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.

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

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