How to simplify / combine function equtions containing `if`?

Question

I have two eqution (blue / green), composed them using a simple if (black).

enter image description here

Blue: (2x)^2 / 2

Green: 0.5 + (1 - (2(1-x))^2) / 2

Black: (x < 0.5) ? (2x)^2 / 2 : 0.5 + (1 - (2(1-x))^2) / 2

See it in action on Fooplot: http://fooplot.com/plot/7knnh3zxis

Question is how can I simplify / extract the if statement?

Seemingly this is a single eqution offset by 0.5, 0.5 from the origin. Can someone shed a light on how to simplify this? I have some other similar equtions I am to simplify.

I'm not a math ninja, so explain with care if you can. :) Sorry if this is a noob question here at math.stackexchange.com, this is the first time I ask here.

UPDATE: I expressed as a single function, than it turned out that this all above can be simplified to a surprisingly simple function, like: -(4(x-0.5)abs((x-0.5))-4(x-0.5)-1)/2

See http://fooplot.com/plot/t5msqna67s

There is not a unique way to simplify a function. For most purpouses the expression you have is the best way to express the function. Why you need a simplification? What's your goal? — Emanuele Paolini, Dec 01 '14 at 11:42
Hmm, actually every combined function will provide less performance, as I wondering... — Geri, Dec 01 '14 at 13:46

score 2 · Answer 1 · answered Dec 01 '14 at 11:42

2

You can't write the whole function you have defined as a polynomial (sum of multiples of $x^n$). Your function is differentiable everywhere and piecewise a polynomial, but that doesn't mean that the function as a whole is a polynomial. To see this, note that your function's second derivative is discontinuous.

Although it's kind of cheating, you could write it as:

$$f(x):=\frac{2x^2}{2}+ U\left(x-\frac{1}{2}\right)\cdot \left[\frac{1}{2} + \frac{1 - (2(1-x))^2}{2} - \frac{2x^2}{2}\right]$$

with 'U' being the 'unit step': $U(x)=\left\{\begin{array}{ll}0 & x<0 \\ 1 & x \geq 0\end{array}\right.$

Note that some polynomials of degree 3 will have a similar shape to your function, but they will go off to infinity faster on their tail ends.

answered Dec 01 '14 at 11:42

Christian Chapman

4,763

Aha, the second derivative is the key I was grasping for. Good call :) – rschwieb Dec 01 '14 at 11:50
Yap, I saw x^3 functions, like 0.5 + ((-x+ 0.5)^3 + 2x) - 1 http://fooplot.com/plot/srw8xwhbdj – Geri Dec 01 '14 at 13:47
I just could not provide exactly the same... – Geri Dec 01 '14 at 13:48

score 1 · Accepted Answer · answered Dec 01 '14 at 11:52

1

If you with extracting mean to replace the "if" with some function then you could use the following function:

$\displaystyle t(x)=\frac{x+|x|}{2\cdot x}$ which is equal to $ \;t(x)=\!\! \left\{ \begin{array}{l} 1 \quad\text{if } x>0\\ 0 \quad\text{if } x<0 \end{array} \right. $

In your example $f(x)=\text{blue}(x)\cdot t(x-0.5)+\text{green}(x)\cdot t(0.5-x)$.

answered Dec 01 '14 at 11:52

Lehs

13,791
4
25
77

Yap, simply "weight" the two eqution actually. I think this is actually the answer, but it is seemingly less performant. – Geri Dec 01 '14 at 13:51
1

@Geri: In a computer program it is less time efficient. And also, $t$ is not defined for $x=0$. – Lehs Dec 01 '14 at 13:56

How to simplify / combine function equtions containing `if`?

2 Answers2