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.
Table of contents
- Cover Page
- Half Title Page
- Title Page
- Copyright Page
- Contents
- Preface
- Acknowledgments
-
Part 1 A New Number Format: The Unum
- Chapter 1 Overview
- Chapter 2 Building up to the unum format
- Chapter 3 The “original sin” of computer arithmetic
-
Chapter 4 The complete unum format
- 4.1 Overcoming the tyranny of fixed storage size
- 4.2 The IEEE Standard float formats
- 4.3 Unum format: Flexible range and precision
- 4.4 How can appending extra bits save storage?
- 4.5 Ludicrous precision? The vast range of unums
- 4.6 Changing environment settings within a computing task
- 4.7 The reference prototype
- 4.8 Special values in a flexible precision environment
- 4.9 Converting exact unums to real numbers
- 4.10 A complete exact unum set for a small utag
- 4.11 Inexact unums
- 4.12 A visualizer for unum strings
- Chapter 5 Hidden scratchpads and the three layers
- Chapter 6 Information per bit
- Chapter 7 Fixed-size unum storage
- Chapter 8 Comparison operations
- Chapter 9 Add/subtract, and the unbiased rounding myth
- Chapter 10 Multiplication and division
- Chapter 11 Powers
- Chapter 12 Other important unary operations
- Chapter 13 Fused operations (single-use expressions)
- Chapter 14 Trial runs: Unums face challenge calculations
- Part 1 Summary
-
Part 2 A New Way to Solve: The Ubox
-
Chapter 15 The other kind of error
- 15.1 Sampling error
- 15.2 The deeply unsatisfying nature of classical error bounds
- 15.3 The ubox approach
- 15.4 Walking the line
- 15.5 A ubox connected-region example: Computing the unit circle area
- 15.6 A definition of answer quality and computing “speed”
- 15.7 Another Kahan booby trap: The “smooth surprise”
- Chapter 16 Avoiding interval arithmetic pitfalls
- Chapter 17 What does it mean to “solve” an equation?
- Chapter 18 Permission to guess
- Chapter 19 Pendulums done correctly
- Chapter 20 The two-body problem (and beyond)
- Chapter 21 Calculus considered evil: Discrete physics
- Chapter 22 The end of error
-
Chapter 15 The other kind of error
- Glossary
-
Appendix A: Glossary of unum functions
- A.1 Environment and bit-extraction functions
- A.2 Constructors
- A.3 Visualization functions
- A.4 Conversion functions
- A.5 Argument validity tests
- A.6 Helper functions for functions defined above
- A.7 Comparison operations
- A.8 Arithmetic operations
- A.9 Fused operations (single-use expressions)
- A.10 Some defined data values
- A.11 Auto-precision functions
- Appendix B: Glossary of ubox functions
-
Appendix C: Algorithm listings for Part 1
- C.1 The set-the-environment function
- C.2 Type-checking functions
- C.3 The unum-to-float converter and supporting functions
- C.4 The u-layer to general interval conversions
- C.5 The real-to-unum or x" conversion function
- C.6 Unification functions and supporting functions
- C.7 The general interval to unum converter
- C.8 Comparison tests and ubound intersection
- C.9 Addition and subtraction functions
- C.10 Multiplication functions
- C.11 Division routines
- C.12 Automatic precision adjustment functions
- C.13 Fused operations (single-use expressions)
- C.14 Square and square root
- C.15 The power function xy and exp(x)
- C.16 Absolute value, logarithm, and trigonometry functions
- C.17 The unum Fast Fourier Transform
- Appendix D: Algorithm listings for Part 2
- For Further Reading
- Index
Product information
- Title: The End of Error
- Author(s):
- Release date: June 2017
- Publisher(s): Chapman and Hall/CRC
- ISBN: 9781482239874
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