Average Error: 4.3 → 0.9
Time: 4.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.32996296778409252 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\left({\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{e^{1 \cdot x} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.32996296778409252 \cdot 10^{-14}:\\
\;\;\;\;\sqrt{\left({\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{e^{1 \cdot x} - \sqrt{1}}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r10493 = 2.0;
        double r10494 = x;
        double r10495 = r10493 * r10494;
        double r10496 = exp(r10495);
        double r10497 = 1.0;
        double r10498 = r10496 - r10497;
        double r10499 = exp(r10494);
        double r10500 = r10499 - r10497;
        double r10501 = r10498 / r10500;
        double r10502 = sqrt(r10501);
        return r10502;
}
↓
double f(double x) {
        double r10503 = x;
        double r10504 = -1.3299629677840925e-14;
        bool r10505 = r10503 <= r10504;
        double r10506 = 2.0;
        double r10507 = exp(r10506);
        double r10508 = 0.5;
        double r10509 = r10508 * r10503;
        double r10510 = pow(r10507, r10509);
        double r10511 = 1.0;
        double r10512 = sqrt(r10511);
        double r10513 = r10510 + r10512;
        double r10514 = r10511 * r10503;
        double r10515 = exp(r10514);
        double r10516 = r10515 - r10512;
        double r10517 = exp(r10503);
        double r10518 = r10517 - r10511;
        double r10519 = r10516 / r10518;
        double r10520 = r10513 * r10519;
        double r10521 = sqrt(r10520);
        double r10522 = 0.5;
        double r10523 = sqrt(r10506);
        double r10524 = r10503 / r10523;
        double r10525 = r10522 * r10524;
        double r10526 = 2.0;
        double r10527 = pow(r10503, r10526);
        double r10528 = r10527 / r10523;
        double r10529 = 0.25;
        double r10530 = 0.125;
        double r10531 = r10530 / r10506;
        double r10532 = r10529 - r10531;
        double r10533 = r10528 * r10532;
        double r10534 = r10523 + r10533;
        double r10535 = r10525 + r10534;
        double r10536 = r10505 ? r10521 : r10535;
        return r10536;
}

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 < -1.3299629677840925e-14

    1. Initial program 0.7

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

    3. Applied *-un-lft-identity0.7

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

    4. Applied add-sqr-sqrt0.7

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

    5. Applied add-sqr-sqrt0.6

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

    6. Applied difference-of-squares0.2

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

    7. Applied times-frac0.2

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

    8. Simplified0.2

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

    10. Applied add-log-exp0.2

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

    11. Applied exp-to-pow0.2

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

    12. Applied sqrt-pow10.0

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

    13. Simplified0.0

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

    15. Applied add-log-exp0.0

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

    16. Applied exp-to-pow0.0

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

    17. Applied sqrt-pow10.0

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

    18. Simplified0.0

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

    19. Taylor expanded around inf 0.0

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

    if -1.3299629677840925e-14 < x

    1. Initial program 36.3

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

    2. Taylor expanded around 0 8.5

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

    3. Simplified8.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.32996296778409252 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\left({\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{e^{1 \cdot x} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))