Answer to Question #344310 in Assembler for 211

Question

Convert the following IEEE single-precision floating point numbers from hex to decimal:

a. 42F48000

b. CAB00000

c. 00700000

d. 3ABC0000

Answer 1

 

42F48000       Hexadecimal representation

0100 0010 1111 1000 0000 0000 000 0000  Binary representation

 

0    10000101    1111000000000000000000

Sign bit:       0

Mantissa:    1111000000000000000000

Exponent:    10000101

The general representation formula is:

(-1)^s * (1+mantissa) * 2^(Exponent – 127)

Conversion:

The value of a IEEE-754 number is computed as:

The sign is stored in bit 32. The exponent can be computed from bits 24-31 by subtracting 127. The mantissa (also known as significand or fraction) is stored in bits 1-23. An invisible leading bit (i.e. it is not actually stored) with value 1.0 is placed in front, then bit 23 has a value of 1/2, bit 22 has value 1/4 etc. As a result, the mantissa has a value between 1.0 and 2:

Mantissa:         1 + 1/2 + 1/4 + 1/8 +1/16 = 1.9375

Exponent:        133 – 127 = 6 -> 2^6 

Sign:                 -1^0 = 1

ANSWER:

1 x 64 x 1.9375= 124

 

 

CAB00000

1100 1010 1011 0000 0000 0000 000 0000  Binary representation

 

1  10010101  0110000000000000000000

Sign bit:       1

Mantissa:    0110000000000000000000

Exponent:    10010101

 

Mantissa:         1 + 1/4 + 1/8 = 1.375

Exponent:        149 – 127 = 22           -> 2^22

Sign:                 -1^0 = 1

 

ANSWER:

-1 x 4194304 x 1.375 = -5767168

 

 

00700000

0000 0000 0111 0000 0000 0000 000 0000  Binary representation

Sign bit:       0

Mantissa:    1110000000000000000000

Exponent:    00000000

 

Mantissa:         1 + 1/2 + 1/4 + 1/8 = 1.875

Exponent:        1 – 127 = -126 -> 2^(-126)

Sign:                 -1^0 = 1

 

ANSWER:

1 x 2^(-126) x 1.875= 1.028557556969…

 

 

3ABC0000

0011 1010 1011 1100 0000 0000 000 0000             Binary representation

 

Sign bit:       0

Mantissa:    0111100000000000000000

Exponent:    01110101

 

Mantissa:         1 +  1/4 + 1/8 + 1/16 +1/32= 1.46875

Exponent:        117 – 127 = -10         -> 2^(-10) = 1/1024

Sign:                 -1^0 = 1

 

ANSWER:

1 x 2^(-10) x 1.46875= 1.001434326171875