A Priori Worst Case Error Bounds for Floating-Point Computations

A new technique for the a priori calculation of rigorous error bounds for floating-point computations is introduced. The theorems given in the paper combined with interval arithmetic lead to the implementation of reliable software routines, which enable the user to compute the desired error bounds automatically by a suitable computer program. As a prominent example, a table-lookup algorithm for calculating the function exp(x)-1 that has been published by P.T.P. Tang (1992) is analyzed using these new tools. The result shows the high quality of the new approach.

[1]  Warren E. Ferguson,et al.  Accurate and monotone approximations of some transcendental functions , 1991, [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic.

[2]  W. Krämer,et al.  Multiple-Precision Computations with Result Verification , 1993 .

[3]  Walter Krämer,et al.  A Computer Oriented Approach to Get Sharp Reliable Error Bounds , 1997, Reliab. Comput..

[4]  Ping Tak Peter Tang.  Table-driven implementation of the exponential function in IEEE floating-point arithmetic , 1989, TOMS.

[5]  G. Corliss,et al.  C-Xsc: A C++ Class Library for Extended Scientific Computing , 1993 .

[6]  Ulrich W. Kulisch,et al.  PASCAL-XSC , 1992, Springer Berlin Heidelberg.

[7]  Abraham Ziv,et al.  Fast evaluation of elementary mathematical functions with correctly rounded last bit , 1991, TOMS.

[8]  Warren E. Ferguson.  Exact computation of a sum or difference with applications to argument reduction , 1995, Proceedings of the 12th Symposium on Computer Arithmetic.

[9]  Ping Tak Peter Tang.  Table-driven implementation of the logarithm function in IEEE floating-point arithmetic , 1990, TOMS.

[10]  U. Kulisch,et al.  Scientific Computing with Automatic Result Verification. , 1994 .

[11]  Ulrich W. Kulisch,et al.  Scientific Computation with Automatic Result Verification , 1988, Computing Supplementum.

[12]  William Kahan,et al.  Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic , 1996 .

[13]  Ping Tak Peter Tang,et al.  Table-lookup algorithms for elementary functions and their error analysis , 1991, [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic.

[14]  Ping Tak Peter Tang.  Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic , 1992, TOMS.