The end of error : unum computing /: unum computing. (2015)

• Record Type: Book
• Further Information: Note: John L. Gustafson.
• Authors: Gustafson, John L
• Contents: A New Number Format: The Unum; Overview; 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"; Intersection 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 Powers; Square; 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 Glossary 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 Index
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2015
• Extent: 1 online resource, illustrations (colour)
• Subjects: 004.01513; Computer arithmetic; Errors
• Languages: English
• ISBNs: 9781482239874
• Related ISBNs: 9781482239867
• Notes: Note: Description based on CIP data; item not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.