A Radical Approach to Computation with Real Numbers

John L. Gustafson

Abstract


If we are willing to give up compatibility with IEEE 754 floats and design a number format with goals appropriate for 2016, we can achieve several goals simultaneously: Extremely high energy efficiency and information-per-bit, no penalty for decimal operations instead of binary, rigorous bounds on answers without the overly pessimistic bounds produced by interval methods, and unprecedented high speed up to some precision. This approach extends the ideas of unum arithmetic introduced two years ago by breaking completely from the IEEE float-type format, resulting in fixed bit size values, fixed execution time, no exception values or “gradual underflow” issues, no wasted bit patterns, and no redundant representations (like “negative zero”). As an example of the power of this format, a difficult 12-dimensional nonlinear robotic kinematics problem that has defied solvers to date is quickly solvable with absolute bounds. Also unlike interval methods, it becomes possible to operate on arbitrary disconnected subsets of the real number line with the same speed as operating on a simple bound.


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References


John L. Gustafson. The End of Error: Unum Computing. Boca Raton: Chapman & Hall/CRC Press, 2015.

IEEE 754-2008 IEEE Standard for Floating-Point Arithmetic. Web publication available at http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4610935, 2008.

Jean-Michel Muller et al. Floating-point arithmetic. Web publication available at

http://perso.ens-lyon.fr/jean-michel.muller/chapitre1.pdf, 2009.

Yousif Ismaill Al-Mashhadany. Inverse kinematics problem (IKP) of 6-DOF robot manipulator by Locally Recurrent Neural Networks (LRNNs). International Conference on Management and Service Science (MASS), January 2010.




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