Floating-point arithmetic

Question 1

What numbers can we store exactly on a computer?

Answer 1

Integers up to some maximum size

Question 2

What is the largest possible number than can be stored using 64-bit?

Answer 2

Assuming one bit is used to store the sign ±, the largest possible number is 2⁶³ - 1

Question 3

What is fixed point representation?

Answer 3

Question 4

What is (10.1)₂

Answer 4

1 x 2¹ + 0 x 2⁰ + 1 x 2^-1 = 2.5

Question 5

With fixed-point numbers are any numbers ever the same?

Answer 5

No - every number has a unique representation

Question 6

What is a problem with fixed-point representation?

Answer 6

Easy to “escape”

Question 7

What is meant by fixed-point representaion being easy to escape?

Answer 7

Numbers like (0.01)₁₀(0.10)₁₀ = (0.001)₁₀ can’t be represented.

Question 8

What is floating-point representation?

Answer 8

Question 9

What is the (0.d₁d₂…d_m)_β in the following called?

Answer 9

• Fraction
• Significand
• Mantissa

Question 10

What is β and e in the following called?

Answer 10

• Base
• Exponent

Question 11

What is one advantage and disadvantage to usinh floating point numbers over fixed point numbers?

Answer 11

• You can represent a much larger range of numbers in a floating-point representation
• However the numbers in floating-point representation are not equally spaced

Question 12

In floating-point numbers if d₁ ≠ 0 then each number in F has a unique representaion and is called?

Answer 12

Normalised

Question 13

What is the IEEE?

Answer 13

A standard for double-precision (64 bit) arithmetic

Question 14

What are the 64 bits used in the IEEE standard?

Answer 14

• 52 bits for the fraction
• 11 for the exponent
• 1 for the sign

Question 15

What is the IEEE representation?

Answer 15

Question 16

What does exponent bias mean in the IEEE standard?

Answer 16

The actual exponents are in range -1022 go 1055

Question 17

What are the exponents -1022 and 1025 used to store in the IEEE standard?

Answer 17

±0 and ±∞ respectively

Question 18

When β = 2, what does the first digit being normalsied mean?

Answer 18

The first digit is normalised to 1, so doesn’t need to be stored in memory

Question 19

Define underflow.

Answer 19

If a calculation falls below the lower non-zero limit (in absolute value it is called underflow.

Question 20

Define overflow

Answer 20

If a calculation falls above the upper limit (in absolute value) it is called overflow, and usually results in a flaoting-point exception

Question 21

Define rounding.

Answer 21

The mapping from ℝ to F is called rounding.

Question 22

What is used to denote rounding?

Answer 22

fl(x)

Question 23

How do you round a number?

Answer 23

Round the nearest number in F to x, if x lies exactly midway between two numbers in F, a method of breakinf ties is required. This is to round to the nearest even digit

Question 24

How do we count significant figures.

Answer 24

Start with the first non-zero digit from the left, and count all digits thereafter, uncluding ginal zeros if they are after the decimal point.

Question 25

What is the equation for the fl(x)?

Answer 25

fl(x) = x(1 + δ)

Question 26

What is the equation for the relative error incurred by rounding?

Answer 26

Question 27

What does δ stand for?

Answer 27

The relative rounding error

Question 28

How do we find an upper bound of |δ|?

Answer 28

Question 29

What is the upper bound of |δ|?

Answer 29

|δ| ≤ ε_M

Question 30

What does ε_M stand for?

Answer 30

Machine epsilon (or unit roundoff)

Question 31

Why is the machine epsilon also called the unit roundoff?

Answer 31

It is the distance between the smallest number in F greater than 1 but not rounded to 1

Question 32

What does ε_M equal?

Answer 32

Question 33

What is the fundamental axiom of floating-point arithmetic?

Answer 33

Question 34

What is the error when we are adding the following two numbers?

Answer 34

Question 35

What is a major cause of error in floating-point calculations?

Answer 35

Loss of significance

Question 36

What is loss of significance?

Answer 36

If x ± y is very close together, then there can be an arbitrarily large relative error in the result compared to the inital values of x and y.

Question 37

Does (a + b) + c = a + (b + c) in floating point arithmetic?

Answer 37

Not always