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abs and its "extra" result
The abs function has, in its usual (most useful) formulation, one extra value in its codomain than just "all non-negative values". That extra value is the most negative integer, which satisfies abs(x)...
View ArticleSquare root of bitwise NOT
The square root of bitwise NOT, if it exists, would be some function f such that f(f(x)) = not x, or in other words, f²(x) = not x. It is similar in concept to the √NOT gate in Quantum Computing, but...
View ArticleInformation on incrementation
Defining incrementJust to avoid any confusion, the operation that this post is about is adding 1 (one) to a value: $$\text{increment}(x) = x + 1$$ Specifically, performing that operation in the domain...
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Why does AND distribute over XOR
blockquote { background: #f9f9f9; border-left: 10px solid #ccc; margin: 1.5em 10px; padding: 0.5em 10px; quotes: "\201C""\201D""\2018""\2019"; } blockquote:before { color: #ccc; content: open-quote;...
View ArticleThe range a sum cannot fall within
.nobr { white-space: nowrap; } Throughout this post, the variables A, B, and R are used, with R defined as R = A + B, and A ≤ B. Arithmetic in this post is unsigned and modulo 2k. Note that A ≤ B is...
View ArticlePartial sums of blsi and blsmsk
blsi is an x86 operation which extracts the rightmost set bit from a number, it can be implemented efficiently in terms of more familiar operations as i&-i. blsmsk is a closely related operation...
View ArticleInteger promotion does not help performance
There is a rule in the C language which roughly says that arithmetic operations on short integer types implicitly convert their operands to normal-sized integers, and also give their result as a...
View ArticleBit-level commutativity and "sum gap" revisited
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } In an earlier post, bit-level commutativity, it was shown that addition is...
View ArticleWeighted popcnt
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } Since I showed the code below on Twitter, and some people understood it...
View ArticleColumn-oriented row reduction
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(Not) transposing a 16x16 bitmatrix
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View Articlegrevmul
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View ArticleSome ways to check whether an unsigned sum wraps
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } When computing x + y, does the sum wrap? There are various ways to find...
View ArticlePropagating bounds through bitwise operations
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Permuting bits with GF2P8AFFINEQB
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } It's no secret that GF2P8AFFINEQB can be tricky to think about, even in...
View ArticlePartial sums of popcount
The partial sums of popcount, aka A000788: Total number of 1's in binary expansions of 0, ..., n can be computed fairly efficiently with some mysterious code found through its OEIS entry (see the link...
View ArticleThe solutions to 𝚙𝚘𝚙𝚌𝚗𝚝(𝚡)
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } popcnt(x) < tzcnt(x) asks the question "does x have fewer set bits than...
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Enumerating all mathematical identities (in fixed-size bitvector arithmetic...
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } Once again a boring-but-specific title, I don't want to clickbait the...
View ArticleImplementing grevmul with GF2P8AFFINEQB
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } As a reminder, grev is generalized bit-reversal, and performs a...
View ArticleMultiplying 64x64 bit-matrices with GF2P8AFFINEQB
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } This is a relatively simple use of GF2P8AFFINEQB. By itself GF2P8AFFINEQB...
View ArticleSharpening a lower bound with KnownBits information
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } I have written about this before, but that was a long time ago, I've had a...
View ArticleSorting the nibbles of a u64
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } I was reminded (on mastodon) of this nibble-sorting technique (it could be...
View ArticleEnumerating identities, part 2
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } Part 2 of Enumerating all mathematical identities (in fixed-size bitvector...
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Histogramming bytes with positional popcount (GF2P8AFFINEQB edition)
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } A while ago, after some back and forth on twitter/X with @corsix, I...
View ArticleBit-permuting 16 u32s at once with AVX-512
.nobr { white-space: nowrap; } blockquote { background-color: #EEE; clear: both; } img { max-width: 100%; max-height: 100%; } The basic trick to apply the same bit-permutation to each of the u32s is...
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