32 bit IEEE 754 format Normalized Numbers and the ... - CITR
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32 bit IEEE 754 format Normalized Numbers and the ... - CITR
1. 7. 32 bit IEEE 754 format exponent s significand. 32 bits. 8 bits. 23 bits. • Sign
Bit: – 0 means positive, 1 means negative. Value of a number is: (-1)s x F x 2E.
32 bit IEEE 754 format 8 bits
s exponent
23 bits
significand 32 bits
• Sign Bit: – 0 means positive, 1 means negative
Value of a number is: (-1)s x F x 2E
exponent
significand
7
Normalized Numbers and the significand • Normalized binary numbers always start with a 1 (the leftmost bit of the significand value is a 1). • Why store the 1 (it’s always there)? • IEEE 754 uses this, so the fraction is 24 bits but only 23 need to be stored. • All numbers must be normalized! 8
Exponent Representation • We need negative and positive exponents. • Could use 2s complement notation – this would make comparison of floating point numbers a bit tricky. • exponent value 11111111 is smaller than 00000000.
• Instead they chose a biased (excess-K) representation. – exponent values are offset by a fixed bias.
9
1
32 bit IEEE 754 exponent • The exponent uses 8 bits. • The bias is 127. – treat the 8 bit exponent as a unsigned integer and subtract 127 from it.
00000001 is the representation for –126 10000000 is the representation for +1 11111110 is the representation for +127 10
Special Exponents • 00000000 is a special case exponent – used for the representation of the floating point number 0 (and other things, depending on the sign and significand).
• 11111111 is also a special case – used in the representation of infinity (and other things, depending on the sign and significand). 11
32 bit IEEE 754 Range • Smallest (positive) normalized number is: 1.00000000000000000000000 x 2-126 • Largest normalized number is: 1.11111111111111111111111 x 2127
12
2
Expression for value of 32 bit IEEE 754 (-1)s x (1+significand) x 2(exponent-127)
Sign Bit
23 bit significand as a fraction
8 bit exponent as unsigned int
13
Double Precision s
11 bits
20 bits
exponent
signif
icand
32 bits
14
64 bit IEEE 754 • exponent is 11 bits – bias is 1023 – range is a little larger than the 32 bit format.
• Significand is 55 bits – plus the leading 1. – accuracy is much better than 32 bit format.
15
3
Example Representations 0.7510
½+¼
0.11 x 20
1.1 x 2-1
0 01111110 100000000000000000000000 s exponent
significand
As unsigned int is 126. 126 – 127 = -1
Leading 1 is not stored!
What number is this?
0 10000001 110000000000000000000000 s exponent
significand 16
Exercises • What is the double precision (64 bit format) representation for the number 128? • Same question for single precision
• What is the single precision format for the number –8.125? • Same question for double precision
17
Comparing Numbers s exponent
significand
• Comparison of normalized floating point numbers: – check sign bits – check exponents. • unsigned integer comparison works. • Larger exponents are represented by larger unsigned ints.
