118 lines
4.6 KiB
Markdown
118 lines
4.6 KiB
Markdown
---
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title: Shifts
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TARGET DECK: Obsidian::STEM
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FILE TAGS: binary
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tags:
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- binary
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---
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## Overview
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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:
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* **Logical**
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* Drops the `k` least significant bits and fills the left end of the result with `k` zeros.
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* This mode is always used when calling `>>` on unsigned data.
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* Sometimes denoted as `>>>` to disambiguate from arithmetic right shifts.
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* **Arithmetic**
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* Drops the `k` least significant bits and fills the left end of the result with `k` copies of the most significant bit.
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* This mode is usually used when calling `>>` on signed data.
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%%ANKI
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Basic
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How is decimal value $2^n$ written in binary?
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Back: As `1` followed by $n$ zeros.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707432641574-->
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END%%
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%%ANKI
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Basic
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What kinds of left shift operations are there?
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Back: Just logical.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589773-->
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END%%
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%%ANKI
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Basic
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What kinds of right shift operations are there?
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Back: Logical and arithmetic
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589784-->
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END%%
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%%ANKI
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Basic
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What is a logical right shift operation?
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Back: One that fills the left end of the result with `k` zeros.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589786-->
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END%%
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%%ANKI
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Basic
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What is an arithmetic right shift operation?
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Back: One that fills the left end of the result with `k` copies of the most significant bit.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589789-->
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END%%
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%%ANKI
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Basic
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What kind of right shift operation is *usually* applied to signed numbers?
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Back: Arithmetic.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589801-->
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END%%
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%%ANKI
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Basic
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What kind of right shift operation is applied to unsigned numbers?
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Back: Logical.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707854589804-->
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END%%
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%%ANKI
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Basic
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What portability issue do shift operations introduce?
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Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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Tags: c
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<!--ID: 1707854589808-->
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END%%
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%%ANKI
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Cloze
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{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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Tags: c
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<!--ID: 1707854589813-->
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END%%
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In C, it is undefined behavior to shift by more than the width $w$ of an integral type. Typically though, only the last $w$ bits are considered in the computation. For example, given `int32_t x`, `(x << 32) = (x << 0)`.
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%%ANKI
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Basic
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Ignoring UB, what *typically* happens when shifting an `int32_t` by `k ≥ 32` bits?
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Back: The shift value is interpreted as `k mod 32`.
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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Tags: c
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<!--ID: 1707873410777-->
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END%%
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%%ANKI
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Basic
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How is $x \bmod 2^k$ equivalently written as a bit mask?
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Back: `x & ((1 << k) - 1)`
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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Tags: c
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<!--ID: 1707873410780-->
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END%%
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## References
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* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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