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FPBIBLIO.TXT version 1.8, 18-August-1994

This is a bibliography of material on floating point arithmetic that I came up with while doing research on a floating-point package of my own. I don't claim it to be anywhere near complete. The material listed is only what I myself possess. My main interest was in software based, binary floating point arithmetic on a microprocessor, so you won't find much material about the hardware used in floating point arithmetic (e.g. adders, carry propagation schemes, higher radix representation for multi- plication and division, etc.) in this list. There is also not too much on non-binary floating point arithmetic. For most fields covered in this bibliography, the important or historically relevant articles should be included. There is also some material on integer arithmetic in this list as some of the methods used with integer arithmetic contain interesting ideas that may be useful in the realization of a floating point arithmetic package. Also, depending on the type of microprocessor used, one may need to implement integer multiplication and division for use in the floating-point package, so articles about this topic are included as well. As I am German, there is a bit of material in German in this bibliography. All German umlauts are written in German LaTeX format, that is "a, "o, "u, "s. These are equivalent to \"a, \"o, \"u, \"s in plain TeX.

Thanks to the people who have helped me with previous versions of this document by sending me papers or additional references:

Steven Sommars (sesv@research.att.com), Jim Kiernan (jmk@teak.cray.com), Warren Ferguson (ferguson@seas.smu.edu), Nhuan Doduc (ndoduc@framentec.fr), K.C. Ng (kwok.ng@eng.sun.com)

Norbert Juffa 460 Navaro Way #201 San Jose, CA 95134 USA

email: norbert@iit.com

Changes from Version 1.7:

o Added Section on multiprecision arithmetic o Added Section on floating-point processors o Added Section on floating-point summation

+++++++++++++++ Bibliography for floating point arithmetic ++++++++++++++++++++


Books, hardware oriented

[Cav] Cavanagh, J.F.: Digital Computer Arithmetic.

     New York, NY: Wiley 1984

[Flo] Flores, I.: The Logic of Computer Arithmetic.

     Englewood Cliffs, NJ: Prentice Hall 1963

[Gos] Gosling, J.B.: Design of Arithmetic Units for Digital Computers.

     London: Macmillan Press 1980

[Hwa] Hwang, K.: Computer Arithmetic.

     New York, NY: Wiley 1979

[Kor] Koren, I.: Computer Arithmetic Algorithms.

     Englewood Cliffs, NJ: Prentice Hall 1993

[Omo] Omondi, A.: Computer Arithmetic: Architecture, Implementation, Analysis.

     Englewood Cliffs, NJ: Prentice Hall 1994

[Sco] Scott, N.R.: Computer Number Systems and Arithmetic.

     Englewood Cliffs, NJ: Prentice Hall 1985

[Sch] Schmid, H.: Decimal Computation.

     New York, NY: Wiley 1974

[Sch2] Schmid, H.: Eletronische Dezimalrechner, Schaltungen und Verfahren.

     M"unchen: R. Oldenburg 1978

[Spa] Spaniol, O.: Arithmetik in Rechenanlagen.

     Stuttgart: Teubner 1976

[Spa2] Spaniol, O.: Computer Arithmetic: Logic and Design

     New York, NY: Wiley 1981

[Ste] Sterbenz, P.H.: Floating Point Computation.

     Englewood Cliffs, NJ: Prentice Hall 1974

[Swa] Swartzlander, E.E. (ed.): Computer Arithmetic.

     Stroudsburg, PA: Dowden, Hutchinson & Ross 1980

[Swa2] Swartzlander, E.E. (ed.): Computer Arithmetic. Vol. 1, 2.

     Los Alamitos, CA: IEEE Computer Society Press 1990

[Was] Waser, S.; Flynn, M.J.: Introduction to Arithmetic for Digital

     Systems Designers.
     Fort Worth, TX: Holt, Rinehart and Winston 1982

Books, software oriented or theory

[Cod] Cody, W.J.; Waite, W.: Software Manual for the Elementary Functions.

     Englewood Cliffs, NJ: Prentice Hall 1980

[Fik] Fike, C.T.: Computer Evaluation of Mathematical Functions.

     Englewood Cliffs, NJ: Prentice Hall 1968

[Har] Hart, J.F.; Cheney, E.W.; Lawson, C.L.; Maehly, H.J.; Mesztenyi, C.K.;

     Rice, J.R.; Thacher, H.G.; Witzgall, C.: Computer Approximations.
     New York, NY: Wiley 1968

[Has] Hastings, C.B.: Approximations for Digital Computers.

     Princeton, NJ: Princeton University Press 1955

[Knu] Knuth, D.E.: The Art of Computer Programming, 2nd Edition.

     Vol. 2: Seminumerical Algorithms.
     Reading, MA: Addison-Wesley 1981

[Lam] Lampe, B.; Jorke, G.; Wengel, N.: Arithmetische Algorithmen der

     Mikrorechentechnik.
     Berlin: Verlag Technik 1989

[Luk1] Luke, Y.L.: Mathematical Functions and their Approximations.

     New York, NY: Academic Press 1975

[Luk2] Luke, Y.L.: Algorithms for the Computation of Mathematical Functions.

     New York, NY: Academic Press 1977

[Lyu] Lyusternik, L.A.; Chervonenkis, O.A.; Yanpolski, A.R.:

     Handbook for Computing Elementary Functions.
     New York, NY: Pergamon Press 1965

[Mor] Morgan, D.: Numerical Methods.

     San Mateo, CA: M&T 1992

[Mos] Moshier, S.L.B.: Methods and Programs for Mathematical Functions.

     Chichester: Ellis Horwood 1989

[Pla] Plauger, P.J.: The Standard C Library.

     Englewoods Cliff, NJ: Prentice Hall 1992

[Pre] Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Vetterling, W.T.:

     Numerical Recipes in C.
     Cambridge: Cambridge University Press 1988

[Ric] Rice, J.R.: The Approximation of Functions. Vol. 1.

     Reading, MA: Addison-Wesley 1964

[Ric2] Rice, J.R.: The Approximation of Functions. Vol. 2.

     Reading, MA: Addison-Wesley 1969

[Wil] Wilkinson, J.H.: Rounding Errors in Algebraic Processes.

     Englewood Cliffs, NJ.: Prentice-Hall 1963

Books, machine specific

[App1] Apple Computers, Inc.: Apple Numerics Manual, 2nd ed.

     Reading, MA: Addison Wesley 1988.

[App2] Apple Computers, Inc.: Inside Macintosh: PowerPC Numerics.

     Reading, MA: Addison-Wesley 1994.

[Kor] Kortemeyer, G.: Coprozessoren Programmierung mit Turbo Pascal und C++.

     Vaterstetten: IWT 1993

[Mot] Motorola, Inc.: 68881/882 Floating-Point Coprocessor User's Manual

     Second Edition.
     Englewood Cliffs, NJ.: Prentice Hall 1989

[Pal1] Palmer, J.F.; Morse, S.P.: The 8087 Primer.

     New York, NY.: Wiley 1984

[Pal2] Palmer, J.F.; Morse, S.P.: Die mathematischen Grundlagen der

     Numerik-Prozessoren 8087/80287
     M"unchen: te-wi Verlag 1985

[Sta] Startz, R.: 8087/80287/80387 for the IBM PC & Compatibles, 3rd ed.

     New York: Brady 1988

[Thi] Thies, K.-D.: Die 8087/80287 numerischen Prozessor Erweiterungen f"ur

     8086/80286 Systeme.
     M"unchen: te-wi 1985

[Thi2] Thies, K.-D.: PC/XT/AT Numerik Buch.

     M"unchen: te-wi 1989

Journal Publications, Conference Papers, Technical Reports, Ph.D. Dissertations, Book Contributions, etc.

[Note: All section headings begin with a two '#' characters so it is easier to find the section using the 'find' feature of editors.]

##1.0 Choice of base, floating point formats ##1.1 Precision and Rounding ##1.2 Determination of parameters of floating point arithmetic ##1.3 IEEE standards for floating point arithmetic ##1.4 Floating point arithmetic, general and implementation issues ##1.5 Floating point packages ##1.6 Floating point processors ##1.7 Test of floating point routines ##2.0 Addition and Subtraction ##2.0.1 Floating-point summation ##2.1 Multiplication ##2.2 Division ##3.0 Elementary functions, general ##3.1 Elementary functions, CORDIC and related algorithms ##3.2 Elementary functions, function approximation ##3.2.1 Polynomial evaluation ##3.3 Square root, general ##3.3.1 Square root, bitoriented, iterative, and table methods of computation ##3.3.2 Square root, Newton's method ##3.4 Sine and Cosine ##3.5 Logarithm ##3.6 Exponential function ##3.7 Arctangent ##3.8 Other transcendental functions ##4.0 Binary-decimal conversion ##5.0 BCD arithmetic ##6.0 Multiple precision arithmetic

##1.0 Choice of base, floating point formats

[1] Brown, W.S.; Richman, P.L.: The Choice of Base.

     Communications of the ACM, Vol. 12, No. 10, October 1969, pp. 560-561

[2] Kreifelts, T,: Optimale Basiswahl f"ur eine Gleitkomma-Arithmetik.

     Computing, Vol. 11, 1973, pp. 353-363

[3] Kuki, H.; Cody, W.J.: A Statistical Study of the Accuracy of Floating

     Point Number Systems.
     Communications of the ACM, Vol. 16, No. 4, April 1973, pp. 223-230

[4] Cody, W.J.: Static and Dynamic Numerical Characteristics of

     Floating-Point Arithmetic.
     IEEE Transactions on Computers, Vol. C-22, No. 6, June 1973, pp. 598-601

[5] Brent, R.P.: On the Precision Attainable with Various Floating-Point

     Number Systems.
     IEEE Transactions on Computers, Vol. C-22, No. 6, June 1973, pp. 601-607

[6] Kreifelts, T.: Optimale Basiswahl f"ur eine Gleitkomma-Arithmetik

     (Berichtigung).
     Computing, Vol. 14, 1975, pp. 313-314

[7] Liddiard, L.A.: Required Scientific Floating Point Arithmetic.

     Proceedings of the 4th Symposium on Computer Arithmetic, Santa Monica,
     CA, USA, 25-27 October 1978, pp. 56-62

[8] Hull, T.E.: Desirable Floating-Point Arithmetic and Elementary Functions

     for Numerical Computation.
     Proceedings of the 4th Symposium on Computer Arithmetic, Santa Monica,
     CA, USA, 25-27 October 1978, pp. 63-69

[9] Agrawal, J.C.; Sehdev, P.S.: Comparison and Evaluation of Floating Point

     Representations in IBM/370 and VAX-11/780.
     Proceedings of the 4th Symposium on Empirical Foundations of Information
     and Software Science, Atlanta, GA, USA, 22-24 October 1986, pp. 353-369

[10] Plauger, P.J.: Properties of floating-point arithmetic.

     Computer Language, Vol. 5, No. 3, March 1988, pp. 17-22

[11] Johnstone, P.; Petry, F.E.: Higher Radix Floating Point Representations.

     Proceedings of the 9th Symposium on Computer Arithmetic, Santa Monica,
     CA, USA, 6-8 September 1989, pp. 128-135

[12] Kalbasi, K.: Can you trust your computer?

     IEEE Potentials, Vol. 9, No. 2, April 1990, pp. 15-18

[13] Ochs, T.: Numeric types, representations, and other fictions.

     Computer Language, Vol. 8, No. 8, August 1991, pp. 93-101

[14] Plauger, P.J.: Floating-Point Arithmetic.

     Embedded Systems Programming, Vol. 4, No. 8, August 1991, pp. 95-99

##1.1 Precision and Rounding

[15] Goldberg, I.B.: 27 Bits Are Not Enough For 8-Digit Accuracy.

     Communications of the ACM, Vol. 10, No. 2, February 1967, pp. 105-106

[16] Hamming, R.W.: On the Distribution of Numbers.

     Bell System Technical Journal, Vol. 49, No. 8.,
     October 1970, pp. 1609-1625

[17] Kaneko, T.; Liu, B.: On Local Roundoff Errors in Floating-Point

     Arithmetic.
     Journal of the Association for Computing Machinery, Vol. 20, No. 3,
     July 1973, pp. 391-398

[18] Tsao, N.: On the Distribution of Significant Digits and Roundoff Errors.

     Communications of the ACM, Vol. 17, No. 5, May 1974, pp. 269-271

[19] Goodman, R.; Feldstein, A.: Round-Off Error in Products.

     Computing, Vol. 15, 1975, pp. 263-273

[20] Kuck, D.J.; Parker, D.S.; Sameh, A.H.: ROM-Rounding: A New Rounding

     Scheme.
     Proceedings of the 3rd Symposium on Computer Arithmetic, Dallas, TX,
     USA, 19-20 October 1975, pp. 67-72

[21] Garner, H.L.: A Survey of Some Recent Contributions to Computer

     Arithmetic.
     IEEE Transactions on Computers, Vol. C-25, No. 12, December 1976,
     pp. 1277-1282

[22] Goodman, R.; Feldstein, A.: Effect of Guard Digits and Normalization

     Options on Floating Point Multiplication.
     Computing, Vol. 18, No. 2, 1977, pp. 93-106

[23] Kent, J.G.: Highlights of a Study of Floating-Point Instructions.

     IEEE Transactions on Computers, Vol. C-26, No. 7, July 1977, pp. 660-666

[24] Kuck, D.J.; Parker, D.S.; Sameh, A.H.: Analysis of Rounding Methods in

     Floating-Point Arithmetic.
     IEEE Transactions on Computers, Vol. C-26, No. 7, July 1977, pp. 643-650

[25] Bustoz, J.; Feldstein, A.; Goodman, R.; Linnainmaa, S.: Improved

     Trailing Digits Estimates Applied to Optimal Computer Arithmetic.
     Journal of the Association for Computing Machinery, Vol. 26, No. 4,
     October 1979, pp. 716-730

[26] Henrich, C.J.: Floating-point arithmetic: can it be trusted?

     Mini-Micro Systems, Vol. 13, No. 11, November 1980, pp. 143-151

[27] Miller, W.: A Remark on Gradual Underflow.

     Computing, Vol. 27, No. 3, 1981, pp. 217-225

[28] Brown, W.S.: A Simple but Realistic Model of Floating-Point Computation.

     ACM Transactions on Mathematical Software, Vol. 7, No. 4, December 1981,
     pp. 445-480

[29] Rump, S.M.: Computer und Rechengenauigkeit.

     Elektronische Rechenanlagen, Vol. 24, No. 6, December 1982, pp. 268-277

[30] Schatte, P.: On Mantissa Distribution in Computing and Benford's Law.

     Journal of Information Processing and Cybernetics, Vol. 24, No. 9, 1988,
     pp. 443-455

[31] Santoro, M.R.; Bewick, G.; Horowitz, M.A.: Rounding Algorithms for IEEE

     Multipliers.
     Proceedings of the 9th Symposium on Computer Arithmetic, Santa Monica,
     CA, USA, 6-8 September 1989, pp. 176-183

[32] Goodman, R.H.: Some Models of Relative Error in Products.

     Applied Numerical Mathematics, Vol. 6, No. 3, March 1990, pp. 209-220

[33] Yoshida, N.; Goto, E.; Ichikawa, S.: Pseudorandom Rounding for Truncated

     Multipliers.
     IEEE Transactions on Computers, Vol. 40, No. 9, September 1991,
     pp. 1065-1067

[34] Ris, F.; Barkmeyer, E.; Schaffert, C.; Farkas, P.: When Floating-Point

     Addition Isn't Commuative.
     SIGNUM Newsletter, Vol. 28, No. 1, January 1993

[35] Kabuo, H.; Taniguchi, T.; Miyoshi, A.; Yamashita, H.; Urano, M.;

     Edamatsu, H.; Kuninobu, S.: Accurate Rounding Scheme for the
     Newton-Raphson Method Using Redundant Binary Representation.
     IEEE Transactions on Computers, Vol. 43, No. 1, January 1994

##1.2 Determination of parameters of floating point arithmetic

[36] Malcolm, M.A.: Algorithms To Reveal Properties of Floating-Point

     Arithmetic.
     Communications of the ACM, Vol. 15, No. 11, November 1972, pp. 949-951

[37] Gentleman, M.W.; Marovich, S.B.: More on Algorithms that Reveal

     Properties of Floating Point Arithmetic Units.
     Communications of the ACM, Vol. 17, No. 5, May 1974, pp. 276-277

[38] Lastman, G.J.: Determination of Floating Point Characteristics for a

     Personal Computer.
     Proceedings 1983 International Electrical, Electronics Conference,
     Vol. 2. Toronto, Ont., Canada, 26-28 September 1983, pp. 424-427

[39] Razaz, M.; Schonfelder, J.L.: Test Procedures for Measurement of

     Floating-Point Characteristics of Computing Environments.
     The Computer Journal, Vol. 31, No. 1, February 1988, pp. 12-16

[40] Cody, W.J.: Algorithm 665. MACHAR: A Subroutine to Dynamically

     Determine Machine Parameters.
     ACM Transactions on Mathematical Software, Vol. 14, No. 4,
     December 1988, pp. 302-311

##1.3 IEEE standards for floating point arithmetic

[41] Ris, F.N.: A Unified Decimal Floating-Point Architecture For the

     Support of High-Level Languages.
     SIGNUM Newsletter, Vol. 11, No. 3, October 1976, pp. 18-22

[42] Coonen, J.; Kahan, W.; Palmer, J.; Pittman, T.; Stevenson, D.:

     A Proposed Standard for Binary Floating Point Arithmetic; Draft 5.11.
     SIGNUM Newsletter, Special Issue, October 1979, pp. 4-12

[43] Kahan, W.; Palmer, J.: On a Proposed Floating-Point Standard.

     SIGNUM Newsletter, Special Issue, October 1979, pp. 13-21

[44] Fraley, B.; Walter, S.: Proposal to Eliminate Denormalized Numbers.

     SIGNUM Newsletter, Special Issue, October 1979, pp. 22-23

[45] Payne, M.; Strecker, W.: Draft Proposal for a Binary Normalized

     Floating Point Standard.
     SIGNUM Newsletter, Special Issue, October 1979, pp. 24-28

[46] Cody, W.: Impact of The Proposed IEEE Floating Point Standard on

     Numerical Software.
     SIGNUM Newsletter, Special Issue, October 1979, pp. 29-30

[47] Feldman, S.I.: The Impact of the Proposed Standard for Floating Point

     Arithmetic on Languages and Systems.
     SIGNUM Newsletter, Special Issue, October 1979, pp. 31-32

[48] Stone, H.S.: Towards a Floating-Point Standard.

     Electro/80 Conference Record, Boston, MA, USA, 13-15 May 1980,
     pp. 18/0/1-5

[49] Payne, M.; Bhandarkar, D.: VAX Floating Point: A Solid Foundation for

     Numerical Computation.
     Electro/80 Conference Record, Boston, MA, USA, 13-15 May 1980,
     pp. 18/1/1-12

[50] Fraley, R.A.; Walther, J.S.: Safe Treatment of Overflow and Underflow

     Conditions.
     Electro/80 Conference Record, Boston, MA, USA, 13-15 May 1980,
     pp. 18/2/1-5

[51] Hough, D.: Applications of a Proposed Standard for Floating-Point

     Arithmetic.
     Electro/80 Conference Record, Boston, MA, USA, 13-15 May 1980,
     pp. 18/3/1-6

[52] Stevenson, D.: IEEE Task 754: A Proposed Standard for Binary

     Floating-Point Arithmetic; Draft 8.0.
     Computer, Vol. 14, No. 3, March 1981, pp. 51-62

[53] Cody, W.J.: Analysis of Proposals for the Floating-Point Standard.

     Computer, Vol. 14, No. 3, March 1981, pp. 63-68

[54] Hough, D.: Applications of the Proposed IEEE 754 Standard for the

     Floating-Point Arithmetic.
     Computer, Vol. 14, No. 3, March 1981, pp. 70-74

[55] Coonen, J.T.: Underflow and the Denormalized Numbers.

     Computer, Vol. 14, No. 3, March 1981, pp. 75-87

[56] Cody, W.J.; Coonen, J.T.; Gay, D.M.; Hanson, K.; Hough, D.;

     Kahan, W.; Karpinski, R.; Palmer, J.; Ris, F.N.; Stevenson, D.:
     A Proposed Radix- and Word-length-independent Standard for
     Floating-point Arithmetic.
     IEEE Micro, Vol. 4, No.4, August 1984, pp. 86-100

[57] Cody, W.J.; Coonen, J.T.; Gay, D.M.; Hanson, K.; Hough, D.;

     Kahan, W.; Karpinski, R.; Palmer, J.; Ris, F.N.; Stevenson, D.:
     A Proposed Radix- and Word-length-independent Standard for
     Floating-point Arithmetic.
     SIGNUM Newsletter, Vol. 20, No. 1, January 1985, pp. 37-51

[58] IEEE: IEEE Standard for Binary Floating-Point Arithmetic.

     SIGPLAN Notices, Vol. 22, No. 2, 1985, pp. 9-25

[59] IEEE Standard for Binary Floating-Point Arithmetic.

     ANSI/IEEE Std 754-1985.
     New York, NY: Institute of Electrical and Electronics Engineers 1985

[60] IEEE Standard for Radix-Independent Floating-Point Arithmetic.

     ANSI/IEEE Std 854-1987.
     New York, NY: Institute of Electrical and Electronics Engineers 1987

[61] Advanced Micro Devices: IEEE floating-point format.

     Microprocessors and Microsystems, Vol. 12, No. 1, January 1988,
     pp. 13-23

[62] Cody, W.J.: Algorithm XXX: Functions to Support the IEEE Standard for

     Binary Floating-Point Arithmetic. Preprint MCS-P90-0789, Mathematics
     and Computer Science Division, Argonne National Laboratory, July 1989

[63] Pan, J.; Levitz, K.N.: A Formal Specification of the IEEE Floating-Point

     Standard with Application to the Verification of Floating-Point
     Coprocessors.
     Conference Record Twenty-Fourth Asilomar Conference on Signals, Systems
     and Computers. Pacific Grove, CA, USA, 5-7 November 1990, pp. 505-510

[64] Cody, W.J.; Coonen, J.T.: Algorithm 722: Functions to Support the IEEE

     Standard for Binary Floating-Point Arithmetic.
     ACM Transactions on Mathematical Software, Vol. 19, No. 4,
     December 1993, pp. 443-451

##1.4 Floating point arithmetic, general issues and implementation issues

[65] Kuki, H.: Mathematical Function Subprograms for Basic System

     Libraries - Objectives, Constraints, and Trade-Off.
     In: Rice, J.R. (ed.): Mathematical Software.
     New York, NY: Academic Press 1971

[66] Aird, T.; Dodson, D.; Houstis, E.; Rice, J.: Statistics on the Use of

     Mathematical Subroutines from a Computer Center Library.
     SIGNUM Newsletter, Vol. 8, No. 4, October 1973, pp. 8-9

[67] Bohlender, G.: Floating-Point Computation of Functions with Maximum

     Accuracy.
     IEEE Transactions on Computers, Vol. C-26, No. 7, July 1977, pp. 621-632

[68] Hashizume, B.: Floating Point Arithmetic.

     BYTE, Vol. 2, No. 11, November 1977, pp. 76-78, 180-188

[69] Andrews, M.: Influence of architecture on numerical algorithms.

     Microprocessors and Microsystems, Vol. 2, No. 3, June 1978, pp. 130-137

[70] Boney, J.: Math in the Real World.

     BYTE, Vol. 3, No. 9, September 1978, pp. 114-119

[71] Bohlender, G.: Genaue Berechnung mehrfacher Summen, Produkte und Wurzeln

     von Gleitkommazahlen und allgemeine Arithmetik in h"oheren
     Programmiersprachen. Dissertation, Universit"at Karlsruhe 1978

[72] Rauch, E.: Einige Aspekte der Auswahl und Realisierung numerischer

     Verfahren in anwendungsorientierten Systemen.
     In: Meinardus, G. (Hrsg.): Approximation in Theorie und Praxis.
     Mannheim: Bibliographisches Institut 1979

[73] Reid, J.: Functions for Manipulating Floating-Point Numbers.

     SIGNUM Newsletter, Vol. 14, No. 4, December 1979, pp. 11-13

[74] Gr"uner, K.: Allgemeine Rechnerarithmetik und deren Implementierung mit

     optimaler Genauigkeit. Dissertation, Universit"at Karlsruhe 1979

[75] Coonen, J.T.: An Implementation Guide to a Proposed Standard for

     Floating-Point Arithmetic.
     Computer, Vol. 13, No. 1, January 1980, pp. 68-79

[76] Brown, W.S.; Feldman, S.I.: Environment Parameters and Basic Functions

     for Floating-Point Computation.
     ACM Transactions on Mathematical Software, Vol. 6, No. 4, December 1980,
     pp. 510-523

[77] Wehringer, A.: Flie"skomma-Arithmetik.

     Elektronikschau 1981, Heft 5, Seiten 34-36

[78] Bohlender, G.; Gr"uner, K.; Wolff von Gudenberg, J.: Realisierung einer

     optimalen Arithmetik.
     Elektronische Rechenanlagen, Vol. 24, No. 2, April 1982, pp. 68-72

[79] Kulisch, U.W.; Miranker, W.L.: The Arithmetic of the Digital Computer:

     A New Approach.
     SIAM Review, Vol. 28, No. 1, March 1986, pp. 1-40

[80] Rump, S.M.: Sichere Ergebnisse auf Rechenanlagen.

     Informatik Spektrum, Vol. 9, No. 3, June 1986, pp. 174-183

[81] Grehan, R.: Floating-Point Without a Coprocessor.

     BYTE, Vol. 13, No. 9, September 1988, pp. 313-319

[82] Grehan, R.: Floating-Point Without a Coprocessor, Part 2.

     BYTE, Vol. 13, No. 10, October 1988, pp. 293-297

[83] Grehan, R.: Floating-Point Revisited.

     BYTE, Vol. 14, No. 4, April 1989, pp. 311-318

[84] Ochs, T.: Floating-point theory and practice.

     Computer Language, Vol. 6, No. 3, March 1989, pp. 67-81

[85] Dritz, K.W.: Rationale for the Proposed Standard for a Generic Package

     of Elementary Functions for Ada.
     Argonne National Laboratory, Mathematics and Computer Science Division
     Report ANL-89/2 Rev. 1, October 1989

[86] Goldberg, D.: Computer Arithmetic.

     In: Hennessy, J.L; Patterson, D.A: Computer Architecture - A
     Quantitative Approach. San Mateo, CA: Morgan Kaufmann 1990

[87] Ochs, T.: A rotten foundation.

     Computer Language, Vol. 8, No. 2, February 1991, pp. 103-107

[88] Ochs, T.: Son of rotten foundation: The sequel.

     Computer Language, Vol. 8., No. 3, March 1991, pp. 85-91

[89] Goldberg, D.: What Every Computer Scientist Should Know About

     Floating-Point Arithmetic.
     ACM Computing Surveys, Vol. 23, No. 1, March 1991, pp. 5-48

[90] Ochs, T.: Numerics for the rest of us.

     Computer Language, Vol. 8, No. 10, October 1991, pp. 113-127

[91] Wichmann, B.A.: A Note on the Use of Floating Point in Critical Systems.

     The Computer Journal, Vol. 35, No. 1, February 1992, pp. 41-44

##1.5 Floating-point packages

[92] Wolff von Gudenberg, J.: Einbettung allgemeiner Rechnerarithmetik in

     Pascal mittels eines Operatorkonzepts und Implementierung der
     Standardfunktionen mit optimaler Genauigkeit.
     Dissertation, Universit"at Karlsruhe 1980

[93] Grappel, R.; Hemenway, J.: Increase Z8000 power with floating-point

     routines.
     EDN, Vol. 25, No. 8, April 1980, pp. 179-185

[94] Vogt, R.; Waser, R.: Arithmetikroutinen f"ur die Me"sdatenverarbeitung.

     Elektronik 1983, Heft 20, Seiten 85-92

[95] Sand, J.R.; Bumgarner, J.O.: Dysan IEEE P-754 Binary Floating Point

     Architecture.
     1983 Rochester Forth Conference, Rochester, NY, USA, 7-11 June 1983,
     pp. 185-194

[96] Rauchwerk, M.D.: A microprocessor-based fast floating point library.

     Conference Proceedings of IEEE SOUTHEASTCON 84, Louisville, KY, USA,
     8-11 April 1984, pp. 488-490

[97] Dietrich, D.; Fischer, R.: Floating-Point-Routinen, entwickelt f"ur

     Mikrorechner.
     Elektroniker (Schweiz) 1984, Heft 8, Seiten 49-54

[98] Lohninger, H.: Gleitkommaarithmetik f"ur den 68000.

     mc 1985, Heft 2, Seiten 58-64

[99] Gross, T.: Floating-Point Arithmetic on a Reduced-Instruction-Set

     Processor.
     Proceedings of the 7th Symposium on Computer Arithmetic, Urbana, IL,
     USA, 4-6 June 1985, pp. 86-92

[100] Lorenz, E.: Aspekte der Implementierung eines Programmpaketes zur

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[164] Papamichalis, P.; Simar Jr., R.: The TMS320C30 Floating-Point Digital

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[170] Benschneider, B.J.; Bowhill, W.J.; Cooper, E.M.; Gavrielov, M.N.;

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     1989 IEEE International Solid-State Circuits Conference Digest of
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[172] Kohn, L.; Fu, S.-W.: A 1,000,000 Transistor Microprocessor.

     1989 IEEE International Solid-State Circuits Conference Digest of
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[174] Ashton, C.: The Am29C327 floating point processor.

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[213] Cody, W.J.: Performance Evaluation of Programs Related to the Real Gamma

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     ACM Transactions on Mathematical Software, Vol. 17, No. 1,
     March 1991, pp. 46-54

[214] Plauger, P.J.: Washing the watchers.

     Computer Language, Vol. 8, No. 9, September 1991, pp. 23-32

[215] Nagal, T.; Hatano, Y.: Performance evaluation of mathematical functions.

     Supercomputer, Vol. 8, No. 8, November 1991, pp. 46-56

[216] Markstein, V.; Markstein, P.; Nguyen, T.; Poole, S.: Wide Format

     Floating-Point Math Libraries.
     Proceedings Supercomputing '91. Albuquerque, NM, USA,
     18-22 November 1991

[217] Cody, W.J.: Algorithm 714 - CELEFUNT: A Portable Test Package for

     Complex Elementary Functions.
     ACM Transactions on Mathematical Software, Vol. 19, No. 1, March 1993,
     pp. 1-21

##2.0 Addition and Subtraction

[218] Sweeney, D.W.: An analysis of floating-point addition.

     IBM Systems Journal, Vol. 4, No. 1, 1965, pp. 31-42

[219] Feldstein, A.; Goodman, R.: Loss of Significance in Floating Point

     Subtraction and Addition.
     IEEE Transactions on Computer, Vol. C-31, No. 4, April 1982, pp. 328-335

2.0.1 Floating-point Summation

[220] Wolfe, J.M.: Reducing Truncation Errors by Programming.

     Communications of the ACM, Vol. 7, No. 6, June 1964, pp. 355-356

[221] Kahan, W.: Further Remarks on Reducing Truncation Errors.

     Communications of the ACM, Vol. 8, No. 1, January 1965, p. 40

[222] Moller, O.: Quasi Double-Precision in Floating-Point Addition.

     BIT, Vol. 5, 1965, pp. 37-50

[223] Moller, O.: Note on Quasi Double-Precision.

     BIT, Vol. 5, 1965, pp. 251-255

[224] Linz, P.: Accurate Floating-Point Summation.

     Communications of the ACM, Vol. 13, No. 6, June 1970, pp. 361-362

[225] Malcolm, M.A.: On Accurate Floating-Point Summation.

     Communications of the ACM, Vol. 14, No. 11, November 1971, pp. 731-736

[226] Gregory, J.: A Comparison of Floating Point Summation Methods.

     Communications of the ACM, Vol. 15, No. 9, September 1972, pp. 838

[227] Linnainmaa, S.: Analysis of Some Known Methods of Improving the Accuracy

     of Floating-Point Sums.
     BIT, Vol. 14, 1974, pp. 167-202

[228] Ozawa, K.: Analysis and Improvement of Kahan's Summation Algorithm.

     Journal of Information Processing, Vol. 6, No. 4, 1983, pp. 226-230

[229] Robertazzi, T.G.; Schwartz, S.C.: Best "ordering" for Floating-point

     Addition.
     ACM Transactions on Mathematical Software, Vol. 14, No. 1, March 1988,
     pp. 101-110

[230] Dunham, C.B.: Summation

     SIGNUM Newsletter,

##2.1 Multiplication

[231] Heising, W.; Rabin, M.O.; Winograd, S.: Multiplication Method.

     IBM Technical Disclosure Bulletin, Vol. 15, No. 4, September 1972,
     pp. 1147-1148

[232] Sheue, A.E.: Two's-Complement Multiplication.

     SIG Micro Newsletter, Vol. 10, No. 1, March 1979, pp. 21-23

[233] Ambikairajah, E.; Carey, M.J.: Technique for Performing Multiplication

     on a 16-bit Microprocessor Using an Extension of Booth's Algorithm.
     Electronics Letters, Vol. 16, No. 2, January 1980, pp. 53-54

[234] Wehringer, A.: Schnelle 16-bit-Multiplikation und Division.

     Elektronikschau 1981, Heft 10, Seiten 36-37

[235] Schatte, P.: The Frequency of Postshifts in Floating-Point

     Multiplication.
     Elektronische Informationsverarbeitung und Kybernetik, Vol. 18, No. 9,
     1982, pp. 523-526

[236] Goodrich, J.L.: Very efficient 8080 program multiplies and divides.

     Electronics, Vol. 55, No. 4, February 1982, pp. 144-145

[237] Baxter, I.: Code replication speeds multiplication.

     EDN, Vol. 28, No.4, February 1983, pp. 261-262

[238] Dyer, D.C.: Z80 routine performs 16-bit multiply.

     EDN, Vol. 28, No. 5, March 1983, p. 144

[239] Coupe, B: Superefficient programs for 8080 and Z80 multiply.

     Electronics, Vol. 56, No. 6, March 1983, pp. 142-143

[240] Iffrig, L.D.: Use less code for fast 8080 multiply.

     EDN, Vol. 28, No. 13, June 1983, p. 293

[241] Robison, A.D.: Use Squares for Fast Multiplication.

     EDN, Vol. 28, No. 21, October 1983, pp. 263+267

[242] Collis, B.: Macros speed 8080, Z80 multiplication.

     EDN, Vol. 28, No. 24, November 1983, p. 225

[243] Lorenz, E; Sandau, R.: M"oglichkeiten der Implementierung

     leistungsf"ahiger Multiplikationsprogramme in Mikrorechnersystemen.
     Nachrichtentechnik Elektronik, Vol. 34, No. 8, 1984, pp. 288-290

##2.2 Division

[244] Pope, D.A.; Stein, M.L.: Multiple Precision Arithmetic.

     Communications of the ACM, Vol. 3, No. 12, December 1960, pp. 652-654

[245] Rabinowitz, P.: Multiple-Precision Division.

     Communications of the ACM, Vol. 4, No. 2, February 1961, p. 98

[246] Cox, A.G.; Luther, H.A.: A Note on Multiple Precision Arithmetic.

     Communications of the ACM, Vol. 4, 1961, p. 353

[247] Stein, M.L.: Divide-and-Correct Methods for Multiple Precision Division.

     Communications of the ACM, Vol. 7, No. 8, August 1964, pp. 472-474

[248] Krishnamurthy, E.V.: On a Divide-and-Correct Method for Variable

     Precision Division.
     Communications of the ACM, Vol. 8, No. 3, March 1965, pp. 179-181

[249] Krishnamurthy, E.V.; Nandi, S.K.: On the Normalization Requirement of

     Divisor in Divide-and-Correct Methods.
     Communications of the ACM, Vol. 10, No. 12, December 1967, pp. 809-813

[250] Collins, G.E.; Musser, D.R.: Analysis of the Pope-Stein Division

     Algorithm.
     Information Processing Letters, Vol. 6, No. 5, October 1977, pp. 151-155

[251] Sanyal, S.: An algorithm for nonrestoring division.

     Computer Design, Vol. 16, No. 5, May 1977, pp. 124-127

[252] Lemaire, C.A.; Svercek, J.C.: Improved Non-restoring Division.

     IBM Technical Disclosure Bulletin, Vol. 23, No. 3, August 1980,
     pp. 1149-1151

[253] Grappel, R.D.: 68000 routine divides 32-bit numbers.

     EDN, Vol. 26, No. 5, March 1981, pp. 161-162

[254] Asai, H.: A Consideration of a Practical Implementation For a New

     Convergence Division.
     Information Processing Letters, Vol. 17, No. 5,
     December 1983, pp. 273-281

[255] McGuire, D.W.: 8048 program computes 16-by-8-bit quotient.

     Electronics, Vol. 56, No. 10, May 1983, pp. 152-153

[256] James, F.V.: An 8085 routine divides 32-bit unsigned numbers.

     Electronics, Vol. 56, No. 22, November 1983, pp. 163-165

[257] Galand, C.: Fast Division.

     IBM Technical Disclosure Bulletin, Vol. 26, No. 3B, August 1983,
     pp. 1537-1539

[258] Pfenninger, E.: Divisionsalgorithmus - einfach und schnell.

     Elektroniker 1984, Heft 15, Seiten 62-64

[259] Mithani, D.; Iyer, S.: Algorithm speeds nonrestoring division in

     uprogrammed systems.
     EDN, February 1985, pp. 199-208

[260] Semba, I: An Algorithm for Division of Large Integers.

     Journal of Information Processing, Vol. 9, No. 3, 1986, pp. 145-147

[261] Lorang, O.: Schnelle Division.

     Elektronik 1986, Heft 22, Seiten 167-168

[262] Jain, V.K.; Landia, D.L.; Alvarez, C.E.: Systolic L-U Decomposition

     Array With a New Reciprocal Cell.
     Proceedings 1989 IEEE International Conference on Computer Design:
     VLSI in Computers & Processor, ICCD '89, Cambridge, MA, USA, 2-4 October
     1989, pp. 460-465

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     Electronic Letters, Vol. 28, No. 3, November 1992, pp. 2151-2152

##5.0 BCD arithmetic

[492] Franklin, J.W.: Zoned Decimal Arithmetic.

     IBM Technical Disclosure Bulletin, Vol. 15, No. 7, December 1972,
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[493] Besslich, P.W.; Raman, S.: Multiplication, Division and Square Root

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[494] Schmid, H.: BCD logic I: BCD - logic of many uses.

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[496] Schmid, H.: BCD logic III: BCD division.

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[497] Schmid, H.: BCD logic IV: BCD decimal-point location.

     Electronic Design, Vol. 21, No. 16, August 1973, pp. 80-84

[498] Schmid, H.: BCD logic V: BCD square root.

     Electronic Design, Vol. 21, No. 17, August 1973, pp. 66-69

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     Electronic Design, Vol. 21, No. 18, September 1973, pp. 118-123

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     Electronic Design, Vol. 21, No. 19, September 1973, pp. 68-73

[501] Chen, T.C.; Ho, I.T.: Storage-Efficient Representation of Decimal Data.

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[502] Chroust, G.: Method of Adding Decimal Numbers by Means of Binary

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[505] Maric, I.; Cucancic, L.: On the Possibilities of the BCD Code

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##6. Multiple Precision Arithmetic

[509] Stroud, A.H.; Secrest, D.: A multiple-precision floating-point

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[510] Blum, B.I.: An Extended Arithmetic Package.

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[511] Tienari, M.; Suokonautio, V.: A Set of Procedures Making Real Arithmetic

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[512] Filho, A.M.S.; Schwachheim, G.: Algorithm 309: Gamma Function with

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[513] Kuki, H.; Ascoly, J.: FORTRAN extended-precision library.

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[514] Dekker, T.J.: A Floating-Point Technique for Extending the Available

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[515] Brent, R.P.: Fast Multiple-Precision Evaluation of Elementary Functions.

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[516] Wyatt Jr., W.T.; Lozier, D.W.; Orser, D.J.: A Portable Extended Precision

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[517] Brent, R.P.: A Fortran Multiple-Precision Arithmetic Package.

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[518] Brent, R.P.: Algorithm 524: MP, A Fortran Multiple-Precision Arithmetic

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[519] Brent, R.P.: Unrestricted Algorithms for Elementary and Special

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[520] Verma, S.B.; Sharan, M.: Multiple Precision Floating-point Computation

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[521] Linnainmaa, S.: Software for Doubled-Precision Floating-Point

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