Transcript Sign Exponent Fraction/Significand

Intro to Computer Org.
Floating Point Math &
Representation
Motivation
Integers are a very limited subset
of the real numbers, even if we
could represent all integers.
 Much of the time, we wish to use
fractional numbers, be they rational
or irrational.

A Human Perspective

Consider a standard fractional
number such as 105.8712.
The decimal indicates the fractional
component.
 We could try storing this one as an
integer and mark the decimal place…

A Human Perspective

But consider a number such as
1.058712 * 1046.


This one can’t be easily represented
by integers due to its magnitude.
What about 1.058712 * 10-46?

All of these are fairly easy to represent
in our writing system.
A Human Perspective

Note that the last two numbers
presented were written in scientific
notation.

Commonly used shorthand that
preserves the most significant details
of any number.
Binary Scientific Notation

Just as we have scientific notation
in the decimal number system, we
can have it in a binary system.

Ex: 1.11010001 * 23
= 111010.001
= 25 + 24 + 23 + 21 + 2-3
= 32 + 16 + 8 + 2 + .125
= 58.125
Decimal -> Binary

We already learned how to
convert integers to binary.
1.
2.
3.
Divide the number by two.
Take the remainder as the next digit,
working right-to-left.
If quotient = 0, stop.
Otherwise, take the quotient as the
new number, return to step 1.
Decimal -> Binary

To convert the fractional part from
decimal to binary:
1.
2.
3.
Multiply the number by two.
Remove the integer part and save it
as the next digit, working left-toright.
If remainder of number = 0, stop.
Otherwise, take the remainder as the
new number, return to step 1.
Decimal -> Binary

Note that the process is almost a
direct inversion of the process for
whole numbers.
Decimal -> Binary
0.625 * 2 = 1.250 =>
0.250 * 2 = 0.500 =>
0.500 * 2 = 1.000 =>

0 is left, so we’re done!
.1___
.10__
.101_
Decimal -> Binary

Because we’re using base 2, some
fractions become repeating.
0.2
 0.4
 0.8
 0.6
 0.2

*
*
*
*
*
2=
2=
2=
2=
2…
0.4
0.80
1.60
1.20
=>
=>
=>
=>
.0______
.00_____
.001____
.0011___
Decimal -> Binary

Because we’re using base 2, some
fractions become repeating.
0.2
 0.4
 0.8
 0.6
 0.2

*
*
*
*
*
2=
2=
2=
2=
2…
0.4
0.80
1.60
1.20
=>
=>
=>
=>
.0______
.00_____
.001____
.0011___
Decimal -> Binary

For conversions in this class, it’s
highly advised to handle each side
of a decimal point separately.
Decimal -> Binary

Let’s now consider 14.625.
14 => 11102.
 0.625 => 0.1012.
 So, 14.625 => 1110.1012.


Final step – normalization
14.625 = 1.1101012 * 23.
 Scientific notation allows only one
leading digit.

Binary Scientific Notation
All the basic mathematical
operations operate in a similar
fashion to their decimal scientific
notation counterparts.
 Read starting in page 195 for more
details on the operations.


Part of section 3.6, the introduction to
floating point.
Binary Scientific Notation

Notable exception – sometimes
division is performed by taking the
reciprocal of the divisor, then
performing multiplication.
Floating Point

The floating point specification has
three pieces.
A sign bit
 The fractional part of the number
 An exponent field


The entire number fits within
32 bits if single-precision
 64 bits if double-precision

Floating Point

The floating point specification has
three pieces.
A sign bit
 The fractional part of the number
 An exponent field

Floating Point – Sign Bit

Floating point represents sign in a
manner identical to that of the
sign-magnitude representation of
integers.
Leading bit = 1 if negative, 0 if
positive.
 Denotable as S in formulas, where
(-1)S gives the sign of the floatingpoint number.

Floating Point

The floating point specification has
three pieces.
A sign bit
 The fractional part of the number



Also called significand.
An exponent field
Floating Point - Significand

With one exception, all binary
numbers in scientific notation share
the same leading digit – 1.


Definition of the notation says leading
digit must be non-zero.
As a result, the leading 1 is not
stored – only the parts after the
“binary point” are.
Floating Point - Significand

Thus, to obtain the part of the
number that represents the
significand, (1. + Fraction) signifies
the need to prefix the bits
appropriately.
Floating Point

The floating point specification has
three pieces.
A sign bit
 The fractional part of the number
 An exponent field

Floating Point – Exponents
Floating point numbers store their
exponents in yet another
representation – one that is
biased.
 Take desired exponent and add the
bias to it.

Single-precision: Add 127 (27-1).
 Double-precision: Add 1023 (210-1).

Floating Point – Exponents

Represent the result as an
unsigned integer.

Why might one wish to do this?
Floating Point – Exponents

Special cases

After biasing the exponent, two values
indicate special conditions.
The minimum possible value (zero)
 The maximum possible value (all ones)




255 in single-precision
2047 in double-precision
The corresponding unbiased
exponents are thus out-of-range.
Floating Point – Exponents
Before covering the special cases,
let’s take a look at standard
floating-point numbers.
 We’ll use single-precision for our
examples.

Single-Precision Floats
Sign
Exponent
Fraction/Significand
1 bit
8 bits
23 bits

Let’s take a look at 14.625.
= 1.1101012 * 23.
 Sign = positive, so we’d represent it with a
zero.

Single-Precision Floats
Sign
Exponent
Fraction/Significand
0
8 bits
23 bits

Let’s take a look at 14.625.
= 1.1101012 * 23.
 Exponent (unbiased) = 3.
 Exponent (biased) = 3 + 127 = 130.
 130 = 100000102.


Sidenote – 2(8-1) – 1 = 127.

Pattern holds for double-precision.
Single-Precision Floats
Sign
Exponent
Fraction/Significand
0
10000010
23 bits

Let’s take a look at 14.625.
= 1.1101012 * 23.
 We ignore the leading 1.
 The part to the right of the binary point is
110101, so we store this.

Single-Precision Floats
Sign
Exponent
Fraction/Significand
0
10000010
1101 0100 0000 0000 0000 000

Let’s take a look at 14.625.
= 1.1101012 * 23.
 We ignore the leading 1.
 The part to the right of the binary point is
110101, so we store this and append zeros
to the end to fill out the remaining bits.

Single-Precision Floats

How about (136.111…) * 2-13?

Let’s work this one on the board.
Floating Point - Zero

One of the special cases of floating
point is when all bits of the
float/double are zero.

A very convenient representation of
zero!
Imprecision of Floating Point

All floating-point numbers are
stored in a scientific notation
format with limited bits.
This gives a tradeoff between range of
numbers representable and the
precision on those numbers.
 Single-precision gives 22 bits for the
significant, giving accuracy to a little
over seven decimal digits.

Imprecision of Floating Point

What is the smallest possible
difference between any two
numbers with the same (binary)
magnitude?
That is, the same power of two in
binary scientific notation.
 It depends on the power of two!

Imprecision of Floating Point
Sign
Exponent
0
3 (unbiased) 22 bits

Let’s take a look at numbers between 8
and 16. (16 not included.)


Fraction/Significand
Unbiased exponent =3.
What would be the smallest possible
difference, looking at the bit patterns?
Imprecision of Floating Point
Sign
Exponent
0
3 (unbiased) 0000 0000 0000 0000 0000 01

Fraction/Significand
The answer, in floating point format, is
all zeros with a one at the end.


Since the exponent is the same for both
numbers, the difference is only in the
significand.
The above difference is the smallest nonzero difference possible.

Warning – the above is before normalization –
there is no leading 1!
Imprecision of Floating Point
Sign
Exponent
0
3 (unbiased) 0000 0000 0000 0000 0000 01

Fraction/Significand
What would this translate to in decimal?

(-1)0 * (1. + 2-22) * 23 = 2-22 + 3 = 2-19


Note that we don’t add the 1 to the fraction
because we’re dealing with a difference between
two floating point numbers here before
normalization.
2-19 = 0.0000019073486328125, or
1.9073486328125 * 10-6.
Imprecision of Floating Point
Note that this number is not
precise to seven actual decimal
places when the number is not in
scientific notation!
 Numerical Analysis studies the
consequences of discrete
representations of real numbers,
equations, and the like.

Other Special Cases
Special case – biased exponent is
all ones.
 Is the significand all zero bits?

If so, the bit pattern represents ±∞.
 If not, it represents NaN. (Not a
number)


Useful for when illegal operations are
performed, such as dividing by zero.
Other Special Cases

Last special case – biased exponent
is zero, but the significand is nonzero.

This represents a denormalized
number.
Allows a gradual loss of precision at the
limits of the exponent range.
 For more details, read section 3.6 of the
text. We won’t worry about it much here.

More examples (On board)

6,706,993,152 – approximate
population of the world.


1110001.012 + 101011.10012


1100011111100010010011000000000002.
Answer in single-precision + decimal
1110001.012 * 101011.10012
Answer in single-precision
 Answer in decimal notation
