Correctly rounded exponential function in double-precision arithmetic

We present an algorithm for implementing correctly rounded exponentials in double-precision floating point arithmetic. This algorithm is based on floating-point operations in the widespread EEE-754 standard, and is therefore more efficient than those using multiprecision arithmetic, while being fully portable. It requires a table of reasonable size and IEEE-754 double precision multiplications and additions. In a preliminary implementation, the overhead due to correct rounding is a 6 times slowdown when compared to the standard library function.

[1]  Pat H. Sterbenz,et al.  Floating-point computation , 1973 .

[2]  Bruce W. Char,et al.  Maple V Library Reference Manual , 1992, Springer New York.

[3]  P. Michael Farmwald.  High bandwidth evaluation of elementary functions , 1981, 1981 IEEE 5th Symposium on Computer Arithmetic (ARITH).

[4]  Marc Daumas,et al.  Division of Floating Point Expansions with an Application to the Computation of a Determinant , 1999, J. Univers. Comput. Sci..

[5]  Jonathan Richard Shewchuk,et al.  Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates , 1997, Discret. Comput. Geom..

[6]  Shane Story,et al.  New algorithms for improved transcendental functions on IA-64 , 1999, Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336).

[7]  Arnaud Tisserand,et al.  Toward Correctly Rounded Transcendentals , 1998, IEEE Trans. Computers.

[8]  Weng-Fai Wong,et al.  Fast Hardware-Based Algorithms for Elementary Function Computations Using Rectangular Multipliers , 1994, IEEE Trans. Computers.

[9]  Douglas M. Priest,et al.  Algorithms for arbitrary precision floating point arithmetic , 1991, [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic.

[10]  Peter Tang,et al.  The Computation of Transcendental Functions on the IA-64 Architecture , 1999 .

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

[12]  Darrall Henderson.  Elementary Functions: Algorithms and Implementation , 2000 .

[13]  M. D. MacLaren.  The Art of Computer Programming. Volume 2: Seminumerical Algorithms (Donald E. Knuth) , 1970 .

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

[15]  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.

[16]  T. J. Dekker,et al.  A floating-point technique for extending the available precision , 1971 .