From 29c036e86130e35a473bd9c0867b53c470a0aa08 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Sat, 11 Feb 2023 15:04:26 -0500 Subject: Implementing bit interleaving and uninterleaving --- docs/implementation/primitive/take.html | 38 ++++++++++++++++++++++++++++ implementation/primitive/take.md | 45 +++++++++++++++++++++++++++++++++ 2 files changed, 83 insertions(+) create mode 100644 docs/implementation/primitive/take.html create mode 100644 implementation/primitive/take.md 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 @@ + + + + BQN: Implementation of Take and Drop + +
(github) / BQN / implementation / primitive
+

Implementation of Take and Drop

+

The function Take 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.

+

Bit interleaving and uninterleaving

+

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 β₯Šπ•©, or undoes this process. Bit interleaving with nonzero bits might be used for ⍉𝕩 when ≠𝕩 is small, or π•©βˆΎΛ˜π•¨ when both arguments have small cells.

+

Careful! 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 β‰₯61, and 59, might not. Beware 59!

+

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 7 ↑ 𝕩 where a cell of 𝕩 is 5 bits. Here are the input and expected result, labelling zeros with . and argument bits with letters:

+
...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE
+...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE
+
+

The number of cells that can be widened at a time is ⌊64÷7, or 9. 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.

+

With the pdep instruction all we need to do is construct the appropriate mask indicating where the output cells should go. Let K m be (2⋆m)-1, that is, a number consisting of m ones in binary. Then the appropriate mask is (K 5) Γ— (K 7Γ—9) Γ· (K 7). The mask K 7Γ—9 has 9 groups of 7 1s, and division by K 7 converts each group to its bottom bit. Then multiplying by K 5 converts each bit to 5 of them.

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Because interleaving and uninterleaving are useful even on short arrays, it's best to precompute the division (K 7Γ—9) Γ· (K 7). Since 9 was computed as ⌊64Γ·7, this value depends only on the width 7 so a table of 64 words is enough. And 7|64, which might be useful for alignment, can be computed from the word as lΒ¬7, where l is the number of leading 0 bits. Other similar schemes are possible.

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On generic hardware these operations take more work. If we have n cells in a word (9 here), then it can be done with ⌈2⋆⁼n steps. Numbering the cells starting at 0 on the right (little-endian) and the steps ending at 0 for interleaving, step j moves cells that have a 1 in bit position j.

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...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE
+...ABCDE................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE
+...ABCDE........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE
+...ABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE
+...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE
+
+

The amount to move is 2⋆j 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 (w<<sh &~ mask) | (w & mask). To go backwards, shift first, like (w &~ mask)>>sh | (w & mask). 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:

+
...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE
+0000000011111111111111111111111111111111111111111111111111111111
+...ABCDE................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE
+1111111100000000000000000000000000001111111111111111111111111111
+...ABCDE........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE
+1111111100000000000000111111111111110000000000000011111111111111
+...ABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE
+0111111100000001111111000000011111110000000111111100000001111111
+...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE
+0001111100111110011111001111100111110011111001111100111110011111
+
+

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 m of all ones, then repeatedly take m ^ (m<<sh) with a series of shifts sh that decrease by factors of 2.

diff --git a/implementation/primitive/take.md b/implementation/primitive/take.md new file mode 100644 index 00000000..37cc9e41 --- /dev/null +++ b/implementation/primitive/take.md @@ -0,0 +1,45 @@ +*View this file with results and syntax highlighting [here](https://mlochbaum.github.io/BQN/implementation/primitive/take.html).* + +# Implementation of Take and Drop + +The function [Take](../../doc/take.md) 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. + +## Bit interleaving and uninterleaving + +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 `β₯Šπ•©`, or undoes this process. Bit interleaving with nonzero bits might be used for `⍉𝕩` when `≠𝕩` is small, or `π•©βˆΎΛ˜π•¨` when both arguments have small cells. + +**Careful!** 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 β‰₯61, and 59, might not. **Beware 59!** + +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 `7 ↑ 𝕩` where a cell of `𝕩` is 5 bits. Here are the input and expected result, labelling zeros with `.` and argument bits with letters: + + ...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE + ...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE + +The number of cells that can be widened at a time is `⌊64Γ·7`, or `9`. 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. + +With the pdep instruction all we need to do is construct the appropriate mask indicating where the output cells should go. Let `K m` be `(2⋆m)-1`, that is, a number consisting of `m` ones in binary. Then the appropriate mask is `(K 5) Γ— (K 7Γ—9) Γ· (K 7)`. The mask `K 7Γ—9` has 9 groups of 7 1s, and division by `K 7` converts each group to its bottom bit. Then multiplying by `K 5` converts each bit to 5 of them. + +Because interleaving and uninterleaving are useful even on short arrays, it's best to precompute the division `(K 7Γ—9) Γ· (K 7)`. Since `9` was computed as `⌊64Γ·7`, this value depends only on the width 7 so a table of 64 words is enough. And `7|64`, which might be useful for alignment, can be computed from the word as `lΒ¬7`, where `l` is the number of leading 0 bits. Other similar schemes are possible. + +On generic hardware these operations take more work. If we have `n` cells in a word (9 here), then it can be done with `⌈2⋆⁼n` steps. Numbering the cells starting at 0 on the right (little-endian) and the steps _ending_ at 0 for interleaving, step `j` moves cells that have a 1 in bit position `j`. + + ...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE + ...ABCDE................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE + ...ABCDE........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE + ...ABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE + ...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE + +The amount to move is `2⋆j` 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 `(w<>sh | (w & mask)`. 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: + + ...................ABCDEabcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE + 0000000011111111111111111111111111111111111111111111111111111111 + ...ABCDE................abcdeABCDEabcdeABCDEabcdeABCDEabcdeABCDE + 1111111100000000000000000000000000001111111111111111111111111111 + ...ABCDE........abcdeABCDEabcdeABCDE........abcdeABCDEabcdeABCDE + 1111111100000000000000111111111111110000000000000011111111111111 + ...ABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE....abcdeABCDE + 0111111100000001111111000000011111110000000111111100000001111111 + ...ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE..abcde..ABCDE + 0001111100111110011111001111100111110011111001111100111110011111 + +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 `m` of all ones, then repeatedly take `m ^ (m<