Average Error: 41.0 → 0.4
Time: 11.0s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5991589303897387 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.5991589303897387 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} + \left(e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)} + 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 <= -4.599158930389739e-07)) {
		VAR = ((double) sqrt((((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / (((double) (((double) exp(((double) (x + x)))) - ((double) (1.0 * 1.0)))) / ((double) (((double) exp(x)) + 1.0))))));
	} else {
		VAR = ((double) (((double) sqrt(2.0)) + ((double) (((double) exp(((double) log(((double) ((((double) pow(x, 2.0)) / ((double) sqrt(2.0))) * ((double) (0.25 - (0.125 / 2.0))))))))) + ((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 < -4.5991589303897387e-7

    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

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

    if -4.5991589303897387e-7 < 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

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

    5. Applied add-exp-log0.6

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

    6. Applied add-exp-log0.6

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

    7. Applied add-exp-log31.8

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

    8. Applied pow-exp31.8

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

    9. Applied div-exp31.8

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

    10. Applied prod-exp31.8

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

    11. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5991589303897387 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

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))))