Average Error: 4.6 → 1.1
Time: 5.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \le 1.6150103447977282:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \le 1.6150103447977282:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\\

\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 ((sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0))) <= 1.6150103447977282)) {
		VAR = sqrt((((sqrt(exp((2.0 * x))) + sqrt(1.0)) * (sqrt(exp((2.0 * x))) - sqrt(1.0))) / (exp(x) - 1.0)));
	} else {
		VAR = (((0.25 * (pow(x, 2.0) / sqrt(2.0))) + (sqrt(2.0) + (0.5 * (x / sqrt(2.0))))) - (0.125 * (pow(x, 2.0) / pow(sqrt(2.0), 3.0))));
	}
	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 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))) < 1.6150103447977282

    1. Initial program 1.7

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

    3. Applied add-sqr-sqrt1.7

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

    4. Applied add-sqr-sqrt1.4

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

    5. Applied difference-of-squares0.5

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

    if 1.6150103447977282 < (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0)))

    1. Initial program 54.3

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

    3. Applied add-cbrt-cube54.3

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

    4. Simplified54.3

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

    5. Taylor expanded around 0 9.9

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \le 1.6150103447977282:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))