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I don't understand the difference between faster by factor versus faster by percent. For example,
Machine A Execution Time: 5.106 ms
Machine B Execution Time: 0.851 ms
Obviously, machine B is faster 4.255 ms. Now, I want to display how much faster B is than A by what factor as well as what percent. As far as I'm concerned:
Machine B is 600% faster than machine A. (5.106 ms / 0.851 ms)
By what factor is B faster than A? I don't see how it's different than percent.
Actually, if you want to use the phrase, "Faster than..." the value should be calculated as $$\frac{\text{difference between values}}{\text{value being compared to}}$$ not $$\frac{\text{value in question}}{\text{value being compared to}}$$
For example, 2 is 100% more than 1. Not 200% more.
I would say that to find the factor B is faster than A, it would be your initial calculation. 600%.
Also note that the phrases "percent faster than" or "...times smaller than" can be problematic and not as straightforward as "...times bigger". Just from an English semantics point of view. But it's generally understood what you mean.
Editing to add @pjs36 comment. When using "by a factor of", you more often see it as a number, not a percentage.
Final Edit: I would say these things.
This could almost be an English question as much as mathematics. I pondered the idea (initiated by conversation about an answer I'm posting on Stack Overflow, comparing run-times for two similar coding methods), and the more I looked into it, the more confused I became - between questions on various Stack sites, Wikipedia, online "faster than" calculators, and other forums like Quora. There are many a few different answers depending on who you ask, and the context.
The basis of the calculation comes down to one rule:
Something can't be more than 100% faster than something else.
The way I figure it, the calculation for FasterThan % is:
$$\frac{({\color{Red} S}lower-{\color{Green} F}aster)}{{\color{Red} S}lower}=\%Faster Than$$
Alternatively, the calculation for SlowerThan % is:
$$\left |\frac{({\color{Green} F}aster)-{\color{Red} S}lower}{{\color{Green} F}aster} \right |=\%Slower Than$$
A (Slower) B (Faster)
--------------- --------------- -------------------------- -------------------------------
100 seconds 90 seconds B is 10% faster than A A is 11.1% slower than B
2 seconds 1 second B is 50% faster than A A is 100% slower than B
1.348 seconds 0.605 seconds B is 55.1% faster than A A is 122.6% slower than B
5.106 seconds 0.851 seconds B is 83.3% faster than A A is 500% slower than B
1 0 B is 100% faster than A A is infinitely slower than B
€ 100.00 € 90.00 B costs 10% less than A A costs 11.1% more than B
Your original answer is correct, even if by accident. "Faster" implies relative speed or velocity ("wow, that car is fast!"). So first, you would express both values in terms of a speed or velocity. Instead of number of seconds (a duration), convert to "executions per second", which more accurately describes your situation (you described that the duration corresponded to an execution).
vA = 1 execution / 5.106 ms = 1 / 0.005106 = 195.848 executions/second vB = 1 execution / 0.851 ms = 1 / 0.000851 = 1175.088 executions/second
So Machine B is (vB/vA = 6.0) times faster than Machine A.
Of course, to calculate vB/vA you get (1/tB)/(1/tA) and you can cancel everything out to get tA/tB to arrive at the same result.
This would still work even you described it in terms of executions per millisecond, since the units will cancel out.
I've been struggling around this problem for 5 hours straight (yes, straight), and I've been getting different answers from different people and sources, so I wrote everything on a paper and tried to get a logical thinking from both results.
Let's forget about time at a moment, let talk about speed.
Faster than implies how much more speed you has, while slower means how much less speed you has.
Now, talking about time, time is inversely proportional to speed, and when we talk about faster than, we are talking about speed, not time (but they are related).
So, if X is faster than Y, X must have more speed than Y, so its percentage must be greater than Y (in regards to faster than percentage), otherwise it means it is slower.
So the formula of @turkeyhundt is correct:
$$ \frac{5.106 - 0.851}{0.851} = 5.0 = {500}\% $$
If you got the greater value at the divider:
$$ \frac{5.106 - 0.851}{5.106} = 0.833... = {83.333...}\% $$
And now the most important part of the question, if I say that B is X% faster than A, I must be able to resolve the A value without knowing it, just from the B and the percentage:
B (0.851) is 500% faster than A:
$$ 0.851 + (0.851 * 5.0) = A $$
$$ 5.106 = A $$
The same as if I said that A (5.106) is 83.333% slower than B:
$$ 5.106 - (5.106 * 0.8333333333333334) = B $$
$$ 0.851 = B $$
The decision between
+or-is just if the result should be greater or lesser than, by evaluating the faster (+) or slower (-) assumption.
I was just thinking as if this problem were a math test, I would not have A or B when questioning about faster or slower than, so that was my conclusion:
If the percentage of A speed is greater than B, A is faster, otherwise, A is slower.
So it does mean that, if you've got something like B is 83% faster than A and A is 500% slower than B, that is probably incorrect, it is the other way around, the greater percentage means faster (more speed), the lesser percentage means slower.
We also have some semantics problem here, faster than is talking about speed, but we also interpret it as time, and they are inversely proportional.
If the question was what percent of time A takes more than B, then you could say A takes 500% more time than B (A = B + 500%), and B takes 83% less time than A (B = A - 83%), which is the thing that @ashleedawg has stated as faster than, but is in fact, how much more or less time one takes. And here you can see, how time and speed are inversely proportional.
Just as an advisory, I'm a software engineer, but not that good at mathematics, just sharing the conclusion that I had after those 5 hours.
Edit:
I've made a mistake by assuming that it should always be greater than 100% because I've not taken into account slight fractional differences, like A taking 1.4s and B taking 1.1s, that would've resulted in B being 27% faster than A, and A 21% slower, so I've fixed the answer.
Also, this does not applies to money and costs, since they are not inversely proportional: more money means more expensive, less money means cheaper. While time and speed are inversely proportional, more speed is equals less time, and less speed is equals to more time.
Just one more thing to keep in mind, and I think that is the reason we confuse this: the less time it takes, faster it is, and because time is decreasing, we calculate as if it were proportional, by having always the greater percentage meaning more time and lesser percentage meaning less time, however, faster than or slower than is not about time, but speed, and again, time and speed are inversely proportional.
