FLOATING-POINT ARITHMETIC • Floating-point representation

74 Pages·2003·0.2 MB·English

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FLOATING-POINT ARITHMETIC
(cid:15) Floating-point representation and dynamic range
(cid:15) Normalized/unnormalized formats
(cid:15) Values represented and their distribution
(cid:15) Choice of base
(cid:15) Representation of signi(cid:12)cand and of exponent
(cid:15) Rounding modes and error analysis
(cid:15) IEEE Standard 754
(cid:15) Algorithms and implementations: addition/subtraction, multiplication and
division
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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VALUES REPRESENTED IN FLPT SYSTEM
A B C D E
- inf b d + inf
[A, B] - negative floating-point numbers (normalized)
[D,E] - positive floating-point numbers (normalized)
(B,b] & [d,D) - denormals
C - zero
> E - positive overflow
< A - negative overflow
(B, C) - negative underflow (normalized)
(C, D) - positive underflow (normalized)
(a)
significand
0.100
0.101
0.110
0.111
denormals
0 1/8 1/4 1/2 1 2
exponent -2 -1 0 1
(b)
Figure 8.1: a) Regions in (cid:13)oating-point representation. b) Example for m = f = 3, r = 2, and (cid:0)2 (cid:20) E (cid:20) 1 (only positive region).
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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Floating-point system
Normalized Unnormalized
A (cid:0)(rm(cid:0)f (cid:0) r(cid:0)f) (cid:2) bEmax
B (cid:0)rm(cid:0)f(cid:0)1 (cid:2) bEmin (cid:0)r(cid:0)f (cid:2) bEmin
C 0
D rm(cid:0)f(cid:0)1 (cid:2) bEmin r(cid:0)f (cid:2) bEmin
E (rm(cid:0)f (cid:0) r(cid:0)f) (cid:2) bEmax
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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DISTRIBUTION FOR b = 2, m = f = 4, and e = 2
Signi(cid:12)cand 2E
1 2 4 8
0.1000 1/2 1 2 4
0.1001 9/16 9/8 9/4 9/2
0.1010 10/16 10/8 10/4 5
0.1011 11/16 11/8 11/4 11/2
0.1100 12/16 12/8 3 6
0.1101 13/16 13/8 13/4 13/2
0.1110 14/16 14/8 14/4 7
0.1111 15/16 15/8 15/4 15/2
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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DISTRIBUTION FOR b = 2, m = f = 3, and e = 3
Signi(cid:12)cand 2E
1 2 4 8 16 32 64 128
0.100 1/2 1 2 4 8 16 32 64
0.101 5/8 5/4 5/2 5 10 20 40 80
0.110 6/8 3/2 3 6 12 24 48 96
0.111 7/8 7/4 7/2 7 14 28 56 112
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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DISTRIBUTION FOR b = 4, m = f = 4, and e = 2
Signi(cid:12)cand 4E
1 4 16 64
0.0100 1/4 1 4 16
0.0101 5/16 5/4 5 20
0.0110 6/16 6/4 6 24
0.0111 7/16 7/4 7 28
0.1000 1/2 2 8 32
0.1001 9/16 9/4 9 36
0.1010 10/16 10/4 10 40
0.1011 11/16 11/4 11 44
0.1100 12/16 3 12 48
0.1101 13/16 13/4 13 52
0.1110 14/16 14/4 14 56
0.1111 15/16 15/4 15 60
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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DISTRIBUTION OF FLPT NUMBERS
(a) b=2, f=4, e=2
E: 1 2 4 8
0 1/2 1 2 3 4 5 6 7
(b) b=2, f=3, e=3
E: 1 2 4 8 16, 32, 64, 128
0 1/2 1 2 3 4 5 6 7 8 ,10,12,14,16,20,24,28,
32,40,48,56, 64,80,96,112
(c) b=4, f=4, e=2
E: 1 4 16, 64
01/41/2 1 2 3 4 5 6 7 8 , 9, ..., 16, 20, 24, ...,60
Figure 8.2: EXAMPLES OF DISTRIBUTIONS OF FLOATING-POINT NUMBERS.
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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REPRESENTATION OF SIGNIFICAND AND EXPONENT
(cid:15) SIGNIFICAND: SM with HIDDEN BIT
(cid:15) EXPONENT: BIASED E = E + B, minE = 0 ) B = (cid:0)E
R R min
(cid:15) Symmetric range (cid:0)B (cid:20) E (cid:20) B ) 0 (cid:20) E (cid:20) 2B (cid:20) 2e (cid:0) 1
R
(cid:15) for 8-bit exponent: B = 127, (cid:0)127 (cid:20) E (cid:20) 128, 0 (cid:20) E (cid:20) 255
R
(cid:15) E = 255 not used
R
(cid:15) SIMPLIFIES COMPARISON OF FLOATING-POINT NUMBERS (same as in
(cid:12)xed-point)
(cid:15) MINIMUM EXPONENT REPRESENTED BY 0 SO THAT
FLOATING-POINT VALUE 0: ALL ZEROS
(0 sign, 0 exponent, 0 signi(cid:12)cand)
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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SPECIAL VALUES AND EXCEPTIONS
(cid:15) Special values - not representable in the FLPT system
{ NAN (Not A Number)
{ In(cid:12)nity (pos, neg)
{ allow computation in presence of special values
(cid:15) Exceptions: result produced not representable - set a (cid:13)ag
{ Exponent over(cid:13)ow
{ Under(cid:13)ow
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic
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ROUNDOFF MODES AND ERROR ANALYSIS
(cid:15) Exact results (inf. precision): x, y, etc.
(cid:15) FLPT number representing x is R (x) with rounding mode mode
mode
(cid:15) Basic relations:
1. If x (cid:20) y then R (x) (cid:20) R (y)
mode mode
2. If x is a FLPT number then R (x) = x
mode
3. If F1 and F2 are two consecutive FLPT numbers then for F1 (cid:20) x (cid:20) F2
x is either F1 or F2
F1 F2
x
Figure 8.3: Relation between x, Rmode(x), and (cid:13)oating-point numbers F1 and F2.
Digital Arithmetic - Ercegovac/Lang 2003 8 { Floating-Point Arithmetic