According to my previous post about chain multiplication of 3 numbers, it's certain the most naive top to bottom recursive factorial algorithm that times itself by n*(n-1)*(n-2)...*2 is slower than the bottom to top 2*3*4....*n.
This post is about the proof of the concept.

T(n) as the time complexity of top to bottom multiplication for n!
B(n) as the time complexity of bottom to top multiplication for n!
The relationship can be presented like this

How do we compare these two values? calculate those 2 functions from 1 to 10000 does show B(n) is faster and grow slower. It is good evidence, still it's not a proof. Let's work on the proof now.
Take off those floor function and the + 1, these have to little significance on the growth. Change log to general log instead of base 2 log. Easier on writing and the properties of each express will hold. Because all log functions have the same logarithmic growth rate. And in this general log, it's not possible to have a number lesser than one since there is a + 1 in the expression we can't see.

Suddenly, the entire expression became easier. It doesn't take a genius to see that:
in the B(n),
log k logarithmically increases
log((k-1)!) = increases at lesser than linearlogarithmic rate but faster than linear rate.
e^n < n! < n^n
n < log(n!) < log(n^n)
n < log(n!) < n log n

In T(n), log(n!/(n-k)!) is increasing and log(n-k) is decreasing
log(n!/(n-k)!) increases like log((k-1)!)
log(n-k) decreases logarithmically

T(n) might actually smaller than B(n)!
Let's put everything inside one log.

Now, we can take off this log sign. Why? because log is a continuous increasing function. so the relationship between both summations is still the same without the log. This same technique can be applied to the log in the exponents. Notice I have removed the B(n) and T(n). B(n) and T(n) are not approximate to the 2 expression right now, but each expression still corresponds to the same greater or lesser relationship between B(n) and T(n).

Now, because I don't have the most amazing brain in the world, I will do some cheat...
I removed the bases, only leave the exponents!
The growth of the exponents is WAY faster than the base(factorial growth VS linear growth), it's ok to ignore the base.
Note I have also changed the first expression's k's initial value to 1.
So everything come down to this comparison

n! is equal to n*(n-1)!, because the 2nd expression got a n!, it is greater than the 1st expression. so calculate factorial by multiply from 2 to n is faster than n to 2.
More rigor proof is required if I want to publish a paper...

Skim to the conclusion is fine.

Let us not assume the schoolbook multiplication's time complexity is O(1), but O(ab). Where a is the first number's binary digit count and b is the second number's binary digit count. Put another way, the complexity of multiplication is the product of the binary digit count of each multiple. of course this could be faster than O(ab), it would accord to how many zeros in the binary format of the number.
for example. 2×1024(10×10000000000), some optimized CPU only did one bit multiplication and one addition. But we can forget that kind of special case since it depend on CPU architecture most of the time.

If one of the multiple is 0, the product is always 0. Every equation after this point is assuming there is only positive integers.

Many people does not know. that chain multiplication's time complexity have something to do with the order of the multiplication. For example.
5×7×3 is in theory, slower than 3×5×7
3×5 = 15, complexity= 2×3 = 6
15×7 = 105, complexity= 4×3 = 12
Total complexity = 6+12 = 18

For 5×7×3
5×7 = 35, complexity= 3×3 = 9
35×3 = 105, complexity= 6×2 = 12
Total complexity = 9+12 = 21

Now, it's easy to formulate some mathematical model.
Before that, we have to define a function that count the binary digit of a number.D(x)

We know that ab and ba have the same time complexity.(O(ab) = O(ba)), so there are only 3 orders to calculate abc:
(ab)c
(ac)b
(bc)a
The complexity formula for each one of them

Expand the formula by replace all D(x) with equals? No, let's try expand D(ab) in the form of D(a) + D(b) + c first.
Look this though and carefully.


ahh, so D(ab) can be either D(a)+D(b)-1 or D(a)+D(b). Assume that D(ab), D(ac) and D(bc) all follow the later equation.
D(a)×D(b)+(D(a)+D(b))×D(c)=D(a)×D(b)+D(a)×D(c)+D(b)×D(c)
D(a)×D(c)+(D(a)+D(c))×D(b)=D(a)×D(b)+D(a)×D(c)+D(b)×D(c)
D(b)×D(c)+(D(b)+D(c))×D(a)=D(a)×D(b)+D(a)×D(c)+D(b)×D(c)

Wow, then all 3 equations is completely the same! Let's call this equation MAX.
To calculates most efficiently, we have to find one or more D(xy) = D(x)+D(y)-1
Suppose all 3 equations can have D(ab) = D(a)+D(b)-1
we will get these 3 results
D(a)×D(b)+D(a)×D(c)+D(b)×D(c)-D(c)=MAX-D(c)
D(a)×D(b)+D(a)×D(c)+D(b)×D(c)-D(b)=MAX-D(b)
D(a)×D(b)+D(a)×D(c)+D(b)×D(c)-D(a)=MAX-D(a)
Above equations states it's better to have the largest number multiply in the end.

Conclusion:
3 numbers, a,b and c.
x and y and z are 3 variables does not equal to each other(unless there is equality among a, b and c) and it's domain are {a, b, c}.
If it's not possible to find a D(xy) = D(x)+D(y)-1. then the order of the multiplication does not have effect on the time complexity.
If it's it is possible to find a D(xy) = D(x)+D(y)-1. then when z is the largest while the previous relation still holds, then it reduce the most on the time complexity.

After Note:
The chain multiplication introduced here is only for 3 numbers, but it can be applied to larger chains by calling this recursively.
I have googled around and I can't find any article about this subject, most likely it's because the time complexity it saves are extremely low(D(n) is approximately log(n)) and to find the perfect order for the chain multiplication would cost a lot more than the multiplication itself.

If I made any error(likely.) please leave a comment so I can admire your skill publicly.

i18n = internationalization
It's a numeronym, there are 18 letters between i and n.
Nice.
Now I have a new name. M3l
Wonder what else people can use for internationalization.

Fate drives me into sick days. I believe it's a fever.
My temperature rose up to 39C, I lay in bed, with the air cond on. covered in a blanket. All I want at that time, is some hot water. The fever drained huge amount of life energy from my body, boil water became a enormous task. So I decide to wait.
My parents already know about this fever. But they have jobs, and they have to work. I'm glad I can finally see my dad at 4 PM. He came back home early.
At the same day, dad and I have to catch a train to Yingtan to see grandma and grandpa. That's a 6 hour trip starting from 6:40PM.
Although my fever is not going down, I manage to stay strong and make to the train station. Actually, everything else isn't very important. Everything between the time I got on the train to the time where I actually going on the train station to head back.
The train got delayed, in the beginning, there is no telling how long the train is going to be delayed. After 4 hours, it's 4 am, I haven't slept all night. How lucky.
Then, on the train, I'm tired and wish I could get to sleep, but some little kid is extremely annoying and asking all kind of questions in the dark.
And that's the time I understand, I was the same as that kid, except, I only create problems to people who care about me.
Ahh.. people can only hurt others who care...
this post have no theme...

I haven't posted for a few days.
so I just let you guys know what I'm working on.
I'm working on proving an factorial algorithm is better than the naive one. Which include finding the complexity of each algorithm(this complexity does not assume multiplication of 2 numbers have the same complexity when the digit of each number change)
You might want to check out my blog after 2 weeks since I will be working on that with all my FREE time.