Transcript Floating Point

Floating Point
Representation and Arithmetic
(see Patterson Chapter 4)
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Outline
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Review of floating point scientific notation
Floating point binary
IEEE Floating Point Standard
Addition in Floating Point
Remarks about multiplication
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Floating Point Notation
• Decimal
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12.4568ten (decimal notation) means
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10*1 + 2 + 4/10 + 5/100 + 6/1000 + 8/10000
In scientific notation
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12.4568 =
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124568 * 10-4 = 1245680 * 10-5 =
12456.8 * 10-3 = 1245.68 * 10-2 =
124.568 * 10-1 =12.4568 * 100
1.24568 * 101
1.24568*101 is an example of normalised
scientific notation.
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Floating Point in Binary
• Binary
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0.010011two =
(0/2) + (1/22) + (0/24) +(1/25) + (1/26)
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0 + 1/4 + 0 + 1/32 + 1/64 =
(0.25 + 0.03125 + 0.015625)ten =
0.296875ten
• In scientific notation
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=
=
10011*2-6 = 1001.1*2-5 =
100.11*2-4
1.0011*2-2 normalised
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Normalised Notation
• In normalised binary scientific notation
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unless the number is 0
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always have 1.sssssss...sss * 2E
sss...sss is the significand
E is the exponent
• The significand s1s2...sn represents
n
i
 s / 2
i
i 1
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Representation
• Note that it is impossible to exactly represent
all decimal numbers in this way (eg 0.3)
• Problem of representation of floating point
numbers in fixed word length
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need to represent
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sign
significand
exponent
in one word (32 bits).
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Representation
31 30
23 22
sign bit exponent
significand: 23 bits F
S
8 bits E
0
• Represents floating point number:
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(-1)S * (1.0+F) * 2E
S is 1 bit (if S=1 then negative)
F is 23 bits
E is 8 bits
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Squeezing out More from the Bits
• Since every non-zero binary f.p. number
(normalised) is of the form:
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1.sss...sss *2E
• We do not have to represent explicitly the 1 in
the word, and can therefore interpret the bitpattern as:
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(-1)S (1 + significand) * 2E
• thus ‘reclaiming’ an extra bit!
• E= 0000 0000 is reserved for zero.
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Requirements
• As far as possible the ALU should be able
to reuse integer machinery in
implementation of f.p.
• Eg, comparison with zero
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easy because of sign bit
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fp numbers can be easily classified as negative,
zero or positive without additional hardware.
• Comparison of two fp numbers x<y not so
straightforward •
how are negative exponents to be formed?
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Bad Example: (1/2) > 2 ???
• Representation of 1/2 is
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0.1two = 1.0*2-1 (normalised)
0
1111 1111
S
E
0000....
0000
significand
Representation of 2 is
» 10two = 1.0*21 (normalised)
0
0000 0001
S
E
0000....
0000
significand
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Representation of Exponent
• Inappropriate to use two’s complement for
the exponent
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Ideally want 0000 0000 to represent most
negative number, 1111 1111 most positive.
Number range:
0111 1111 = 127ten
1111 1111
positive 1111 1110
.......
use this for 20 0111 1111
0111 1110
...
negative
0000 0000
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Biased Representation
(IEEE FP Standard)
• The ‘bias’ 127 represents 0
• 128 to 255 represent positive exponents
• 1 to 127 represent negative exponents
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(remember 0 is reserved for the entire number
being zero).
• The actual exponent is therefore:
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E - bias
• (-1)S * (1 + significand) * 2E-bias
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Example 1
• Represent 0.3125ten = 5/16
• 5/16 = 1/4 + 1/16 = 0.0101two
=
1.01*2-2
• S=0
• E = ???
• -2
= E-bias
• E = 125ten
= E-127
= 0111 1101two
• Significand = 010.…000
• 0
0111 1101
010000...000
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Example 2
• What does
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0 0111 1101
represent?
010000...000
• S=0
• E = 0111 1101 = 125ten
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Exponent = E-bias = 125-127 = -2
• Significand = 1/4
• (-1)S(1+sig.)2E-bias = (1 + 1/4)*(1/4) = 5/16
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Addition of FP Numbers
• Given two numbers:
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normalise them both
adjust the floating point of the smaller number to
match the larger one
Add them together
renormalise
check for underflow/overflow of exponent
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if so then break;
round significand to required number of bits
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might need renormalisation (eg, 11111 round to 4 bits).
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Addition Example
• 0.5 + 2.75 = 3.25
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0.1two + 10.11two
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1.0*2-1 + 1.011*21
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0.010*21 + 1.011*21
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1.101*21 (already normalised)
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(1 + (1/2) + (1/8)) * 2
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3.25
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Remarks
• The IEEE FP standard represents floats
in 32 bits, higher precision represented
across two words (doubles).
• Multiplication is relatively easy, since the
exponents add, and the significands can
be done with integer multiplication.
• There can be huge pitfalls in reliably
transferring floating point code to
different hardware!
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Summary
• FP scientific notation
• normalised representation in binary
• Bias to represent -ve to +ve range in
exponent
• Addition
• Notice how a 32-bit binary string can
represent many different entities in memory.
• Memory architectures NEXT.
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