Average Error: 41.0 → 0.4
Time: 8.5s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2211091000924002 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.2211091000924002 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\

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

\end{array}
double code(double x) {
	return ((double) sqrt((((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) (((double) exp(x)) - 1.0)))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -1.2211091000924002e-05)) {
		VAR = ((double) sqrt((((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) cbrt(((double) pow((((double) (((double) exp(((double) (x + x)))) - ((double) (1.0 * 1.0)))) / ((double) (((double) exp(x)) + 1.0))), 3.0)))))));
	} else {
		VAR = ((double) (((double) (((double) (0.5 * (x / ((double) sqrt(2.0))))) + ((double) sqrt(2.0)))) + ((double) exp(((double) log(((double) ((((double) pow(x, 2.0)) / ((double) sqrt(2.0))) * ((double) (0.25 - (0.125 / 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 < -1.2211091000924002e-5

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Simplified0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Applied cbrt-undiv0.0

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

    9. Simplified0.0

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

    if -1.2211091000924002e-5 < x

    1. Initial program 61.6

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

    2. Taylor expanded around 0 0.6

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

    3. Simplified0.6

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)}\]
    4. Using strategy rm

    5. Applied add-exp-log0.6

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \color{blue}{e^{\log \left(0.25 - \frac{0.125}{2}\right)}}\]

    6. Applied add-exp-log0.6

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \frac{{x}^{2}}{\color{blue}{e^{\log \left(\sqrt{2}\right)}}} \cdot e^{\log \left(0.25 - \frac{0.125}{2}\right)}\]

    7. Applied add-exp-log31.9

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \frac{{\color{blue}{\left(e^{\log x}\right)}}^{2}}{e^{\log \left(\sqrt{2}\right)}} \cdot e^{\log \left(0.25 - \frac{0.125}{2}\right)}\]

    8. Applied pow-exp31.9

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \frac{\color{blue}{e^{\log x \cdot 2}}}{e^{\log \left(\sqrt{2}\right)}} \cdot e^{\log \left(0.25 - \frac{0.125}{2}\right)}\]

    9. Applied div-exp31.9

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{e^{\log x \cdot 2 - \log \left(\sqrt{2}\right)}} \cdot e^{\log \left(0.25 - \frac{0.125}{2}\right)}\]

    10. Applied prod-exp31.9

      \[\leadsto \left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{e^{\left(\log x \cdot 2 - \log \left(\sqrt{2}\right)\right) + \log \left(0.25 - \frac{0.125}{2}\right)}}\]

    11. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2211091000924002 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{\sqrt{2}} + \sqrt{2}\right) + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))