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Sunday, May 3, 2026

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Saturday, May 2, 2026

Every float on one pagevitaut.net https://vitaut.net/posts/2026/every-float/ - In my previous post about Żmij , a high-performance binary-to-decimal floating-point conversion library, I drew a small diagram of a rounding interval around a single floating-point value. That worked well enough for the local picture, but it doesn’t say much about the global one. Where do the irregular intervals at powers of two come from? What does the subnormal range actually look like? And how do the binades (the $[2^e, 2^{e+1})$ slices of the real line on which all FP numbers share an exponent) fit together? So I tried to draw that instead: the entire set of representable numbers, laid out so the interesting structure is visible at a glance. A small format that is actually used To draw all the floats you really need a small format. Luckily small formats have recently escaped the textbook: 8-bit floating-point formats are now used in production for AI training and inference. The one we will look at here is E4M3 : 1 sign bit, 4 exponent bits, 3 significand bits, bias 7, with subnormals and a single NaN slot. It is specified in FP8 Formats for Deep Learning by NVIDIA, Arm and Intel, three companies that famously agree on very little, and is supported in hardware by GPUs such as the H100, H200 and B200, where it is the workhorse format for low-precision inference. E4M3 has only 256 encodings, so we can comfortably show every value at once and still have room to think. The usual picture is a single line The traditional way to visualize floating-point numbers is to put them on a real number line. For E4M3 the result is accurate but not very useful: most of the action is squeezed near zero, huge gaps open up near the maximum, and there is nothing on the page that tells you why the spacing changes the way it does. (You can see for yourself in the linear number line panel at the bottom of the embedded explorer further down.) The structure that makes floating-point floating-point, the partition into binades, is exactly the thing the linear axis hides. Zooming in helps a bit, but it doesn’t scale: at any zoom level you only ever see a slice of one or two binades, and the relationship between the binary representation and the decimal positions is still mostly invisible. A 2-D picture, suggested by an LLM After some back and forth with an LLM, a much better idea came up: forget the linear axis, just plot every value by its exponent. The first sketch was crude but got the point across, which was more than I had any right to expect from a vibe-coding session: Log₂ scale (this is the "real" structure) E = -9 • • • • • • • (subnormals) E = -6 • • • • • • • • E = -5 • • • • • • • • E = -4 • • • • • • • • E = -3 • • • • • • • • E = -2 • • • • • • • • E = -1 • • • • • • • • E = 0 • • • • • • • • E = 1 • • • • • • • • E = 2 • • • • • • • • E = 3 • • • • • • • • E = 4 • • • • • • • • E = 5 • • • • • • • • E = 6 • • • • • • • • E = 7 • • • • • • • • 👉 Every exponent bucket has exactly 8 evenly spaced values 👉 That's why FP behaves like a logarithmic number system The reason this works is that every finite floating-point value can be written as $c \cdot 2^{e_2}$ with $c$ an integer in a small range. For E4M3 normals $c \in \{8, \ldots, 15\}$ and $e_2 = E - 10$; for subnormals $c \in \{1, \ldots, 7\}$ and $e_2 = -9$. So every value sits at integer coordinates $(c, e_2)$: the rows are binades, the columns are integer significands, and within each row the dots are linearly spaced. The mysterious “logarithmic spacing” of floating-point numbers is just what you see when you stack these linear rows and look at them from the side. Vibe-coded into something usable A few more iterations and the sketch turned into the interactive explorer below ( standalone page , source ). Click any dot to inspect the value, toggle subnormals and NaN, or scrub through the encoding directly: Two axes, two scales: the binary exponent $e_2$ runs down the left, and the matching decimal exponent $e_{10} = \lfloor e_2 \log_{10} 2 \rfloor$ down the right. A few things to look at: The horizontal line through each row is the binade, extended by half a cell on each side. Those half-cells are exactly the rounding interval : real numbers between them round back to the dot in the middle (modulo rounding-mode tie-breaks I’m glossing over). Crossing each row are vertical decimal ticks at two scales: minor every $10^{e_{10}}$, major every $10^{e_{10}+1}$. They are the only two decimal grids that matter in that binade. Anything coarser misses the rounding interval, anything finer just adds digits. So shortest-decimal conversion is one row’s worth of work: pick a dot, find the coarsest tick that lands inside its rounding interval. Reading the shortest decimal by hand Let’s pick a value and walk through it. Set the encoding to 51 in the explorer (or click the dot at row $e_2 = -4$, column $c = 11$). The bits are 0 0110 011 , i.e. $E = 6$, $M = 3$, so $$ v = (8 + 3) \cdot 2^{6 - 10} = 11 \cdot 2^{-4} = 0.6875. $$ Its row is $e_2 = -4$, with $e_{10} = \lfloor -4 \log_{10} 2 \rfloor = -2$. So on this row: minor ticks are at multiples of $10^{-2} = 0.01$, major ticks are at multiples of $10^{-1} = 0.1$. The dot’s neighbors in the binade are at $10 \cdot 2^{-4} = 0.625$ on the left and $12 \cdot 2^{-4} = 0.75$ on the right. The half-cells around the dot therefore span $(0.65625, 0.71875)$, the rounding interval. Now eyeball the ticks in that interval: The major tick at $0.7$ sits right inside it. (Several minor ticks $0.66, 0.67, \ldots, 0.71$ are also inside, but we don’t care: the major tick already gives the shortest answer.) Therefore the shortest decimal that round-trips through E4M3 to $0.6875$ is simply 0.7 . No tables, no big-integer arithmetic, just one dot and the ticks on its row. This is exactly what Schubfach (and Dragonbox, and Żmij) compute for double , just at a much larger scale and with a lot more arithmetic to keep track of: pick the coarsest decimal grid whose spacing still fits inside the rounding interval, then round the value to that grid. Where the special cases go Subnormals turn out to be just the bottom row $e_2 = -9$, with the column index running $1\ldots 7$ instead of $8\ldots 15$ and the leftmost half-cell extending a little further than usual. The irregular rounding intervals at powers of two show up at the leftmost normal column ($c = 8$ in every row except the bottom), where the left half-cell is shorter than the right because the predecessor sits in the binade below and is only half a ULP away. Even the e10 vs e10 + 1 decision in Schubfach has an obvious reading: “does the major tick fit inside the interval, or do we have to fall back to minor ticks?”. The full explorer is a single HTML/JS/SVG file with no build step; if you want to fork it, adapt it to a different format ( E5M2 ? bfloat16 ? Whatever your hardware vendor invents next quarter?), or just read how it’s wired together, the source is in the website’s repo . Happy floatspotting! window.MathJax = { tex: { inlineMath: [['$', '$'], ['\\(', '\\)']], displayMath: [['$$', '$$'], ['\\[', '\\]']] }, svg: { fontCache: 'global' } }; - https://vitaut.net/posts/2026/every-float/ -📝vitaut.net

Friday, May 1, 2026

Thursday, April 30, 2026

_Adventure:_ Is there light in the cobble crawl?The original _Colossal Cave Adventure_ consists basically of a Fortran source file and a textual data file. These files would often travel from one installation to another via paper printouts: printed out at one site, typed in by hand at another. The lines of WOOD0350's Fortran source (intentionally or not) never exceed 80 columns regardless of your tab stop. But the data file fits within 80 columns only with a tab stop of four. With an eight-space tab stop, four lines of the data file exceed 80 columns:📝Arthur O’Dwyer

Wednesday, April 29, 2026

My 7-min “lightning talk” is online: Why C++ is growing, and why C++26 will likely be adopted quicklyAt the London C++ meetup last month, I participated on a panel where each panelist gave a short introductory presentation. My 7-minute intro (aka “lightning talk”) just got posted — you can view it here. The one-sentence blurb: “C++ is accelerating, and C++26 is built for what developers need now.” Also check out the longer … Continue reading My 7-min “lightning talk” is online: Why C++ is growing, and why C++26 will likely be adopted quickly →📝Sutter’s Mill
Figma to Qt 1.0 Is Here: The Most Reliable Way to Bring Your Design From Figma to DeviceWhen a perfectly crafted design leaves Figma and enters a development pipeline, things often get compromised. Spacing shifts, colors change, and details deviate through multiple layers of interpretation and unexpected limitations. You will see timelines lagging, design becoming unclear, and user experience losing priority. Figma to Qt is built to ensure your GUI designs get from Figma to device . Version 1.0 is now available for free download in the Figma Community .📝Qt Blog
One Map Key, One Lookup@media only screen and (max-width: 600px) { .body { overflow-x: auto; } .post-content table, .post-content td { width: auto !important; white-space: nowrap; } } This article was adapted from a Google Tech on the Toilet (TotT) episode. You can download a printer-friendly version of this TotT episode and post it in your office. By Roman Govsheev Can you spot the wasted CPU cycles in the map usage? if employee_id in employees: mail_to(employees[employee_id].email_address) The redundant lookup caused the waste by performing a check ( in ) and a fetch ( [] ) as two separate operations when one is sufficient. Every lookup involves a cost —whether it's computing a hash and scanning buckets or performing an O (log n ) traversal. These costs add up quickly. But avoiding them isn’t just “premature optimization”—it’s about writing cleaner, more robust code that stays efficient at scale and prevents potential race conditions. Instead of paying this cost twice, perform the lookup once and reuse the result: if ( employee := employees.get(employee_id)) is not None: mail_to( employee .email_address) Assigning the search result to a variable avoids a second lookup. This efficiency is native to Go via the “comma ok” idiom ( val, ok := map[key] ) and C++ using map.find(key) , both handling retrieval and existence in a single pass. The same inefficiency applies when counting or initializing default. Stop checking for presence; instead, use idioms that handle missing keys automatically at the container level : The redundant way The efficient way If key not in counts: counts[key] = 1 else: counts[key] += 1 counts = defaultdict(int) # Initializes 0 automatically # ... other logic ... counts[key] += 1 Here are some details depending on which language you use: C++: operator[] returns a reference to the value—automatically inserting a default (like 0) if the key is missing—allowing the increment to happen in place. Java: Use map.computeIfAbsent() to perform retrieval and updates in a single call. This is more concise and, on concurrent collections, has the potential to be thread-safe—preventing the “check-then-act” race conditions common with separate contains and put calls. Python: Use collections.defaultdict to handle defaults at the container level, which pushes the logic into optimized C code for better performance and robustness. Note that the += operation (shown later in the above code sample) still involves both a read and a write operation. Go: Use val, ok := map[key] to handle retrieval and existence in one memory access.📝Google Testing Blog

Tuesday, April 28, 2026

_Adventure:_ Walking on the ceilingOn 2012-12-01 I wrote to Don Woods (in a postscript to a production update on [_Colossal Cave: The Board Game_](https://boardgamegeek.com/boardgame/121751/colossal-cave-the-board-game)): > By the way, I just noticed last week that in "Adventure", in the Hall > of the Mountain King, the directions NORTH and LEFT are synonyms, as > are SOUTH and RIGHT... as are WEST and FORWARD! West being forward > makes sense, if the Hall of Mists is back to the east; but for the > rest I suppose the adventurer must be walking on the ceiling. :) This > little mixup is present all the way back to > [Crowther's code](https://github.com/Quuxplusone/Advent/blob/master/CROW0005/advdat.77-03-11#L249-L253). > I just thought it was funny that nobody had commented on it before, as > far as I know.📝Arthur O’Dwyer

Monday, April 27, 2026