Average Error: 41.2 → 0.3
Time: 5.4s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.2840672600851088 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(x \cdot \frac{\frac{1}{2}}{\sqrt{2}} + \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \sqrt[3]{x \cdot \frac{x}{\sqrt{2}}}\right)\right) \cdot \frac{3}{16}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.2840672600851088 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} + \left(x \cdot \frac{\frac{1}{2}}{\sqrt{2}} + \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \sqrt[3]{x \cdot \frac{x}{\sqrt{2}}}\right)\right) \cdot \frac{3}{16}\right)\\

\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 <= -7.284067260085109e-16)) {
		VAR = ((double) sqrt(((double) (((double) (((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) (x * ((double) (0.5 / ((double) sqrt(2.0)))))) + ((double) (((double) (((double) cbrt(((double) (x * ((double) (x / ((double) sqrt(2.0)))))))) * ((double) (((double) cbrt(((double) (x * ((double) (x / ((double) sqrt(2.0)))))))) * ((double) cbrt(((double) (x * ((double) (x / ((double) sqrt(2.0)))))))))))) * 0.1875))))));
	}
	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 < -7.2840672600851088e-16

    1. Initial program 0.9

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

    2. Simplified0.5

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{e^{x} - 1}}}\]
    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. Applied associate-/r/0.0

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

    6. Simplified0.0

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

    if -7.2840672600851088e-16 < x

    1. Initial program 62.5

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

    2. Simplified62.3

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

    3. Taylor expanded around 0 0.5

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

    4. Simplified0.5

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

    6. Applied add-cube-cbrt0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.2840672600851088 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(x \cdot \frac{\frac{1}{2}}{\sqrt{2}} + \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \left(\sqrt[3]{x \cdot \frac{x}{\sqrt{2}}} \cdot \sqrt[3]{x \cdot \frac{x}{\sqrt{2}}}\right)\right) \cdot \frac{3}{16}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))