Average Error: 4.6 → 0.8
Time: 19.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.780812387076979337394696627937232713719 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\sqrt{1} + \sqrt[3]{\left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(0.5 \cdot \frac{x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\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({\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)}}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r1330462 = 2.0;
        double r1330463 = x;
        double r1330464 = r1330462 * r1330463;
        double r1330465 = exp(r1330464);
        double r1330466 = 1.0;
        double r1330467 = r1330465 - r1330466;
        double r1330468 = exp(r1330463);
        double r1330469 = r1330468 - r1330466;
        double r1330470 = r1330467 / r1330469;
        double r1330471 = sqrt(r1330470);
        return r1330471;
}
↓
double f(double x) {
        double r1330472 = x;
        double r1330473 = -6.780812387076979e-12;
        bool r1330474 = r1330472 <= r1330473;
        double r1330475 = 1.0;
        double r1330476 = sqrt(r1330475);
        double r1330477 = 2.0;
        double r1330478 = exp(r1330477);
        double r1330479 = 2.0;
        double r1330480 = r1330472 / r1330479;
        double r1330481 = pow(r1330478, r1330480);
        double r1330482 = r1330481 * r1330481;
        double r1330483 = r1330482 * r1330481;
        double r1330484 = cbrt(r1330483);
        double r1330485 = r1330476 + r1330484;
        double r1330486 = sqrt(r1330485);
        double r1330487 = r1330481 - r1330476;
        double r1330488 = exp(r1330472);
        double r1330489 = r1330488 - r1330475;
        double r1330490 = r1330487 / r1330489;
        double r1330491 = sqrt(r1330490);
        double r1330492 = r1330486 * r1330491;
        double r1330493 = sqrt(r1330477);
        double r1330494 = 0.5;
        double r1330495 = r1330472 / r1330493;
        double r1330496 = r1330494 * r1330495;
        double r1330497 = r1330472 * r1330472;
        double r1330498 = r1330497 / r1330493;
        double r1330499 = 0.25;
        double r1330500 = 0.125;
        double r1330501 = r1330500 / r1330477;
        double r1330502 = r1330499 - r1330501;
        double r1330503 = r1330498 * r1330502;
        double r1330504 = r1330496 + r1330503;
        double r1330505 = r1330493 + r1330504;
        double r1330506 = r1330474 ? r1330492 : r1330505;
        return r1330506;
}

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. Simplified0.1

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

    11. Applied add-log-exp0.1

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

    12. Applied exp-to-pow0.1

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

    13. Applied sqrt-pow10.0

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

    15. Applied add-log-exp0.0

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

    16. Applied exp-to-pow0.0

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

    17. Applied sqrt-pow10.0

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

    19. Applied add-cbrt-cube0.0

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

    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

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

  4. Final simplification0.8

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

Reproduce

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