## An algorithmic sketch: inplace_merge

March 13th, 2016

One of the things I like to do in my spare time is study the STL algorithms. It is easy to take them for granted and easy, perhaps, to imagine that they are mostly trivial. And some are: I would think that any decent interview candidate ought to be able to write find correctly. (Although even such a trivial-looking algorithm is based on a thorough understanding of iterator categories.)

Uncovering the overlooked

But there are some algorithms that are non-trivial, and some are important building blocks. Take inplace_merge. For brevity, let’s consider the version that just uses operator< rather than being parametrized on the comparison. The one easily generalizes to the other in a way that is not important to the algorithm itself.

template <typename BidirectionalIterator> void inplace_merge(BidirectionalIterator first, BidirectionalIterator middle, BidirectionalIterator last);

It merges two consecutive sorted ranges into one sorted range. That is, if we have an input like this: $x_0 \: x_1 \: x_2 \: ... \: x_n \: y_0 \: y_1 \: y_2 \: ... \: y_m$

Where $\forall i < n, x_i \leq x_{i+1}$ and $\forall j < m, y_j \leq y_{j+1}$

We get this result occupying the same space: $r_0 \: r_1 \: r_2 \: ... \: r_{n+m}$

Where $\forall i < n+m, r_i \leq r_{i+1}$ and the new range is a permutation of the original ranges. In addition, the standard states a few additional constraints:

• inplace_merge is stable - that is, the relative order of equivalent elements is preserved
• it uses a BidirectionalIterator which shall be ValueSwappable and whose dereferent (is that a word?) shall be MoveConstructible and MoveAssignable
• when enough additional memory is available, (last-first-1) comparisons. Otherwise an algorithm with complexity N log(N) (where N = last-first) may be used

Avenues of enquiry

Leaving aside the possible surprise of discovering that an STL algorithm may allocate memory, some thoughts spring to mind immediately:

• Why does inplace_merge need a BidirectionalIterator?
• How much memory is required to achieve O(n) performance? Is a constant amount enough?

And to a lesser extent perhaps:

• Why are merge and inplace_merge not named the same way as other algorithms, where the more normal nomenclature might be merge_copy and merge?
• What is it with the algorists' weasel-word "in-place"?

It seems that an O(n log n) algorithm should be possible on average, because in the general case, simply sorting the entire range produces the desired output. Although the sort has to be stable, which means something like merge sort, which leads us down a recursive rabbit hole. Hm.

At any rate, it's easy to see how to achieve inplace_merge with no extra memory needed by walking iterators through the ranges:

template <typename ForwardIt> void naive_inplace_merge( ForwardIt first, ForwardIt middle, ForwardIt last) { while (first != middle && middle != last) { if (*middle < *first) { std::iter_swap(middle, first); auto i = middle; std::rotate(++first, i, ++middle); } else { ++first; } } }

After swapping (say) $x_0$ and $y_0$, the ranges look like this: $y_0 \: x_1 \: x_2 \: ... \: x_n \: x_0 \: y_1 \: y_2 \: ... \: y_m$

And the call to rotate fixes up $x_1 \: ... \: x_n \: x_0$ to be ordered again. From there we proceed as before on the ranges $x_0 \: ... \: x_n$ and $y_1 \: ... \: y_m$.

This algorithm actually conforms to the standard! It has O(n) comparisons, uses no extra memory, and has the advantage that it works on ForwardIterator! But unfortunately, it's O(n²) overall in operations, because of course, rotate is O(n). So how can we do better?

Using a temporary buffer

If we have a temporary buffer available that is equal in size to the smaller of the two ranges, we can move the smaller range to it, move the other range up if necessary, and perform a "normal" merge of the two ranges into the original space:

template <typename BidirIt> void naive_inplace_merge2( BidirIt first, BidirIt middle, BidirIt last) { using T = typename std::iterator_traits<BidirIt>::value_type;   auto d1 = std::distance(first, middle); auto d2 = std::distance(middle, last);   auto n = std::min(d1, d2); auto tmp = std::make_unique<char[]>(n * sizeof(T)); T* begint = reinterpret_cast<T*>(tmp.get()); T* endt = begint + n;   if (d1 <= d2) { std::move(first, middle, begint); std::merge(begint, endt, middle, last, first); } else { std::move(middle, last, begint); auto i = std::move_backward(first, middle, last); std::merge(i, last, begint, endt, first); } }

This is essentially the algorithm used by STL implementations if buffer space is available. And this is the reason why inplace_merge requires BidirectionalIterator: because move_backward does.

(This isn't quite optimal: the std::move_backward can be mitigated with reverse iterators and predicate negation, but the BidirectionalIterator requirement remains. Also, strictly speaking, std::merge is undefined behaviour here because one of the input ranges overlaps the output range, but we know the equivalent loop is algorithmically safe.)

Provisioning of the temporary buffer is also a little involved because we don't know that elements in the range are default constructible (and perhaps we wouldn't want to default-construct our temporaries anyway). So to deal correctly with non-trivial types here, std::move should actually be a loop move-constructing values. And when std::inplace_merge is used as a building block for e.g. std::stable_sort, it would also be nice to minimize buffer allocation rather than having an allocation per call. Go look at your favourite STL implementation for more details.

Thinking further

The literature contains more advanced algorithms for merging if a suitably-sized buffer is not available: the basis for the STL's choice is covered in Elements of Programming chapter 11, and in general the work of Dudzinski & Dydek and of Kim & Kutzner seems to be cited a lot.

But I knew nothing of this research before tackling the problem, and attempting to solve it requiring just ForwardIterator.

I spent a couple of evenings playing with how to do inplace_merge. I covered a dozen or more A4 sheets of squared paper with diagrams of algorithms evolving. I highly recommend this approach! After a few hours of drawing and hacking I had a really good idea of the shape of things. Property-based testing came in very handy for breaking my attempts, and eventually led me to believe that a general solution on the lines I was pursuing would either involve keeping track of extra iterators or equivalently require extra space. Keeping track of iterators seemed a messy approach, so an extra space approach is warranted.

How much extra space? Consider the "worst case": $x_0 \: x_1 \: x_2 \: ... \: x_n \: y_0 \: y_1 \: y_2 \: ... \: y_m$

Assume for the moment that $m \leq n$. When $y_m < x_0$, we need extra space to hold all of $x_0 \: ... \: x_m$. If $n \leq m$ then we will need extra space for $x_0 \: ... \: x_n$ to likewise move them out of the way. Either way, the number of units of extra space we need is min(n, m).

As we move elements of $x$ into temporary storage, we can see that in general at each stage of the algorithm we will have a situation something like this (using $Z$ to mean a moved-from value): $... \: x_i \: ... \: x_n \: Z_0 \: ... \: Z_{j-1} \: y_j \: ... \: y_m$

With some values of $x$ moved into temporary storage: $x_{i-t} \: ... \: x_{i-1}$

The temporary storage here is a queue: we always push on to the end and pop from the beginning, since the elements in it start, and remain, ordered. Since we know an upper bound on the number of things in the queue at any one time, it can be a ring buffer (recently proposed) over a regular array of values.

Sketching the start

From this, we can start sketching out an algorithm:

1. Allocate a buffer of size min(m, n) - call it tmp
2. We'll walk the iterators along the x (first) and y (middle) ranges
3. The output iterator o will start at first
4. The next x to consider will either be in-place in the x range, or (if tmp is not empty) in tmp - call it xlow
5. If *y < *xlow move *x to tmp, move *y to o, inc y
6. Otherwise, if *xlow is in tmp, move *x to tmp and *xlow from tmp to o
7. inc o, inc x
8. if y < last and o < middle, goto 4
9. deal with the end(s) of the ranges

Dealing with the end

This gets us as far as exhausting the smaller range: after this, we will be in one of two situations.

Situation 1. If we have exhausted the $y$ range, things look like this: $... \: x_i \: ... \: x_n \: Z_0 \: ... \: Z_m$

With values of $x$ in temporary storage: $x_{i-t} \: ... \: x_{i-1}$

To fix up this situation, we can repeatedly swap the tmp range with the equivalent x range until we reach middle (i.e $Z_0$), and then simply move the remaining tmp values into place.

I originally wrote a loop repeatedly swapping the values in tmp right up to the end; but I realised that would involve swapping a moved-from object, which would be wrong (it might work… until it doesn’t). Moved-from objects should either be destroyed or made whole (assigned to); nothing else.

Situation 2. The possibility is that we have exhausted the $x$ range, in which case things look like this: $... \: Z_0 \: ... \: Z_{n-i} \: y_j \: ... \: y_m$

With values of $x$ in temporary storage: $x_i \: ... \: x_n$

To fix up this situation, we can just do a regular merge on the remaining y range and tmp, outputting starting at middle (i.e $Z_0$). (With the same proviso as before about undefined behaviour with overlapping ranges.) We know that it will be safe to do a loop equivalent to merge, because we have exactly the required amount of space before $y_j$ to fit $x_i \: ... \: x_n$. This is the same as the STL’s normal buffered merge strategy.

Final thoughts

I tackled this exercise from scratch, knowing nothing about actual implementations of inplace_merge. This algorithm does some extra housekeeping under the hood, but:

• it does the minimum number of comparisons
• each element is moved at most twice: into tmp and out again
• it needs only ForwardIterator

Optimization and benchmarking under differing conditions of range size, comparison and move cost is left as an exercise to the reader…

I cannot recommend Elements of Programming enough. I am partway through reading it; after this exercise I skipped to chapter 11 to see what it said. Every time I dive into the STL algorithms, I am re-impressed by the genius of Alex Stepanov: Paul McJones’ recent talk The Concept of Concept explains this well, in particular the key role of concepts in the STL in service of algorithmic purity. Alex knew about concepts from the start: it’s taken C++ over 2 decades to catch up.

After doing this work, I discovered a recent proposal that calls for weakening the iterator categories of inplace_merge and related algorithms.

An implementation of this algorithm is on github. It’s just a sketch, written for clarity rather than optimality. This has been a fun exercise.

## ELI5: monoids

February 29th, 2016

(Resulting from my claim that “a child of 8 can understand monoids…”)

Wikipedia says: “In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.”

Wolfram says: A monoid is a set that is closed under an associative binary operation and has an identity element I ∈ S such that for all a ∈ S, Ia = aI = a.

Mathematics has to be precise, which is why it uses jargon. But what do these concise definitions mean in everyday language? Consider adding up numbers.

• The set is the whole numbers (and we need zero). 0, 1, 2, 3 etc.
• The associative binary operation is addition.
• “binary” just means it’s a thing you do to two numbers.
• “associative” means it doesn’t matter what order you group things in. 1 + 2 + 3 gives the same answer whether you add 1 and 2 first and then add 3, or add 2 and 3 first and then add the answer to 1.
• The set being “closed” under addition means that when you add two numbers you get another number – you don’t get some other kind of thing. (You might think this is obvious, but in maths it has to be stated.)
• The identity element is 0 – the thing that doesn’t make any difference when you add it. Anything plus zero is itself.

So adding whole numbers is a monoid. A mathematician would say that the non-negative integers form a monoid under addition. The important thing is that the numbers aren’t a monoid on their own; it’s the combination of the set (0, 1, 2, 3…) and the operation (+) that makes the monoid. If we chose another operation, we could get another monoid. Think about multiplication, for instance.

It turns out that lots of things behave the same way as addition on numbers, which is why the notion of a monoid is very useful to mathematicians and computer scientists.

## Lameness Explained

December 9th, 2015

OK, more than one person wanted explanations of The C++ <random> Lame List, so here are some of my thoughts, if only to save people searching elsewhere.

1. Calling rand() is lame because it’s an LCG with horrible randomness properties, and we can do better. And if you’re not calling rand(), there’s no reason to call srand().
2. Using time(NULL) to seed your RNG is lame because it doesn’t have enough entropy. It’s only at a second resolution, so in particular, starting multiple processes (e.g. a bringing up bunch of servers) at the same time is likely to seed them all the same.
3. No, rand() isn’t good enough even for simple uses, and it’s easy to do the right thing these days. The lower order bits of rand()‘s output are particularly non-random, and odds are that if you’re using rand() you’re also using % to get a number in the right range. See item 6.
4. In C++14 random_shuffle() is deprecated, and it’s removed in C++17, which ought to be reason enough. If you need more reason, one version of it is inconvenient to use properly (it uses a Fisher-Yates/Knuth shuffle so takes an RNG that has to return random results in a shifting range) and the other version of it can use rand() under the hood. See item 1.
5. default_random_engine is implementation-defined, but in practice it’s going to be one of the standard generators, so why not just be explicit and cross-platform-safe (hint: item 10)?. Microsoft’s default is good, but libc++ and libstdc++ both use LCGs as their default at the moment. So not much better than rand().
6. It is overwhelmingly likely that whatever RNG you use, it will output something in a power-of-two range. Using % to get this into the right range probably introduces bias. Re item 3, consider a canonical simple use: rolling a d6. No power of two is divisible by 6, so inevitably, % will bias the result. Use a distribution instead. STL (and others) have poured a lot of time into making sure they aren’t biased.
7. random_device is standard, easy to use, and should be high quality randomness. It may not be very well-performing, which is why you probably want to use it for seeding only. But you do want to use it (mod item 8).
8. Just know your platform. It might be fine in desktop-land, but random_device isn’t always great. It’s supposed to be nondeterministic and hardware based if that’s available… trust but verify, as they say.
9. Not handling exceptions is lame. And will bite you. I know this from experience with random_device specifically.
10. The Mersenne twisters are simply the best randomness currently available in the standard.
11. Putting mt19937 on the stack: a) it’s large (~2.5k) and b) you’re going to be initializing it each time through. So probably not the best. See item 17 for an alternative.
12. You’re just throwing away entropy if you don’t seed the generator’s entire state. (This is very common, though.)
13. Simply, uniform_int_distribution works on a closed interval (as it must – otherwise it couldn’t produce the maximum representable value for the given integral type). If you forget this, it’s a bug in your code – and maybe one that takes a while to get noticed. Not good.
14. Forgetting ref() around your generator means you’re copying the state, which means you’re probably not advancing the generator like you thought you were.
15. seed_seq is designed to seed RNGs, it’s that simple. It tries to protect against poor-quality data from random_device or whatever variable-quality source of entropy you have.
16. Not considering thread safety is always lame. Threads have been a thing for quite a while now.
17. thread_local is an easy way to get “free” thread safety for your generators.
18. You should be using a Mersenne twister (item 10) so just use the right thing for max(). Job done.

If you want more, see rand() Considered Harmful (a talk by Stephan T Lavavej), or The bell has tolled for rand() (from the Explicit C++ blog), or see Melissa O’Neill’s Reddit thread, her talk on PCG, and the associated website.

And of course, cppreference.com.

## The C++ <random> Lame List

December 7th, 2015

Network programmers of a certain age may remember the Windows Sockets Lame List.

I previously wrote a short “don’t-do-that-do-this” guide for modern C++ randomness, and I was recently reading another Reddit exchange featuring STL, author of many parts of Microsoft’s STL implementation, when it struck me that use of C++ <random> needs its own lame list to discourage using the old and busted C parts and encourage the using the new C++ hotness. So here, in no particular order, and with apologies to Keith Moore (wherever he may be) is an incomplete lame list for use of <random>.

1. Calling rand() or srand(). Lame.
2. Using time(NULL) to seed an RNG. Inexcusably lame.
3. Claiming, “But rand() is good enough for simple uses!” Dog lame.
4. Using random_shuffle() to permute a container. Mired in a sweaty mass of lameness.
5. Using default_random_engine. Nauseatingly lame.
6. Using % to get a random value in a range. Lame. Lame. Lame. Lame. Lame.
7. Not using random_device to seed an RNG. Violently lame.
8. Assuming that random_device is going to do the right thing on your platform. Uncontrollably lame.
9. Failing to handle a possible exception from the construction or use of random_device. Totally lame.
10. Using anything in the standard but mt19937 or mt19937_64 as a generator. Intensely lame.
11. Putting mt19937 on the stack. In all my years of observing lameness, I have seldom seen something this lame.
12. Seeding mt19937 with only one 32-bit word rather than its full state_size. Pushing the lameness envelope.
13. Forgetting that uniform_int_distribution works on a closed interval. Thrashing in a sea of lameness.
14. Passing random_device or a generator to generate_n() by value, forgetting to wrap it with ref(). Glaringly lame.
15. Failing to use seed_seq to initialize a generator’s state properly. Indescribably lame.
16. Not considering thread safety when using a generator. Floundering in an endless desert of lameness.
17. Using a global generator without making it thread_local. Suffocating in self lameness.
18. Using RAND_MAX instead of mt19937::max(). Perilously teetering on the edge of a vast chasm of lameness.

This list will undoubtedly grow as I continue to write lame code…

November 4th, 2015

When I was young, I read lots of books with titles (or at least subheadings) along the lines of, “Amaze Your Friends and Confound Your Enemies” – a lot of them were filled with tricks and oddities like the Birthday Paradox, or the old saw about the elephant from Denmark.

It turns out that if you read and continue to read enough of this kind of thing, you can continue to amaze your friends well into adulthood! A lot of times this means appearing to be good at mental arithmetic. And the trick to being good at mental arithmetic is not to be especially fast at rote calculation. It lies in a web of knowledge about numbers.

I often think that number theory is poorly served by the mathematical curriculum almost everywhere. Kids learn times tables and are introduced to prime numbers, and then I think the number theory track more or less stops, and a student doesn’t meet it again until university-level maths. Which is a shame, because there are many interesting and fun problems in number theory that can be easily stated and understood by a ten-year-old but which still remain unsolved. Also, a more continuous (ha!) grounding in number theory would give us a better understanding of some very important features of the modern world – an obvious example being cryptography.

I was lucky enough, as a 12-13 year old, to have a recreational maths class at school for a year where we tackled problems together. It was distinctly constructivist in nature – whether by design or not – and it was a really fun class because we were typically all trying to solve a hard problem over the course of a few weeks. “Copying” other people’s work and building on it towards a solution was par for the course – as in real life! We took inspiration from the writings of people like Martin Gardner, Sam Loyd, Raymond Smullyan, Henry Dudeney and Eric Emmet. The teacher would ramp up the difficulty of problems as we went – and in mathematics there is almost always a way, having solved one problem, to remove a constraint or make it more general in some way to provide a step up to the next level.

A typical class exchange:

Teacher: Who can tell me how many squares there are on a chessboard?
Student A: 64!
Teacher: Ah yes. Correct. But I see you’re only counting the squares that are one square in size, as it were. I think there might be more squares there…
Student A: …
Student B: Wait a minute…
Class: *realization* *time passes, working-out*
Class: 204!
Teacher: Correct. For your chessboard there. But my chessboard has n squares on a side, not 8.
Class: *argh* *more time passes, probably a week*
Class: Um… (a few tries, and then)… n(n+1)(2n+1)/6 ?
Teacher: Right! But… hm… my chessboard got broken, now it’s not square any more, it’s n by m. Oh, and I want to know how many rectangles are on it.
Class: *mind blown*

Associations – it’s what brains do

Anyway, the human brain is fantastically good at constructing associations. And being “good at maths” is about having lots of those associations when it comes to numbers. Mathematicians often have this. Computer people have this facility, when it comes to powers of 2, and it looks astounding to muggles when it comes out in another context (eg Biology class):

Teacher (thinking he is asking a hard question): This germ divides in two every hour. If we start with just one germ here, how many will we have after a day?
Nerdette in the back row (instantly): 16,777,216
Rest of the class: How the hell…?

When you have enough associations, they start to overlap and provide multiple ways to an answer. I was recently out with a group of friends and at the end of the night we came to pay the bill. There were seven of us, and the bill was $195. I immediately knew it was about$28 each, and I didn’t have to calculate, because of some associations:

• 196 is 14 squared, which is 7 * 28. Immediate answer. (Square numbers up to 20 or so – very useful to know.)
• In the UK weight is often measured in stones. 14lbs = 1 stone, and I know that 98lbs = 7 stones or 7*14 = 98. And as 98*2 = 196, because (100-2)*2 = 200-4 = 196, so 14*2 = 28. Corroboration by overlapping association.

“Wizardry” with 1/7

Another second’s thought provides the exact amount per person, because 1/7 is a useful and interesting fraction to know. It has a recurring 6-digit pattern, and it cycles. Once you know the six digits of 1/7, it’s trivial to figure out/remember the other fractions:

• 1/7 = 0.142857142857142857…
• 2/7 = 0.285714…
• 3/7 = 0.428571…
• 4/7 = 0.571428…
• 5/7 = 0.714285…
• 6/7 = 0.857142…

This is a fun trick for kids: compute 142857 * 2, 142857 * 3, etc and see how the digits cycle. Then try 142857 * 7… and you get 999999. Neat.

Anyway, 28*7=196 but the bill is $195, which means actually everyone pays$28, less 1/7 of a dollar. So the exact figure is $27 and 85.714285… cents. Fun for kids When numbers are your friends, it’s easy to look like you’re a wizard. And it’s really just about forming those associations. When I see 41, I think of Euler’s famous expression x2 + x + 41 which is prime for every x from 0 to 39. When I see 153, I think, “Hello, 13 + 53 + 33!” And similarly for many other numbers and mathematical techniques, thanks to all that reading and playing with numbers as a child. These days I entertain my kids by having them do fun math tricks like: Enter any three-digit number into your calculator (say 456) Multiply by 7 Now multiply again by 11 Now multiply again by 13 The result is the original number “doubled up” (say 456456) – because 7 * 11 * 13 = 1001 I’m teaching them how to amaze their friends and confound their enemies! ## Compile-time RNG tricks October 15th, 2015 (Start at the beginning of the series – and all the source can be found in my github repo) Compile-time random number generation is quite useful. For instance, we could generate GUIDs (version 4 UUIDs): namespace cx { struct guid_t { uint32_t data1; uint16_t data2; uint16_t data3; uint64_t data4; }; template <uint64_t S> constexpr guid_t guidgen() { return guid_t { cx::pcg32<S>(), cx::pcg32<S>() >> 16, 0x4000 | cx::pcg32<S>() >> 20, (uint64_t{8 + (cx::pcg32<S>() >> 30)} << 60) | uint64_t{cx::pcg32<S>() & 0x0fffffff} << 32 | uint64_t{cx::pcg32<S>()} }; } } #define cx_guid \ cx::guidgen<cx::fnv1(__FILE__ __DATE__ __TIME__) + __LINE__> There are situations right now where one might have scripts (probably Python or some such language) in one’s build chain that do this type of generation of C++ code, for example to introduce randomness to each build in the name of security, or simply to mark the build uniquely. It would be nice not to have to maintain extra non-C++ code and extra steps in the build chain. Is PCG32 secure? Maybe secure enough… Security you say? We have a source of random numbers that changes every build… we could transparently “encrypt” string literals and decode them at point of use. Might be useful to someone. To wrangle with a string in a constexpr manner, I need to treat it as an array, and then encrypt (I’ll just xor) each character with my random byte stream. I can figure out how long a string literal is at compile time, store the random seed as a template parameter, and use an index_sequence expansion to do the encryption. template <uint64_t S> constexpr char encrypt_at(const char* s, size_t idx) { return s[idx] ^ static_cast<char>(pcg32_output( pcg32_advance(S, idx+1)) >> 24); } template <size_t N> struct char_array { char data[N]; }; template <uint64_t S, size_t ...Is> constexpr char_array<sizeof...(Is)> encrypt( const char *s, std::index_sequence<Is...>) { return {{ encrypt_at<S>(s, Is)... }}; } That takes care of constructing an encrypted array from a string literal. The rest is just wrapping it up in an encrypted_string class and providing a sane runtime decryption function (we could use the existing C++11 constexpr functions, but they are strangely formulated for runtime use – maybe a more natural formulation would be easier to optimize). And we can give the class a conversion to string. inline std::string decrypt(uint64_t S, const char* s, size_t n) { std::string ret; ret.reserve(n); for (size_t i = 0; i < n; ++i) { S = pcg32_advance(S); ret.push_back(s[i] ^ static_cast<char>(pcg32_output(S) >> 24)); } return ret; } template <uint64_t S, size_t N> class encrypted_string { public: constexpr encrypted_string(const char(&a)[N]) : m_enc(encrypt<S>(a, std::make_index_sequence<N-1>())) {} constexpr size_t size() const { return N-1; } operator std::string() const { return decrypt(S, m_enc.data, N-1); } private: const char_array<N-1> m_enc; }; template <uint64_t S, size_t N> constexpr encrypted_string<S, N> make_encrypted_string( const char(&s)[N]) { return encrypted_string<S, N>(s); } #define CX_ENCSTR_RNGSEED \ uint64_t{cx::fnv1(__FILE__ __DATE__ __TIME__) + __LINE__} #define cx_make_encrypted_string \ cx::make_encrypted_string<CX_ENCSTR_RNGSEED> The more observant and pedantic among you may have noticed that strictly, I’ve strayed into C++14 territory here, using std::index_sequence. But I think I’m still in the spirit of C++11 constexpr. And more importantly, VS2015 supports std::integer_sequence. Anyway, let’s exercise this code and check it does the right thing: int main(int, char* []) { constexpr auto enc = cx_make_encrypted_string("I accidentally the string"); cout << string(enc) << endl; return 0; } Here’s the result: $ ./test_constexpr I accidentally the string $ And the plaintext string doesn’t appear in the binary: $ objdump -s -j .rodata ./test_constexpr   ./test_constexpr: file format elf64-x86-64   Contents of section .rodata: 404700 01000200 d31e337d 58244d8a 48e52178 ......3}X\$M.H.!x 404710 139d483c 5113d3f1 0aec79a3 80626173 ..H<Q.....y..bas 404720 69635f73 7472696e 6700 ic_string.

Not bad. Per-compile obfuscation on string literals, with zero memory overhead and only a modest cost at the point of use.

## Compile-time counters, revisited

October 14th, 2015

(Start at the beginning of the series – and all the source can be found in my github repo)

Some time ago I read a blog post showing how to make a compile-time counter: a constexpr function that would return monotonically increasing integers. When I first read it I didn’t really take the time to understand it fully, but now that I was on a compile-time computation kick, I decided to grok it fully.

Without getting too far into the nitty-gritty (go read the other blog post if you’re interested), the technique relies on using template instantiations to “bring-to-life” functions that affect future template instantiations. Thus we have a flag that declares (but does not yet define) a friend function:

template <int N> struct flag { friend constexpr int adl_flag (flag<N>); };

And then a recursive reader function template that uses ADL and SFINAE to match against the (as-yet-undefined) friend function(s), bottoming out at zero:

template <int N, int = adl_flag(flag<N>{})> constexpr int reader(int, flag<N>) { return N; }   template <int N> constexpr int reader( float, flag<N>, int R = reader(0, flag<N-1>{})) { return R; }   constexpr int reader(float, flag<0>) { return 0; }

And finally, a writer that, when instantiated with the result of calling reader, instantiates the friend function, making the next call to reader terminate at one level higher:

template <int N> struct writer { friend constexpr int adl_flag (flag<N>) { return N; } static constexpr int value = N; };

This is a curiosity, right? A foible of C++, a fairy tale told by wizened programmers to fresh graduates to simultaneously impress and revolt them, no? Could anything useful be done with this? Well, C++ is full of such tales, and it’s a short hop from can’t-look-away-revolting to established feature – after all, template metaprogramming was discovered pretty much by accident…

In fact, while at CppCon, I met up with Ansel and Barbara from CopperSpice, who are using a very similar technique to do away with the Qt Metaobject Compiler.

Max recursion depth, we meet again

My first thought was that this technique suffers at the hands of my old enemy, maximum recursion depth. In this case, maximum template instantiation depth, which despite a standard-recommended 1024, is frequently just 256 – lower than the recommended constexpr recursion depth of 512. So let’s do something about that.

Well, one quick-and-dirty way to do this is to compute the count in two halves: lower bits and upper bits, and then stick them together. When we reach the max on the lower bits, we’ll roll over one of the upper bits. So we have two flags representing the high and low, with the low flag also parameterized on the high bits:

template <int H, int L> struct flag1 { friend constexpr int adl_flag1(flag1<H, L>); }; template <int H> struct flag2 { friend constexpr int adl_flag2(flag2<H>); };

And two readers: the low bits reader is in a struct to avoid partial specialization of a function, because it’s effectively parameterized on the high bits as well as the low bits.

template <int H> struct r1 { template <int L, int = adl_flag1(flag1<H, L>{})> static constexpr int reader(int, flag1<H, L>) { return L; } template <int L> static constexpr int reader( float, flag1<H, L>, int R = reader(0, flag1<H, L-1>{})) { return R; } static constexpr int reader(float, flag1<H, 0>) { return 0; } };   template <int H, int = adl_flag2(flag2<H>{})> constexpr int reader(int, flag2<H>) { return H; } template <int H> constexpr int reader( float, flag2<H>, int R = reader(0, flag2<H-1>{})) { return R; } constexpr int reader(float, flag2<0>) { return 0; }

The low bits writer looks much the same as before, and the high bits writer is specialized on a bool indicating whether or not to instantiate the friend function, which we only do when the low bits roll over:

template <int H, bool B> struct writehi { friend constexpr int adl_flag2(flag2<H>) { return H; } static constexpr int value = H; };   template <int H> struct writehi<H, false> { static constexpr int value = H; };

The writer can then write both the high and low bits accordingly:

template <int H, int L> struct writer { static constexpr int hi_value = writehi<H+1, L == MAX>::value; static constexpr int lo_value = writelo<H, (L & BIT_MASK)>::value; static constexpr int value = (H << BIT_DEPTH) + L; };

Using this approach we can easily increase the maximum number that we can get out of our counter from 256 to 16k or so, which is enough for one translation unit, for me.

Random acts of compiler abuse

Now, a counter is OK, as far as that goes, but something more useful might be nice… how about random numbers? But surely only a madman would try to implement a Mersenne Twister in C++11 constexpr-land. (This despite the fact that I did SHA256 string hashing.) No, these days when I think of random numbers, I think of Melissa O’Neill and her excellent PCG32. If you haven’t seen her video, go watch it. I’ll wait here. PCG32 has some distinct advantages for constexpr implementation:

• It’s easy to implement
• It’s fast
• It’s not a lot of code (seriously, ~10 lines)
• It’s easy to implement
• It’s understandable
• It’s easy to implement

Here’s a simple implementation of the whole 32-bit affair:

constexpr uint64_t pcg32_advance(uint64_t s) { return s * 6364136223846793005ULL + (1442695040888963407ULL | 1); }   constexpr uint64_t pcg32_advance(uint64_t s, int n) { return n == 0 ? s : pcg32_advance(pcg32_advance(s), n-1); }   constexpr uint32_t pcg32_xorshift(uint64_t s) { return ((s >> 18u) ^ s) >> 27u; }   constexpr uint32_t pcg32_rot(uint64_t s) { return s >> 59u; }   constexpr uint32_t pcg32_output(uint64_t s) { return (pcg32_xorshift(s) >> pcg32_rot(s)) | (pcg32_xorshift(s) << ((-pcg32_rot(s)) & 31)); }

And now we can use exactly the same pattern as the constexpr counter, except the writer, instead of giving us an integer, will give us a random number (and we also plumb through the random seed S as a template parameter):

template <uint64_t S, int H, int L> struct writer { static constexpr int hi_value = writehi<S, H+1, L == MAX>::value; static constexpr int lo_value = writelo<S, H, (L & BIT_MASK)>::value; static constexpr uint32_t value = pcg32_output(pcg32_advance(S, (H << BIT_DEPTH) + L)); };

There is probably a way to improve this by storing the actual PCG-computed value in a template instantiation, rather than simply using the integer to pump the PCG every time as I am doing. But it’s a proof-of-concept and works well enough for now. A simple macro will give us a suitable seed for our compile-time RNG by hashing a few things together:

#define cx_pcg32 \ cx::pcg32<cx::fnv1(__FILE__ __DATE__ __TIME__) + __LINE__>

And now we have a compile-time RNG that is seeded differently every compile, every file, every line. There are potentially a lot of template instantiations – maybe using a lot of memory in the compiler – but we can do some useful things with this.

## More string hashing with C++11 constexpr

October 14th, 2015

(Start at the beginning of the series – and all the source can be found in my github repo)

So FNV1 was easy, and Murmur3 wasn’t too much harder; for a challenge and to see how far I could go, I decided to try to compute an MD5 string hash using C++11 constexpr.

This was significantly harder. I broke out my copy of Applied Cryptography 2e, found a reference implementation of MD5 in C, read through RFC 1321 and the pseudocode on the Wikipedia page.

Few, few the bird make her nest

I built up MD5 piece by piece, pulling out parts of the reference implementation to check that I’d got each building block right before moving on. The actual round function primitives were the easy part. As usual for a hash function, they are a mixture of bitwise functions, shifts, rotates, adds. These types of things make for trivial constexpr functions, for example:

constexpr uint32_t F(uint32_t X, uint32_t Y, uint32_t Z) { return (X & Y) | (~X & Z); }   constexpr uint32_t rotateL(uint32_t x, int n) { return (x << n) | (x >> (32-n)); }   constexpr uint32_t FF(uint32_t a, uint32_t b, uint32_t c, uint32_t d, uint32_t x, int s, uint32_t ac) { return rotateL(a + F(b,c,d) + x + ac, s) + b; }

There are similar functions for the other low-level primitives of the MD5 round functions, conventionally called F, G, H and I.

Now, MD5 works on buffer chunks of 512 bits, or 16 32-bit words. So assuming we have a string long enough, it’s easy, if a bit long-winded, to convert a block of string into a schedule that MD5 can work on:

struct schedule { uint32_t w; };   constexpr schedule init(const char* buf) { return { { word32le(buf), word32le(buf+4), word32le(buf+8), word32le(buf+12), word32le(buf+16), word32le(buf+20), word32le(buf+24), word32le(buf+28), word32le(buf+32), word32le(buf+36), word32le(buf+40), word32le(buf+44), word32le(buf+48), word32le(buf+52), word32le(buf+56), word32le(buf+60) } }; }

Seconds out, round one

It’s not pretty, but such are the constructs that may arise when you only have C++11 constexpr to work with. The output of MD5 is going to be 4 32-bit words of hash (denoted A, B, C and D in the literature), and in the main loop, which happens for each 512 bits of the message, there are four rounds, each round having 16 steps. After each step, the 4 words are rotated so that A becomes the new B, B becomes the new C, etc. So a round step is fairly easy – here’s the round 1 step which uses the F primitive:

struct md5sum { uint32_t h; };   constexpr md5sum round1step(const md5sum& sum, const uint32_t* block, int step) { return { { FF(sum.h, sum.h, sum.h, sum.h, block[step], r1shift[step&3], r1const[step]), sum.h, sum.h, sum.h } }; }

As you can see, there are some constants (r1shift and r1const) for each stage of round 1. Rotating the words after each round step is also easy:

constexpr md5sum rotateCR(const md5sum& sum) { return { { sum.h, sum.h, sum.h, sum.h } }; }   constexpr md5sum rotateCL(const md5sum& sum) { return { { sum.h, sum.h, sum.h, sum.h } }; }

So now we are able to put together a complete round, which recurses, calling the round step function and rotating the output until we’re done after 16 steps.

constexpr md5sum round1(const md5sum& sum, const uint32_t* msg, int n) { return n == 16 ? sum : rotateCL(round1(rotateCR(round1step(sum, msg, n)), msg, n+1)); }

Rounds 2 through 4 are very similar, but instead of using the F primitive, they use G, H and I respectively. A complete MD5 transform for one 512-bit block looks like this (with a helper function that sums the MD5 result parts):

constexpr md5sum sumadd(const md5sum& s1, const md5sum& s2) { return { { s1.h + s2.h, s1.h + s2.h, s1.h + s2.h, s1.h + s2.h } }; }   constexpr md5sum md5transform(const md5sum& sum, const schedule& s) { return sumadd(sum, round4( round3( round2( round1(sum, s.w, 0), s.w, 0), s.w, 0), s.w, 0)); }

So far so good. This works, as long as we’re processing the complete 512-bit blocks contained in the message. Now to consider how to finish off. The padding scheme for MD5 is as follows:

• Append a 1-bit (this is always done, even if the message is a multiple of 512 bits)
• Add as many 0-bits as you need to, to make up 448 bits (56 bytes)
• Append a 64-bit value of the original length in bits to the 448 bits to make a final 512-bit block

This gets messy in C++11 constexpr-land, but suffice to say that I wrote a leftover function analogous to init that could deal with padding. Now, finally, the complete MD5 calculation, which has three conditions:

1. As long as there is a 64-byte (512-bit) block to work on, recurse on that.
2. If the leftover is 56 bytes or more, pad it without the length in there and recurse on an “empty block”.
constexpr md5sum md5update(const md5sum& sum, const char* msg, int len, int origlen) { return len >= 64 ? md5update(md5transform(sum, init(msg)), msg+64, len-64, origlen) : len >= 56 ? md5update(md5transform(sum, leftover(msg, len, origlen, 64)), msg+len, -1, origlen) : md5transform(sum, leftover(msg, len, origlen, 56)); }

Woot!

I don’t mind admitting that when I finally got this working, I did a happy dance around my apartment. Even though the code has ugly parts. But of course the thrill of achievement soon gives way to the thirst for more, and MD5 isn’t exactly today’s choice for hash algorithms, even if it is still often used where cryptographic strength isn’t paramount. So I started thinking about SHA256.

Of course, cryptography proceeds largely by incrementally twiddling algorithms when they are found lacking: adding rounds, beefing up functions, etc. And so the NSA’s SHA series proceeded from Ron Rivest’s MD series. In particular, the block size and padding schemes are identical. That was a huge leg-up on SHA256, since I’d already solved the hardest part and I could reuse it.

SHA256 produces a larger digest, and obviously has different magic numbers that go into the rounds, but structurally it’s very similar to MD5. Again I used Wikipedia’s pseudocode as a reference. The only real thing that I needed to do over MD5, besides change up some trivial maths, was the schedule extend step. SHA256 copies the 16 words of the message to a 64-word block to work on, and extends the 16 words into the remaining 48 words. Computing a single value in the schedule, w[i], is easily written in a recursive style, where n represents the “real” values we already have, initially 16:

constexpr uint32_t extendvalue(const uint32_t* w, int i, int n) { return i < n ? w[i] : extendvalue(w, i-16, n) + extendvalue(w, i-7, n) + s0(extendvalue(w, i-15, n)) + s1(extendvalue(w, i-2, n)); }

And to extend all the values, we can then do:

constexpr schedule sha256extend(const schedule& s) { return { { s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, s.w, extendvalue(s.w, 16, 16), extendvalue(s.w, 17, 16), // // and so on... // extendvalue(s.w, 62, 16), extendvalue(s.w, 63, 16) } }; }

When I did this initially, I ran into the compiler’s max step limit for a constexpr computation. Different from the max recursion limit, the max step limit is a guideline for how many expressions should be evaluated within a single constant expression. Appendix B suggests 220. If a compiler did memoization on extendvalue I suspect this would be fine, but evidently clang doesn’t, so to compromise, I split the function into 3: sha256extend16, sha256extend32 and sha256extend48, each of which extends by 16 words at a time. And that worked.

After extending the schedule and changing some of the maths functions, the rest was easy – practically the same as for MD5. Now I was done with string hashing.

For the next experiments with compile-time computation, I wanted to understand a strange thing I’d seen online…

## C++11 compile-time string hashing

October 13th, 2015

(Start at the beginning of the series – and all the source can be found in my github repo)

Now that I was used to writing C++11-style constexpr, I decided to try some compile-time string hashing. It turns out that FNV1 is very easy to express in a move-down-the-string recursive style:

namespace detail { constexpr uint64_t fnv1(uint64_t h, const char* s) { return (*s == 0) ? h : fnv1((h * 1099511628211ull) ^ static_cast<uint64_t>(*s), s+1); } } constexpr uint64_t fnv1(const char* s) { return true ? detail::fnv1(14695981039346656037ull, s) : throw err::fnv1_runtime_error; }

That wasn’t difficult enough

So far so good. How about some other hash functions? On a previous project I had used MurmurHash, so I decided to give that a go.

(Coincidentally, a blog post has recently surfaced implementing 32-bit Murmur3 using C++14 constexpr. But I was sticking to C++11 constexpr.)

Implementing such a big function from cold in C++11 constexpr was bound to be difficult, so I scaffolded. I used the freely-available runtime implementation for testing, and initially I converted it to C++14 constexpr, which was fairly easy. After doing that, and with Wikipedia as a pseudocode guide I was ready to start breaking down (or building up) the functionality, C++11-style. At the base of 32-bit Murmur3 is the hash round function which is done for each 4-byte chunk of the key:

constexpr uint32_t murmur3_32_k(uint32_t k) { return (((k * 0xcc9e2d51) << 15) | ((k * 0xcc9e2d51) >> 17)) * 0x1b873593; }   constexpr uint32_t murmur3_32_hashround( uint32_t k, uint32_t hash) { return (((hash^k) << 13) | ((hash^k) >> 19)) * 5 + 0xe6546b64; }

And this can easily be put into a loop, with a helper function to constitute 4-byte chunks (the somewhat strange formulation of word32le here is because it is used differently in some other code):

constexpr uint32_t word32le(const char* s, int len) { return (len > 0 ? static_cast<uint32_t>(s) : 0) + (len > 1 ? (static_cast<uint32_t>(s) << 8) : 0) + (len > 2 ? (static_cast<uint32_t>(s) << 16) : 0) + (len > 3 ? (static_cast<uint32_t>(s) << 24) : 0); }   constexpr uint32_t word32le(const char* s) { return word32le(s, 4); }   constexpr uint32_t murmur3_32_loop( const char* key, int len, uint32_t hash) { return len == 0 ? hash : murmur3_32_loop( key + 4, len - 1, murmur3_32_hashround( murmur3_32_k(word32le(key)), hash)); }

So this is the first part of Murmur3 that will process all the 4-byte chunks of key. What remains is to deal with the trailing bytes (up to 3 of them) and then finalize the hash. To deal with the trailing bytes, I chain the end functions (_end0 through _end3) and “jump in” at the right place. There is probably a more elegant way to do this, but…

constexpr uint32_t murmur3_32_end0(uint32_t k) { return (((k*0xcc9e2d51) << 15) | ((k*0xcc9e2d51) >> 17)) * 0x1b873593; }   constexpr uint32_t murmur3_32_end1(uint32_t k, const char* key) { return murmur3_32_end0( k ^ static_cast<uint32_t>(key)); }   constexpr uint32_t murmur3_32_end2(uint32_t k, const char* key) { return murmur3_32_end1( k ^ (static_cast<uint32_t>(key) << 8), key); }   constexpr uint32_t murmur3_32_end3(uint32_t k, const char* key) { return murmur3_32_end2( k ^ (static_cast<uint32_t>(key) << 16), key); }   constexpr uint32_t murmur3_32_end(uint32_t hash, const char* key, int rem) { return rem == 0 ? hash : hash ^ (rem == 3 ? murmur3_32_end3(0, key) : rem == 2 ? murmur3_32_end2(0, key) : murmur3_32_end1(0, key)); }

Finalizing the hash is a very similar affair, with 3 stages:

constexpr uint32_t murmur3_32_final1(uint32_t hash) { return (hash ^ (hash >> 16)) * 0x85ebca6b; } constexpr uint32_t murmur3_32_final2(uint32_t hash) { return (hash ^ (hash >> 13)) * 0xc2b2ae35; } constexpr uint32_t murmur3_32_final3(uint32_t hash) { return (hash ^ (hash >> 16)); }   constexpr uint32_t murmur3_32_final(uint32_t hash, int len) { return murmur3_32_final3( murmur3_32_final2( murmur3_32_final1(hash ^ static_cast<uint32_t>(len)))); }

And that’s all there is to it: all that remains is to stitch the pieces together and provide the driver function:

constexpr uint32_t murmur3_32_value(const char* key, int len, uint32_t seed) { return murmur3_32_final( murmur3_32_end( murmur3_32_loop(key, len/4, seed), key+(len/4)*4, len&3), len); }   constexpr uint32_t murmur3_32(const char *key, uint32_t seed) { return true ? murmur3_32_value(key, strlen(key), seed) : throw err::murmur3_32_runtime_error; }

Doing strlen non-naively

One more thing: strlen crept in there. This is of course a constexpr version of strlen. The naive way to do this is to step down the string linearly, as seen in the FNV1 algorithm. With a max recursive depth of 512, I thought this fairly limiting… it might not be uncommon to have 1k string literals that I want to hash?

So the way I implemented strlen was to measure the string by chunks, constraining a max recursion depth of say 256. Instead of a plain linear recursion, it’s basically recursing for every 256 bytes of string length, and then recursing on the chunk inside that. So the max depth is 256 + 256 = 512, and we can deal with strings close to 64k in size (depending on how deep we are already when calling strlen). There’s a chance that strlen might be made constexpr in the future – I think it’s already implemented as an intrinsic in some compilers. So maybe I can throw away that code someday.

Anyway, now we have a complete implementation of 32-bit Murmur3 in C++11 constexpr style, and the tests go something like this (and can be checked online):

static_assert(murmur3_32("hello, world", 0) == 345750399, "murmur3 test 1"); static_assert(murmur3_32("hello, world1", 0) == 3714214180, "murmur3 test 2"); static_assert(murmur3_32("hello, world12", 0) == 83041023, "murmur3 test 3"); static_assert(murmur3_32("hello, world123", 0) == 209220029, "murmur3 test 4"); static_assert(murmur3_32("hello, world1234", 0) == 4241062699, "murmur3 test 5"); static_assert(murmur3_32("hello, world", 1) == 1868346089, "murmur3 test 6");

OK. That was a lot of code for a blog post. But after the trivial FNV1 and the only slightly harder Murmur3, I wanted more of a challenge. What about MD5? Or SHA-256? That’s for the next post.

## More constexpr floating-point computation

October 13th, 2015

(Start at the beginning of the series – and all the source can be found in my github repo)

In the last post, I covered my first forays into writing C++11-style constexpr floating point math functions. Once I’d done some trig functions, exp, and floor and friends, it seemed like the rest would be a piece of cake. Not entirely…

But there was some easy stuff. With trunc sorted out, fmod was just:

constexpr float fmod(float x, float y) { return y != 0 ? x - trunc(x/y)*y : throw err::fmod_domain_error; }

And remainder was very similar. Also, fmax, fmin and fdim were trivial: I won’t bore you with the code.

Euler was quite good at maths

Now I come from a games background, and there are a few maths functions that get used all the time in games. I already have sqrt. What I want now is atan and atan2. To warm up, I implemented the infinite series calculations for asin and acos, each of which are only defined for an input |x| <= 1. That was fairly straightforward, broadly in line with what I'd done before for the series expansions of sin and cos.

Inverting tan was another matter: but a search revealed a way to do it using asin: $arctan(x) = arcsin(\frac{x}{\sqrt{x^2+1}})$

Which I went with for a while. But then I spotted the magic words:

“Leonhard Euler found a more efficient series for the arctangent” $arctan(z) = \frac{z}{1+z^2} \sum_{n=0}^{\infty} \prod_{k=1}^n \frac{2kz^2}{(2k+1)(1+z^2)}$

More efficient (which I take here to mean more quickly converging)? And discovered by Euler? That’s good enough for me.

namespace detail { template <typename T> constexpr T atan_term(T x2, int k) { return (T{2}*static_cast<T>(k)*x2) / ((T{2}*static_cast<T>(k)+T{1}) * (T{1}+x2)); } template <typename T> constexpr T atan_product(T x, int k) { return k == 1 ? atan_term(x*x, k) : atan_term(x*x, k) * atan_product(x, k-1); } template <typename T> constexpr T atan_sum(T x, T sum, int n) { return sum + atan_product(x, n) == sum ? sum : atan_sum(x, sum + atan_product(x, n), n+1); } } template <typename T> constexpr T atan(T x) { return true ? x / (T{1} + x*x) * detail::atan_sum(x, T{1}, 1) : throw err::atan_runtime_error; }

And atan2 follows from atan relatively easily.

It’s big, it’s heavy, it’s wood

Looking over what I’d implemented so far, the next important function (and one that would serve as a building block for a few others) was log. So I programmed it using my friend the Taylor series expansion.

Hm. It turns out that a naive implementation of log is quite slow to converge. (Yes, what a surprise.) OK, a bit of searching yields fruit: “… use Newton’s method to invert the exponentional function … the iteration simplifies to:” $y_{n+1} = y_n + 2 \frac{x - e^{y_n}}{x + e^{y_n}}$

“… which has cubic convergence to ln(x).”

Result! I coded it up.

namespace detail { template <typename T> constexpr T log_iter(T x, T y) { return y + T{2} * (x - cx::exp(y)) / (x + cx::exp(y)); } template <typename T> constexpr T log(T x, T y) { return feq(y, log_iter(x, y)) ? y : log(x, log_iter(x, y)); } }

It worked well. A week or so later, an article about C++11 constexpr log surfaced serendipitously, and on reading it I realised my implementation wasn’t very well tested, so I added some tests. The tests revealed numerical instability for larger input values, so I took a tip from the article and used a couple of identities to constrain the input: $ln(x) = ln(\frac{x}{e}) + 1$
and $ln(x) = ln(e.x) - 1$

Coding these expressions to constrain the input proper to detail::log seemed to solve the instability issues.

Are we done yet?

Now I was nearing the end of what I felt was a quorum of functions. To wit:

• sqrt, cbrt, hypot
• sin, cos, tan
• floor, ceil, trunc, round
• asin, acos, atan, atan2
• fmod, remainder
• exp, log, log2, log10

With a decent implementation of exp and log it was easy to add the hyperbolic trig functions and their inverses. The only obviously missing function now was pow. Recall that I already had ipow for raising a floating-point value to an integral power, but I had been puzzling over a more general version of pow to allow raising to a floating-point power for a while. And then I suddenly realised the blindingly obvious: $x^y = (e^{ln(x)})^y = e^{ln(x).y}$

And for the general case, pow was solved.

template <typename T> constexpr T pow(T x, T y) { return true ? exp(log(x)*y) : throw err::pow_runtime_error; }

I had now reached a satisfying conclusion to maths functions and I was getting fairly well-versed in writing constexpr code. Pausing only to implement erf (because why not?), I turned my attention now to other constexpr matters.