Average Error: 41.0 → 0.4
Time: 4.9s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.3880608955832627 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\sqrt[3]{{\left(\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(e^{\log \left(x \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{3}{16}\right)\right)} + \frac{x}{\sqrt{2}} \cdot \frac{1}{2}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.3880608955832627 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\sqrt[3]{{\left(\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} + \left(e^{\log \left(x \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{3}{16}\right)\right)} + \frac{x}{\sqrt{2}} \cdot \frac{1}{2}\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 <= -9.388060895583263e-06)) {
		VAR = ((double) sqrt((((double) (((double) pow(((double) exp(x)), 2.0)) - 1.0)) / ((double) cbrt(((double) pow((((double) (((double) pow(((double) exp(x)), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (((double) exp(x)) + 1.0))), 3.0)))))));
	} else {
		VAR = ((double) (((double) sqrt(2.0)) + ((double) (((double) exp(((double) log(((double) (x * ((double) ((x / ((double) sqrt(2.0))) * 0.1875)))))))) + ((double) ((x / ((double) sqrt(2.0))) * 0.5))))));
	}
	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 < -9.3880608955832627e-6

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied flip--0.0

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

    5. Simplified0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Applied cbrt-undiv0.0

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

    10. Simplified0.0

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

    if -9.3880608955832627e-6 < x

    1. Initial program 61.6

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

    2. Simplified61.3

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

    3. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\frac{1}{4} \cdot \frac{{x}^{2}}{\sqrt{2}} + \sqrt{2}\right)\right) - \frac{1}{8} \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    4. Simplified0.6

      \[\leadsto \color{blue}{\sqrt{2} + \left(\left(\frac{x}{\sqrt{2}} \cdot x\right) \cdot \frac{3}{16} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)}\]
    5. Using strategy rm

    6. Applied add-exp-log0.6

      \[\leadsto \sqrt{2} + \left(\left(\frac{x}{\sqrt{2}} \cdot x\right) \cdot \color{blue}{e^{\log \frac{3}{16}}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    7. Applied add-exp-log31.9

      \[\leadsto \sqrt{2} + \left(\left(\frac{x}{\sqrt{2}} \cdot \color{blue}{e^{\log x}}\right) \cdot e^{\log \frac{3}{16}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    8. Applied add-exp-log31.9

      \[\leadsto \sqrt{2} + \left(\left(\frac{x}{\color{blue}{e^{\log \left(\sqrt{2}\right)}}} \cdot e^{\log x}\right) \cdot e^{\log \frac{3}{16}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    9. Applied add-exp-log31.9

      \[\leadsto \sqrt{2} + \left(\left(\frac{\color{blue}{e^{\log x}}}{e^{\log \left(\sqrt{2}\right)}} \cdot e^{\log x}\right) \cdot e^{\log \frac{3}{16}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    10. Applied div-exp31.9

      \[\leadsto \sqrt{2} + \left(\left(\color{blue}{e^{\log x - \log \left(\sqrt{2}\right)}} \cdot e^{\log x}\right) \cdot e^{\log \frac{3}{16}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    11. Applied prod-exp31.9

      \[\leadsto \sqrt{2} + \left(\color{blue}{e^{\left(\log x - \log \left(\sqrt{2}\right)\right) + \log x}} \cdot e^{\log \frac{3}{16}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    12. Applied prod-exp31.9

      \[\leadsto \sqrt{2} + \left(\color{blue}{e^{\left(\left(\log x - \log \left(\sqrt{2}\right)\right) + \log x\right) + \log \frac{3}{16}}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]

    13. Simplified0.6

      \[\leadsto \sqrt{2} + \left(e^{\color{blue}{\log \left(x \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{3}{16}\right)\right)}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.3880608955832627 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\sqrt[3]{{\left(\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(e^{\log \left(x \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{3}{16}\right)\right)} + \frac{x}{\sqrt{2}} \cdot \frac{1}{2}\right)\\ \end{array}\]

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))))