Daniel Lemire's blog

JSON is a standard format for data interchange. It is effectively a tiny subset of JavaScript made of objects and arrays. It looks as follows {"key":1, "text":[1.0,2.0]}.

Many programming languages include a JSON library in their standard libraries: C#, Go, Java (soon), Python, JavaScript, etc. The Go implementation is convenient, but not especially fast.

Go 1.27 makes a new JSON package (encoding/json/v2) available by default in its standard library. The two APIs look almost the same:

import (
    json    "encoding/json"
    jsonv2  "encoding/json/v2"
)
b, err := json.Marshal(v)
b, err  = jsonv2.Marshal(v)
err = json.Unmarshal(b, &v)
err = jsonv2.Unmarshal(b, &v)

The two are not directly comparable as they differ with respect to Unicode validation, case sensitivity, etc. So it is not a drop-in replacement.

However, the legacy API (encoding/json) has also been reimplemented on top of the new engine. You can use the legacy API with either the new engine or the old one (GOEXPERIMENT=nojsonv2) through a flag. So we have three possibilities.

1. json (legacy) — encoding/json built with GOEXPERIMENT=nojsonv2, the original implementation
2. json (Go 1.27) — encoding/json as of 1.27, v1 API on the v2 backend
3. json/v2 — encoding/json/v2

I used the usual simdjson documents: twitter.json (632 kB, nested objects with short string keys), canada.json (2.25 MB, one large array of coordinates), and citm_catalog.json (1.73 MB, nested objects with numeric keys). I parse them into any (interface{}), which is the general-purpose path.

I ran this on an Apple M4 Max and on an Intel Xeon Gold 6548N (Emerald Rapids) using Go 1.27.0, on a single core (GOMAXPROCS=1), reporting the median of eight runs.

When unmarshalling, the legacy API on the new backend is faster than the original on twitter.json (172 MB/s to 203 MB/s) and on citm_catalog.json (186 MB/s to 241 MB/s), but slower on canada.json (128 MB/s down to 106 MB/s). When marshalling, it is up to twice as fast: 198 MB/s to 374 MB/s on twitter.json. So merely upgrading to Go 1.27, without changing a line of code, should make marshalling faster.

Switching to the new API helps more. Compared to the original implementation, encoding/json/v2 unmarshals 1.5x to 2.3x faster and marshals 1.2x to 3x faster. Compared to the Go 1.27 legacy API, unmarshalling gains another 1.8x to 2x, while marshalling gains much less (1.0x to 1.7x): part of the remaining difference is that json/v2 does less work during marshalling.

Thus far, I was unmarshalling into any, meaning that I assumed that I did not know the structure of the document. I also round-trip a slice of 10,000 small structs:

type Record struct {
    ID     int      `json:"id"`
    Name   string   `json:"name"`
    Email  string   `json:"email"`
    Active bool     `json:"active"`
    Score  float64  `json:"score"`
    Tags   []string `json:"tags"`
}

The schema is specified: the JSON must be [{"id":..., "name":...}, {"id":..., "name":...}...]. I still get faster unmarshalling with the new API, but the legacy API with the legacy engine is faster when marshalling.

The original implementation is faster. The Go 1.27 release notes said that marshal performance is broadly at parity with the previous implementation. For my test, it is not the case.

So unmarshalling gets faster across the board with encoding/json/v2, and marshalling gets faster for any, but it is about 1.5x slower for typed structs in my tests.

The code is available.

In Java, you find the location of a substring using indexOf.

String haystack = "The quick brown fox jumps over the lazy dog";
String needle = "fox";
int index = haystack.indexOf(needle);

Naively, you might implement indexOf by a loop inside a loop, like so.

int naiveIndexOf(String haystack, String needle) {
    for (int i = 0; i <= haystack.length() - needle.length(); i++) {
        int j = 0;
        for (; j < needle.length() 
          && haystack.charAt(i + j) == needle.charAt(j); j++) {}
        if (j == needle.length()) { return i; }
    }
    return -1;
}

The Java implementation is much more sophisticated, and it is highly accelerated.

However, there are pathological cases where the Java implementation can be slow. What do I mean? Well, you do expect that the search will be more and more expensive as the size of the string grows. Right? So if you search through a 1 kilobyte string and then search through a 10 kilobyte search, you would not be surprised if the latter takes ten times slower.

But what of the substring? If you search for short substrings (fox in my example), the everything is fine. But what if you search longer and longer substrings (fox jumps or fox jumps over)? If it gets more expensive when both the string and the substring get longer, then you have what we call a quadratic complexity. In other words, it is slow.

In Java, if n is the length of your string and m is the length of the substring, then the complexity of indexOf is O(n·m). And if you look at my naive implementation (naiveIndexOf) then you see that in the worst case, it might do up to close to haystack.length() * needle.length() comparisons, that is, it is O(n·m).

The exact implementation of the indexOf function depends on your CPU and Java version. I am using OpenJDK 25 on Apple Silicon (ARM). For my purposes, I will use as a haystack of n copies of a and for the needle, the same thing, but ending with a different letter.

// n > m
String haystack = "a".repeat(n);
String needle = "a".repeat(m - 1) + "b";

I measured OpenJDK 25 on an Apple M4 Max. The haystack is one megabyte. Numbers are nanoseconds per haystack character.

At m = 4096, a single indexOf over one megabyte takes 1.1 seconds.

Can you do better against such adversarial inputs? The textbook solution is the Two-Way algorithm of Crochemore and Perrin (1991). The implementation is simple and your favourite AI can code it for you in any programming language.

Two-Way stays at about 0.3 ns/character no matter how long the needle is. At m = 4096 it is about 3500 times faster than indexOf on the first-character adversary.

So, should you switch to Two-Way for everything? No. On random text, the indexOf function is much faster than Two-Way.

And Two-Way has to do non-trivial work before the search begins. So it has additional fixed overhead. It would lose most of the time in the real world, sometimes by a wide margin.

Should you worry about this? No. The indexOf function in Java is fine.

If an adversary can control the needle (substring), then make sure to reject long needles. Most of the time, we search for short sequences (say, less than 80 characters). If you are worried about your system crashing, you will put bounds on inputs in any case.

The Java source is available.

Further reading: Crochemore, M., & Perrin, D. (1991). Two-way string-matching. Journal of the ACM, 38(3), 650–674.

We are all familiar with IP addresses such as 192.168.0.1. They are typically written as four numbers in the range 0 to 255 inclusive, separated by dots. In C#, you can parse them with the standard library using IPAddress.TryParse.

Pedantic people are quick to point out that IP addresses can take different forms: they can be IPv6 or IPv4 and there are many weird ways to write an IPv4 address. But for the purpose of performance optimization, we care about the common case. The common case is strings such as 192.168.0.1 or 12.121.244.111.

Our processors are capable of data parallelism, meaning that they have instructions (called SIMD) that can process several bytes at once, at least 16 bytes, sometimes more. A few years ago, I showed that you can parse IPv4 addresses with SIMD. I have been revisiting this idea with AVX-512, the instruction set that recent x64 (AMD/Intel) processors support. I expect that all Intel and AMD processors made in the near future will have great support for AVX-512, and it is already the case for server processors and recent AMD processors.

So I wondered, could we do it in C#? People are sometimes surprised that I care about C#. Isn’t that more Microsoft slop? No. Not at all. C# and .NET are very reasonable, portable systems.

Plus you can write fast code in C#. I have two optimized libraries that I hope the Microsoft .NET team will one day adopt in the standard .NET library: an optimized Utf8Utility.GetPointerToFirstInvalidByte function used internally to validate Unicode strings (in the SimdUnicode library) and a fast base64 decoding library. I love working with .NET C#.

As of .NET 10, we have AVX-512 support, including masked loads. What are masked loads and why do they matter? Suppose that I give you a string that is no longer than 16 bytes, but could be shorter. If you load data in a SIMD register, you normally have to load the full register width (so 8, 16, 32, 64 bytes). So what do you do when it is not possible? You can pad the input string or pull other tricks, but it gets dirty. A nice approach is to have masked loads where you, say, load the full register (say 16 bytes), but you indicate which bytes you want to be loaded from memory with a mask. So if you use 0b10011 as a mask, then only the first, second, and fifth bytes are loaded from memory. This makes it possible to initialize a 16-byte register with a string that has between 0 and 16 bytes, while never reading beyond the string. I have an article entitled Modern vector programming with masked loads and stores if you want to know more.

To make things trickier, C#, like Java and JavaScript, defaults to UTF-16, meaning that each character, even if it is an ASCII character like A or 1, uses two bytes. The ASCII codepoint value occupies the least significant bits of a 16-bit word.

So what we need to do is to selectively load from a 32-byte input, and then drop the unnecessary zero bytes. The gist of it looks as follows in C#.

unsafe bool TryParseAvx512(ReadOnlySpan<char> s, out uint ip) {
        int len = s.Length;
        fixed (char* cp = s)
        {
            // next two lines are a trick to load just the first len characters
            Vector256<ushort> charMask = Vector256.LessThan(CharLaneIndex, Vector256.Create((ushort)len));
            Vector256<ushort> chars = Avx512BW.VL.MaskLoad((ushort*)cp, charMask, Vector256.Create((ushort)'0'));
            // check that everything is ASCII otherwise, it is not an IP!
            if (Avx512BW.VL.CompareGreaterThan(chars, Vector256.Create((ushort)0x7F)).ExtractMostSignificantBits() != 0)
            {
                return false;
            }
            // There we go, we have the address as ASCII
            // in a 16-byte register.
            Vector128<byte> str = Avx512BW.VL.ConvertToVector128Byte(chars);
            // ...
        }
}

This looks a bit difficult to read, but that’s fine. Most people never need to worry about such code.

Then we use a somewhat fancy trick where we locate the dots, and use the fact that there are only 81 ways to position the dots. We then move the bytes, do a dot product and validate. It is the same routine as the C++ code. It is not trivial, but I am working on a formal paper to document the tricks used.

The pedantic people will say: wait, there are other ways to write IP addresses !!! Ok fine. We handle them with a fallback, like so.

if (TryParseAvx512(s, out uint ip))
{
    address = new IPAddress(ip);
    return true;
}
return IPAddress.TryParse(s, out address);

What about the cases where your processor does not support AVX-512? C# makes this dead easy. You can just guard it with one if:

if (Avx512BW.VL.IsSupported) { ... }

To benchmark this, I generated 10,000 random 32-bit addresses and parsed the resulting strings 20 million times, constructing an IPAddress each time. On a relatively recent Intel processor (Intel Xeon Gold 6548N, Emerald Rapids) running .NET 10, I get the following.

So the AVX-512 approach is about three times faster than the standard library. My routine itself does not take fourteen nanoseconds; there is other overhead.

As usual, the C# source is available.

Like most programming languages, Go has both stack allocations, whose lifetime is limited to the current function, and dynamic (or heap) allocations.

The name stack comes from the fact that the memory management is somewhat trivial. There is typically one stack per thread (or goroutine in Go). When a function needs memory, it simply appends data to the stack. When the function returns, the memory is dropped from the end of the stack. So the memory last allocated is deallocated first.

Heap memory is potentially considerably more complex. For one thing, it is meant to be accessible by several threads (or goroutines). An object can be allocated by one function and later reclaimed after an entirely different function, possibly running on a different thread (or goroutine), has dropped the last reference to it. Unlike the stack, there is no prescribed order for allocating and reclaiming heap memory. In Go, the garbage collector does the reclaiming.

Typically, stack allocations have a size known at compile time. Many systems give each thread a fixed-size stack, although Go grows goroutine stacks as needed.

There are many ways in Go to do a heap allocation. A common one is when you allocate a slice, as in this instance where you allocate memory for 100 integers:

x := make([]int, 100)

If the slice x is not entirely local to a function, Go will typically just allocate it on the heap. It will do so similarly when a function returns a pointer. For example, in the following instance, I assign the value 1 to a local integer variable, but I return a pointer to it.

func f() *int {
  x := 1
  return &x
}

In C/C++, this would be quite bad. You should get a warning such as address of local variable 'x' returned. In Go, the variable x will typically get allocated on the heap.

In many Go programs, we end up doing a lot of heap allocations of small objects. It can become a bottleneck in some cases. Think about when you are maintaining a tree or a linked list where each value (node) is an object that must live on the heap. If the data structure is highly dynamic, you will be constantly allocating these small objects.

Memory allocation on the heap is usually not done at arbitrary sizes. You often cannot get exactly, say, 13 bytes. In Go, small allocations are rounded up to a size class: 8 bytes, 16 bytes, 24 bytes, 32 bytes, and so forth. There is also some overhead to each heap allocation, from rounding and from allocator metadata.

The compiler knows the size of the object, but prior to Go 1.27, Go would call a generic function when doing a heap allocation. This generic function would then look up the size class and take the corresponding path. Starting with 1.27, for small objects (under 80 bytes), Go relies on dedicated functions.

It is easy to benchmark in Go. A basic benchmark might look as follows.

type Node struct {
    value int64
    next  *Node
}
var sink any
func BenchmarkAllocNode16(b *testing.B) {
    for b.Loop() {
        sink = &Node{}
    }
}

On my MacBook, the results are quite telling. Go 1.27 is nearly twice as fast!

This will not help all software, just the components that do many small allocations.

The code is available.

AI-assisted programming is fast evolving and there is a tension between ‘we no longer need to understand the code’ and ‘what is my purpose as a programmer’. I recorded a short video on this topic with how I think the tension can result in conflicts.

When a compiler optimizes your program, it has to guess. Which functions are worth inlining? Which side of a branch is the common one? Which method does this interface call actually reach? At compile time it cannot know, so it uses heuristics. Profile-guided optimization (PGO) replaces the guessing with measurement: you run your program, record where it spends its time, and hand that recording back to the compiler for a second build.

PGO is a common feature of compiler systems. Google applied PGO to Chrome under Windows in 2016, reporting gains of up to 15%. I expect all mainstream Web browsers to be built with PGO.

There are now fancier techniques than mere heuristics with PGO. You can use AI to recognize patterns and so forth. But they are not always widely available.

Go has supported PGO since version 1.20. You collect a profile, and pass it to the compiler.

A CPU profile is a statistical record of where a program spends its time. While the program runs, the Go runtime interrupts it about a hundred times a second and writes down the call stack at that instant. After a few seconds you have thousands of such samples, and counting them tells you which functions were executing and who called them. In Go you produce one by wrapping the work you care about:

f, _ := os.Create("cpu.pprof")
pprof.StartCPUProfile(f)   // from runtime/pprof
defer pprof.StopCPUProfile()

The compiler reads the call-stack counts and uses them for two things above all: inlining call sites that turn out to be hot, and devirtualizing interface calls whose target is nearly always the same concrete type.

I took three JSON documents that I wanted to parse:

• twitter.json (632 kB), a nest of small objects with short string keys
• canada.json (2.25 MB), essentially one enormous array of floating-point coordinates
• citm_catalog.json (1.73 MB), deeply nested objects with numeric keys

I parse each of them with the standard library’s encoding/json into an interface{}. The baseline, with no profile, parses at 112 MB/s for twitter.json, 74 MB/s for canada.json and 116 MB/s for citm_catalog.json.

The procedure is three commands:

go build -o bench .                        # ordinary build
./bench -profile cpu.pprof -train twitter.json   # collect a CPU profile
go build -pgo=cpu.pprof -o bench_pgo .     # build again, with the profile

I did it three times, profiling each document on its own, and then measured all three documents against each of the three builds.

Each panel of the figure is one document being parsed, and the three bars inside it are the three PGO builds: the binary trained on twitter.json, the one trained on canada.json, and the one trained on citm_catalog.json. Bar height is the speed gain over the ordinary, profile-free build of that same document, in percent, so zero means PGO changed nothing and a bar below the axis means the PGO build was slower. The green bar in each panel is the matched case, where the profile was collected on the very document being measured.

The gains are modest. The best result is canada.json at +4.7%, and most differences are in the 2–3% range. Profiling one document usually helps the others, but not reliably. Profiling twitter.json gave a decent improvement everywhere: +3.1%, +2.0%, +2.8%. But profiling canada.json bought 4.7% on canada.json and essentially nothing anywhere else. Interestingly, profiling citm_catalog.json produced a mere +0.8% on its own document while helping twitter.json more.

A 3% speedup is not exciting in isolation, but it may come nearly for free. Observe how you may get slightly negative results for cases you did not train for. That’s expected generally, but the effect is modest in the case of Go because its optimizations are themselves modest in the first pace. That is, you are not getting a much an effect, but the process is less likely to backfire for other workloads.

The code is available.

C++26 adds a new container to the standard library: std::hive. It is meant to occupy the ground between std::vector and std::list. Like a vector, it keeps its elements in contiguous blocks of memory, so scanning it does not require you to chase a pointer for every element. Like a list, it never moves an element once it has been inserted: your pointers, references and iterators stay valid, and you may erase any element in constant time without disturbing the others.

Internally, a hive is a linked list of blocks. Each block carries a skipfield: a small integer per slot that tells the iterator how many erased slots to jump over.

No standard library ships std::hive yet to my knowledge. Fortunately there is an implementation (plf::hive by Matt Bentley) as a single header file that you can use today.

I use elements of type uint64_t, GCC 16.1 with -O3 -march=native, on an Intel Xeon Gold 6548N (Emerald Rapids), pinned to one core. Numbers are nanoseconds per element, along with the cycles and instructions retired per element.

We start from an empty container and append a million values. The container is then destroyed.

A std::list needs one allocation per element, and glibc’s malloc and free together cost over 200 instructions per element. It is an order of magnitude behind everyone else. That is not news.

The interesting comparison is vector against hive. A hive is about twice the cost of a vector, and it needs twice the instructions. This is the price of the skipfield: every insertion writes an element and a skipfield entry, and maintains the block bookkeeping. Note that calling reserve on a hive does not help in my experiments.

Next we iterate over the the container and sum the values.

A hive iterates no faster than a linked list here, slightly slower, in fact, and about eight times slower than a vector. (Update: Joseph Garvin points out that I measure the happy case for the std::list in this instance where all the entries were allocated in sequence. The worst case scenario for std::list when the nodes are all over the heap can be much slower.)

The vector loop retires one instruction per element and finishes in 0.78 cycles: the processor is executing several elements at once. This is possible because the std::vector implementation benefits from autovectorization: the compiler recognizes that it can load several words at once in wide (SIMD). Further, it does not have to check the bitfield like the std::hive data structure.

We can check this. Walk the same container with two independent iterators, one starting halfway in, and count the cost per element visited:

The vector does not care: it was already throughput-bound. The hive and the list get nearly twice as fast per element, because two independent chains can be in flight at once. Hive iteration is latency-bound, exactly like list iteration. It merely has better locality.

That locality does show up when the data gets big. At ten million elements the list falls apart while the hive holds steady:

Erasing is what a hive is for, so it would be unfair not to look. I erase half the elements at scattered positions using std::remove_if:

The hive wins, but by less than you might expect, and at ten million elements the ordering reverses (1.3 ns for the vector against 2.5 for the hive). std::remove_if is a single streaming pass, and streaming passes are cheap. Of course the vector moved every surviving element and invalidated every pointer into it, which is precisely what a hive promises not to do.

Memory, measured by asking glibc how many bytes it has handed out, per live element:

A hive costs about a byte per element over a vector, for a payload of eight bytes, when the vector is packed tight. A list costs more due to the overhead of the linked list.

A vector built by push_back has a capacity that typically exceeds its size. Thus even if you have 8-byte entries, you will use, on average, more than 8 bytes per entry even for large vectors. You can recover the excess capacity with the shrink_to_fit method.

What should we conclude?

The std::hive data structure is not a faster vector. But it is a much better std::list. It gives you the same guarantees that make people reach for a list, stable references, cheap erasure anywhere, while using less memory.

My source code is available.

When your program asks for memory that is not in cache, the processor has to go to RAM. That trip costs on the order of 100 nanoseconds. On a 3 GHz core, that is about 300 cycles of doing nothing.

Memory latency has not improved in ten years. The 2016 Broadwell answers a random access in 100 ns. The 2025 Turin, with DDR5-6400 and every advantage of a decade of progress, takes 140 ns. It got worse.

The good news is that a modern core does not have to sit still. It can issue a second request before the first one comes back, and a third, and a tenth. The number of requests a single core can keep in flight is its memory-level parallelism. It is one of the most important numbers in software performance, and one of the least advertised: you will not find it on a spec sheet.

Thankfully, memory-level parallelism has improved a lot. To measure it, I use my testingmlp benchmark. The idea is a pointer chase. We build a 1 GiB array containing a single random cycle covering every element: each element holds the index of the next. Following the cycle is inherently serial. Each load has to complete before you know the address of the next one, so a single chase measures pure memory latency and nothing else. Then we run several such chases at once, from different starting points on the same cycle. We call these lanes. With two lanes, the core has two independent loads to work on. With twenty, twenty. We increase the number of lanes and watch the throughput. When adding a lane stops helping, we have found the limit. As my metric, I use the total estimated bandwidth.

I ran experiments on the Amazon cloud (AWS). The bandwidth shape is the same everywhere: a steep, nearly linear climb as we add lanes, then a knee, then a plateau. 

How did it evolve over time? Intel went from 10 to 30, meaning that a single Intel core can sustain 30 memory requests at once in practice. AMD went from 15 to 58. Graviton went from 6 to 19.

Intel was flat for a long time. Broadwell and Cascade Lake both sit at 10 concurrent misses. Ice Lake doubled it to 20. Granite Rapids is at 30. Intel has roughly tripled in a decade, with all the gain arriving in the last two generations.

AMD started ahead and stayed ahead, then jumped. Naples was already at 15 in 2018, when Intel was at 10. Milan reached 22. And then Turin does something different in kind: 58 concurrent cache lines from a single core.

Graviton 1 was a toy: 6 concurrent misses. Graviton 2 doubled it, Graviton 3 went to 17, and then Graviton 4 essentially stood still at 18. Graviton 5 only reaches 19. But look at the latency panel: since 2017, Graviton 5 is the only chip in this entire collection that made a random access faster than its predecessor. AWS advertised better DRAM latency for Graviton 5, and that claim holds up.

So who wins? On bandwidth and memory-level parallelism, it is AMD, and it is not close. The Zen 5 core in the m8a instances sustains 58 concurrent cache-line fetches and 24.5 GiB/s of random-access throughput from one core. AMD is roughly twice as fast as Intel.

The raw output, the system information from each machine, and the scripts are in the usual place.

Note that Apple Silicon does even better, but it is another category.

Every week, I discuss with people who want to get a PhD. For years, I have been advising people not to pursue a PhD. It may come as a surprise to some.

You would expect people with a PhD to earn more money. Individuals who complete doctorates tend to have higher cognitive abilities and greater motivation. But smarter people tend to earn more, period.

So do people with a PhD earn more?

Historically, PhD holders earn more, but the bulk of the observed advantage is concentrated among those who get a professorship after the PhD. And there is no certain path from the PhD to a professorship. We have been producing many more PhDs than we have professorship, for decades. And the disparity is ever growing.

When I entered university at the beginning of the 1990s, about 0.5% of the Canadian population had a PhD. This has nearly tripled today, and it is fast increasing. Something of the order of one person out of 80 has a PhD. Comparatively, there is roughly one professor or university-level instructor per 900 people. With a fast aging population, we simply do not need many more professors and instructors than we already have.

There are specific fields where some jobs are difficult to get without a PhD. Machine learning is one such example. Many people in the industry have a PhD, and they tend to select those who also have a PhD. Further, there is a somewhat direct relationship between the work you might do during your PhD, if you are any good, and the actual work you might do later. It is much less clear in a lot of other disciplines.

The most significant economic cost of a PhD is not tuition but the years of delayed full-time earnings and career progression. In the tech industry, it is typical to award half a year of experience for each year spent on a PhD. This means that even though the individual starting with a PhD might earn more starting out, they are not necessarily getting a higher lifetime income.

Benjamin et al. (2025) find that the early-career benefits a PhD can be effectively zero:

In the short run, pursuing a PhD entails substantial opportunity costs. Early-career earnings for PhD graduates are significantly lower than those of individuals with master’s or professional degrees, reflecting prolonged enrolment and delayed entry into the labour market. These costs are especially high for non-completers, particularly those who exit the program after several years without earning a credential. Over the lifecycle, earnings do eventually recover (and surpass those of bachelor’s and master’s graduates) but only under specific conditions. The most favourable long-run outcomes are concentrated among those who secure academic employment and remain in full-time work late into life. This “double premium,” combining higher earnings and longer careers, plays a central role in shaping the average return to a PhD. Outside academia, PhD holders resemble master’s graduates in both earnings and employment patterns.

Thus, the financial case for a PhD is narrower than people assume. If you fail to get a professorship, or you want an early retirement, you may very well end up with a poor outcome. And it is not getting better over time.

References

• Altonji, J. G., & Zhu, Z. (2025). Returns to specific graduate degrees: Estimates using Texas administrative records (NBER Working Paper No. 33530). National Bureau of Economic Research. https://www.nber.org/papers/w33530
• Benjamin, D., Miloucheva, B., & Vigezzi, N. (2025). The opportunity cost of a PhD: Spending your twenties (Working Paper No. 802). University of Toronto, Department of Economics. https://www.economics.utoronto.ca/public/workingPapers/tecipa-802.pdf
• Cooper, P. Is grad school worth it? A comprehensive return on investment analysis. Foundation for Research on Equal Opportunity. https://freopp.org/whitepapers/is-grad-school-worth-it-a-comprehensive-return-on-investment-analysis/

In 1945, Vannevar Bush published a report entitled Science: The Endless Frontier. His thesis was that prosperity follows from basic research. The report was highly influential in the United States and elsewhere. It led to the creation of an entirely new government bureaucracy.

With this report, Bush popularized the linear model of innovation: innovation (such as medical cures) flows sequentially from basic research to applied research to development to production and diffusion. Grow basic research, and the rest will follow.

When Bush wrote his report, basic research was not usually supported directly by the state. We did not have a large basic research infrastructure. And yet, the West had just lived through an unprecedented period of rapid scientific progress: the theory of evolution, electromagnetism, radio communication, special and general relativity, quantum mechanics, nuclear technology, rockets, the combustion engine, and more. We would get the invention of the transistor only two years after Bush’s report. We also did not have today’s peer-review mechanism.

Even though Bush’s report has been viewed as a piece of genius that unlocked a golden era of scientific prosperity, I believe that the linear model of innovation is hopelessly naïve. I believe the thesis that a large bureaucracy delivering funding to other bureaucracies (such as universities) is how we get innovation is absurd. Except perhaps in the domain of computing (“bits”), we have been largely stagnant technologically since about the 1970s. So Bush’s model failed over time. To be clear, it might have worked for a while by encouraging more young people to study engineering and science. It may also have shone a favorable light on a few enterprising professors who got to promote useful ideas.

If you visit a research lab today in a leading university, what you are most likely to see is a boring bureaucracy that caters to whatever is politically favorable at the moment—a bureaucracy that plays it safe and avoids controversy. You see young people seeking well-paid jobs, going through the motions with often little genuine interest in, say, curing cancer. We have never published so many research papers—the volume has been growing exponentially ever since Bush wrote his report—but it is doubtful that this is how technological breakthroughs are achieved.

The evidence is overwhelming that shoddy science is widespread. We have a severe reproducibility crisis: if you redo an experiment (even a highly cited one), you are likely to fail to reproduce the results. This affects psychology, medicine, and many other fields. The system does not particularly care because the incentives to get things right are not there. As long as the work is politically aligned, solidity of the results seems secondary.

There was a TV show (The Big Bang Theory) where the main character, Sheldon Cooper—an awkward genius—gets to work on crazy ideas. That is how Bush imagined it: fund young people like Sheldon Cooper, and you will get extraordinary breakthroughs. In the real world, Sheldon would not get very far on campus. I have met misfits like him. When they are incapable of playing the political game, the system crushes them. But even if that were not the case, extraordinary intelligence needs to be applied to the right problems to be of value. You could have a ChatGPT that is brighter than any of us in every possible way, and it could still be deployed simply to fill out forms faster and better than we do—it may not cure cancer.

The American government has just released what might be considered an update to Bush’s report. Michael Kratsios wrote a report entitled Science, A New Golden Age. The report states outright that the linear model no longer holds. It states what I have argued for vehemently: innovation is not a linear process. Take large-language models, for example, which can be used by engineers and scientists to further their research. I have also argued that the success of large-language models today has as much to do with the users as with the researchers.

At this point, some people engage in the following type of rhetoric: if we had not invented calculus, we would not have AI today; therefore, calculus caused AI. But you could also say that the subsidized nail factory in the Soviet Union, which made overpriced and bad nails, was necessary to hold Landau’s house together, and that without those nails we would not have the theory of Landau levels. The causality argument goes in all directions.

Innovation is the result of a complex system. We see that the United States and, more recently, China are innovative countries. In 2026, you do not go to France for the latest advances. The evidence is overwhelming that scientific and technological progress depends as much on culture as on anything else. It is not something to be managed by bureaucrats.

One of the cultural ingredients that seems essential is meritocracy. You must put the people who are good at building on top of your hierarchy. This does not happen magically. You need a set of incentives in which rewarding the wrong people is costly.

What does Kratsios propose? Many interesting ideas that, I expect, could renew our culture. He proposes to break out of the Cold War–era funding model. Today, the research funding mechanism is centered around the government giving money to the university bureaucracy. The grant might be in the name of one professor, but the recipient is still the university. In the new model, instead of funding universities, the government would assign money directly to individuals in various ways (short grants, prizes, and so forth). This would shift power away from administrators toward individuals who know how to get things done. It would also neutralize some of the political power of the current mandarin class of scientists who control access to the top positions.

The report recommends restoring permissionless innovation. It is sometimes poorly understood how limited the system has become. I once had a graduate student undertake interviews with practitioners. This required an ethics approval which, in her case, took a few months to obtain. Again, the system has built up political structures that seek to block innovation it does not like. They need to be torn down, the sooner the better.

The report has many other interesting recommendations. One that I particularly like is an AI-guided agenda. We need to hook up our brand-new AIs to experimental devices. We are not going to cure aging with chatbots. We need experiments on a massive scale.

Will Kratsios’s vision move from report to reality? History shows that cultural and institutional change is never easy. Yet the stakes could not be higher. By embracing meritocracy, permissionless innovation, and ambitious AI-augmented experimentation, we have a genuine chance to escape decades of stagnation and rekindle the spirit of discovery that once defined the West. The opportunity is before us. It must not be squandered.

I have been on X for a long time. Like most people who post regularly, I have a gut feeling for what might interest people. I post in the morning. Longer posts seem to do better.

But gut feelings are not measurements. And until recently, digging into your own posting data meant either clicking around the web UI or writing custom scripts. Neither is particularly friendly when you want to ask ad hoc questions with an AI assistant.

X recently launched hosted MCP servers: official endpoints that AI tools can connect to. MCP is a protocol for plugging tools into language models: the model can search posts, manage bookmarks, fetch trends, and so on. In practice, I connected an AI coding agent to the X MCP server and simply started asking questions about my account.

I spent a session exploring about two months of my own activity. Here is what I found interesting.

Over roughly sixty days (mid-May through mid-July 2026), I published on the order of 435 posts that were not pure retweets of other people—mostly a mix of original posts, replies, and a few X Articles. The agent pulled them through the MCP tools, kept the public metrics (likes, views, reposts), and ran simple analyses.

I asked for every post to be binned by local hour of day (America/Toronto, Eastern time), and for each hour: how many posts, and the min / median / max view count.

My posting is heavily skewed toward the morning:

The 9 a.m. hour is both my busiest and, among busy hours, my strongest by median views. The overall median across all hours was only about 188 views, so most of what I write is quiet. The distribution is heavy-tailed: a few posts get tens or hundreds of thousands of impressions; the rest are background noise.

I then binned posts by character length in steps of 25 characters (using the text as returned by the API, including short t.co URLs).

The bulk of my writing is short, often a reply of a few dozen characters:

Under about 175 characters, the median stays at zero or one like. Around full-length posts (roughly the old 280-character regime and a bit beyond), engagement jumps. The 300–325 character band is where a large fraction of my “serious” posts live, and the median likes there are an order of magnitude higher than for short replies.

I also asked the AI to identify the posts that had the most likes, the following types of posts were liked:

• AI vs. “experts” claiming models are nowhere near human intelligence
• Go adding SIMD-style data-parallelism to the standard library
• SIMD-accelerated data processing talks and library notes (JSON, string→integer maps, vulnerability-report fatigue)
• Nvidia hardware, university AI-cheating, C++ contracts

The interesting part is the workflow. I did not export a CSV by hand and open Excel. I asked an agent, connected to X’s MCP server. If AI agents can do this for one account’s metrics, they can do it for bug trackers, logs, paper drafts, and codebases.

Suppose that you have a configuration file in JSON. Something like this:

{ "width": 1920, "height": 1080, "fullscreen": true,
  "title": "My Game", "volume": 0.8 }

Normally you ship this file alongside your program, open it at startup, read it, and parse it. That is a lot of work for data that never changes. What if the file is fixed at build time? Could the compiler read it, parse it, and bake the result directly into the executable as a constant?

With C++26, the answer is yes. We need two new ingredients, all of which are usable right now with the latest version of the GCC compiler (16).

1. #embed to pull the file into the program at compile time,
2. A software library supporting static reflection like simdjson.

Let me show you how far we can take this.

The new #embed directive reads a file and expands it into a comma-separated list of byte values. To read the file data.json at compile time and keep it around as a constant, we write:

constexpr const char json_data[] = {
#embed "data.json"
    , 0
};

I use constexpr because I want the compiler to be allowed to inspect these bytes during constant evaluation. The trailing , 0 simply appends a null terminator, so the array can be treated as an ordinary C string.

There is no run-time input/output of any kind. The bytes are part of the program.

But embedded bytes are not yet useful by themselves. What I really want is a typed C++ object. In my example, the target type is this configuration struct:

struct Window {
  int         width;
  int         height;
  bool        fullscreen;
  std::string title;
  double      volume;
};

The traditional way to populate such a struct from JSON is to write, by hand, one line per field: read "width", store it into width, read "height", store it into height, and so on. It is tedious. And because it runs at startup, a malformed file becomes a run-time error, discovered by your users rather than by you.

Recent versions of simdjson can parse JSON at compile time using C++26 static reflection. The entry point is simdjson::compile_time::parse_json, and it does something I still find slightly magical: it reads the JSON and, from the keys it finds, and synthesises the struct type for you.

#define SIMDJSON_STATIC_REFLECTION 1
#include "simdjson.h"
constexpr const char json_data[] = {
#embed "data.json"
    , 0
};
constexpr auto window = simdjson::compile_time::parse_json<json_data>();

The variable window is a value computed entirely by the compiler. Its type is generated from the document: it has a width and a height (both 64-bit integers), a bool fullscreen, a double volume, and a title. From here on I write window.width and it behaves like any ordinary field.

How do I know the parsing really happened at compile time? Because I can assert things about the result that the compiler must check before the program even exists:

static_assert(window.width      == 1920);
static_assert(window.height     == 1080);
static_assert(window.fullscreen == true);

If I corrupt the JSON — delete a brace, misspell true, leave a trailing comma — the program no longer compiles, and the error points at the parse_json line. The broken file is caught at build time, on my machine, instead of at startup on someone else’s.

Because window is a genuine compile-time constant, any computation over it is a constant too. Consider this function:

int  screen_area()   { return window.width * window.height; }

Compiled with -O3, there is no multiplication, no field access, and certainly no parsing left — only the answers, as immediate values (here on my macBook):

screen_area:    mov  w0, #0xa400        // 0x1fa400 = 2073600
                movk w0, #0x1f, lsl #16
                ret

The JSON has vanished from the binary. It was read and parsed exactly once, by the compiler, and all that survives is the number 2073600.

Because static reflection is so new, when building with GCC 16, you need to pass the flags -std=c++26 -freflection: the -freflection flag is necessary to activate compile-time reflection You must also set the simdjson macro SIMDJSON_STATIC_REFLECTION=1 before importing the simdjson.h. It is a temporary safeguard.

The source code to reproduce these examples is available.

Reference: P2996 — Reflection for C++26 and the simdjson library.

Credit: The simdjson implementation is joint work with Francisco Geiman Thiesen.

The keyword in politics these days is ‘sovereign’.

What few will admit is that it is effectively the adoption of the American strategy: Make America Great Again. In other words, reindustrialization of key sectors of the economy. The UK used to be a computing champion. Our chip designs (ARM) originated from the UK. Canada had BlackBerry, everyone was using Canadian phones.

Like Canada, many countries have progressively slid into financialization. Huge banks and bank-related businesses, surrounded by emptied factories.

Part of it was the doing of economists who promoted globalization. We are going to make our best CPUs in Taiwan, because they have a comparative advantage (whatever that means).

Another part is the rise of the managerial class, or our version of the technocracy: the summum of the status game is to make PowerPoint presentations in a nice office. Everyone has 2 or 3 university degrees. And if you don’t have many degrees, what is wrong with you?

I think that this is breaking apart for a few reasons.

One of them is Trump. And I don’t mean bombastic statements, bad hair color or overly long ties… But rather the realization that globalization might leave your people economically better off for a time because they have cheap stuff… But it also leaves your people with few skills. You can fund a robotics factory near Montreal, and you’ll find 2000 people with robotics PhDs, but nobody actually knows how to build robots. The analogy is thus: salary is not everything (to the great chagrin of economists). I have left a much better paying job. My previous job meant that I had to sit in an office and do little concrete. I would have had a great and early retirement… but I would never have developed my skills nor would I have built anything. And that’s what we did at the country level. Great total compensation, but a dead-end skill-wise.

Another factor, I believe, is the COVID era (2020-2023) and its final outcome: empty offices, closed coffee shops. What happened at work was illegible. Lots of people in offices. Certainly, something important was happening. Entire businesses and government organizations have now migrated partially or entirely to a pajama party of some kind. Netflix in the middle of the workday is no longer a dream, but a reality.

Another element is educational misalignment. A country like Canada has the most schooled population in history. You cannot throw a rock without hitting someone with a PhD. Meanwhile, we are not making robots or microchips. You can’t even pay with your phone in the Montreal subway. It is a project for another decade, maybe. We are using a push strategy: push more people with degrees into the economy and you are going to get a fancier economy. Won’t work.

Finally, the AI breakthrough of 2022 is the final nail in the proverbial coffin. My country (Canada) claimed for decades that we were the AI powerhouse. All these PDFs online can’t lie, can they? Canada basically invited modern AI, didn’t it? We did. On paper. On paper we did a great many things. In practice? Few know how to build anything.

In a country like Canada, the population has not yet caught up. They blame the orange man for whatever trouble they see. And the politicians promise to do what they must: shower money to build tech sovereignty. It won’t work. They will try again. It won’t work again. The cycle makes things worse because it sustains a managerial class that is great at politics but terrible at building.

Meanwhile, you can’t escape preference falsification. People will use ChatGPT, Claude, Grok, Gemini. And if Elon produces robots, they’ll want them.

Trump will leave office in two years… Canada and the UK will still be flat-lined economically. The USA will still surge ahead.

Countries like Canada and the UK will have to realize that it is industry and know-how first. Build stuff and the wealth will follow. Stop the virtue signaling. Stop the credentialism. Build.

Our 64-bit Intel and AMD processors have evolved over decades. When you compile a Go program for a 64-bit Intel or AMD processor, the compiler targets, by default, a nearly 20-year-old instruction set. The binary that comes out runs on essentially any x64 chip, but it also leaves on the table every instruction that was added since 2003.

We often refer to microarchitecture levels. Each level bundles a set of instruction-set extensions that you can assume are present:

In my view, this ladder is already slightly obsolete. It was frozen around 2020, and the hardware has moved on. We would need to add the latest AVX-512 sub-extensions (VBMI, VBMI2, VNNI, BF16, FP16, VPOPCNTDQ, and so on), which recent server and consumer chips support but which v4 does not require. While v1 through v4 are a useful common language, a realistic “use everything this CPU offers” target today would need at least a v5, and arguably the whole scheme should be replaced by finer-grained feature detection.

In any case, the Go toolchain exposes this v1 through v4 ladder via the GOAMD64 environment variable. Setting GOAMD64=v3 tells the compiler it may use everything up to and including AVX2. The default is v1, the lowest common denominator.

This raises an obvious question. If I take a real, performance-sensitive library and recompile it at each level, how much do I actually gain? I picked Roaring Bitmaps, a compressed bitset data structure used in databases and search engines.

A Roaring Bitmap stores a set of 32-bit integers. It splits the 32-bit space into chunks of 65,536 values, keyed by the high 16 bits, and stores each chunk in a container that holds only the low 16 bits. A container comes in one of three shapes, and the library always keeps whichever is smallest:

• an array container: a sorted list of 16-bit values, used when the chunk is sparse (a few thousand elements at most);
• a bitmap container: a flat 8 KB bit vector (65,536 bits, one per possible value), used when the chunk is dense;
• a run container: a list of [start, length] intervals, used when the set bits cluster into consecutive runs.

I fetched the latest release of the library, then ran its own benchmark suite four times, once per level, collecting eight samples each. I did this on a single Intel Xeon Gold 6548N (Emerald Rapids, which supports all four levels, including AVX-512) under Go 1.26.2 and Roaring v2.18.2.

A population count (or popcount, also called the Hamming weight) is simply the number of bits set to 1 in a machine word. Roaring leans on it constantly: the cardinality of a bitmap container, how many values it holds, is the sum of the population counts of its 1024 64-bit words. Modern x86 chips have a dedicated popcnt instruction that does this in a single operation, but it only became available at the v2 level (SSE4.2, 2008). Without it, the compiler has to fall back to a multi-instruction bit-twiddling sequence.

The clearest single result is population count: counting the number of set bits in a bitmap container. The v1 baseline cannot use the popcnt instruction, so Go emits a software fallback. The moment we move to v2, popcnt becomes available and the time is cut almost in half:

That is a 43% reduction, and it is free: no source change, just a compiler flag. Notice, though, that v3 and v4 do nothing more. A single popcnt instruction is already optimal; as far as the Go compiler is concerned, AVX2 and AVX-512 have nothing to add.

Population count is the easy win. What about the rest of the library?

Another clear win is building a container from a dense bitmap. The FromDense array benchmark takes a raw 8 KB bit vector and constructs the most compact container for it: it popcounts every word to learn the cardinality, then scans out the positions of the set bits. That word-at-a-time popcount-and-scan loop is exactly what the compiler can auto-vectorize once 256-bit registers are available, so the gains keep coming past v2:

v2 already cuts 21% by using scalar popcnt/tzcnt instructions, and v3 (AVX2) nearly doubles that to a 38% reduction. As with popcount, v4 adds nothing.

Set operations show the same pattern. The IntersectionCardinality benchmark counts how many values two bitmaps have in common: for bitmap containers, it ANDs the words pairwise and population-counts the result, without ever materializing the intersection. Here v2 does essentially nothing (the scalar popcnt is already in the inner loop), but v3 lets the compiler widen the AND-and-count loop to 256-bit registers, cutting the time by 22%:

Takeaways:

1. On modern hardware, everyone should be using v2 or better. The resulting binary will run in any data center and on any non-ancient laptop.
2. The v3 level might be worth investigating.
3. The v4 level should have helped in some of my benchmarks, but it did not. I suspect that the Go compiler is just not great at it.

(Obviously: run your own benchmarks.)

Where is your intelligence located? In your brain?

It is a simplistic answer. A better model is that your intelligence is embodied.

Consider a cook working at an expensive restaurant. He has all his favorite knives and cooking instructions, placed exactly where he wants them. His kitchen is part of his intelligence, of his skills. The same cook working in your kitchen can probably cook better than you do, but he can’t reproduce the same meals he would prepare in his favorite kitchen.

We often assess computer programmers using whiteboard tests. It is an endless source of complaints. Programmers rightly point out that it forces them out of their element. They are just not as good when you take away their laptop. It is not an excuse, it is a real issue: you are cutting them off from part of what makes them so intelligent.

To sum it up, the model of intelligence as a brain in a jar, disconnected from anything else, is ridiculous.

If you accept the idea of embodied intelligence, then many actions that we view as a consequence of our intelligence are actually part of our intelligence. First and foremost, language. Our ability to talk or write to each other means that I am not limited by my own person. Have you ever heard of human beings isolated in small tribes making technological breakthroughs? Nah. Progress requires lots of people communicating together. Up until a few decades ago, progress required cities. Today I am less certain than it does, as I can more and more communicate with anyone in the world from anywhere. But language is still critical, we have not invented anything better. Similarly, having hands and the ability to build sophisticated tools (like laptops) allows us to extend our intelligence.

At the end of 2022, we got a breakthrough technology: ChatGPT. It built on several pre-existing ideas such as (large) language models, neural networks, and so forth. That’s the ‘GPT’ part. But an important, if underappreciated, part of the breakthrough was the ‘Chat’ component. Someone had the idea of connecting a large language model with a chat interface. Maybe this came naturally and obviously to people building this system, but it should not be assumed to be trivial or unimportant.

Language is a key component of our intelligence, and, thus, it makes sense that it would be pivotal for machine intelligence.

We embodied the AI software in a chat box.

The next step was what we call today ‘agentic AI’. We keep the chat box, but we add the ability for the AI software to interact with tools, and to make plans to use them. In effect, we give the AI more agency: it can do stuff and learn from the results as they happen. It is starting to resemble a human being with hands and tools.

I was talking with a colleague this week. My colleague is all in on the AI revolution. He uses his AI to help him write better and faster, and to get his data analysis done faster, without so much help from technical experts.

But my colleague was not aware of the agentic AI approach. I tried to explain on the phone. What does it mean to give the AI access to tools? Is this only about saving the effort of copying and pasting the AI’s response?

I ended up making a video where I start an AI in a shell within something called RStudio. It is an environment people use to program in R, to do data analysis. I don’t use R or RStudio, but thanks to the AI, I was able to build an entire climate research project in a few minutes, complete with the retrieval of the data from the web.

How did the AI do it? I recorded it. It tried a few things, initially struggling to download the data. At some point, it finds out that it needs new R packages, so it installs them, and once they are installed, it can proceed to generate figures, verifying that it works.

Agentic AI greatly extends machine intelligence by improving the embodiment of AI.

I believe that it is not yet understood as it should be.

In Montreal, the most established professor in the field of AI is Yoshua Bengio. He started his own non-trivial enterprise a few years ago (Element AI). His latest venture is Law Zero, which aims to create a Scientist AI. The first goal of this project is to build AI without the agentic component. It should be a disembodied AI that has no goal of its own, no agency.

I fear that Bengio suffers from what Kevin Kelly called Thinkism. Let me quote from Kelly’s 2008 essay.

No intelligence, no matter how super duper, can figure out how human body works simply by reading all the known scientific literature in the world and then contemplating it. No super AI can simply think about all the current and past nuclear fission experiments and then come up with working nuclear fusion in a day. Between not knowing how things work and knowing how they work is a lot more than thinkism. There are tons of experiments in the real world which yields tons and tons of data that will be required to form the correct working hypothesis. Thinking about the potential data will not yield the correct data. Thinking is only part of science; maybe even a small part. (…) Thinkism is not enough. Without conducting experiments, building prototypes, having failures, and engaging in reality, an intelligence can have thoughts but not results. It cannot think its way to solving the world’s problems. (…) The Singularity is an illusion that will be constantly retreating — always “near” but never arriving. We’ll wonder why it never came after we got AI. Then one day in the future, we’ll realize it already happened. The super AI came, and all the things we thought it would bring instantly — personal nanotechnology, brain upgrades, immortality — did not come. Instead other benefits accrued, which we did not anticipate, and took long to appreciate. Since we did not see them coming, we look back and say, yes, that was the Singularity.

I believe that University professors are especially prone to thinkism. They view intelligence as being centered on what is happening in their brain. When you live in an ivory tower, it is easy to dismiss the real world as the core source of intelligence. Further, they are often people who did quite well in school where thinkism is naturally prevalent.

I have been a professor most of my life. However, I tire quickly of talking with other professors. What I most enjoy is working with people who have new tools that they apply in the real world. Unsurprisingly, I spent most of my time working with software that people deploy in the real world.

What Kelly is saying is that a high degree of intelligence is not enough to do much of anything. The real world is not the final stage of your thinking process. It is maybe the most important part of it.

And thus, when you connect your AI with the real world, giving it the ability of running experiments (as virtually all software developers do today), you get impressive results that go much beyond what AI software can do on its own.

Agency is not a feature. Agency is primary.

Every machine connected to the Internet has an address called an IP address. Originally, these addresses were 32-bit integers (IPv4), giving a theoretical maximum of about four billion distinct addresses. We are all familiar with these addresses (e.g., 192.168.0.0). There was a big fuss about how we would run out of addresses. It never happened because we don’t actually need every device to have its own unique address. Your home router needs an address, but every device in your home does not need a worldwide unique address.

Nevertheless, the range was extended to cover 128 bits (IPv6). An IPv6 address is conventionally written as eight groups of four hexadecimal digits separated by colons. For example:

2001:0db8:85a3:0000:0000:8a2e:0370:7334

Because addresses often contain runs of zeros, the format allows two shortcuts:

• Leading zeroes within a group may be omitted: 2001:db8:85a3:0:0:8a2e:370:7334.
• A single run of all-zero groups may be replaced by ::: 2001:db8:85a3::8a2e:370:7334.

The double-colon trick can appear only once in an address, and can match one or more zero groups. Hence ::1 is the loopback address (all zeros except the last group), and :: is the unspecified address (all zeros).

IPv6 also accepts an embedded IPv4 address in the last 32 bits, written in the usual dotted-decimal form. This is mostly used for IPv4-mapped IPv6 addresses such as ::ffff:192.168.1.1. The longest possible textual form is 45 characters:

0000:0000:0000:0000:0000:ffff:255.255.255.255

So a parser must accept hexadecimal groups, the compressed form, and an optional IPv4 tail. It is more involved than parsing IPv4.

The standard C function for the job is inet_pton, available on essentially every system.

Can we do better?

A few years ago, I showed that you could parse IPv4 addresses really fast. Can we do the same with IPv6?

The trick is to use data parallelism: we invoke the so-called SIMD instructions that all our processors support. These instructions can process potentially dozens of bytes at once.

Shreesh Adiga gave it a try with AVX-512, the powerful instruction set supported by recent Intel server processors and all new AMD CPUs. The idea is to load the entire string into a 512-bit register, find the colons with a single comparison, compute the spacing between them to drive a byte-level expand, translate hex digits via a permute, and finish with a multiply-accumulate that combines the hex digits into bytes. Almost the whole parser is branch-free, meaning that there are few if clauses.

I put together a small benchmark that generates random IPv6 addresses with inet_ntop (so the addresses are written in their canonical, compressed form) and parses each one with both inet_pton and the AVX-512 routine. The benchmark runs on an Intel Xeon Gold 6548N CPU @ 2.8 GHz (Emerald Rapids) with GCC, compiled with -march=native -O3.

The AVX-512 routine is about 12 times faster than inet_pton, parsing more than 70 million addresses per second on a single core. It uses eight times fewer instructions, and runs them at a higher throughput (2.45 instructions per cycle versus 1.56).

The source code used for this benchmark is available on my blog repository.

Update. Peter Fors points out that the step in my benchmark, where I sum
up the bytes, adds some overhead especially under GCC. Thus I underestimate the speed slightly.

In software programming, the product between two integers is often computed to a fixed number of bits with overflow. Consider 8-bit integers. If you multiply 127 by 127, you get back the number 1 as an 8-bit unsigned integer, with an overflow. The actual full product is 16129. To represent 16129, you typically use 16 bits of precision.

Thus we have the notion of the full product. The full product of two 32-bit integers is typically represented using 64 bits. The question that preoccupied me is what fraction of all 64-bit integers can be written as the product of two 32-bit integers.

You might wonder why you would care?

We often design hash functions: they are special functions that take an input and generate a random-looking output. Several years ago I designed a very fast hash function called clhash. It is a super-fast hash function for strings having a few hundred bytes or more. If you don’t know about clhash, check it out. It is interesting in its own right.

This clhash hash function uses a type of multiplication typical of cryptographic applications. I was trying to argue that our approach had benefits compared with techniques based on standard multiplications. Let me illustrate. A simple hash function for 32-bit integers could take the least significant bits and multiply them with the most significant bits.

// simpleHighLowHash is a simple (and weak) 32-bit hash
// that multiplies the high 16 bits by the low 16 bits.
func simpleHighLowHash(x uint32) uint32 {
    high := uint16(x >> 16)
    low := uint16(x & 0xFFFF)
    return uint32(high) * uint32(low)
}

Maybe you’d want the hash function to be uniform: all possible 32-bit hash values should be equally probable. It is only possible in this instance if the hash function can produce all 32-bit hash values, which is not the case.

The great mathematician Erdös showed that the proportion of all 2n-bit values that can be generated by the product of two n-bit values goes to zero as n becomes large. This means that if you have, say, 10000000-bit integers multiplying 10000000-bit integers, you’d expect relatively few 20000000-bit integers to be produced. But what about practical cases like 32-bit integers or 64-bit integers?

You can just brute-force the problem easily up to the multiplication of 16-bit integers into 32-bit products. At that point, slightly one out of five 32-bit numbers is a product between two 16-bit integers. About 80% of all 32-bit integers are never produced by this hash. However, the running time grows exponentially, and brute force won’t scale all the way to 32 bits.

So what do we do about the 32-bit case? That is, what do you do when you multiply two 32-bit integers to produce a 64-bit product? What fraction of 64-bit values can the following function produce?

func simpleHighLowHash(x uint64) uint64 {
    high := uint32(x >> 32)
    low := uint32(x & 0xFFFFFFFF)
    return uint64(high) * uint64(low)
}

Can we get an exact result?

Yes!!!

Webster and his colleagues built the math to allow us to scale up the exact computation. He was kind enough to publish his code.

There are 3,215,709,724,700,470,902 64-bit (unsigned) integers that can be written as a product of two 32-bit integers. That’s about 17% of all possible values.

What about actually computing a pair of integers given their product? One approach consists of computing its full prime factorization, and then using those factors to build all possible divisors that are strictly less than 2^32, starting with a set of candidates containing only 1 and iteratively multiplying existing candidates by each prime factor (only keeping products that stay below 2^32). We can avoid adding duplicates to our set by processing unique prime factors with their multiplicity. Finally, we select the maximum such candidate m as the largest divisor under 2^32, compute the corresponding leftover n / m, and report whether a valid split into two 32-bit factors exists. In general, the answer (if it exists) is not unique: this returns the pair where one value is maximized. In Python, the code might look as follows.

for p in factor_multiplicities:
    new_candidates = []
    for c in candidates:
        for i in range(factor_multiplicities[p] + 1):
            if c * (p ** i) < 2**32:
                new_candidates.append(c * (p ** i))
    for new_c in new_candidates:
        candidates.append(new_c)
m = max(candidates)
print(f"Maximum candidate: {m}")
leftover = n // m
print(f"Leftover: {leftover}")
if leftover >= 2**32:
    print("Leftover is too large, cannot find a suitable candidate.")

You might be able to come up with a more efficient algorithm. I find it interesting to consider that if you pick a value at random, it will usually fail! That is, most 64-bit integers cannot be written as the product of two 32-bit integers.

Converting a 64-bit integer to its decimal string representation is a mundane task that shows up everywhere: logging, JSON serialization, CSV output, debug prints, etc. In C++, you might use std::to_chars, sprintf, or some library routine.

How do these functions work? At a high level, they repeatedly divide by ten. Start with your integer k. Divide it by ten, use the remainder as the last digit (it is between 0 and 9 inclusively). You then add the code point value of the character 0 to get the ASCII digit. To go faster, you can divide by 100 and use a lookup table so that the value between 0 and 99 inclusively is mapped to a string.

So far so good. Unfortunately, even with all these optimizations, this string generation may become a performance bottleneck. Can you do better?

Let us assume that you have a recent AMD processor or an Intel server. Then you have powerful data-parallel instructions (AVX-512) that can multiply eight 64-bit integers at once. We often refer to these instructions as SIMD (single instruction multiple data). My colleague Jaël Champagne Gareau and I recently published a new paper on exactly this problem. The title says it all: Converting an Integer to a Decimal String in Under Two Nanoseconds.

When you write n / 100 in code, an optimizing compiler converts the operation to a multiplication followed by a shift. It is often described as a multiplicative inverse. Generally, you can replace the division of n by d with the division of c * n or c * n + c by m for convenient integers c and m chosen so that they approximate the reciprocal: c/m ~= 1/d. We often call c * n + c a fused multiply-add. Picking m to be a power of two means that the division by m is just a shift. Then you can get the remainder of the division by using the remainder of the division by m, multiplied by d and divided again by m, which is essentially a multiplication followed by a shift (Lemire et al., 2021).

We can put this to good use with the Integer Fused Multiply-Add (IFMA) instructions available on recent Intel and AMD processors. They essentially allow you to compute eight instances of (c * n + c)/m in one instruction. The expression (c * n + c)/m gives you the division, but we need the remainder, so instead we pick (c * n + c)%m which we need to multiply by the divisor.

The fun thing with AVX-512 instructions is that they can use a different c and a different divisor for each of the eight operations. Using Intel intrinsic functions, our core routine which converts a value smaller than 10^8 to eight digits looks as follows:

__m512i to_string_avx512ifma_8digits(uint64_t n) {
  __m512i bcstq_l   = _mm512_set1_epi64(n);
  constexpr uint64_t twoto52 = 0x10000000000000ULL; // 2^52
  __m512i ifma_const = _mm512_setr_epi64(
    twoto52 / 100000000, twoto52 / 10000000,
    twoto52 / 1000000, twoto52 / 100000,
    twoto52 / 10000, twoto52 / 1000, 
    twoto52 / 100, twoto52 / 10
  );
  __m512i zmmTen    = _mm512_set1_epi64(10);
  __m512i asciiZero = _mm512_set1_epi64('0');
  __m512i lowbits_l  = _mm512_madd52lo_epu64(ifma_const, 
    bcstq_l, ifma_const); // ifma_const * bcstq_l + ifma_const
  __m512i highbits_l = _mm512_madd52hi_epu64(asciiZero, 
    zmmTen, lowbits_l);
  return highbits_l;
}

It compiles down to two multiplication-add instructions: vpmadd52huq. That’s it. Two instructions to generate eight digits.

It works by broadcasting n across all eight 64-bit lanes of a __m512i vector (bcstq_l). It then prepares a vector of carefully chosen multiplicative inverses (ifma_const) that represent the reciprocals of 10^8, 10^7, … The magic happens in the single _mm512_madd52lo_epu64 instruction, which simultaneously performs eight fused multiply-adds: each lane computes (ifma_const[i] * n + ifma_const[i]) using 52-bit low-half multiplication, effectively extracting the quotient when dividing by the corresponding power of ten. A second _mm512_madd52hi_epu64 instruction (with a vector of ten and a vector of '0') then isolates the digit values and adds the ASCII '0' offset in the high 52 bits, producing eight packed digit characters in a single 512-bit register.

If all your integers require eight digits, you are done. But in the general case, putting this to good use requires a bit of effort.

Thankfully, even if you, say, need only six digits, you can do the full 8-digit computation and then use a masked store if you want to store only six digits, ignoring the two leftovers. That is, instruction sets like AVX-512 allow you to write only some of the data to memory, which is quite convenient.

We have two variants. One is branch-heavy and does well on homogeneous data (numbers with similar digit lengths). The other is branch-light and better for mixed workloads. A quick profiling step can pick the right one for your dataset.

Our implementation is consistently 1.4–2× faster than the best competitors and 2–4× faster than std::to_chars across a wide range of inputs. What I find interesting is that even if the std::to_chars implementation is not at all naive, you can do significantly better in many ways. James Anhalt’s approach (jeaiii) is also quite fast on modern hardware.

Further reading
– The paper: doi:10.1002/spe.70079
– The benchmarks are on GitHub (fully reproducible)
– Shortly after our paper came online, Barend Erasmus created a software library implementing our proposed approach. I am not certain that Barend includes both the homogeneous and heterogeneous approaches.