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| author | Marshall Lochbaum <mwlochbaum@gmail.com> | 2023-02-11 15:04:26 -0500 |
|---|---|---|
| committer | Marshall Lochbaum <mwlochbaum@gmail.com> | 2023-02-11 15:04:26 -0500 |
| commit | 29c036e86130e35a473bd9c0867b53c470a0aa08 (patch) | |
| tree | f4aabc1adae724d899154a105a6794153b5b58b9 /docs/implementation | |
| parent | 68fd472402e3b192402a2a23673fc1e6f83fe94b (diff) | |
Implementing bit interleaving and uninterleaving
Diffstat (limited to 'docs/implementation')
| -rw-r--r-- | docs/implementation/primitive/take.html | 38 |
1 files changed, 38 insertions, 0 deletions
diff --git a/docs/implementation/primitive/take.html b/docs/implementation/primitive/take.html new file mode 100644 index 00000000..032d9952 --- /dev/null +++ b/docs/implementation/primitive/take.html @@ -0,0 +1,38 @@ +<head> + <link href="../../favicon.ico" rel="shortcut icon" type="image/x-icon"/> + <link href="../../style.css" rel="stylesheet"/> + <title>BQN: Implementation of Take and Drop</title> +</head> +<div class="nav">(<a href="https://github.com/mlochbaum/BQN">github</a>) / <a href="../../index.html">BQN</a> / <a href="../index.html">implementation</a> / <a href="index.html">primitive</a></div> +<h1 id="implementation-of-take-and-drop"><a class="header" href="#implementation-of-take-and-drop">Implementation of Take and Drop</a></h1> +<p>The function <a href="../../doc/take.html">Take</a> on multidimensional arrays can be an important utility for working with arrays that have an odd cell shape. For example, a sorting algorithm on 25-bit cells would be very hard to write, but it's fast to expand each cell to 32 bits, sort, and trim back to 25 bits.</p> +<h2 id="bit-interleaving-and-uninterleaving"><a class="header" href="#bit-interleaving-and-uninterleaving">Bit interleaving and uninterleaving</a></h2> +<p>When the argument and result cells fit in a machine word, Take performs an operation I call bit interleaving if the width increases, or bit uninterleaving if it decreases. That's because it inserts some number of zero bits between every few bits of <code><span class='Function'>⥊</span><span class='Value'>𝕩</span></code>, or undoes this process. Bit interleaving with nonzero bits might be used for <code><span class='Function'>⍉</span><span class='Value'>𝕩</span></code> when <code><span class='Function'>≠</span><span class='Value'>𝕩</span></code> is small, or <code><span class='Value'>𝕩</span><span class='Function'>∾</span><span class='Modifier'>˘</span><span class='Value'>𝕨</span></code> when both arguments have small cells.</p> +<p><strong>Careful!</strong> A cell under 64 bits wide might not fit into any single machine word. For example, 57 bits starting at a 6-bit offset span 9 bytes. The first bit is bit 6 of byte 0, and the last is bit 0 of byte 8. Assuming the entire array is byte aligned, each cell always fits in a word for sizes ≤58, and 60. Cell sizes ≥61, and 59, might not. <strong>Beware 59!</strong></p> +<p>Interleaving can be implemented with pdep, and uninterleaving with pext, in the BMI2 instructions. And these operations can be performed generically with a series of shifts and masks. Consider <code><span class='Number'>7</span> <span class='Function'>↑</span> <span class='Value'>𝕩</span></code> where a cell of <code><span class='Value'>𝕩</span></code> is 5 bits. Here are the input and expected result, labelling zeros with <code><span class='Value'>.</span></code> and argument bits with letters:</p> +<pre><span class='Value'>...................</span><span class='Function'>ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span> +</pre> +<p>The number of cells that can be widened at a time is <code><span class='Function'>⌊</span><span class='Number'>64</span><span class='Function'>÷</span><span class='Number'>7</span></code>, or <code><span class='Number'>9</span></code>. In some cases I suppose it'd be possible to pack in one more by letting the leading zeros run past the top bit; that sounds complicated.</p> +<p>With the pdep instruction all we need to do is construct the appropriate mask indicating where the output cells should go. Let <code><span class='Function'>K</span> <span class='Value'>m</span></code> be <code><span class='Paren'>(</span><span class='Number'>2</span><span class='Function'>⋆</span><span class='Value'>m</span><span class='Paren'>)</span><span class='Function'>-</span><span class='Number'>1</span></code>, that is, a number consisting of <code><span class='Value'>m</span></code> ones in binary. Then the appropriate mask is <code><span class='Paren'>(</span><span class='Function'>K</span> <span class='Number'>5</span><span class='Paren'>)</span> <span class='Function'>×</span> <span class='Paren'>(</span><span class='Function'>K</span> <span class='Number'>7</span><span class='Function'>×</span><span class='Number'>9</span><span class='Paren'>)</span> <span class='Function'>÷</span> <span class='Paren'>(</span><span class='Function'>K</span> <span class='Number'>7</span><span class='Paren'>)</span></code>. The mask <code><span class='Function'>K</span> <span class='Number'>7</span><span class='Function'>×</span><span class='Number'>9</span></code> has 9 groups of 7 1s, and division by <code><span class='Function'>K</span> <span class='Number'>7</span></code> converts each group to its bottom bit. Then multiplying by <code><span class='Function'>K</span> <span class='Number'>5</span></code> converts each bit to 5 of them.</p> +<p>Because interleaving and uninterleaving are useful even on short arrays, it's best to precompute the division <code><span class='Paren'>(</span><span class='Function'>K</span> <span class='Number'>7</span><span class='Function'>×</span><span class='Number'>9</span><span class='Paren'>)</span> <span class='Function'>÷</span> <span class='Paren'>(</span><span class='Function'>K</span> <span class='Number'>7</span><span class='Paren'>)</span></code>. Since <code><span class='Number'>9</span></code> was computed as <code><span class='Function'>⌊</span><span class='Number'>64</span><span class='Function'>÷</span><span class='Number'>7</span></code>, this value depends only on the width 7 so a table of 64 words is enough. And <code><span class='Number'>7</span><span class='Function'>|</span><span class='Number'>64</span></code>, which might be useful for alignment, can be computed from the word as <code><span class='Value'>l</span><span class='Function'>¬</span><span class='Number'>7</span></code>, where <code><span class='Value'>l</span></code> is the number of leading 0 bits. Other similar schemes are possible.</p> +<p>On generic hardware these operations take more work. If we have <code><span class='Value'>n</span></code> cells in a word (9 here), then it can be done with <code><span class='Function'>⌈</span><span class='Number'>2</span><span class='Function'>⋆</span><span class='Modifier'>⁼</span><span class='Value'>n</span></code> steps. Numbering the cells starting at 0 on the right (little-endian) and the steps <em>ending</em> at 0 for interleaving, step <code><span class='Value'>j</span></code> moves cells that have a 1 in bit position <code><span class='Value'>j</span></code>.</p> +<pre><span class='Value'>...................</span><span class='Function'>ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span> +</pre> +<p>The amount to move is <code><span class='Number'>2</span><span class='Function'>⋆</span><span class='Value'>j</span></code> times the difference between the argument and result widths. To move the appropriate cells but not others, we need to blend with a mask, as in <code><span class='Paren'>(</span><span class='Value'>w</span><span class='Function'><<</span><span class='Value'>sh</span> <span class='Value'>&~</span> <span class='Value'>mask</span><span class='Paren'>)</span> <span class='Function'>|</span> <span class='Paren'>(</span><span class='Value'>w</span> <span class='Value'>&</span> <span class='Value'>mask</span><span class='Paren'>)</span></code>. To go backwards, shift first, like <code><span class='Paren'>(</span><span class='Value'>w</span> <span class='Value'>&~</span> <span class='Value'>mask</span><span class='Paren'>)</span><span class='Function'>>></span><span class='Value'>sh</span> <span class='Function'>|</span> <span class='Paren'>(</span><span class='Value'>w</span> <span class='Value'>&</span> <span class='Value'>mask</span><span class='Paren'>)</span></code>. Interleaving leaves some junk that needs to be cleared out with a final mask (same as the one used for pdep), and likewise uninterleaving requires the initial word to be cleaned with that mask. Here are the masks interleaved (heh) with results:</p> +<pre><span class='Value'>...................</span><span class='Function'>ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE</span> +<span class='Number'>0000000011111111111111111111111111111111111111111111111111111111</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE</span> +<span class='Number'>1111111100000000000000000000000000001111111111111111111111111111</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE</span> +<span class='Number'>1111111100000000000000111111111111110000000000000011111111111111</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE</span> +<span class='Number'>0111111100000001111111000000011111110000000111111100000001111111</span> +<span class='Value'>...</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span><span class='Value'>..abcde..</span><span class='Function'>ABCDE</span> +<span class='Number'>0001111100111110011111001111100111110011111001111100111110011111</span> +</pre> +<p>The masks aren't very quick to generate, so it's best to do it once for all cells and save them. One way is to start with a mask <code><span class='Value'>m</span></code> of all ones, then repeatedly take <code><span class='Value'>m</span> <span class='Value'>^</span> <span class='Paren'>(</span><span class='Value'>m</span><span class='Function'><<</span><span class='Value'>sh</span><span class='Paren'>)</span></code> with a series of shifts <code><span class='Value'>sh</span></code> that decrease by factors of 2.</p> |