From 06afced5c7f4e8fe51a291d64ed73565988c272b Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 13:09:45 +0200 Subject: [PATCH 1/9] blog post about prospero challenge --- images/2025-glitchy-prospero.png | Bin 0 -> 18042 bytes images/2025-image-octree-zoomed-in.png | Bin 0 -> 412354 bytes images/2025-image-octree-zoomed-out.png | Bin 0 -> 46763 bytes images/2025-image-prospero-dataflow.png | Bin 0 -> 352393 bytes posts/2025/04/prospero-in-rpython.md | 720 ++++++++++++++++++++++++ 5 files changed, 720 insertions(+) create mode 100644 images/2025-glitchy-prospero.png create mode 100644 images/2025-image-octree-zoomed-in.png create mode 100644 images/2025-image-octree-zoomed-out.png create mode 100644 images/2025-image-prospero-dataflow.png create mode 100644 posts/2025/04/prospero-in-rpython.md diff --git a/images/2025-glitchy-prospero.png b/images/2025-glitchy-prospero.png new file mode 100644 index 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The challenge is to render a 1024x1024 of +a quote from The Tempest by Shakespeare. The input is a mathematical formula +with 7866 operations, which is evaluated once per pixel. + +What made the challenge particularly enticing for me personally was the fact +that the formula is basically a trace in +[SSA-form](https://en.wikipedia.org/wiki/Static_single-assignment_form) – a +linear sequence of operations, where every variable is assigned exactly once. +The challenge is to evaluate the formula as fast as possible. I tried a number +of ideas how to speed up execution and will talk about them in this somewhat +meandering post. Most of it follows Matt's implementation +[Fidget](https://github.com/mkeeter/fidget) very closely. There are two points +of difference: + +- I tried to add more peephole optimizations, but they didn't end up helping + much. +- I implemented a "demanded information" optimization that removes a lot of + operations by only keeping the sign of the result. This optimization ended up + being useful. + +Most of the prototyping in this post was done in RPython (a statically typable +subset of Python2, that can be compiled to C), but I later rewrote the program +in C to get better performance. All the code [can be found on +Github](https://github.com/cfbolz/pyfidget/). + +## Input program + +The input program is a sequence of operations, like this: + +``` +_0 const 2.95 +_1 var-x +_2 const 8.13008 +_3 mul _1 _2 +_4 add _0 _3 +_5 const 3.675 +_6 add _5 _3 +_7 neg _6 +_8 max _4 _7 +... +``` + +The first column is the name of the result variable, the second column is the +operation, and the rest are the arguments to the operation. `var-x` is a +special operation that returns the x-coordinate of the pixel being rendered, +and equivalently for `var-y` the y-coordinate. The sign of the result gives the +color of the pixel, the absolute value is not important. + +## A baseline interpreter + +To run the program, I first parse them and replace the register names with +indexes, to avoid any dictionary lookups at runtime. +Then I implemented a simple interpreter for the SSA-form +input program. The interpreter is a simple register machine, where every +operation is executed in order. The result of the operation is stored into a +list of results, and the next operation is executed. This was the slow baseline +implementation of the interpreter but it's very useful to compare the optimized +versions against. + +This is roughly what the code looks like + +```python +class DirectFrame(object): + def __init__(self, program): + self.program = program + self.next = None + + def run_floats(self, x, y, z): + self.setxyz(x, y, z) + return self.run() + + def setxyz(self, x, y, z): + self.x = x + self.y = y + self.z = z + + def run(self): + program = self.program + num_ops = program.num_operations() + floatvalues = [0.0] * num_ops + for op in range(num_ops): + func, arg0, arg1 = program.get_func_and_args(op) + if func == OPS.const: + floatvalues[op] = program.consts[arg0] + continue + farg0 = floatvalues[arg0] + farg1 = floatvalues[arg1] + if func == OPS.var_x: + res = self.x + elif func == OPS.var_y: + res = self.y + elif func == OPS.var_z: + res = self.z + elif func == OPS.add: + res = self.add(farg0, farg1) + elif func == OPS.sub: + res = self.sub(farg0, farg1) + elif func == OPS.mul: + res = self.mul(farg0, farg1) + elif func == OPS.max: + res = self.max(farg0, farg1) + elif func == OPS.min: + res = self.min(farg0, farg1) + elif func == OPS.square: + res = self.square(farg0) + elif func == OPS.sqrt: + res = self.sqrt(farg0) + elif func == OPS.exp: + res = self.exp(farg0) + elif func == OPS.neg: + res = self.neg(farg0) + elif func == OPS.abs: + res = self.abs(farg0) + else: + assert 0 + floatvalues[op] = res + return self.floatvalues[num_ops - 1] + + def add(self, arg0, arg1): + return arg0 + arg1 + + def sub(self, arg0, arg1): + return arg0 - arg1 + + def mul(self, arg0, arg1): + return arg0 * arg1 + + def max(self, arg0, arg1): + return max(arg0, arg1) + + def min(self, arg0, arg1): + return min(arg0, arg1) + + def square(self, arg0): + val = arg0 + return val*val + + def sqrt(self, arg0): + return math.sqrt(arg0) + + def exp(self, arg0): + return math.exp(arg0) + + def neg(self, arg0): + return -arg0 + + def abs(self, arg0): + return abs(arg0) +``` + +Running the naive interpreter on the prospero image file is super slow, since +it performs 7866 * 1024 * 1024 float operations, plus the interpretation overhead. + +## Using Quadtrees to render the picture + +The approach that Matt describes in his really excellent +[talk](https://www.youtube.com/watch?v=UxGxsGnbyJ4) is to use +[quadtrees](https://en.wikipedia.org/wiki/Quadtree): recursively subdivide the +image into quadrants, and evaluate the formula in each quadrant. For every +quadrant you can simplify the formula by doing a range analysis. After a few +recursion steps, the formula becomes significantly smaller, often only a few +hundred or a few dozen operations. + +At the bottom of the recursion you either reach a square where the range +analysis reveals that the sign for all pixels is determined, then you can fill +in all the pixels of the quadrant. Or you can evaluate the (now much simpler) +formula in the quadrant by executing it for every pixel. + +This is an interesting use case of JIT compiler/optimization techniques because +it requires the optimizer to execute really quickly, since it is an essential +part of the performance of the algorithm. The optimizer runs literally hundreds +of times to render a single image, and if the algorithm is used for 3D models +it becomes even more crucial. + +## Writing a simple optimizer + +To implement the quadtree recursion is straightforward. Since the program has +no control flow the optimizer is very simple to write. I've written a couple of +blog posts on how to easily write optimizers for linear sequences of +operations, and I'm using the approach described in these [Toy +Optimizer](https://pypy.org/categories/toy-optimizer.html) posts. The interval +analysis is basically an [abstract +interpretation](https://pypy.org/posts/2024/08/toy-knownbits.html) of the +operations. The optimizer does a sequential forward pass over the input +program. For every operation, the output interval is computed. The optimizer +also performs optimizations based on the computed intervals, which helps in +reducing the number of operations executed (I'll talk about this further down). + +Here's a sketch of the Python code that does the optimization: + +```python +class Optimizer(object): + def __init__(self, program): + self.program = program + num_operations = program.num_operations() + self.resultops = ProgramBuilder(num_operations) + self.intervalframe = IntervalFrame(self.program) + # old index -> new index + self.opreplacements = [0] * num_operations + self.index = 0 + + def get_replacement(self, op): + return self.opreplacements[op] + + def newop(self, func, arg0=0, arg1=0): + return self.resultops.add_op(func, arg0, arg1) + + def newconst(self, value): + const = self.resultops.add_const(value) + self.intervalframe.minvalues[const] = value + self.intervalframe.maxvalues[const] = value + #self.seen_consts[value] = const + return const + + def optimize(self, a, b, c, d, e, f): + program = self.program + self.intervalframe.setxyz(a, b, c, d, e, f) + numops = program.num_operations() + for index in range(numops): + newop = self._optimize_op(index) + self.opreplacements[index] = newop + return self.opreplacements[numops - 1] + + def _optimize_op(self, op): + program = self.program + intervalframe = self.intervalframe + func, arg0, arg1 = program.get_func_and_args(op) + assert arg0 >= 0 + assert arg1 >= 0 + if func == OPS.var_x: + minimum = intervalframe.minx + maximum = intervalframe.maxx + return self.opt_default(OPS.var_x, minimum, maximum) + if func == OPS.var_y: + minimum = intervalframe.miny + maximum = intervalframe.maxy + return self.opt_default(OPS.var_y, minimum, maximum) + if func == OPS.var_z: + minimum = intervalframe.minz + maximum = intervalframe.maxz + return self.opt_default(OPS.var_z, minimum, maximum) + if func == OPS.const: + const = program.consts[arg0] + return self.newconst(const) + arg0 = self.get_replacement(arg0) + arg1 = self.get_replacement(arg1) + assert arg0 >= 0 + assert arg1 >= 0 + arg0minimum = intervalframe.minvalues[arg0] + arg0maximum = intervalframe.maxvalues[arg0] + arg1minimum = intervalframe.minvalues[arg1] + arg1maximum = intervalframe.maxvalues[arg1] + if func == OPS.neg: + return self.opt_neg(arg0, arg0minimum, arg0maximum) + if func == OPS.min: + return self.opt_min(arg0, arg1, arg0minimum, arg0maximum, arg1minimum, arg1maximum) + ... + + def opt_default(self, func, minimum, maximum, arg0=0, arg1=0): + self.intervalframe._set(newop, minimum, maximum) + return newop + + def opt_neg(self, arg0, arg0minimum, arg0maximum): + # peephole rules go here, see below + minimum, maximum = self.intervalframe._neg(arg0minimum, arg0maximum) + return self.opt_default(OPS.neg, minimum, maximum, arg0) + + @symmetric + def opt_min(self, arg0, arg1, arg0minimum, arg0maximum, arg1minimum, arg1maximum): + # peephole rules go here, see below + minimum, maximum = self.intervalframe._max(arg0minimum, arg0maximum, arg1minimum, arg1maximum) + return self.opt_default(OPS.max, minimum, maximum, arg0, arg1) + + ... +``` + +The resulting optimized traces are then simply interpreted at the bottom of the +quadtree recursion. Matt talks about also generating machine code from them, +but when I tried to use PyPy's JIT for that it was simply way too slow at +producing machine code. + + +## Testing soundness of the interval abstract domain + +To make sure that my interval computation in the optimizer is correct, I +implemented a hypothesis-based property based test. It checks the abstract +transfer functions of the interval domain for soundness. It does so by +generating random concrete input values for an operation, random intervals that +surround the random concrete values, then performs the concrete operation to +get the concrete output, and checks that the abstract transfer function applied +to the input intervals gives an interval that contains the concrete output. + +For example, the random test for the `square` operation would look like this: + +```python +from hypothesis import given, strategies, assume +from pyfidget.vm import IntervalFrame, DirectFrame +import math + +regular_floats = strategies.floats(allow_nan=False, allow_infinity=False) + +def make_range_and_contained_float(a, b, c): + a, b, c, = sorted([a, b, c]) + return a, b, c + +frame = DirectFrame(None) +intervalframe = IntervalFrame(None) + +range_and_contained_float = strategies.builds(make_range_and_contained_float, regular_floats, regular_floats, regular_floats) + +def contains(res, rmin, rmax): + if math.isnan(rmin) or math.isnan(rmax): + return True + return rmin <= res <= rmax + + +@given(range_and_contained_float) +def test_square(val): + a, b, c = val + rmin, rmax = intervalframe._square(a, c) + res = frame.square(b) + assert contains(res, rmin, rmax) +``` + +This test generates a random float `b`, and two other floats `a` and `c` such +that the interval `[a, c]` contains `b`. The test then checks that the result +of the `square` operation on `b` is contained in the interval `[rmin, rmax]` +returned by the abstract transfer function for the `square` operation. + + +## Peephole rewrites + +The only optimization that Matt does in his implementation is a peephole +optimization rule that removes `min` and `max` operations where the intervals +of the arguments don't overlap. In that case, the optimizer statically can know +which of the arguments will be the result of the operation. I implemented this +peephole optimization in my implementation as well, but I also added a few more +peephole optimizations that I thought would be useful. + + +```python +class Optimizer(object): + + def opt_neg(self, arg0, arg0minimum, arg0maximum): + # new: add peephole rule --x => x + func, arg0arg0, _ = self.resultops.get_func_and_args(arg0) + if func == OPS.neg: + return arg0arg0 + minimum, maximum = self.intervalframe._neg(arg0minimum, arg0maximum) + return self.opt_default(OPS.neg, minimum, maximum, arg0) + + @symmetric + def opt_min(self, arg0, arg1, arg0minimum, arg0maximum, arg1minimum, arg1maximum): + # Matt's peephole rule + if arg0maximum < arg1minimum: + return arg0 # we can use the intervals to decide which argument will be returned + # new one by me: min(x, x) => x + if arg0 == arg1: + return arg0 + func, arg0arg0, arg0arg1 = self.resultops.get_func_and_args(arg0) + minimum, maximum = self.intervalframe._max(arg0minimum, arg0maximum, arg1minimum, arg1maximum) + return self.opt_default(OPS.max, minimum, maximum, arg0, arg1) + + ... +``` + +However, it turns out that all my attempts at adding other peephole +optimization rules were not very useful. Most rules never fired, and the ones +that did only had a small effect on the performance of the program. The only +peephole optimization that I found to be useful was the one that Matt describes +in his talk. Matt's `min`/`max` optimization were 96% of all rewrites that my +peephole optimizer applied for the `prospero.vm` input. The remaining 4% of +rewrites were (the percentages are of that 4%): + +``` +--x => x 4.65% +(-x)**2 => x ** 2 0.99% +min(x, x) => x 20.86% +min(x, min(x, y)) => min(x, y) 52.87% +max(x, x) => x 16.40% +max(x, max(x, y)) => max(x, y) 4.23% +``` + +In the end it turned out that having these extra optimization rules made the +total runtime of the system go up. Checking for the rewrites isn't free, and +since they apply so rarely they don't pay for their own cost in terms of +improved performance. + +There are some further rules that I tried that never fired at all: + +``` +a * 0 => 0 +a * 1 => a +a * a => a ** 2 +a * -1 => -a +a + 0 => a +a - 0 => a +x - x => 0 +abs(known positive number x) => x +abs(known negative number x) => -x +abs(-x) => abs(x) +(-x) ** 2 => x ** 2 +``` + +This investigation is clearly way too focused on a single program and should be +re-done with a larger set of example inputs, if this were an actually serious +implementation. + +## Demanded Information Optimization + +TODO Max: I am having a hard time understanding the demanded bits thing from your localized-to-prospero explanation. I think some of it is that I am wondering "what about the other uses of operation A that operation B only needs one bit from" and part of it is that the sign operations are implicit here + +LLVM has an information called 'demanded bits'. It is a backwards analysis that +allows you to determine which bits of a value are actually used in the final +result. This information can then be used in peephole optimizations. For +example, if you have an expression that computes a value, but only the last +byte of that value is used in the final result, you can optimize the expression +to only compute the last byte. + +Here's an example. Let's say we first byte-swap a 64-bit int, and then mask off the last byte: + +```c +uint64_t byteswap_then_mask(uint64_t a) { + return byteswap(a) & 0xff; +} +``` + +In this case, the "demanded bits" of the `byteswap(a)` expression are +`0b0...011111111`, which inversely means that we don't care about the upper 56 +bits. Therefore the whole expression can be optimized to `a >> 56`. + +For Fidget, we can observe that for the resulting pixel values, the value of +the result is not used at all, only its sign. Essentially, every program ends +implicitly with a `sign` operation that returns `0.0` for negative values and +`1.0` for positive values. For clarity, I will show this `sign` operation in +the rest of the section, even if it's not actually in the real code. + + +This makes it possible to simplify certain min/max +operations further. Here is an example of a program, together with the +intervals of the variables: + +``` +x var-x # [0.1, 1] +y var-y # [-1, 1] +m min x y # [-1, 1] +out sign m +``` + +This program can be optimized to: + +``` +y var-y +out sign m +``` + +Because that expression has the same result as the original expression: if `x > +0.1`, for the result of `min(x, y)` to be negative then `y` needs to be negative. + +Another, more complex, example is this: + +``` +x var-x # [1, 100] +y var-y # [-10, 10] +z var-z # [-100, 100] +m1 min x y # [-10, 10] +m2 max z out # [-10, 100] +out sign m2 +``` + +Which can be optimized to this: + +``` +y var-y +z var-z +m2 max z y +out sign m2 +``` + +This is because the sign of `min(x, y)` is the same as the sign of `y` if `x > +0`, and the sign of `max(z, min(x, y))` is thus the same as the sign of `max(z, +y)`. + +To implement this optimization, I do a backwards pass over the program after +the peephole optimization forward pass. For every `min` call I encounter, where +one of the arguments is positive, I can optimize the `min` call away and +replace it with the other argument. For `max` calls I simplify their arguments +recursively. + +The code looks roughly like this: + +```python +def work_backwards(resultops, result, minvalues, maxvalues): + def demand_sign_simplify(op): + func, arg0, arg1 = resultops.get_func_and_args(op) + if func == OPS.max: + narg0 = demand_sign_simplify(arg0) + if narg0 != arg0: + resultops.setarg(op, 0, narg0) + narg1 = demand_sign_simplify(arg1) + if narg1 != arg1: + resultops.setarg(op, 1, narg1) + if func == OPS.min: + if minvalues[arg0] > 0.0: + return demand_sign_simplify(arg1) + if minvalues[arg1] > 0.0: + return demand_sign_simplify(arg0) + narg0 = demand_sign_simplify(arg0) + if narg0 != arg0: + resultops.setarg(op, 1, narg0) + narg1 = demand_sign_simplify(arg1) + if narg1 != arg1: + resultops.setarg(op, 1, narg1) + return op + return demand_sign_simplify(result) +``` + +In my experiment, this optimization lets me remove 25% of all operations in +prospero, at the various levels of my octree. I'll briefly look at performance +results further down. + +## Further ideas about the demanded sign simplification + +There is another idea how to short-circuit the evaluation of expressions that I +tried briefly but didn't pursue to the end. Let's go back to the first example +of the previous subsection, but with different intervals: + +``` +x var-x # [-1, 1] +y var-y # [-1, 1] +m min x y # [-1, 1] +out sign m +``` + +Now we can't use the "demanded sign" trick in the optimizer, because neither +`x` nor `y` are known positive. However, during *execution* of the program, if +`x` turns out to be negative we can end the execution of this trace +immediately, since we know that the result must be negative. + +So I experimented with adding `return_early_if_neg` flags to all operations +with this property. The interpreter then checks whether the flag is set on an +operation and if the result is negative, it stops the execution of the program +early: + +``` +x var-x[return_early_if_neg] +y var-y[return_early_if_neg] +m min x y +out sign m +``` + +This looked pretty promising, but it's also a trade-off because the cost of +checking the flag and the value isn't zero. Here's a sketch to the change in the interpreter: + + +```python +class DirectFrame(object): + ... + def run(self): + program = self.program + num_ops = program.num_operations() + floatvalues = [0.0] * num_ops + for op in range(num_ops): + ... + if func == OPS.var_x: + res = self.x + ... + else: + assert 0 + if program.get_flags(op) & OPS.should_return_if_neg and res < 0.0: + return res + floatvalues[op] = res + return self.floatvalues[num_ops - 1] +``` + +I implemented this in the RPython +version, but didn't end up porting it to C, because it interferes with SIMD. + +## Dead code elimination + +Matt performs dead code elimination in his implementation by doing a single +backwards pass over the program. This is a very simple and effective +optimization, and I implemented it in my implementation as well. The dead code +elimination pass is very simple: It starts by marking the result operation as +used. Then it goes backwards over the program. If the current operation is +used, its arguments are marked as used as well. Afterwards, all the operations +that are not marked as used are removed from the program. The PyPy JIT actually +performs dead code elimination on traces in exactly the same way (and I don't +think we ever explained how this works on the blog), so I thought it was worth +mentioning. + +Matt also performs register allocation as part of the backwards pass, but I +didn't implement it because I wasn't too interested in that aspect. + +## Random testing of the optimizer + +To make sure I didn't break anything in the optimizer, I implemented a random +test that generates random input programs and checks that the output of the +optimizer is equivalent to the input program. The test generates random +operations, random intervals for the operations and a random input value within +that interval. It then runs the optimizer on the input program and checks that +the output program has the same result as the input program. This is again +implemented with `hypothesis`. Hypothesis' test case minimization feature is +super useful for finding optimizer bugs. It's just not fun to analyze a problem +on a many-thousand-operation input file, but Hypothesis often generated reduced +test cases that were only a few operations long. + +## Visualizing programs + +It's actually surprisingly annoying to visualize `prospero.vm` well, because +it's quite a bit too large to just feed it into Graphviz. I made the problem +slightly easier by grouping several operations together, where only the first +operation in a group is used as the argument for more than one operation +further in the program. This made it slightly more managable for Graphviz. But +it still wasn't a big enough improvement to be able to visualize all of +`prospero.vm` in its unoptimized form at the top of the octree. + +Here's a visualization of the optimized `prospero.vm` at one of the octree +levels: + +![graph vizualization of a part of the input program](/images/2025-image-prospero-dataflow.png) + +The result is on top, every node points to its arguments. The `min` and `max` +operations form a kind of "spine" of the expression tree, because they are +unions and intersection in the constructive solid geometry sense. + +I also wrote a function to visualize the octree recursion itself, the output +looks like this: + +![graph visualization of the octree recursion, zoomed out](/images/2025-image-octree-zoomed-out.png) + +![graph visualization of the octree recursion, zoomed in](/images/2025-image-octree-zoomed-in.png) + +Green nodes are where the interval analysis determined that the output must be +entirely outside the shape. Yellow nodes are where the octree recursion +bottomed out. + +## C implementation + +To achieve even faster performance, I decided to rewrite the implementation in +C. While RPython is great for prototyping, it can be challenging to control +low-level aspects of the code. The rewrite in C allowed me to experiment with +several techniques I had been curious about: + +- [`musttail` optimization](https://blog.reverberate.org/2021/04/21/musttail-efficient-interpreters.html) for the interpreter. +- SIMD (Single Instruction, Multiple Data): Using Clang's + [`ext_vector_type`](https://clang.llvm.org/docs/LanguageExtensions.html#vectors-and-extended-vectors), I process eight pixels at once using AVX (or some other + SIMD magic that I don't properly understand). +- Efficient struct packing: I packed the operations struct into just 8 + bytes by limiting the maximum number of operations to 65,536, with the idea + of making the optimizer faster. + +I didn't rigorously study the performance impact of each of these techniques +individually, so it's possible that some of them might not have contributed +significantly. However, the rewrite was a fun exercise for me to explore these +techniques. The code can be found +[here](https://github.com/cfbolz/pyfidget/blob/main/pyfidget/experiments.c). + +## Testing the C implementation + +At various points I had bugs in the C implementation, leading to a fun glitchy +version of prospero: + +![glitchy prospero](/images/2025-glitchy-prospero.png) + +To find these bugs, I used the same random testing approach as in the +RPython version. I generated random input programs as strings in Python and +checked that the output of the C implementation was equivalent to the output of +the RPython implementation (simply by calling out to the shell and reading the +generated image, then comparing pixels). This helped ensure that the C +implementation was +correct and didn't introduce any bugs. It was surprisingly tricky to get this +right, for reasons that I didn't expect. At lot of them are related to the fact +that in C I used `float` and Python uses `double` for its (Python) `float` +type. This made the random tester find weird floating point corner cases where +rounding behaviour between the widths was different. + +I solved those by using `double` in C when running the random tests by means of +an `IFDEF`. + +It's super fun to watch the random program generator produce random images, here are a few: + + + +## Performance + +Some very rough performance results on my laptop (an AMD Ryzen 7 PRO 7840U with +32 GiB RAM running Ubuntu 24.04), comparing the RPython version, the C version +(with and without demanded info), and Fidget (in `vm` mode, its JIT made things +worse for me), both for 1024x1024 and 4096x4096 images: + +| Implementation | 1024x1024 | 4096x4096 | +|----------------------|-----------|-----------| +| RPython | 26.8ms | 75.0ms | +| C (no demanded info) | 24.5ms | 45.0ms | +| C (demanded info) | 18.0ms | 37.0ms | +| Fidget | 10.8ms | 57.8ms | + +The demanded info seem to help quite a bit, which was nice to see. + +## Conclusion + +That's it! I had lots of fun with the challenge and have a whole bunch of other +ideas I want to try out, but I should actually go back to working on my actual +projects now ;-). From 61822c3f9f95646bf744a28412f99c5c46b3e3b1 Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 13:14:12 +0200 Subject: [PATCH 2/9] typos --- posts/2025/04/prospero-in-rpython.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index 4a866a6aa..5d45d716d 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -626,14 +626,14 @@ It's actually surprisingly annoying to visualize `prospero.vm` well, because it's quite a bit too large to just feed it into Graphviz. I made the problem slightly easier by grouping several operations together, where only the first operation in a group is used as the argument for more than one operation -further in the program. This made it slightly more managable for Graphviz. But +further in the program. This made it slightly more manageable for Graphviz. But it still wasn't a big enough improvement to be able to visualize all of `prospero.vm` in its unoptimized form at the top of the octree. Here's a visualization of the optimized `prospero.vm` at one of the octree levels: -![graph vizualization of a part of the input program](/images/2025-image-prospero-dataflow.png) +![graph visualization of a part of the input program](/images/2025-image-prospero-dataflow.png) The result is on top, every node points to its arguments. The `min` and `max` operations form a kind of "spine" of the expression tree, because they are From 87fd538f09ea03fe1fb290e2152c0b1a44b7fce6 Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:21:54 +0200 Subject: [PATCH 3/9] Update posts/2025/04/prospero-in-rpython.md Co-authored-by: Matti Picus --- posts/2025/04/prospero-in-rpython.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index 5d45d716d..cba87ed9a 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -13,7 +13,7 @@ Recently I had a lot of fun playing with the [Prospero Challenge](https://www.mattkeeter.com/projects/prospero/) by [Matt -Keeter](https://www.mattkeeter.com/). The challenge is to render a 1024x1024 of +Keeter](https://www.mattkeeter.com/). The challenge is to render a 1024x1024 image of a quote from The Tempest by Shakespeare. The input is a mathematical formula with 7866 operations, which is evaluated once per pixel. From 59c4ae95e7a1dfca033805e04e98c12ffac2cc1e Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:22:23 +0200 Subject: [PATCH 4/9] Update posts/2025/04/prospero-in-rpython.md Co-authored-by: Matti Picus --- posts/2025/04/prospero-in-rpython.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index cba87ed9a..f9aab95c9 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -69,8 +69,8 @@ Then I implemented a simple interpreter for the SSA-form input program. The interpreter is a simple register machine, where every operation is executed in order. The result of the operation is stored into a list of results, and the next operation is executed. This was the slow baseline -implementation of the interpreter but it's very useful to compare the optimized -versions against. +implementation of the interpreter but it's very useful to compare against the optimized +versions. This is roughly what the code looks like From 2edf1cf09d2db532cf3bcca93397c829f1bff57b Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:22:39 +0200 Subject: [PATCH 5/9] Update posts/2025/04/prospero-in-rpython.md Co-authored-by: Matti Picus --- posts/2025/04/prospero-in-rpython.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index f9aab95c9..3b3f9bb3b 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -181,8 +181,8 @@ analysis reveals that the sign for all pixels is determined, then you can fill in all the pixels of the quadrant. Or you can evaluate the (now much simpler) formula in the quadrant by executing it for every pixel. -This is an interesting use case of JIT compiler/optimization techniques because -it requires the optimizer to execute really quickly, since it is an essential +This is an interesting use case of JIT compiler/optimization techniques, +requiring the optimizer itself to execute really quickly since it is an essential part of the performance of the algorithm. The optimizer runs literally hundreds of times to render a single image, and if the algorithm is used for 3D models it becomes even more crucial. From 551b1e494ea57b0ffc85c3723a659d236cce1959 Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:27:50 +0200 Subject: [PATCH 6/9] Update posts/2025/04/prospero-in-rpython.md Co-authored-by: Matti Picus --- posts/2025/04/prospero-in-rpython.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index 3b3f9bb3b..ef568db32 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -184,7 +184,7 @@ formula in the quadrant by executing it for every pixel. This is an interesting use case of JIT compiler/optimization techniques, requiring the optimizer itself to execute really quickly since it is an essential part of the performance of the algorithm. The optimizer runs literally hundreds -of times to render a single image, and if the algorithm is used for 3D models +of times to render a single image. If the algorithm is used for 3D models it becomes even more crucial. ## Writing a simple optimizer From a58518f995a94209b59a75c3f68938adb809a67a Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:29:38 +0200 Subject: [PATCH 7/9] Apply suggestions from code review thanks @mattip! Co-authored-by: Matti Picus --- posts/2025/04/prospero-in-rpython.md | 15 +++++++-------- 1 file changed, 7 insertions(+), 8 deletions(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index ef568db32..84fe0f158 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -189,7 +189,7 @@ it becomes even more crucial. ## Writing a simple optimizer -To implement the quadtree recursion is straightforward. Since the program has +Implementing the quadtree recursion is straightforward. Since the program has no control flow the optimizer is very simple to write. I've written a couple of blog posts on how to easily write optimizers for linear sequences of operations, and I'm using the approach described in these [Toy @@ -291,7 +291,7 @@ class Optimizer(object): The resulting optimized traces are then simply interpreted at the bottom of the quadtree recursion. Matt talks about also generating machine code from them, -but when I tried to use PyPy's JIT for that it was simply way too slow at +but when I tried to use PyPy's JIT for that it was way too slow at producing machine code. @@ -300,9 +300,9 @@ producing machine code. To make sure that my interval computation in the optimizer is correct, I implemented a hypothesis-based property based test. It checks the abstract transfer functions of the interval domain for soundness. It does so by -generating random concrete input values for an operation, random intervals that +generating random concrete input values for an operation and random intervals that surround the random concrete values, then performs the concrete operation to -get the concrete output, and checks that the abstract transfer function applied +get the concrete output, and finally checks that the abstract transfer function applied to the input intervals gives an interval that contains the concrete output. For example, the random test for the `square` operation would look like this: @@ -425,7 +425,7 @@ implementation. TODO Max: I am having a hard time understanding the demanded bits thing from your localized-to-prospero explanation. I think some of it is that I am wondering "what about the other uses of operation A that operation B only needs one bit from" and part of it is that the sign operations are implicit here -LLVM has an information called 'demanded bits'. It is a backwards analysis that +LLVM has an static analysis pass called 'demanded bits'. It is a backwards analysis that allows you to determine which bits of a value are actually used in the final result. This information can then be used in peephole optimizations. For example, if you have an expression that computes a value, but only the last @@ -609,7 +609,7 @@ didn't implement it because I wasn't too interested in that aspect. ## Random testing of the optimizer -To make sure I didn't break anything in the optimizer, I implemented a random +To make sure I didn't break anything in the optimizer, I implemented a test that generates random input programs and checks that the output of the optimizer is equivalent to the input program. The test generates random operations, random intervals for the operations and a random input value within @@ -716,5 +716,4 @@ The demanded info seem to help quite a bit, which was nice to see. ## Conclusion That's it! I had lots of fun with the challenge and have a whole bunch of other -ideas I want to try out, but I should actually go back to working on my actual -projects now ;-). +ideas I want to try out, thanks Matt for this interesting puzzle. From 29c35f581ae51bf63496ad6fe2584066743e4401 Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 14:33:00 +0200 Subject: [PATCH 8/9] it's not really about fidget anyway --- posts/2025/04/prospero-in-rpython.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index 84fe0f158..e299c17de 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -444,7 +444,7 @@ In this case, the "demanded bits" of the `byteswap(a)` expression are `0b0...011111111`, which inversely means that we don't care about the upper 56 bits. Therefore the whole expression can be optimized to `a >> 56`. -For Fidget, we can observe that for the resulting pixel values, the value of +For the Prospero challenge, we can observe that for the resulting pixel values, the value of the result is not used at all, only its sign. Essentially, every program ends implicitly with a `sign` operation that returns `0.0` for negative values and `1.0` for positive values. For clarity, I will show this `sign` operation in From 8c07c011a1a4d20c4f06b1c47e3ba04683907dd9 Mon Sep 17 00:00:00 2001 From: CF Bolz-Tereick Date: Wed, 9 Apr 2025 17:09:09 +0200 Subject: [PATCH 9/9] change date --- posts/2025/04/prospero-in-rpython.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/posts/2025/04/prospero-in-rpython.md b/posts/2025/04/prospero-in-rpython.md index e299c17de..7e0896ebb 100644 --- a/posts/2025/04/prospero-in-rpython.md +++ b/posts/2025/04/prospero-in-rpython.md @@ -1,8 +1,8 @@