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Book Description

The Future of Numerical Computing

Written by one of the foremost experts in high-performance computing and the inventor of Gustafson’s Law, The End of Error: Unum Computing explains a new approach to computer arithmetic: the universal number (unum). The unum encompasses all IEEE floating-point formats as well as fixed-point and exact integer arithmetic. This new number type obtains more accurate answers than floating-point arithmetic yet uses fewer bits in many cases, saving memory, bandwidth, energy, and power.

A Complete Revamp of Computer Arithmetic from the Ground Up

Richly illustrated in color, this groundbreaking book represents a fundamental change in how to perform calculations automatically. It illustrates how this novel approach can solve problems that have vexed engineers and scientists for decades, including problems that have been historically limited to serial processing.

Suitable for Anyone Using Computers for Calculations

The book is accessible to anyone who uses computers for technical calculations, with much of the book only requiring high school math. The author makes the mathematics interesting through numerous analogies. He clearly defines jargon and uses color-coded boxes for mathematical formulas, computer code, important descriptions, and exercises.

1. Cover Page
2. Half Title Page
3. Title Page
5. Contents
6. Preface
7. Acknowledgments
8. Part 1 A New Number Format: The Unum
1. Chapter 1 Overview
2. Chapter 2 Building up to the unum format
3. Chapter 3 The “original sin” of computer arithmetic
4. Chapter 4 The complete unum format
5. Chapter 5 Hidden scratchpads and the three layers
2. 5.2 The unum layer
3. 5.3 The math layer
4. 5.4 The human layer
5. 5.5 Moving between layers
6. 5.6 Summary of conversions between layers in the prototype
7. 5.7 Are floats “good enough for government work”?
6. Chapter 6 Information per bit
1. 6.1 Information as the reciprocal of uncertainty
2. 6.2 “Unifying” a bound to a single unum
3. 6.3 Unification in the prototype
4. 6.4 Can ubounds save storage compared with traditional floats?
7. Chapter 7 Fixed-size unum storage
1. 7.1 The Warlpiri unums
2. 7.2 The Warlpiri ubounds
3. 7.3 Hardware for unums: Faster than float hardware?
8. Chapter 8 Comparison operations
1. 8.1 Less than, greater than
2. 8.2 Equal, nowhere equal, and “not nowhere equal”
3. 8.3 Intersection
9. Chapter 9 Add/subtract, and the unbiased rounding myth
1. 9.1 Re-learning the addition table … for all real numbers
2. 9.2 “Creeping crud” and the myth of unbiased rounding
3. 9.3 Automatic accuracy control and a simple unum math test
10. Chapter 10 Multiplication and division
1. 10.1 Multiplication requires examining each quadrant
2. 10.2 Hardware for unum multiplication
3. 10.3 Division introduces asymmetry in the arguments
11. Chapter 11 Powers
1. 11.1 Square
2. 11.2 Square root
3. 11.3 Nested square roots and “ULP straddling”
4. 11.4 Taxing the scratchpad: Integers to integer powers
5. 11.5 A practice calculation of xy at low precision
6. 11.6 Practical considerations and the actual working routine
7. 11.7 Exp(x) and &#8220;The Table-Maker&#8217;s Dilemma&#8221;) and “The Table-Maker’s Dilemma”
12. Chapter 12 Other important unary operations
13. Chapter 13 Fused operations (single-use expressions)
1. 13.1 Standardizing a set of fused operations
2. 13.2 Fused multiply-add and fused multiply-subtract
3. 13.3 Solving the paradox of slow arithmetic for complex numbers
4. 13.4 Unum hardware for the complete accumulator
5. 13.5 Other fused operations
14. Chapter 14 Trial runs: Unums face challenge calculations
15. Part 1 Summary
9. Part 2 A New Way to Solve: The Ubox
1. Chapter 15 The other kind of error
2. Chapter 16 Avoiding interval arithmetic pitfalls
3. Chapter 17 What does it mean to “solve” an equation?
1. 17.1 Another break from traditional numerical methods
2. 17.2 A linear equation in one unknown, solved by inversion
3. 17.3 “Try everything!” Exhaustive search of the number line
4. 17.4 The universal equation solver
5. 17.5 Solvers in more than one dimension
6. 17.6 Summary of the ubox solver approach
4. Chapter 18 Permission to guess
5. Chapter 19 Pendulums done correctly
6. Chapter 20 The two-body problem (and beyond)
1. 20.1 A differential equation with multiple dimensions
2. 20.2 Ubox approach: The initial space step
3. 20.3 The next starting point, and some state law enforcement
4. 20.4 The general space step
5. 20.5 The three-body problem
6. 20.6 The n-body problem and the galaxy colliders
7. Chapter 21 Calculus considered evil: Discrete physics
8. Chapter 22 The end of error
10. Glossary
11. Appendix A: Glossary of unum functions
12. Appendix B: Glossary of ubox functions
13. Appendix C: Algorithm listings for Part 1
14. Appendix D: Algorithm listings for Part 2