Question

This problem covers floating-point IEEE format.

(a) Assuming single precision IEEE 754 format, what is the binary pattern for decimal number -6.16?

(b) Assuming single precision IEEE 754 format, what decimal number is represented by this word: 0 01111100 01100000000000000000000

(Hint: remember to use the biased form of the exponent.)

Answer 1

a)
-6.16
Converting 6.16 to binary
   Convert decimal part first, then the fractional part
   > First convert 6 to binary
   Divide 6 successively by 2 until the quotient is 0
       > 6/2 = 3, remainder is 0
       > 3/2 = 1, remainder is 1
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 110
   So, 6 of decimal is 110 in binary
   > Now, Convert 0.16000000 to binary
       > Multiply 0.16000000 with 2.    Since 0.32000000 is < 1. then add 0 to result
       > Multiply 0.32000000 with 2.    Since 0.64000000 is < 1. then add 0 to result
       > Multiply 0.64000000 with 2.    Since 1.28000000 is >= 1. then add 1 to result
       > Multiply 0.28000000 with 2.    Since 0.56000000 is < 1. then add 0 to result
       > Multiply 0.56000000 with 2.    Since 1.12000000 is >= 1. then add 1 to result
       > Multiply 0.12000000 with 2.    Since 0.24000000 is < 1. then add 0 to result
       > Multiply 0.24000000 with 2.    Since 0.48000000 is < 1. then add 0 to result
       > Multiply 0.48000000 with 2.    Since 0.96000000 is < 1. then add 0 to result
       > Multiply 0.96000000 with 2.    Since 1.92000000 is >= 1. then add 1 to result
       > Multiply 0.92000000 with 2.    Since 1.84000000 is >= 1. then add 1 to result
       > Multiply 0.84000000 with 2.    Since 1.68000000 is >= 1. then add 1 to result
       > Multiply 0.68000000 with 2.    Since 1.36000000 is >= 1. then add 1 to result
       > Multiply 0.36000000 with 2.    Since 0.72000000 is < 1. then add 0 to result
       > Multiply 0.72000000 with 2.    Since 1.44000000 is >= 1. then add 1 to result
       > Multiply 0.44000000 with 2.    Since 0.88000000 is < 1. then add 0 to result
       > Multiply 0.88000000 with 2.    Since 1.76000000 is >= 1. then add 1 to result
       > Multiply 0.76000000 with 2.    Since 1.52000000 is >= 1. then add 1 to result
       > Multiply 0.52000000 with 2.    Since 1.04000000 is >= 1. then add 1 to result
       > Multiply 0.04000000 with 2.    Since 0.08000000 is < 1. then add 0 to result
       > Multiply 0.08000000 with 2.    Since 0.16000000 is < 1. then add 0 to result
       > Multiply 0.16000000 with 2.    Since 0.32000000 is < 1. then add 0 to result
       > Multiply 0.32000000 with 2.    Since 0.64000000 is < 1. then add 0 to result
       > Multiply 0.64000000 with 2.    Since 1.28000000 is >= 1. then add 1 to result
       > Multiply 0.28000000 with 2.    Since 0.56000000 is < 1. then add 0 to result
       > Multiply 0.56000000 with 2.    Since 1.12000000 is >= 1. then add 1 to result
       > Multiply 0.12000000 with 2.    Since 0.24000001 is < 1. then add 0 to result
       > Multiply 0.24000001 with 2.    Since 0.48000002 is < 1. then add 0 to result
       > Multiply 0.48000002 with 2.    Since 0.96000004 is < 1. then add 0 to result
       > Multiply 0.96000004 with 2.    Since 1.92000008 is >= 1. then add 1 to result
       > Multiply 0.92000008 with 2.    Since 1.84000015 is >= 1. then add 1 to result
       > Multiply 0.84000015 with 2.    Since 1.68000031 is >= 1. then add 1 to result
       > Multiply 0.68000031 with 2.    Since 1.36000061 is >= 1. then add 1 to result
       > Multiply 0.36000061 with 2.    Since 0.72000122 is < 1. then add 0 to result
       > Multiply 0.72000122 with 2.    Since 1.44000244 is >= 1. then add 1 to result
       > Multiply 0.44000244 with 2.    Since 0.88000488 is < 1. then add 0 to result
       > Multiply 0.88000488 with 2.    Since 1.76000977 is >= 1. then add 1 to result
       > Multiply 0.76000977 with 2.    Since 1.52001953 is >= 1. then add 1 to result
       > Multiply 0.52001953 with 2.    Since 1.04003906 is >= 1. then add 1 to result
       > Multiply 0.04003906 with 2.    Since 0.08007812 is < 1. then add 0 to result
       > Multiply 0.08007812 with 2.    Since 0.16015625 is < 1. then add 0 to result
       > Multiply 0.16015625 with 2.    Since 0.32031250 is < 1. then add 0 to result
       > Multiply 0.32031250 with 2.    Since 0.64062500 is < 1. then add 0 to result
       > Multiply 0.64062500 with 2.    Since 1.28125000 is >= 1. then add 1 to result
       > Multiply 0.28125000 with 2.    Since 0.56250000 is < 1. then add 0 to result
       > Multiply 0.56250000 with 2.    Since 1.12500000 is >= 1. then add 1 to result
       > Multiply 0.12500000 with 2.    Since 0.25000000 is < 1. then add 0 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 to result
       > Multiply 0.50000000 with 2.    Since 1.00000000 is >= 1. then add 1 to result
       > This is equal to 1, so, stop calculating
   0.16000000000000014 of decimal is .001010001111010111000010100011110101110000101001 in binary
   so, 6.16 in binary is 110.001010001111010111000010100011110101110000101001
-6.16 in simple binary => 110.001010001111010111000010100011110101110000101001
so, -6.16 in normal binary is 110.001010001111010111000010100011110101110000101001 => 1.10001010001111010111 * 2^2

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+2=129) => 10000001
   Divide 129 successively by 2 until the quotient is 0
       > 129/2 = 64, remainder is 1
       > 64/2 = 32, remainder is 0
       > 32/2 = 16, remainder is 0
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10000001
   So, 129 of decimal is 10000001 in binary
frac/significant bits are 10001010001111010111000

so, -6.16 in single-precision format is 1 10000001 10001010001111010111000
in hexadecimal it is 0xC0C51EB8

b)
0 01111100 01100000000000000000000
sign bit is 0(+ve)
exp bits are 01111100
   => 01111100
   => 0x2^7+1x2^6+1x2^5+1x2^4+1x2^3+1x2^2+0x2^1+0x2^0
   => 0x128+1x64+1x32+1x16+1x8+1x4+0x2+0x1
   => 0+64+32+16+8+4+0+0
   => 124
in decimal it is 124
so, exponent/bias is 124-127 = -3
frac bits are 011

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.011 * 2^-3
1.011 in decimal is 1.375
   => 1.011
   => 1x2^0+0x2^-1+1x2^-2+1x2^-3
   => 1x1+0x0.5+1x0.25+1x0.125
   => 1+0.0+0.25+0.125
   => 1.375
so, 1.375 * 2^-3 in decimal is 0.171875
so, 00111110001100000000000000000000 in IEEE-754 single precision format is 0.171875
Answer: 0.171875