What is machine epsilon for this 9-bit floating point number? As examples, the 9

Question

What is machine epsilon for this 9-bit floating point number?

As examples, the 9-bit floating-point number with a binary representation of converts to its decimal via (-1)^1 times 2^1-3 times 0.10001 = -1 times 2 times 0.10001 = -0.0010001 = -(2^-3 + 2^-7) _10 = (1/8 + 1/128) = -17/128 = 0.1328125. While the floating-point number converts to decimal via (-1)^0 times 2^6-3 times 1.10111 = 1 times 2^3 times 1.10111 = 1101.11 = (2^3 + 2^2 + 2^0 + 2^-1 + 2^-2) _10 = 8 + 4 + 1 + 0.5 + 0.25 = 13.75 Assignment (show how you calculate these): What is machine epsilon _m for this 9-bit floating-point number?

Explanation / Answer

The machine epsilon, mach is a measure of the accuracy of a floating point representation and is found by calculating the difference between 1 and the next number that can be represented. For example, assume a 10-bit hypothetical computer where the first bit is used for the sign of the number, the second bit for the sign of the exponent, the next four bits for the exponent and the next four for the mantissa.

We represent 1 as

0 0 0 0 0 0 0 0 0 0

and the next higher number that can be represented is

0 0 0 0 0 0 0 0 0 1

The difference between the two numbers is

(1. 0001)2 × 2(0000)2 .(1. 0000)2 × 2 (0000)2 = (0 0001)2

= (1*2-4)10

= (0.0625)10

The machine epsilon is mach = .0 0625 .

The machine epsilon, mach is also simply calculated as two to the negative power of the number of bits used for mantissa. As far as determining accuracy, machine epsilon, mach is an upper bound of the magnitude of relative error that is created by the approximate representation of a number.

mach=2-4 =0.0625

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.