Average Error: 59.1 → 0.2
Time: 10.1s
Precision: 64
$\sqrt{e^{x} - 1}$
$\begin{array}{l} \mathbf{if}\;x \le 1.59252664246739995129731326528599311132 \cdot 10^{-4}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x + x} - 1 \cdot 1}}{\sqrt{e^{x} + 1}}\\ \end{array}$
\sqrt{e^{x} - 1}
\begin{array}{l}
\mathbf{if}\;x \le 1.59252664246739995129731326528599311132 \cdot 10^{-4}:\\
\;\;\;\;\sqrt{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{e^{x + x} - 1 \cdot 1}}{\sqrt{e^{x} + 1}}\\

\end{array}
double f(double x) {
double r1064816 = x;
double r1064817 = exp(r1064816);
double r1064818 = 1.0;
double r1064819 = r1064817 - r1064818;
double r1064820 = sqrt(r1064819);
return r1064820;
}


double f(double x) {
double r1064821 = x;
double r1064822 = 0.00015925266424674;
bool r1064823 = r1064821 <= r1064822;
double r1064824 = 2.0;
double r1064825 = pow(r1064821, r1064824);
double r1064826 = 0.5;
double r1064827 = 0.16666666666666666;
double r1064828 = r1064821 * r1064827;
double r1064829 = r1064826 + r1064828;
double r1064830 = r1064825 * r1064829;
double r1064831 = r1064830 + r1064821;
double r1064832 = sqrt(r1064831);
double r1064833 = r1064821 + r1064821;
double r1064834 = exp(r1064833);
double r1064835 = 1.0;
double r1064836 = r1064835 * r1064835;
double r1064837 = r1064834 - r1064836;
double r1064838 = sqrt(r1064837);
double r1064839 = exp(r1064821);
double r1064840 = r1064839 + r1064835;
double r1064841 = sqrt(r1064840);
double r1064842 = r1064838 / r1064841;
double r1064843 = r1064823 ? r1064832 : r1064842;
return r1064843;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < 0.00015925266424674

1. Initial program 60.3

$\sqrt{e^{x} - 1}$
2. Taylor expanded around 0 0.0

$\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(x + \frac{1}{6} \cdot {x}^{3}\right)}}$
3. Simplified0.0

$\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x}}$

## if 0.00015925266424674 < x

1. Initial program 6.9

$\sqrt{e^{x} - 1}$
2. Using strategy rm
3. Applied flip--9.6

$\leadsto \sqrt{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}$
4. Applied sqrt-div9.7

$\leadsto \color{blue}{\frac{\sqrt{e^{x} \cdot e^{x} - 1 \cdot 1}}{\sqrt{e^{x} + 1}}}$
5. Simplified9.3

$\leadsto \frac{\color{blue}{\sqrt{e^{x + x} - 1 \cdot 1}}}{\sqrt{e^{x} + 1}}$
3. Recombined 2 regimes into one program.
4. Final simplification0.2

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.59252664246739995129731326528599311132 \cdot 10^{-4}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x + x} - 1 \cdot 1}}{\sqrt{e^{x} + 1}}\\ \end{array}$

# Reproduce

herbie shell --seed 1
(FPCore (x)
:name "sqrt(exp(x) - 1)"
:precision binary64
(sqrt (- (exp x) 1)))