December 2019

[Dieter]

Originally Posted by LaurV

Well, you are right, and I was a bit stupid, haha... Of course, if you have 61 values, you add them and divide the result by 61. If you have 89, you add them and divide the result by 89. If you have M82589933 values, you add them and divide the result by M82589933. When you integrate, that is how those little rectangles work... Grrr...


However, we solved this puzzle already, after half hour of playing with MS Excel, with a much smaller epsilon than the required one. It is quite simple, if you consider two things, starting from the mentioned theorem, you need to find the largest degree polynomial that you can write down with 15 operations (no powering, you must write x^n as x*x*x... n times, hehe) and then one of the two things that you have to consider is the fact that the function is symmetric in x therefore you can double the degree of the polynomial, and yet use the same amount of operations, and the second one, you will find by yourself... hehe...

I don‘t know if it is allowed to write
y=x*x, f(x)=(a*y+b)*y+...,
because we have to find „ an expression with no more than 15 operations ...“.
In my understanding it has to be one expression (ein geschlossener Ausdruck).

[Dr Sardonicus]

I have submitted a solution to the December 2019 "Ponder This."

[what]

So brackets are not allowed, ex. x*(5*x+2) would have to be written as 5*x*x+2*x?

[EdH]

Originally Posted by what

So brackets are not allowed, ex. x*(5*x+2) would have to be written as 5*x*x+2*x?

I think parentheses are allowed, but not separate sections. Their example has parentheses, but is a single expression.

[what]

so far I got 0.0005108359660008244 with 11 operations using fourier approximation with some tweaking. I think i need a different approach

[LaurV]

Nobody said anything about separate expressions.
Of course, you have to write everything in one expression, y=f(x), where you expand f.

Parenthesis are allowed, see their example.

[what]

Originally Posted by LaurV

Well, you are right, and I was a bit stupid, haha... Of course, if you have 61 values, you add them and divide the result by 61. If you have 89, you add them and divide the result by 89. If you have M82589933 values, you add them and divide the result by M82589933. When you integrate, that is how those little rectangles work... Grrr...


However, we solved this puzzle already, after half hour of playing with MS Excel, with a much smaller epsilon than the required one. It is quite simple, if you consider two things, starting from the mentioned theorem, you need to find the largest degree polynomial that you can write down with 15 operations (no powering, you must write x^n as x*x*x... n times, hehe) and then one of the two things that you have to consider is the fact that the function is symmetric in x therefore you can double the degree of the polynomial, and yet use the same amount of operations, and the second one, you will find by yourself... hehe...

"therefore you can double the degree of the polynomial, and yet use the same amount of operations"
how do you do so?

[what]

Originally Posted by what

"therefore you can double the degree of the polynomial, and yet use the same amount of operations"
how do you do so?

nevermind its easy. I wasnt thinking

[matzetoni]

I chose a non-polynomial approach and got a MSE of 1.4107e-05 with 11 operations by optimizing coefficients of the ansatz function.

[LaurV]

I may be able to prove that the best you can do in such a way that all constants that you use are integers that fit in 16-bit word (two's complement. i.e. from -32768 to +32767), is about 2.3e-5 (which is a viable solution) regardless of arrangement of operations.

[Max0526]

Nine operations did it for me (plot 1). I sent the solution yesterday. The coefficients were decimal though. The plot goes off beyond [-1, 1] of course (plot 2).
The fifteen operations produce the MSE ten times better than expected (plot 3).