Average Error: 4.4 → 0.9
Time: 5.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.328422371476003 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.328422371476003 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -1.328422371476003e-05)) {
		VAR = sqrt((((sqrt(exp((2.0 * x))) + sqrt(1.0)) * cbrt(pow((sqrt(exp((2.0 * x))) - sqrt(1.0)), 3.0))) / (exp(x) - 1.0)));
	} else {
		VAR = sqrt(((x * (1.0 + (0.5 * x))) + 2.0));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.328422371476003e-05

    1. Initial program 0.1

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{e^{x} - 1}}\]

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{e^{x} - 1}}\]

    5. Applied difference-of-squares0.0

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]
    6. Using strategy rm

    7. Applied add-cbrt-cube0.0

      \[\leadsto \sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}}{e^{x} - 1}}\]

    8. Simplified0.0

      \[\leadsto \sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}}{e^{x} - 1}}\]

    if -1.328422371476003e-05 < x

    1. Initial program 32.9

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 6.7

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified6.7

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.328422371476003 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))