Hello, my name is Claudio and I am a researcher here in Brazil, an enthusiast and big fan of the Wolfram / Alpha platform. While testing the numerical recognition capability of the algorithm, I found something intriguing. I would like to explain in detail and send feedback information to you. What I found: Aligning three points in a parabola such that:
Point 1 = (1,r)
Point 2 = (2,s)
Point 3 = (3,t) ,
The resulting parabola equation is:
y=((r-2*s+t)/2)*x^2+((-5*r+8*s-3*t)/2)*x+(3*r-3*s+t)
Then, integrating in the intervals 1 to 3 and then dividing by 2, this is the integral mean of this parabola in that interval. I will call M this mean:
M = (r+4*s+t)/6
Well, if we use for example:
r = (sqrt(5)+1)/2 ,
s = E ,
t = Pi ,
The average M will be:
M = (1+sqrt(5)+8*e+2*pi)/12 ,
M = 2.60545899269597817135139668384...
Then the intriguing fact begins. When I input this number for Wolfram / Alpha recognition, it generates possible, even more complex, forms using E, Pi, log variables. Wolfram / Alpha generates:
(7+10*sqrt(3)+4*e+8*pi+2*pi^2-3*log(3)+log(8))/(2*sqrt(2)+5*e-10*pi+4*pi^2+log(324))
And it was not able to recognize a simpler form, only with variables with E and Pi, as the original number: (1+sqrt(5)+8e+2pi)/12 . I hope I've been able to explain what I found. I believe that the algorithm skips a simpler step in recognition to try to find a more complex one without having the need, causing a not so precise recognition. I thought you'd like to know that. Please, I would appreciate myself some feedback info. Thanks.