Why is Desmos so insanely fast?

h1a8 · Jul 12, 2023

Why is Desmos (graphing calculator app for Android, ios, windows, etc) far faster than everything else in calculating?
I've done the same calculations in Mathematica, Maple, hp prime pro app on windows (ios and Android), various calculator apps on ios and Android, etc

This is one of the things I calculated.

$$
\sum_{x=1}^{1000000} \sqrt[3]{ e^{\sin{[\tan^{-1}{(\ln{3x^2+2x-3}) }] }}}
$$

Desmos gave an answer in LESS THAN A SECOND on a mid spec Android phone. It took Mathematica, Maple, etc with 12th generation i7 processor many seconds to compute. All other math program apps in Android and ios (including TI Nspire and many more) took longer. Is it in the code? Superior algorithm?

Baluncore · Jul 12, 2023

Maybe Desmos does the computation in the cloud, not on your phone.

h1a8 · Jul 12, 2023

Baluncore said:

Maybe Desmos does the computation in the cloud, not on your phone.

Thanks
That would make a lot of sense.
Any idea how I would post the expression in Latex form for this forum?

Edit: It still calculates the same speed while in airplane mode lol. I don't think it's the cloud.

Baluncore · Jul 12, 2023

h1a8 said:

Edit: It still calculates the same speed while in airplane mode lol. I don't think it's the cloud.

Is the answer correct?
Single or double precision?
What answer does it return?

The LaTeX guide is at the bottom left of your post edit window.
Scroll down.

pbuk · Jul 12, 2023

Baluncore said:

Maybe Desmos does the computation in the cloud, not on your phone.

Exactly the opposite: Desmos does everything on the client.

And that's why it's fast - it doesn't send any data anywhere and wait for a busy server to get around to processing it, it just gets on with it locally. And contrary to the commonly held myth, modern JavaScript engines are really fast.

h1a8 · Jul 12, 2023

Baluncore said:

Is the answer correct?
Single or double precision?
What answer does it return?

The LaTeX guide is at the bottom left of your post edit window.
Scroll down.

Yes I check all others (mathematica, maple, etc)
The solution it gives is 1,395,279.57136

I looked at the guide. No dice. Keeps posting the actual code (probably wrong) I stead of the conversion.

Ibix · Jul 12, 2023

h1a8 said:

Keeps posting the actual code (probably wrong) I stead of the conversion.

Known bug that appears to be very hard to fix. However, if there's already ##\LaTeX## on the page (which there now is) then the code will render. You may need to refresh the page for it to work for you.

Baluncore · Jul 12, 2023

I get the same; 1,395,279.571359562
in 0.206 sec on an old slow pentium.

There are quick-cuts, for example, avoid the cube root by dividing by three before computing the Exp. There may be others.

pbuk · Jul 12, 2023

I'm not sure what might be wrong with your Maple/Mathematica etc. set-ups but Wolfram Alpha turns it round in 112 ms ignoring network latency.

pbuk · Jul 12, 2023

Baluncore said:

Single or double precision?

All JavaScript results are IEEE 754 double precision (64 bit).

h1a8 · Jul 12, 2023

pbuk said:

I'm not sure what might be wrong with your Maple/Mathematica etc. set-ups but Wolfram Alpha turns it round in 112 ms ignoring network latency.

How did you get the time 112 ms?

pbuk · Jul 12, 2023

h1a8 said:

How did you get the time 112 ms?

It's the "results" path that delivers the response: 118 ms this time. The rest of the time loading the page is downloading fonts, graphs etc.

Edit: I suppose it may have cached the calculations earlier, but this is in the right ballpark - here it is on my desktop (a bit newer i7 than @Baluncore's).

Code:

> node vs-desmos
Result is 1395279.5713595615 in 139 ms

JavaScript:

const f = () => {
  let sum = 0;
  for (let x = 1; x <= 1e6; ++x) {
    sum += Math.pow(
      Math.exp(Math.sin(Math.atan(Math.log(3 * x * x + 2 * x - 3)))),
      1 / 3
    );
  }
  return sum;
};
const start = performance.now();
const result = f();
const time = performance.now() - start;
console.log(`Result is ${result} in ${Math.round(time)} ms`);

Baluncore · Jul 12, 2023

pbuk said:

(a bit newer i7 than Baluncore's)

My i7 was built in 2012, it still runs FreeBasic on 32 bit, Win 7.

Avoid cube root; Exp(x)^(1/3) = Exp( x/3 )
Avoid two transcendentals; Sin( Atan( x ) = Abs( x ) / Sqrt( x^2 + 1 )

pbuk · Jul 12, 2023

Baluncore said:

Avoid cube root; Exp(x)^(1/3) = Exp( x/3 )
Avoid two transcendentals; Sin( Atan( x ) = Abs( x ) / Sqrt( x^2 + 1 )

Premature optimization, but since you mentioned it:

Avoid cube root; 139 ms -> 57 ms.
Avoid two transcendentals; 57 ms -> 34 ms.

Why are we doing this?

Baluncore · Jul 12, 2023

pbuk said:

Why are we doing this?

I was wondering if there was an opportunity for Desmos to do some unusual algebraic optimisation prior to the computation. I am not saying it does, just wondering if the example might collapse through some trapdoor in a benchmark.

pbuk · Jul 12, 2023

Now here is something surprising:

The timing of 139 ms above is from Node JS v20.
I tried it in Edge, which is supposed to be the same V8 engine, and it was 120 ms. Chrome just the same.
Firefox 103 ms.

Why is Desmos so insanely fast?

1. Why is Desmos considered to be insanely fast?

2. How does Desmos achieve such fast performance?

3. Does Desmos prioritize speed over accuracy?

4. Are there any limitations to Desmos' speed?

5. Can Desmos' speed be improved even further?

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