacus: a technique for automated behavioral synthesis of approximate computing circuits. In: Proceedings of the conference on design, automation & test in Europe. European Design and Automation Association, Leuven, p 361Google Scholar
  21. 21.
    Mazahir S, Hasan O, Hafiz R, Shafique M, Henkel J (2017) Probabilistic error modeling for approximate adders. IEEE Trans Comput 66(3):515–530MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liang J, Han J, Lombardi F (2013) New metrics for the reliability of approximate and probabilistic adders. IEEE Trans Comput 62(9):1760–1771MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ayub MK, Hasan O, Shafique M (2017) Statistical error analysis of low power approximate adders. In: Design automation conference. ACM, New YorkGoogle Scholar
  24. 24.
    Chan W-TJ, Kahng A, Kang S, Kumar R, Sartori J (2013) Statistical analysis and modeling for error composition in approximate computation circuits. In: Proceedings of IEEE 31st international conference computing design, pp 47–53Google Scholar
  25. 25.
    Venkatesan R, Agarwal A, Roy K, Raghunathan A (2011) MACACO: modeling and analysis of circuits for approximate computing. In: Proceedings of international conference on computer-aided design, pp 667–673Google Scholar
  26. 26.
    Han J, Orshansky M (2013) Approximate computing: an emerging paradigm for energy-efficient design. In: 18th IEEE European test symposium, pp 1–6Google Scholar
  27. 27.
    Mazahir S, Hasan O, Shafique M (2017) Adaptive approximate computing in arithmetic datapaths. IEEE Design Test 35: 65–74CrossRefGoogle Scholar
  28. 28.
    Almurib HA, Kumar T, Lombardi F (2016) Inexact designs for approximate low power addition by cell replacement. In: Design, automation & test in Europe. IEEE, Piscataway, pp 660–665Google Scholar
  29. 29.
    Mazahir S, Hasan O, Hafiz R, Shafique M (2017) Probabilistic error analysis of approximate recursive multipliers. IEEE Trans Comput 66(11):1982–1990MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lin C-H, Lin C (2013) High accuracy approximate multiplier with error correction. In: Proceedings of IEEE 31st international conference computing design, pp 33–38Google Scholar
  31. 31.
    Snigdha FS, Sengupta D, Hu J, Sapatnekar SS (2016) Optimal design of jpeg hardware under the approximate computing paradigm. In: Design automation conference. ACM, New York, pp 106:1–106:6Google Scholar
  32. 32.
    Mazahir S, Hasan O, Hafiz R, Shafique M, Henkel J (2016) An area-efficient consolidated configurable error correction for approximate hardware accelerators. In: Proceedings of IEEE/ACM 53rd design automation conferenceGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sana Mazahir
    • 1
  • Muhammad Kamran Ayub
    • 2
  • Osman Hasan
    • 2
  • Muhammad Shafique
    • 3
  1. 1.Georgia Institute of TechnologyAtlantaUSA
  2. 2.School of Electrical Engineering and Computer ScienceNational University of Sciences and Technology (NUST)IslamabadPakistan
  3. 3.Institute of Computer EngineeringVienna University of Technology (TU Wien)ViennaAustria

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