Average Error: 4.4 → 0.8
Time: 5.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.11558615421983944 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} \cdot {\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.11558615421983944 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\left({\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} \cdot {\left(\sqrt{e^{2}}\right)}^{\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}}\\

\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 r10646 = 2.0;
        double r10647 = x;
        double r10648 = r10646 * r10647;
        double r10649 = exp(r10648);
        double r10650 = 1.0;
        double r10651 = r10649 - r10650;
        double r10652 = exp(r10647);
        double r10653 = r10652 - r10650;
        double r10654 = r10651 / r10653;
        double r10655 = sqrt(r10654);
        return r10655;
}
↓
double f(double x) {
        double r10656 = x;
        double r10657 = -1.1155861542198394e-16;
        bool r10658 = r10656 <= r10657;
        double r10659 = 2.0;
        double r10660 = exp(r10659);
        double r10661 = sqrt(r10660);
        double r10662 = 0.5;
        double r10663 = r10662 * r10656;
        double r10664 = pow(r10661, r10663);
        double r10665 = r10664 * r10664;
        double r10666 = 1.0;
        double r10667 = sqrt(r10666);
        double r10668 = r10665 + r10667;
        double r10669 = pow(r10660, r10663);
        double r10670 = r10669 - r10667;
        double r10671 = exp(r10656);
        double r10672 = r10671 - r10666;
        double r10673 = r10670 / r10672;
        double r10674 = r10668 * r10673;
        double r10675 = sqrt(r10674);
        double r10676 = 0.5;
        double r10677 = sqrt(r10659);
        double r10678 = r10656 / r10677;
        double r10679 = r10676 * r10678;
        double r10680 = 2.0;
        double r10681 = pow(r10656, r10680);
        double r10682 = r10681 / r10677;
        double r10683 = 0.25;
        double r10684 = 0.125;
        double r10685 = r10684 / r10659;
        double r10686 = r10683 - r10685;
        double r10687 = r10682 * r10686;
        double r10688 = r10677 + r10687;
        double r10689 = r10679 + r10688;
        double r10690 = r10658 ? r10675 : r10689;
        return r10690;
}

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 < -1.1155861542198394e-16

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied add-sqr-sqrt0.8

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

    5. Applied add-sqr-sqrt0.7

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

    6. Applied difference-of-squares0.3

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

    7. Applied times-frac0.3

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

    8. Simplified0.3

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

    10. Applied add-log-exp0.3

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

    11. Applied exp-to-pow0.3

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

    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. Using strategy rm

    20. Applied add-sqr-sqrt0.0

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

    21. Applied unpow-prod-down0.0

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

    if -1.1155861542198394e-16 < x

    1. Initial program 36.7

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

    2. Taylor expanded around 0 8.4

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

    3. Simplified8.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.11558615421983944 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} \cdot {\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))