About Floating Point

What are the function of floating point?

To representation of non-integral numbers including very small and very large numbers.

5.98 x 10 ⁷ = Significant digits × base ^exponent

Example

Actual	Floating Point
0.0000000478	4.78 × 10-8
0.00000001	0.1 × 10-7
-1000000000	-1.0 × 109
-0.00111	-1.11× 2-3

IEEE 754-1985 was an industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087.

IEEE Floating-Point Format

Single Precision

Double Precision

x = (-1)^S × (1+Fraction) × 2^{(Exponent-Bias)}

sign = 0, because the number is positive. (1 indicates negative.)

biased exponent = actual exponent + bias

Floating Point Example

From decimal to floating number

• Convert -1313.3125 to IEEE 32-bit floating point format.

a.      The integral part is 1313₁₀ = 10100100001₂.

b.     The fractional:

0.3125	× 2 =	0.625	0	Generate 0 and continue.
0.625	× 2 =	1.25	1	Generate 1 and continue with the rest.
0.25	× 2 =	0.5	0	Generate 0 and continue.
0.5	× 2 =	1.0	1	Generate 1 and nothing remains.

c.      So 1313.3125₁₀ = 10100100001.0101₂.

d.     Normalize: 10100100001.0101₂ = 1.01001000010101₂ × 2¹⁰.

e.      Fraction is 01001000010101000000000,

exponent is 10 + 127 = 137 = 10001001₂,

sign bit is 1 because it is negative number.

Then you will get the answers as follow:

Binary 32 bits

Sign [1bit]	Exponent [8bits]	Fraction [23bits]
1 (-)	10001001	01001000010101 000000000

Binary 64 bits

Sign [1bit]	Exponent [11bits]	Fraction [52bits]
1 (-)	10000001001	01001000010101 0000000000000000 0000000000000000 000000

From floating point to decimal

a.      Separate:

01000100001101100001000000000000 ₂

Sign [1bit]	Exponent [8bits]	Fraction [23bits]
0(+)	10001000	011011000010 00000000000

b.     Exponent: 10001000₂ = 136₁₀; 136 − 127 = 9.

c.      Denormalize: 1.01101100001₂ × 2⁹ = 1011011000.01.

d.     Convert:

Exponents      2⁹    2⁸   2⁷   2⁶ 2⁵  2⁴  2³ 2² 2¹  2⁰ 2^-1  2^-2
Place Values 512 256 128 64 32 16  8   4   2   1  0.5 0.25

 Bits               1      0    1   1   0   1   1   0   0  0 . 0     1

Value            512     +128+64  +16 +8                   +0.25=728.25

e.      Sign: positive because the sign 1bit is 0

Result: 01000100001101100001000000000000 ₂ is 
728.25.

Floating point addition

is analogous to addition using scientific notation. For example, to add 2.25x 10^0 to 1.340625x 10^2 :

1.     Shift the decimal point of the smaller number to the left until the exponents are equal. Thus, the first number becomes 0.0225x 10^2 .

2.     Add the numbers with decimal points aligned:

3.     Normalize the result. = 1.363125x10^2

Floating Point Multiplication

Multiply the following two numbers in scientific notation by hand:

1.110 × 10¹⁰ × 9.200 × 10^-5

1.   Add the exponents to find

New Exponent = 10 + (-5) = 5

If we add biased exponents, bias will be added twice. Therefore we need to subtract it once to compensate: (10 + 127) + (-5 + 127) = 259 259 - 127 = 132 which is (5 + 127) = biased new exponent

2.   Multiply

1.110 × 9.200 = 10.212000

Can only keep three digits to the right of the decimal point, 
so the result is

10.212 × 10⁵

3.   Normalise the result

1.0212 × 10⁶

4.   Round it

1.021 × 10⁶

 Published by 

SITI NURHASTINI BINTI ROSALI  ( B031310320 )