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Floating point representation - II

Representation of floating point numbers:

In the IEEE Single-precision representation of a real number, one bit used to represent sing , and it is set 0 for positive number and 1 for negative one. A representation of the exponent is stored in next 8bits and the remaining twenty-three bits are occupied by a representation of the mantissa of the number

Here are some examples:
How to represent real numbers in floating point format:

Examples:
1. Representing 23/4 in single precision floating point number.
=> 23/4 = 5.75
Converting above real number to binary form
=> 101.11 (5 in binary 101, .(2^-1 + 2^-2) = .75)

Representing above binary to SP floating point format (32bit)
[(-1)^S x 2^(E – 127) x 1.M]
=> 1.0111 x 2^2 relating this to above given equation

(Numeric ‘1’ before decimal point is called hidden bit as it is by default given in representation).

Sing S = 0; No. of bits used to represent exponent = 1
Exponent (E – 127) = 2 i.e. E = 129; No. of bits used to represent exponent = 8
Mantissa M = 0111000…. ; No. of bits used to represent Mantissa = 23

Finally 5.75 in SP floating point representations is as shown below 0|10000001|01110000000000000000000

Note: What if the fraction part of a real number cannot be expressed as sum of powers of two (as in the above example .75 = (1/2 + 1/4) ex: 7/5 is exactly 1.4, .4 cannot be expressed in terms of sums of power two, 7/5 has infinity binary expansion 1.011001100110011001100.
In a single precision representation, the expansion is rounded off at the twenty-third digit after the binary point.

2. Extracting real number from SP floating point number representation
11000100000100110000000000000000

1|10001000|0100110000000000000000
S|-----E----|-------------M---------------|

Sign = 1 i.e (-1)1 = -1 negative number
Exponent (10001000) = 127 + e, 136 = 127 + e i.e. exponent = 9;
Mantissa = 1.01001100000000000000000

i.e. Mantissa = (one plus, plus no one halves, plus one quarter (/14), plus no one eight, plus no one sixteenth, plus one thirty second, plus one sixty fourth,…all zeros)

=> (1 + 1/4 + 1/32 + 1/64) = X
=> (64 + 16 + 2 + 1) = X x 64;
=> X = 83/64;

So the complete number = -(83/64) x 29 = -664.00;
Introduction to Floating point representation IEEE 754
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FLOATING POINT REPRESENTATION

An Introduction to IEE 754 floating point number representation:
IEEE 754 Floating point is the most common representation used to store real numbers in a computer.
Index
1. What are floating point numbers
2. How to represent a floating point number.
a. Single and Double precisions
3.Data ranges

What are floating point number:

The term ‘floating point’ refers to the decimal point (or) binary point in a real number.
The decimal point placed in between group of digits is called floating point numbers.

Ex: 28.36, 0.00124

Floating point number representation IEEE 754 standard:

Floating point number contains three components
1. Sign bit - Represents the sign of floating point number.
2. Exponent - Represents the magnitude of the exponent (explained at later part) (Ex: 1.086 * 10^6)
3. Mantissa - Represents the precision bits of the number. (Ex: 1.086 * 10^6)

Single precision floating point representation (32-bit):

Fig 1: SP floating point representation

S = 1bit; E = 8 bit - Min = 126, Max = 127, Bias = 127; M = 23bits.

31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
S  ------ E-8bit------ -----------------------------M-23bit-------------------

Double precision floating point representation (64-bit):

Fig 2: DP floating point representation

S = 1bit; E = 12 bit - Min = 1022, Max = 1023, Bias = 1023; M = 23bits.

Normalized floating point representation:

Components of a floating point representation

In general to maximize the representable numbers, FP numbers are typically stored in ‘normalized’ form. This puts a radix point after a non-zero digit. Fig 1 & 2 are normalized representation of floating point (FP) numbers.

In order to represent in more optimized way number is represented with base 2, since only non-zero value possible is 1, this is implicitly stored.

Demoralized floating point representation:

If the exponent is all zeros and the fraction part is non-zero’s, then the value is ‘denormalized’ number. Which does not have any assumed leading but as 1 before decimal point