Finite precision number systems and arithmetic:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2010
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
133 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | XV, 699 S. graph. Darst. |
| ISBN: | 9780521761352 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804145591877369856 |
|---|---|
| adam_text | Titel: Finite precision number systems and arithmetic
Autor: Kornerup, Peter
Jahr: 2010
CONTENTS
Preface page xi
1 Radix polynomial representation 1
1.1 Introduction 1
1.2 Radix polynomials 2
1.3 Radix-/3 numbers 7
1.4 Digit symbols and digit strings 10
1.5 Digit sets for radix representation 14
1.6 Determining a radix representation 20
1.7 Classifying base-digit set combinations 31
1.8 Finite-precision and complement representations 35
1.8.1 Finite-precision radix-complement representations 38
1.9 Radix-/? approximation and roundings 43
1.9.1 Best radix-/? approximations 43
1.9.2 Rounding into finite representations 47
1.10 Other weighted systems 50
1.10.1 Mixed-radix systems 50
1.10.2 Two-level radix systems 52
1.10.3 Double-radix systems 52
1.11 Notes on the literature 54
2 Base and digit set conversion 59
2.1 Introduction 59
2.2 Base/radix conversion 60
2.3 Conversion into non-redundant digit sets 66
2.4 Digit set conversion for redundant systems 75
2.4.1 Limited carry propagation 77
2.4.2 Carry determined by the right context 80
Contents
2.4.3 Conversion into a contiguous digit set 83
2.4.4 Conversion into canonical, non-adjacent form 92
2.5 Implementing base and digit set conversions 95
2.5.1 Implementation in logic 103
2.5.2 On-line digit set conversion 108
2.6 The additive inverse 111
2.7 Notes on the literature 115
3 Addition 119
3.1 Introduction 119
3.2 How fast can we compute? 120
3.3 Digit addition 125
3.4 Addition with redundant digit sets 129
3.5 Basic linear-time adders 136
3.5.1 Digit serial and on-line addition 144
3.6 Sub-linear time adders 147
3.6.1 Carry-skip adders 148
3.6.2 Carry-select adders 149
3.6.3 Carry-look-ahead adders 151
3.7 Constant-time adders 159
3.7.1 Carry-save addition 159
3.7.2 Borrow-save addition 163
3.8 Addition and overflow in finite precision systems 165
3.8.1 Addition in redundant digit sets 165
3.8.2 Addition in radix-complement systems 169
3.8.3 l s complement addition 171
3.8.4 2 s complement carry-save addition 173
3.9 Subtraction and sign-magnitude addition 177
3.9.1 Sign-magnitude addition and subtraction 180
3.9.2 Bit-serial subtraction 183
3.10 Comparisons 184
3.10.1 Equality testing 185
3.10.2 Ordering relations 188
3.10.3 Leading zeroes determination 192
3.11 Notes on the literature 200
* Multiplication 207
4.1 Introduction 207
4.2 Classification of multipliers 208
4.3 Recoding and partial product generation 211
4.3.1 Radix-2 multiplication 212
• 4.3.2 Radix-4 multiplication 214
4.3.3 Hieh-radix mulüDlication 217
Contents
4.4 Sign-magnitude and radix-complement multiplication 220
4.4.1 Mapping into unsigned operands 221
4.4.2 2 s complement operands 222
4.4.3 The Baugh and Wooley scheme 222
4.4.4 Using a recoded multiplier 224
4.5 Linear-time multipliers 227
4.5.1 The classical iterative multiplier 228
4.5.2 Array multipliers 229
4.5.3 LSB-first serial/parallel multipliers 231
4.5.4 A pipelined serial/parallel multiplier 237
4.5.5 Least-significant bit first (LSB-first) serial/serial
multipliers 241
4.5.6 On-line or most-significant bit first (MSB-first)
multipliers 248
4.6 Logarithmic-time multiplication 252
4.6.1 Integer multipliers with overflow detection 258
4.7 Squaring 262
4.7.1 Radix-2 squaring 263
4.7.2 Recoded radix-4 squaring 264
4.7.3 Radix-4 squaring by operand dual recoding 266
4.8 Notes on the literature 269
5 Division 275
5.1 Introduction 275
5.2 Survey of division and reciprocal algorithms 277
5.2.1 Digit-serial algorithms 279
5.2.2 Iterative refinement algorithms 281
5.2.3 Resource requirements 282
5.2.4 Reciprocal look-up algorithms 283
5.3 Quotients and remainders 284
5.3.1 Integer quotient, remainder pairs 284
5.3.2 Radix-ß quotient, remainder pairs 287
5.3.3 Converting between radix-ß quotient, remainder
pairs 289
5.4 Deterministic digit-serial division 292
5.4.1 Restoring division 293
5.4.2 Robertson diagrams 297
5.4.3 Non-restoring division 298
5.4.4 Binary SRT division 305
5.5 SRT division 307
5.5. t Fundamentals of SRT division 308
5.5.2 Digit selection 310
viii Contents
5.5.3 Exploiting symmetries 321
5.5.4 Digit selection by direct comparison 324
5.5.5 Digit selection by table look-up 325
5.5.6 Architectures for SRT division 326
5.6 Multiplicative high-radix division 329
5.6.1 Short reciprocal division 330
5.6.2 Prescaled division 334
5.6.3 Prescaled division with remainder 338
5.6.4 Efficiency of multiplicative high radix division 342
5.7 Multiplicative iterative refinement division 344
5.7.1 Newton-Raphson division 346
5.7.2 Convergence division 350
5.7.3 Postscaled division 354
5.7.4 Efficiency of iterative refinement division 359
5.8 Table look-up support for reciprocals 361
5.8.1 Direct table look-up 363
5.8.2 Ulp accurate and monotonie reciprocal
approximations 370
5.8.3 Bipartite tables 375
5.8.4 Linear and quadratic interpolation 383
5.9 Notes on the literature 390
Square root 398
6.1 Introduction 398
6.2 Roots and remainders 400
6.3 Digit-serial square root 402
6.3.1 Restoring and non-restoring square root 404
6.3.2 SRT square root 407
6.3.3 Combining SRT square root with division 409
6.4 Multiplicative high-radix square root 416
6.4.1 Short reciprocal square root 419
6.4.2 Prescaled square root 422
6.5 Iterative refinement square root 426
6.5.1 Newton-Raphson square root 428
6.5.2 Newton-Raphson root-reciprocal 432
6.5.3 Convergence square root 434
6.5.4 Exact and directed one-ulp roots 437
6.6 Notes on the literature 443
Floating-point number systems 447
7.1 Introduction 447
7.2 Floating-point factorization and normalization 450
7.2.1 Floatine-Doint number factorization 450
Contents ix
7.2.2 Finite precision floating-point number systems 452
7.2.3 Distribution of finite precision floating-point
numbers 454
7.2.4 Floating-point base conversion and equivalent digits 457
7.3 Floating-point roundings 459
7.3.1 Precise roundings 460
7.3.2 One-ulp roundings and tails 463
7.3.3 Inverses of the rounding mappings 464
7.4 Rounded binary sum and product implementation 470
7.4.1 Determining quasi-normalized rounding intervals 472
7.4.2 Rounding from quasi-normalized rounding intervals 477
7.4.3 Implementing floating-point addition and subtraction 479
7.5 Quotient and square root rounding 483
7.5.1 Prenormalizing rounded quotients and roots 484
7.5.2 Quotient rounding using remainder sign 485
7.5.3 Rounding equivalence of extra accurate quotients 487
7.5.4 Precisely rounded division in Qpß 488
7.5.5 Precisely rounded square root 493
7.5.6 On-the-fly rounding 495
7.6 The IEEE standard for floating-point systems 498
7.6.1 Precision and range 499
7.6.2 Operations on floating-point numbers 507
7.6.3 Closure 513
7.6.4 Floating-point encodings 516
7.7 Notes on the literature 522
Modular arithmetic and residue number systems 528
8.1 Introduction 528
8.2 Single-modulus integer systems and arithmetic 529
8.2.1 Determining the residue a m 532
8.2.2 The multiplicative inverse 534
8.2.3 Implementation of modular addition and
multiplication 540
8.2.4 Multioperand modular addition 544
8.2.5 ROM-based addition and multiplication 547
8.2.6 Modular multiplication for very large moduli 549
8.2.7 Modular exponentiation 557
8.2.8 Inheritance and periodicity modulo 2* 558
8.3 Multiple modulus (residue) number systems 564
8.4 Mappings between residue and radix systems 569
8.5 Base extensions and scaling 578
8.5.1 Mixed-radix base extension 579
x Contents
8.5.2 CRT base extension 579
8.5.3 Scaling 582
8.6 Sign and overflow detection, division 584
8.6.1 Overflow 584
8.6.2 Sign determination and comparison 584
8.6.3 The core function 586
8.6.4 General division 600
8.7 Single-modulus rational systems 604
8.8 Multiple-modulus rational systems 613
8.9 p-adic expansions and Hensel codes 618
8.9.1 p-adic numbers 618
8.9.2 Hensel codes 621
8.10 Notes on the literature 623
9 Rational arithmetic 633
9.1 Introduction 633
9.2 Numerator-denominator representation systems 634
9.3 The mediant rounding 641
9.4 Arithmetic on fixed- and floating-slash operands 647
9.5 A binary representation of the rationals based on continued
fractions 656
9.6 Gosper s Algorithm 666
9.7 Bit-serial arithmetic on rational operands 672
9.8 The RPQ cell and its operation 683
9.9 Notes on the literature 686
Author index 691
Index 693
|
| any_adam_object | 1 |
| author | Kornerup, Peter 1939- Matula, David W. 1937- |
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| dewey-full | 513 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 513 - Arithmetic |
| dewey-raw | 513 |
| dewey-search | 513 |
| dewey-sort | 3513 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 9780521761352 |
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| spelling | Kornerup, Peter 1939- Verfasser (DE-588)14291875X aut Finite precision number systems and arithmetic Peter Kornerup ; David W. Matula Cambridge [u.a.] Cambridge Univ. Press 2010 XV, 699 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 133 Literaturangaben Arithmetic Foundations Computerarithmetik (DE-588)4135485-0 gnd rswk-swf Rechnen (DE-588)4048716-7 gnd rswk-swf Computer (DE-588)4070083-5 gnd rswk-swf Zahlensystem (DE-588)4117700-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Computerarithmetik (DE-588)4135485-0 s DE-604 Zahlensystem (DE-588)4117700-9 s Rechnen (DE-588)4048716-7 s Computer (DE-588)4070083-5 s Matula, David W. 1937- Verfasser (DE-588)142918776 aut Encyclopedia of mathematics and its applications 133 (DE-604)BV000903719 133 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469438&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kornerup, Peter 1939- Matula, David W. 1937- Finite precision number systems and arithmetic Encyclopedia of mathematics and its applications Arithmetic Foundations Computerarithmetik (DE-588)4135485-0 gnd Rechnen (DE-588)4048716-7 gnd Computer (DE-588)4070083-5 gnd Zahlensystem (DE-588)4117700-9 gnd |
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| title | Finite precision number systems and arithmetic |
| title_auth | Finite precision number systems and arithmetic |
| title_exact_search | Finite precision number systems and arithmetic |
| title_full | Finite precision number systems and arithmetic Peter Kornerup ; David W. Matula |
| title_fullStr | Finite precision number systems and arithmetic Peter Kornerup ; David W. Matula |
| title_full_unstemmed | Finite precision number systems and arithmetic Peter Kornerup ; David W. Matula |
| title_short | Finite precision number systems and arithmetic |
| title_sort | finite precision number systems and arithmetic |
| topic | Arithmetic Foundations Computerarithmetik (DE-588)4135485-0 gnd Rechnen (DE-588)4048716-7 gnd Computer (DE-588)4070083-5 gnd Zahlensystem (DE-588)4117700-9 gnd |
| topic_facet | Arithmetic Foundations Computerarithmetik Rechnen Computer Zahlensystem Lehrbuch |
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