I’ve seen a number of questions about simulations and animations in javascript, which often involve calculating the hypotenuse:

```
hypot = Math.sqrt(x*x + y*y);
```

Since cartesian coordinates are the weapon of choice in most of these engines, these calculations are needed to find the distance between pairs of points, etc. So any speedup in calculating the hypotenuse could be a great help to many projects.

To that end, can you see a faster method than the simple implementation above? I found an approximation which was marginally faster in Chrome, but turned out to be much slower in Firefox, based on this approximation function in SuperCollider.

Edit 2015-08-15: I’ve switched the accepted answer to being the Math.hypot one; I suspect the pragmatic approach at present would be to use Math.hypot or a synthesized hypot function if not available, and to compare against the square (per sch’s answer) if that is sufficient and Math.hypot is not available.

An important point that many do not know:

Reference and complete text: John D. Cook – http://www.johndcook.com/blog/2010/06/02/whats-so-hard-about-finding-a-hypotenuse/

You can look to equality of

`x`

and`y`

. If are equals you can calculate hypotenuse as`(x + y)/sqrt(2)`

where`sqrt(2)`

is a constant.So this method can be used for case where x = y. For other cases it can be used with maximum imprecision of ~41%. This is a big error. But when you specify allowable error limits you can use this method. For example, if define allowable error to 5% you can get that

`b`

must be between`0.515*a`

and`1.942*a`

.So if you don’t need perfect imprecision of your calculations, you can improve performance of calculations with range of values.

By analogy you can look to equality of

`x`

or`y`

to`zero`

. And with some accuracy calculate hypotenuse more faster for this cases.P.S. I’ve read about this in the one russian article.

Often, you don’t need to compute the square root and

`hypot^2 = x*x + y*y`

is enough. This is the case for example if you want to compare the distances and don’t need the actual values.Performance of

`Math.sqrt()`

and`Math.hypot()`

depends on javascript running environment.I just run this code in Node.js, Chrome and Firefox:

with the following results:

Node.js v14.15.4:

Chrome 88.0.4324.150:

Firefox 85.0.2:

which means V8 has awful implementation of Math.hypot(). Actually I wouldn’t be much surprised if it also depended on CPU architecture/model.

Note that in my example I feed integer numbers to

`Math.sqrt()`

and`Math.hypot()`

. Another test revealed that with floating-point numbers Node.js runs test code 20% slower than on integers. In Chrome`Math.sqrt()`

performance is exactly the same and`Math.hypot()`

runs like 3% slower. Firefox: no difference.In ECMAScript ES6 you can use

Math.hypot:Edit: You can run this test on a blank tab, are 2 million operations with both methods, the results are very good, it is 24% faster.Note: In this test, it’s used ES6’sMath.hypot.