---
tags: computer-arch
---
# Quiz1 of Computer Architecture (2021 Fall)
> Solutions
## Problem `A`
[Two's Complement](https://en.wikipedia.org/wiki/Two%27s_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](https://en.wikipedia.org/wiki/Two%27s_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](https://en.wikipedia.org/wiki/Two%27s_complement) when adding numbers of opposite signs?
> * A01 = ?
==No==.
> Overflow errors only occur when the correct result of the addition falls outside the range of $[−(2^{n−1}), 2^{n−1} − 1]$. Adding numbers of opposite signs will not result in numbers outside of this range.
2. If you interpret a $N$ bit [Two's Complement](https://en.wikipedia.org/wiki/Two%27s_complement) number as an unsigned number, would negative numbers be smaller than positive numbers?
> * A02 = ?
==No==
> In Two's Complement, the MSB is always `1` for a negative number. This means ALL Two's Complement negative numbers will be larger than the positive numbers.
3. If you interpret an $N$ bit [Bias notation](https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/comps-programming/number_systems.pdf) number as an unsigned number (assume there are negative numbers for the given bias), would negative numbers be smaller?
> * A03 = ?
==Yes==
> In bias notation, we add a bias to the unsigned interpretation to create the value. This means that negative numbers will stay smaller than positive numbers. This is unlike Two's Complement.
---
## 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 = ?
==0x7F==
> Unsigned overflow will occur for `c > 0x80`. Signed overflow can only happen if c is negative (also `> 0x80`). Largest is therefore, `0x7F`
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 = ?
==0xF==
> For signed overflow, need `(+) + (+) = (−)`. For no unsigned overflow, need no carryout from MSB. The first `(−)` encoding we can reach from `0x71` is `0x80`. `0x80 – 0x71 = 0xF`.
---
## Problem `C`
According to IEEE 754 Floating Point Standard, the value of a real number can be represented in scientific binary notation as:
$$
Value = (-1)^{sign} \times Mantissa_{2} \times 2^{Exponent} = (-1)^S \times 1.M_2 \times 2^{E-bias}
$$
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 `3145728.125`~10~ (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 = ?
==No==
> Could only represent $2^{21} + 2^{20}$. Not enough bits in the mantissa to hold $2^{-3}$, which caused rounding.
2. What is the decimal value of float `0x80000000`? (Answer with leading `+` and `-`)
> * C02 = ?
==`-0`==
3. What is the decimal value of float `0xFF94BEEF`? (Answer with leading `+` and `-`)
> * C03 = ?
==`NaN`==
4. What is the decimal value of float `0x41180000`? (Answer with leading `+` and `-`)
> * C04 = ?
==`+9.5`==
> 0x41180000 = 0b 0|100 0001 0|001 1000 0…0.
> S = 0, E = $128+2 = 130$ $\to$ Exponent = E – bias = 3, Mantissa = 1.0011~2~
> $$
> 1.0011_2 \times 2^3 = 1001.1_2 = 8 + 1 + 0.5 = 9.5
> $$
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)](https://youtu.be/SMLcrWgE2sk))
> * C05 = ?
==0x00000001==
> 0x00000001 = $2^{−23} \times 2^{−126}$
6. What is the smallest positive normalized value that can be stored using a single precision float?
> * C06 = ?
==0x00800000==
> 0x00800000 = $2^{−126}$
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 = ?
==`+39.5625`==
---
## Problem `D`
Floating Point Mathematical Properties
* Not associative: $(2 + 2^{50}) – 2^{50} \neq 2 + (2^{50} – 2^{50})$
* Not distributive: $100 \times (0.1 + 0.2) \neq 100 \times 0.1 + 100 \times 0.2$
* Not cumulative: $2^{25} + 1 + 1 + 1 + 1 \neq 2^{25} + 4$
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 = ?
==No, Rounding error / Overflow== (後者只要一個解釋符合就給分)
---
## Problem `E`
Compute the decimal result of the following arithmetic expressions involving **6-bit** [Two's Complement](https://en.wikipedia.org/wiki/Two%27s_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 = ?
==Overflow==
> Adding together we get 0b1011101, however since we are working with 6-bit numbers we truncate the first digit to get 0b011101 = 29. Since we added two negative numbers and ended up with a positive number, this results in an overflow.
2. Input: `0xFF − 0xAA`
> * E02 = ?
==Impossible==
> This is not possible, as these hex numbers would need 8 bits to represent and we are working with 6 bit numbers.
3. Input: `0x3B + 0x06`
> * E03 = ?
==1==
> Converting to binary, we get 0b111011 + 0b000110 = (after truncating) 0b000001 = 1.
> Despite the extra truncated bit, this is not an overflow as `-5 + 6` indeed equals `1`
---
## Problem `F`
1. How do you write the bitwise [exclusive-nor (XNOR)](https://en.wikipedia.org/wiki/XNOR_gate) operator in C program?
> * F01 = ?
==`x == y`== (只要有比較就給分)
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 = ?
==`x & (x - 1) == 0`==
3. The following function `absf` returns absolute value of a single precision float. What is the value of `F03`? Answer in HEX.
```cpp
#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 = ?
==`0x7fffffff`==
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