2Mul, 135,318 2MultFMA, 152 2Sum, 129, 318
accumulator, 314
accurate step, 396
ACL2,471
addition, 246
of binary floating-point in hardware, 288
of binary floating-point in software, 329
of integers, 274
of integers in decimal, 275
of signed zeros, 247
subnormal handling, 251, 294
additive range reduction, 378
Al-Khwarizmi, 167
algebraic function, 421,424
algebraic number, 420
a (smallest normal number), 17,153
alternate exception-handling attributes, 93,95
argument reduction, 378
arithmetic formats, 80
ARRE (average relative representation
error), 30
attributes, 93
Babai's algorithm, 528
Babylonians, 4
backward error, 186
bad cases for the TMD, 409
base, 13
basic formats, 56, 80
BCD (binary coded decimal), 83
Benford's law, 29
bias, 57, 82, 84,85
biased exponent, 58-60, 84-86, 245
big-endian, 61
binade, 415
binary 128, 82
binaryl6, 82
binary32,16, 82
binary64, 82
binding, 109
bipartite table method, 287
block floating-point, 313
Booth recoding, 281
breakpoint, 21, 406
Briggs, 375
Burger and Dybvig conversion algo- rithm, 44
C programming language, 209
C++ programming language, 220
C99, 210
cancellation, 124,193
canonical encoding, 83, 85
carry-ripple adder, 274
carry-select adder, 279
carry-skip adder, 276, 277
catastrophic cancellation, 124, 378
CENA, 182
Chebyshev
polynomials, 389
theorem, 391
dinger conversion algorithms, 46
close path, 249
closest vector problem, 525
Cody, 29
Cody and Waite reduction algorithm, 379
cohort, 14, 83, 97, 240
combination field, 83, 84
comparisons, 65, 99
comparison predicates, 65, 99
CompensatedDotProduct algorithm, 202
compensated algorithms, 182
compensated polynomial evaluation, 203
compensated summation, 192
component of an expansion, 503
compound adder, 278, 299
compression of expansions, 508
condition numbers, 187
continued fractions, 382, 521, 522
contracted expressions, 214
convergent (continued fractions), 522
conversion algorithms, 43
Coq, 472
CORDIC algorithm, 375
correctly rounded function, 21, 22
CRlibm, 381
CVP, see closest vector problem
data dependency, 273
DblMult, 178
decimal addition, 275
decimal arithmetic in hardware, 272
decimal division, 309
decimal multiplication, 283
decimal encoding, 85
declet, 83
degree of an algebraic number, 421
Dekker, 126,135
Dekker product, 125,135,473
delay, 273
denormal number, 15
directed rounding modes, 22
division, 262
SRT algorithms, 308
by a constant, 312
by digit recurrence, 263, 306
in decimal, 309
in hardware, 305
division by zero, 25, 67,101
double-double numbers, 403
double-word addition, 497
double-word multiplication, 498
double precision, 56, 57, 61, 64, 65, 71, 82
double rounding, 75, 77,114
DSP (digital signal processing), 297, 316
dynamic range, 30
elementary function, 421
^max/ 14
^min/ 14
end-around carry adder, 279
endianness, 61
Ercegovac, 263
ErrFma, 176
Estrin's method, 394
Euclidean lattice, 446, 524
exactly rounded function, 21
exceptions, 25, 66, 74, 95,100, 475
exclusion lemma, 162, 422, 423
exclusion zone, 162
expansion, 503
Expansion-Sum algorithm, 505
exponent bias, 85
extendable precision, 80, 92
extended formats, 56
extended precision, 71, 72, 80, 92, 94
extremal exponents, 14
faithful arithmetic, 22,131
faithful result, 22,179,311
faithful rounding, 22
far path, 249
Fast-Expansion-Sum algorithm, 506
Fast2Sum, 126,127
<fenv.h>, 210
field-programmable gate array, 271, 279, 287
fixed point, 313, 314, 317,477
FLIP, 321
<float.h>, 210
FMA, 51,104, 254, 472
binary implementation, 259
decimal, 259
hardware implementation, 302
subnormal handling, 258, 305
FORTRAN, 223
FPGA, see field-programmable gate array FPGA specific operators, 309
fraction, 16, 56
full adder, 275
fused multiply-add, see FMA
gamma function, 409 7 notation, 184
Gappa, 474
Gay conversion algorithms, 44, 46
Goldschmidt iteration, 160
GPGPU (general-purpose graphics pro- cessing units), 108
GPU (graphical processing unit), 108,271
graceful underflow, 17
gradual underflow, 17, 53
Grow-Expansion algorithm, 505
Haar condition, 392
hardness to round, 409
Harrison, 119
Heron iteration, 167
hidden bit convention, 16
Higham, 190
HOL Light, 472
Horner algorithm, 185, 394, 473
running error bound, 189
IeeeCC754,116
IEEE 754-1985 standard, 5, 55, 56
IEEE 754-2008 standard, 79
IEEE 854-1987 standard, 6, 70
ILP (instruction-level parallelism), 328
implicit bit convention, 16, 29, 30
inclusion property, 51
inexact exception, 25, 69, 102, 103, 245,
265, 267
infinitary, 25
infinitely precise significand, 15, 21, 407,
422, 423
insertion summation, 191
integer multiplication, 281
integer powers, 177
integral significand, 14, 16, 83, 86, 422, 423
interchange formats, 80
interval arithmetic, 51, 475
INTLAB, 510
invalid operation exception, 20, 25, 67,
69, 82,100
is normal bit, 245, 295, 301, 303, 322
IteratedProductPower, 178
iterated products, 37
Java, 227
A-fold summation algorithm, 196
Kahan, 5, 8,17, 32,126,405
Karatsuba's complex multiplication, 144
Knuth, 129
Kulisch, 316
L2 polynomial approximations, 376
L°° polynomial approximations, 376
Lang, 263
language, 205
large accumulator, 314
largest finite number, 16,17, 67,102
latency, 273
leading bit convention, 16
leading-zero anticipation, 286, 291, 295, 303
leading-zero counter, 284, 289-291, 293, 319
by monotonic string conversion, 285
combined with shifter, 286
tree-based, 285
least squares polynomial approxima- tions, 376
left-associativity, 207
LIA-2, 25
Lindemann's theorem, 429
Liouville's theorem, 424
little-endian, 61
LLL algorithm, 442, 446, 524
logarithmic distribution, 29
logB, 98,101
look-up table, 279, 287, 311
LOP (leading one predictor), see leading
zero anticipation LSB (least significant bit), 290
LUT, see look-up table LZA, see leading zero anticipation LZC, see leading zero counter
MACHAR, 111
machine epsilon, 39
Malcolm, 119
mantissa, 14
Markstein, 169
Mars Climate Orbiter, 8
<math.h>, 210
Matula, 40,41
minimal polynomial, 421
minimax polynomial approximations, 376
minimax rational approximations, 376
modified Booth recoding, 281
Moller, 129
monotonic conversion, 64
MPCHECK, 116
MRRE (maximum relative representa- tion error), 29
MSB (most significant bit), 285
multipartite table method, 287
multiplication, 251
of binary floating-point in hardware, 296
by a constant, 311
by a floating-point constant, 312
by an arbitrary precision constant,
171, 312
digit by integer, 280
in decimal, 283
of integers, 281
subnormal handling, 252, 301
multiplicative range reduction, 378
NaN (Not a Number), 20, 25, 58, 65, 67, 69, 70, 74, 82, 85, 98, 100, 212, 221, 230, 464
Nesterenko, 431
Newton-Raphson iteration, 155, 160,
167, 264, 513
nonadjacent expansion, 504
noncanonical encodings, 84
nonoverlapping expansions, 504
'P-nonoverlapping, 504 5-nonoverlapping, 504
nonoverlapping floating-point numbers, 504
P-nonoverlapping, 504 5-nonoverlapping, 504
normalized representation, 15
normal number, 15,16
normal range, 23
norm (computation of), 26, 310
Q (largest finite FP number), 16
orthogonal polynomials, 389
Oughtred, 4
output radix conversion, 43
overflow, 25, 67,101
in addition, 248
parallel adders, 277
Paranoia, 112,122
partial carry save, 277
payload, 98,100
Payne and Hanek reduction algorithm,
381, 382
pipeline, 273
pole, 25
pow function, 216
precision, 13
preferred exponent, 83,240,249,253,259, 265, 267
preferred width attributes, 93, 95
prefix tree adders, 278
Priest, 503
programming language, 205
PVS, 472
quad-word addition, 502
quad-word renormalization, 500
quadratic convergence, 156
quantum, 14,16, 33
quantum exponent, 14
quick step, 395
quiet NaN, 58, 67, 69, 70, 82, 98,100,101, 212, 221
radix, 13
radix conversion, 40,43, 246
range reduction, 151, 378, 379
RD (round down), see round toward -oo read-only memory, 287
reconfigurable circuits, see field-
programmable gate array RecursiveDotProduct algorithm, 185
RecursiveSum algorithm, 184
relative backward error, 186
relative error, 23, 37
remainder, 63
Remez algorithm, 376,391
reproducibility attributes, 93, 97
RN , see round to nearest ROM, see read-only memory round bit, 21, 243
round digit, 243
rounding, 241
a value with unbounded exponent
range, 241
in decimal with binary encoding, 246
by injection, 298
division, 241
in binary, 243
in decimal, 243
square root, 241, 472
rounding breakpoint, 21,406
rounding direction attributes, 20, 93,94
rounding modes, 20
roundTiesToAway, 95
roundTiesToEven, 94, 95
roundTowardNegative, 94
roundTowardPositive, 94
roundTowardZero, 94
round toward -boo, 20, 52
round toward -oo, 20, 52
round toward zero, 21
round to nearest, 21
round to nearest even, 21, 62
RU (round up), see round toward +oc
Rump, 12
running error bounds, 181
RZ, see round toward zero
Scale-Expansion algorithm, 507
scaleB, 98
SETUN computer, 5
Shewchuk, 126,129, 503
shift-and-add algorithms, 375
shortest vector problem, 525
signaling NaN, 58,67,69, 70,82,100,212,
217, 221
signed infinities, 20
signed zeros, 20
significand, 3, 4,13,14,16
significand alignment, 248
single precision, 16, 56-58, 61, 64, 65, 71, 82
slide rule, 4
SLZ algorithm, 443
smallest normal number, 16,17
smallest subnormal number, 17,153
SoftFloat, 116,321
square root, 265
SRT division, 263, 308
SRTEST, 116
SSE2, see SSE
SSE (Streaming SIMD Extension), 53, 76, 106
standard model of floating-point arith- metic, 183
status flag, 66
Steele and White conversion algorithm,
41,44
Sterbenz's lemma, 122
sticky bit, 21, 243
strongly nonoverlapping expansion, 505
subnormal number, 15-17, 58, 122-124,
128,133,135
subnormal range, 23
subtraction, 246
SVP, see shortest vector problem
6 notation, 184
table-based methods, 287
Table Maker's Dilemma, 179,405-407
tabulated differences, 432
TestFloat, 116
three-distance theorem, 437,438
tie-breaking rule, 21
Torres y Quevedo, Leonardo, 4
trailing significand, 16, 56, 59, 60, 81, 82, 84, 85
transcendental function, 421
transcendental number, 421,429
trap, 19, 66-69, 74
trap handler, 66, 68, 69, 74
Tuckerman test, 169
two-length configurations, 438
TwoMul, 135, 318
TwoMultFMA, 152
TwoSum, 129, 318
UCBTest, 116
ulp (unit in the last place), 14, 32, 37, 43, 169, 382
Goldberg definition, 33
Harrison definition, 32
underflow, 18, 25, 68,102
in addition, 248
unit roundoff, 25, 39,183
unordered, 65
unsigned infinity, 20
value-changing optimization attributes, 93, 96
Veltkamp splitting, 132,133
VLIW, 328
VLSI (very large-scale integration), 270, 271, 287
Waldschmidt, 431
weight function, 376
Weil height, 430
worst cases for the TMD, 409
write-read cycle, 41
YBC 7289, 4
Z3 computer, 4, 20 Ziv, 407
zero Zuse, 4, 20
in the binary interchange formats, 82
in the decimal interchange formats, 85
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