Hacker's Delight

"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." --Josh Bloch (Praise for the first edition) In Hacker’s Delight, Second Edition, Hank Warren once again compiles an irresistible collection of programming hacks: timesaving techniques, algorithms, and tricks that help programmers build more elegant and efficient software, while also gaining deeper insights into their craft. Warren’s hacks are eminently practical, but they’re also intrinsically interesting, and sometimes unexpected, much like the solution to a great puzzle. They are, in a word, a delight to any programmer who is excited by the opportunity to improve. Extensive additions in this edition include A new chapter on cyclic redundancy checking (CRC), including routines for the commonly used CRC-32 code A new chapter on error correcting codes (ECC), including routines for the Hamming code More coverage of integer division by constants, including methods using only shifts and adds Computing remainders without computing a quotient More coverage of population count and counting leading zeros Array population count New algorithms for compress and expand An LRU algorithm Floating-point to/from integer conversions Approximate floating-point reciprocal square root routine A gallery of graphs of discrete functions Now with exercises and answers  

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.