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Henry S. Warren, Jr., has had a fifty-year career with IBM, spanning from the IBM 704 to the PowerPC and beyond. He has worked on various military command and control systems and on the SETL (SET Language) project under Jack Schwartz. Since 1973, Hank has been with IBM’s Research Division, focusing on compilers and computer architectures. He currently works on a supercomputer project aimed at an exaflop. Hank received his Ph.D. in computer science from the Courant Institute at New York University.
Preface xv
CHAPTER 1: INTRODUCTION 1
1–1 Notation 1
1–2 Instruction Set and Execution Time Model 5
CHAPTER 2: BASICS 11
2–1 Manipulating Rightmost Bits 11
2–2 Addition Combined with Logical Operations 16
2–3 Inequalities among Logical and Arithmetic Expressions 17
2–4 Absolute Value Function 18
2–5 Average of Two Integers 19
2–6 Sign Extension 19
2–7 Shift Right Signed from Unsigned 20
2–8 Sign Function 20
2–9 Three-Valued Compare Function 21
2–10 Transfer of Sign Function 22
2–11 Decoding a “Zero Means 2**n” Field 22
2–12 Comparison Predicates 23
2–13 Overflow Detection 28
2–14 Condition Code Result of Add, Subtract, and Multiply 36
2–15 Rotate Shifts 37
2–16 Double-Length Add/Subtract 38
2–17 Double-Length Shifts 39
2–18 Multibyte Add, Subtract, Absolute Value 40
2–19 Doz, Max, Min 41
2–20 Exchanging Registers 45
2–21 Alternating among Two or More Values 48
2–22 A Boolean Decomposition Formula 51
2–23 Implementing Instructions for all 16 Binary Boolean Operations 53
CHAPTER 3: POWER-OF-2 BOUNDARIES 59
3–1 Rounding Up/Down to a Multiple of a Known Power of 2 59
3–2 Rounding Up/Down to the Next Power of 2 60
3–3 Detecting a Power-of-2 Boundary Crossing 63
CHAPTER 4: ARITHMETIC BOUNDS 67
4–1 Checking Bounds of Integers 67
4–2 Propagating Bounds through Add’s and Subtract’s 70
4–3 Propagating Bounds through Logical Operations 73
CHAPTER 5: COUNTING BITS 81
5–1 Counting 1-Bits 81
5–2 Parity 96
5–3 Counting Leading 0’s 99
5–4 Counting Trailing 0’s 107
CHAPTER 6: SEARCHING WORDS 117
6–1 Find First 0-Byte 117
6–2 Find First String of 1-Bits of a Given Length 123
6–3 Find Longest String of 1-Bits 125
6–4 Find Shortest String of 1-Bits 126
CHAPTER 7: REARRANGING BITS AND BYTES 129
7–1 Reversing Bits and Bytes 129
7–2 Shuffling Bits 139
7–3 Transposing a Bit Matrix 141
7–4 Compress, or Generalized Extract 150
7–5 Expand, or Generalized Insert 156
7–6 Hardware Algorithms for Compress and Expand 157
7–7 General Permutations, Sheep and Goats Operation 161
7–8 Rearrangements and Index Transformations 165
7–9 An LRU Algorithm 166
CHAPTER 8: MULTIPLICATION 171
8–1 Multiword Multiplication 171
8–2 High-Order Half of 64-Bit Product 173
8–3 High-Order Product Signed from/to Unsigned 174
8–4 Multiplication by Constants 175
CHAPTER 9: INTEGER DIVISION 181
9–1 Preliminaries 181
9–2 Multiword Division 184
9–3 Unsigned Short Division from Signed Division 189
9–4 Unsigned Long Division 192
9–5 Doubleword Division from Long Division 197
CHAPTER 10: INTEGER DIVISION BY CONSTANTS 205
10–1 Signed Division by a Known Power of 2 205
10–2 Signed Remainder from Division by a Known Power of 2 206
10–3 Signed Division and Remainder by Non-Powers of 2 207
10–4 Signed Division by Divisors ≥ 2 210
10–5 Signed Division by Divisors ≤ –2 218
10–6 Incorporation into a Compiler 220
10–7 Miscellaneous Topics 223
10–8 Unsigned Division 227
10–9 Unsigned Division by Divisors ≥ 1 230
10–10 Incorporation into a Compiler (Unsigned) 232
10–11 Miscellaneous Topics (Unsigned) 234
10–12 Applicability to Modulus and Floor Division 237
10–13 Similar Methods 237
10–14 Sample Magic Numbers 238
10–15 Simple Code in Python 240
10–16 Exact Division by Constants 240
10–17 Test for Zero Remainder after Division by a Constant 248
10–18 Methods Not Using Multiply High 251
10–19 Remainder by Summing Digits 262
10–20 Remainder by Multiplication and Shifting Right 268
10–21 Converting to Exact Division 274
10–22 A Timing Test 276
10–23 A Circuit for Dividing by 3 276
CHAPTER 11: SOME ELEMENTARY FUNCTIONS 279
11–1 Integer Square Root 279
11–2 Integer Cube Root 287
11–3 Integer Exponentiation 288
11–4 Integer Logarithm 291
CHAPTER 12: UNUSUAL BASES FOR NUMBER SYSTEMS 299
12–1 Base –2 299
12–2 Base –1 + i 306
12–3 Other Bases 308
12–4 What Is the Most Efficient Base? 309
CHAPTER 13: GRAY CODE 311
13–1 Gray Code 311
13–2 Incrementing a Gray-Coded Integer 313
13–3 Negabinary Gray Code 315
13–4 Brief History and Applications 315
CHAPTER 14: CYCLIC REDUNDANCY CHECK 319
14–1 Introduction 319
14–2 Theory 320
14–3 Practice 323
CHAPTER 15: ERROR-CORRECTING CODES 331
15–1 Introduction 331
15–2 The Hamming Code 332
15–3 Software for SEC-DED on 32 Information Bits 337
15–4 Error Correction Considered More Generally 342
CHAPTER 16: HILBERT’S CURVE 355
16–1 A Recursive Algorithm for Generating the Hilbert Curve 356
16–2 Coordinates from Distance along the Hilbert Curve 358
16–3 Distance from Coordinates on the Hilbert Curve 366
16–4 Incrementing the Coordinates on the Hilbert Curve 368
16–5 Non-Recursive Generating Algorithms 371
16–6 Other Space-Filling Curves 371
16–7 Applications 372
CHAPTER 17: FLOATING-POINT 375
17–1 IEEE Format 375
17–2 Floating-Point To/From Integer Conversions 377
17–3 Comparing Floating-Point Numbers Using Integer Operations 381
17–4 An Approximate Reciprocal Square Root Routine 383
17–5 The Distribution of Leading Digits 385
17–6 Table of Miscellaneous Values 387
CHAPTER 18: FORMULAS FOR PRIMES 391
18–1 Introduction 391
18–2 Willans’s Formulas 393
18–3 Wormell’s Formula 397
18–4 Formulas for Other Difficult Functions 398
ANSWERS TO EXERCISES: 405
APPENDIX A: ARITHMETIC TABLES FOR A 4-BIT MACHINE 453
APPENDIX B: NEWTON’S METHOD 457
APPENDIX C: A GALLERY OF GRAPHS OF DISCRETE FUNCTIONS 459
C–1 Plots of Logical Operations on Integers 459
C–2 Plots of Addition, Subtraction, and Multiplication 461
C–3 Plots of Functions Involving Division 463
C–4 Plots of the Compress, SAG, and Rotate Left Functions 464
C–5 2D Plots of Some Unary Functions 466
Bibliography 471
Index 481