Average Error: 4.3 → 0.2
Time: 5.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0012905509290539541:\\ \;\;\;\;\sqrt{e^{\log \left(\frac{e^{2 \cdot x} - 1}{e^{x} - 1}\right)}}\\ \mathbf{elif}\;x \le 5.8401109832474551 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{e^{0.25 \cdot {x}^{2} + \left(\log 2 + 0.5 \cdot \left(x - \frac{{x}^{2}}{{2}^{2}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0012905509290539541:\\
\;\;\;\;\sqrt{e^{\log \left(\frac{e^{2 \cdot x} - 1}{e^{x} - 1}\right)}}\\

\mathbf{elif}\;x \le 5.8401109832474551 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{e^{0.25 \cdot {x}^{2} + \left(\log 2 + 0.5 \cdot \left(x - \frac{{x}^{2}}{{2}^{2}}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) (((double) exp(x)) - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.0012905509290539541)) {
		VAR = ((double) sqrt(((double) exp(((double) log(((double) (((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) (((double) exp(x)) - 1.0))))))))));
	} else {
		double VAR_1;
		if ((x <= 5.840110983247455e-07)) {
			VAR_1 = ((double) sqrt(((double) exp(((double) (((double) (0.25 * ((double) pow(x, 2.0)))) + ((double) (((double) log(2.0)) + ((double) (0.5 * ((double) (x - ((double) (((double) pow(x, 2.0)) / ((double) pow(2.0, 2.0))))))))))))))));
		} else {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (((double) sqrt(((double) exp(((double) (2.0 * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) exp(((double) (2.0 * x)))))) - ((double) sqrt(1.0)))))) / ((double) (((double) exp(x)) - 1.0))))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if x < -0.0012905509290539541

    1. Initial program 0.0

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

    3. Applied add-exp-log64.0

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

    4. Applied add-exp-log64.0

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

    5. Applied div-exp64.0

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

    6. Simplified0.0

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

    if -0.0012905509290539541 < x < 5.840110983247455e-07

    1. Initial program 38.1

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

    3. Applied add-exp-log52.4

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

    4. Applied add-exp-log52.4

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

    5. Applied div-exp52.4

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

    6. Simplified38.1

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

    7. Taylor expanded around 0 0.2

      \[\leadsto \sqrt{e^{\color{blue}{\left(\log 2 + \left(0.25 \cdot {x}^{2} + 0.5 \cdot x\right)\right) - 0.5 \cdot \frac{{x}^{2}}{{2}^{2}}}}}\]

    8. Simplified0.2

      \[\leadsto \sqrt{e^{\color{blue}{0.25 \cdot {x}^{2} + \left(\log 2 + 0.5 \cdot \left(x - \frac{{x}^{2}}{{2}^{2}}\right)\right)}}}\]

    if 5.840110983247455e-07 < x

    1. Initial program 9.6

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

    3. Applied add-sqr-sqrt9.6

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

    4. Applied add-sqr-sqrt9.3

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

    5. Applied difference-of-squares5.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0012905509290539541:\\ \;\;\;\;\sqrt{e^{\log \left(\frac{e^{2 \cdot x} - 1}{e^{x} - 1}\right)}}\\ \mathbf{elif}\;x \le 5.8401109832474551 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{e^{0.25 \cdot {x}^{2} + \left(\log 2 + 0.5 \cdot \left(x - \frac{{x}^{2}}{{2}^{2}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))