Simple upper bound for $t(2\Phi(\alpha/t) - 1)$, where $\alpha > 0$ and $t \in (0, 1)$

Question

Let $\alpha > 0$ and $ t \in (0, 1)$. For simplicity, take $\alpha=1$. Let $\Phi$ be the normal cumulative distribution function.

Of course, the core of the problem is the term $t\Phi(\alpha/t)$. Note that this function is nondecreasing.

Question. Is there an obvious simple upper bound for $t(2\Phi(\alpha/t)-1)$ ?

Important note. By "simple", I mean things things like simple polynomials, exponentials (e.g $C_1e^{-C_2t^2}$), but not things like exponential divided by polynomial, etc., which would be obtained, for example using Wilk's ratio.

This leads me to my second question

Question. Is it possible to use symbolic computing tools like wolframalpha to solve such problems ? If yes, how ?

Answer 1

$$t \left(2 \Phi \left(\frac{\alpha }{t}\right)-1\right)=t\, \text{erf}\left(\frac{\alpha }{t\sqrt{2} }\right)$$ and we know that $$\text{erf}(x) \leq \sqrt{1-e^{-\frac{4 }{\pi }x^2}}$$ making $$t\, \text{erf}\left(\frac{\alpha }{t\sqrt{2} }\right)\leq t\, \sqrt{1-e^{-\frac{2 \alpha ^2}{\pi t^2}}}$$