Digital Design Session 6

Digital Design – © 2020 Mouli Sankaran Email: mouli.sankaran@yahoo.com 2
Session 6: Focus
 Number Representations
◦ Scientific Notation
◦ Normalized numbers
 Floating-point Representation
◦ IEEE 754 Format
◦ Single Precision (32 bits)
 Exponent and Significand
 Biased Exponent
 Range of floating point numbers
◦ Reference: Video Lecture on this topic

Number Representations

Integers and Real Numbers
 Integers: the universe is infinite but discrete
◦ No fractions
◦ No numbers in between consecutive integers, e.g., 6 and 7
◦ A countable (finite) number of items within a finite range
◦ Referred to as fixed-point numbers
 Real numbers – the universe is infinite and
continuous
◦ Fractions are represented by decimal notation
 Rational numbers, e.g., 5/2 = 2.5
 Irrational numbers, e.g., 22/7 = 3.14159265 . . .
◦ Infinite numbers exist, even in the smallest range
◦ Referred to as floating-point numbers

Wide Range of Numbers
 A large number:
7,564,000,000,000,000 = 7.56 × 1015
 A small number:
0.000000000000007564 = 7.564 × 10 –15

Definitions
Scientific Notation
 Scientific notation is a way of writing numbers that are too big or
too small, to be conveniently written in decimal form
Scientific Notation: (many unnormalized forms for the same no.)
 10.0ten  10-10 , 0.1ten  10-8 and 0.01ten  10-7
 0.524×105 and 52.4×103
Normalized Scientific Notation
 Real number is written with one nonzero decimal digit before the
decimal point
Normalized Scientific Notation: (only one normalized form)
 1.0ten  10-9
 5.24×104

Floating-Point Numbers
 Binary Numbers
 Base 2
 Binary point – multiplication by 2 moves the point to the
right
 Normalized scientific notation
 e.g., 1.101two = 1 * 2 0 + 1 * 2 -1 + 0 * 2 -2 + 1 * 2-3
= 1 + 0.5 + 0 + 0.125
= 1.625
 Similar to the decimal numbers shown in scientific notation,
binary numbers can also be written in scientific notation …
 1.0two  2-1 => 0.5ten
 1.xxxxxxtwo  2yyyy (general format)

Advantages of Scientific Notation
 Simplifies exchange of data
 Simplifies arithmetic algorithms (because of
standardized format)
 Increases accuracy of the numbers that can be stored
in a word
◦ Since unnecessary leading zeros are replaced by digits to the
right of the binary point
◦ 0.00000101110  1.0111 * 2-6

Floating-Point
General Format

Floating Point Numbers
 General format
±1.bbbbbtwo×2eeee
or (-1)S × (1+F) × 2E
 Where
 S = Sign, 0 for positive, 1 for negative
 F = Fraction (or mantissa) as a binary integer, 1+F is called
Significand – Controls the Precision
 E = Exponent as a binary integer, positive or negative (two’s
complement representation) – Controls the Range
 To have more bits into the significand, IEEE 754 makes the
leading 1-bit of normalized binary numbers implicit

Floating-Point General Format
 General format
±1.bbbbbtwo×2eeee
or (-1)S × (1+F) × 2E
 Fraction, is a value between 0 and 1, placed in the
Fraction (F) field [bbbbb…]
 Exponent, is a value placed in the exponent field,
computing the value as 2E [eeee….]
 As mentioned earlier, the leading value 1 of the
significant is implicit and not specifically stored

Binary to Decimal Conversion
Binary (-1)S (1.b1b2b3b4) × 2E
Decimal(-1)S × (1 + b1×2-1 + b2×2-2 + b3×2-3 + b4×2-4) × 2E
Example: -1.1100 × 2-2 (binary) = - (1 + 2-1 + 2-2) ×2-2
= - (1 + 0.5 + 0.25)/4
= - 1.75/4
= - 0.4375 (decimal)

Floating-Point Representations

William Morton (Velvel) Kahan
One of the foremost experts on floating-point
computations.
A primary architect of the Intel 8087 floating-
point coprocessor and IEEE 754 floating-
point standard.
Architect of the IEEE 754 floating point standard
b. 1933, Canada
Professor of Computer Science, UC-Berkeley

Single Precision
Floating-Point Representation

Single Precision Representation (32 bits)
 – 126 ≤ E ≤ +127 (biased notation)
 Range of values are from 2-126 to (2-2-23) 2+127
 Range of magnitudes, 1.175 ×10-38 to 3.403 ×1038
bit 31
S E: 8-bit Exponent F: 23-bit Fraction
bits 0-22bits 23-30
Note: Normal limits of exponents are between -128 and +127 in 2’s complement form
Format: (-1)S (1.b1b2b3 ...b23) × 2E

Bias to the Exponent
 Exponent: 8 bits wide (in Single precision)
 Range of values: 0x00 to 0xFF (0 to 255 in decimal)
 Reserved: 0x00 and 0xFF
 Other values of Exponent: 0x01 to 0xFE (1 to 254 in decimal)
 Actual Exponent value = Value in Exponent field - Bias
 = 01 – 127 => -126 (min value)
 = 254 – 127 => +127 (max value)
 Bias (127) helps in performing integer comparison of floating
point representation (leaving out the sign bit)
 Since zero value (0.0) has no leading 1, it is given the reserved
exponent value (00) so that hardware won’t attach a leading 1 to
it
bits 0-22bits 23-30 Format:
(-1)S (1.b1b2…b23) × 2E

Range of 32-bit Integers
Expressible numbers
-231 231-10
 Signed Integers: 2’s complement Format
 Unsigned Integers:
Expressible numbers
0 232-1

IEEE 754: 32-bit Format
Negative
Overflow
Positive
Overflow
Expressible negative
numbers
Expressible positive
numbers
0-2-126 2-126
Positive underflow
Negative underflow
(2 – 2-23)×2+127- (2 – 2-23)×2+127
Positive zeroNegative zero + ∞– ∞
±1.0000 0000 0000 0000 0000 000 x 2-126
±1.1111 1111 1111 1111 1111 111 x 2+127
bits 0-22bits 23-30

Overflow and Underflow
Overflow
 It means that a positive exponent which is too large to fit in the
exponent field
Underflow
 This is an event where a non-zero fraction has become too
small that it cannot be represented
 This situation occurs when the negative exponent is too large
to fit into the exponent field
Format: (-1)S (1.b1b2…b23) × 2E

IEEE 754 Numerical Types
Biased Exponent
Max: 1111 11112
b31 b30 ………..…….b23 b22 b21………..……………….……………….b 0

Normalized Numbers

Positive and Negative Zeros
0 00000000 00000000000000000000000
Exponent Fraction
 Positive zero
1 00000000 00000000000000000000000
Exponent Fraction
 Negative zero

Denormalized Numbers
22 031
Negative
Overflow
Positive
Overflow
Expressible negative
numbers
Expressible positive
numbers
0-2-126 2-126
Positive underflowNegative underflow
(2 – 2-23)×2+127- (2 – 2-23)×2+127
Positive zeroNegative zero + ∞– ∞
±1.0000 0000 0000 0000 0000 000 x 2-126
±1.1111 1111 1111 1111 1111 111 x 2+127

Positive and Negative Infinities
22 031
0 1111111100000000000000000000000
Exponent Fraction
+ ∞ (+ infinity)
1 1111111100000000000000000000000
Exponent Fraction
- ∞ (- infinity)

Double Precision
Floating-Point Representation

Double Precision Representation (64 bits)
S E: 11-bit Exponent F: 52-bit Fraction +
bits 0-19bits 20-30
bit 63
Continuation of 52-bit Fraction
bits 0-31
Note: Extended Precision: 80 bits wide
 Double-precision values represent numbers, infinities and NaNs
analogous to single-precision values
 If 0 < exponent < 0x7FF, the value is a normalized number and
is equal to:
 −1S × 2exponent−1023 × (1.fraction)
 Range of values of double precision is from 2.225 x 10-308 to
1.798 x 10308
Note: Half Precision: 16 bits wide
IEEE 754 - 2008

Video Lecture on
Floating Point Representation
 You can listen to my lecture through this link below
too:
◦ Video Lecture on Floating Point Representation by
Mouli

Session 6: Summary
 Number Representations
◦ Scientific Notation
◦ Normalized numbers
 Floating-point Representation
◦ IEEE 754 Format
◦ Single Precision (32 bits) – float type in C
 Exponent and Significand
 Biased Exponent
 Range of floating point numbers
◦ Double Precision (64 bits) – double type in C
◦ Reference: Video Lecture on this topic

References
Ref 1 Ref 2

Digital Design Session 6

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Digital Design Session 6