forthcoming book on elementary transcendental functions

[David G. Hough at validgh]

>From USENET:

From: jmmulleraens-lyon.fr (Jean-Michel Muller)
Newsgroups: comp.arch.arithmetic
Subject: Re: SquareRoot Arithmetic
Date: 10 Jan 1997 15:10:14 GMT
Organization: Ecole Normale Superieure de Lyon, France

In the past there used to be two classes of methods for evaluating
the elementary functions:

- Polynomial or rational approximations (see for instance ref. CW80 and
Har68 below), that were mainly used in software. Some packages were so
good (and the basic arithmetic was sometimes so bad) that in ACGSST86 a
machine is cited for which  X**2 was more frequently exactly rounded
than X*X

- Cordic-like methods, used for instance in the 8087 and in some of its
Intel followers (probably till the 486). See Nav83 for a presentation
of the implementation on the
8087.

Now, it seems that table-based methods are favoured (this might change
if redundant Cordic-like methods become used). Those methods use tables to
reduce the argument to a very small interval, on which a small degree
polynomial approximation suffices. This is interesting since small degree
implies fast and accurate computations. To give results that are
accurate up to the very last bits, these methods require very careful
implementation and clever tricks.
The major works in this domain are by Gal (see e.g. Gal86) and Tang
(see e.g. Tan91). Both of them proposed implementation sof their methods
suited for the IEEE F.P. arithmetic.

I am finishing a book presenting the major methods for computing the
elementary functions. It should appear around may 1997.


aBOOK{Har68,
        AUTHOR             = {J.F. Hart et al.},
        PUBLISHER          = {Wiley},
        TITLE              = {Computer Approximations},
        YEAR               = {1968},
        KEYWORDS           = {elementary functions,polynomial approximations,rational approximations}
}

aBOOK{CW80,
        AUTHOR             = {W. Cody and W. Waite},
        PUBLISHER          = {Prentice-Hall Inc},
        TITLE              = {Software Manual for the Elementary Functions},
        YEAR               = {1980},
        KEYWORDS           = {elementary functions,exponential,floating-point arithmetic,logarithm,polynomial approximations,square root,trigonometric functions}
}

aARTICLE{ACGSST86,
        AUTHOR             = {R.C.~Agarwal and J.C.~Cooley and F.G.~Gustavson and J.B.~Shearer and G.~Slishman and B.~Tuckerman},
        JOURNAL            = {IBM Journal of Research and Development},
        KEY                = {ACGSST86},
        MONTH              = mar,
        NUMBER             = {2},
        PAGES              = {126--144},
        TITLE              = {New Scalar and Vector Elementary Functions for the {IBM} System/370},
        VOLUME             = {30},
        YEAR               = {1986}
}

aARTICLE{Nav83,
        AUTHOR             = {Rafi Nave},
        JOURNAL            = {Microprocessing and Microprogramming},
        KEY                = {Nav83},
        PAGES              = {221--225},
        TITLE              = {Implementation of Transcendental Functions on a Numerics Processor},
        VOLUME             = {11},
        YEAR               = {1983}
}

aINPROCEEDINGS{Gal86,
        AUTHOR             = {S.~Gal},
        BOOKTITLE          = {Accurate Scientific Computations. Lecture Notes in Computer Science, No 253},
        EDITOR             = {Springer-Verlag},
        JOURNAL            = {Accurate Scientific Computations. Lecture Notes in Computer Science},
        KEY                = {Gal86},
        PAGES              = {1--16},
        TITLE              = {Computing elementary functions: a new approach for achieving high accuracy and good performance},
        VOLUME             = {235},
        YEAR               = {1986}
}

aINPROCEEDINGS{Tan91,
        AUTHOR             = {P.T.P. Tang},
        BOOKTITLE          = {proceedings of the 10th IEEE Symposium on Computer Arithmetic},
        EDITOR             = {P. Kornerup and D. Matula},
        MONTH              = {June},
        PAGES              = {232-236},
        PUBLISHER          = {IEEE Computer Society Press},
        TITLE              = {Table lookup algorithms for elementary functions and their error analysis},
        YEAR               = {1991},
        KEYWORDS           = {accuracy control,elementary functions,table lookup algorithms},
        LOCATION           = {Grenoble (France)}
}

 
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Jean-Michel Muller, CNRS, Lab. LIP, Ecole Normale Superieure de Lyon
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