Juffa floating-point bibliography, second edition
David Hough
sun!Eng!David.Hough
Fri Sep 18 08:12:38 PDT 1992
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 to 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 TeX format, that is "a, "o, "u, "s.
Thanks to the people who have helped me with the second version of this
document by sending me papers or additional references:
Steven Sommars (sesvaatt.com), Jim Kiernan(jmkacray.com), Warren Ferguson
(fergusonaseas.smu.edu), Nhuan Doduc (doducaframentec.fr)
Norbert Juffa
Wielandtstr. 14
7500 Karlsruhe 1
Germany
email: S_JUFFAaIRAVCL.IRA.UKA.DE
+++++++++++++++ Bibliography for floating point arithmetic ++++++++++++++++++++
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Books
[Cav] Cavanagh, J.F.: Digital Computer Arithmetic.
New York, NY: Wiley 1984
[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
[Flo] Flores, I.: The Logic of Computer Arithmetic.
Englewood Cliffs, NJ: Prentice Hall 1963
[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
[Hwa] Hwang, K.: Computer Arithmetic.
New York, NY: Wiley 1979
[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, 2.
Reading, MA: Addison-Wesley 1964 (Vol. 1), 1969 (Vol. 2)
[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
[Spa] Spaniol, O.: Arithmetik in Rechenanlagen.
Stuttgart: Teubner 1976
[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
Journal Publications, Conference Papers, Technical Reports,
Ph.D. Dissertations, Book Contributions, etc.
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] Ochs, T.: Numeric types, representations, and other fictions.
Computer Language, Vol. 8, No. 8, August 1991, pp. 93-101
1.1 Precision and Rounding
[13] 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
[14] Hamming, R.W.: On the Distribution of Numbers.
Bell System Technical Journal, Vol. 49, No.8,
October 1970, pp. 1609-1625
[15] 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
[16] Tsao, N.: On the Distribution of Significant Digits and Roundoff Errors.
Communications of the ACM, Vol. 17, No. 5, May 1974, pp. 269-271
[17] Goodman, R.; Feldstein, A.: Round-Off Error in Products.
Computing, Vol. 15, 1975, pp. 263-273
[18] 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
[19] 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
[20] Goodman, R.; Feldstein, A.: Effect of Guard Digits and Normalization
Options on Floating Point Multiplication.
Computing, Vol. 18, No. 2, 1977, pp. 93-106
[21] 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
[22] 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
[23] 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
[24] Henrich, C.J.: Floating-point arithmetic: can it be trusted?
Mini-Micro Systems, Vol. 13, No. 11, November 1980, pp. 143-151
[25] Miller, W.: A Remark on Gradual Underflow.
Computing, Vol. 27, No. 3, 1981, pp. 217-225
[26] Rump, S.M.: Computer und Rechengenauigkeit.
Elektronische Rechenanlagen, Vol. 24, No. 6, December 1982, pp. 268-277
[27] 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
[28] 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
[29] Goodman, R.H.: Some Models of Relative Error in Products.
Applied Numerical Mathematics, Vol. 6, No. 3, March 1990, pp. 209-220
[30] Kalbasi, K.: Can you trust your computer?
IEEE Potentials, Vol. 9, No. 2, April 1990, pp. 15-18
[31] Yoshida, N.; Goto, E.; Ichikawa, S.: Pseudorandom Rounding for Truncated
Multipliers.
IEEE Transactions on Computers, Vol. 40, No. 9, September 1991,
pp. 1065-1067
1.2 Determination of parameters of floating point arithmetic
[32] Malcolm, M.A.: Algorithms To Reveal Properties of Floating-Point
Arithmetic.
Communications of the ACM, Vol. 15, No. 11, November 1972, pp. 949-951
[33] 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
[34] 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
[35] 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
[36] 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
[37] 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
[38] Kahan, W.; Palmer, J.: On a Proposed Floating-Point Standard.
SIGNUM Newsletter, Special Issue, October 1979, pp. 13-21
[39] Fraley, B.; Walter, S.: Proposal to Eliminate Denormalized Numbers.
SIGNUM Newsletter, Special Issue, October 1979, pp. 22-23
[40] Payne, M.; Strecker, W.: Draft Proposal for a Binary Normalized
Floating Point Standard.
SIGNUM Newsletter, Special Issue, October 1979, pp. 24-28
[41] Cody, W.: Impact of The Proposed IEEE Floating Point Standard on
Numerical Software.
SIGNUM Newsletter, Special Issue, October 1979, pp. 29-30
[42] 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
[43] 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
[44] 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
[45] IEEE: IEEE Standard for Binary Floating-Point Arithmetic.
SIGPLAN Notices, Vol. 22, No. 2, 1985, pp. 9-25
[46] IEEE Standard for Binary Floating-Point Arithmetic.
ANSI/IEEE Std 754-1985.
New York, NY: Institute of Electrical and Electronics Engineers 1985
[47] IEEE Standard for Radix-Independent Floating-Point Arithmetic.
ANSI/IEEE Std 854-1987.
New York, NY: Institute of Electrical and Electronics Engineers 1987
[48] Advanced Micro Devices: IEEE floating-point format.
Microprocessors and Microsystems, Vol. 12, No. 1, January 1988,
pp. 13-23
[49] 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
1.4 Floating point arithmetic, general and implementation issues
[50] 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
[51] 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
[52] Bohlender, G.: Floating-Point Computation of Functions with Maximum
Accuracy.
IEEE Transactions on Computers, Vol. C-26, No. 7, July 1977, pp. 621-632
[53] Hashizume, B.: Floating Point Arithmetic.
BYTE, Vol. 2, No. 11, November 1977, pp. 76-78, 180-188
[54] Andrews, M.: Influence of architecture on numerical algorithms.
Microprocessors and Microsystems, Vol. 2, No. 3, June 1978, pp. 130-137
[55] Boney, J.: Math in the Real World.
BYTE, Vol. 3, No. 9, September 1978, pp. 114-119
[56] Bohlender, G.: Genaue Berechnung mehrfacher Summen, Produkte und Wurzeln
von Gleitkommazahlen und allgemeine Arithmetik in h"oheren
Programmiersprachen. Dissertation, Universit"at Karlsruhe 1978
[57] 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
[58] Reid, J.: Functions for Manipulating Floating-Point Numbers.
SIGNUM Newsletter, Vol. 14, No. 4, December 1979, pp. 11-13
[59] Gr"uner, K.: Allgemeine Rechnerarithmetik und deren Implementierung mit
optimaler Genauigkeit. Dissertation, Universit"at Karlsruhe 1979
[60] Coonen, J.T.: An Implementation Guide to a Proposed Standard for
Floating-Point Arithmetic.
Computer, Vol. 13, No. 1, January 1980, pp. 68-79
[61] 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
[62] Wehringer, A.: Flie"skomma-Arithmetik.
Elektronikschau 1981, Heft 5, Seiten 34-36
[63] Bohlender, G.; Gr"uner, K.; Wolff von Gudenberg, J.: Realisierung einer
optimalen Arithmetik.
Elektronische Rechenanlagen, Vol. 24, No. 2, April 1982, pp. 68-72
[64] 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
[65] Rump, S.M.: Sichere Ergebnisse auf Rechenanlagen.
Informatik Spektrum, Vol. 9, No. 3, June 1986, pp. 174-183
[66] Grehan, R.: Floating-Point Without a Coprocessor.
BYTE, Vol. 13, No. 9, September 1988, pp. 313-319
[67] Grehan, R.: Floating-Point Without a Coprocessor, Part 2.
BYTE, Vol. 13, No. 10, October 1988, pp. 293-297
[68] Grehan, R.: Floating-Point Revisited.
BYTE, Vol. 14, No. 4, April 1989, pp. 311-318
[69] Ochs, T.: Floating-point theory and practice.
Computer Language, Vol. 6, No. 3, March 1989, pp. 67-81
[70] 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
[71] Goldberg, D.: Computer Arithmetic.
In: Hennessy, J.L; Patterson, D.A: Computer Architecture - A
Quantitative Approach. San Mateo, CA: Morgan Kaufmann 1990
[72] Ochs, T.: A rotten foundation.
Computer Language, Vol. 8, No. 2, February 1991, pp. 103-107
[73] Ochs, T.: Son of rotten foundation: The sequel.
Computer Language, Vol. 8., No. 3, March 1991, pp. 85-91
[74] Goldberg, D.: What Every Computer Scientist Should Know About
Floating-Point Arithmetic.
ACM Computing Surveys, Vol. 23, No. 1, March 1991, pp. 5-48
[75] 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 and processors
[76] Wolff von Gudenberg, J.: Einbettung allgemeiner Rechnerarithmetik in
Pascal mittels eines Operatorkonzepts und Implementierung der
Standardfunktionen mit optimaler Genauigkeit.
Dissertation, Universit"at Karlsruhe 1980
[77] Grappel, R.; Hemenway, J.: Increase Z8000 power with floating-point
routines.
EDN, Vol. 25, No. 8, April 1980, pp. 179-185
[78] Cassola, R.L.: Floating Point Algorithm Design.
Computer Design, Vol. 21, No. 6, June 1982, pp. 107-114
[79] Vogt, R.; Waser, R.: Arithmetikroutinen f"ur die Me"sdatenverarbeitung.
Elektronik 1983, Heft 20, Seiten 85-92
[80] 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
[81] Teufel, T.: Ein optimaler Gleitkommaprozessor.
Dissertation, Universit"at Karlsruhe 1984
[82] 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
[83] Dietrich, D.; Fischer, R.: Floating-Point-Routinen, entwickelt f"ur
Mikrorechner.
Elektroniker (Schweiz) 1984, Heft 8, Seiten 49-54
[84] Look, H.W.: Compatible software and hardware impelentations permitted by
IEEE standards for binary floating-point arithmetic.
AFIPS Conference Proceedings of the 1984 NCC, Las Vegas, NV, USA,
9-12 July 1984, pp. 101-105
[85] Lohninger, H.: Gleitkommaarithmetik f"ur den 68000.
mc 1985, Heft 2, Seiten 58-64
[86] 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
[87] Lorenz, E.: Aspekte der Implementierung eines Programmpaketes zur
schnellen und flexiblen Ausf"uhrung von arithmetischen Operationen mit
dem U880.
Nachrichtentechnik Elektronik, Vol. 35, No. 5, 1985, pp. 179-181
[88] Dutta, U.; Bhattacharya, D.; Sarma, A.D.: Implementation of Multibyte
Floating Point Arithmetic in 8-bit Microprocessor.
Mechanical Engineering Bulletin (India), Vol. 17, No. 3, September 1986,
pp. 104-113
[89] Himmer"oder, H.-J.; Toschke, R.M.:
c't-KAT-Ce. Ein 68000-Einplatinenrechner, Teil 3: REAL-Arithmetik.
c't 1987, Heft 1, Seiten 152-158
[90] Lange, E.: Implementation and Test of the ACRITH Facility in a
System /370.
IEEE Transactions on Computers, Vol. C-36, No. 9, September 1987,
pp. 1088-1096
[91] Davila, J.M.; Phillips, A.J.; Tabak, D.: Floating Point Arithmetic on
a RISC.
Microprocessing and Microprogramming, Vol. 23, No. 1-5,
March 1988, pp. 179-184
[92] V"olzke, H.: Flie"skomma-Arithmetik und IEEE-Spezifikationen.
Teil 1: Standards und Strukturen.
mc 1988, Heft 10, Seiten 123-129
[93] V"olzke, H.: Flie"skomma-Arithmetik und IEEE-Spezifikationen.
Teil 2: Entwurf eines Fliesskommapakets.
mc 1988, Heft 11, Seiten 78-95
[94] V"olzke, H.: Flie"skomma-Arithmetik und IEEE-Spezifikationen.
Teil 3: Die verwendeten Algorithmen.
mc 1988, Heft 12, Seiten 95-108
[95] V"olzke, H.: Flie"skomma-Arithmetik und IEEE-Spezifikationen.
Teil 4: Die Konvertierungsroutinen.
mc 1989, Heft 1, Seiten 66-73
[96] V"olzke, H.: Flie"skomma-Arithmetik und IEEE-Spezifikationen.
Teil 5: Ein- und Ausgabefunktionen.
mc 1989, Heft 2, Seiten 65-71
1.6 Test of floating point routines
[97] Cody, W.J.: Performance testing of function subroutines.
AFIPS Conference Proceedings, Vol. 34, SJCC 1969, pp. 759-763
[98] Karpinski, R.: Paranoia: A Floating-Point Benchmark.
BYTE, Vol. 10, No. 2, February 1985, pp. 223-235
[99] Schryer, N.L.: A Test Of a Computer's Floating-Point Arithmetic Unit.
Bell Laboratories, Computer Science Technical Report 89
[100] Spafford, E.H.; Flaspohler, J.C.: A Report on the Accuracy of Some
Floating-Point Math Functions on Selected Computers.
Georgia Institute of Technology, Technical Report GIT-SERC-86/02,
GIT-ICS-85/06.
[101] Du Croz, J.: FPV - a Floating-Point Validation Package.
Software Quality Assurance, Reliability, and Testing. London, UK,
9-10 December 1986, pp. 47-55
[102] Tang, P.T.P.: Testing Computer Arithmetic by Elementary Number Theory.
Preprint MCS-P84-0889, Mathematics and Computer Science Division,
Argonne National Laboratory, August 1989
[103] Cody, W.J.; Stoltz, L.: The Use of Taylor Series to Test Accuracy of
Function Programs.
ACM Transactions on Mathematical Software, Vol. 17, No. 1,
March 1991, pp. 56-63
[104] Plauger, P.J.: Washing the watchers.
Computer Language, Vol. 8, No. 9, September 1991, pp. 23-32
[105] Nagal, T.: Performance evaluation of mathematical functions.
Supercomputer, Vol. 8, No. 8, November 1991, pp. 46-56
2.0 Addition and Subtraction
[106] Sweeney, D.W.: An analysis of floating-point addition.
IBM Systems Journal, Vol. 4, No. 1, 1965, pp. 31-42
[107] 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
2.1 Multiplication
[108] Heising, W.; Rabin, M.O.; Winograd, S.: Multiplication Method.
IBM Technical Disclosure Bulletin, Vol. 15, No. 4, September 1972,
pp. 1147-1148
[109] Sheue, A.E.: Two's-Complement Multiplication.
SIG Micro Newsletter, Vol. 10, No. 1, March 1979, pp. 21-23
[110] 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
[111] Wehringer, A.: Schnelle 16-bit-Multiplikation und Division.
Elektronikschau 1981, Heft 10, Seiten 36-37
[112] Schatte, P.: The Frequency of Postshifts in Floating-Point
Multiplication.
Elektronische Informationsverarbeitung und Kybernetik, Vol. 18, No. 9,
1982, pp. 523-526
[113] Goodrich, J.L.: Very efficient 8080 program multiplies and divides.
Electronics, Vol. 55, No. 4, February 1982, pp. 144-145
[114] Baxter, I.: Code replication speeds multiplication.
EDN, Vol. 28, No.4, February 1983, pp. 261-262
[115] Dyer, D.C.: Z80 routine performs 16-bit multiply.
EDN, Vol. 28, No. 5, March 1983, p. 144
[116] Coupe, B: Superefficient programs for 8080 and Z80 multiply.
Electronics, Vol. 56, No. 6, March 1983, pp. 142-143
[117] Iffrig, L.D.: Use less code for fast 8080 multiply.
EDN, Vol. 28, No. 13, June 1983, p. 293
[118] Robison, A.D.: Use Squares for Fast Multiplication.
EDN, Vol. 28, No. 21, October 1983, pp. 263+267
[119] Collis, B.: Macros speed 8080, Z80 multiplication.
EDN, Vol. 28, No. 24, November 1983, p. 225
[120] 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
[121] Pope, D.A.; Stein, M.L.: Multiple Precision Arithmetic.
Communications of the ACM, Vol. 3, No. 12, December 1960, pp. 652-654
[122] Cox, A.G.; Luther, H.A.: A Note on Multiple Precision Arithmetic.
Communications of the ACM, Vol. 4, 1961, p. 353
[123] Stein, M.L.: Divide-and-Correct Methods for Multiple Precision Division.
Communications of the ACM, Vol. 7, No. 8, August 1964, pp. 472-474
[124] 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
[125] 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
[126] 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
[127] Sanyal, S.: An algorithm for nonrestoring division.
Computer Design, Vol. 16, No. 5, May 1977, pp. 124-127
[128] Lemaire, C.A.; Svercek, J.C.: Improved Non-restoring Division.
IBM Technical Disclosure Bulletin, Vol. 23, No. 3, August 1980,
pp. 1149-1151
[129] Grappel, R.D.: 68000 routine divides 32-bit numbers.
EDN, Vol. 26, No. 5, March 1981, pp. 161-162
[130] 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
[131] McGuire, D.W.: 8048 program computes 16-by-8-bit quotient.
Electronics, Vol. 56, No. 10, May 1983, pp. 152-153
[132] James, F.V.: An 8085 routine divides 32-bit unsigned numbers.
Electronics, Vol. 56, No. 22, November 1983, pp. 163-165
[133] Galand, C.: Fast Division.
IBM Technical Disclosure Bulletin, Vol. 26, No. 3B, August 1983,
pp. 1537-1539
[134] Mithani, D.; Iyer, S.: Algorithm speeds nonrestoring division in
uprogrammed systems.
EDN, February 1985, pp. 199-208
[135] Semba, I: An Algorithm for Division of Large Integers.
Journal of Information Processing, Vol. 9, No. 3, 1986, pp. 145-147
[136] Lorang, O.: Schnelle Division.
Elektronik 1986, Heft 22, Seiten 167-168
[137] Parker, A; Hamblen, J.O.: Optimal value for the Newton-Raphson division
algorithm.
Information Processing Letters, Vol. 42, No. 3, May 1992, pp. 141-144
[138] Ozawa, K.: A Fast O(n**2) Division Algorithm for Multiple-Precision
Floating-Point Numbers.
Journal of Information Processing, Vol. 14, No. 3, 1991, pp. 354-356
3.0 Elementary functions, general
[139] Filippi, S.: Die Berechnung einiger elementarer transzendenter
Funktionen mit Hilfe des Richardson-Algorithmus.
Computing, Vol. 1, 1966, pp. 127-132
[140] Kogbetliantz, E.G.: Generation of Elementary Functions.
In: Ralston, A.; Wilf, H.S. (ed.): Mathematical Methods for Digital
Computers.
New York, NY: Wiley 1966
[141] Rothmaier, B.: Dokumentation der Standardfunktionen des Betriebssystems
Hydra X8. Interner Bericht Nr. 70/8 Institut f"ur Informatik.
Universit"at Karlsruhe 1970
[142] Rothmaier, B.: Die Berechnung der elementaren Funktionen mit beliebiger
Genauigkeit. Dissertation, Universit"at Karlsruhe 1971
[143] Cody, W.J.: Software for the Elementary Functions.
In: Rice, J.R. (ed.): Mathematical Software.
New York, NY: Academic Press 1971
[144] Frankowski, K.S.: A Realistic Model for Error Estimates in the
Evaluation of Elementary Functions.
Proceedings of the 4th Symposium on Computer Arithmetic, Santa Monica,
CA, USA, 25-27 October 1978, pp. 70-74
[145] Brent, R.P.: Unrestricted Algorithms for Elementary and Special
Functions.
Information Processing 80, Proceedings of the IFIP Congress 80, Tokyo,
Japan, 6-9 October 1980, pp. 613-619
[146] Jung, C.: Berechnung der reellen und reellintervallwertigen Standard-
funktionen mit maximaler Genauigkeit in einem hexadezimalen Gleit-
kommaformat.
Diplomarbeit, Institut f"ur Angewandte Mathematik, Universit"at
Karlsruhe, September 1983
[147] Nave, R.: Implementation of Transcendental Functions on a Numerics
Processor.
Microprocessing and Microprogramming, Vol. 11, No. 3-4,
March-April 1983, pp. 221-225
[148] Longo, S.A.: Calculating Functions using Ada.
Journal of Pascal, Ada & Modula-2, Vol. 3, No. 4, July-August 1984,
pp. 34-36
[149] Butterfield, J.: Math and Tables.
Compute, Vol. 6, No. 9, September 1984, pp. 134-135
[150] Wolff von Gudenberg, J.: Berechnung maximal genauer Standardfunktionen
mit einfacher Mantissenl"ange.
Elektronische Rechenanlagen, Vol. 26, No. 5, October 1984, pp. 230-238
[151] Borwein, J.M.; Borwein, P.B.: The Arithmetic-Geometric Mean and Fast
Computation of Elementary Functions.
SIAM Review, Vol. 26, No. 3, July 1984, pp. 351-366
[152] Gal, S.: Computing Elementary Functions: A New Approach for Achieving
High Accuracy and Good Performance.
In: Accurate Scientific Computations. Lecture Notes in Computer
Science 235
New York, NY: Springer 1985
[153] Agarwal, R.C.; Cooley, J.W.; Gustavson, F.G.; Shearer, J.B.;
Slishman, G.; Tuckerman, B.: New scalar and vector elementary functions
for the IBM System /370.
IBM Journal Research & Development, Vol. 30, No. 2, March 1986,
pp. 126-144
[154] Braune, K.: Hochgenaue Standardfunktionen f"ur reelle und komplexe
Punkte und Intervalle in beliebigen Gleitpunktrastern.
Dissertation, Universit"at Karlsruhe 1987.
[155] Thompson, P.: Implementing an Elementary Function Library.
SIGNUM Newsletter, Vol. 22, No. 2, April 1987, pp. 2-5
[156] Koopman, P.: Transcendental Functions.
Forth Dimensions, Vol. 9, No. 4, December 1987, pp. 21-22
[157] Markstein, P.W.: Computation of elementary functions on the IBM RISC
System/6000 processor.
IBM Journal Research & Development, Vol. 34, No. 1, January 1990,
pp. 111-119
[158] Silverstein, J.D.; Sommars, S.E.; Tao, Y.C.: The UNIX System Math
Library, a Status Report.
Proceedings of the Winter 1990 USENIX Conference, Washington, DC, USA,
22-26 January 1990, pp. 117-131
[159] Kiernan, J.M.; Blachowiak, T.B.: Fast, Accurate Elementary Functions For
the Cray Y-MP Computer.
Proceedings Cray User Group, Spring 1990, pp. 243-252
[160] Koren, I.; Zinaty, O.: Evaluating Elementary Functions in a Numerical
Coprocessor Based on Rational Approximations.
IEEE Transactions on Computers, Vol. C-39, No. 8, August 1990,
pp. 1030-1037
[161] Dunham, C.B.: Feasibility of "Perfect" Function Evaluation.
SIGNUM Newsletter, Vol. 25, No. 4, October 1990, pp. 25-26
[162] Gal, S.; Bachelis, B.: An Accurate Elementary Mathematical Library for
the IEEE Floating Point Standard.
ACM Transactions on Mathematical Software, Vol. 17, No. 1,
March 1991, pp. 26-45
[163] Ziv, Abraham: Fast Evaluation of Elementary Mathematical Functions with
Correctly Rounded Last Bit.
ACM Transaction on Mathematical Software, Vol. 17, No. 3,
September 1991, pp. 410-423
[164] Chance, R.J.: The effect of processor architecture on an efficient
floating point table look-up algorithm.
Microprocessors and Microsystems, Vol. 15, No. 8,
October 1991, pp. 411-415
[165] Tang, P.T.P.: Table-Lookup Algorithms for Elementary Functions and Their
Error Analysis.
Proceedings of the 10th IEEE Symposium on Computer Arithmetic, Grenoble,
France, 26-28 June 1991, pp. 232-236
[166] Ferguson, W.E.; Brightman, T.: Accurate and Monotone Approximations of
Some Transcendental Functions.
Proceedings of the 10th IEEE Symposium on Computer Arithmetic, Grenoble,
France, 26-28 June 1991, pp. 237-244
3.1 Elementary functions, CORDIC and related algorithms
[167] Wensley, J.H.: A Class of Non-Analytical Iterative Processes.
The Computer Journal, Vol. 1, 1958, pp. 163-167
[168] Volder, J.E.: The CORDIC Trigonometric Computing Technique.
IRE Transactions on Electronic Computers, Vol. EC-8, No. 5,
September 1959, pp. 330-334
[169] Daggett, D.H.: Decimal-Binary Conversions in CORDIC.
IRE Transactions on Electronic Computers, Vol. EC-8, No. 5,
September 1959, pp. 335-339
[170] Cantor, D.; Estrin, G.; Turn, R.: Logarithmic and Exponential Function
Evaluation in a Variable Structure Digital Computer.
IRE Transactions on Electronic Computers, Vol. EC-11,
April 1962, pp. 155-164
[171] Meggitt, J.E.: Pseudo Division and Pseudo Multiplication Processes.
IBM Journal Research & Development, Vol. 6, No. 2, April 1962,
pp. 210-226
[172] Specker, W.H.: A Class of Algorithms for Ln x, Exp x, Sin x, Cos x,
Tan**-1 x, and Cot**-1 x.
IEEE Transactions on Electronic Computers, Vol. EC-14, No. 1,
February 1965, pp. 85-86
[173] Linhardt, R.J.; Miller, H.S.: Digit-by-Digit Transcendental-Function
Computation.
RCA Review, Vol. 30, June 1969, pp. 209-247
[174] Lugish, B.G. de: A Class of Algorithms for Automatic Evaluation of
Certain Elementary Function in a Binary Computer.
University of Illinois, Department of Computer Science Report No. 399,
June 1970
[175] Walther, J.S.: A unified algorithm for elementary functions.
AFIPS Conference Proceedings, Vol. 38, SJCC 1971, pp. 379-385
[176] Sarkar, B.P.; Krishnamurthy, E.V.: Economic Pseudodivision Processes for
Obtaining Square Root, Logarithm, and Arctan
IEEE Transactions on Computers, Vol. C-20, No. 12, December 1971,
pp. 1589-1593
[177] Chen, T.C.: Automatic Computation of Exponentials, Logarithms,
Ratios and Square Roots.
IBM Journal Research & Development, Vol. 16, No. 4, July 1972,
pp. 380-388
[178] Ercegovac, M.D.: Radix-16 Evaluation of Certain Elementary Functions.
IEEE Transactions on Computers, Vol. C-22, No. 6, June 1973, pp. 561-566
[179] Baker, P.W.: More Efficient Radix-2 Algorithms for Some
Elementary Functions.
IEEE Transactions on Computers, Vol. C-24, No. 11, November 1975,
pp. 1049-1054
[180] Andrews, M.; Mraz, T.: Unified elementary function generator.
Microprocessors and Microsystems, Vol. 2, No. 5,
October 1978, pp. 270-273
[181] Rodrigues, M.R.D.; Zurawski, J.H.P.; Gosling, J.B.: Hardware evaluation
of mathematical functions.
IEE Proceedings, Part E, Vol. 128, No. 4, July 1981, pp. 155-164
[182] V"olz, H.: CORDIC und "ahnliche Algorithmen der elementaren Funktionen
mit besonderer Eignung f"ur Mikrorechner.
Nachrichtentechnik Elektronik, Vol. 33, No. 12, 1983, pp. 506-510
[183] Naseem, A.; Fisher, P.D.: A Modified CORDIC Algorithm.
Proceedings 1984 IEEE International Conference on Computer Design: VLSI
in Computers. Port Chester, NY, USA, 8-11 October 1984, pp. 684-688
[184] Abruzzo, J.: Applicability of CORDIC Algorithm to Arithmetic Processing.
Conference Record 18th Asilomar Conference on Circuits, Systems, and
Computers. Pacific Grove, CA, USA, 5-7 November 1984, pp. 79-86
[185] Naseem, A.; Fisher, P.D.: The Modified CORDIC Algorithm.
Proceedings of the 7th Symposium on Computer Arithmetic, Urbana, IL,
USA, 4-6 June 1985, pp. 144-152
[186] Bittner, L.: Weiteres "uber Pseudo-Divisionsverfahren zur Berechnung der
Standardfunktionen und gewisser Nicht-Standardfunktionen.
Zeitschrift angewandte Mathematik und Mechanik, Vol. 65, No. 12, 1985,
pp. 605-612
[187] Vachss, R.: The Cordic Magnification Function.
IEEE Micro, Vol. 7, No. 5, October 1987, pp. 83-84
[188] Hu, Y.H.; Naganathan, S.: Angle Recoding Method for Efficient
Implementation of the CORDIC Algorithm.
1989 IEEE International Symposium on Circuits and Systems, Vol. 1,
Portland, OR, USA, 8-11 May 1989, pp. 175-178
[189] Ahmed, H.M.: Efficient Elementary Function Generation with Multipliers.
Proceedings of the 9th Symposium on Computer Arithmetic, Santa Monica,
CA, USA, 6-8 September 1989, pp. 52-59
[190] Jarvis, P.: Implementing Cordic Algorithms.
Dr. Dobb's Journal, Vol. 15, No. 10, October 1990, pp. 152-158
[191] Takagi, N.; Asada, T.; Yajima, S.: Redundant CORDIC Methods with a
Constant Scale Factor for Sine and Cosine Computation.
IEEE Transactions on Computers, Vol. C-40, No. 9, September 1991,
pp. 989-995
[192] Wesner, J.: Ein Tick schneller.
mc 1992, Heft 2, Seiten 80-86
3.2 Elementary functions, function approximation
[193] Spielberg, K.: Efficient Continued Fraction Approximations To Elementary
Functions.
Mathematics of Computation, Vol. 15, 1961, pp. 409-417
[194] Spielberg, K.: Representation of Power Series in Terms of Polynomials,
Rational Approximations and Continued Fractions.
Journal of the Association for Computing Machinery, Vol. 8,
1961, pp. 613-627
[195] Phillips, G.M.: Estimate of the maximum error in best polynomial
approximations.
The Computer Journal, Vol. 11, 1968, pp. 110-111
[196] Mohn, K.; Roman, R.V.: An interactive polynomial approximation
algorithm.
The Computer Journal, Vol. 19, No. 1, pp. 74-78
[197] Derenzo, S.: Approximations for Hand Calculators Using Small Integer
Coefficients.
Mathematics of Computation, Vol. 31, No. 137, January 1977, pp. 214-225
[198] Ruckdeschel, F.: Functional Approximations.
BYTE, Vol. 3, No. 11, November 1978, pp. 34-46
[199] Moshier, S.L.: Computer Approximations.
BYTE, Vol. 11, No. 4, April 1986, pp. 161-178
[200] Trefethen, L.N.; Gutknecht, M.H.: Pad‚, Stable Pad‚, and Chebyshev-Pad‚
Approximation.
In: Mason, J.C.; Cox, M.G. (ed.): Algorithms for Approximation.
Oxford: Clarendon Press 1987
[201] Breuer, P.T.: A New Method for Real Rational Uniform Approximation.
In: Mason, J.C.; Cox, M.G. (ed.): Algorithms for Approximation.
Oxford: Clarendon Press 1987
[202] Dunham, C.: Applications of Best Approximation.
SIGNUM Newsletter, Vol. 26, No. 2, April 1991, pp. 2-10
[203] Plauger, P.J.: Approximating functions.
Computer Language, Vol. 8, No. 6, June 1991, pp. 17-25
[204] Plauger, P.J.: Economizing polynomials.
Computer Language, Vol. 8, No. 7, July 1991, pp. 21-27
[205] Ochs, T.: Numerics for the rest of us.
Computer Language, Vol. 8, No. 10, October 1991, pp. 113-127
3.2.1 Polynomial evaluation
[206] Knuth, D.E.: Evaluation of Polynomials By Computer.
Communications of the ACM, Vol. 5, No. 12, December 1962, pp. 595-599
[207] Eisman, S.H.: Polynomial Evaluation Revisited.
Communications of the ACM, Vol. 6, No. 7, July 1963, 384-385
[208] Eve, J.: The Evaluation of Polynomials.
Numerische Mathematik, Vol. 6, 1964, pp. 17-21
[209] Fike, C.T.: Methods of Evaluating Polynomial Approximations in Function
Evaluation Routines.
Communications of the ACM, Vol. 10, No. 3, March 1967, pp. 175-178
[210] Reimer, M.: Auswertungsalgorithmen fast-optimaler numerischer
Stabilit"at f"ur Polynome.
Computing, Vol. 17, No. 4, 1977, pp. 289-296
[211] Kusterer, R.; Reimer, M.: Stable Evaluation of Polynomials in Time
log n.
Mathematics of Computation, Vol. 33, No. 147, July 1979, pp. 1019-1031
[212] Dunham, C.B.: Perturbation Analysis of Horner's Method for Nice Cases.
SIGNUM Newsletter, Vol. 24., No. 2-3, April 1989, pp. 8-9
3.3 Square root, general
[213] Andrews, M.; McCormick, S.F.; Taylor, G.D.: Evaluation of Functions on
Microcomputers: Square Root.
Computers & Mathematics with Applications, Vol. 4,
No. 4, 1978, pp. 359-367
[214] Alt, H.: Square Rooting Is as Difficult as Multiplication.
Computing, Vol. 21, No. 3, 1979, pp. 221-232
[215] Andrews, M.: Mathematical Microprocessor Software: A ûx Comparison.
IEEE Micro, Vol. 2, No. 3, May 1982, pp. 63-79
[216] Dietrich, D.: Verfahren zur L"osung von Quadratwurzeln f"ur
Mikrorechnerprozeduren.
Elektroniker (Schweiz) 1983, Heft 8, Seiten EL-1 - EL-6
[217] Moler, C.; Morrison, D.: Replacing Square Roots by Pythagorean Sums.
IBM Journal Research & Development, Vol. 27, No. 6, November 1983,
pp. 577-581
[218] Montuschi, P.; Mezzalama, M.: Survey of square rooting algorithms.
IEE Proceedings, Part E, Vol. 137, No. 1, January 1990, pp. 31-40
[219] Crenshaw, J.W.: Square roots are simple?
Embedded Systems Programming, Vol. 4, No. 11, November 1991, pp. 30-52
3.3.1 Square root, bitoriented, iterative, and table methods of computation
[220] Lenaerts, E.H.: Automatic Square Rooting.
Electronic Engineering, Vol. 27, July 1955, pp. 287-289
[221] Bemer, R.W.: A Machine Method for Square-Root Computation.
Communications of the ACM, Vol. 1, No. 1, 1958, pp. 6-7
[222] Metze, G.: Minimal Square Rooting.
IEEE Transactions on Electronic Computers, Vol. EC-14, April 1965,
pp. 181-185
[223] Egbert, W.E.: Personal Calculator Algorithms I: Square Roots.
Hewlett-Packard Journal, Vol. 28, No. 9, May 1977, pp. 22-24
[224] Alexander, V.L.: Square Root Routine.
IBM Technical Disclosure Bulletin, Vol. 20, No. 3, August 1977, p. 1222
[225] Murphy, T.R.; Rickard, P.L.: Square-Root Procedure for Floating-Point
Numbers.
IBM Technical Disclosure Bulletin, Vol. 21, No. 2, July 1978, p. 785
[226] Waldecker, D.E.: Nonrestoring Square Root with Simplified Answer
Generation.
IBM Technical Disclosure Bulletin, Vol. 22, No. 11, April 1980,
pp. 4807-4808
[227] Peng, H.: Algorithms for extracting square roots and cube roots.
Proceedings of the 5th Symposium on Computer Arithmetic, Ann Arbor,
MI, USA, 18-19 May 1981, pp. 121-126
[228] Grote, H.: Code improves on a square-root routine.
EDN, Vol. 26, No. 11, May 1981, pp. 198-200
[229] Bice, P.K.: Algorithm adds square root to micro's arithmetic capability.
Electronic Design, Vol. 29, No. 11, May 1981, p. 146
[230] Kunz, W.: Quadratwurzel mit dem æP Z80.
Elektronik 1981, Heft 7, Seiten 109-110
[231] Rix, P.: Universeller Quadratwurzel-Algorithmus.
Elektronik 1982, Heft 23, Seiten 81-82
[232] Conover, B.; Gustafson. D.L.: An Algorithm for High Speed Square Roots.
1985 Region 5 IEEE Conference Digest. Lubbock, TX, USA, 13-15 March
1985, pp. 19-21
[233] Sreedharan, J.; Dhurkadas, A.: 8086 algorithm solves square roots.
EDN, Vol. 30, No. 7, April 1985, p. 272
[234] Enzmann, K.: Wurzelziehen durch sukzessive Approximation.
Elektronik 1985, Heft 18, Seite 92
[235] Majerski, S.: Square-Rooting Algorithms for High-Speed Digital Circuits.
IEEE Transactions on Computers, Vol. C-34, No. 8, August 1985,
pp. 724-733
[236] Grappel, R.D.: Square-root algorithm is fast and simple.
EDN, Vol. 31, No. 8, April 1986, pp. 248-248
[237] Campbell, R.A: NS32000 Square Roots.
Dr. Dobb's Journal, Vol. 11, No. 3, March 1986, pp. 122-123 + 106
[238] Wollenberg, R.; Milnikel, R.: Schnelles Radizierverfahren durch
Tabellenzugriff.
Elektronik 1986, Heft 6, Seiten 79-82
[239] Pfenninger, E.: Schneller Quadratwurzel-Algorithmus.
Elektronik 1986, Heft 22, Seiten 179-180
[240] B"ottner, H.: Quadratwurzel-Algorithmus.
mc 1987, Heft 5, Seite 58
[241] Johnson, K.C.: ALGORITHM 650: Efficient Square Root Implementation
on the 68000.
ACM Transactions on Mathematical Software, Vol. 13, No. 2,
June 1987, pp. 138-151
[242] Rolfe, T.J.: On a Fast Integer Square Root Algorithm.
SIGNUM Newsletter, Vol. 22, No.4, October 1987, pp. 6-11
[243] Barany, T.E.: Fast evaluation of integer roots in microcontroller
systems.
Microprocessor and Microsystems, Vol. 12, No. 6, July-August 1988,
pp. 341-344
[244] Milnikel, R.; Wollenberg, R.: Schnelles Radizierverfahren f"ur
Gleitkommazahlen im IEEE-Format.
Elektronik 1988, Heft 8, Seiten 114-122
[245] Johnson, K.R.: An Iterative Method for Approximating Square Roots.
Mathematics Magazine, Vol. 62, No. 4, October 1989, pp. 253-259
[246] Peter, O.: Prozessor zieht Wurzeln.
c't 1990, Heft 1, Seiten 300-306
[247] Hashemian, R.: Square Rooting Algorithms for Integer and Floating-Point
Numbers.
IEEE Transactions on Computers, Vol. C-39, No. 8,
August 1990, pp. 1025-1029
[248] Hamza, K.M.; Abdul-Karim, M.A.H.: Microprocessor Based Direct Square
Root Extractor.
Modelling, Simulation & Control A, Vol. 34, No. 1, 1991, pp. 45-48
3.3.2 Square root, Newton's method
[249] Gower, J.C.: A Note on an Iterative Method for Root Extraction.
The Computer Journal, Vol. 1, 1958, pp. 142-143
[250] Eve, J.: Starting approximation for the iterative calculation of square
roots.
The Computer Journal, Vol. 6, 1963, pp. 274-276
[251] Cody, W.J.: Double-Precision Square Root for The CDC-3600.
Communications of the ACM, Vol. 7, No. 12, December 1964, pp. 715-718
[252] Swarztrauber, P.N.: On the Double-Precision Square Root Routine (letter
to the editor).
Communications of the ACM, Vol. 8, No. 4, April 1965, p. 202
[253] King, R.: On the Double-Precision Square Root Routine (letter to the
editor).
Communications of the ACM, Vol. 8, No. 4, April 1965, p. 202
[254] James, W.; Jarratt, P.: The Generation of Square Roots on a Computer
with Rapid Multiplication compared with Division.
Mathematics of Computation, Vol. 19, 1965, pp. 497-500
[255] Fike, C.T.: Starting Approximations for Square Root Calculation on IBM
System/360.
Communications of the ACM, Vol. 9, No. 4, April 1966, pp. 297-299
[256] Moursund, D.G.: Optimal Starting Values for Newton-Raphson
Calculation of ûx.
Communications of the ACM, Vol. 10, No. 7, July 1967, pp. 430-432
[257] Fike, C.T.: A Rational Approximation Optimal by Moursund's Criterion
(letter to the editor).
Communications of the ACM, Vol. 10, No. 11, November 1967, pp. 683-684
[258] Holzwarth, A.: Ein Verfahren zur Bestimmmung bester Tschebyscheff-
Approximationen der Quadratwurzelfunktion.
Computing, Vol. 4, 1969, pp. 168-177
[259] King, R.F.; Phillips, D.L.: The Logarithmic Error and Newton's Method
for the Square Root.
Communications of the ACM, Vol. 12, No. 2, February 1969, pp. 87-88
[260] Sterbenz, P.H.; Fike, C.T.: Optimal Starting Approximations for
Newton's Method.
Mathematics of Computation, Vol. 23, 1969, pp.313-318
[261] Wilson, M.W.: Optimal Starting Approximations for Generating
Square Root for Slow or No Divide.
Communications of the ACM, Vol. 13, No. 9, September 1970, pp. 559-560
[262] Rothmaier, B.: Die Berechnung der Quadratwurzel nebst Schranken auf
Dualmaschinen. Interner Bericht Nr. 70/17, Institut f"ur Informatik.
Universit"at Karlsruhe 1970.
[263] Ninomiya, I: Best Rational Starting Approximations and Improved Newton
Iteration for the Square Root.
Mathematics of Computation, Vol. 24, No. 110, April 1970, pp. 391-404
[264] Phillips, D.L.: Generalized Logarithmic Error and Newton's Method for
the mth Root.
Mathematics of Computation, Vol. 24, No. 110, April 1970, pp. 383-389
[265] Ramamoorthy, C.V.; Goodman, J.R.; Kim, K.H.: Some Properties of
Iterative Square-Rooting Methods Using High-Speed Multiplication.
IEEE Transactions on Computers, Vol. C-21, No. 8, August 1972,
pp. 837-847
[266] Pedersen, P.W.: Hvordan beregner man kvadratroden?
Elektronik (Denmark), No. 4, April 1980, pp. 18-21
[267] Meinardus, G.; Taylor, G.D.: Optimal Partitioning of Newton's Method for
Calculating Roots.
Mathematics of Computation, Vol. 35, No. 152, October 1980,
pp. 1221-1230
[268] Fredette, G.: 68000 routine extracts square roots.
EDN, Vol. 26, No. 16, August 1981, pp. 185-194
[269] Chamrad, V.: A Fast Floating-Point Square-Rooting Routine for the
8080/8085 Microprocessors.
Kybernetika, Vol. 19, No. 4, 1983, pp. 335-344
[270] Chen, S.G.; Hsieh, P.Y.: Fast Computation of the Nth Root.
Computers & Mathematics with Applications, Vol. 17, No. 10, 1989,
pp. 1423-1427
[271] Yeyios, A.K.: On two sequences of algorithms for approximating square
roots.
Journal of Computational and Applied Mathematics, Vol. 40,
No. 1, June 1992, pp. 63-72
3.4 Sine and Cosine
[272] Kogbetliantz, E.G.: Computation of Sin N, Cos N, and Mth Root of N Using
an Electronic Computer.
IBM Journal Research & Development, Vol. 3, No. 2, April 1959,
pp. 147-152
[273] Steer, D.G.; Penstone, S.R.: Digital Hardware for Sine-Cosine Functions.
IEEE Transaction on Computers, Vol. C-26, No. 12, December 1977,
pp. 1283-1286
[274] Colquhoun, D.G.: A Fast Approximation to the Sine Function.
Software - Practice and Experience, Vol. 7, No. 2, March-April 1977,
pp. 227-229
[275] Egbert, W.E.: Personal Calculator Algorithms II: Trigonometric
Functions.
Hewlett-Packard Journal, Vol. 28, No. 10, June 1977, pp. 17-20
[276] Wang, J.Y.: The Evaluation of Periodic Functions with Large Input
Arguments.
SIGNUM Newsletter, Vol. 13, No. 4, December 1978, pp. 7-9
[277] Ganesan, K.; Augustine, A.: 8086 generates sines and cosines.
EDN, Vol. 26, No. 6, March 1981, pp. 186-188
[278] Sewell, B.T.: A Rapid Lookup Table Method for Trigonometric Functions.
Software - Practice and Experience, Vol. 12, No. 10,
October 1982, pp. 885-887
[279] Payne, M.H.; Hanek, R.N.: Radian Reduction for Trigonometric Functions.
SIGNUM Newsletter, Vol. 18, No. 1, January 1983, pp. 19-24
[280] Payne, M.H.; Hanek, R.N.: Degree Reduction for Trigonometric Functions.
SIGNUM Newsletter, Vol. 18, No. 2, April 1983, pp. 18-19
[281] Murray, J.T.: Sin/Cos Functions via Approximations Plus Error
Compensation.
IBM Technical Disclosure Bulletin, Vol. 26, No. 10A, March 1984,
pp. 4967-4968
[282] Helyer, R.: Sine and cosine calculations.
Microprocessors and Microsystems, Vol. 2, No. 5,
October 1985, p. 284
[283] Campbell, R.A.: In Search of a Sine.
Dr. Dobb's Journal, Vol. 11, No. 12, December 1986, pp. 30-32
[284] Obermaier, A.: Sin und cosin mit Integerarithmetik.
mc 1987, Heft 6, Seiten 108-112
[285] Juffa, N.: Schnelle Algorithmen f"ur Sin und Cos.
mc 1988, Heft 5, Seiten 105-107
[286] Tang, P.T.P.: Some Software Implementations of the Functions Sine and
Cosine. Report ANL-90/3. Argonne National Laboratory, April 1990
[287] Ochs, T.: This sine is also right!
Computer Language, Vol. 9, No. 1, January 1992, pp. 89-93
3.5 Logarithm
[288] Bemer, R.W.: A Subroutine Method for Calculating Logarithms.
Communications of the ACM, Vol. 1, No. 5, May 1958, pp. 5-7
[289] Spielberg, K.: Polynomial and Continued-Fraction Approximations for
Logarithmic Functions.
Mathematics of Computation, Vol. 16, 1962, pp. 205-217
[290] Egbert, W.E.: Personal Calculator Algorithms IV: Logarithmic Functions.
Hewlett-Packard Journal, Vol. 29, No. 8, April 1978, pp. 29-32
[291] Horna, O.A.: Fast algorithms for the computation of binary logarithms.
COMSAT Technical Review, Vol. 10, No. 1, Spring 1980, pp. 91-101
[292] McCormick, S.F.; Taylor, G.D.; Pryor, D.V.: Evaluation of Functions on
Microcomputers: ln(x).
Computers & Mathematics with Applications, Vol. 8, No. 5, 1982,
pp. 389-392
[293] Sasaki, T.; Kanada, Y.: Practically Fast Multiple Precision Evaluation
of LOG (X).
Journal of Information Processing, Vol. 5, No. 4, 1982, pp. 247-250
[294] Maenner, R.: A Fast Integer Binary Logarithm of Large Arguments.
IEEE Micro, Vol. 7, No. 6, December 1987, pp. 41-45
[295] Tang, P.T.P: Table-Driven Implementation of the Logarithm Function in
IEEE Floating-Point Arithmetic.
ACM Transactions on Mathematical Software, Vol. 16, No. 4,
December 1990, pp. 378-400
[296] Kostopoulos, D.K.: An Algorithm for the Computation of Binary
Logarithms.
IEEE Transactions on Computers, Vol. 40, No. 11, November 1991,
pp. 1267-1270
3.6 Exponential function
[297] Kogbetliantz, E.G.: Computation of e**n for -inf<N<inf Using an
Electronic Computer.
IBM Journal Research & Development, Vol. 1, No. 2, April 1957,
pp. 110-115
[298] Spielberg, K.: Computation of e**x with the use of large tables.
IBM Systems Journal, Vol. 5, No. 2, 1966, pp. 102-114
[299] Wrathall, C.; Chen, T.C.: Convergence Guarantee and Improvements for
a fast Hardware Exponential and Logarithm Evaluation Scheme.
Proceedings of the 4th Symposium on Computer Arithmetic, Santa Monica,
CA, USA, 25-27 October 1978, pp. 175-182
[300] Andrews, M; Jaeger, D.; McCormick, S.F.; Taylor, G.D.: Evaluation of
Functions on Microcomputers: exp(x).
Computers & Mathematics with Applications, Vol. 7, No. 6, 1981,
pp. 503-508
[301] Hantler, S.L.; Karp, A.H.: Exponential by Sequential Squaring.
IBM Technical Disclosure Bulletin, Vol. 25, No. 1, June 1982,
pp. 171-173
[302] Karp, A.H.: Exponential and Logarithm by Sequential Squaring.
IEEE Transactions on Computers, Vol. C-33, No. 5, May 1984, pp. 462-464
[303] Grappel, R.D.: Fast subroutine calculates exponentials.
EDN, Vol. 30, No. 10, May 1985, p. 231
[304] Tang, P.T.P.: Table-Driven Implementation of the Exponential
Function in IEEE Floating-Point Arithmetic.
ACM Transactions on Mathematical Software, Vol. 15, No. 2,
June 1989, pp. 144-157
[305] Tang, P.T.P.: Table-driven Implementation of the Expm1 Function in
IEEE Floating-Point Arithmetic.
Preprint MCS-P144-0390, Mathematics and Computer Science Division.
Argonne National Laboratory, March 1990
[306] Tang, P.T.P.: Table-Driven Implementation of the Expm1 Function In
IEEE Floating-Point Arithmetic.
ACM Transactions on Mathematical Software, Vol. 18, No. 2, June 1992,
pp. 211-222
3.7 Arctangent
[307] Luke, Y.L.: On the Computation of log Z and arc tan Z.
Mathematical Tables and Other Aids to Computation, Vol. 11, 1957,
pp. 16-18
[308] Kogbetliantz, E.G.: Computation of Arctan N for -inf<N<+inf Using an
Electronic Computer.
IBM Journal Research & Development, Vol. 2, No. 1, January 1958,
pp. 43-53
[309] Perlin, I.E.; Garrett, J.R.: High Precision Calculation of Arcsin x,
Arccos x, and Arctan x.
Mathematics of Computation, Vol. 14, 1960, pp. 270-274
3.8 Other transcendental functions
[310] Clark, N.W.; Cody, W.J.: Self-contained exponentation.
AFIPS Conference Proceedings, Vol. 35, FJCC 1969, pp. 701-706
[311] Clark, N.W.; Cody, W.J.; Kuki, H.: Self-Contained Power Routines.
In: Rice, J.R. (ed.): Mathematical Software.
New York, NY: Academic Press 1971
[312] Kogbetliantz, E.G.: Computation of Arcsin N for 0<N<1 Using an
Electronic Computer.
IBM Journal Research & Development, Vol. 2, No. 3, July 1958,
pp. 218-222
[313] Egbert, W.E.: Personal Calculator Algorithms III: Inverse
Trigonometric Functions.
Hewlett-Packard Journal, Vol. 29, No. 3, November 1977, pp. 22-23
[314] Morrison, D.R.: A Method for Computing Certain Inverse Functions.
Mathematical Tables and other Aids to Computation,
Vol. 10, 1965, pp. 202-208
[315] Clenshaw, C.W.; Miller, G.F.; Woodger, M.: Algorithms for Special
Functions I.
Numerische Mathematik, Vol. 4, 1963, pp. 403-419
[316] Miller, G.F.: Algorithms for Special Functions II.
Numerische Mathematik, Vol. 7, 1965, pp. 194-196
[317] Kr"amer, W.: Inverse Standardfunktionen f"ur reelle und komplexe
Intervallargumente mit a priori Fehlerabsch"atzungen f"ur beliebige
Datenformate.
Dissertation, Universit"at Karlsruhe 1987
[318] Preston, F.S.: A New Algorithm for the Tangent.
IEEE Transactions on Computers, Vol. C-27, No. 2, February 1978, p. 167
[319] Filho, A.M.S.; Schwachheim, G.: Algorithm 309: Gamma Function with
Arbitrary Precision.
Communications of the ACM, Vol. 10, No. 8, August 1967, pp. 511-512
4.0 Binary-decimal conversion
[320] Matula, D.W.: In-and-Out Conversions.
Communications of the ACM, Vol. 11, No. 1, January 1968, pp. 47-50
[321] Loevenbruck, A.P.: Conversion of Number Representations.
IBM Technical Disclosure Bulletin, Vol. 15, No. 7, December 1972,
pp. 2148-2151
[322] Schreiber, F.A., Stefanelli, R.: Two Methods for Fast Binary-BCD
Conversion.
Proceedings of the 4th Symposium on Computer Arithmetic, Santa Monica,
CA, USA, 25-27 October 1978, pp. 200-207
[323] Havender, J.W.: Decimal-to-binary floating point number conversion
mechanism.
IBM Technical Disclosure Bulletin, Vol. 23, No. 2, July 1980,
pp. 706-708
[324] Havender, J.W.: Decimal-to-binary Number Conversion.
IBM Technical Disclosure Bulletin, Vol. 23, No. 3, August 1980,
pp. 1126-1127
[325] Griffiths, L.K.: Binary-to-Decimal Conversion.
IBM Technical Disclosure Bulletin, Vol. 24, No. 1A, June 1981,
pp. 237-238
[326] Murugesan, S.; Agrawal, V.K.: Algorithm converts fractions to BCD.
EDN, Vol. 28, No. 19, September 1983, pp. 245-246
[327] Agrawal, V.K.; Murugesan, S.: Algorithm converts BCD fractions to
binary.
EDN, Vol. 29, No. 13, June 1984, pp. 278-280
[328] Nowak, M.: Hex-BCD-Umwandlung.
mc 1986, Heft 10, Seite 68
[329] Henning, D.: Konvertierung bin"arer in dezimale Gleitkommazahlen f"ur
me"stechnische Anwendungen.
Radio Fernsehen Elekronik, Vol. 35, No. 11, 1986, pp. 731-733
[330] Clinger, W.D.: How to Read Floating Point Numbers Accurately.
Proceedings of ACM SIGPLAN '90 Conference on Programming Language Design
and Implementation. White Plains, NY, 20-22 June 1990, pp. 92-101
[331] Steele, G.L.; White, J.L.: How to Print Floating-Point Numbers
Accurately.
Proceedings of ACM SIGPLAN '90 Conference on Programming Language Design
and Implementation. White Plains, NY, 20-22 June 1990, pp. 112-123
5.0 BCD arithmetic
[332] Franklin, J.W.: Zoned Decimal Arithmetic.
IBM Technical Disclosure Bulletin, Vol. 15, No. 7, December 1972,
pp. 2123-2124
[333] Schmid, H.: BCD logic II: BCD multiplication.
Electronic Design, Vol. 21, No. 14, July 1973, pp. 62-69
[334] Schmid, H.: BCD logic III: BCD division.
Electronic Design, Vol. 21, No. 15, July 1973, pp. 86-92
[335] Schmid, H.: BCD logic IV: BCD decimal-point location.
Electronic Design, Vol. 21, No. 16, August 1973, pp. 80-84
[336] Schmid, H.: BCD logic V: BCD square root.
Electronic Design, Vol. 21, No. 17, August 1973, pp. 62-69
[337] Schmid, H.: BCD logic VI: BCD logarithms and exponentials.
Electronic Design, Vol. 21, No. 18, September 1973, pp. 118-123
[338] Schmid, H.: BCD logic VII: BCD trig and hyperbolic functions.
Electronic Design, Vol. 21, No. 19, September 1973, pp. 68-73
[339] Chroust, G.: Method of Adding Decimal Numbers by Means of Binary
Arithmetic.
IBM Technical Disclosure Bulletin, Vol. 23, No. 10, March 1981,
pp. 4525-4526
[340] Bohannan, J.W.; Bradley, D.J.; Kummer, D.A.; Saenz, J.A.: Multiplication
Algorithm for Packed BCD Numbers.
IBM Technical Disclosure Bulletin, Vol. 25, No. 4, September 1982,
p. 2225
[341] Wingert, J.A.: Improved Table-assisted Addition and Multiplication
Methods.
IBM Technical Disclosure Bulletin, Vol. 25, No. 9, February 1983,
pp. 4742-4743
[342] Maric, I.; Cucancic, L.: On the Possibilities of the BCD Code
Application in the Floating-Point Arithmetic Algorithms.
International Journal of Mini and Microcomputers, Vol. 5, No. 2, 1983,
pp. 19-22
[343] Ahmad, M.: Implementable Decimal Arithmetic Algorithms for
Micro/Minicomputers.
Microprocessing and Microprogramming, Vol. 19, No. 2,
February 1987, pp. 119-128
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