Quiz1 of Computer Architecture (2021 Fall)

General Information

• You are allowed to read lecture materials.
  • That is, an open book exam.
• You shall not disclose your answer during the quiz.
• Each answer has 5 points.
• 09:10 ~ 10:20AM on Oct 5, 2021
• Fill Google Form to answer

Problem A

Two's Complement is the standard for representing signed integers:

• The most significant bit (MSB) has a negative value; all others have positive values (same as unsigned)
• Binary addition is performed the same way for signed and unsigned
• The bit representation for the negative (additive inverse) of a two's complement number can be found by:
  • flipping all the bits and adding 1 (i.e.-x = ~x + 1).

The above "number wheel" showing the relationship between 4-bit numerals and their Two's Complement interpretations is shown on the right:

• The largest number is 7 whereas the smallest number is -8
• There is a nice symmetry between numbers and their negative counterparts except for -8

This section is designed as a conceptual check for you to determine if you conceptually understand and have any misconceptions about this topic. Please answer Yes / No to the following questions:

1. Is it possible to get an overflow error in Two's Complement when adding numbers of opposite signs?

• A01 = ?

2. If you interpret a
   $N$ bit Two's Complement number as an unsigned number, would negative numbers be smaller than positive numbers?

• A02 = ?

3. If you interpret an
   $N$ bit Bias notation number as an unsigned number (assume there are negative numbers for the given bias), would negative numbers be smaller?

• A03 = ?

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Problem B

Arithmetic overflow occurs when the result of a calculation can't be represented in the current encoding scheme (i.e., it lies outside of the representable range of values), resulting in an incorrect value.

• Unsigned overflow: the result lies outside of
  $[UMin,UMax]$; an indicator of this is when you add two numbers and the result is smaller than either number.
• Signed overflow: the result lies outside of
  $[TMin,TMax]$; an indicator of this is when you add two numbers with the same sign and the result has the opposite sign.

1. Find the largest 8-bit unsigned numeral c (answer in HEX) such that c + 0x80 causes NEITHER signed nor unsigned overflow in 8 bits.

• B01 = ?

2. Find the smallest 8-bit numeral c (answer in HEX) such that c + 0x71 causes signed overflow, but NOT unsigned overflow in 8 bits.

• B02 = ?

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Problem C

According to IEEE 754 Floating Point Standard, the value of a real number can be represented in scientific binary notation as:

$$Value=(-1{)}^{sign}\times Mantiss{a}_2\times 2^{Exponent}=(-1{)}^S\times 1.M_2\times 2^{E-bias}$$

The binary representation for floating point values uses three fields:

• S: encodes the sign of the number (0 for positive, 1 for negative)
• E: encodes the exponent in biased notation with a bias of
  $2^{w-1}-1$
• M: encodes the mantissa (or significand, or fraction) – stores the fractional portion, but does not include the implicit leading 1.

1. Let’s say that we want to represent the number 3145728.125₁₀ (broken down as
   $2^{21}+2^{20}+2^{-3}$). Is it enough to represent this number single precision floating point? (Please answer Yes / No)

• C01 = ?

2. What is the decimal value of float 0x80000000? (Answer with leading + and -)

• C02 = ?

3. What is the decimal value of float 0xFF94BEEF? (Answer with leading + and -)

• C03 = ?

4. What is the decimal value of float 0x41180000? (Answer with leading + and -)

• C04 = ?

5. What is the smallest positive value that can be stored using a single precision float? Answer in HEX value. (video: How to Calculate Smallest Float Value in IEEE 754 Standard (Single Precision))

• C05 = ?

6. What is the smallest positive normalized value that can be stored using a single precision float?

• C06 = ?

7. What is the decimal value of float 0xFF800000? (Answer with leading + and -)

• C07 = ?

8. What is the decimal value of float 0x421E4000? (Answer with leading + and -)

• C08 = ?

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Problem D

Floating Point Mathematical Properties

• Not associative:
  $(2+2^{50})–2^{50}\ne 2+(2^{50}–2^{50})$
• Not distributive:
  $100\times (0.1+0.2)\ne 100\times 0.1+100\times 0.2$
• Not cumulative:
  $2^{25}+1+1+1+1\ne 2^{25}+4$

If x and y are variable type float, will the expression (x + 2 * y) - y == x + y always be evaluated as true? (Please answer Yes / No with explanation.)

D01 = ?

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Problem E

Compute the decimal result of the following arithmetic expressions involving 6-bit Two's Complement numbers as they would be calculated on a computer. Do any of these result in an overflow? Are all these operations possible?

1. Input: 0b100011 + 0b111010

• E01 = ?

2. Input: 0xFF − 0xAA

• E02 = ?

3. Input: 0x3B + 0x06

• E03 = ?

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Problem F

1. How do you write the bitwise exclusive-nor (XNOR) operator in C program?

• F01 = ?

2. Given x as an unsigned integer, please use bitwise operators and +, -, == to check if x is a power of 2. Write down the expression in C without any branches (i.e., if, else, ? :, do, while, for, goto)

• F02 = ?

3. The following function absf returns absolute value of a single precision float. What is the value of F03? Answer in HEX.

#include <stdint.h>
float absf(float x) {
  uint32_t mask = F03;
  union {
    uint32_t i;
    float f;
  } u = {.f = x};
  u.i &= mask;
  return u.f;
}

• F03 = ?