A growing collection of interactive tutorials, guides and quizzes about maths, algorithms, performance, and programming languages. https://wordsandbuttons.online/ Sun, 21 Sep 2025 08:51:46 +0000 Sun, 30 Dec 2018 15:40:00 +0000 1800 A guessing game with C++ operators, context, and disassembly. https://wordsandbuttons.online/challenge_your_performance_intuition_with_cpp_operators.html 46c480f5-b10f-4cc9-a1b7-acc781056309 Tue, 25 Dec 2018 10:58:15 +0000 A folow up on a guessing game with C++ operators. Let's redo a few rounds. The benchmark is the same, the questions are the same. The compiler is the same. The only thing that changes is the platform. This is now CHIP with ARMv7. https://wordsandbuttons.online/using_logical_operators_for_logical_operations_is_good.html 57e102d8-cb60-48ae-ae0c-5d8995ab5d88 Sun, 30 Dec 2018 15:44:00 +0000 The same as https://wordsandbuttons.online/programmers_introduction_to_linear_equations.html only in Ukrainian. https://wordsandbuttons.online/ua/programmers_introduction_to_linear_equations.html ff3b144f-3808-4e7a-9d70-b0d26728fcf8 Sat, 5 Jan 2019 08:24:00 +0000 The algorithm that takes a distance function and makes a smooth contour out of it in three steps. https://wordsandbuttons.online/the_simplest_possible_smooth_contouring_algorithm.html f73660bb-ae4b-4d83-8418-8f7c3350b2c3 Tue, 15 Jan 2019 18:20:00 +0000 It's about 100 lines of Python code that generate a linear solver in LLVM intermediate language outperforming C and C++ solutions. Originally published in mid 2018, now completetly rewritten. https://wordsandbuttons.online/outperforming_everything_with_anything.html 1bcc8b28-2349-4bee-b778-8d879d2cbefb Sat, 19 Jan 2019 09:05:00 +0000 Vectorization is when you take some minecraft-style raster image and make a crisp vector picture out of it. The algorithm describes shows how bilinear interpolation, polynomial approximation, differential analysis, and iterative algorithms work together to solve a practical problem. https://wordsandbuttons.online/simple_image_vectorization.html e99ce59c-a731-4d5e-9977-1e2fd4c571bd Thu, 7 Mar 2019 19:50:00 +0000 Dot and cross vector products are absolutely essential for everything three-dimensional. Whatever 3D thing you do: graphics, printing, physical simulation, — as soon as you want anything beyond what your framework provides, you have to do some vector magic, and these are your most basic spells. The more embarrassing it is for me to admit. Up until very recently, I couldn't remember which way the cross product goes. https://wordsandbuttons.online/interactive_mnemonics_for_dot_and_cross_vector_products.html 336eac33-7496-4d6e-8b11-1111f950f352 Tue, 26 Mar 2019 18:00:00 +0000 “Lexical” since it doesn't need true syntax analysis, primitive tokenization and filtering are enough. And it's “differential” because it aims to highlight the difference between lexemes. Ideally, the smaller the lexical difference, the greater the color difference should be. It works with any languages. And it saves your traffic. https://wordsandbuttons.online/lexical_differential_highlighting_instead_of_syntax_highlighting.html 9eee5c1c-c3a7-43ac-a35a-8817343b2183 Thu, 11 Apr 2019 18:00:00 +0000 Unforeseen floating point error is the source of the most unpleasant bugs. The bugs that come and go unpredictably. The bugs that don't reproduce on unit-tests and lay low through the integration phase only to be seen by your most important customer. https://wordsandbuttons.online/estimating_floating_point_error_the_easy_way.html 59cc6722-9a9e-4047-8159-5418a62eb302 Sat, 11 May 2019 08:45:00 +0000 I still think that reexplaining some obscure concepts with different words (and buttons) might help someone understand floating point numbers better. To do fewer mistakes, to make things faster, to create better software in general. https://wordsandbuttons.online/yet_another_floating_point_tutorial.html 13b8a239-48dc-468d-a700-57336a64a738 Thu, 16 May 2019 05:40:00 +0000 It's a Python library that does the boring part of math for you. Moreover, it does it fast, accurate, and without angst. In other words, it is everything I'm not. https://wordsandbuttons.online/sympy_makes_math_fun_again.html 3cea5d53-0c1a-486e-badc-c85101827677 Tue, 25 Jun 2019 18:05:00 +0000 Usually, when you want to make a parametric curve, you should go for a polynomial spline. But sometimes, for the reasons mentioned only in the second half of this page to keep you intrigued, you are not satisfied with polynomials. You have to look elsewhere. This page shows you one possible option apart from polynomials and it is an old-school parametric curve made from arcs and line segments stitched together. https://wordsandbuttons.online/circles_and_lines_vs_polynomial_splines.html 4cc82d82-827a-4d15-9b0b-5eac9c54a59e Thu, 22 Aug 2019 6:40:00 +0000 Let's play a game. There will be twelve rounds, each round is a match between two solutions. Or rather between their execution times. Using your intuition and best judgment, please estimate their relative performance. https://wordsandbuttons.online/challenge_your_performance_intuition_with_cpp_magic_squares.html 203a70f2-bc43-4cda-af10-baaa5286418b Mon, 26 Aug 2019 10:55:00 +0000 I stole exactly three ideas from other people, made them work together, and gave that compilation a funny name. I had my fun playing with the concept but the conclusion I had to put into my thesis was: “SWInE is rather promising”. In academic language, this means “useless”. However, the three ideas behind the algorithm are prominent. And if an algorithm helps show them in action, then maybe this is what makes the algorithm worthy. https://wordsandbuttons.online/swine_simplicial_weight_interpolation_and_extrapolation.html 8689596b-7314-4711-9160-63c360b45ec6 Sun, 13 Oct 2019 13:05:00 +0000 If the operation you want to speed-up already runs in a few nano-seconds, your reasoning about algorithmic complexity probably wouldn't apply. The most effective algorithms become mediocre while the useless rise from the oblivion to shine and amaze. One of these algorithms is the index sort. It has unconditional O(n2) complexity and this makes it probably the least performant sorting algorithm you never heard about. It's useless on the macro-scale. But on the nano-scale, it excels. https://wordsandbuttons.online/challenge_your_performance_intuition_with_nanosecond_sorting.html 4fb603c2-6457-42e5-bd9c-b04296720518 Tue, 5 Nov 2019 6:25:00 +0000 When you write in C++ you actually write in two languages at once. First is C++, and the second one is the invisible Prolog. If written properly, the second program is helpful. However, if being neglected, it turns your code into an untangleable mess of incomprehencibles really-really fast. https://wordsandbuttons.online/logic_programming_in_cpp.html 489a9e82-d56b-433a-b647-4a1fe8ade56e Tue, 13 Nov 2019 17:50:00 +0000 SWInE is a promising alternative to splines. This page guides you from a simple weight interpolation to spatial local interpolation and extrapolation in 3 steps. https://wordsandbuttons.online/swine_simplicial_weight_interpolation_and_extrapolation.html 4d9bc22a-df02-4963-9e6f-9de6f30c4d76 Sat, 16 Nov 2019 5:55:00 +0000 Interactive mnemonics and changeable snippets.TL&DR: Sine is like S and cosine is like C. You can make things like O or 8 by using them both. https://wordsandbuttons.online/sine_and_cosine.html 8f438a3e-f866-4814-acd2-d8a0b2a44d77 Mon, 9 Dec 2019 11:40:00 +0000 It generates a seamless black and white honeycomb texture. There are a few options in the UI but if you need more, there is some changeable code too. If you need more freedom, just get this thing on Github. https://wordsandbuttons.online/honeycomb_texture_generator.html 5d51b56c-4587-464e-a09e-0c3a3d7ce090 Fri, 27 Dec 2019 08:10:00 +0000 Why would you care about homogeneous coordinates, whatever they are? Well, if you work with geometry: 3D-graphics, image processing, physical simulation, — the answer is obvious. Knowing the mathematics behind your framework enables you to write more efficient code. But even if you don’t work with geometry at all, you still might enjoy learning about projective space and its link to linear algebra. It is an outstanding example of mathematical magic: you pay with a small complication, you gain an enormous simplification in return. https://wordsandbuttons.online/interactive_guide_to_homogeneous_coordinates.html b6bf429a-7405-4d18-9fa7-c87131611c29 Sat, 11 Jan 2020 17:40:00 +0000 The page contains interactive plots to play with different strategies and distributions. https://wordsandbuttons.online/binary_search.html 6bd16767-e6c0-40fc-8dc1-93978d8b0721 Wed, 12 Feb 2020 17:20:00 +0000 People do it with different feelings. Sometimes with pride for a clever solution, sometimes with guilt for an obscure hack. Which is odd since there is nothing particularly hacky about it. The ISO/IEC 14882 is ok with it. The IEEE 754 is ok with it. So why is it a hack and not a norm? I'm talking, of course, about storing return codes in floating-point NaNs. https://wordsandbuttons.online/error_codes_are_not_numbers_but_they_are_can_we_exploit_that.html 37571a45-dcde-4568-a82e-1eab340251bf Thu, 27 Feb 2020 18:50:00 +0000 It's about a feature that cleans up dirt stains from aged newspapers. I called it darning because of how it works. https://wordsandbuttons.online/image_darning.html 0739f36d-d2fd-4735-8425-73ac7848e209 Sat, 28 Mar 2020 15:50:00 +0000 Can’t say that running away from complexity was a smart thing to do. It was definitely not the most productive way to create software. The journey, however, has taught me a few things so it was not a complete waste of time either. This book is a reflection of these lessons. It consists of small essays each about one particular language and one particular insight. They can be read in order but they can be read separately. I encourage you to start from the languages you’re the least interested in. I also encourage you to try the actual languages too. I hope you’ll find this book not too boring, not too trivial, and if nothing else, not too long. https://wordsandbuttons.online/SYTYKC.pdf 20d605f3-c637-4034-8e72-3eaf40a21423 Tue, 21 Apr 2020 15:05:00 +0000 With arctangent used as a scale, we can show any function on a screen. And not just a fragment of it but the whole function. https://wordsandbuttons.online/arctangent_scale_its_like_the_logarithmic_scale_but_infinite.html 58d9e19c-0072-412a-93d3-ee82ae399569 Sat, 9 May 2020 22:50:00 +0000 An interactive explanation of how polynomial transformations such as trilinear or biquadratic or even linear-cubic work, and how to craft your own that fits your particular task the best. https://wordsandbuttons.online/biwhatever_transformations.html 89f837ab-22fc-4c88-a891-15eda1b28d2f Tue, 16 Jun 2020 14:34:50 +0000 This interactive guide shows how homogeneous coordinates actually make geometry simpler and not more complicated. It explains the extra coordinate, the matrices, the generalized transformations. Most of what you need to know about projective geometry as a practicing programmer is here. https://wordsandbuttons.online/interactive_guide_to_homogeneous_coordinates.html d155e541-e3ed-4e8a-8a8f-aee8f49ff888 Sat, 27 Jun 2020 09:52:30 +0000 This depicts an old idea of using multifocal lemniscates to draw arbitrary curves. In somewhat limited scope, this may be indeed an economical alternative to splines. https://wordsandbuttons.online/can_we_use_lemniscates_for_ultra_cheap_vector_gaphics.html 6bdab756-f51c-447f-9e18-6b88347c08ef Fri, 17 Jul 2020 10:06:03 +0000 This is a brief introduction into polynomials. From how to make a polynomial run through your set of points to how to make it into a spline. https://wordsandbuttons.online/programmers_guide_to_polynomials_and_splines.html 439932a9-54bc-4586-9531-6549f371844c Tue, 28 Jul 2020 04:40:40 +0000 NURBS stands for the non-uniform rational basis spline. There are three separate concepts. This guide walks you through these concepts one by one. https://wordsandbuttons.online/nurbs_is_just_an_acronym.html 1e40bfa4-e852-4200-9ab0-cd6724c1d831 Fri, 14 Aug 2020 06:11:27 +0000 This explains approximation and interpolation, how to use polynomials for that, and how to make both concepts work together. https://wordsandbuttons.online/polynomial_approximation_and_interpolation.html 6e522621-2957-4ada-8eb0-60220c3808db Fri, 21 Aug 2020 10:30:06 +0000 This explains why the division by zero in floating point numbers is ok. https://wordsandbuttons.online/why_is_it_ok_to_divide_by_0_0.html 1c0efcd5-b41b-4e20-b3a7-8376b0a286c5 Sat, 22 Aug 2020 10:48:09 +0000 This shows how computeable intervals written in rational bounds may not only account for the input error but keep computational error under control as well. https://wordsandbuttons.online/yet_another_alternative_to_floating_point_numbers.html b6c23e2d-7647-4f1c-a3f9-ccf5dda86f49 Sat, 31 Oct 2020 13:39:20 +0000 A simple image processing algorithm that cleans up dirt stains from old newspapers. I called it “darning” because of how it works. https://wordsandbuttons.online/image_darning.html f7ba7547-7974-45ce-9403-cf5a6c425b8d Tue, 17 Nov 2020 19:25:57 +0000 Non-Boolean logics are rare but not extinct. Interval logic is one example. Sometimes, you can implement a logic you want within total order or partial order but sometimes even that isn't enough and you need an even more general relation. With operator overloading, you have the freedom to go there but you also have less assurance when working within the total order. https://wordsandbuttons.online/partial_order_and_non_boolean_logic.html 3469e844-7f0d-4d78-b9a3-26dba11c333a Sat, 26 Dec 2020 07:25:33 +0000 An interactive explanation of how polynomial transformations such as trilinear or biquadratic or even linear-cubic work, and how to craft your own. Now with a practical example! https://wordsandbuttons.online/biwhatever_transformations.html c7f3e03c-9e30-414d-bceb-bd6ea79f26b6 Sun, 27 Dec 2020 07:50:51 +0000 This shows the global properties of polynomials, their derivatives, and explains how the Maclaurine and Taylor series work all with animated plots in arctangent scale. https://wordsandbuttons.online/trippy_polynomials_in_arctangent_scale.html 08b3302d-87bf-4dc4-9ff4-5362a6f8060d Sun, 10 Jan 2021 09:41:01 +0000 This explains Lagrange polynomial: why does it run through all the points, what is the basis polynomial, and how come it's a polynomial in the first place. https://wordsandbuttons.online/lagrange_polynomial_as_a_gateway_drug_to_basis_splines.html bbc3e5e1-f2a4-416d-a93b-33718d51155f Mon, 18 Jan 2021 16:19:49 +0000 An interactive explanation of how iterative algorithms work. This explains convergence and the exit condition problem on an oversimplified linear system solver. https://wordsandbuttons.online/interactive_introduction_to_iterative_algorithms.html c5ec09ce-7659-4056-b6fc-2637b2cdf3b7 Sun, 24 Jan 2021 09:50:21 +0000 Marching cubes and dual contouring are often used for mesh generation. This explanation shows how they work, what are their differences, similarities, and limitations. https://wordsandbuttons.online/interactive_explanation_of_marching_cubes_and_dual_contouring.html 74baf289-5b55-45d0-99f5-7e1b2f510855 Sun, 24 Jan 2021 09:50:27 +0000 An alternative to polynomial splines. Smooth parametric curves made from arcs and line segments. https://wordsandbuttons.online/circles_and_lines_vs_polynomial_splines.html 4ba640fc-96c6-459c-8ded-62b6d8c88a94 Sun, 24 Jan 2021 09:50:40 +0000 Most of what you should know about floating-point numbers put together in an interactive tutorial with quests and puzzles. https://wordsandbuttons.online/yet_another_floating_point_tutorial.html a71b1dcf-d06a-4e71-ad3f-91432abee0fd Tue, 26 Jan 2021 15:47:41 +0000 A brief introduction to mathematical analysis with a little SymPy on the side. The page explains how to disassemble a function, and how to assemble it back from the derivatives. https://wordsandbuttons.online/mathematical_analysis_explained_with_python_blood_and_tnt.html 157ef698-a959-44e0-9a8f-f7c0acc13e60 Sat, 30 Jan 2021 15:35:06 +0000 This is an introduction to linear equation systems. It explains linear dependency, under- and over-specification, direct and iterative solvers. The guide should give you enough knowledge to find a proper solution for your task but not enough to implement one efficiently yourself. https://wordsandbuttons.online/programmers_introduction_to_linear_equations.html 98aff1f0-97f7-4722-8c23-c985a63d75d4 Fri, 12 Feb 2021 19:18:12 +0000 An introduction into symbolic computations in Python. Don't worry, it's much simpler than it sounds. It's about making Python do your math for you with very little investment in the technology. https://wordsandbuttons.online/sympy_makes_math_fun_again.html a07dca57-dc49-4923-ace3-388704fcd687 Sat, 13 Feb 2021 08:16:58 +0000 This explains the geometry of complex numbers. Explains conformal transformations, introduces analytic functions, and shows that analytic complex functions are conformal. And using the connection between the geometry and the analysis explains it all in just some five minutes. https://wordsandbuttons.online/complex_numbers_and_conformal_mapping.html 319a4c5d-216e-431a-b943-69c8dd79d874 Mon, 08 Mar 2021 08:59:50 +0000 One more interactive quiz. This time, it's all about the sine function. Which one is faster and when? https://wordsandbuttons.online/challenge_your_performance_intuition_with_cpp_sine.html 8aeba7a9-8ba8-4f3e-8b2a-7ad5bae648e5 Sat, 27 Mar 2021 13:57:15 +0000 Component redundancy is used heavily in safety-critical and mission-critical systems for reliability improvement. But outside this niche, it's surprisingly little known in the world of software. Which is a shame since it's a simple but economical idea. It costs nothing to keep in mind, and it saves you a lot on hotfixes and emergency repairs. https://wordsandbuttons.online/what_can_we_learn_from_sexaplication_on_nuclear_power_plants.html c53a016f-74ca-4a04-9609-c36d51d8b43e Sat, 10 Apr 2021 08:42:48 +0000 Component redundancy is used heavily in safety-critical and mission-critical systems for reliability improvement. But outside this niche, it's surprisingly little known in the world of software. Which is a shame since it's a simple but economical idea. It costs nothing to keep in mind, and it saves you a lot on hotfixes and emergency repairs. https://wordsandbuttons.online/redundant_stories_about_redundancy.html 0a37c0f0-2a41-47c5-9a17-3a904c24df1d Mon, 12 Apr 2021 05:08:15 +0000 An interactive quiz featuring several obscure high-level languages and assembly variants. https://wordsandbuttons.online/can_you_tell_an_assembly_language_when_you_see_one.html 7b8dedd4-34fb-4c36-b385-ecae2387e754 Fri, 23 Apr 2021 19:16:33 +0000 This shows how a picture can be turned into an HTML table. With this, you can not only have nicer tables or uglier pictures, but you can have something that is both at the same time. https://wordsandbuttons.online/a_picture_is_worth_a_thousand_tags.html 90f8f336-fa6d-435b-a81b-4fd64e2abfe4 Sat, 08 May 2021 10:19:15 +0000 An interactive quiz about microoptimizations in CUDA. 10 rounds, two pieces of code per each, you get to guess which is the faster. https://wordsandbuttons.online/check_if_your_performance_intuition_still_works_with_cuda.html 349d4449-bc3a-4bf4-97ba-27fd34253a10 Tue, 01 Jun 2021 15:12:23 +0000 This is not an interactive page, it has zero didactic value. It does explain why Words and Buttons has been stagnant for a while though. I have been writing a book. Why? Well, visit and see. https://wordsandbuttons.online/why.html 1260788f-41d4-4f85-9aa8-938def81acae Thu, 29 Dec 2022 09:41:00 +0000 This is a very concise introduction to x86 disassembly. It shows that you probably have all the skills to start reading disassembly right now even if you thought it was too complicated for you. It really isn't, see it for yourself. https://wordsandbuttons.online/you_dont_have_to_learn_assembly_to_read_disassembly.html ca421761-ff8f-4db4-bc69-9d71a5ad9867 Fri, 03 Mar 2023 17:13:12 +0000 An introduction into symbolic computations in Python. Don't worry, it's much simpler than it sounds. It's about making Python do your math for you with very little investment in the technology. https://wordsandbuttons.online/sympy_makes_math_fun_again.html 792c0433-af93-4c23-a77d-21f60cbe23f9 Fri, 03 Mar 2023 17:13:53 +0000 Type inference works like logic deduction so any program in a statically typed language is two programs. The first one is the thing you sell, and the second – is a model that undergoes some sort of verification every time you run a compiler. This second program, although, often being written unknowingly, is not a free bonus, it’s something you have to pay for in several ways. https://wordsandbuttons.online/static_typing_isnt_free.html 4826838a-f6df-4377-bfd8-727ba68fcf7e Wed, 28 Jun 2023 08:29:12 +0000 This page uses polynomial modeling to show why software engineering tasks are often impossible to estimate. https://wordsandbuttons.online/either_your_estimates_suck_or_your_job_does.html 76becea2-10a3-4f32-9786-acacd2680cd0 Tue, 19 Sep 2023 15:06:15 +0000 This interactive guide shows how homogeneous coordinates actually make programming geometry simpler and not more complicated. The page explains the extra coordinate, matrices, and generalized transformations. Most of what you need to know about projective geometry as a practicing programmer is here. https://wordsandbuttons.online/interactive_guide_to_homogeneous_coordinates.html cbf1c894-f375-4104-9066-ee8e465af80c Tue, 17 Oct 2023 04:42:36 +0000 Rational interpolation is a step forward from polynomial interpolation towards rational splines. With rational interpolation, you can build functions that run through a set of points and also have vertical asymptotes whenever you want. With this capability, you can now model functions like logarithms better. https://wordsandbuttons.online/rational_interpolation.html b11ee945-1296-47c6-967f-42ba2b774b71 Tue, 05 Mar 2024 06:24:39 +0000 All the “C++ killers”, even these which I wholeheartedly love and respect like Rust, Julia, and D, help you write more features with fewer bugs, but they don't much help when you need to squeeze the very last FLOPS from the hardware you rent. As such, they don’t have a competitive advantage over C++. Or, for that matter, even over each other. Most of them, for instance, Rust, Julia, and Clang even share the same backend. You can’t win a car race if you all share the same car. So, which technologies do hold a competitive advantage over C++ or, speaking generally, all the traditional ahead-of-time compilers? https://wordsandbuttons.online/the_real_cpp_killers.html a290c4ad-feea-4f1d-b3fc-8a118018f6c4 Wed, 20 Mar 2024 16:56:46 +0000 An image interpolation that gives us a continuous and smooth image, where every interpolated value only depends on the four neighboring pixel values. The image becomes smooth, but sharp features remain sharp. https://wordsandbuttons.online/a_smooth_and_sharp_image_interpolation.html 30adce54-fe7b-480b-a379-f5e7d24a8cd1 Mon, 29 Apr 2024 16:25:00 +0000 An interactive demo of bisection search and golden ratio search algorithms. There is also a motivation to learn them both. Spoiler alert! One converges better, and the other has a better computational cost. https://wordsandbuttons.online/why_learn_about_the_golden_section_search.html 8264cbfd-9285-43ea-a60f-ae2e64d820a3 Sun, 29 Jun 2025 15:39:22 +0000 An interactive demo of the Gauss-Seidel method for solving linear equation systems. When and why does it work? And why, when you flip the equations, it suddenly stop working? You can now play and see. https://wordsandbuttons.online/gauss_seidel_visually_explained.html 6a44eb93-0d18-45f5-9483-286cb4d45747 Sun, 21 Sep 2025 08:51:46 +0000