Answer to Question #326826 in Assembler for Nick

Question

Convert the following decimal numbers to IEEE single-precision format. Give the results as eight hexadecimal digits.

a. 10

b. 5/6

c. -10/15

d. 0.75 

Answer 1

Convert the following decimal numbers to IEEE single-precision format.

Give the results as eight hexadecimal digits.

 

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:

 

a.     10

The general representation formula is:

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

 Sign

   bit        Exponent                   Mantissa

    1       10000010    01000000000000000000000

 (0)^1        2^3                     1 + 1/4

1 * 8 * 1.25 = 10

 

BIN:      0100 0001 0010 0000 0000 0000 0000 0000

HEX:      0x 41200000

 

b.     5/6 = 0.833333333

 

Sign

bit         Exponent                   Mantissa

0       01111110     10101010101010101010101

 (-1)^0        2^(-1)                     1 + 1/2+1/8  + 1/32 + 1/128 +..

1 * 0.5 * 1.6666 = 0.8333333

BIN:      0011 1111 0101 0101 0101 0101 0101 0101

HEX:      0x3f555555

 

 

 

c.      -10/15=0.6666666666

 

bit         Exponent                   Mantissa

1       01111110     01010101010101010101010

 (-1)^1        2^(-1)                     1 + 1/4+1/16  + 1/64 + 1/256 +..

-1*0.5*1.33333 = 0.6666666

 

BIN:      1011 1111 0010 1010 1010 1010 1010 1010

HEX:      0xbf2aaaaa

 

 

d.     0.75 

 

bit         Exponent                   Mantissa

1       01111110     10000000000000000000000

 (-1)^0        2^(-1)                     1 + 1/2

1*0.5*1.5 = 0.75

 

 

BIN:              0011 1111 0100 0000 0000 0000 0000 0000

HEX:             0x3f400000