2. Let fl(x) denote the machine floating-point version of a number x. Consider a computer using...

 

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2. Let fl(x) denote the machine floating-point version of a number x. Consider a computer using a binary floating point representation with 4 bits of precision in the significand. Assume that rounding is is used in going from a number x, which is outside the computer, to its floating point approximation, which is stored in the computer. (Rounding a significand that has more than 4 binary digits is done by chopping to 4 digits if its fifth digit is zero. Otherwise, if the fifth digit is 1, chop to 4 digits and then add 1 to the fourth digit.) Let x = 17/16. Find the binary expansion of x, and then determine fl(x) (the representation of x inside our computer). Show that x- X fl(x) <2-4 Let x 17/4. Determine fl(x) and show that x- fl(x) <2-4 These calculations are special cases of the following fact: if a number x outside the computer is represented using a binary floating point number with n binary digits (bits) of precision, then the absolute value of the relative error in the floating point representation satisfies x-fl(x) <2-n if rounding is used. The relative error is the quantity inside the absolute value sign. 2. Let fl(x) denote the machine floating-point version of a number x. Consider a computer using a binary floating point representation with 4 bits of precision in the significand. Assume that rounding is is used in going from a number x, which is outside the computer, to its floating point approximation, which is stored in the computer. (Rounding a significand that has more than 4 binary digits is done by chopping to 4 digits if its fifth digit is zero. Otherwise, if the fifth digit is 1, chop to 4 digits and then add 1 to the fourth digit.) Let x = 17/16. Find the binary expansion of x, and then determine fl(x) (the representation of x inside our computer). Show that x- X fl(x) <2-4 Let x 17/4. Determine fl(x) and show that x- fl(x) <2-4 These calculations are special cases of the following fact: if a number x outside the computer is represented using a binary floating point number with n binary digits (bits) of precision, then the absolute value of the relative error in the floating point representation satisfies x-fl(x) <2-n if rounding is used. The relative error is the quantity inside the absolute value sign.