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Hacker's Delight
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Table of Contents

Foreword xiii 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

About the Author

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.

Reviews

"This is the first book that promises to tell the deep, dark secrets of computer arithmetic, and it delivers in spades. It contains every trick I knew plus many, many more. A godsend for library developers, compiler writers, and lovers of elegant hacks, it deserves a spot on your shelf right next to Knuth. In the ten years since the first edition came out, it's been absolutely invaluable to my work at Sun and Google. I'm thrilled with all of the new material in the second edition." - Joshua Bloch

"When I first saw the title, I figured that the book must be either a cookbook for breaking into computers (unlikely) or some sort of compendium of little programming tricks. It's the latter, but it's thorough, almost encyclopedic, in its coverage. The second edition covers two new major topics and expands the overall collection with dozens of additional little tricks, including one that I put to use right away in a binary search algorithm: computing the average of two integers without risking overflow. This hacker is indeed delighted!" - Guy Steele

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