# Alternate expression for $e^x-1$

So if the following function is evaluated with the floating-point arithmetic, we get poor results for certain range of values of $x$. Therefore, I need to provide an alternate function that can be used for those values of $x$. The function is: $$f(x)=e^x-1$$ So how would I make an alternate expression or function for this. The range of this function is $y\gt -1$, $y\in\mathbb{R}$. So would I use the Taylor series for this....?

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Have you learned anything from the comments and answers to your other question that would help you with this? math.stackexchange.com/questions/108678/… – Gerry Myerson Feb 13 '12 at 3:21
i do understand that...but this is a slightly different function....and Conjugate would not work for this...So I am thinking Taylor series would work. – A_1_615 Feb 13 '12 at 3:28
When you say does not work for certain values of $x$, you should also tell which ones and what exactly you mean by 'does not work'! Also, you write $f$, but talk about $y$. – Aryabhata Feb 13 '12 at 3:32
The advanced HP calculators (48, 49, 50) have built in functions for exp(x)-1 and ln(1+x) for exactly this reason: improved accuracy around x = 0. – marty cohen Feb 13 '12 at 5:28
Yes, I read the post, and I also read your comment: "...and Conjugate would not work for this..." so, yes, you sure as heck did mention "Conjugate" here. – Gerry Myerson Feb 14 '12 at 4:50

\begin{align*} \exp\,x-1&=\left(\exp\frac{x}{2}\right)^2-1\\ &=\left(\exp\frac{x}{2}\right)\left(\exp\frac{x}{2}-\exp\left(-\frac{x}{2}\right)\right)\\ &=2\left(\exp\frac{x}{2}\right)\left(\frac12\left(\exp\frac{x}{2}-\exp\left(-\frac{x}{2}\right)\right)\right)\\ &=2\left(\exp\frac{x}{2}\right)\left(\sinh\frac{x}{2}\right) \end{align*}
@A_1_615: if as you claim you know how to find the series expansion of $\exp\,x$, then it's just as simple as subtracting $1$ from your series... – J. M. Feb 14 '12 at 1:00