ISBN: 3540665722
TITLE: Pi - Unleashed
AUTHOR: Arndt, Jrg; Haenel, Christoph
TOC:

1. The State Of Pi Art 1
2. How Random Is pi? 21
2.1 Probabilities 21
2.2 Is pi normal? 21
2.3 So is pi not normal? 24
2.4 The 163 phenomenon 25
2.5 Other statistical results 28
2.6 The Intuitionists and pi 30
2.7 Representation of continued fractions 32
3. Shortcuts To pi 35
3.1 Obscurer approaches to pi 35
3.2 Small is beautiful 37
3.3 Squeezing pi through a sieve 38
3.4 pi and chance (Monte Carlo methods) 39
3.5 Memorabilia 44
3.6 Bit for bit 47
3.7 Refinements 49
3.8 The pi room in Paris 50
4. Approximations For pi And Continued Fractions 51
4.1 Rational approximations 51
4.2 Other approximations 55
4.3 Youthful approximations 63
4.4 On continued fractions 64
5. Arcus Tangens 69
5.1 John Machin's arctan formula 69
5.2 Other arctan formulae 72
6. Spigot Algorithms 77
6.1 The spigot algorithm in detail 78
6.2 Sequence of operations 80
6.3 A faster variant 82
6.4 Spigot algorithm for e 84
7. Gauss And pi 87
7.1 The pi AGM formula 87
7.2 The Gauss AGM algorithm 90
7.3 Schnhage variant 92
7.4 History of a formula 94
8. Ramanujan And 103
8.1 Ramanujan's series 103
8.2 Ramanujan's unusual biography 105
8.3 Impulses 110
9. The Borweins And pi 113
10. The BBP Algorithm 117
10.1 Binary modulo exponentiation 120
10.2 A C program on the BBP series 123
10.3 Refinements 126
11. Arithmetic 131
11.1 Multiplication 131
11.2 Karatsuba multiplication 132
11.3 FFT multiplication 135
11.4 Division 145
11.5 Square root 146
11.6 nth root 149
11.7 Series calculation 150
12. Miscellaneous 153
12.1 A pi quiz 153
12.2 Let numbers speak 154
12.3 A proof that pi = 2 155
12.4 The big change 155
12.5 Almost but not quite 156
12.6 Why always more? 158
12.7 pi and hyperspheres 158
12.8 Vite  Wallis = Osler 160
12.9 Squaring the circle with holes 162
12.10 An (in)finite funnel 164
13. The History Of pi 165
13.1 Antiquity 167
13.2 Polygons 17 0
13.3 Infinite expressions 185
13.4 High-performance algorithms 198
13.5 The hunt for single pi digits 203
Table: History of pi in the pre-computer era 205
Table: History of pi in the computer era 206
Table: History of digit extraction records 207
14. Historical Notes. 209
14.1 The earliest squaring the circle in history? 209
14.2 A pi law 211
14.3 The Bieberbach story 213
15. The Future: pi Calculations On The Internet 215
15.1 The binsplit algorithm 215
15.2 The pi project on the Internet 219
16. pi Formula Collection223
17. Tables 239
17.1 Selected constants to 100 places (base 10) 239
17.2 Digits 0 to 2,500 of pi (base 10) 240
17.3 Digits 2,501 to 5,000 of pi (base 10) 241
17.4 Digits 0 to 2,500 of pi (base 16) 242
17.5 Digits 2,501 to 5,000 of pi (base 16) 243
17.6 Continued fraction elements 0 to 1,000 of pi 244
17.7 Continued fraction elements 1,001 to 2,000 of pi 245
A. Documentation For The hfloat Library 247
A.1 What hfloat is (good for) 247
A.2 Compiling the library 248
A.3 Functions of the hfloat library 248
A.4 Using hfloats in your own code 250
A.5 Computations with extreme precision 250
A.6 Precision and radix 251
A.7 Compiling & running the pi-example-code 253
A.8 Structure of hfloat 253
A.9 Organisation of the files 254
A.10 Distribution policy & no warranty 255
Bibliography 257
Index 265
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