1st Edition

The End of Error Unum Computing

By John L. Gustafson Copyright 2015
    442 Pages 191 Color Illustrations
    by Chapman & Hall

    438 Pages
    by Chapman & Hall

    416 Pages 191 Color Illustrations
    by Chapman & Hall

    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.

    A New Number Format: The Unum
    Fewer bits. Better answers
    Why better arithmetic can save energy and power

    Building up to the unum format
    A graphical view of bit strings: Value and closure plots
    Negative numbers
    Fixed point format
    Floating point format, almost
    What about infinity and NaN? Improving on IEEE rules

    The "original sin" of computer arithmetic
    The acceptance of incorrect answers
    "Almost infinite" and "beyond infinity"
    No overflow, no underflow, and no rounding
    Visualizing ubit-enabled numbers

    The complete unum format
    Overcoming the tyranny of fixed storage size
    The IEEE Standard float formats
    Unum format: Flexible range and precision
    How can appending extra bits save storage?
    Ludicrous precision? The vast range of unums
    Changing environment settings within a computing task
    The reference prototype
    Special values in a flexible precision environment
    Converting exact unums to real numbers
    A complete exact unum set for a small utag
    Inexact unums
    A visualizer for unum strings

    Hidden scratchpads and the three layers
    The hidden scratchpad
    The unum layer
    The math layer
    The human layer
    Moving between layers
    Summary of conversions between layers in the prototype
    Are floats "good enough for government work"?

    Information per bit
    Information as the reciprocal of uncertainty
    "Unifying" a bound to a single ULP
    Unification in the prototype
    Can ubounds save storage compared with traditional floats?

    Fixed-size unum storage
    The Warlpiri unums
    The Warlpiri ubounds
    Hardware for unums: Faster than float hardware?

    Comparison operations
    Less than, greater than
    Equal, nowhere equal, and "not nowhere equal"

    Add/subtract, and the unbiased rounding myth
    Re-learning the addition table … for all real numbers
    "Creeping crud" and the myth of unbiased rounding
    Automatic error control and a simple test of unum math

    Multiplication and division
    Multiplication requires examining each quadrant
    Hardware for unum multiplication
    Division introduces asymmetry in the arguments

    Square root
    Nested square roots and "ULP straddling"
    Taxing the scratchpad: Integers to integer powers
    A practice calculation of xy at low precision
    Practical considerations and the actual working routine
    Exp(x) and "The Table-Maker’s Dilemma"

    Other important unary operations
    Scope of the prototype
    Absolute value
    Natural logarithm, and a mention of log base 2
    Trig functions: Ending the madness by degrees

    Fused operations (single-use expressions)
    Standardizing a set of fused operations
    Fused multiply-add and fused multiply-subtract
    Solving the paradox of slow arithmetic for complex numbers
    Unum hardware for the complete accumulator
    Other fused operations

    Trial runs: Unums face challenge calculations
    Floating point II: The wrath of Kahan
    Rump’s royal pain
    The quadratic formula
    Bailey’s numerical nightmare
    Fast Fourier Transforms using unums

    A New Way to Solve: The Ubox
    The other kind of error
    Sampling error
    The deeply unsatisfying nature of classical error bounds
    The ubox approach
    Walking the line
    A ubox connected-region example: Computing the unit circle area
    A definition of answer quality and computing "speed"
    Another Kahan booby trap: The "smooth surprise"

    Avoiding interval arithmetic pitfalls
    Useless error bounds
    The wrapping problem
    The dependency problem
    Intelligent standard library routines
    Polynomial evaluation without the dependency problem
    Other fused multiple-use expressions

    What does it mean to "solve" an equation?
    Another break from traditional numerical methods
    A linear equation in one unknown, solved by inversion
    "Try everything!" Exhaustive search of the number line
    The universal equation solver
    Solvers in more than one dimension
    Summary of the ubox solver approach

    Permission to guess
    Algorithms that work for floats also work for unums
    A fixed-point problem
    Large systems of linear equations
    The last resort

    Pendulums done correctly
    The introductory physics approach
    The usual numerical approach
    Space stepping: A new source of massive parallelism
    It’s not just for pendulums

    The two-body problem (and beyond)
    A differential equation with multiple dimensions
    Ubox approach: The initial space step
    The next starting point, and some state law enforcement
    The general space step
    The three-body problem
    The n-body problem and the galaxy colliders

    Calculus considered evil: Discrete physics
    Continuum versus discrete physics
    The discrete version of a vibrating string
    The single-atom gas
    Structural analysis

    The end of error


    For further reading

    Appendix A: Glossary of unum functions
    Appendix B: Glossary of ubox functions
    Appendix C: Algorithm listings for Part 1
    Appendix D: Algorithm listings for Part 2



    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 Gustafson’s Law, for which he won the inaugural ACM Gordon Bell Prize. He is also a recipient of the IEEE Computer Society’s Golden Core Award. Find more details on his website.

    "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 Gustafson’s 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. Unum’s 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 Gustafson’s] 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