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Unformatted text preview: m −3 to 3.
By adding bias (=3), we can move the range so it is
from 0 to 6:
true biased MSB 3 0 0 2 → 1 0 1 → 2 0 0 → 3 0 1 → 4 1 2 → 5 1 3 January 24, 2013 → → 6 1 Lecture 3 AWB Floating point  bias 11 With the bias, ﬂoating point numbers can be treated as
unsigned integers for which comparing is easy.
• For non positive exponents the MSB = 0
• For positive exponents the MSB = 1
(Bias ”reverses” the meaning of the MSB in two’s complement). Adding and subtracting bias is less costly than having
separate ways for comparing unsigned and ﬂoating
point. January 24, 2013 Lecture 3 AWB Floating point  bias 12 Introducing bias decreases the set of valid normalized
numbes.
This allows for representation of ’special’ symbols like
NaN, Inf, denormalize, etc.
• NaN  for example when dividing 0 by 0,
• Inf  for example when product is larger than the max
number,
• Denormalize  no leading 1 in the mantissa:
x = (1 2s)2c · (0.m) January 24, 2013 Lecture 3 AWB Floating point  special numbers 13 Number Sign bit 0 don’t care all 0’s all 0’s ∞ 0 all 1’s all 0’s −∞ 1 all 1’s all 0’s NaN X all 1’s nonzero denormals 0,1 all 0’s nonzero January 24, 2013 Exponent bits Fraction bits Lecture 3 AWB Floating point arithmetic 14 Floating point used in scientiﬁc computations
• abraviated as f l arithmetic
• very involved
• performed by a separate (from binary adder) circuit
• read and understand f l addition on page 257 in text
More things to remember
• biased exponent
• special sysmbols January 24, 2013 Lecture 3 AWB Logic gates 15 ab
Full Adder cout (a + b + c)20 =
cout 21 + s20 FA cin s Q: What is inside the FA module?
A: circuits built from logic gates.
Logic gates represent Boolean operations performed
on binary variables. January 24, 2013 Lecture 3 AWB Boolean algebra 16 A set of two elements X = {0, 1} ({TRUE, {FALSE)
endowed with operations AND, OR and NOT.
Operations are deﬁned by Truth Tables.
AND 1 OR 0 1 NOT 0 0 0 0 0 1 0 1 1 TT: 0
0 1 1 1 1 1 0 Gates: January 24, 2013 Lecture 3 AWB Boolean expressions 17 A, B ∈ {0, 1} are called binary variables.
Boolean expression is a logical expression that
evaluates to TRUE or FALSE (0 or 1).
Boolean expressions are obtained from binary variables
or other Boolean expression by combining them by
Boolean operators. January 24, 2013 Lecture 3 AWB Boolean expressions 18 Boolean expression Circuit
A Y = A AND B . B Y A = 0, B = 1 ⇒ Y = ?
A Y = (A OR (NOT B )) AND C B
C A = 1, B = 1, C = 0 ⇒ Y = ? January 24, 2013 Lecture 3 AWB Alternative notation 19 A AND B ↔ A·B A OR B ↔ NOT A ↔ A+B
¯
A Y = (A OR (NOT B )) AND C January 24, 2013 Lecture 3 ¯
↔ (A + B ) · C AWB Other binary operations 20 NAND 0 1 NOR 0 1 XOR 0 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 1 1 0 A Y B (A · B ) January 24, 2013 A Y B (A + B ) Lecture 3 a s b A⊕B AWB Some properties 21 ’·’ 0 1 ’ +’ 0 1 NOT 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 (P1) Idempotency: x + x =...
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