The document explains MIPS floating-point representation, detailing the forms of binary floating-point representation, specifically normalized format and IEEE 754 standards for single and double precision. It includes examples demonstrating the conversion of decimal numbers to binary representation using single and double precision formats, as well as half precision. Key components such as sign bit, exponent, and fraction/mantissa are defined, along with their respective biases.

1. MIPS FLOATING POINT
REPRESENTATION

2. BINARY FLOATING POINT REPRESENTATION
(-1)S x 1.F x 2E
Normalized Floating Point Number
s = Sign Bit ( 0 for +ve and 1 for –ve)
F = Fraction / Mantissa
E = Exponent
Eg: +1.01010011 x 2-7
s = 0
F = 01010011
E = -7

3. BINARY FLOATING POINT REPRESENTATION
(-1)S x 1.F x 2E
Normalized Floating Point Number
s = Sign Bit ( 0 for +ve and 1 for –ve)
F = Fraction / Mantissa
E = Exponent
Eg: -1.11010 x 25
s = 1
F = 11010
E = 5

4. IEEE 754 BINARY FLOATING POINT REPRESENTATION
Single Precision Representation Double Precision Representation
A floating-point value represented in a single 32-bit word. A floating-point value represented in two 32-bitwords.
31 30……......23 22……………………………….0
S E’ Fraction
63 62……......52 51……………………………….0
S E’ Fraction
E’ = E + 127 E’ = E + 1023
1 8 23 1 11 52

5. EXAMPLE 1
Show the IEEE 754 binary representation of the number -0.7510 in single precision
Binary Representation
0.75 x 2 = 1.50 1
0.50 x 2 = 1.00 1
-0.75 = -0.11
Normalization
-0.11 = -1.1 x 2-1
(-1)S x 1.F x 2E s = 1, F = 1, E = -1, E’ = E + 127 = -1 + 127 = 126
31 30……......23 22……………………………….0
1 111 1110
1 8 23
0 1000 0000 0000 0000 0000 000

6. EXAMPLE 1
Show the IEEE 754 binary representation of the number -0.7510 in Double precision
Binary Representation
0.75 x 2 = 1.50 1
0.50 x 2 = 1.00 1
-0.75 = -0.11
Normalization
-0.11 = -1.1 x 2-1
(-1)S x 1.F x 2E s = 1, F = 1, E = -1, E’ = E + 1023 = -1 + 1023 = 1022
63 62……......52 51……………………………….0
1 111 1111 110
1 11 52
0
1 000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
0000
0000

7. EXAMPLE 2
Show the IEEE 754 binary representation of the number +0.6010 in single precision
+0.6 = +0.1001
Normalization
+0.6 = +1.0011001 x 2-1
(-1)S x 1.F x 2E s = 0, F = 0011001, E = -1
E’ = E + 127 = -1 + 127 = 126
31 30……......23 22……………………………….0
0 111 1110
1 8 23
0 001 1001 1001 1001 1001 1001 1001 1001
0.40 x 2 = 0.80 0
0.80 x 2 = 1.60 1
0.60 x 2 = 1.20 1
0.20 x 2 = 0.40 0
0.20 x 2 = 0.40 0
0.60 x 2 = 1.20 1
Binary Representation
. . . .
. .
001 1001 1001 1001 1001 1010
G R S Rounding Action
0 0 0 Truncate
0 0 1 Truncate
0 1 0 Truncate
0 1 1 Truncate
1 0 0 Round to Even
1 0 1 Round Up
1 1 0 Round Up
1 1 1 Round Up

8. OTHER IEEE 754 BINARY
FLOATING POINT REPRESENTATION
Half Precision Representation
A floating-point value represented in a 16-bit half word.
15 14……......10 9……………………………….0
S E’ Fraction
E’ = E + 15
1 5 10

9. IEEE 754-2008 contains a half precision that is only 16 bits wide. The left most bit is still the sign bit, the
exponent is 5 bits wide and has a bias of 15, and the mantissa is 10 bits long. A hidden 1 is assumed. Write
down the bit pattern to represent 1.5625 x 10-1 assuming a version of this format, which uses an excess-16
format to store the exponent.
0.6250 x 2 = 1.250 1
0.250 x 2 = 0.50 0
0.31250 x 2 = 0.6250 0
0.15625 x 2 = 0.31250 0
Binary Representation
0.15625 = +0.00101
1.5625 x 10-1 = 0.15625
0.50 x 2 = 1.00 1
(-1)S x 1.F x 2E s = 0, F = 01, E = -3,
E’ = E + 15 = -3 + 15 = 12
15 14……......10 9……………….0
0 1100
1 5 10
0 01 0000 0000
+1.01 x 2-3
Computer Organization and Design by David A Patterson – Page no. 239
+1.01 x 2-3