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Two's-complement multiplication.

c-declarations
Joshua Potter 2024-03-04 20:37:56 -07:00
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"Basic": [

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---
title: "2024-03-04"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [x] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [x] Log Work Hours (Max 3 hours)
* Continued working my way through unsigned and two's-complement multiplication.
* 101weiqi (serial numbers)
* Q-329986
* Q-55201
* Q-114780
* Q-13894
* Q-254641
* Setup better nvim-dap integration. Made sure this also worked on Fedora.

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@ -895,34 +895,26 @@ In two's-complement encoding, truncating $x$ to $k$ bits is equal to $U2T_k(T2U_
%%ANKI
Basic
What is the $k$-truncation of $w$-bit two's-complement $x$?
Back: $U2T_k(T2U_w(x) \bmod 2^k)$
Back: $U2T_k(x \bmod 2^k)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708701087974-->
END%%
%%ANKI
Cloze
Two's-complement $k$-truncation of $w$-bit $x$ is {$U2T_k$}$(${$T2U_w(x) \bmod 2^k$}$)$.
Two's-complement $k$-truncation of $w$-bit $x$ is {$U2T_k$}$(${$x \bmod 2^k$}$)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708701087985-->
END%%
%%ANKI
Basic
What is the purpose of $U2T_k$ in two's-complement truncation expression $U2T_k(T2U_w(x) \bmod 2^k)$?
What is the purpose of $U2T_k$ in two's-complement truncation expression $U2T_k(x \bmod 2^k)$?
Back: To reinterpret the sign bit correctly.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708702794304-->
END%%
%%ANKI
Basic
What is the purpose of $T2U_w$ in two's-complement truncation expression $U2T_k(T2U_w(x) \bmod 2^k)$?
Back: To ensure $x$ is encoded with the right "type".
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708702794309-->
END%%
%%ANKI
Basic
Why isn't $T2U_w$ in two's-complement truncation $U2T_k(T2U_w(x) \bmod 2^k)$ strictly necessary?
@ -1367,119 +1359,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1709492205974-->
END%%
### Multiplication
Unsigned multiplication, denoted with the $*_w^u$ operator, is defined as follows: $$x *_w^u y = (x \cdot y) \bmod 2^w$$
%%ANKI
Basic
What does $*_w^u$ denote?
Back: Unsigned multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205977-->
END%%
%%ANKI
Basic
What is the result of $x *_w^u y$?
Back: $(x \cdot y) \bmod 2^w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205981-->
END%%
%%ANKI
Basic
*Why* does $x *_w^u y = (x \cdot y) \bmod 2^w$ (at least in C)?
Back: Because unsigned multiplication is *defined* to be the result truncated to $w$ bits.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492205984-->
END%%
%%ANKI
Basic
How do $+_w^u$ and $*_w^u$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^u\; y = (x \;\square\; y) \bmod 2^w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492205988-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^u y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205991-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^u y$?
Back: $w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205995-->
END%%
Similarly, two's-complement multiplication is defined as follows: $$x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$$
%%ANKI
Basic
What does $*_w^t$ denote?
Back: Two's-complement multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205998-->
END%%
%%ANKI
Basic
What is the result of $x *_w^t y$?
Back: $U2T_w((x \cdot y) \bmod 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206002-->
END%%
%%ANKI
Basic
How do $+_w^t$ and $*_w^t$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^t\; y = U2T_w((x \;\square\; y) \bmod 2^w)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492206006-->
END%%
%%ANKI
Basic
How can we write $x *_w^t y$ in terms of unsigned multiplication?
Back: $x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206012-->
END%%
%%ANKI
Basic
How is the following expressed more simply (i.e. using more standard algebra)? $$x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$$
Back: $x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206017-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206024-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206031-->
END%%
### Shifting
Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
@ -1581,7 +1460,7 @@ Tags: c17
<!--ID: 1707854589813-->
END%%
In C, it is undefined behavior to shift by more than the width $w$ of an integral type.
In C, it is undefined behavior to shift by more than the width $w$ of an integral type or by a negative value.
%%ANKI
Basic
@ -1664,6 +1543,301 @@ Tags: c17
<!--ID: 1707873410780-->
END%%
### Multiplication
Unsigned multiplication, denoted with the $*_w^u$ operator, is defined as follows: $$x *_w^u y = (x \cdot y) \bmod 2^w$$
%%ANKI
Basic
Given decimal integers $m$ and $n$, how many digits exist in $m \cdot n$?
Back: At most the number of digits in $m$ plus the number of digits in $n$.
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221438-->
END%%
%%ANKI
Basic
Given binary integers $m$ and $n$ of width $w$, how many bits exist in $m \cdot n$?
Back: At most $2w$.
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221442-->
END%%
%%ANKI
Basic
What does $*_w^u$ denote?
Back: Unsigned multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205977-->
END%%
%%ANKI
Basic
How do you multiply $10_2 \cdot 10_2$ to a $4$-bit unsigned result by hand?
Back:
```
10
x 10
-----
00
+ 10
-----
0100
```
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221444-->
END%%
%%ANKI
Basic
What is the result of $x *_w^u y$?
Back: $(x \cdot y) \bmod 2^w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205981-->
END%%
%%ANKI
Basic
*Why* does $x *_w^u y = (x \cdot y) \bmod 2^w$ (at least in C)?
Back: Because unsigned multiplication is *defined* to be the result truncated to $w$ bits.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492205984-->
END%%
%%ANKI
Basic
How do $+_w^u$ and $*_w^u$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^u\; y = (x \;\square\; y) \bmod 2^w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492205988-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^u y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205991-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^u y$?
Back: $w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205995-->
END%%
%%ANKI
Basic
Given unsigned `x`, what arithmetic operation is equivalent to `x << k`?
Back: $x *_w^u 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428810-->
END%%
%%ANKI
Basic
What bitwise operation is equivalent to $x *_w^u 2^k$?
Back: `x << k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428815-->
END%%
%%ANKI
Basic
How is `unsigned x` equivalently modified without using multiplication?
```c
x = x * pow(2, k);
```
Back:
```c
x = (x << k);
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709570428818-->
END%%
Similarly, two's-complement multiplication is defined as follows: $$x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$$
%%ANKI
Basic
What does $*_w^t$ denote?
Back: Two's-complement multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205998-->
END%%
%%ANKI
Basic
What is the result of $x *_w^t y$?
Back: $U2T_w((x \cdot y) \bmod 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206002-->
END%%
%%ANKI
Basic
How do you multiply $10_2 \cdot 01_2$ to a $4$-bit two's-complement result by hand?
Back:
```
1110
x 0001
-------
1110
+ 0000
-------
1110
```
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221447-->
END%%
%%ANKI
Basic
What pre-processing step is done when multiplying to a $w$-bit two's-complement result by hand?
Back: Sign extend the factors to width $w$.
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221449-->
END%%
%%ANKI
Basic
When performing two's-complement multiplication by hand, why prefer multiplying by a positive value?
Back: Sign extension of a positive value yields `0`s.
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
<!--ID: 1709563221452-->
END%%
%%ANKI
Basic
How do $+_w^t$ and $*_w^t$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^t\; y = U2T_w((x \;\square\; y) \bmod 2^w)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709492206006-->
END%%
%%ANKI
Basic
How can we write $x *_w^t y$ in terms of unsigned multiplication?
Back: $x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206012-->
END%%
%%ANKI
Basic
How is the following expressed more simply (i.e. using more standard algebra)? $$x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$$
Back: $x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206017-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206024-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206031-->
END%%
%%ANKI
Basic
Given two's-complement `x`, what arithmetic operation is equivalent to `x << k`?
Back: $x *_w^t 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428822-->
END%%
%%ANKI
Basic
What bitwise operation is equivalent to $x *_w^t 2^k$?
Back: `x << k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428825-->
END%%
%%ANKI
Basic
Assuming two's-complement encoding, how is `int x` equivalently modified without using multiplication?
```c
x = x * pow(2, k);
```
Back:
```c
x = (x << k);
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709570428828-->
END%%
%%ANKI
Basic
How can we rewrite $x \cdot 1101_2$ as an expression of *only* `<<` and `+`?
Back: `(x << 3) + (x << 2) + (x << 0)`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709570428832-->
END%%
%%ANKI
Basic
*Why* is $x \cdot 13$ equal to `(x << 3) + (x << 2) + (x << 0)`?
Back: Because the binary representation of $13$ is $1101_2$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709570428836-->
END%%
%%ANKI
Basic
How can we rewrite $x \cdot 1100_2$ as an expression of *only* `<<` and `-`?
Back: `(x << 4) - (x << 2)`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17
<!--ID: 1709570428839-->
END%%
%%ANKI
Basic
Convert $x \cdot 11011100_2$ to an expression containing `-`. How many `-` operators are there?
Back: $2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428844-->
END%%
%%ANKI
Basic
Convert $x \cdot K$ to an expression excluding `-`. The number of `+` operators correspond to what?
Back: One less than the number of `1`s in $K$'s binary representation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428848-->
END%%
%%ANKI
Basic
Convert $x \cdot K$ to an expression containing `-`. The number of `-` operators correspond to what?
Back: The number of runs of `1`s in $K$'s binary representation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428851-->
END%%
### Division
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.