# Math Keeps Changing, Hacker News This is a written version of a talk that I gave at WaffleJS in February, which itself was an expansion of a Twitter conversation from October. Okay, so it starts with my delayed math education. As part of my Computer Science program, I had access to world-class math professors, access that I mostly wasted. I did not like math: the topics were so removed from practice, and I was already frustrated by the highly theoretical, and – I thought at the time and mostly still do – out-of-touch CS program.

Unfortunately, a few years after graduating, I got the hunger for math. Seeing how I could apply just a little bit of math knowledge to great effect in my work & hobbies had me inspired. But I had no clear way of learning it.

But I started it in 2019. In tech years that’s a really long time ago. Between then and now, there have been 8 LTS releases of Node . JavaScript and its environments have radically changed. 2017 was before the introduction of React or the first commit to Babel.

So what I noticed over the years was that tests kept breaking when I updated Node. I’d have a test like:

``` (equal)    ss    gamma   (  .  ,   .  );      That would work in Node v . and break in Node v . And this is not some complex method: gamma is implemented with arithmetic, Math.pow, Math.sqrt, and Math.sin.
fpm_start( "true" );

Arithmetic
```

So I know what you might be thinking: arithmetic. JavaScript, on Twitter, gets a lot of heat for this behavior:

`  (0.1   0.2=0.)      As I wrote in  JavaScript wats, dissected , this is the behavior of every popular programming language, even stodgy pedantic ones like Haskell. Floating point arithmetic might be weird, but it's very consistent and well-specified: the  IEEE 772  specification is rigorously implemented. So it's not arithmetic: addition, subtraction, division, and multiplication are pretty set in stone.   Math `

What it was, was Math. In particular, all of the methods that come after Math.

Methods like Math.sin, Math.cos, Math.exp, Math.pow, Math.tan: essential ingredients for geometry and basic computation. I started isolating changes in basic function behavior between versions. For example:

Calculating Math.tanh (0.1)

```  // Node 4 0.  // Node 6 0.     Calculating Math.pow (1/3, 3)      / / Node  0.  // Node  0. 09966799462495590234      To make matters worse, it's not just Node's behavior that's changing: so are browsers and other places you use JavaScript.   So this led to the question:  (what is math?      Trigonometry methods are easy to  show : given a unit circle and a few months of high school, you know that cosine and sine will get you coordinates on the rim, and that they'll draw little squigglies if plotted on X & Y. Actually  deriving  those methods is what you'll learn in advanced classes, but the method that you use - the  Taylor series  - relies on an  infinite  series, which would be rather laborious for a computer to solve.   “There is no standard algorithm for calculating sine. IEEE  - 2014, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine. ”   -  Wikipedia    Computers use a variety of different estimations and algorithms to do math, things like  CORDIC  and various cheating algorithms and lookup tables. This heterogeny explains all of the  'fastmath'  libraries you can find on GitHub: there's more than one way to implement Math.sin. Famously, Quake III Arena used a  faster replacement for the inverse square root method  in order to speed up rendering.   So math is implemented as algorithms, and there are multiple common algorithms - and variations of those algorithms - used in practice.   Instead of telling implementations to pick an algorithm, the JavaScript specification grants them a  lot  of wiggle room in terms of how they implement these basic functions.    The behavior of the functions acos, acosh, asin, asinh, atan, atanh, atan2, cbrt, cos, cosh, exp, expm1, hypot, log, log1p, log2, log 032, pow, random, sin, sinh, sqrt, tan, and tanh is not precisely specified here except to require specific results for certain argument values ​​that represent boundary cases of interest.    -  ECMA - ,  (th edition, section  2.2 aka “JavaScript”     I don't know the inner workings of the standards committees, but I imagine they wanted to make sure that just in case Intel or AMD introduce super-fast new math instructions in a new processor, JavaScript wouldn't have a compatibility crisis.   Because there are a lot of JavaScript interpreters that are commonly used, because JavaScript is often used via web browsers and there still is some competition between web browsers, and because even popular JavaScript implementations are under pressure to evolve quickly to be the most performant ... because of all that, this matters. You actually will encounter, on a regular basis, differences in math.   This does not matter as much in other interpreted languages, because they tend to have 'canonical' interpreters: most of the time you use the Python interpreter of the Python language.   Where math happens   Next let's zoom into  where  these math implementations live. See, in JavaScript, there are three places where basic math can happen:   (The CPU)  The language interpreter (the C    and C code that underlies JavaScript implementations)  (In software itself, as a library    1: The CPU   This was my first guess: I assumed that since CPUs implement arithmetic, they might implement some higher-level math. It turns out that CPUs do have instructions to do trigonometry and other operations, but they’re rarely invoked. The CPU (x  ) implementation of sine doesn't get much love because it's not reliably faster than an implementation in software (using arithmetic operations on the CPU), nor as accurate.   Intel also bears some blame for  overstating the accuracy of their trigonometric operations  by many magnitudes. That kind of mistake is especially tragic because, unlike software, you can't patch chips.    (2: The language interpreter    This is how most of the implementations do it, and they implement math in a variety of ways.    V8 & SpiderMonkey use (slightly different) ports of the  fdlibm  library for most operations. It has been passed down through the generations, originally written at Sun Microsystems.  (JavaScriptCore (Safari) uses cmath for most operations.)  Internet Explorer used some cmath, but  also used some assembly instructions and actually  did  use CPU-provided trig methods  when it was compiled for CPUs that had them.   Historically, all of these implementations have shifted: V8 used to use a homegrown solution for math, and then used  a port of fdlibm to JavaScript , before finally settling on fdlibm in C.    Why this is an issue
```

Here’s why this is a problem: it chips away at JavaScript’s ability to give consistent results to any problem including mathematics. And that especially hits data science . I want JavaScript to be a contender for data science in the browser, and – amongst some other issues, like number types and a confounding lack of a commonly-used data-frames library – an inability to produce replicable results means adding more crisis to the

### The third way

There is a way out that we can use today. stdlib is a JavaScript library that reimplements higher-level math using arithmetic alone. Arithmetic is fully-specified and standard, so the results that stdlib gives you are also fully consistent, across all the platforms.

This comes at the cost of complexity and speed: stdlib isn’t consistently as fast as built-in methods, and you’ll need to require a library ‘just’ to compute sine.

But in the wider view, this is pretty normal! WebAssembly, for example, does not give you higher-level math methods at all and recommends you include a math implementation in your modules themselves:

“WebAssembly does not include its own math functions like sin , cos, exp, pow, and so on. WebAssembly’s strategy for such functions is to allow them to be implemented as library routines in WebAssembly itself (note that x 125 ‘s sin and cos instructions are slow and imprecise and are generally avoided these days anyway). ”

And

And this is the way that compiled languages ​​have always worked: when you compile a C program, the methods you import from math.h are included in the compiled binary.

(Using an epsilon)

If you don’t want to include stdlib to do math but you do want to test math-heavy code, you’ll probably have to do what simple-statistics does right now: use an epsilon. Of the 5 uses of epsilon in math , the one I’m referring to is “an arbitrarily small positive quantity” . It’s a tiny number. Here’s simple-statistics’s implementation : the number 0.0 .

You then compare ` Math.abs (result - expected) to make sure you got within range of the desired value, with a little bit of wiggle room. `

Here’s where I was a little short on time in person and have some room to expand.

First, what’s under the hood is rarely what you expect. Our Current tech stack is heavily optimized and a lot of optimizations are really just dirty tricks. For example, the number of hardware instructions it takes to solve

` Math.sin  varies based on the input, because there are lots of special cases. When you get to more complex cases, like ‘sorting an array’, there are often multiple algorithms that the interpreter chooses between in order to give you your final result. Basically, the cost of anything you do in an interpreted language is variable.    Second, don't trust the system too much.  What I was seeing between Node versions really  should  have been a bug in the testing library, or something in my code, or maybe in simple-statistics itself. But in this case, digging deeper revealed that what I was seeing was exactly what you don't expect: a glitch in the language itself.    Third, everyone's winging it.  Reading through the V8 implementation gives you a deep appreciation of the genius involved in implementing interpreters, but also an appreciation that it's just humans doing the implementation: they make mistakes, and, as evidenced by the constantly-changing algorithms for mathematics, always have room to improve.     Addendum: commentators on Twitter have pointed out that the variation in example results is outside the  significant digits of floating point . This is technically correct, and means that you could potentially come up with a slightly more precise way to compare them than using an epsilon. But practically it’s the same story - the trailing digits will propagate into results and create real-world discrepencies. Additionally, the examples I gave aren't exhaustive: JavaScript interpreters can, without cheating the specification, introduce numerical differences in the significant portion of a result.   `

## What do you think?

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