Floating Point Representations CDA 3101 Discussion Session 02

Question 1 Converting the binary number to decimal, if the binary is Unsigned? 2 ’ s complement? Single precision floating-point?

Question 1.1 Converting bin (unsigned) to dec * * … + 1* * *2 2 =

Question 1.2 Converting bin (2 ’ s complement) to dec * * … + 1* * *2 2 =

Question 1.3 Converting bin (Single precision FP) to dec Sign bit : 1 Exponent : = 73 Fraction : =1* * … + 1* * *2 -21 = (-1) S * (1.Fraction) * 2 (Exponent - 127) = (-1) 1 * ( ) * 2 ( ) = * = * S(1)Biased Exponent(8)Fraction (23)

Question 2 Show the IEEE 754 binary representation for the floating-point number in single ­ precision and double ­ precision

Question 2.1 Converting to single-precision FP Step1: Covert fraction 0.1 to binary (multiplying by 2) 0.1*2 = 0.2, 0.2*2 = 0.4, 0.4*2 = 0.8, 0.8*2 = 1.6, 0.6*2 = 1.2, 0.2*2 = 0.4, 0.4*2 = 0.8, 0.8*2 = 1.6, 0.6*2 = 1.2, … … … * 2 -4 Step2: Express in single precision format (-1) S * (1.Fraction) * 2 (Exponent +127) = (-1) 0 * ( ) * 2 (-4+127)

Question 2.2 Converting to double-precision FP Step1: Covert fraction 0.1 to binary (multiplying by 2) 0.1*2 = 0.2, 0.2*2 = 0.4, 0.4*2 = 0.8, 0.8*2 = 1.6, 0.6*2 = 1.2, 0.2*2 = 0.4, 0.4*2 = 0.8, 0.8*2 = 1.6, 0.6*2 = 1.2, … … … * 2 -4 Step2: Express in double precision format (-1) S * (1.Fraction) * 2 (Exponent +1023) = (-1) 0 * ( ) * 2 ( )

Question 3 Convert the following single-precision numbers into decimal a b

Question 3.1 Converting bin (Single precision FP) to dec Sign bit : 0 Exponent : = Infinity Fraction : = 0 Infinity S(1)Biased Exponent(8)Fraction (23)

Question 3.2 Converting bin (Single precision FP) to dec Sign bit : 0 Exponent : = 0 Fraction : =1*2 -22 = (-1) S * (0.Fraction) * = (-1) 0 * ( ) * = * S(1)Biased Exponent(8)Fraction (23)

Question 4 Consider the 80-bit extended-precision IEEE 754 floating point standard that uses 1 bit for the sign, 16 bits for the biased exponent and 63 bits for the fraction (f). Then, write (i) the 80- bit extended-precision floating point representation in binary and (ii) the corresponding value in base-10 positional (decimal) system of a.the third smallest positive normalized number b.the largest (farthest from zero) negative normalized number c.the third smallest positive denormalized number that can be represented.

Question 4.1 The third smallest positive normalized number Bias: = Sign: 0 Biased Exponent: Fraction (f): 61 zeros followed by 10 Decimal Value: (-1) 0 *2 ( ) *( ) =

Question 4.2 The largest (farthest from zero) negative normalized number Sign: 1 Biased Exponent: Fraction: 63 ones Decimal Value: (-1) 1 *2 ( ) *( … ) = ( )2 -63 = (approx.)

Question 4.3 The third smallest positive denormalized number Sign: 0 Biased Exponent: Fraction: 61 zeros followed by 11 Decimal Value: (-1) 0 * *( ) = 3*