# What does -1.13 times faster mean?

I'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books.

The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I would have thought that "1 times faster" means "no faster", "2 times faster" means "twice as fast", and "0.5 times faster" means "half as fast"/"twice as slow". Have they just got completely confused in that graph, or is it me?

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If 20 percent more means 1.2 times as much, then I would think that "x times faster" means (1 + x) times as fast. The Google Books page isn't loading for me, so I can't actually answer your question; could you post a screenshot? – Qiaochu Yuan Dec 2 '10 at 17:34
Decrufted the Google Books link. Can you try again, @Qiaochu? – J. M. Dec 2 '10 at 17:39
@Qiaochu There you go. – Skilldrick Dec 2 '10 at 17:39
The answerers have taken pretty good cracks at this question already. But in case you're looking for someone else to express their disapproval at this very careless use of mathematical language, please accept a hearty "Boo!" on my behalf. – Pete L. Clark Dec 3 '10 at 5:31
Thanks for all the responses everybody (and especially your Boo!, @Pete :)). I actually work in publishing, drawing graphs like this, so I think I can guess what happened. The author supplied two execution times for each bar, and the publisher decided that the graph needed to show relative speeds. So the illustrator applied some dodgy maths to the numbers and came up with these values (and the author never looked at the graph to see if it made sense). – Skilldrick Dec 3 '10 at 18:13

Sorry for the confusion, but you got it absolutely right.

2 times faster means two times as fast
A takes 100 ms
B takes 200 ms
so A is two times faster than B

This particular graph is probably the most confusing taken out of the flow of the previous ones in the chapter. This one shows an optimization that most often you shouldn't bother. Especially when there are cases of stuff being 100 times faster. And especially when it's not consistent among browsers.

The point was not to worry about micro-optimizations (unless it's critical to you and you've done all else). So 20% faster (or 1.2 times faster) is probably not worth it most of the time when there are other optimization that will make something 10 times faster.

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thanks for this! [For those who don't know, this is the author of the chapter...] – Skilldrick Dec 3 '10 at 22:04

I would say $-1.13$ faster means 1.13 slower. For example $-1.13$ faster than 160 mph is $-180.8$ mph, i.e. 180.8 going in the opposite direction.

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Wouldn't 1.13 times slower be 0.88 times faster? – Skilldrick Dec 2 '10 at 17:38
I am thinking in terms of "slope". So 2 times faster means $y=2x$. $-2$ faster means $y=-2x$. – TCL Dec 2 '10 at 17:53
How does that notion of direction make sense in terms of execution speed? I'm with @Skilldrick here -- the intended interpretation is probably -1.13 = 1/1.13 but basically the graph is wrong. – walkytalky Dec 2 '10 at 18:29
Our everyday language is sometimes ambiguous. Perhaps the person who says $-1.13$ faster has to define what he/she is saying. – TCL Dec 2 '10 at 21:42
@walkytalky, in addition to identifying $-x=1/x$ it seems $1$, $0$ and $-1$ are identified also. – Douglas S. Stones Dec 2 '10 at 23:56

There are many abuses like this in the world. I have seen things on sale that were 400% OFF!!! Would they give me 3x the original price if I took it? Maybe -1.13 times faster means that you send them 113% as many bits as you received in the base case.

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Suppose browser $A$ finishes loading a given page in five seconds, while browser $B$ completes in half the time.

How many times faster is $B$ than $A$? Well, twice, of course. But what about the inverse? How many "times faster" is $B$ than $A$?

One way to answer this is to assume that "$-x$ times faster than" just means "$1/x$ times the speed of". The intuitive problem with this is determining how much faster something is moving compared to something moving slower than it. It's just a really awkward way of putting it, but basically we can look at the number of "times faster" as a negative or a fractional value.

They're both more or less valid interpretations.

Now if we ask the question, how many times slower than $B$ is $A$, we get the original answer -- it's twice as slow.

So how many times faster is it? It depends on which interpretation we're using. It's correct to say that it is -2 times "as fast" since it is twice as slow; but also (and more intuitively) correct to say the speed of $A$ is 0.5 times the speed of $B$.

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I suppose Skilldrick is right, and $-x$ times faster is $x$ times slower or $1/x$ times faster. For example, if A takes 2 seconds and B is $2$ times faster, B takes 1 second; if C is $-2$ times faster, it takes 4 seconds. And A is $0$ times faster than itself.

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Strictly speaking, $-1.13$ times faster would mean that the program is doing things in reverse order. And Opera would completely stand still. Since that can't be what they really mean, it must be an abuse of language as has already been pointed out.

Could it be though that the original scale was logarithmic and that the person then just interpreted the numbers incorrectly? Which would mean that the graph is correct, but the person who added the 'times faster' doesn't get the concept of logarithm?

So, $-1.13$ should be $a^{-1.13}$ in some suitable base, probably base $10$, $2$ or $e$?

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The scale isn't logarithmic, as the accompanying paragraph demonstrates. – Yuval Filmus Dec 3 '10 at 5:19
The accompanying paragraph flatout contradicts the barplot. I think the whole thing doesn't make much sense. – Raskolnikov Dec 3 '10 at 10:11