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Begin notes on unsigned arithmetic.

c-declarations
Joshua Potter 2024-02-24 11:34:58 -07:00
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@ -822,6 +822,163 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708701087982-->
END%%
## Arithmetic
### Addition
Addition of two unsigned or two two's-complement numbers operate in much the same way as grade-school arithmetic. Digits are added one-by-one and overflows "carried" to the next summation. Overflows are truncated; the final carry bit is discarded in the underlying bit adder.
%%ANKI
Basic
*Why* is adding $w$-bit integral types equal to $w$-bit truncation?
Back: The underlying bit adder discards any final carry bit.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678721-->
END%%
%%ANKI
Basic
Why should you generally prefer `x < y` over `x - y < 0`?
Back: The former avoids possible underflows.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678725-->
END%%
%%ANKI
Basic
How is `x - y < 0` rewritten more safely?
Back: `x < y`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678728-->
END%%
%%ANKI
Basic
What hardware-level advantage does two's-complement introduce over other signed encodings?
Back: The same circuits can be used for unsigned and two's-complement addition.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678732-->
END%%
%%ANKI
Basic
What representational-level advantage does two's-complement introduce over other signed encodings?
Back: `0` is encoded in only one way.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678736-->
END%%
Unsigned addition of $w$-bit integral types, denoted $+_w^u$, behaves like so:
$$x +_w^u y = \begin{cases}
x + y & \text{if } x + y < 2^w \\
x + y - 2^w & \text{otherwise}
\end{cases}$$
This is more simply expressed as $x +_w^u y = (x + y) \bmod 2^w$.
%%ANKI
Basic
What kind of overflows does unsigned addition potentially exhibit?
Back: Positive overflow.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678739-->
END%%
%%ANKI
Basic
Why is unsigned addition overflow *not* UB?
Back: Because the C standard enforces unsigned encoding of `unsigned` data types.
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: 1708799678742-->
END%%
%%ANKI
Basic
What does $+_w^u$ denote?
Back: Unsigned addition 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: 1708799678745-->
END%%
%%ANKI
Basic
Unsigned addition overflow is equivalent to what bit-level manipulation tactic?
Back: Truncation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678748-->
END%%
%%ANKI
Basic
What is the result of $x +_w^u y$?
Back: $(x + y) \bmod 2^w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678751-->
END%%
%%ANKI
Basic
*Why* does $x +_w^u y = (x + y) \bmod 2^w$?
Back: Because discarding any carry bit is equivalent to truncating the sum 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.
<!--ID: 1708799678755-->
END%%
%%ANKI
Cloze
Without using modulo arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678758-->
END%%
%%ANKI
Cloze
Without using modulo arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678761-->
END%%
%%ANKI
Basic
How do you detect whether unsigned addition $s \coloneqq x +_w^u y$ overflowed?
Back: Overflow occurs if and only if $s < x$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678765-->
END%%
%%ANKI
Basic
How would you complete the body of this function?
```c
/* Determine whether arguments can be added without overflow */
int uadd_ok(unsigned x, unsigned y);
```
Back:
```c
return (x + y) >= x;
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678769-->
END%%
%%ANKI
Basic
Does unsigned overflow detection depend on the left or right operand of $s \coloneqq x +_w^u y$?
Back: Either.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678772-->
END%%
%%ANKI
Basic
Why can we compare $s$ to $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
Back: Because unsigned addition is commutative.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678776-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.