Average Error: 40.7 → 0.4
Time: 6.3s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.4460959501863895 \cdot 10^{-06}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}}\right)\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.4460959501863895 \cdot 10^{-06}:\\
\;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} + \left(\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}}\right)\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{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 <= -6.4460959501863895e-06)) {
		VAR = ((double) sqrt((((double) (((double) pow(((double) exp(x)), 2.0)) - 1.0)) / (((double) (((double) pow(((double) exp(x)), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (((double) exp(x)) + 1.0))))));
	} else {
		VAR = ((double) (((double) sqrt(2.0)) + ((double) (((double) (((double) ((((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) sqrt(((double) sqrt(2.0))))) * ((double) (x * (((double) cbrt(x)) / ((double) sqrt(((double) sqrt(2.0))))))))) * 0.1875)) + ((double) (0.5 * (x / ((double) sqrt(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 < -6.44609595018638952e-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

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

    if -6.44609595018638952e-6 < x

    1. Initial program 61.5

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

    2. Simplified61.1

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

    3. Taylor expanded around 0 0.6

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

    4. Simplified0.6

      \[\leadsto \color{blue}{\sqrt{2} + \left(\left(\frac{x}{\sqrt{2}} \cdot x\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.6

      \[\leadsto \sqrt{2} + \left(\left(\frac{x}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot x\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    7. Applied sqrt-prod0.6

      \[\leadsto \sqrt{2} + \left(\left(\frac{x}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot x\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    8. Applied add-cube-cbrt0.6

      \[\leadsto \sqrt{2} + \left(\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}} \cdot x\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    9. Applied times-frac0.6

      \[\leadsto \sqrt{2} + \left(\left(\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}}\right)} \cdot x\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    10. Applied associate-*l*0.6

      \[\leadsto \sqrt{2} + \left(\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot x\right)\right)} \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    11. Simplified0.6

      \[\leadsto \sqrt{2} + \left(\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(x \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}}\right)}\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4460959501863895 \cdot 10^{-06}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\frac{{\left(e^{x}\right)}^{2} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt{2}}}\right)\right) \cdot 0.1875 + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))