Do the following conversions:: IEEE standard floating point representation of -6

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Do the following conversions:: IEEE standard floating point representation of -64.25 Do the following conversions:: IEEE standard floating point representation of -64.25

Explanation / Answer

IEEE floating point numbers have three basic components: thesign, the exponent, and the mantissa. The following figure shows the layout for single (32-bit) anddouble (64-bit) precision floating-point values. The number of bitsfor each field are shown (bit ranges are in square brackets): Sign Exponent Fraction Bias Single Precision 1[31] 8[30-23] 23 [22-0] 127 Double Precision 1[63] 11[62-52] 52[51-0] 1023 Sign bit : '0' denotes a positive number. '1' denotes negativenumber. The Exponent : The exponent field needs to represent both positiveand negative exponents. To do this, a bias is added tothe actual exponent in order to get the stored exponent. For IEEEsingle-precision floats, this value is 127. Thus, an exponent ofzero means that 127 is stored in the exponent field. A stored valueof 200 indicates an exponent of (200-127), or 73. For reasonsdiscussed later, exponents of -127 (all 0s) and +128 (all 1s) arereserved for special numbers. For double precision, the exponentfield is 11 bits, and has a bias of 1023. The exponent's base is two. Mantissa : it also known as the significand, representsthe precision bits of the number. It is composed of an implicitleading bit and the fraction bits. In order to maximize thequantity of representable numbers, floating-point numbers aretypically stored in normalized form. This basically putsthe radix point after the first non-zero digit. For base two, since the only possible non-zero digit is 1.Thus, we can just assume a leading digit of 1, and don't need torepresent it explicitly. Mantissa = 1.f, where f is thefield of fraction bits. Now Consider the given number -64.25 Since it is a negative number Sign bit =1; 64.25 = 1000000. 0100000000000000 ( 23 bit representation) This can be written as 1.0000000100000000000000*26 Mantissa or Significand = 1.0000000100000000000000 exponent - 127 =6 => exponent field in binary =10000101 Combining all this we have"11000010110000000100000000000000" In Hexa decimal it comes to be 'C2808000'. Using the same procedure we represent in Double Precision also,that comes out to be "11000000010110000000100000000000000000000000000000000000000000000" [ Hereexponent field of 11 bits value 1029, as exponent -1023 =6 =>exponent =1029. Number 64.25 represented as 52 bit binarynumber.] .................................................................

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