Average Error: 4.9 → 0.8
Time: 12.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-281509160522293}{18446744073709551616}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{\sqrt[3]{\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)} \cdot \sqrt[3]{e^{x} - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-281509160522293}{18446744073709551616}:\\
\;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{\sqrt[3]{\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)} \cdot \sqrt[3]{e^{x} - 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r19833 = 2.0;
        double r19834 = x;
        double r19835 = r19833 * r19834;
        double r19836 = exp(r19835);
        double r19837 = 1.0;
        double r19838 = r19836 - r19837;
        double r19839 = exp(r19834);
        double r19840 = r19839 - r19837;
        double r19841 = r19838 / r19840;
        double r19842 = sqrt(r19841);
        return r19842;
}
↓
double f(double x) {
        double r19843 = x;
        double r19844 = -281509160522293.0;
        double r19845 = 1.8446744073709552e+19;
        double r19846 = r19844 / r19845;
        bool r19847 = r19843 <= r19846;
        double r19848 = 2.0;
        double r19849 = r19848 * r19843;
        double r19850 = exp(r19849);
        double r19851 = sqrt(r19850);
        double r19852 = 1.0;
        double r19853 = sqrt(r19852);
        double r19854 = r19851 + r19853;
        double r19855 = r19851 - r19853;
        double r19856 = exp(r19843);
        double r19857 = r19856 - r19852;
        double r19858 = r19857 * r19857;
        double r19859 = cbrt(r19858);
        double r19860 = cbrt(r19857);
        double r19861 = r19859 * r19860;
        double r19862 = r19855 / r19861;
        double r19863 = r19854 * r19862;
        double r19864 = sqrt(r19863);
        double r19865 = r19852 / r19848;
        double r19866 = r19865 * r19843;
        double r19867 = r19852 + r19866;
        double r19868 = r19843 * r19867;
        double r19869 = r19868 + r19848;
        double r19870 = sqrt(r19869);
        double r19871 = r19847 ? r19864 : r19870;
        return r19871;
}

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 < -1.5260642170642034e-05

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.0

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

    8. Simplified0.0

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

    10. Applied add-cbrt-cube0.0

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

    11. Simplified0.0

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

    13. Applied add-cube-cbrt0.0

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

    14. Applied unpow-prod-down0.0

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

    15. Applied cbrt-prod0.0

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

    16. Simplified0.0

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

    17. Simplified0.0

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

    if -1.5260642170642034e-05 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 5.9

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified5.9

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-281509160522293}{18446744073709551616}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{\sqrt[3]{\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)} \cdot \sqrt[3]{e^{x} - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))