Floating Point Arithmetic

Question 1

How are integers represented in computing?

Answer 1

Integers can be represented exactly using a fixed number of bits.

Question 2

What is the largest unsigned integer that can be represented with n bits?

Answer 2

(2^n) - 1

Question 3

How are negative integers represented in computing?

Answer 3

Using two’s complement notation.

Question 4

Why do we need floating point representation?

Answer 4

Many scientific calculations involve non-integer values.

e.g., mass of an electron

Question 5

What key feature makes floating point representation different from integer representation?

Answer 5

The decimal point can float, allowing for a wide range of values.

Question 6

What components make up a floating point number?

Answer 6

1. Significand/Mantissa
2. Exponent
3. Base

Question 7

How would 9.109 × 10−31 be represented in floating point?

Answer 7

• Significand: 9.109
• Base: 10
• Exponent: -31

Question 8

What is the most widely adopted standard for floating point arithmetic? What does it specify?

Answer 8

• IEEE 754
  It specificies:
• Number representations (e.g., single precision, double precision)
• How operations like addition, subtraction, multiplication, and division behave

Question 9

What are the two most commonly used floating point precisions in C?

Answer 9

• Single Precision: 32-bit (float)
• Double Precision: 64-bit (double)

Question 10

Why is double precision preferred for scientific computing?

Answer 10

It provides greater accuracy and range compared to single precision

Question 11

What happens if a number exceeds single precision limits?

Answer 11

It is represented as infinity (inf)

Question 12

How many bits does double precision floating point use? How are they split?

Answer 12

64-bits (8-bytes):
- 52 bits for the mantissa
- 11 bits for the exponent
- 1 bit for the sign

Question 13

What does it mean for a floating point number to be normalised?

Answer 13

The exponent bits are not all 0s and not all 1s

Question 14

How is a normalised floating point number represented in IEEE 754?

Answer 14

x=±(1.b{1}.b{2}…b{52}) x 2^(a{1}.a{2}…a{11})-1023
where:
- b{1}, b{2}, …b{52} are the mantissa bits
- a{1}, a{2}, …a{11} are the exponent bits

Question 15

What is the smallest normalised double precision number? What is the largest?

Answer 15

• Small: 1 x 2^(1−1023) ≈ 10^(−308)
• Large: (2-2^-52) x 2^(2046-1023) ≈ 10^308

Question 16

Why are floating point numbers not always exact?

Answer 16

Because they have finite precision, leading to rounding errors.

Question 17

What is machine epsilon?

Answer 17

• The smallest difference between 1 and the next representable number
• For double precision, approximately: 2^(−52) ≈ 10^(−16)

Question 18

What are the five floating point exceptions in IEEE 754?

Answer 18

1. Overflow (Result is too large) → inf or -inf
2. Underflow (Result too small) → 0 or subnormal
3. Divide by Zero → inf or -inf
4. Invalid → NaN
5. Inexact → Rounded result

Question 19

What is peak performance (R_{peak})?

Answer 19

The theoretical maximum performance of a system, measured in FLOP/s.

Question 20

How is R_{peak} used in HPC rankings?

Answer 20

It is listed in Top500 rankings, but R_{max} is used for final ranking.

Question 21

What system factors determine R_{peak}?

Answer 21

1. Number of sockets
2. Number of processor cores per socket
3. Clock frequency
4. Number of operations per cycle

Question 22

What modern features can increase the number of operations per cycle?

Answer 22

• Vector instructions: operates on multiple numbers at once
• Fused Multiply-Add operations: can multiply and add in one instruction

Question 23

Given:
- 2 sockets
- 6 cores per socket
- 2.8 GHz clock speed
- 4 operations per cycle

What is the compute node R_{peak}?

Answer 23

R_{peak} = 2×6×2.8×4 = 134.4GFLOP/s

Question 24

If a cluster has 192 compute nodes, where each compute node’s R_{peak} = 134.4 GFLOP/s, what is its peak performance?

Answer 24

192×134.4GFLOP/s = 25.2TFLOP/s