A couple of years ago I wrote about random floating point numbers. In that article I was mainly concerned about how neat the code is, and I didn’t pay attention to its performance. Recently, a comment from Oliver Hunt and a blog post from Alisa Sireneva prompted me to wonder if I made an unwarranted assumption. So I wrote a little benchmark, which you can find in pcg-dxsm.git. As a brief recap, there are two basic ways to convert a random integer to a floating point number between 0.0 and 1.0: Use bit fiddling to construct an integer whose format matches a float between 1.0 and 2.0; this is the same span as the result but with a simpler exponent. Bitcast the integer to a float and subtract 1.0 to get the result. Shift the integer down to the same range as the mantissa, convert to float, then multiply by a scaling factor that reduces it to the desired range. This produces one more bit of randomness than the bithacking conversion. (There are other less basic ways.) My benchmark has 2 x 2 x 2 tests: bithacking vs multiplying 32 bit vs 64 bit sequential integers vs random integers Each operation is isolated from the benchmark loop by putting it in a separate translation unit (to prevent the compiler from inlining) and there is a fence instruction (ISB SY on ARM, MFENCE on AMD) in the loop to stop the CPU from overlapping successive iterations. I ran the benchmark on my Apple M1 Pro and my AMD Ryzen 7950X. In the table below, the leftmost column is the number of random bits. The top half measures sequential numbers, the bottom half is random numbers. The times are nanoseconds per operation, which includes the overheads of the benchmark loop and function call. arm amd 23 12.15 11.22 24 13.37 11.21 52 12.11 11.02 53 13.38 11.20 23 14.75 12.62 24 15.85 12.81 52 16.78 14.23 53 18.02 14.41 The times vary a little from run to run but the difference in speed of the various loops is reasonably consistent. I think my conclusion is that the bithacking conversion is about 1ns faster than the multiply conversion on my ARM box. There’s a subnanosecond difference on my AMD box which might indicate that the conversion takes different amounts of time depending on the value? Dunno.

In hot weather I like to drink my coffee in an iced latte. To make it, I have a very large Bialetti Moka Express. Recently when I got it going again after a winter of disuse, it took me a couple of attempts to get the technique right, so here are some notes as a reminder to my future self next year. It’s worth noting that I’m not fussy about my coffee: I usually drink pre-ground beans from the supermarket, with cream (in winter hot coffee) or milk and ice. basic principle When I was getting the hang of my moka pot, I learned from YouTube coffee geeks such as James Hoffmann that the main aim is for the water to be pushed through the coffee smoothly and gently. Better to err on the side of too little flow than too much. I have not had much success trying to make fine temperature adjustments while the coffee is brewing, because the big moka pot has a lot of thermal inertia: it takes a long time for any change in gas level to have any effect on on the coffee flow. routine fill the kettle and turn it on put the moka pot’s basket in a mug to keep it stable fill it with coffee (mine needs about 4 Aeropress scoops) tamp it down firmly [1] when the kettle has boiled, fill the base of the pot to just below the pressure valve (which is also just below the filter screen in the basket) insert the coffee basket, making sure there are no stray grounds around the edge where the seal will mate screw on the upper chamber firmly put it on a small gas ring turned up to the max [2] leave the lid open and wait for the coffee to emerge immediately turn the gas down to the minimum [3] the coffee should now come out in a steady thin stream without spluttering or stalling when the upper chamber is filled near the mouths of the central spout, it’ll start fizzing or spluttering [4] turn off the gas and pour the coffee into a carafe notes If I don’t tamp the grounds, the pot tends to splutter. I guess tamping gives the puck better integrity to resist channelling, and to keep the water under even pressure. Might be an effect of the relatively coarse supermarket grind? It takes a long time to get the pot back up to boiling point and I’m not sure that heating it up slower helps. The main risk, I think, is overshooting the ideal steady brewing state too much, but: With my moka pot on my hob the lowest gas flow on the smallest rings is just enough to keep the coffee flowing without stalling. The flow when the coffee first emerges is relatively fast, and it slows to the steady state several seconds after I turn the heat down, so I think the overshoot isn’t too bad. This routine turns almost all of the water into coffee, which Hoffmann suggests is a good result, and a sign that the pressure and temperature aren’t getting too high.

TIL (or this week-ish I learned) why big-sigma and big-pi turn up in the notation of dependent type theory. I’ve long been aware of the zoo of more obscure Greek letters that turn up in papers about type system features of functional programming languages, μ, Λ, Π, Σ. Their meaning is usually clear from context but the reason for the choice of notation is usually not explained. I recently stumbled on an explanation for Π (dependent functions) and Σ (dependent pairs) which turn out to be nicer than I expected, and closely related to every-day algebraic data types. sizes of types The easiest way to understand algebraic data types is by counting the inhabitants of a type. For example: the unit type () has one inhabitant, (), and the number 1 is why it’s called the unit type; the bool type hass two inhabitants, false and true. I have even seen these types called 1 and 2 (cruelly, without explanation) in occasional papers. product types Or pairs or (more generally) tuples or records. Usually written, (A, B) The pair contains an A and a B, so the number of possible values is the number of possible A values multiplied by the number of possible B values. So it is spelled in type theory (and in Standard ML) like, A * B sum types Or disjoint union, or variant record. Declared in Haskell like, data Either a b = Left a | Right b Or in Rust like, enum Either<A, B> { Left(A), Right(B), } A value of the type is either an A or a B, so the number of possible values is the number of A values plus the number of B values. So it is spelled in type theory like, A + B dependent pairs In a dependent pair, the type of the second element depends on the value of the first. The classic example is a slice, roughly, struct IntSlice { len: usize, elem: &[i64; len], } (This might look a bit circular, but the idea is that an array [i64; N] must be told how big it is – its size is an explicit part of its type – but an IntSlice knows its own size. The traditional dependent “vector” type is a sized linked list, more like my array type than my slice type.) The classic way to write a dependent pair in type theory is like,      Σ len: usize . Array(Int, len) The big sigma binds a variable that has a type annotation, with a scope covering the expression after the dot – similar syntax to a typed lambda expression. We can expand a simple example like this into a many-armed sum type: either an array of length zero, or an array of length 1, or an array of length 2, … but in a sigma type the discriminant is user-defined instead of hidden. The number of possible values of the type comes from adding up all the alternatives, a summation just like the big sigma summation we were taught in school. ∑ a ∈ A B a When the second element doesn’t depend on the first element, we can count the inhabitants like, ∑ A B = A*B And the sigma type simplifies to a product type. telescopes An aside from the main topic of these notes, I also recently encountered the name “telescope” for a multi-part dependent tuple or record. The name “telescope” comes from de Bruijn’s AUTOMATH, one of the first computerized proof assistants. (I first encountered de Bruijn as the inventor of numbered lambda bindings.) dependent functions The return type of a dependent function can vary according to the argument it is passed. For example, to construct an array we might write something like, fn repeat_zero(len: usize) -> [i64; len] { [0; len] } The classic way to write the type of repeat_zero() is very similar to the IntSlice dependent pair, but with a big pi instead of a big sigma:      Π len: usize . Array(Int, len) Mmm, pie. To count the number of possible (pure, total) functions A ➞ B, we can think of each function as a big lookup table with A entries each containing a B. That is, a big tuple (B, B, … B), that is, B * B * … * B, that is, BA. Functions are exponential types. We can count a dependent function, where the number of possible Bs depends on which A we are passed, ∏ a ∈ A B a danger I have avoided the terms “dependent sum” and “dependent product”, because they seem perfectly designed to cause confusion over whether I am talking about variants, records, or functions. It kind of makes me want to avoid algebraic data type jargon, except that there isn’t a good alternative for “sum type”. Hmf.

About half a year ago I encountered a paper bombastically titled “the ultimate conditional syntax”. It has the attractive goal of unifying pattern match with boolean if tests, and its solution is in some ways very nice. But it seems over-complicated to me, especially for something that’s a basic work-horse of programming. I couldn’t immediately see how to cut it down to manageable proportions, but recently I had an idea. I’ll outline it under the “penultimate conditionals” heading below, after reviewing the UCS and explaining my motivation. what the UCS? whence UCS out of scope penultimate conditionals dangling syntax examples antepenultimate breath what the UCS? The ultimate conditional syntax does several things which are somewhat intertwined and support each other. An “expression is pattern” operator allows you to do pattern matching inside boolean expressions. Like “match” but unlike most other expressions, “is” binds variables whose scope is the rest of the boolean expression that might be evaluated when the “is” is true, and the consequent “then” clause. You can “split” tests to avoid repeating parts that are the same in successive branches. For example, if num < 0 then -1 else if num > 0 then +1 else 0 can be written if num < 0 then -1 > 0 then +1 else 0 The example shows a split before an operator, where the left hand operand is the same and the rest of the expression varies. You can split after the operator when the operator is the same, which is common for “is” pattern match clauses. Indentation-based syntax (an offside rule) reduces the amount of punctuation that splits would otherwise need. An explicit version of the example above is if { x { { < { 0 then −1 } }; { > { 0 then +1 } }; else 0 } } (This example is written in the paper on one line. I’ve split it for narrow screens, which exposes what I think is a mistake in the nesting.) You can also intersperse let bindings between splits. I doubt the value of this feature, since “is” can also bind values, but interspersed let does have its uses. The paper has an example using let to avoid rightward drift: if let tp1_n = normalize(tp1) tp1_n is Bot then Bot let tp2_n = normalize(tp2) tp2_n is Bot then Bot let m = merge(tp1_n, tp2_n) m is Some(tp) then tp m is None then glb(tp1_n, tp2_n) It’s probably better to use early return to avoid rightward drift. The desugaring uses let bindings when lowering the UCS to simpler constructions. whence UCS Pattern matching in the tradition of functional programming languages supports nested patterns that are compiled in a way that eliminates redundant tests. For example, this example checks that e1 is Some(_) once, not twice as written. if e1 is Some(Left(lv)) then e2 Some(Right(rv)) then e3 None then e4 Being cheeky, I’d say UCS introduces more causes of redundant checks, then goes to great effort to to eliminate redundant checks again. Splits reduce redundant code at the source level; the bulk of the paper is about eliminating redundant checks in the lowering from source to core language. I think the primary cause of this extra complexity is treating the is operator as a two-way test rather than a multi-way match. Splits are introduced as a more general (more complicated) way to build multi-way conditions out of two-way tests. There’s a secondary cause: the tradition of expression-oriented functional languages doesn’t like early returns. A nice pattern in imperative code is to write a function as a series of preliminary calculations and guards with early returns that set things up for the main work of the function. Rust’s ? operator and let-else statement support this pattern directly. UCS addresses the same pattern by wedging calculate-check sequences into if statements, as in the normalize example above. out of scope I suspect UCS’s indentation-based syntax will make programmers more likely to make mistakes, and make compilers have more trouble producing nice error messages. (YAML has put me off syntax that doesn’t have enough redundancy to support good error recovery.) So I wondered if there’s a way to have something like an “is pattern” operator in a Rust-like language, without an offside rule, and without the excess of punctuation in the UCS desugaring. But I couldn’t work out how to make the scope of variable bindings in patterns cover all the code that might need to use them. The scope needs to extend into the consequent then clause, but also into any follow-up tests – and those tests can branch so the scope might need to reach into multiple then clauses. The problem was the way I was still thinking of the then and else clauses as part of the outer if. That implied the expression has to be closed off before the then, which troublesomely closes off the scope of any is-bound variables. The solution – part of it, at least – is actually in the paper, where then and else are nested inside the conditional expression. penultimate conditionals There are two ingredients: The then and else clauses become operators that cause early return from a conditional expression. They can be lowered to a vaguely Rust syntax with the following desugaring rules. The 'if label denotes the closest-enclosing if; you can’t use then or else inside the expr of a then or else unless there’s another intervening if. then expr ⟼ && break 'if expr else expr ⟼ || break 'if expr else expr ⟼ || _ && break 'if expr There are two desugarings for else depending on whether it appears in an expression or a pattern. If you prefer a less wordy syntax, you might spell then as => (like match in Rust) and else as || =>. (For symmetry we might allow && => for then as well.) An is operator for multi-way pattern-matching that binds variables whose scope covers the consequent part of the expression. The basic form is like the UCS, scrutinee is pattern which matches the scrutinee against the pattern returning a boolean result. For example, foo is None Guarded patterns are like, scrutinee is pattern && consequent where the scope of the variables bound by the pattern covers the consequent. The consequent might be a simple boolean guard, for example, foo is Some(n) && n < 0 or inside an if expression it might end with a then clause, if foo is Some(n) && n < 0 => -1 // ... Simple multi-way patterns are like, scrutinee is { pattern || pattern || … } If there is a consequent then the patterns must all bind the same set of variables (if any) with the same types. More typically, a multi-way match will have consequent clauses, like scrutinee is { pattern && consequent || pattern && consequent || => otherwise } When a consequent is false, we go on to try other alternatives of the match, like we would when the first operand of boolean || is false. To help with layout, you can include a redundant || before the first alternative. For example, if foo is { || Some(n) && n < 0 => -1 || Some(n) && n > 0 => +1 || Some(n) => 0 || None => 0 } Alternatively, if foo is { Some(n) && ( n < 0 => -1 || n > 0 => +1 || => 0 ) || None => 0 } (They should compile the same way.) The evaluation model is like familiar shortcutting && and || and the syntax is supposed to reinforce that intuition. The UCS paper spends a lot of time discussing backtracking and how to eliminate it, but penultimate conditionals evaluate straightforwardly from left to right. The paper briefly mentions as patterns, like Some(Pair(x, y) as p) which in Rust would be written Some(p @ Pair(x, y)) The is operator doesn’t need a separate syntax for this feature: Some(p is Pair(x, y)) For large examples, the penultimate conditional syntax is about as noisy as Rust’s match, but it scales down nicely to smaller matches. However, there are differences in how consequences and alternatives are punctuated which need a bit more discussion. dangling syntax The precedence and associativity of the is operator is tricky: it has two kinds of dangling-else problem. The first kind occurs with a surrounding boolean expression. For example, when b = false, what is the value of this? b is true || false It could bracket to the left, yielding false: (b is true) || false Or to the right, yielding true: b is { true || false } This could be disambiguated by using different spellings for boolean or and pattern alternatives. But that doesn’t help for the second kind which occurs with an inner match. foo is Some(_) && bar is Some(_) || None Does that check foo is Some(_) with an always-true look at bar ( foo is Some(_) ) && bar is { Some(_) || None } Or does it check bar is Some(_) and waste time with foo? foo is { Some(_) && ( bar is Some(_) ) || None } I have chosen to resolve the ambiguity by requiring curly braces {} around groups of alternative patterns. This allows me to use the same spelling || for all kinds of alternation. (Compare Rust, which uses || for boolean expressions, | in a pattern, and , between the arms of a match.) Curlies around multi-way matches can be nested, so the example in the previous section can also be written, if foo is { || Some(n) && n < 0 => -1 || Some(n) && n > 0 => +1 || { Some(0) || None } => 0 } The is operator binds tigher than && on its left, but looser than && on its right (so that a chain of && is gathered into a consequent) and tigher than || on its right so that outer || alternatives don’t need extra brackets. examples I’m going to finish these notes by going through the ultimate conditional syntax paper to translate most of its examples into the penultimate syntax, to give it some exercise. Here we use is to name a value n, as a replacement for the |> abs pipe operator, and we use range patterns instead of split relational operators: if foo(args) is { || 0 => "null" || n && abs(n) is { || 101.. => "large" || ..10 => "small" || => "medium" ) } In both the previous example and the next one, we have some extra brackets where UCS relies purely on an offside rule. if x is { || Right(None) => defaultValue || Right(Some(cached)) => f(cached) || Left(input) && compute(input) is { || None => defaultValue || Some(result) => f(result) } } This one is almost identical to UCS apart from the spellings of and, then, else. if name.startsWith("_") && name.tailOption is Some(namePostfix) && namePostfix.toIntOption is Some(index) && 0 <= index && index < arity && => Right([index, name]) || => Left("invalid identifier: " + name) Here are some nested multi-way matches with overlapping patterns and bound values: if e is { // ... || Lit(value) && Map.find_opt(value) is Some(result) => Some(result) // ... || { Lit(value) || Add(Lit(0), value) || Add(value, Lit(0)) } => { print_int(value); Some(value) } // ... } The next few examples show UCS splits without the is operator. In my syntax I need to press a few more buttons but I think that’s OK. if x == 0 => "zero" || x == 1 => "unit" || => "?" if x == 0 => "null" || x > 0 => "positive" || => "negative" if predicate(0, 1) => "A" || predicate(2, 3) => "B" || => "C" The first two can be written with is instead, but it’s not briefer: if x is { || 0 => "zero" || 1 => "unit" || => "?" } if x is { || 0 => "null" || 1.. => "positive" || => "negative" } There’s little need for a split-anything feature when we have multi-way matches. if foo(u, v, w) is { || Some(x) && x is { || Left(_) => "left-defined" || Right(_) => "right-defined" } || None => "undefined" } A more complete function: fn zip_with(f, xs, ys) { if [xs, ys] is { || [x :: xs, y :: ys] && zip_with(f, xs, ys) is Some(tail) => Some(f(x, y) :: tail) || [Nil, Nil] => Some(Nil) || => None } } Another fragment of the expression evaluator: if e is { // ... || Var(name) && Map.find_opt(env, name) is { || Some(Right(value)) => Some(value) || Some(Left(thunk)) => Some(thunk()) } || App(lhs, rhs) => // ... // ... } This expression is used in the paper to show how a UCS split is desugared: if Pair(x, y) is { || Pair(Some(xv), Some(yv)) => xv + yv || Pair(Some(xv), None) => xv || Pair(None, Some(yv)) => yv || Pair(None, None) => 0 } The desugaring in the paper introduces a lot of redundant tests. I would desugar straightforwardly, then rely on later optimizations to eliminate other redundancies such as the construction and immediate destruction of the pair: if Pair(x, y) is Pair(xx, yy) && xx is { || Some(xv) && yy is { || Some(yv) => xv + yv || None => xv } || None && yy is { || Some(yv) => yv || None => 0 } } Skipping ahead to the “non-trivial example” in the paper’s fig. 11: if e is { || Var(x) && context.get(x) is { || Some(IntVal(v)) => Left(v) || Some(BoolVal(v)) => Right(v) } || Lit(IntVal(v)) => Left(v) || Lit(BoolVal(v)) => Right(v) // ... } The next example in the paper compares C# relational patterns. Rust’s range patterns do a similar job, with the caveat that Rust’s ranges don’t have a syntax for exclusive lower bounds. fn classify(value) { if value is { || .. -4.0 => "too low" || 10.0 .. => "too high" || NaN => "unknown" || => "acceptable" } } I tend to think relational patterns are the better syntax than ranges. With relational patterns I can rewrite an earlier example like, if foo is { || Some(< 0) => -1 || Some(> 0) => +1 || { Some(0) || None } => 0 } I think with the UCS I would have to name the Some(_) value to be able to compare it, which suggests that relational patterns can be better than UCS split relational operators. Prefix-unary relational operators are also a nice way to write single-ended ranges in expressions. We could simply write both ends to get a complete range, like >= lo < hi or like if value is > -4.0 < 10.0 => "acceptable" || => "far out" Near the start I quoted a normalize example that illustrates left-aligned UCS expression. The penultimate version drifts right like the Scala version: if normalize(tp1) is { || Bot => Bot || tp1_n && normalize(tp2) is { || Bot => Bot || tp2_n && merge(tp1_n, tp2_n) is { || Some(tp) => tp || None => glb(tp1_n, tp2_n) } } } But a more Rusty style shows the benefits of early returns (especially the terse ? operator) and monadic combinators. let tp1 = normalize(tp1)?; let tp2 = normalize(tp2)?; merge(tp1, tp2) .unwrap_or_else(|| glb(tp1, tp2)) antepenultimate breath When I started writing these notes, my penultimate conditional syntax was little more than a sketch of an idea. Having gone through the previous section’s exercise, I think it has turned out better than I thought it might. The extra nesting from multi-way match braces doesn’t seem to be unbearably heavyweight. However, none of the examples have bulky then or else blocks which are where the extra nesting is more likely to be annoying. But then, as I said before it’s comparable to a Rust match: match scrutinee { pattern => { consequent } } if scrutinee is { || pattern => { consequent } } The || lines down the left margin are noisy, but hard to get rid of in the context of a curly-brace language. I can’t reduce them to | like OCaml because what would I use for bitwise OR? I don’t want presence or absence of flow control to depend on types or context. I kind of like Prolog / Erlang , for && and ; for ||, but that’s well outside what’s legible to mainstream programmers. So, dunno. Anyway, I think I’ve successfully found a syntax that does most of what UCS does, but much in a much simpler fashion.