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The largest number representable in 64 bits
A rewrite of this blog post with many new insights and updates appeared in Jan 2026
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 417 posts.
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Modelling a Basic Income with Python and Monte Carlo Simulation
Basic Income is a popular policy floating around the internet lately. The idea consists of declaring that the government will pay each living adult a fixed amount of money every month - an amount comparable to a minimum wage job. This policy would then replace all current welfare programs - welfare, unemployment …
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Stochastic Superoptimization
An “optimizer” should really be called an “improver”, and a “superoptimizer” should be called an “optimizer” 🙂
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The Existential Risk of Mathematical Error
Mathematical mistake/error-rates limit our understanding of rare risks and ability to defend against them.
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Noether's Theorem in a Nutshell (2020)
Noether's theorem is an amazing result which lets physicists get conserved quantities from symmetries of the laws of nature. Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, and so on.
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Brainfuck Optimization Strategies
In almost every computation a great variety of arrangements for the succession of the processes is possible, and various considerations must influence the selection amongst them for the purposes of a Calculating Engine. One essential object is to choose that arrangement which shall tend to reduce to a minimum the time…
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Selection bias and bombers (2008)
During WWII, statistician Abraham Wald told the RAF to add armor where it didn't seem planes needed it. But he was right.
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Why are amplitudes complex?
[By prior agreement, this post will be cross-posted on Microsoft’s Q# blog, even though it has nothing to do with the Q# programming language. It does, however, contain many examples th…
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Fast Fibonacci Algorithms (2015)
Definition: The Fibonacci sequence is defined as \(F(0) = 0\), \(F(1) = 1\), and \(F(n) = F(n-1) + F(n-2)\) for \(n ≥ 2\). So the sequence (starting with \(F(0)\)) is 0, 1, 1, 2, 3, 5, 8, 13, 21, ….
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The Opposite of a Bloom Filter
A Bloom filter is a data structure that may report it contains an item that it does not (a false positive), but is guaranteed to report correctly if it contains the item (“no false negatives”). The opposite of a Bloom filter is a data structure that may report a false negative, but can never report a false positive…
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R.J. Lipton - A proof that P is not equal to NP
A serious proof that claims to have resolved the P=NP question. Vinay Deolalikar is a Principal Research Scientist at HP Labs who has done important research in various areas of networks. He also h…
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Math Every Day
Math Every Day Stevey's Drunken Blog Rants™
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Oh My Gosh, It’s Covered in Rule 30s
New British train station design identified as cellular automata rule 30. Generating rule-based patterns. And a website that lets you try it yourself.
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The Kolmogorov option
Andrey Nikolaevich Kolmogorov was one of the giants of 20th-century mathematics. I’ve always found it amazing that the same man was responsible both for establishing the foundations of …
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Elliptic Curve Cryptography Explained
Fang-Pen Lin's blog about programming
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Debunking NIST's calculation of the Kyber-512 security level
[Sidney Harris cartoon used with permission. Copyright holder: ScienceCartoonsPlus.com .]
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A generalised solution to distributed consensus
A generalised solution to distributed consensus Howard & Mortier, arXiv’19 This is a draft paper that Heidi Howard recently shared with the world via Twitter, and here’s the accompanying …
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Real-world dynamic programming: seam carving
Dynamic programming has a reputation as a technique you learn in school, then only use to pass interviews at software companies. Indeed, most developers do not regularly work on problems where dynamic programming is needed. Ultimately, dynamic programming is a technique for efficiently solving problems that can be…
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Fibonacci Hashing: The Optimization That the World Forgot
I recently posted a blog post about a new hash table, and whenever I do something like that, I learn at least one new thing from my comments. In my last comment section Rich Geldreich talks about h…
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Convolution Is Fancy Multiplication
Like making engineering students squirm? Have them explain convolution and (if you're barbarous) the convolution theorem. They'll mutter something about…
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A new fast hash table in response to Google’s new fast hash table
Hi, I wrote my new favorite hash table. This came about because last year I wrote the fastest hash table (I still make that claim) and this year one of the organizers of the C++Now conference asked…
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P vs. NP
For those who just want the survey itself, not the backstory, it’s here. (Note: Partly because of the feedback I’ve gotten on this blog, it’s now expanded to 121 pages!) Upda…
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Beautiful branchless binary search
I read a blog post by Alex Muscar, “Beautiful Binary Search in D”. It describes a binary search called “Shar’s algorithm”. I’d never heard of it and it’s i…
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Simpson’s Paradox (2016)
A riveting tale of how good data can lead to bad decisions.