Contents
How to read this book
Part 1
A New Number Format: The Unum
1.1 Fewer bits. Better answers
1.2 Why better arithmetic can save energy and power
Chapter 2 Building up to the unum format
2.1 A graphical view of bit strings: Value and closure plots
2.4 Floating point format, almost
2.5 What about infinity and NaN? Improving on IEEE rules
Chapter 3 The “original sin” of computer arithmetic
3.1 The acceptance of incorrect answers
3.2 “Almost infinite” and “beyond infinity”
3.3 No overflow, no underflow, and no rounding
3.4 Visualizing ubit-enabled numbers
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
4.8 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
4.12 A visualizer for unum strings
Chapter 5 Hidden scratchpads and the three layers
5.1.1 The Cray-1 supercomputer
5.1.2 The original definition of the C language
5.1.3 Coprocessors also try to be helpful
5.1.4 IBM almost gets it right: The fused multiply-add
5.1.5 Kulisch gets it right: The exact dot product (EDP)
5.2.2 Processor design for the u-layer
5.3.1 The general bound, or “gbound”
5.3.2 Who uses NaN? Sun Microsystems gets a nasty surprise
5.3.3 An explosion of storage demand in the scratchpad?
5.3.4 Traditional interval arithmetic. Watch out.
5.3.6 Why endpoints must be marked open or closed
5.4.1 Numbers as people perceive them
5.5.1 Defining the conversion function
5.5.2 Testing the conversion function
5.6 Summary of conversions between layers in the prototype
5.7 Are floats “good enough for government work”?
6.1 Information as the reciprocal of uncertainty
6.2 “Unifying” a bound to a single unum
6.3 Unification in the prototype
6.3.1 The option of lossy compression
6.3.3 Much ado about almost nothing and almost infinite
6.4 Can ubounds save storage compared with traditional floats?
Chapter 7 Fixed-size unum storage
7.3 Hardware for unums: Faster than float hardware?
7.3.1 General comments about the cost of handling exceptions
7.3.2 Unpacked unum format and the “summary bits” idea
Chapter 8 Comparison operations
8.1.1 Conceptual definition of ordering for general intervals
8.1.2 Hardware design for “less than” and “greater than” tests
8.2 Equal, nowhere equal, and “not nowhere equal”
8.2.1 Conceptual definition of “equal” for general intervals
8.2.2 Hardware approach for equal and “not nowhere equal”
Chapter 9 Add/subtract, and the unbiased rounding myth
9.1 Re-learning the addition table … for all real numbers
9.1.1 Examples and tests of data motion
9.1.2 Three-dimensional visualization of ubound addition
9.1.3 Hardware design for unum addition and subtraction
9.2 “Creeping crud” and the myth of unbiased rounding
9.3 Automatic accuracy control and a simple unum math test
Chapter 10 Multiplication and division
10.1 Multiplication requires examining each quadrant
10.2 Hardware for unum multiplication
10.2.1 Multiplies that fit the standard hardware approach
10.2.2 Extended precision and the “complete accumulator”
10.3 Division introduces asymmetry in the arguments
10.3.1 The “bad boy” of arithmetic
10.3.2 Hardware for unum division
11.3 Nested square roots and “ULP straddling”
11.4 Taxing the scratchpad: Integers to integer powers
11.5 A practice calculation of xy at low precision
11.6 Practical considerations and the actual working routine
11.6.1 Why the power function can be fast
11.6.2 The prototype power function
11.6.3 A challenge to the defenders of floats
11.7 Exp(x) and “The Table-Maker’s Dilemma”
11.7.1 Another dilemma caused by the dishonesty of rounding
11.7.2 The prototype exponential function
Chapter 12 Other important unary operations
12.3 Natural logarithm, and a mention of log base 2
12.4 Trig functions: Ending the madness by degrees
Chapter 13 Fused operations (single-use expressions)
13.1 Standardizing a set of fused operations
13.2 Fused multiply-add and fused multiply-subtract
13.3 Solving the paradox of slow arithmetic for complex numbers
13.4 Unum hardware for the complete accumulator
13.4.1 Cost within the unpacked unum environment
13.4.2 Fused dot product and fused sums in the prototype
13.4.3 Consistent results for parallel computing
13.5.1 Not every pairing of operations should be fused
13.5.5 Fused norm, fused root dot product, and fused mean
Chapter 14 Trial runs: Unums face challenge calculations
14.1 Floating point II: The wrath of Kahan
14.4 Bailey’s numerical nightmare
14.5 Fast Fourier Transforms using unums
Part 2
A New Way to Solve: The Ubox
Chapter 15 The other kind of error
15.2 The deeply unsatisfying nature of classical error bounds
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
16.4 Intelligent standard library routines
16.5 Polynomial evaluation without the dependency problem
16.6 Other fused multiple-use expressions
Chapter 17 What does it mean to “solve” an equation?
17.1 Another break from traditional numerical methods
17.2 A linear equation in one unknown, solved by inversion
17.2.1 Inversion with unums and intervals
17.2.2 Ubounds as coefficients
17.3 “Try everything!” Exhaustive search of the number line
17.3.1 The quadratic formula revisited
17.3.2 To boldly split infinities: Try everything
17.3.3 Highly adjustable parallelism
17.4 The universal equation solver
17.4.1 Methods that USUALLY work
17.4.2 The general failure of the inversion method
17.4.3 Inequality testing; finding extrema
17.4.4 The “intractable” fallacy
17.5 Solvers in more than one dimension
17.6 Summary of the ubox solver approach
Chapter 18 Permission to guess
18.1 Algorithms that work for floats also work for unums
18.3 Large systems of linear equations
Chapter 19 Pendulums done correctly
19.1 The introductory physics approach
19.2 The usual numerical approach
19.3 Space stepping: A new source of massive parallelism
19.4 It’s not just for pendulums
Chapter 20 The two-body problem (and beyond)
20.1 A differential equation with multiple dimensions
20.1.1 Setting up a traditional simulation
20.1.2 Trying a traditional method out on a single orbit
20.1.3 What about interval arithmetic methods?
20.2 Ubox approach: The initial space step
20.2.1 Solving the chicken-and-egg puzzle
20.2.2 First, bound the position
20.3 The next starting point, and some state law enforcement
20.6 The n-body problem and the galaxy colliders
Chapter 21 Calculus considered evil: Discrete physics
21.1 Continuum versus discrete physics
21.2 The discrete version of a vibrating string
Appendix A: Glossary of unum functions
A.1 Environment and bit-extraction functions
A.6 Helper functions for functions defined above
A.9 Fused operations (single-use expressions)
Appendix B: Glossary of ubox functions
B.2 Neighbor-finding functions
B.3 Ubox creation by splitting
B.5 Ubox bounds, width, volume
B.7 Fused polynomial evaluation and acceleration formula
Appendix C: Algorithm listings for Part 1
C.1 The set-the-environment function
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.12 Automatic precision adjustment functions
C.13 Fused operations (single-use expressions)
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
D.1 ULP manipulation functions
D.2 Neighbor-finding functions
D.3 Ubox creation by splitting
D.5 Ubox bounds, width, volumes