Average Error: 4.6 → 0.8
Time: 19.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.780812387076979337394696627937232713719 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\sqrt{1} + \sqrt[3]{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}\right)}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{2}} \cdot 0.5 + \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \sqrt{2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.780812387076979337394696627937232713719 \cdot 10^{-12}:\\
\;\;\;\;\sqrt{\sqrt{1} + \sqrt[3]{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}\right)}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r1057352 = 2.0;
        double r1057353 = x;
        double r1057354 = r1057352 * r1057353;
        double r1057355 = exp(r1057354);
        double r1057356 = 1.0;
        double r1057357 = r1057355 - r1057356;
        double r1057358 = exp(r1057353);
        double r1057359 = r1057358 - r1057356;
        double r1057360 = r1057357 / r1057359;
        double r1057361 = sqrt(r1057360);
        return r1057361;
}
↓
double f(double x) {
        double r1057362 = x;
        double r1057363 = -6.780812387076979e-12;
        bool r1057364 = r1057362 <= r1057363;
        double r1057365 = 1.0;
        double r1057366 = sqrt(r1057365);
        double r1057367 = 2.0;
        double r1057368 = exp(r1057367);
        double r1057369 = 2.0;
        double r1057370 = r1057362 / r1057369;
        double r1057371 = pow(r1057368, r1057370);
        double r1057372 = r1057371 * r1057371;
        double r1057373 = r1057371 * r1057372;
        double r1057374 = cbrt(r1057373);
        double r1057375 = r1057366 + r1057374;
        double r1057376 = sqrt(r1057375);
        double r1057377 = r1057371 - r1057366;
        double r1057378 = exp(r1057362);
        double r1057379 = r1057378 - r1057365;
        double r1057380 = r1057377 / r1057379;
        double r1057381 = sqrt(r1057380);
        double r1057382 = r1057376 * r1057381;
        double r1057383 = sqrt(r1057367);
        double r1057384 = r1057362 / r1057383;
        double r1057385 = 0.5;
        double r1057386 = r1057384 * r1057385;
        double r1057387 = r1057383 / r1057362;
        double r1057388 = r1057362 / r1057387;
        double r1057389 = 0.25;
        double r1057390 = 0.125;
        double r1057391 = r1057390 / r1057367;
        double r1057392 = r1057389 - r1057391;
        double r1057393 = r1057388 * r1057392;
        double r1057394 = r1057386 + r1057393;
        double r1057395 = r1057394 + r1057383;
        double r1057396 = r1057364 ? r1057382 : r1057395;
        return r1057396;
}

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 < -6.780812387076979e-12

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied add-sqr-sqrt0.5

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

    5. Applied add-sqr-sqrt0.4

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

    6. Applied difference-of-squares0.1

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

    7. Applied times-frac0.1

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

    8. Applied sqrt-prod0.1

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

    10. Applied add-log-exp0.1

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

    11. Applied exp-to-pow0.1

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

    12. Applied sqrt-pow10.0

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

    14. Applied add-log-exp0.0

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

    15. Applied exp-to-pow0.0

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

    16. Applied sqrt-pow10.0

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

    18. Applied add-cbrt-cube0.0

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

    if -6.780812387076979e-12 < x

    1. Initial program 36.3

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

    2. Taylor expanded around 0 6.8

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

    3. Simplified6.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.780812387076979337394696627937232713719 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\sqrt{1} + \sqrt[3]{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}\right)}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{2}} \cdot 0.5 + \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \sqrt{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))