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2019-12-06, 12:44   #12
Dieter

Oct 2017

89 Posts

Quote:
 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).

 2019-12-06, 16:43 #13 Dr Sardonicus     Feb 2017 Nowhere 66478 Posts I have submitted a solution to the December 2019 "Ponder This."
 2019-12-06, 20:03 #14 what   Dec 2019 Kansas 24 Posts So brackets are not allowed, ex. x*(5*x+2) would have to be written as 5*x*x+2*x? Last fiddled with by what on 2019-12-06 at 20:14
2019-12-06, 20:46   #15
EdH

"Ed Hall"
Dec 2009

2·3·13·43 Posts

Quote:
 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.

 2019-12-06, 23:47 #16 what   Dec 2019 Kansas 24 Posts so far I got 0.0005108359660008244 with 11 operations using fourier approximation with some tweaking. I think i need a different approach Last fiddled with by what on 2019-12-06 at 23:55
 2019-12-07, 02:35 #17 LaurV Romulan Interpreter     Jun 2011 Thailand 19×461 Posts 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.
2019-12-07, 02:45   #18
what

Dec 2019
Kansas

100002 Posts

Quote:
 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?

Last fiddled with by what on 2019-12-07 at 02:46

2019-12-07, 04:59   #19
what

Dec 2019
Kansas

24 Posts

Quote:
 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

 2019-12-08, 01:05 #20 matzetoni     Feb 2019 73 Posts I chose a non-polynomial approach and got a MSE of 1.4107e-05 with 11 operations by optimizing coefficients of the ansatz function.
 2019-12-11, 04:54 #21 LaurV Romulan Interpreter     Jun 2011 Thailand 19·461 Posts 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. Last fiddled with by LaurV on 2019-12-11 at 04:55
 2019-12-13, 22:41 #22 Max0526   "Max" Jun 2016 Toronto 701 Posts 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). Attached Thumbnails       Last fiddled with by Max0526 on 2019-12-13 at 22:52 Reason: added two more plots

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