The end of error: unum computing

Machine generated contents note: ch. 1 Overview --
1.1. Fewer bits. Better answers --
1.2. Why better arithmetic can save energy and power --
ch. 2 Building up to the unum format --
2.1. A graphical view of bit strings: Value and closure plots --
2.2. Negative numbers --
2.3. Fixed point format --
2.4. Floating point format, almost --
2.5. What about infinity and NaN? Improving on IEEE rules --
ch. 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 undertow, and no rounding --
3.4. Visualizing ubit-enabled numbers --
ch. 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 --
ch. 5 Hidden scratchpads and the three layers --
5.1. The hidden scratchpad --
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. The unum layer --
5.2.1. The ubound --
5.2.2. Processor design for the u-layer --
5.3. The math 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. --
5.3.5. Significance arithmetic --
5.3.6. Why endpoints must be marked open or closed --
5.4. The human layer --
5.4.1. Numbers as people perceive them --
5.4.2. Demands and concessions --
5.5. Moving between layers --
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"? --
ch. 6 Information per bit --
6.1. Information as the reciprocal of uncertainty --
6.2. "Unifying" a bound to a single ULP --
6.3. Unification in the prototype --
6.3.1. The option of lossy compression --
6.3.2. Intelligent unification --
6.3.3. Much ado about almost nothing and almost infinite --
6.4. Can ubounds save storage compared with traditional floats? --
ch. 7 Fixed-size unum storage --
7.1. The Warlpiri unums --
7.2. The Warlpiri ubounds --
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 --
ch. 8 Comparison operations --
8.1. Less than, greater than --
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" --
8.3. Intersection --
ch. 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 --
ch. 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" --
10.2.3. Really big multiplies --
10.3. Division introduces asymmetry in the arguments --
10.3.1. The "bad boy" of arithmetic --
10.3.2. Hardware for unum division --
ch. 11 Powers --
11.1. Square --
11.2. Square root --
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 --
ch. 12 Other important unary operations --
12.1. Scope of the prototype --
12.2. Absolute value --
12.3. Natural logarithm, and a mention of log base 2 --
12.4. Trig functions: Ending the madness by degrees --
ch. 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. Other fused operations --
13.5.1. Not every pairing of operations should be fused --
13.5.2. Fused product --
13.5.3. Fused add-multiply --
13.5.4. Fused product ratio --
13.5.5. Fused norm, fused root dot product, and fused mean --
ch. 14 Trial runs: Unums face challenge calculations --
14.1. Floating point II: The wrath of Kahan --
14.2. Rump's royal pain --
14.3. The quadratic formula --
14.4. Bailey's numerical nightmare --
14.5. Fast Fourier Transforms using unums --
Summary --
ch. 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" --
ch. 16 Avoiding interval arithmetic pitfalls --
16.1. Useless error bounds --
16.2. The wrapping problem --
16.3. The dependency problem --
16.4. Intelligent standard library routines --
16.5. Polynomial evaluation without the dependency problem --
16.6. Other fused multiple-use expressions --
ch. 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 --
ch. 18 Permission to guess --
18.1. Algorithms that work for floats also work for unums --
18.2. A fixed-point problem --
18.3. Large systems of linear equations --
18.4. The last resort --
ch. 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 --
ch. 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 --
20.4. The general space step --
20.5. The three-body problem --
20.6. The n-body problem and the galaxy colliders --
ch. 21 Calculus considered evil: Discrete physics --
21.1. Continuum versus discrete physics --
21.2. The discrete version of a vibrating string --
21.3. The single-atom gas --
21.4. Structural analysis --
ch. 22 The end of error --
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 --
B.1. ULP-related manipulations --
B.2. Neighbor-finding functions --
B.3. Ubox creation by splitting --
B.4. Coalescing ubox sets --
B.5. Ubox bounds, width, volume --
B.6. Test operations on sets --
B.7. Fused polynomial evaluation and acceleration formula --
B.8. The try-everything solver --
B.9. The guessing function --
C.1. The set-the-environment function --
C.2. Type-checking functions. Note continued: 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 --
D.1. ULP manipulation functions --
D.2. Neighbor-finding functions --
D.3. Ubox creation by splitting --
D.4. Coalescing ubox sets --
D.5. Ubox bounds, width, volumes --
D.6. Test operations on sets --
D.7. Fused polynomial evaluation and acceleration formula --
D.8. The try-everything solver --
D.9. The guess function.