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E-raamat: The End of Error: Unum Computing

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The Future of Numerical Computing

Written by one of the foremost experts in high-performance computing and the inventor of Gustafsons 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.

Arvustused

"The author of the present book believes that it is time to supplement the century-old floating point arithmetic with something better: unum arithmetic. The book covers various operations with unum arithmetic and topics like polynomial evaluation, solving equations, two-body problem, etc. The appendices give a glossary of unum functions, ubox functions, and some algorithm listings." Zentralblatt MATH 1320

"This book is an extraordinary reinvention of computer arithmetic and elementary numerical methods from the ground up. Unum arithmetic is an extension of floating point in which it is also possible to represent the open intervals between two floating point numbers. This leads to arithmetic that is algebraically much cleaner, without rounding error, overflow underflow, or negative zero, and with clean and consistent treatment of positive and negative infinity and NaN. These changes are not just marginal technical improvements. As the book fully demonstrates, they lead to what can only be described as a radical re-foundation of elementary numerical analysis, with new methods that are free of rounding error, fully parallelizable, fully portable, easier for programmers to master, and often more economical of memory, bandwidth, and power than comparable floating point methods. The book is exceptionally well written and produced and is illustrated on every page with full-color diagrams that perfectly communicate the material. Anyone interested in computer arithmetic or numerical methods must read this book. It is surely destined to be a classic." David Jefferson, Center for Advanced Scientific Computing, Lawrence Livermore National Laboratory

"John Gustafsons book The End of Error presents the ideas of computer arithmetic in a very easy-to-read and understandable form. While the title is provocative, the content provides an illuminating discussion of the issues. The examples are engaging, well thought out, and simple to follow." Jack Dongarra, University Distinguished Professor, University of Tennessee

"John Gustafson presents a bold and brilliant proposal for a revolutionary number representation system, unum, for scientific (and potentially all other) computers. Unums main advantage is that computing with these numbers gives scientists the correct answer all the time. Gustafson is able to show that the universal number, or unum, encompasses all standard floating-point formats as well as fixed-point and exact integer arithmetic. The book is a call to action for the next stage: implementation and testing that would lead to wide-scale adoption." Gordon Bell, Researcher Emeritus, Microsoft Research

"Reading more and more in [ John Gustafsons] book became a big surprise. I had not expected such an elaborate and sound piece of work. It is hard to believe that a single person could develop so many nice ideas and put them together into a sketch of what perhaps might be the future of computing. Reading [ this] book is fascinating." Ulrich Kulisch, Karlsruhe Institute of Technology, Germany

Preface xiii
Acknowledgments xvii
How to read this book xix
Part 1 A New Number Format: The Unum
Chapter 1 Overview
1(6)
1.1 Fewer bits. Better answers
2(3)
1.2 Why better arithmetic can save energy and power
5(2)
Chapter 2 Building up to the unum format
7(18)
2.1 A graphical view of bit strings: Value and closure plots
8(2)
2.2 Negative numbers
10(4)
2.3 Fixed point format
14(3)
2.4 Floating point format, almost
17(5)
2.5 What about infinity and NaN? Improving on IEEE rules
22(3)
Chapter 3 The "original sin" of computer arithmetic
25(10)
3.1 The acceptance of incorrect answers
25(3)
3.2 "Almost infinite" and "beyond infinity"
28(2)
3.3 No overflow, no undertow, and no rounding
30(3)
3.4 Visualizing ubit-enabled numbers
33(2)
Chapter 4 The complete unum format
35(20)
4.1 Overcoming the tyranny of fixed storage size
35(1)
4.2 The IEEE Standard float formats
36(2)
4.3 Unum format: Flexible range and precision
38(2)
4.4 How can appending extra bits save storage?
40(1)
4.5 Ludicrous precision? The vast range of unums
41(1)
4.6 Changing environment settings within a computing task
42(1)
4.7 The reference prototype
43(4)
4.8 Special values in a flexible precision environment
47(1)
4.9 Converting exact unums to real numbers
48(2)
4.10 A complete exact unum set for a small utag
50(1)
4.11 Inexact unums
51(2)
4.12 A visualizer for unum strings
53(2)
Chapter 5 Hidden scratchpads and the three layers
55(30)
5.1 The hidden scratchpad
56(6)
5.1.1 The Cray-1 supercomputer
57(1)
5.1.2 The original definition of the C language
58(1)
5.1.3 Coprocessors also try to be helpful
59(1)
5.1.4 IBM almost gets it right: The fused multiply-add
60(1)
5.1.5 Kulisch gets it right: The exact dot product (EDP)
61(1)
5.2 The unum layer
62(3)
5.2.1 The ubound
62(2)
5.2.2 Processor design for the u-layer
64(1)
5.3 The math layer
65(9)
5.3.1 The general bound, or "gbound"
65(2)
5.3.2 Who uses NaN? Sun Microsystems gets a nasty surprise
67(1)
5.3.3 An explosion of storage demand in the scratchpad?
68(1)
5.3.4 Traditional interval arithmetic. Watch out.
69(3)
5.3.5 Significance arithmetic
72(1)
5.3.6 Why endpoints must be marked open or closed
72(2)
5.4 The human layer
74(4)
5.4.1 Numbers as people perceive them
74(1)
5.4.2 Demands and concessions
75(3)
5.5 Moving between layers
78(4)
5.5.1 Defining the conversion function
78(3)
5.5.2 Testing the conversion function
81(1)
5.6 Summary of conversions between layers in the prototype
82(1)
5.7 Are floats "good enough for government work"?
83(2)
Chapter 6 Information per bit
85(8)
6.1 Information as the reciprocal of uncertainty
85(1)
6.2 "Unifying" a bound to a single ULP
86(1)
6.3 Unification in the prototype
87(4)
6.3.1 The option of lossy compression
87(1)
6.3.2 Intelligent unification
88(2)
6.3.3 Much ado about almost nothing and almost infinite
90(1)
6.4 Can ubounds save storage compared with traditional floats?
91(2)
Chapter 7 Fixed-size unum storage
93(10)
7.1 The Warlpiri unums
93(2)
7.2 The Warlpiri ubounds
95(3)
7.3 Hardware for unums: Faster than float hardware?
98(5)
7.3.1 General comments about the cost of handling exceptions
98(1)
7.3.2 Unpacked unum format and the "summary bits" idea
99(4)
Chapter 8 Comparison operations
103(8)
8.1 Less than, greater than
103(3)
8.1.1 Conceptual definition of ordering for general intervals
103(2)
8.1.2 Hardware design for "less than" and "greater than" tests
105(1)
8.2 Equal, nowhere equal, and "not nowhere equal"
106(3)
8.2.1 Conceptual definition of "equal" for general intervals
106(3)
8.2.2 Hardware approach for equal and "not nowhere equal"
109(1)
8.3 Intersection
109(2)
Chapter 9 Add/subtract, and the unbiased rounding myth
111(16)
9.1 Re-learning the addition table... for all real numbers
111(6)
9.1.1 Examples and tests of data motion
114(1)
9.1.2 Three-dimensional visualization of ubound addition
115(1)
9.1.3 Hardware design for unum addition and subtraction
116(1)
9.2 "Creeping crud" and the myth of unbiased rounding
117(4)
9.3 Automatic accuracy control and a simple unum math test
121(6)
Chapter 10 Multiplication and division
127(14)
10.1 Multiplication requires examining each quadrant
128(4)
10.2 Hardware for unum multiplication
132(4)
10.2.1 Multiplies that fit the standard hardware approach
132(3)
10.2.2 Extended precision and the "complete accumulator"
135(1)
10.2.3 Really big multiplies
135(1)
10.3 Division introduces asymmetry in the arguments
136(5)
10.3.1 The "bad boy" of arithmetic
136(3)
10.3.2 Hardware for unum division
139(2)
Chapter 11 Powers
141(14)
11.1 Square
141(2)
11.2 Square root
143(2)
11.3 Nested square roots and "ULP straddling"
145(1)
11.4 Taxing the scratchpad: Integers to integer powers
146(1)
11.5 A practice calculation of xy at low precision
147(2)
11.6 Practical considerations and the actual working routine
149(3)
11.6.1 Why the power function can be fast
149(1)
11.6.2 The prototype power function
150(1)
11.6.3 A challenge to the defenders of floats
151(1)
11.7 Exp(x) and "The Table-Maker's Dilemma"
152(3)
11.7.1 Another dilemma caused by the dishonesty of rounding
152(1)
11.7.2 The prototype exponential function
153(2)
Chapter 12 Other important unary operations
155(4)
12.1 Scope of the prototype
155(1)
12.2 Absolute value
155(1)
12.3 Natural logarithm, and a mention of log base 2
156(1)
12.4 Trig functions: Ending the madness by degrees
156(3)
Chapter 13 Fused operations (single-use expressions)
159(14)
13.1 Standardizing a set of fused operations
159(1)
13.2 Fused multiply-add and fused multiply-subtract
160(1)
13.3 Solving the paradox of slow arithmetic for complex numbers
161(1)
13.4 Unum hardware for the complete accumulator
162(5)
13.4.1 Cost within the unpacked unum environment
162(2)
13.4.2 Fused dot product and fused sums in the prototype
164(1)
13.4.3 Consistent results for parallel computing
165(2)
13.5 Other fused operations
167(6)
13.5.1 Not every pairing of operations should be fused
167(1)
13.5.2 Fused product
167(2)
13.5.3 Fused add-multiply
169(1)
13.5.4 Fused product ratio
169(2)
13.5.5 Fused norm, fused root dot product, and fused mean
171(2)
Chapter 14 Trial runs: Unums face challenge calculations
173(20)
14.1 Floating point II: The wrath of Kahan
173(6)
14.2 Rump's royal pain
179(2)
14.3 The quadratic formula
181(3)
14.4 Bailey's numerical nightmare
184(4)
14.5 Fast Fourier Transforms using unums
188(5)
Summary
193(2)
Part 2 A New Way to Solve: The Ubox
Chapter 15 The other kind of error
195(20)
15.1 Sampling error
195(2)
15.2 The deeply unsatisfying nature of classical error bounds
197(2)
15.3 The ubox approach
199(2)
15.4 Walking the line
201(2)
15.5 A ubox connected-region example: Computing the unit circle area
203(7)
15.6 A definition of answer quality and computing "speed"
210(2)
15.7 Another Kahan booby trap: The "smooth surprise"
212(3)
Chapter 16 Avoiding interval arithmetic pitfalls
215(18)
16.1 Useless error bounds
215(1)
16.2 The wrapping problem
215(4)
16.3 The dependency problem
219(2)
16.4 Intelligent standard library routines
221(1)
16.5 Polynomial evaluation without the dependency problem
222(9)
16.6 Other fused multiple-use expressions
231(2)
Chapter 17 What does it mean to "solve" an equation?
233(24)
17.1 Another break from traditional numerical methods
233(1)
17.2 A linear equation in one unknown, solved by inversion
234(7)
17.2.1 Inversion with unums and intervals
236(1)
17.2.2 Ubounds as coefficients
237(4)
17.3 "Try everything!" Exhaustive search of the number line
241(4)
17.3.1 The quadratic formula revisited
241(1)
17.3.2 To boldly split infinities: Try everything
242(3)
17.3.3 Highly adjustable parallelism
245(1)
17.4 The universal equation solver
245(6)
17.4.1 Methods that USUALLY work
245(2)
17.4.2 The general failure of the inversion method
247(3)
17.4.3 Inequality testing; finding extrema
250(1)
17.4.4 The "intractable" fallacy
251(1)
17.5 Solvers in more than one dimension
251(5)
17.6 Summary of the ubox solver approach
256(1)
Chapter 18 Permission to guess
257(16)
18.1 Algorithms that work for floats also work for unums
257(4)
18.2 A fixed-point problem
261(9)
18.3 Large systems of linear equations
270(2)
18.4 The last resort
272(1)
Chapter 19 Pendulums done correctly
273(14)
19.1 The introductory physics approach
273(1)
19.2 The usual numerical approach
274(2)
19.3 Space stepping: A new source of massive parallelism
276(6)
19.4 It's not just for pendulums
282(5)
Chapter 20 The two-body problem (and beyond)
287(24)
20.1 A differential equation with multiple dimensions
287(7)
20.1.1 Setting up a traditional simulation
288(4)
20.1.2 Trying a traditional method out on a single orbit
292(2)
20.1.3 What about interval arithmetic methods?
294(1)
20.2 Ubox approach: The initial space step
294(8)
20.2.1 Solving the chicken-and-egg puzzle
294(2)
20.2.2 First, bound the position
296(6)
20.3 The next starting point, and some state law enforcement
302(3)
20.4 The general space step
305(3)
20.5 The three-body problem
308(2)
20.6 The n-body problem and the galaxy colliders
310(1)
Chapter 21 Calculus considered evil: Discrete physics
311(16)
21.1 Continuum versus discrete physics
311(2)
21.2 The discrete version of a vibrating string
313(3)
21.3 The single-atom gas
316(5)
21.4 Structural analysis
321(6)
Chapter 22 The end of error
327(6)
Glossary 333(4)
Appendix A: Glossary of unum functions 337(6)
A.1 Environment and bit-extraction functions
337(1)
A.2 Constructors
337(1)
A.3 Visualization functions
337(1)
A.4 Conversion functions
338(1)
A.5 Argument validity tests
338(1)
A.6 Helper functions for functions defined above
338(1)
A.7 Comparison operations
339(1)
A.8 Arithmetic operations
339(1)
A.9 Fused operations (single-use expressions)
340(1)
A.10 Some defined data values
341(1)
A.11 Auto-precision functions
342(1)
Appendix B: Glossary of ubox functions 343(4)
B.1 ULP-related manipulations
343(1)
B.2 Neighbor-finding functions
343(1)
B.3 Ubox creation by splitting
343(1)
B.4 Coalescing ubox sets
344(1)
B.5 Ubox bounds, width, volume
344(1)
B.6 Test operations on sets
344(1)
B.7 Fused polynomial evaluation and acceleration formula
345(1)
B.8 The try-everything solver
345(1)
B.9 The guessing function
345(2)
Appendix C: Algorithm listings for Part 1 347(36)
C.1 The set-the-environment function
348(1)
C.2 Type-checking functions
349(2)
C.3 The unum-to-float converter and supporting functions
351(1)
C.4 The u-layer to general interval conversions
352(1)
C.5 The real-to-unum or x conversion function
353(3)
C.6 Unification functions and supporting functions
356(2)
C.7 The general interval to unum converter
358(1)
C.8 Comparison tests and ubound intersection
359(3)
C.9 Addition and subtraction functions
362(1)
C.10 Multiplication functions
363(2)
C.11 Division routines
365(2)
C.12 Automatic precision adjustment functions
367(1)
C.13 Fused operations (single-use expressions)
367(5)
C.14 Square and square root
372(1)
C.15 The power function xy and exp(x)
373(5)
C.16 Absolute value, logarithm, and trigonometry functions
378(4)
C.17 The unum Fast Fourier Transform
382(1)
Appendix D: Algorithm listings for Part 2 383(18)
D.1 ULP manipulation functions
383(2)
D.2 Neighbor-finding functions
385(2)
D.3 Ubox creation by splitting
387(2)
D.4 Coalescing ubox sets
389(3)
D.5 Ubox bounds, width, volumes
392(1)
D.6 Test operations on sets
393(1)
D.7 Fused polynomial evaluation and acceleration formula
394(4)
D.8 The try-everything solver
398(1)
D.9 The guess function
399(2)
For Further Reading 401(2)
Index 403
Dr. John L. Gustafson is an applied physicist and mathematician. He is a former Director at Intel Labs and former Chief Product Architect at AMD. A pioneer in high-performance computing, he introduced cluster computing in 1985 and first demonstrated scalable massively parallel performance on real applications in 1988. This became known as Gustafsons Law, for which he won the inaugural ACM Gordon Bell Prize. He is also a recipient of the IEEE Computer Societys Golden Core Award. Find more details on his website.
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