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Urgent Please Explain and show the difference between IEEE 16, 32, 64, 128-bit floating-point numbers.

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Programming

Urgent Please

Explain and show the difference between IEEE 16, 32, 64, 128-bitfloating-point numbers.

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4FloatingPoint Number RepresentationA floatingpoint number or real number can represent a verylarge 1231088 or a very small1231088 value It could also represent verylarge negative number 1231088 and very smallnegative number 1231088 as well as zero asillustratedA floatingpoint number is typically expressed in the scientificnotation with a fraction F and anexponent E of a certain radixr in the form of FrE Decimalnumbers use radix of 10 F10E while binary numbersuse radix of 2 F2ERepresentation of floating point number is not unique Forexample the number 5566 can be represented as5566101 05566102005566103 and so on The fractional part can benormalized In the normalized form there is only a singlenonzero digit before the radix point For example decimal number1234567 can be normalized as1234567102 binary number 10101011Bcan be normalized as 10101011B23It is important to note that floatingpoint numbers suffer fromloss of precision when represented with a fixed number ofbits eg 32bit or 64bit This is because there areinfinite number of real numbers even within a small rangeof says 00 to 01 On the other hand a nbit binarypattern can represent a finite 2ndistinct numbers Hence not all the real numbers can berepresented The nearest approximation will be used insteadresulted in loss of accuracyIt is also important to note that floating number arithmetic isvery much less efficient than integer arithmetic It could be speedup with a socalled dedicated floatingpoint coprocessorHence use integers if your application does not requirefloatingpoint numbersIn computers floatingpoint numbers are represented inscientific notation of fraction F andexponent E with a radix of 2 inthe form of F2E Both E andF can be positive as well as negative Moderncomputers adopt IEEE 754 standard for representing floatingpointnumbers There are two representation schemes 32bitsingleprecision and 64bit doubleprecisionThe IEEE 754 standard specifies a binary16 ashaving the following formatSign bit 1 bitExponent width 5 bitsSignificand precision 11 bits 10 explicitly storedThe format is laid out as followsThe format is assumed to have an implicit lead bit with value 1unless the exponent field is stored with all zeros Thus only 10bits of the significand appear in the memory format but the totalprecision is 11 bits In IEEE 754 parlance there are 10 bits ofsignificand but there are 11 bits of significand precisionlog10211 3311 decimal digits or 4digits slightly less than 5 units in the last placeExponent encodingeditThe halfprecision binary floatingpoint exponent is encodedusing an offsetbinary representation with the zero offset being15 also known as exponent bias in the IEEE 754 standardEmin 000012 011112 14Emax 111102 011112 15Exponent bias 011112 15Thus as defined by the offset binary representation in orderto get    See Answer
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