Simplification of Expression with Standard Normal CDF

Question

Let $\Phi(x)$ be the CDF of the standard normal distribution. For $a,x\in\mathbb{R}$, can the following expression:

$$(\Phi(x)-\Phi(-x))(1-(\Phi(x)-\Phi(-x)))$$

be simplified into something a littler nicer to work with? What is the simplest form of this expression?

Answer 1

As Youem points out, $\Phi(-x)=1-\Phi(x)$ but this substitution saves minimal time:

$$\begin{align*} (\Phi(x)-\Phi(-x))\cdot(1-(\Phi(x)-\Phi(-x))) &=(\Phi(x)-(1-\Phi(x)))\cdot(1-(\Phi(x)-(1-\Phi(x))))\\\\ &=(2\Phi(x)-1)\cdot(-2\Phi(x)+2)\\\\ &=-4\Phi(x)^2+6\Phi(x)-2 \end{align*}$$

The original expression is more interpretable as well.